Datasets:

sail

/

regmix-data

like 4

Follow

Sea AI Lab 70

--- abstract: 'We study the structure of accretion disks around supermassive black holes in the radial range $30\sim 100$ gravitational radii, using a three dimensional radiation magneto-hydrodynamic simulation. For typical conditions in this region of Active Galactic Nuclei (AGN), the Rosseland mean opacity is expected to be larger than the electron scattering value. We show that the iron opacity bump causes the disk to be convective unstable. Turbulence generated by convection puffs up the disk due to additional turbulent pressure support and enhances the local angular momentum transport. This also results in strong fluctuations in surface density and heating of the disk. The opacity drops with increasing temperature and convection is suppressed. The disk cools down and the whole process repeats again. This causes strong oscillations of the disk scale height and luminosity variations by more than a factor of $\approx 3-6$ over a few years’ timescale. Since the iron opacity bump will move to different locations of the disk for black holes with different masses and accretion rates, we suggest that this is a physical mechanism that can explain the variability of AGN with a wide range of amplitudes over a time scale of years to decades.' author: - 'Yan-Fei Jiang(姜燕飞)' - Omer Blaes bibliography: - 'citations.bib' title: Opacity Driven Convection and Variability in Accretion Disks around Supermassive Black Holes --- [UTF8]{}[gbsn]{} Introduction ============ The standard model of geometrically thin, radiatively efficient accretion disks [@SS73; @NOV73] provides a good, observationally tested, first-order model for the soft states of X-ray binaries (both black hole and neutron star, e.g. @DON07). Accounting for proper opacities and the effects of irradiation, it has also been developed into a testable model that successfully explains the outbursts observed in dwarf novae and low mass X-ray binaries [@LAS01]. However, its application to bright active galactic nuclei (AGN) and quasars, where the model still predicts the existence of a geometrically thin disk, has always been problematic (e.g. @KOR99 [@ANT13]). Beyond the most basic prediction of thermal emission in the ultraviolet [@SHI78; @MAL83], the model generally fails to provide a good description of the far ultraviolet spectral energy distribution (e.g. @SHA05 [@LAO14]). While it successfully predicts that shorter wavelength optical/ultraviolet photons originate from smaller distances from the black hole than longer wavelength photons, consistent with the measured trends in continuum reverberation mapping [@EDE15] and microlensing campaigns [@Morganetal2010; @Blackburneetal2011; @Mosqueraetal2013], it is [*not*]{} consistent with the absolute emission region sizes. One of the most glaring observational discrepancies is the extreme variability observed in what are now called changing look quasars (e.g. @LAW18 and references therein). Optical emission from these sources are observed to increase and/or decrease by factors of two to ten on time-scales of a few years to a decade (e.g. @LAM15 [@MacLeodetal2016; @RUA16; @Yangetal2018; @Dexteretal2019]), accompanied by a resurgence and/or loss of the broad emission lines. These changing look AGNs appear to be the tail of a continuous distribution of quasar properties where the large amplitude variability is likely caused by physical processes in the disk [@Rumbaughetal2018; @Luoetal2020]. However, this variability is far shorter than the “viscous" time scale (required to move mass radially from the outer edge of the disk to the inner region) of the standard disk model, but is not inconsistent with the thermal time scale (required to heat or cool the disk). These two time scales differ by a factor of the square of the ratio of the radius $R$ to the local disk scale height $H$. One possible resolution is therefore to suppose that the actual accretion flow is geometrically thick $H/R\sim1$ in the optically emitting regions, perhaps because of vertical support by magnetic fields [@DEX19]. The fact that the standard model of accretion disks does such a poor job of explaining observations suggests that one should look at ways in which AGN accretion disks necessarily differ from their counterparts in X-ray binaries and cataclysmic variables. One very important difference is that accretion disks in bright AGN have thermal pressures which are hugely dominated by radiation pressure. Moreover, their ultraviolet temperatures mean that they are subject to large opacities, the effects of which are to make radiation pressure forces even stronger. One important aspect of this is the likely presence of line-driven outflows from the disk [@PRO00; @LAO14]. But even within the Rosseland photosphere of the disk, opacity effects can be extremely important. Sophisticated one-dimensional models of the vertical structure of the disk generally exhibit density inversions due to enhancements of Rosseland opacity with declining outward temperature (see, e.g., Figure 9 of @HUB00). Such density inversions are also commonly seen in one dimensional models of radiation pressure supported massive star envelopes [@JOS73; @PAX13], which have similar density and temperature conditions to AGN disks. The density inversion can be either driven by the opacity peak due to hydrogen and helium ionization fronts as studied by @HUB00, or the iron opacity bump around the temperature $1.8\times 10^5$ K [@JIA18]. In this paper, we focus on the hotter opacity bumps due to irons. Density inversions due to the opacity peaks are of course unstable, and simulations of these inversions show considerable convective turbulence [@JIA15]. However, under conditions of moderate optical depth (optical depth per pressure scale height less than the ratio of the speed of light to the gas sounds speed), convective heat transport is inefficient and leads to a porous structure with large density fluctuations associated with shocks. Moreover, the time-averaged structure still maintains a vertical density inversion. The large density fluctuations can also trigger even larger enhancements in opacity due to helium recombination, possibly triggering bursty outflows in massive stars [@JIA18]. AGN disks are generally expected to be in this low optical depth, rapid radiative diffusion regime, and so similar behavior might be expected. Accretion disks are considerably more complex compared to stars, however, because of the coupling between the thermal state of the disk and angular momentum transport. The standard accretion disk model with an $\alpha$ stress prescription proportional to total thermal pressure is unstable to thermal and “viscous" instabilities [@LIG74; @SHA76]. Modern radiation MHD simulations of MRI turbulence appear to confirm the thermal instability [@JIA13; @MIS16], although it can be stabilized if the disk is largely supported by magnetic rather than thermal pressure [@SAD16]. Local shearing box simulations show that stability can also be achieved under AGN conditions by the iron opacity bump [@JIA16]. Local shearing box simulations of MRI turbulence in white dwarf [@HIR14; @COL18; @SCE18] and protostellar [@HIR15] accretion disks show that convection can significantly enhance magnetorotational (MRI) turbulence. It seems reasonable to expect that the opacity-driven convection from unstable density inversions in AGN disks might also lead to interesting variations in MRI turbulent stresses. Large turbulent kinetic energy densities might also provide support against the vertical tidal gravity of the black hole similar to what is found in massive stars [@JIA18]. In this paper, we report the results of an initial investigation of these opacity effects using global radiation MHD simulations of AGN accretion disks. This work builds on previous efforts to simulate the near-black hole regions of AGN accretion disks, beginning with super-Eddington flows (@JIA14 [@JIA19b]; see also @SN16) and sub-Eddington, magnetically supported flows [@JIA19a]. The temperatures in these previous simulations were all too high for the iron opacity bump to be present, and the flux mean opacity was dominated by electron scattering. Here, using very similar initial conditions to those in @JIA19a, we simulate a region of the disk further out from the black hole (beyond thirty gravitational radii) where we achieve low enough temperatures that interesting opacity effects take place. The simulations exhibit a high degree of variability in luminosity, intermittent episodes of convection, and rapid and variable radial diffusion of mass, which can all be traced to the effects of variable opacity in these highly radiation pressure dominated flows. This paper is organized as follows. In section 2 we briefly review the simulation methods and the physics that is included, as well as the spatial grid and initial conditions. We provide a detailed description of the physical behavior exhibited in the simulation in section 3. In section 4 we discuss the potential applications of our work to explaining observations of AGN, as well as other accretion-powered sources, and we summarize our conclusions in section 5. Simulation Setup {#sec:setup} ================ We solve the ideal MHD equations coupled with the time-dependent, frequency-integrated radiative transfer equation for intensities over discrete angles in the same way as described in [@JIA19b]. We use spherical polar coordinates in [Athena++]{} (Stone et al. 2020, submitted) covering the radial range from $30r_g$ at the inner radial boundary to $1.2\times 10^4r_g$ at the outer radial boundary, where $r_g$ is the gravitational radius of the black hole. At each radius, the simulation domain includes the whole sphere with polar angle $\theta\in[0,\pi]$ and azimuthal angle $\phi\in[0,2\pi)$. We are most interested in the radial range around $50-100r_g$ where we anticipate that the disk will have enhanced Rosseland opacity due to the iron bump. We therefore do not extend the simulation domain all the way down to the innermost stable circular orbit (ISCO) to avoid the very small time step that would be necessary to simulate these innermost regions. The simulations we have done in the past covering the inner $\sim 30r_g$ do not show any interesting opacity effects because the density is too low and the temperature is too high there [@JIA19a]. The effects of having an inner boundary all the way to the ISCO will be studied in future investigations. Boundary and Initial Conditions ------------------------------- We used static mesh refinement to resolve the inner disk near the midplane region. The root level has resolution $\Delta r/r=\Delta \theta=\Delta \phi=0.098$ and we use four levels of refinement with the finest level covering the region $(r,\theta,\phi)\in [30r_g,400r_g]\times [1.48,1.66]\times [0,2\pi)$, corresponding to a resolution of $\Delta r/r=\Delta \theta=\Delta \phi=6.1\times 10^{-3}$. We use 80 discrete angles in each cell to resolve the angular distribution of the radiation field. The simulation is done for a black hole with mass $\mbh=5\times 10^8\msun$ located at $r=0$. We use the Pseudo-Newtonian potential [@PaczynskiWiita1980] $\phi=-G\mbh/(r-2r_g)$ for the black hole, which is actually pretty close to the Newtonian formula in the radial range we simulate. Here $G$ is the gravitational constant, $r_g\equiv G\mbh/c^2$ is the gravitational radius and $c$ is the speed of light. For the inner boundary condition at $r=30r_g$, we simply allow all gas and radiation to flow inward but do not allow anything to come out of the inner boundary. In particular, we are neglecting any effects due to irradiation from the inner disk inside $30r_g$ in this simulation. For the outer radial boundary condition at $1.2\times 10^4r_g$, we only allow gas and radiation to leave the simulation box but do not allow anything to come in. We initialize the simulation with a rotating torus centered at $400r_g$. The inner and outer edges of the torus are at $226r_g$ and $1050r_g$ respectively. For the entire region in which we are interested (inside $\sim 100r_g$), the density is initialized to be the floor value ($10^{-17}$ g/cm$^3$) in order to minimize the artifacts of the initial condition on the properties of the accretion disk. The torus structure is the same as we used in [@JIA19b]. The initial temperature at the center of the torus is $3.96\times 10^5\ \ K$ and it drops to $2\times 10^4\ \ K$ at the inner and outer edges. The ratio of radiation pressure to gas pressure varies from $150$ to $450$ inside the torus. We initialize the $\phi$ component of the vector potential to be proportional to density and set other components to zero. This results in a big loop of magnetic field initially in the torus, which has a locally net poloidal component through the midplane. The initial magnetic pressure near the inner edge of the torus is about $70$ times the gas pressure and $\sim 10^{-3}$ of the radiation pressure. Due to the limited simulation duration that we can afford to do, only gas near the inner edge of the torus will have enough time to flow towards the inner region and form a disk with self-consistent structure. ![The Rosseland mean (solid) and Planck mean (dashed) opacities used in the simulation as a function of gas temperature for four different fixed densities ranging from $3.6\times 10^{-8}$ g/cm$^3$ to $3.6\times 10^{-12}$ g/cm$^3$, as indicated in the legend. The Planck mean opacities are significantly larger than the Rosseland mean values for $T<3\times 10^5 K$. []{data-label="opacity"}](opacitytable.pdf){width="1.0\hsize"} The Opacity ----------- For typical temperatures and densities in accretion disks around supermassive black holes, the relevant opacity is not dominated by electron scattering and free-free absorption opacity as is commonly assumed in classical accretion disk models for X-ray binaries [@JIA16]. Instead, contributions from many lines increase the effective continuum opacity significantly. Due to the changing ionization states of different species, the effective Rosseland mean and Planck mean opacities also show complicated dependencies on gas density and temperature. This is potentially interesting as it can drive hydrodynamic instabilities [@HEA72; @BlaesSocrates2003]. To capture these opacities accurately, we adopt the OPAL opacity table[^1] [@iglesias96] for the Rosseland and Planck means. These opacities are shown in Figure \[opacity\] as a function of temperature for four different densities, assuming solar metallicity. They are very similar to the opacities of protostellar disks for $T>10^4$ K [@ZhuJiangStone2019] and the envelopes of massive stars [@JIA15]. Figure \[opacity\] shows that the Planck mean opacity is typically orders of magnitude larger than the Rosseland mean opacity for the temperature range $10^4\sim 5\times 10^5$ K. The enhancement in Rosseland opacity around $T=1.8\times 10^5$ K is the iron opacity bump [@PAX13; @JIA15], while the increase below $10^5$ K is due to ionization of hydrogen and helium. Note that the iron opacity bump is typically just a few times larger than the electron scattering value, but we will show that this can have a dramatic effect on the structure and dynamics of quasar accretion disks. In the simulation, the opacity in each cell is calculated with bilinear interpolation of the opacity table based on local gas temperature and density. {width="\textwidth"} {width="\textwidth"} {width="\textwidth"} Results {#sec:result} ======= We will describe our simulation results using the following set of fiducial numbers to scale all the quantities: density $\rho_0=1.0\times 10^{-8} ~{\rm g/cm}^3$, temperature $T_0=2\times 10^5$ K, gas pressure and energy densities $P_0=2.77\times 10^5~{\rm dyn/cm}^2$, length $r_g=7.42\times 10^{13}~{\rm cm}$ and velocity $v_0=5.26\times 10^6~{\rm cm/s}$. The fiducial time unit is $r_g/c=7.8\times 10^{-5}~{\rm yr}$. Notice that for Keplerian rotation, the orbital period at the inner boundary $30r_g$ is $1.03\times 10^3r_g/c=8.03\times 10^{-2}~{\rm yr}$. The magnetic field has the fiducial unit $B_0=2.64\times 10^3~{\rm Gauss}$. Simulation History ------------------ Representative snapshots of the azimuthally-averaged poloidal distribution of density, as well as the distribution of density and radiation energy density in the equatorial plane, are shown in Figures \[fig:rhov2Dsnapshots\] and \[fig:rhomidsnapshots\]. It is immediately obvious that the density distributions undergo significant, non-monotonic variability. The vertical scale height of the disk goes through cycles of expansion and contraction, and radially narrow clumps of density form, dissolve, and reform. This clump formation process often begins with the formation of a nonaxisymmetric ($m=2$) density structure which generally transforms into an axisymmetric ring before eventually diffusing away. Note that this $m=2$ density pattern is not well-correlated with the radiation energy density in Figure \[fig:rhomidsnapshots\]. At first sight these variations might appear to be consistent with the predicted behavior of thermal/viscous instabilities in radiation pressure, electron-scattering dominated classical alpha disks [@SHA76]. This would predict runaway heating or cooling and also an anti-diffusion clumping process [@LIG74]. However, the situation here is more complex because both the clump formation and the vertical expansion and contraction are episodes of finite duration and always reverse. Figure \[fig:sigmakappa\] shows the evolution of shell-integrated surface mass density and shell-averaged (mass-weighted) opacity as a function of radius and time. Enhancements in surface density are clearly highly correlated with enhanced opacity over the electron scattering value. The iron opacity bump is playing a critical role in driving the density variability. ![Evolution of shell-averaged surface mass density (in unit of $\rho_0r_g$) and ratio of Rosseland opacity to Thomson opacity ($\kappa_{\rm es}$) as a function of radius (in units of gravitational radius $r_g$) and time (in units of $10^4G\mbh/c^3\equiv0.78$ years). []{data-label="fig:sigmakappa"}](ST_sigma_kappa.pdf){width="\columnwidth"} ![(Top) Evolution of shell-averaged radiation pressure (black), magnetic pressure (red), turbulent kinetic energy density (green) and gas pressure (blue) at radius 50 gravitational radii. These are all scaled with the fiducial pressure unit $P_0=2.77\times 10^5~{\rm dyn/cm}^2$. (Middle) Evolution of shell-averaged Maxwell stress $S_m$ and Reynolds stress $S_h$ at the same radius. The Reynolds stress is smoothed over the neighboring $100$ data points to reduce noise. Both $S_h$ and $S_m$ are scaled with $P_0$. (Bottom) Evolution of shell-averaged Rosseland mean opacity (scaled with the electron scattering value) at the same radius. []{data-label="fig:pressureskappa"}](his_r_50.pdf){width="\columnwidth"} ![History of the fluctuation components of the shell-averaged radiation pressure (black), magnetic pressure (red), turbulent kinetic energy density (green) and opacity (blue) at $r=50r_g$. The fluctuation components are calculated as the difference between the instantaneous values and the time averaged values between $5\times 10^5r_g/c$ and $10^6 r_g/c$.[]{data-label="fig:fluctuation"}](Fluctuation_his_r_50.pdf){height="0.9\columnwidth" width="0.9\columnwidth"} ![Evolution of shell-averaged turbulent kinetic energy due to radial, polar, and azimuthal motions at $r=50$ gravitational radii.[]{data-label="fig:Ekinhistory"}](Ek_his_50.pdf){width="\columnwidth"} ![Evolution of azimuthally-averaged density (top), Rosseland opacity (middle), and radiation entropy per unit mass (bottom) as a function of height and time at $r=50$ gravitational radii. The blue dashed lines show the location of the Rosseland photospheres. []{data-label="fig:rhokappaentropyhistory"}](ST_rho_kappa_entropy_50.pdf){width="\columnwidth"} ![Evolution of azimuthally-averaged polar (top panel) and azimuthal (bottom panel) magnetic field components as a function of height and time at $r=50$ gravitational radii. []{data-label="fig:B2B3history"}](STBthetaBphi_50.pdf){width="\columnwidth"} To see why, it is helpful to examine the evolution of various quantities at a particular radius. Figure \[fig:pressureskappa\] shows the evolution of various shell-averaged pressures, energy densities and stresses, as well as the opacity, at $r=50$ gravitational radii. The turbulent kinetic energy density is calculated as $E_k=\rho\left[(v_r-\overline{v_r})^2+(v_{\theta}-\overline{v_{\theta}})^2+(v_{\phi}-\overline{v_{\phi}})^2\right]$/2, where $v_r,v_{\theta}$ and $v_{\phi}$ are the three velocity components and $\overline{v_r},\overline{v_{\theta}}$ and $\overline{v_{\phi}}$ are averaged values (mass weighted) along the azimuthal direction. The dominant form of thermal pressure is radiation pressure, with gas pressure always being completely negligible. However, there are also significant, and sometimes dominant, contributions from turbulent kinetic energy density and magnetic pressure. The temporal relationship of these quantities is shown more clearly in Figure \[fig:fluctuation\]. After $t=50\times10^4r_g/c$, the pressures and energy densities form a repeating cyclic pattern with large turbulent kinetic energy followed by magnetic pressure followed by radiation pressure. These cycles are clearly correlated with the opacity, with an enhancement in opacity followed in time by an enhancement in turbulent kinetic energy density. Figure \[fig:Ekinhistory\] shows how the three components of turbulent velocity (radial, polar, and azimuthal) contribute to the turbulent kinetic energy as a function of time. While MRI turbulence is typically dominated by radial and azimuthal motions, here the epochs of large turbulent kinetic energy density are dominated by polar (i.e. vertical in the disk midplane regions) motions, with radial motions making a secondary contribution. It is clear that these motions are due to vertical convection driven by the epochs of enhanced opacity. Opacity Driven Convection ------------------------- Figure \[fig:rhokappaentropyhistory\] shows the evolution of azimuthally-averaged density, opacity, and specific entropy at radius $50 r_g$, but now also as a function of height (represented by the polar angle $\theta$ near the disk midplane). This provides more detail on why the opacity is driving convection: the creation of unstable vertical density inversions which are buoyantly unstable (note the drop in specific entropy as one enters the inversion from below). The formation of these inversions is due to the presence of the iron opacity bump. In an optically thick, radiation pressure supported disk, hydrostatic equilibrium requires the temperature to drop vertically away from the midplane. If the midplane is on the high temperature side of the iron opacity bump, then opacity can increase vertically outward, increasing the radiation pressure force for a given vertical radiation flux. This can overcome the downward gravitational force, requiring a large increase in density in order to have a compensating gas pressure gradient force to maintain hydrostatic equilibrium. This is exactly the same thing that happens in one-dimensional hydrostatic models of massive star envelopes, and these density inversions trigger convection [@JIA15]. If conditions are optically thick enough that the photon diffusion speed is much less than the sound speed in the gas alone, then convection is efficient at transporting heat and generally wipes out the density inversion. However, when photon diffusion is fast, convection is inefficient and the density inversion can survive in a time-averaged sense [@JIA15]. This is the regime in which low density AGN accretion disks exist, which is why we can still see the density inversions in Figure \[fig:rhokappaentropyhistory\] in spite of the convective turbulence. Very similar behavior to that present in massive star envelopes is therefore happening here in AGN accretion disks. However, the situation is even more interesting here, because the convection is also altering the MRI dynamo and MRI stresses. Enhanced convective turbulence can act to increase the magnetic energy density and to enhance MRI stresses [@HIR14; @SCE18], and this is evident in Figure \[fig:pressureskappa\]: peaks in turbulent kinetic energy are always followed by enhanced magnetic energy and enhanced Maxwell stress. However, this increases turbulent dissipation of accretion power, which therefore increases the temperature and radiation pressure. Again, we are on the high temperature side of the iron opacity bump here, so as the shell-averaged radiation pressure increases, the shell-averaged opacity goes down. (Figure \[fig:rhokappaentropyhistory\] shows that these changes in shell-averaged opacity are reflected in the actual opacities near the disk midplane during these epochs.) This then shuts off convection, which reduces MRI turbulent stress and dissipation, which in turn causes temperature and radiation pressure to decrease, resulting in opacity increasing again and launching another cycle of convection. Maxwell stresses generally dominate angular momentum transport in simulations of MRI turbulence, but Figure \[fig:pressureskappa\] shows that here Reynolds stresses can also be large, and even at times negative (i.e. driving inward angular momentum transport). The negative Reynolds stresses are only present during the epochs of enhanced turbulent kinetic energy, i.e. when convection is peaking. The nonaxisymmetric structures evident in Figure \[fig:rhomidsnapshots\] probably contribute to the epochs of enhanced Reynolds stress. Figure \[fig:B2B3history\] provides more detail on how convection is affecting the magnetic field in the midplane regions of the disk. Azimuthal field reversals are commonly observed in vertically stratified simulations of MRI turbulence (the so-called “butterfly diagram"; @BRA95 [@DAV10; @HOG16]) and such field reversals are occuring here too. However, they are happening on much longer time scales than the usual $\sim10$ orbital period time scale ($\sim2\times10^4~r_g/c$ at $r=50r_g$). In fact, the polarity of the field maintains a consistent sign during epochs of strong convection, with $\sim10$ orbital period field reversals happening only between the convective epochs, e.g. at $\simeq62\times10^4r_g/c$ in Figure \[fig:B2B3history\]. This is exactly the behavior that is observed in vertically stratified shearing box simulations of MRI turbulence with convection [@COL17]. The poloidal component of magnetic field $B_{\theta}$ is also enhanced due to convection but with a random sign near the disk midplane. It was this enhancement of vertical field which was suggested to be the reason behind the enhanced MRI turbulent stresses in strong convection by @HIR14. ![Time and azimuthally averaged vertical profiles of radiation pressure ($P_r$), magnetic pressure ($P_B$) and kinetic term ($\rho v_{\theta}^2$) at radius $50r_g$. The time average is done between $4\times 10^5r_g/c$ and the end of the simulation. All the pressure terms are in unit of $P_0$. The stair step pattern in the profiles is due to prolongation of the data in the region with lower resolutions. []{data-label="fig:verticalsupport"}](vertical_pressure.pdf){width="\columnwidth"} ![Correlations between the shell averaged total stress (Maxwell plus Reynolds) and radiation pressure at $r=50$ gravitational radii for two oscillation cycles as indicated at the top of each panel. Each data point is color-coded according to the turbulent kinetic energy density. All the variables are scaled with the fiducial pressure unit $P_0$. []{data-label="fig:StressPr"}](Stress_pressure.pdf){width="\columnwidth"} Turbulent Pressure Support in the Disk -------------------------------------- In accretion disks without strong convection, turbulent pressure caused by the MRI turbulence is typically much smaller than the thermal pressure. The disk is usually supported against vertical gravity by gas pressure, radiation pressure or even magnetic pressure [@HIR06; @BegelmanPringle2007; @JIA13; @JIA19a]. However, as shown in Figure \[fig:pressureskappa\], the turbulent kinetic energy density can be comparable to the radiation pressure in this simulation, which is another characteristic property of radiation pressure dominated convection in the rapid diffusion regime [@JIA15; @JIA18]. Therefore, the kinetic term $\rho {{\mbox{\boldmath $v$}}}{{\mbox{\boldmath $v$}}}$ in the momentum equation can in principle provide additional support against gravity in the vertical direction. To check this, we plot the time and azimuthally averaged profiles of $P_r$, $P_B$ and $\rho v_{\theta}^2$ along the $\theta$ direction at radius $50r_g$ in Figure \[fig:verticalsupport\]. The gas pressure is completely negligible here and we neglect it. The gradient of $\rho v_{\theta}^2$ is clearly much larger than the radiation pressure gradient and it balances more than $75\%$ of the gravitational force near the disk midplane in this time-average. This provides an alternative or additional explanation as to why the stress is increased when convection is on compared with the suggested mechanism proposed by [@HIR14]. Since the typical size of MRI turbulent eddies in the disk is ultimately limited by the disk scale height, the larger the disk scale height, the larger the stress can be. This is also the original argument of the $\alpha$ disk model [@SS73], where the scale height is determined by the thermal pressure and thus the stress is assumed to be proportional to the thermal pressure. Here strong convection-driven turbulent pressure can itself support the disk, allowing a higher stress than we would expect from radiation pressure alone. If we still calculate an effective $\alpha$ as the ratio of stress and radiation pressure, it will be significantly larger when convection is on. Correlations between Stress and Pressure ---------------------------------------- As mentioned in the Introduction, a radiation pressure supported accretion disk in the classical $\alpha$ disk model is thermally unstable [@SHA76], because the total heating rate changes more rapidly with radiation pressure ($P_r^2$) than the change of the total cooling rate ($P_r$). Although the accretion disk structures we find here, as well as the physics we are simulating, are much more complicated than those in the $\alpha$ disk model, it is still interesting to check how the stress varies with the radiation pressure while intermittent convection is operating in the disk. The shell averaged total stress as a function of the shell averaged radiation pressure at $50r_g$ for two oscillation cycles (within the time intervals $[5,6.5]\times 10^5~r_g/c$ and $[6.5,7.8]\times 10^5~r_g/c$) is shown in Figure \[fig:StressPr\]. Each data point is color coded with the corresponding turbulent kinetic energy density. When the disk oscillates, the stress and pressure form closed loops in this plot. When convection is active, as indicated by the large turbulent kinetic energy density, stress increases rapidly while $P_r$ increases more slowly. This heating then reduces the opacity, turning off convection and decreasing $E_k$. The stress then decreases while $P_r$ continues to increase further, presumably because of the dissipation of the convective turbulent kinetic energy and magnetic energy. Finally, both stress and $P_r$ decrease at roughly the same rate. This confirms that when convection is on, stress follows turbulent kinetic energy density closely. The heating rate increases more rapidly than the change of the cooling rate and that is why the disk heats up. When convection is off, the heating and cooling rate have roughly the same dependence on the radiation pressure. This is perhaps why the disk does not undergo a runaway collapse during the phase when it cools down, which is similar to what [@JIA16] found. ![Space-time diagrams of the local mass accretion rate $\dot{M}$ (top panel, in unit of the Eddington mass accretion rate $\Medd$), and derivatives of the total stress $-r^{1/2}\partial \left(r^3 S\right)/\partial r$. Negative and positive values of $\dot{M}$ mean inward and outward accretion correspondingly. Both $S$ and $\dot{M}$ are smoothed over the neighboring 100 data points in time to reduce the noise. The Eddington accretion rate is defined as $10\Ledd/c^2$ with $\Ledd$ to be the Eddington luminosity.[]{data-label="fig:mdotstress"}](Mdot_stress.pdf){width="\columnwidth"} ![Correlations between surface density $\Sigma$ and shell averaged total stress at $r=50$ gravitational radii for the convective cycle between $[65,78]\times 10^4~r_g/c$. Each data point is color coded with the corresponding turbulent kinetic energy density. []{data-label="fig:stresssigma"}](Stress_sigma_Ekin_r50_65_78.pdf){width="\columnwidth"} Radial Mass Diffusion and Clumping ---------------------------------- With the assumption that angular momentum transport is dominated by local turbulent stresses, the vertically averaged equations of mass and angular momentum conservation can be used to write an equation for surface density evolution [@BAL99]: $$\frac{\partial\Sigma}{\partial t}=-\frac{1}{2\pi R}\frac{\partial\dot{M}}{\partial R}, \label{eq:dsigmadt}$$ where $R$ is the cylindrical polar coordinate radius and the accretion rate (assumed negative for inflow) is given by $$\dot{M}=-\frac{2\pi}{\ell^\prime}\frac{\partial}{\partial R}(R^2W_{R\phi}). \label{eq:mdot}$$ Here $\ell^\prime$ is the radial specific angular momentum gradient and $W_{R\phi}$ is the vertically integrated turbulent stress. In viscous or alpha-disk theory, these two equations can be combined to give a radial mass diffusion equation [@LYN74; @LIG74], but we choose not to do that as it is the radial gradients in stress that most clearly drive the clumping of surface mass density observed in our simulation. That this is so may be seen in Figure \[fig:mdotstress\]. The upper panel shows a space-time plot of the shell-averaged mass accretion rate $\dot{M}$, and it is clear that radial gradients of this quantity match very well the clumping pattern observed in the surface density evolution in Figure \[fig:sigmakappa\], in accordance with equation (\[eq:dsigmadt\]). Of course, this had to be true as it merely tests mass conservation in [Athena++]{}. Less trivial is equation (\[eq:mdot\]), which relies on the assumption that all angular momentum transport is done through local turbulent stresses rather than non-local processes (e.g. the spiral waves that are evident in Figure \[fig:rhomidsnapshots\]). If this is true, then a plot of $-r^{1/2}(\partial/\partial r)(r^3S)$, where $S$ is the shell-averaged Maxwell plus turbulent Reynolds stress, should resemble the pattern in accretion rate. This is plotted in the lower panel of Figure \[fig:mdotstress\], and does indeed approximately match the accretion rate behavior shown in the upper panel. It is therefore radial gradients in the turbulent stresses that are largely responsible for the clumping. These radial gradients can be strong enough that mass can actually sometimes diffuse outward, as is evident in the upper panel of Figure \[fig:mdotstress\]. Note from the bottom panel of this Figure that clumping is occurring because there is a radially local deficit of stress. Even though convection enhances the stress overall, high opacity is actually anticorrelated in time with stress in the convective cycles shown in Figure \[fig:pressureskappa\], and this produces the local deficit. As convection kicks in and the stress is enhanced, and the opacity drops, the clump diffuses away. Note that the derivation of the pure “viscous" instability associated with electron-scattering and radiation pressure dominated classical alpha-disk accretion models relies on an assumption of local thermal equilibrium in order to derive an inverse relationship between stress and surface mass density [@LIG74]. This results in an effective negative diffusion coefficient in the radial mass diffusion equation that results from combining equations (\[eq:dsigmadt\]) and (\[eq:mdot\]) [@PRI81]. This analysis can be generalized to include departures from thermal equilibrium [@SHA76], but the coupling with varying opacity in the convective cycles clearly makes things far more complicated here. We have attempted to analyze the local behavior of stress as a function of surface density, just as we did with stress as a function of radiation pressure in Figure \[fig:StressPr\]. There are epochs where loops in such a diagram form and there is some evidence of stress being inversely proportional to surface density when there is no convection present, e.g. the bottom of the loop in Figure \[fig:stresssigma\] which shows the cycle between $[65,78]\times 10^4r_g/c$ at $r=50r_g$. However, this inverse trend is broken by the onset of iron opacity-driven convection, and this behavior is not always generic. It is therefore unclear that such a classical analysis is appropriate in the presence of these complex convective cycles. Resolution ---------- To check how well the MRI turbulence is resolved during different phases of the oscillation cycles in the simulation, we calculate the ratios between the wavelength of the fastest growing MRI mode and the cell sizes along the polar and azimuthal directions, i.e. the quality factors $Q_{\theta}$ and $Q_{\phi}$ [@HAW11; @SOR12]. These are widely used in non-radiative ideal MHD simulations and indicate that MRI turbulence is fully resolved when $Q_{\phi}\gtrsim 25, Q_{\theta}\gtrsim 6$ or both of them are larger than 10. Following [@JIA19a] (section 3.1), we also use them as a check for our radiation MHD simulations. For the first representative snapshot shown in Figure \[fig:rhov2Dsnapshots\], $Q_{\phi}$ stays around $40$ near the disk midplane for radii smaller than $\approx 55r_g$ and then drops to $11$ inside the high density clump. Similarly, $Q_{\theta}$ stays around $7$ until reaching the high density clump, where it drops to $2$. At time $t=8\times 10^5r_g/c$ when the disk expands, $Q_{\phi}$ varies from 30 to 100 over the whole radial range from $30r_g$ to $100r_g$, while $Q_{\theta}$ varies from $20$ to $\approx 3$. When the disk collapses at $t=6.5\times 10^5r_g/c$, $Q_{\phi}$ varies from 30 inside $45r_g$ to $10$ from $45r_g$ to $\approx 100r_g$. The averaged $Q_{\theta}$ varies from 6 to 3 over the same radial range. This suggests that MRI turbulence is reasonably well-resolved in this simulation and we have the worst resolution when the disk collapses, which is not surprising. Fortunately, the accretion does not stop during the collapsing phase as the opacity-driven oscillation cycle continues. ![History of the total luminosity $L_r$ (scaled with the Eddington luminosity $L_{\rm Edd}$) coming from the disk. The top panel shows the luminosity if we only include the disk inside $60r_g$ while the bottom panel shows the luminosity inside $80r_g$. []{data-label="fig:Lumhistory"}](LC_history.pdf){width="\columnwidth"} Lightcurve Variability ---------------------- The disk oscillation cycles driven by the opacity also cause the total luminosity coming from the photosphere to vary significantly with time. The total luminosities in the simulation emerging from radii inside $60r_g$ and $80r_g$, respectively, are shown in Figure \[fig:Lumhistory\]. Normal MRI turbulence without convection can cause the luminosity to vary by a factor of $\sim 2$ over the local thermal time scale. Smaller amplitude variability over the local dynamical time scale can also show up in the luminosity when the optical depth across the disk is low enough (see Figure 1 of @JIA19a). However, with convection driven oscillations in the disk, the luminosity can vary by a factor of $\approx 3- 6$ over the local thermal time scale, which is roughly a few years in this radial range. Discussion ========== The time scale of luminosity variations depicted in Figure \[fig:Lumhistory\] are remarkably consonant with those observed in changing look quasars, and we also see amplitudes of variation by as much as a factor of four. In addition, the variations in scale height of the photosphere depicted in Figure \[fig:rhokappaentropyhistory\] occur on comparable time scales and will effect the ability of this region of the disk to intercept and reprocess radiation from the very inner disk, as well as shadow larger radii of the disk. The typical variability time scale driven by convection is determined by the local thermal time scale at the radial range where the iron opacity bump is located. Since the thermal time scale is roughly related to the local dynamical time scale by $1/\alpha$, the variability time scale will change when the iron opacity bump moves to different radii for different black hole masses and mass accretion rates. For a fixed mass accretion rate in Eddington units, the disk temperature will typically decrease with increasing black hole mass at fixed $r/r_g$. This means the iron opacity bump, which is roughly at a fixed temperature around $1.8\times 10^5$ K, will move closer to the black hole. At the same time, the local dynamical time scale will also increase linearly with black hole mass for a fixed $r/r_g$. The combination of the two effects makes the thermal time scale at the location where the iron opacity is located very insensitive to the black hole mass. In fact, for the classical inner accretion disk in the @SS73 model, the disk midplane temperature scales as $T\propto(r/r_g)^{-3/8}M^{-1/4}$, and is even independent of the accretion rate. This then gives a thermal time scale for fixed midplane temperature independent of both black hole mass and accretion rate, though it is very sensitive to the temperature of the iron opacity bump as well as the alpha parameter: $t_{\rm thermal}\propto \alpha^{-2}T^{-4}$. Real accretion disks will be more complicated than this classic model, as we have tried to demonstrate here. Even so, this suggests that the rapid luminosity variation time scales that we have found in this one simulation may be quite common across many AGNs. However, the amplitude of variability driven by this mechanism may depend on the mass accretion rate and black hole mass when the iron opacity bump moves to different radii from the central black hole. When it is further away from the black hole and when the surface density is smaller due to lower accretion rate, we expect the variability amplitude will get smaller. This is perhaps one reason why changing look AGNs only make up 10 percent of the quasar population. AGN variability has been widely parameterized with stochastic models such as the Auto-Regressive Moving Average (ARMA) approach [@Kellyetal2009; @Kellyetal2014; @Morenoetal2019]. Such modeling provides valuable information on the physical properties of the disk such as the typical timescales associated with the variability. Although the lightcurve from our simulation is still very preliminary and only extends over a short period of time, we have tried ARMA modeling of the luminosity as well as the history of magnetic energy density from the simulation using the [*statsmodels*]{} package [@seabold2010statsmodels]. A lower order ARMA(2,1) model can fit the simulation data very well. Both luminosity and magnetic energy density have the rise timescale of variability around one year. But the variability amplitude for magnetic energy density is larger. This preliminary fitting demonstrates that the simulation data shares some similar stochastic properties to observed AGN lightcurves and it might therefore be possible to constrain the physical properties of the observed system. However, more detailed comparison is beyond the scope of this paper and will be studied in the future. Note from Figure [\[opacity\]]{} that all the effects we have been discussing in this paper are driven by small (factors of $3-5$) enhancements in the Rosseland opacity over the electron scattering value in the iron opacity bump. This is of course due to the huge dominance of radiation pressure over gas pressure in quasar accretion disks, so that even small variations in opacity can have enormous consequences. We assumed solar abundances for our simulation, whereas AGN accretion disks are likely to have significantly supersolar metallicities (@Fieldsetal2007 [@Aravetal2007]). The variability amplitude driven by convection will likely increase with larger metallicity for a given black hole mass and accretion rate, which can be compared with observed properties of AGNs [@JIA16]. One dimensional models of cooler annuli further out in the disk can themselves exhibit density inversions [@HUB00]. Those inversions are due to ionization transitions of hydrogen and helium, as the models of @HUB00 did not include any metals. It is possible that the convective effects we have explored here also happen in these regions of the disk, though if anything they are likely to be more dramatic, as is the case in massive stars when hydrogen and helium opacity effects come into play [@JIA18]. The convective cycles we have observed here share many similarities to those observed in stratified shearing box simulations of local patches of disks in cataclysmic variables [@HIR14; @COL17; @SCE18; @COL18] and protostellar disks [@HIR15]. This includes the intermittency of convection[^2] and the enhanced stresses and persistent magnetic polarity during epochs of convection. This is the first time that these effects have been confirmed in a global simulation. However, this is in a very different regime of radiation pressure dominated flows, and the enhanced vertical support caused by the convective turbulence itself may be a contributing factor to the enhanced stresses. This possible new mechanism for convection-driven enhancement of turbulent stresses cannot explain the enhanced stresses observed in shearing box simulations of gas-pressure dominated disks [@HIR14; @HIR15; @SCE18; @COL18]. There the turbulent kinetic energies were always much less than the thermal pressure, and did not contribute significantly to vertical support of the disk. In fact, convective epochs typically had smaller vertical scale heights than radiative epochs. The large turbulent kinetic energies which are present here are due to radiation pressure dominated convection in a regime where the photon diffusion speed is larger than the gas sound speed. In fact, we also observe such supersonic convection in this regime in massive star envelopes [@JIA18]. In that case, the energy source for these supersonic motions is the flow of heat from the core of the star. The turbulent velocity is much smaller than the radiation sound speed deep in the star and can become comparable to the local radiation sound speed near the photosphere. This is possible because the pressure scale height is $\approx 20\%$ of the stellar radius and the size of the turbulent eddies are comparable to the background radiation pressure scale height. A buoyant fluid element accelerated in the deeper regions can reach a velocity comparable to the local radiation sound speed when it moves to a larger distance. For the case of accretion disks, the energy driving convection ultimately arises from the accretion power. As we have shown in Figure \[fig:mdotstress\], the effects of convection are also not strictly local in radius, as radial mass motions result from the change of angular momentum transporting stresses. The convection is therefore able to tap into the free energy stored in differential rotation, and reach supersonic velocities. We have not explained the formation of the $m=2$ nonaxisymmetric density structures that are evident in Figure \[fig:rhomidsnapshots\]. One possibility is that when a surface density peak is formed in a localized radial range as shown in Figure \[fig:sigmakappa\], the disk is potentially subject to the Rossby wave instability, which has been widely studied for protoplanetary disks [@Lovelaceetal1999; @Lietal2001; @Lyra2012]. This instability can create vortices and excites high frequency waves and even spiral density structures in the disk. [@Lovelaceetal1999] show that the sufficient condition for the Rossby wave instability is that the the inverse potential vorticity multiplied by the entropy function $S$, which is $S^{2/\Gamma}\Sigma/\left(\bfnabla\times {{\mbox{\boldmath $v$}}}\right)_{z}$, has a local maximum as a function of radius in the disk. Here $\Gamma$ is the adiabatic index. It is currently unclear how this instability criterion can be generalized to the radiation pressure dominated regime with realistic 3D structures as in our simulation. Nevertheless, we checked this criterion using $\Gamma=4/3$ and the radiation entropy per unit mass. Indeed, this function does show a local maximum at the location where the high density clump is located. In fact, this function already shows local extrema before the density clumps and spiral patterns can be clearly seen in the disk as shown in Figure \[fig:rhomidsnapshots\] and \[fig:sigmakappa\]. We will leave the detailed study of Rossby wave instability in AGN disks for future studies. But this suggests that it is one possible mechanism to explain the nonaxisymmetric structures we have found in the simulation. A major caveat of our simulation is that it is so expensive that we can only afford to run the simulation for a few thermal timescales for the inner $\sim 60r_g$. The time-averaged mass accretion rate is not a constant as a function of radius, which is necessary if the disk is in steady state. This could either be because the simulation time is not long enough, or a steady state disk is simply not possible when convection driven oscillation are operating. We hope to investigate this further with future calculations of longer duration. Acknowledgements {#acknowledgements .unnumbered} ================ Resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center. OB acknowledges the support of a 2019 JILA visting fellowship which enabled very helpful conversations with Jason Dexter on changing look quasars and with Ben Brown, Evan Anders, and Imogen Cresswell on the physics of supersonic convection. We also thank Matt Coleman, Shigenobu Hirose, Yue Shen and all the members from the Horizon Collaboration for useful discussions. The Center for Computational Astrophysics at the Flatiron Institute is supported by the Simons Foundation. This research was also supported in part by the National Science Foundation under Grant No. NSF PHY-1748958 [^1]: The Planck mean opacity is from the TOPS opacity at this website [ https://aphysics2.lanl.gov/apps/]( https://aphysics2.lanl.gov/apps/) [^2]: Persistent convection has only been observed in local simulations of helium-dominated accretion disks by @COL18.

--- abstract: 'Nuclear matter under astrophysical conditions is explored with time-dependent and static Hartree-Fock calculations. The focus is in a regime of densities where matter segregates into liquid and gaseous phases unfolding a rich scenario of geometries, often called nuclear pasta shapes (e.g. spaghetti, lasagna). Particularly the appearance of the different phases depending on the proton fraction and the transition to uniform matter are investigated. In this context the neutron background density is of special interest, because it plays a crucial role for the type of pasta shape which is built. The study is performed in two dynamical ranges, once for hot matter and once at temperature zero to investigate the effect of cooling.' author: - 'B. Schuetrumpf$^1$, K. Iida$^3$, J. A. Maruhn$^1$, P.-G. Reinhard$^2$' bibliography: - 'ProtonFraction.bib' title: Nuclear Pasta Matter for Different Proton Fractions --- Introduction ============ Nuclear matter in supernova cores and proto-neutron stars covers a wide range of density, temperature, and proton fraction, and hence exhibits various interesting properties such as, e.g., liquid-gas mixing [@Lamb]. It is believed that after the bounce of the inner core, cooling and deleptonization lead to drastic changes in the temperature and proton fraction of each matter element inside the core [@Bethe; @Suzuki]. Just below normal nuclear density, matter is likely to disentangle into exotic shapes, which resemble pastas like spaghetti and lasagna and can affect neutrino transport [@Ravenhall; @Hashimoto; @Lassaut; @Pethick; @Chamel; @Horowitz2004; @Horowitz20042; @Sonoda2007]. Steadily increasing computing power allows meanwhile to simulate pasta matter by fully three-dimensional, symmetry unrestricted time-dependent Hartree-Fock (TDHF) calculations. This has led to a revival of investigations of structure and dynamics of pasta matter within TDHF [@NewtonStone; @Sebille; @Sebille2011; @Pais; @Schuetrumpf2013a] as well as classical and quantum molecular dynamics calculations [@Sonoda2008; @Schneider2013]. In this contribution, we address the question how the map of pasta shapes changes with the proton fraction and how the neutron excess is distributed between the nuclear liquid and a gaseous neutron background. Nuclear pasta matter for different values of the proton fraction has been studied so far using classical molecular dynamics [@Dorso], the TDHF-like DYWAN model[@Sebille2011], and a Thomas-Fermi approximation based on a relativistic mean-field model [@Okamoto]. Here we aim at a fully quantum mechanical TDHF description. We thus extend our previous TDHF calculations for proton fraction 1/3 [@Schuetrumpf2013a] to the general case of arbitrary proton fractions. Thereby we concentrate on the regime of low densities up to saturation density of symmetric nuclear matter. We consider phases at high temperature in the MeV range which are relevant for core-collapse super-novae as well as phases at zero temperature which have relevance to materials that deleptonize in proto-neutron stars. For the calculations done here the Skyrme-TDHF code as explained in [@Maruhn2014] is used. For the astrophysical calculations periodic boundary conditions are assumed. For the interaction the Skyrme force SLy6 is taken [@Chabanat]. Additionally to the proton charge distribution we take a uniform electron background to keep the charge neutrality. Simulation and basic quantities =============================== Static and dynamic properties of inhomogeneous nuclear matter are computed with the TDHF code. For the astrophysical calculations performed here, periodic boundary conditions are applied, thus simulating infinite matter approximately with a periodic lattice of simulation boxes. The imposed periodicity exerts a constraint on the system which, in turn, may shift a bit the transition points from one phase to another. We have checked the impact of box size earlier [@Schuetrumpf2013a] and find that effects are not dramatic for the simple geometries discussed here, although box size can become an issue when dealing with involved shapes as, e.g., gyroids [@Schuetrumpf2014a]. The actual simulation box uses a grid spacing of 1 fm and 16$\times$16$\times$16 grid points thus having a box length $l=16$ fm and a box volume $V=l^3$. We vary the number of $\alpha$-particles $N_\alpha$, and of additional neutrons $N_{n+}$. The Coulomb field in a periodic simulation is given by the Ewald sum [@Ewa21a]. The Fourier representation of kinetic energy allows a very elegant computation in Fourier space. Total charge neutrality is achieved by assuming a homogeneous electron background. This approximation ignores screening effects from the electrons, for detailed discussions see [@Mar05b; @Watanabe2003a; @Dorso2012; @Alcain]. Although screening lengths can vary in a wide range and are occasionally of the order of structure size, the net effect of electron screening is found to be small [@Mar05b; @Watanabe2003a], thus justifying the homogeneous approximation for the present exploration. The initial state is prepared in similar fashion as in [@Schuetrumpf2013a]. From the $N_p$ protons together with $N_p$ neutrons, we form $N_\alpha=N_p/2$ $\alpha$-particles. These are distributed stochastically over the whole space of the simulation box while keeping a minimal distance of ${3.5\rm\,fm}$ between the centers of the $\alpha$-particles to avoid too large overlaps between them. The $\alpha$-particles are initialized at rest. The remaining $N_n-N_p$ neutrons are initialized as plane waves filling successively the states with the lowest kinetic energies with degenerate pairs of spin-up and spin-down particles. Finally, all nucleons are ortho-normalized. The then following time evolution of the system is done by standard TDHF propagation [@Maruhn2014]. This initial state is far above the ground state. During the first few fm/c of dynamical evolution, the system quickly evolves into a fluctuating thermal state whose average properties (shapes, densities, energies) remain basically constant. The emerging temperatures end up in a range well representing pasta matter in core-collapse supernovae. They lie between 2 and 7 MeV depending on proton fraction $X_p$ and density $\rho$ where lower $X_p$ and larger $\rho$ are found to be associated with lower temperatures. For the present exploratory stage, we use just this broad temperature range as representatives of hot matter. In a second step, we cool down the system for each $X_p$ and density $\rho$ to a locally stable zero-temperature state. To that end, we start from a given dynamical state and perform a static calculation with standard techniques of the code [@Maruhn2014]. The starting configuration is taken from some time point in the late phases of the TDHF simulation. The precise time is unimportant for the final result. The whole procedure (random initialization, evaluation of thermal state, cooling) is done twice for each setting of $X_p$ and $\rho$. This gives some clue on the stability of a configuration. A word is in order about the analysis of non-homogeneous structures of infinite systems in a finite simulation box. It is known that the imposed boundary conditions have an impact on the structures due to symmetry violation and possibly spatial mismatch [@Hoc81aB; @All87]. A study within classical molecular dynamics without Coulomb interactions implies that this may be particularly critical for extended, inhomogeneous structures [@Gimenez]. However, the numerical expense of fully quantum-dynamical simulations sets limits on the affordable box sizes. To check its effect, we have studied for the case of $X_p=1/3$ also larger boxes up to 26 fm and find the same structures. Thus we confine the exploratory survey here to the one box size 16 fm. In order to characterize the matter, we will use a few global properties. The total number of protons and neutrons is $$N_p=2N_\alpha \quad,\quad N_n=2N_\alpha+N_{n+}$$ where $N_\alpha$ is the number of initial $\alpha$ particles (strictly related to $N_p$) and $N_{n+}$ the number of excess neutrons (not absorbed into initial $\alpha$ particles). The key parameter is the proton fraction $X_p$ $$X_p=\frac{N_p}{N_p+N_n}=\frac{2N_\alpha}{2N_\alpha+N_{n+}}\leq\frac{1}{2} \quad.$$ The latter inequality means that we confine studies to neutron rich systems which is the typical scenario in astrophysical environment. There are several further measures related to densities. The most important one is the mean density $$\rho = \frac{N_p+N_n}{V} = \frac{4N_\alpha+N_{n+}}{V} = \frac{A}{V}$$ where $V$ is the box volume and $A=N_p+N_n$ is the total nucleon number. It is only at high densities that matter stays homogeneous. Usually, matter segregates into dense regions of nuclear liquid with density $\rho_l$ and a dilute gas phase with density $\rho_g$, consisting here of a neutron gas. These two phases fill correspondingly volumes $V_l$ and $V_g$ with $V_l+V_g=V$. These volumes are defined with the help of the Gibbs dividing surface [@Shchukin]. A threshold density $\rho_\mathrm{thr}$ is set and all regions with $\rho(\mathbf{r})>\rho_\mathrm{thr}$ are added to $V_l$ and all other to $V_g$. The threshold is determined such that $$\rho_l\cdot V_l+\rho_g\cdot V_g = A \quad.$$ From these volumes, the volume fraction $u_l$ of the liquid phase can be deduced as $$u_l = \frac{V_l}{V} = \frac{\rho-\rho_g}{\rho_l-\rho_g} \quad. \label{eq:Gibbs}$$ Since $\rho$ is always smaller than $\rho_l$ for pasta shapes and the gas density consists of the background densities, the volume fraction becomes smaller for larger neutron background densities and the type of pasta shape which is formed depends strongly on this volume fraction. The volume fraction $u_l$ is the most important parameter to determine the geometry of the system. With steadily increasing $u_l$, the system marches through a series of geometrical phases. For the lowest $u_l$ the individual shapes are the sphere, which corresponds to finite nuclei with large zones of vacuum around. In the next stage, the nuclei fuse to cylinder-like shapes, the “rod” structure, also denoted as “spaghetti”. Further increasing $u_l$ leads to planar meshes of orthogonal rods, “rod(2)”, followed by three-dimensional grids of rods, denoted “rod(3)”. Equally densely packed is the “slab” structure, parallel planes completely filled with matter, also called in a more appetizing manner “lasagna”. From then on the picture is reversed. We have more matter than voids. The slab and rod(3) are symmetric under exchange of matter and voids, thus residing at the turning point. Further increasing $u_l$ then yields the phases of “rod(2) bubble”, “rod bubble”, and “sphere bubble”, where bubble denotes that the gas phase has the shape of the pasta. A further increase ends up in homogeneous matter. These are the phases which we will study in the following. Results =======  Binding energy per particle $E/A$ as function of total density for asymmetric nuclear matter at zero temperature for various proton fractions $X_p$ as indicated. The ground states are indicated by black boxes for those $X_p$ where a local minimum could be found. The curves have been computed for the Skyrme force SLy6.](EoS-protfrac.eps){width="\linewidth"}  Equilibrium density as function of proton fraction $X_p$ for different Skyrme interactions (explanations see text). Each curve stops at some low $X_p$ because no equilibrium (minimum of energy as function of $\rho$) could be found for smaller $X_p$. ](vary-asymnucmat-rvsfr.eps){width="0.8\linewidth"} {width="0.95\linewidth"} Equation of state of homogeneous matter --------------------------------------- As a first step, we look at the binding energy curves of homogeneous asymmetric nuclear matter for the SLy6 interaction as shown in Fig. \[fig:EoS-protfrac\]. Pure neutron matter ($X_p=0$) is unbound, a feature which is well known. Unbound matter persists for small $X_p$ up to about $X_p=0.09$. For $X_p=0.1$ we are able to find a local minimum at non-zero density which, however, is a metastable state. A well bound ground state (negative energy) emerges then from $X_p=0.13$ on. Binding and equilibrium densities increase very quickly above $X_p=0.13$. We thus have a clear distinction between low proton content $X_p$ and moderate or large one with a proton fraction around a critical point $X_p\approx 0.13$. Once substantial binding sets on above the critical point, the equilibrium densities saturate quickly around $\rho_\mathrm{nm}=0.12-0.16$ fm$^{-3}$. Just recently a study appeared of the equation-of-state of asymmetric nuclear matter based on classical molecular dynamics simulations [@Lop14a]. Although the formal framework is very different, the classical results at temperatures as low as 2 MeV show the same trends as seen here at zero temperature, e.g., the shape of the binding curves and the drift of the equilibrium density with $X_p$. The sequence of equations of state for different $X_p$ looks very similar for all Skyrme interactions which we have studied. It can be characterized by the sequence of equilibrium points shown as black boxes in Fig. \[fig:EoS-protfrac\]. Fig. \[fig:vary-asymnucmat\] shows this sequence of equilibrium points for a great variety of Skyrme interactions. The upper panel shows results for widely used interactions found in the literature, SkM\* [@Bar82a], SLy6 [@Chabanat], SkI3 [@Rei95a], and UNEDF0 [@Kor10]. The lower panel shows results from a set of interactions in which nuclear matter properties have been systematically varied with respect to standard values set in SV-bas [@Klu09a], SV-mas07 with lower effective mass, SV-K218 with smaller incompressibility, SV-sym34 with larger symmetry energy, and SV-kap00 with smaller Thomas-Reiche-Kuhn sum rule enhancement factor (equivalent to isovector effective mass). All of the interactions show very similar behavior. The largest deviations are seen for the interactions with untypically large symmetry energy, SkI3 in the upper panel and SV-sym34 in the lower one. Even including these, all interactions show the same trends and the binding energies in dependence on the proton fraction are almost identical. Therefore we expect the results in this work obtained with SLy6 representative for all reasonable Skyrme interactions. Varying the interaction may shift slightly the borders between phases. But the overall sequence should be robust. The map of pasta shapes ----------------------- The map of pasta shapes obtained from the dynamical simulations under various initial conditions is shown in the upper panel of Fig. \[fig:phadiag\_X\_P\]. For every value of $X_p$ and mean density $\rho$, two calculations with different initial conditions are performed. These yield in most cases the same phase. There are few cases where two different final states are reached. This is indicated by showing two different colors in a cell. The value $X_p=0.3$ is replaced by $X_p=1/3$ to establish contact with the previous study [@Schuetrumpf2013a] which worked exclusively at this proton fraction. To fill the resulting larger gap towards $X_p=0.2$, we also show a lower value $X_p=0.29$. At first glance, one sees a jump between $X_p=0.1$ and the larger $X_p$ to the extent that $X_p=0.1$ has a much larger range of pure nuclear matter on the side of high $\rho$. This reflects the “phase transition” around $X_p\approx 0.13$ observed in the equation of state in Fig. \[fig:EoS-protfrac\]. The transition between pure nuclear matter and structured matter is, in fact, predominantly a competition between the spherical bubbles (black boxes) and homogeneous matter. All other shapes show steady, often moderate, changes with $X_p$. The sequence in which the shapes appear and disappear with increasing density is practically the same at all $X_p$. There is a slight trend, however, that for an increasing proton fraction the different pasta shapes appear at lower mean densities. Take, e.g. the border between rod and rod(2): it is above a mean density of $\rho=0.05{\rm\,fm^{-3}}$ for $X_p=0.1$ and moves to about $0.04{\rm\,fm^{-3}}$ for symmetric matter ($X_p=0.5$). In Ref. [@Okamoto] the trend that for low proton fractions pasta matter appears in a smaller range in density is also clearly visible. The densities for the transitions between normal nuclei, pasta matter, and uniform nuclear matter are in detail different, like in other approaches. These numbers seem to be very sensitive to the equation of state used for the calculations. It is interesting to compare the present results at $X_p=0.33$ to our earlier simulations [@Schuetrumpf2013a]. The sequence of shapes is the same. But the middle region shows more slab configurations. This is probably an effect of different initialization. The earlier calculation allowed a smaller minimal distance between the $\alpha$-particles leading to an initial state which contains more clustered $\alpha$-particles obviously driving the slab phase. The appearance of shapes thus seems to depend to some extent on the initial state. This indicates that the barriers between the different shapes are sufficiently high to stabilize a shape even though it is not necessarily the minimum configuration. As a further check of the stability of the dynamical configurations, we have cooled each of them down to temperature zero actually starting the static iteration from the final stage of dynamical evolution. The resulting map of shapes is shown in the lower panel of Fig. \[fig:phadiag\_X\_P\]. Most of the dynamical configurations persist when cooled down. There are only a few small shifts of borders between shapes and, not surprisingly, a few more slabs pop up which corrects somewhat the underestimation of slabs. A subtle difference in detail ought to be mentioned. The border to uniform nuclear matter is shifted to smaller mean densities for $X_p=0.1$ but to higher mean densities for all higher proton fractions $X_p>0.1$. The jump in the border between uniform nuclear matter and spherical bubbles thus becomes even more pronounced. In fact, the larger jump for the cooled configurations is closer to the sharp transition indicated in the equation of state, see Fig. \[fig:EoS-protfrac\]. Increasing temperature weakens this transition. Neutron background ------------------  Upper panel: Neutron background densities ($\rho_{\rm nbg}$) of the series of calculations of Fig. \[fig:phadiag\_X\_P\]. Full symbols show results from dynamical calculations and open symbols from zero-temperature configurations, marked with the additional index C in the legend. The case $X_{p,C}=0.5$ is not shown because it did not produce a background. The shaded area indicates the region where $\rho_g$ is only vaguely defined. Lower panel: Neutron Fermi energies for the cooled Pasta shapes in dependence on the mean density for different proton fractions are shown.](X_P_nbg.eps){width="\columnwidth"} We have matter with very different neutron contents. It is thus interesting to look for the gaseous neutron background density $\rho_g$ under the different conditions. In order to find $\rho_g$, we follow the strategy of Ref. [@Schuetrumpf2013a] and compute the distribution of volumes of the neutron density, $v(\rho_\mathrm{ref})=\int d^3r\, \delta(\rho_\mathrm{ref} -\rho_n(\mathbf{r}))$. This displays typically a clear peak at low $\rho_\mathrm{ref}$. A Gauss fit to this low density peak in the neutron distribution then yields $\rho_g$. No clear peak is visible for higher mean densities. In this case, the edge where the curves starts to differ from zero was taken as an indicator. The proton background densities, if present at all, are very small and can be ignored. The results are shown in the upper panel of Fig. \[fig:X\_P\_nbg\]. The shaded area at high $\rho$ indicates the region where the determination of $\rho_g$ becomes uncertain. Naturally, $\rho_g$ is very large for the smallest proton fraction $X_p=0.1$ and drops quickly with increasing $X_p$. The values from cooled configurations are systematically lower than for the dynamical ones. This reflects the fact that cold bound systems can accommodate more extra neutrons than hot ones. For $\rho\gtrsim 0.1{\rm\,fm^{-3}}$ (shaded area) the tails of the density distributions in the voids of the rod bubble or the sphere bubble cover very little space and thus do not reach small values for the density. Therefore in these configurations a large neutron background is present. While the neutron background is present for all proton fractions in the finite temperature calculations, it vanishes completely for $X_p>0.29$ for the zero temperature calculations. In order to corroborate this result, we show in the lower panel of Fig. \[fig:X\_P\_nbg\] the neutron Fermi energies $\epsilon_{F,n}$ for the cooled configurations. For $X_p=0.1$ and $X_p=0.2$, the $\epsilon_{F,n}$ stay significantly above zero, confirming that a large amount of neutrons are unbound. The case of $X_p=0.29$ is transitional to the extent that the $\epsilon_{F,n}$ are close to zero and therefore, the neutron background is only marginally present. For $X_p=0.33$ the Fermi energies are slightly below zero and thus there is no neutron background present any more. For even higher $X_p$ the Fermi energies drop further below zero and yield a strongly bound system. For finite temperature calculations the distribution around the Fermi edge is softened and therefore a small neutron background is present even for large proton fractions. Liquid phase ------------  Upper panel: Liquid phase densities for two representative values of $X_p$. The cooled calculations are marked with the additional index C. Lower panel: Liquid phase occupied volume fractions $u_l$ for two $X_p$. The color bars on the right indicate a pasta shape with the color code as explained in Fig. \[fig:phadiag\_X\_P\]. The bars mark the intervals of $u_l$ for which the pasta shape associated with this color appears.](rho_l-u.eps){width="\linewidth"} The liquid densities $\rho_l$ are also computed from the distribution of the total density $v(\rho_\mathrm{ref})$, but now fitting the high density peaks or flanks. The results for $\rho_l$ are shown in the upper panel of Fig. \[fig:rho\_l\]. Only graphs for two values of $X_p$ are shown because there is very little variation of the $\rho_l$ for $X_p=0.2...0.5$. The line for $X_p=0.4$ is taken to represent all lines for $X_p\geq0.2$. Visibly different is the case $X_p=0.1$ where $\rho_l$ is lower than for the other $X_p$ for both cases, cooled and dynamic. This complies well with the equilibrium densities shown in Fig. \[fig:EoS-protfrac\] which also shows this marked difference between low $X_p\leq0.1$ and larger $X_p>0.13$. The actual values for $\rho_l$ can differ from the case of homogeneous matter, because the liquid phase is not as neutron rich as the proton fraction indicates due to the separate neutron background and due to surface and Coulomb effects. The liquid phase density decreases with increasing mean density and converges to the mean density below nuclear saturation density. Not surprisingly, the cooled calculations show higher liquid densities because they are better bound. The effect is much more pronounced for high $X_p$. From $\rho_l$ and $\rho_g\approx\rho_{\rm nbg}$ the volume fraction $u_l$ can be derived using Eq. (\[eq:Gibbs\]). The result is displayed in the lower panel of Fig. \[fig:rho\_l\]. Again, we take one representative $X_p=0.4$ for the whole group $X_p\geq 0.2$. Most curves show an increasing slope, so they grow faster than linearly. This correlates to the fact that the liquid density decreases with increasing density (upper panel of Fig. \[fig:rho\_l\]) which results in a larger volume fraction. Very interesting is the behavior of the curves for $X_p=0.1$. There are two counteracting effects. First, we have a low liquid density which results in a larger volume fraction. Second, $\rho_g$ is very large which reduces the volume fraction. The figure shows that for low mean densities the background neutron effect dominates, resulting in lower volume fraction than for the other $X_p$. For higher mean densities the small values for $\rho_l$ dominate and the $u_l$ for $X_p=0.1$ crosses the other lines making the transition from low $u_l$ to high ones significantly steeper than for the larger $X_p$. This can also be seen in Fig. \[fig:phadiag\_X\_P\]: The sequence of non-trivial shapes (rod until spherical-bubble) is compressed to a smaller range of $\rho$ than for $X_p\geq 0.2$. The impact of the type of the Skyrme force on these results can be estimated from Fig. \[fig:vary-asymnucmat\]. The main difference is that the liquid densities would be different especially for SkI3 and $a_{\rm sym}=34{\rm\,MeV}$. Therefore the values in Eq. \[eq:Gibbs\] change slightly and the transition points from one to another pasta structure would change. The overall structure of the map of pasta should be the same for all interactions. Conclusion ========== Using time-dependent Hartree-Fock simulations and static Hartree-Fock calculations, we have investigated the scenarios of the various “pasta” configurations in nuclear matter under astrophysical conditions. Thereby, we have explored the appearance of the different shapes (sphere, rod, slab, ..., bubble) in dependence on a given mean density and proton fraction. The dynamical simulations produce thermal states with temperatures in the range 2–7 MeV. The static calculations start from the thermal states and serve to check the stability of geometry under changing thermal conditions. We find a clear distinction between a regime of low proton fractions $X_p\leq 0.13$ and high ones. The results for $X_p=0.2...0.5$ show practically the same shapes as function of mean density $\rho$ while the sequence of non-trivial geometries is compressed to a smaller density range for $X_p=0.1$. The sequence as such is similar in all cases, starting from isolated spheres (nuclei) at very low density, proceeding over rods, planar meshes of orthogonal rods, triaxial meshes of rods, slabs, to the reciprocal profiles of planar meshes of rod-like bubbles, linear bubble-rods, spherical bubbles, and finally homogeneous matter. The special properties of the small low-density regime are corroborated from the equation of state of homogeneous matter. Matter is unbound for low $X_p$. Binding sets on at around $X_p=0.13$ and develops very quickly to strong binding with an almost constant equilibrium density. Generally, the matter segregates into high-density regions of nuclear quantum liquid and low-density regions of a gas of background nucleons, predominantly neutrons. Its density increases with mean density $\rho$ and decreases with increasing $X_p$. There is also a significant differences between thermal and cooled state to the extent that the cooled state contains much less neutron gas, for $X_p>0.29$ even no gas at all. These observations are corroborated by the neutron Fermi energies which are above zero in the cases of largest neutron gas background. The volume fraction $u_l$ (liquid volume divided by total volume) is the most important parameter determining the pasta shapes. Each shape is found to have a certain interval of $u_l$ for which it can appear. The trend $u_l(\rho)$ shows also a clear distinction between $X_p=0.1$ and all larger $X_p$. The curve is steeper for the low $X_p$ which is related to the compressed sequence of shapes at $X_p=0.1$. All in all, the results are rather robust in the region of larger proton fraction $X_p=0.2...0.5$. This means earlier investigations done for one proton fraction, mostly $X_p=1/3$ apply in this whole range. The region of low proton fractions is different and generally more sensitive to the system parameters. This work was supported by the Bundesministerium für Bildung und Forschung (BMBF) under contract number 05P12RFFTB and by Grants-in-Aid for Scientific Research on Innovative Areas through No. 24105008 provided by MEXT.

--- abstract: | We study path integration on a quantum computer that performs quantum summation. We assume that the measure of path integration is Gaussian, with the eigenvalues of its covariance operator of order $j^{-k}$ with $k>1$. For the Wiener measure occurring in many applications we have $k=2$. We want to compute an $\e$-approximation to path integrals whose integrands are at least Lipschitz. We prove: - Path integration on a quantum computer is tractable. - Path integration on a quantum computer can be solved roughly $\e^{-1}$ times faster than on a classical computer using randomization, and exponentially faster than on a classical computer with a worst case assurance. - The number of quantum queries needed to solve path integration is roughly the square root of the number of function values needed on a classical computer using randomization. More precisely, the number of quantum queries is at most $4.22\, \e^{-1}$. Furthermore, a lower bound is obtained for the minimal number of quantum queries which shows that this bound cannot be significantly improved. - The number of qubits is polynomial in $\e^{-1}$. Furthermore, for the Wiener measure the degree is $2$ for Lipschitz functions, and the degree is $1$ for smoother integrands. author: - | J. F. Traub\ [ Computer Science, Columbia University]{}\ H. Woźniakowski\ [ Computer Science, Columbia University and]{}\ [ Institute of Applied Mathematics, University of Warsaw]{} title: 'Path Integration on a Quantum Computer[^1]' --- Introduction ============ Although quantum computers currently exist only as prototypes in the laboratory, we believe it is important to study theoretical aspects of quantum computation and to investigate its potential power. There will be additional incentives to try to build quantum computers if it can be shown that there are substantial speed-ups for a variety of problems. To date there have been two major algorithms for discrete problems on quantum computers that are significantly better than on classical computers: Shor’s factorization and Grover’s data search algorithms, see [@G96; @G98; @S94; @S98]. But numerous problems in science and engineering have [*continuous*]{} mathematical models. Examples include high dimensional integrals, path integrals, partial differential and integral equations, and continuous optimization. Continuous problems are usually solved numerically; they can only be solved to within uncertainty $\e$. The computational complexity of these problems on classical computers is often known; for a recent survey see [@TW98]. Complexity is defined to be the minimal number of function values and arithmetic operations needed to solve the problem to within $\e$. For many continuous problems defined on functions of $d$ variables, the complexity in the worst case deterministic setting is exponential in $\e^{-1}$ or in $d$. In the latter case, the problem is said to suffer from the “curse of dimensionality” and is computationally intractable. For some continuous problems the curse of dimensionality can be vanquished by weakening the worst case deterministic assurance to a stochastic assurance, such as in the randomized setting. Monte Carlo is a prime example of an algorithm in the randomized setting. A start has been made toward solving continuous problems on quantum computers in recent papers [@AW99; @H01a; @H01b; @HNa; @HNb; @N01; @NSW]. They study multivariate integration and approximation. The major technical tool in these papers is the quantum summation algorithm of Brassard, Høyer, Mosca and Tapp that is based on Grover’s iterate, see [@BHMT00; @G96]. The essence of the results of Heinrich and Novak, [@H01b; @HNa; @N01], is that intractability in the worst case setting of multivariate integration in a Sobolev space is [*broken*]{} by the use of the quantum summation algorithm. That is, we have an [*exponential*]{} speed-up of quantum algorithms over deterministic algorithms with a worst case assurance. Furthermore, there is roughly a [*quadratic*]{} speed-up of quantum algorithms over randomized algorithms run on a classical computer. Our paper is a continuation of the idea of using quantum summation for continuous problems. Summation is often required for continuous problems. Algorithms such as Monte Carlo and Quasi-Monte Carlo are used for a variety of continuous problems and they require the summation of many terms. In the worst case setting, the number of terms $n$ is often an exponential function of $\e^{-1}$. However, if we perform summation on a quantum computer, this is [*not*]{} a show stopper, since the cost of the quantum summation algorithm depends only logarithmically on $n$. Hence, as long as $n$ is a single exponential function of $\e^{-1}$ the quantum cost is [*polynomial*]{}, and the problem becomes [*tractable*]{} on a quantum computer. In this paper we show that quantum summation is a powerful tool for computing also path integrals. Path integrals may be viewed as integration of functions of [*infinitely*]{} many variables. They occur in many fields, including quantum physics and chemistry, differential equations, and financial mathematics. Efficient algorithms for approximating path integrals are therefore of great interest. However, and perhaps not surprisingly, path integration is [*intractable*]{} on a classical computer in the worst case setting for integrands with finite smoothness as shown in [@WW96]. Fortunately, the worst case complexity of path integration is only a single exponential function in $\e^{-1}$ if the measure of path integration is Gaussian and the eigenvalues of the covariance operator are of order $j^{-k}$ for $k>1$. For the Wiener measure, which appears in many applications, we have $k=2$. That is why when we use the quantum summation algorithm, path integration becomes [*tractable*]{} on a quantum computer. More precisely, for functions having smoothness $r$, see the precise definition of the class $F_r$ in Section 3, path integrals can be computed using of order - $\e^{-1}$ quantum queries, - $\e^{-(k+\gamma(r))/(k-1)}\,\log \e^{-1}$ quantum operations, and - $\e^{-(1+\gamma(r))/(k-1)}\,\log\,\e^{-1}$ qubits. Here $\gamma(1)=1$ and $\gamma(r)=0$ for $r\ge2$. For the Wiener measure, we have more specific bounds, which we present in Theorem 2. We stress that the cost of a quantum query depends on a particular applications and may be very large. We know more precise bounds on the number of quantum queries. To explain them we comment on two types of errors for path integration. The first error occurs when we replace the original problem by finite dimensional Gaussian integration, and then the second one when we approximate the finite dimensional problem by a finite sum. For simplicity, we assume that the both errors are bounded by $\e/2$ so that the total error is at most $\e$. Hence, we need to apply the quantum summation algorithm with error $\e/2$. To get a $\delta$-error, the quantum summation algorithms requires at most $2.11\delta^{-1}$ quantum queries and this bound is in general sharp, see [@KW]. Hence, for $\delta=\e/2$ we need at most $4.22\e^{-1}$ quantum queries. Obviously, we may reduce the number of quantum queries by choosing a different splitting of the two errors for path integration. So, if the first error is, say, $a\e$ and the second is $(1-a)\e$ for some $a\in (0,1)$, the number of required quantum queries is at most $2.11\e^{-1}/(1-a)$, and for small $a$ it is roughly at most $2.11\e^{-1}$. This can be achieved at the expense of increasing quantum operations and qubits. We also study the question what is the minimal number of quantum queries for solving path integration by an arbitrary quantum algorithm. Similarly as in [@N01], we show that path integration is no easier than a specific summation problem. Then using the lower bound of [@NW98], we conclude that the minimal number of quantum queries is at least of order $\e^{-1+\a}$ for any $\a\in(0,1)$, see Theorem 3. This means that the number of quantum queries used by the algorithm presented in this paper cannot be significantly improved. We stress that the number of qubits is polynomial in $\e^{-1}$. Furthermore, for the Wiener measure the degree is $2$ for $r=1$, and $1$ for $r\ge2$. Hence, if $\e$ is relatively large we do not need too many qubits to solve path integration on a quantum computer. This is important since the number of qubits will be a limiting resource for the foreseeable future. From these bounds and from the known complexity bounds in the worst case and randomized settings, we conclude that - Path integration on a quantum computer can be solved roughly $\e^{-1}$ times faster than on a classical computer using randomization, and exponentially faster than on a classical computer with a worst case assurance. - The number of quantum queries is the square root of the number of function values needed on a classical computer using randomization. We outline the remainder of this paper. In Section 2 we briefly discuss the complexity of summation in the worst case and randomized settings, and the quantum summation algorithm for computing the arithmetic mean of $n$ numbers, each from the interval $[-1,1]$. In Section 3 we define path integration precisely, while in Section 4 we explain a computational approach to path integration. In Section 5 we summarize what is known about the complexity of path integration on a classical computer in the worst case and randomized settings. We also outline an algorithm of Curbera, [@C00], which requires exponentially many function values in the worst case setting, and which is the basis for the quantum path integration algorithm. In Section 6 we discuss path integration on a quantum computer and summarize the advantages of the quantum algorithm. In Section 7 we prove that the upper bound on the number of quantum queries presented in Section 6 is essentially minimal. In the Appendix we present the proof of how many variables must be used to approximate path integrals to within $\e$. Quantum Summation Algorithm =========================== Sums occur frequently in scientific computation. For example, when Monte Carlo or Quasi-Monte Carlo are used to approximate a $d$-dimensional integral, we compute $n^{-1}\sum_{i=1}^nf(x_i)$, where the $x_i$ are $d$-dimensional vectors that are chosen randomly (for Monte Carlo) or deterministically (for Quasi-Monte Carlo), see e.g., [@N92]. As we shall see in Section 5, such algorithms can be also used for approximating path integrals. In fact, for many linear problems it is known that linear algorithms enjoy many optimality properties, see e.g., [@N88; @TWW88; @TW98]. Linear algorithms have the form $\sum_{i=1}^na_if(x_i)$ for coefficients $a_i$ that are sometimes, but not always, equal to $n^{-1}$. Let $y_i=a_if(x_i)n$. Then for all these applications we wish to compute $$\label{sum} S_n(y)\,=\,n^{-1}\,\sum_{i=1}^ny_i.$$ In this paper we restrict ourselves to the case when $|y_i|\le 1$ for $i=1,2,\dots,n$. More general conditions on the $y_i$ of the form $\left(n^{-1}\sum_{i=1}^n|y_i|^p\right)^{1/p}\le 1$ with $p\in[1,\infty]$ are considered in [@H01a; @H01b; @HNa; @HNb]. We are interested in applications where $n$ is huge. We wish to approximate $S_n$ to within $\e$ for $\e\in (0,{1\over 2})$. The terms $y_i$ are not stored or computed in advance. We assume that for a given index $i$ we have a subroutine that computes $y_i$. This assumption is typical for scientific problems where, as explained above, $y_i$ depends on the function value $f(x_i)$. Before we discuss quantum computation of $S_n$, we briefly mention summation complexity results in the worst case and randomized settings on a classical computer, see [@N88; @N01]. The worst case complexity, $\comp^{{\rm wor}}(n,\e)$, is defined as the minimal number of operations needed to compute an $\e$-approximation to $S_n$ for all $|y_i|\le 1$ using deterministic algorithms. The randomized complexity, $\comp^{{\rm ran}}(n,\e)$, is defined analogously when we permit randomized algorithms. It is known that $$\comp^{{\rm wor}}(n,\e)\,\approx\,n\,(1-\e),$$ and if $n\gg \e^{-2}$, $$\comp^{{\rm ran}}(n,\e)\,\approx\,\e^{-2}.$$ Hence, in the worst case setting we must add essentially all $n$ numbers, whereas in the randomized setting it is enough to add only $\e^{-2}$ terms and this, of course, can be achieved by the Monte Carlo algorithm that chooses $\e^{-2}$ samples from the set $\{y_1,y_2,\dots,y_n\}$, each with probability $n^{-1}$, and computes their arithmetic mean. This speed-up is significant. We now turn to what is known about summation on a quantum computer. We wish to compute $QS_n(y,\e)$ which approximates $S_n(y)$ to within $\e$ with probability at least ${3\over 4}$. That is, $QS_n(y,\e)$ is a random variable for which the inequality $|S_n(y)-QS_n(y,\e)|\le \e$ holds with probability at least ${3\over 4}$. The performance of a quantum algorithm can be summarized by the number of quantum queries, quantum operations and qubits which are used, see [@BHMT00; @G98; @H01a; @N01] for precise definitions of quantum computation and quantum algorithms. Here, we only mention that the quantum algorithm obtains information on the terms $y_i$ by using only quantum queries. The number of quantum operations is defined as the total number of bit operations performed by the quantum algorithm, and the number of qubits is defined as $k$ if all quantum computations are performed in the Hilbert space of dimension $2^k$. Since the number of qubits will be a limiting resource for the foreseeable future, it is important to seek algorithms which require as few qubits as possible. Brassard, Høyer, Mosca and Tapp, see [@BHMT00], presented a quantum algorithm $QS_n$ that solves the summation problem. Their algorithm is based on Grover’s iterate, see [@BHMT00; @G98], and uses quantum Fourier and Walsh-Hadamard transforms that can be implemented by well known quantum gates. Assume that $n\gg \e^{-1}$. Then the algorithm $QS_n$ uses of order $$\begin{aligned} \e^{-1}&\qquad& \mbox{quantum queries},\\ \e^{-1}\,\log\,n &\qquad& \mbox{quantum operations, }\\ \log\,n\, &\qquad& \mbox{qubits}. \end{aligned}$$ More precise bounds are known about the number of quantum queries. In [@KW], it is shown that the quantum algorithm $QS_n$ uses at most $\e^{-1}\,2.10\dots$ quantum queries and this bound is sharp for small $\e$ and large $\e n$. Due to the lower bound of Nayak and Wu, see [@NW98], the number of quantum queries of [*any*]{} quantum algorithm that solves the summation problem must be at least of order $\e^{-1}$. Hence, the algorithm $QS_n$ uses almost the minimal number of quantum queries. (In this paper $\log$ denotes $\log_2$.) We can run the quantum algorithm $QS_n$ several times to increase the probability of success. If we want to solve the problem with probability $1-\delta$, then we should run $QS_n$ roughly $\log\,\delta^{-1}$ times and take the median as our final result. Then the number of queries and quantum operations is multiplied by $\log\,\delta^{-1}$, but the number of qubits stays the same. Of course, these quantum results are of interest only if $\e^{-1}$ is significantly less than $n$. Fortunately, this is the case for a number of important problems. Indeed, this paper will supply one more such problem, namely, path integration. So far we assumed that we summed numbers from the interval $[-1,1]$. The interval $[-1,1]$ is taken only for simplicity. If we have the interval $[-M,M]$ then we can rescale the summands to $y_i/M$, and multiply the computed result by $M$. This corresponds to the previous problem over the interval $[-1,1]$ with $\e/M$. Note, however, that for large $M$, and $n>M/\e$, the quantum cost is of order $M/\e$, which is significantly larger than $1/\e$. Definition of Path Integration ============================== We now define path integrals studied in this paper, see also [@WW96]. Let $X$ be an infinite dimensional separable Banach space equipped with a probability measure $\mu$. We assume that $\mu$ is a zero mean Gaussian measure, see e.g., [@V87]. The space $X$ can be embedded in the Hilbert space $H=L_2([0,1])$ for which the embedding $\mbox{Im}:X\to H$ is a continuous linear operator. The inner product of $H$ is denoted by $\il\cdot,\cdot\ir_H$. Then the measure $\nu\,=\,\mu\,\mbox{Im}^{-1}$ is also a zero mean Gaussian measure on the Hilbert space $H$. Let $C_\nu$ be the covariance operator of $\nu$, i.e., $C_\nu:H\to H$ and $$\il C_\nu h_1,h_2\ir_H\,=\,\int_H\il h,h_1 \ir_H\il h,h_2 \ir_H\,\nu(dh) \qquad \forall\,h_1,h_2\in H.$$ The operator $C_\nu$ is self adjoint, nonnegative definite and has a finite trace. We can assume that there exists an orthonormal system $\{\eta_i\}$ from $\mbox{Im}(X)$, $\il\eta_i,\eta_j\ir_H\,=\,\delta_{i,j}$, for which $$C_\nu\,\eta_i\,=\,\l_i\,\eta_i,$$ $$\label{eig} \l_1\,\ge\,\l_2\,\ge\,\cdots\,\ge\,0\quad\mbox{and}\quad \sum_{i=1}^\infty\l_i\,<\,+\infty.$$ We illustrate this definition by the important example of the space $X=C([0,1])$ of continuous functions defined on $[0,1]$ with the sup norm, $\|x\|=\max_{t\in[0,1]}|x(t)|$. The space $C([0,1])$ is equipped with the classical Wiener measure $\mu=w$. The measure $w$ is a zero mean Gaussian measure with covariance function $\min(t,u)$. That is, $$\begin{aligned} \int_{C([0,1])}x(t)\,w(dx)\,&=&\,0\qquad \forall\,t\in [0,1],\\ \int_{C([0,1])}x(t)\,x(u)\,w(dx)\,&=&\,\min(t,u) \qquad \forall\,t,u\in [0,1].\end{aligned}$$ For the Wiener measure $w$, we have $\mbox{Im}(x)=x$ and $$\eta_i\,=\,\sqrt{2}\,\sin\left(\frac{2i-1}{2}\pi x\right),\qquad \l_i\,\,=\,\frac{4}{\pi^2(2i-1)^2}.$$ We return to the case of general $X$ and $\mu$. Let $F$ be a class of real-valued $\mu$-integrable functions defined on $X$. The [*path integration*]{} problem is defined as approximating integrals of $f$ from $F$, $$\label{int} I(f)\,:=\,\int_Xf(x)\,\mu(dx)\,=\,\int_Hf(\mbox{Im}^{-1}x)\,\nu(dx) ,\quad \forall\,f\in F.$$ If only finitely many eigenvalues $\l_i$ of $C_\nu$ are positive, then the measure $\nu$ is concentrated on a finite dimensional subspace of $H$ and path integration reduces to a finite dimensional Gaussian integration. To preserve the main feature of the path integration problem, which is integration over an infinite dimensional space, we assume that all eigenvalues $\l_i$ are positive. The element $x$ from $H$ can be written as $x=\sum_{i=1}^{\infty}t_i\eta_i$, with $t_i= \il x,\eta_i\ir_H$. Therefore the integrand $f$ in (\[int\]) depends on infinitely many variables $t_i$. That is why the path integration problem can be viewed as integration of functions having infinitely many variables. In this paper we will consider the classes $F_r$ of functions whose $r-1$ times Frechet derivatives exist and are bounded, and whose $r$th Frechet derivatives satisfy the Lipschitz condition. More precisely, for a non-negative integer $i$, let $\|f^{(i)}\|\,=\,\sup_{x\in X}\|f^{(i)}(x)\|$. Here, $f^{(i)}(x)$ is an $i$-linear form from $X^i$ to ${\mathbb R}$, and its norm is defined as $\|f^{(i)}(x)\|\,=\, \sup_{\|x_j\|_X\le1}|f^{(i)}(x)x_1x_2\,\cdots\,x_i|$. Obviously, $\|f^{(0)}\|=\|f\|=\sup_{x\in X}|f(x)|$. Let $r$ be a positive integer. For positive numbers $K_0,K_1\dots,K_r$, define $\|f\|_{r-1}=\max_{0\le i\le r-1}\|f^{(i)}\|/K_i$. The class $F_r$ is defined as $$ F\_r={f:  f\_[r-1]{}1, f\^[(r-1)]{}(x)-f\^[(r-1)]{}(y)K\_[r]{}(x-y)\_H,  x,yX}. For $r=1$, the class $F_1$ consists of bounded Lipschitz functions. The values of $f$ are bounded by $K_0$, and the Lipschitz constant by $K_1$. For $r\ge 2$, the class $F_r$ consists of bounded smooth functions. All functions from $F_r$ are $r-1$ times Frechet differentiable, their $i$th derivatives are bounded by $K_i$ for $i=0,1,\dots,r-1$, and the $(r-1)$st derivatives satisfy the Lipschitz condition with the constant $K_r$. Note that for any $f\in F_r$, the path integral $I(f)$ is well defined since $f$ is continuous and bounded. From $|f(x)|\le K_0$ we have $|I(f)|\le K_0$. If $K_0$ is large we can use a different estimate on $I(f)$. We have $I(f)=f(0)+ I(f-f(0))$ and $|I(f-f(0))|\le K_1\int_H\|x\|\nu(dx) \le K_1\left(\sum_{j=1}^\infty\l_j\right)^{1/2}$. Hence, $|I(f)|\le |f(0)| + K_1\left(\sum_{j=1}^\infty\l_j\right)^{1/2}$. This estimate can be better than the previous one for large $K_0$. As we shall see in the next sections, path integration for the class $F_r$ is intractable in the worst case setting. We stress that for other classes of functions, path integration can be tractable even in the worst case setting. An example is provided for the class of smooth integrands occurring in the Feynman-Kac formula, see [@KL; @PWW]. Computational Approach to Path Integration ========================================== We want to approximate $I(f)$ to within $\eps$ for all $f \in F$. The approximate computation of $I(f)$ consists of two steps, see [@WW96]. The first is to approximate the infinite dimensional integration $I$ by a $d$-dimensional integration $I_d$, where $d=d(\e,F)$ is chosen as the minimal integer for which the error of this approximation is at most, say, ${\e\over 2}$. The second step is to compute an approximation to $I_d$ with error at most ${\e\over 2}$. Clearly, we should expect that $d(\e,F)$ would go to infinity as $\e$ goes to zero. More precisely we proceed as follows. Let $f_d:{\mathbb R}^d\to {\mathbb R}$ be defined for $t\,=\,[t_1,t_2,\dots,t_d]\in {\mathbb R}^d$ as $$\label{1580} f_d(t\,)\,=\, f\left(\mbox{Im}^{-1}(t_1\eta_1\,+\,t_2\eta_2\,+\,\cdots\,+\,t_d\eta_d)\right).$$ Define $$\label{intd} I_d(f)\,=\, \frac1{(2\pi)^{d/2}}\,\frac1{\sqrt{\l_1\l_2\cdots\l_d}} \int_{{\mathbb R}^d}f_d(t\,)\,\exp\left(-t_1^2/(2\l_1)-\cdots -t_d^2/(2\l_d)\right)\ d\, t.$$ Observe that $I_d$ is a finite dimensional Gaussian integral with the eigenvalues $\l_i$ as variances. Note that the eigenvalues $\l_i$ tend to zero. Indeed, since $a=\sum_{i=1}^\infty\l_i\,<\,+\infty$ and $\l_i$ are non-increasing then $\l_i\,\le\,a/i$ for all $i$. Hence, we have decreasing dependence on the successive variables $t_i$ in (\[intd\]). For continuous $f$, we have $$I(f)\,=\,\lim_dI_d(f_d).$$ As outlined above, we want to choose the minimal $d=d(\e,F)$ such that $|I(f)-I_d(f)|\le {\e\over 2}$ $\forall f\in F$, and then to compute an ${\e\over 2}$-approximation to a finite-dimensional integral $I_d(f_d)$. We now find $d(\e,F_r)$ for a family of eigenvalues $\l_j$ of the covariance operator $C_\nu$. The family includes the eigenvalues of $C_\nu$ for the Wiener measure. \[thm-1\] Suppose $\l_j$ is of order $j^{-k}$ with $k>1$. Then $$\begin{aligned} \underline{c}_1\,\e^{-2/(k-1)}\,\le\,d(\e,F_{1})\,&\le&\, \overline{c}_1\,\e^{-2/(k-1)},\\ \underline{c}_r\,\e^{-1/(k-1)}\,\le\,d(\e,F_{r})\,&\le&\, \overline{c}_r\,\e^{-1/(k-1)} \ \ \mbox{for}\ r\ge 2,\end{aligned}$$ where $\underline{c}_r$ and $\overline{c}_r$, for $r=1,2,\dots$, are positive numbers independent of $\e$ and depending only on the global parameters $K_i,r,k$ and the trace $\sum_{i=1}^\infty\l_i$. In particular, if $\l_j=aj^{-k}$ with $a>0$ then $$\begin{aligned} d(\e,F_1)\,&\le&\,1\,+\, \left(\frac{4aK_1^2}{k-1}\right)^{1/(k-1)}\,\left(\frac1{\e} \right)^{2/(k-1)},\\ d(\e,F_r)\,&\le&\,1\,+\, \left(\frac{aK_2}{k-1}\right)^{1/(k-1)}\,\left(\frac1{\e}\right)^{1/(k-1)} \qquad \mbox{for}\ \ r\ge2. \end{aligned}$$ For the Wiener measure, $\l_j=4/(\pi^2(2i-1)^2)$, we have $$\begin{aligned} d(\e,F_1)\,&\le&\, \bigg\lceil\left(\frac1{\pi^2}\, \frac{2K_1}{\e}\right)^2\,+\frac12\bigg\rceil,\\ d(\e,F_r)\,&\le&\,\bigg \lceil \frac{K_2}{\pi^2\e}\,+\,\frac12\bigg \rceil \qquad \mbox{for}\ \ r\ge2. \end{aligned}$$ The proof of this theorem is given in the Appendix. We stress that the upper bounds on $d(\e,F_r)$ in Theorem 1 depend only on $K_1$ for $r=1$, and on $K_2$ for $r\ge2$, i.e., on the Lipschitz constants for $f$ or $f^{\prime}$, respectively. This means that we can even take all the remaining $K_i=\infty$ and the upper bounds on $d(\e,F_r)$ still hold. On the other hand, the lower bounds depend on all of the $K_i$. The dependence is weak since they only affect the multiplicative factors of the power of $\e^{-1}$, and the power of $\e^{-1}$ does not depend on $K_i$. Path Integration on a Classical Computer ======================================== In this section we discuss approximation of path integrals on a classical computer. We assume the [*real number*]{} model of computation, which is usually used for the analysis of scientific computing problems, see [@TPT] for the rationale. We assume, in particular, that we can perform arithmetic operations (addition, subtraction, multiplication, division), and comparisons of real numbers. We assume that these operations are performed exactly and each costs unity. To approximate path integrals we must have information concerning the integrands $f\in F$. This information may be supplied by function values $f(x_i)$ for some $x_1,x_2,\dots,x_n$, where $n=n(\e,F)$ will be chosen depending on the error demand $\e$ and the class $F$. As outlined in the previous section, we will need to know $f(x)$ for $x$ belonging to a finite dimensional subspace $X_d=\mbox{span}(\mbox{Im}^{-1}\eta_1,\mbox{Im}^{-1}\eta_2, \dots,\mbox{Im}^{-1}\eta_d)$ with $d=d(\e,F)$. We therefore assume that we can compute values of $f(x)$ for $x\in X_d$ and the cost of one such evaluation is $\c_d$. Usually $\c_d\gg 1$. Furthermore, we will sometimes assume that the cost $\c_d$ depends linearly on $d$, i.e., $\c_d\,=\,\c\,d$; however, this assumption is not essential to the analysis. For a more complete discussion of the real number model of computation with function values, see [@N95; @TWW88]. Let $A(f)$ be any algorithm for approximation of path integrals. The algorithm $A$ uses a finite number $n_1$ of function values at points $x_i$ and a finite number $n_2$ of arithmetic operations and comparisons to compute $A(f)$. The cost of computing $A(f)$ is $\c_dn_1+n_2$. In the [*worst case*]{} setting, the error and cost of $A$ are defined by its worst performance over the class $F$. In the [*randomized*]{} setting, the algorithm $A$ may use randomly chosen samples $x_i$, and its error and cost are defined by the expected error with respect to the distribution generating the random samples for a worst $f$ from $F$. By the worst case or randomized [*complexity*]{}, we mean the minimal cost that is needed to compute an $\e$-approximation for all $f\in F$, see [@TWW88] for precise definitions. We now briefly discuss the worst case and randomized complexities of path integration for the classes $F_r$. We begin with the worst case setting. We first state the result of Bakhvalov, see e.g., [@N88; @TWW88], which states that the worst case complexity of multivariate integration over the unit cube $[0,1]^d$ for $r$-times differentiable functions is of order $\c_d\e^{-d/r}$. For path integration $d$ is an increasing function of $\e^{-1}$, and as shown in Theorem 1, it goes to infinity polynomially in $\e^{-1}$ as $\e$ goes zero. This suggest that the worst case complexity, $\comp^{{\rm wor}}(\e,F_r)$, of path integration in the class $F_r$ is exponential[^2] in $\e^{-1}$. A formal proof may be found in [@WW96] for any $r$, and more precise complexity bounds in [@C00] for $r=1$. Thus path integration is [*intractable*]{} for the class $F_r$ in the worst case setting. This means that the cost of any algorithm for solving this problem must be exponential. Yet, as we shall see, such algorithms will be useful for quantum computation. We now sketch such an algorithm. We first consider the case $r=1$ and then show that an easy modification of the same algorithm can be also used for $r\ge2$. We assume[^3] that $\l_j=\Theta(j^{-k})$. From Theorem 1 we know that it is enough to compute an ${\e\over 2}$-approximation to the integral $I_d(f)$ with $d=d(\e,F_1)$ given in Theorem 1. We have $$\label{1570} I_d(f)\,=\,\int_{{\mathbb R}^d}f_d(t)\,\nu_d(dt),$$ where $\nu_d$ is a Gaussian measure on ${\mathbb R}^d$ with mean zero and with the diagonal covariance matrix $\mbox{diag}(\l_1,\l_2,\dots,\l_d)$. This problem has been studied in [@C00]. Based on that paper we describe an algorithm $S_n$ with worst case error at most ${\e\over 2}$. We opt here for simplicity of the presentation of $S_n$ at a slight expense of its cost. Let $n=m^d$ for the minimal odd integer $m$ for which $$m\,\ge\, \frac{4\,K_1\,\left(\pi\sum_{i=1}^d\l_i\right)^{1/2}}{\e}.$$ For $x\ge0$, let $\psi(x)=\sqrt{2/\pi}\int_0^x\exp(-t^2/2)dt$ be the probability integral, and let $\psi^{-1}$ be its inverse. We note that it is easy to compute $\psi^{-1}(t)$ numerically for any $t\in {\mathbb R}$. As in Lemma 1 of [@C00], for $i=1,2,\dots,d$ define the points $t_{i,j}:$ $$\begin{aligned} t_{i,0}\,&=&\,-\infty,\\ t_{i,j}\,&=&\,(3\l_i)^{1/2}\,\psi^{-1}\left((j-1/2)/m\right), \qquad j=1,2,\dots,m,\\ t_{i,m+1}\,&=&\,\infty.\end{aligned}$$ Then take $t^*_{i,j}=t_{i,j}$ if $|t_{i,j}|\le |t_{i,j+1}|$, and $t^*_{i,j}=t_{i,j+1}$ otherwise. For the integer vector $\vec j=[j_1,j_2,\dots,j_d]$, with $j_i=1,2,\dots,m$, define the sample points $$x_{\vec j}\,=\,[t^*_{1,j_1},t^*_{2,j_2},\dots,t^*_{d,j_d}].$$ Then the algorithm takes the simple form $$\label{alg} S_n(f_d)\,=\,n^{-1}\,\sum_{{\vec j}} f\left(x_{\vec j }\right).$$ Curbera proved in [@C00] that the worst case error of $S_n$ is at most $$e^{{\rm wor}}\left(S_n\right)\,\le\, 2\,K_1\,\left(\pi\,\sum_{i=1}^d\l_i \right)^{1/2}\,m^{-1} \,\le \,{\e\over 2},$$ where the last inequality holds due to the choice of $m$. The cost of $S_n$ is $(\c_d+1)n$ where $$\label{n1} n\,=\,n(\e^{-1})\,=\, m^d\,\le \left(\frac{\a}{\e}\right)^{\overline{c_1}\e^{-2/(k-1)}},$$ where $\a=4\,K_1\,\left(\pi\,\sum_{i=1}^\infty\l_i\right)^{1/2}+2\e$, and $\overline{c_1}$ is from Theorem 1. We now consider the case $r\ge 2$. As shown in Theorem 1, we can now restrict ourselves to the integrals $I_d(f)$ for $d=d(\e,F_r)$. We stress that $d(\e,F_r)$ is much less than $d(\e,F_1)$ for small $\e$. Observe that all functions $f$ from $F_r$ also belong to $F_1$ since they satisfy the Lipschitz condition with the constant $K_1$. Hence we can use the algorithm $S_n$ with the important difference that now $d=d(\e,F_r)$. Hence, we compute an ${\e\over 2}$-approximation by the algorithm $S_n$ with cost $(\c_d+1)n$, where $$\label{n2} n\,=\,n(\e^{-1})\,=\,m^{d(\e,F_r)}\,\le\, \left(\frac{\a}{\e}\right)^{\overline{c_2}\e^{-1/(k-1)}},$$ with $\overline{c_2}$ from Theorem 1. We now justify why it is enough to apply the algorithm $S_n$ for the class $F_r$ for any $r\ge2$. The reason is that for path integration the smoothness parameter $r$ is not as important as for finite dimensional integration. Indeed, since the exponent of $\e^{-1}$ for the worst case complexity of path integration is unbounded (as $\e\to 0$) for any fixed $r$, it does not help much to divide by $r$. As we shall see in the next section, for quantum computation the logarithm of the worst case complexity is important and $r$ can only effect a multiplicative factor. The most important property is how fast $d(\e,F_r)$ goes to infinity. As we know from Theorem 1, the influence of $r$ is significant here since we have different formulas for $d(\e,F_r)$ for $r=1$ and $r\ge2$. However, for $r\ge2$, the use of more efficient algorithms than $S_n$ can only improve the multiplicative factor of the logarithm of the worst case complexity. We turn to the randomized setting for $\l_i=\Theta(j^{-k})$. The randomized complexity, $\comp^{{\rm ran}}(\e,F_r)$, can be easily obtained by applying results of Bakhvalov for finite dimensional integration. The analysis in [@WW96] yields $$\label{rand0} \comp^{{\rm ran}}(\e,F_r)\,=\, \Theta\left((\c_d+1)\, \e^{-2(1+o(1))}\right) \qquad \mbox{as}\ \ \e\to 0,$$ where $d=d(\e,F_r)$ and the factors in the $\Theta$ notation depend at most quadratically on $K_0$ and $K_1$. Hence, in the randomized setting we have roughly quadratic dependence in $\e^{-1}$ on the number of function values. If $\c_d=\c\,d$ then $$\label{rand} \comp^{{\rm ran}}(\e,F_r)\,=\, \Theta\left(\c\,\left(\frac1{\e}\right)^{\frac{1+\gamma(r)}{k-1}\,+\, 2\,+o(1)}\right)$$ where $\gamma(1)=1$ and $\gamma(r)=0$ for $r\ge 2$. Hence, the randomized complexity of path integration depends polynomially on $\e^{-1}$, and therefore the path integration problem is [*tractable*]{} in the randomized setting. In fact, the upper bound can be achieved by the Monte Carlo algorithm with randomized error at most ${\e\over 2}$ and with the cost proportional to $\c_d\, K_0^2\e^{-2}$ randomized evaluations of a function of $d=d(\e,F_r)$ variables, where $d(\e,F_r)$ is given by Theorem 1. Note, however, that if $k$ goes to one then the degree of $\e^{-1}$ in the randomized complexity goes to infinity. The reason is that in this case we have to compute function values of very many variables. On the other hand, for the Wiener measure we have $k=2$, and the degree of $\e^{-1}$ is roughly $3+\gamma(r)$. Path Integration on a Quantum Computer ====================================== We now analyze path integration on a quantum computer. The idea behind solving path integration on a quantum computer is quite simple. (However, the analysis is not so simple.) We will apply analogous techniques for other problems in future papers. Start with an algorithm that computes an $\e$-approximation to path integration in the [*worst*]{} case setting and that requires summation of the form of (\[sum\]). We run this algorithm on a quantum computer using the quantum summation algorithm of Section 2. Obviously, $n$ is now a function of $\e^{-1}$. For path integration for the class $F_r$ we know that $n$ is an exponential function of $\e^{-1}$ and is bounded by (\[n1\]) for $r=1$, and by (\[n2\]) for $r\ge 2$. However, the exponential dependence on $\e^{-1}$ is now [*not*]{} so essential since the cost of the quantum summation algorithm $QS_n$ depends only logarithmically on $n$. Since $\log\,n$ is a polynomial in $\e^{-1}$ we conclude that path integration on a quantum computer can be solved at cost [*polynomial*]{} in $\e^{-1}$. That is, intractability of path integration in the worst case setting is [*broken*]{} on a quantum computer by the use of the quantum summation algorithm. For other intractable problems in the worst case setting for which the worst case complexity can be achieved by summation of $n$ numbers, intractability will be broken as long as $n$ is a single exponential function of $\e^{-1}$, i.e., $n(\e^{-1})\le 2^{p(\e^{-1})}$ with $p$ being a polynomial. Then the quantum cost will be polynomial in $\e^{-1}$, and the problem will be tractable on a quantum computer. This idea will [*not*]{} work if $n(\e^{-1})$ is a double exponential function (or worse) of $\e^{-1}$ since then the logarithm of $n(\e^{-1})$ will be still an exponential function of $\e^{-1}$. We now provide details of this idea for path integration for the class $F_r$ with eigenvalues $\l_j=\Theta(j^{-k})$. We take the algorithm $S_n$ defined by (\[alg\]) with $n$ given by (\[n1\]) for $r=1$ and by (\[n2\]) for $r\ge2$. The algorithm $S_n$ already has the form (\[sum\]) required by the summation algorithm. However, the summands $f(x_{\vec j})$ are not necessarily in the interval $[-1,1]$. The function $f$ belongs to $F_r$ and therefore its values are bounded by $K_0$. Hence, it is enough to scale the problem by running the quantum summation algorithm for $y_{\vec j}=f(x_{\vec j})/K_0$, replace $\e$ by $\e/K_0$, and multiply the computed result by $K_0$. The cost of an algorithm on a quantum computer using $m$ qubits is defined as on a classical computer with the cost of a quantum query taken as $\c_d+m$ since $f(x_{\vec j})$’s are computed and $m$ qubits are processed by a quantum query. Using the results of quantum summation from Section 2 applied for $S_n$ we obtain the following theorem. \[thm2\] Consider path integration for the class $F_r$ with the eigenvalues $\l_j=\Theta(j^{-k})$. Using the quantum summation algorithm $QS_n$ to compute an $\e/K_0$-approximation to $S_n$, we compute an $\e$-approximation for path integrals with probability at least ${3\over 4}$ and of order $$\begin{aligned} \e^{-1}&\qquad& \mbox{quantum queries},\\ \e^{-((k+\gamma(r))/(k-1)}\,\log\,\e^{-1} &\qquad& \mbox{quantum operations, }\\ \e^{-((1+\gamma(r))/(k-1)}\,\log\,\e^{-1}\, &\qquad& \mbox{qubits}, \end{aligned}$$ where $\gamma(1)=1$ and $\gamma(r)=0$ for $r\ge 2$. If $\c_d=\c\,d$, then the cost of this algorithm is of order $$\left(\frac1{\e}\right)^{\frac{1+\gamma(r)}{k-1}\,+1} \left(\c\,+\,\log\,\frac1{\e} \right).$$ For the Wiener measure the results are more precise. The algorithm requires (neglecting ceilings for simplicity) at most $$\begin{aligned} 2K_0\,\e^{-1}&\qquad& \mbox{quantum queries},\\ \frac{2K_0}{\e}\,d^{{\rm up}}(\e,F_r)\log \,\frac{16\,K_0\,K_1} {\e} &\qquad& \mbox{quantum operations, }\\ d^{{\rm up}}(\e,F_r)\,\log \,\frac{16\,K_0\,K_1}{\e} &\qquad& \mbox{qubits}.\end{aligned}$$ If $\c_d=\c\,d$, then the cost of this algorithm is at most $$\frac{2K_0\,d^{{\rm up}}(\e,F_r)}{\e}\,\left(\c\,+\, 2\,\log\,\frac{16\,K_0\,K_1}{\e}\right).$$ Here $d^{{\rm up}}(\e,F_r)$ is an upper bound on $d(\e,F_r)$ given by $$d^{{\rm up}}(\e,F_r)\,=\, \bigg\lceil\left(\frac{K_0\beta(r)}{\pi^2\e}\right)^{1+\gamma(r)}\,+\,\frac12 \bigg \rceil\,$$ with $\beta(r)=2K_1$ for $r=1$, and $\beta(r)=K_2$ for $r\ge2$. If we want to increase the probability of computing an $\e$-approximation to path integration then, as explained in Section 2, we can run the quantum algorithm for $QS_n$ roughly $\log\,\delta^{-1}$ times and take the median as the final result. Then the probability of success is at least $1-\delta$. Obviously the cost is then multiplied by $\log\,\delta^{-1}$ but the number of qubits stays the same. We compare ${\rm cost}(QS_n)$, the cost of the quantum algorithm, with the worst case complexity of path integration. The essence of Theorem 2 is that ${\rm cost}(QS_n)$ depends polynomially on $\e^{-1}$. Since the worst case complexity is exponential in $\e^{-1}$, the use of quantum summation breaks intractability of the worst case setting. Note that we have [*exponential speed-up*]{}, i.e., ${\rm comp}^{{\rm wor}}(\e,F_r)/ {\rm cost}(QS_n)$ is exponential in $\e^{-1}$. We now compare ${\rm cost}(QS_n)$ with the randomized complexity of path integration. As discussed in Section 5, path integration is tractable in the randomized setting and its randomized complexity is characterized by (\[rand0\]) and (\[rand\]). Comparing the formulas for the randomized complexity with ${\rm cost}(QS_n)$ we see that the ratio of the number of quantum queries used by the quantum algorithms to the number of function values used by the best randomized algorithm is roughly $\e^{-1}$. If we compare ${\rm cost}(QS_n)$ to the randomized complexity we see that the speed-up is roughly of order $\e^{-1}$. That is, we solve path integration on a quantum computer roughly $\e^{-1}$ times cheaper than on a classical computer using randomization. We summarize our results in the following corollary. Consider path integration for the class $F_r$ with $\l_j=\Theta(j^{-k})$. Then - Path integration on a quantum computer is tractable. - Path integration on a quantum computer can be solved roughly $\e^{-1}$ times faster than on a classical computer using randomization, and exponentially faster than on a classical computer with a worst case assurance. - The number of quantum queries is the square root of the number of function values needed on a classical computer using randomization. - The number of qubits is polynomial in $\e^{-1}$. Furthermore, for the Wiener measure the degree is $2$ for $r=1$, and $1$ for $r\ge2$. Lower Bounds on the Number of Quantum Queries ============================================= We now study lower bounds on the minimal number $\qq$ of quantum queries needed to compute an $\e$-approximation with probability $3/4$ for path integration for the class $F_r$. From Theorem 2 we know that $\qq$ is at most of order $\e^{-1}$. We show that this bound cannot be significantly improved. Consider path integration for the class $F_r$ with all positive eigenvalues $\l_i$. Then $$\lim_{\e\to 0}\,\e^{1-\a}\,\qq\ =\ \infty\qquad \forall\,\a\in (0,1).$$ [[*Proof:*]{} ]{}The proof consists of two steps. The first one is to reduce path integration to a finite dimensional Gaussian integration which is no harder than the original problem. The second step is essentially the same as in Novak’s papers, see [@N88; @N01], and reduces the finite dimensional Gaussian integration problem to summation for which the lower bound of Nayak and Wu, see [@NW98], applies. In the first step of the proof, for a given $\a\in (0,1)$ we take an integer $d>r(1-\a)/\a$. (Hence, $d$ is large for small $\a$.) The path integration problem for the class $F_r$ is no harder if we assume some additional properties of functions $f$ from $F_r$. We have, see (\[int\]), $$I(f)\,=\,\int_Hf(\mbox{Im}^{-1}x)\,\nu(dx)$$ where $x=\sum_{i=1}^\infty t_i\eta_i$. Let us now assume that $f$ depends only on the first $d$ components $t_1,t_2,\dots,t_d$, and call this class $F_{r,d}$. Obviously, $F_{r,d}$ is a subclass of $F_r$ and therefore path integration for $F_{r,d}$ is no harder than for the class $F_r$. For the class $F_{r,d}$, the path integration problem reduces to a finite dimensional Gaussian integration problem. That is, for $f\in F_{r,d}$ we have $$I_d(f)\,=\,\int_{{\mathbb R}^d}f_d(t)\,\nu_d(dt),$$ where $\nu_d$ is the Gaussian measure given by (\[1570\]), and $f_d$ is given by (\[1580\]). The functions $f_d$ from $F_{r,d}$ are $r-1$ times continuously differentiable and $$\|f_d\|=\sup_{t\in{\mathbb R}^d}|f_d(t)|\,\le\, K_0.$$ Furthermore, their $r-1$ partial derivatives satisfy the Lipschitz condition. More precisely, there exists a positive number $\beta_1$ depending only on $d,r$ and $K_0,K_1,\dots,K_r$ such that $$\left|D^if_d(t)-D^if_d(y)\right|\,\le\,\beta_1\|t-y\|_{\infty}\quad \forall\,t,y\in {\mathbb R}^d,$$ where $D^i$ runs through the set of all partial derivatives of order $r-1$. This shows that the class $F_{r,d}$ is closely related to the class $F^{r-1,1}_d$ studied by Novak, see [@N01], $$F^{r-1,1}_d\,=\,\{f:[0,1]^d\to{\mathbb R}\,|\, f\in C^{r-1}([0,1]^d),\, |D^if(t)-D^if(y)|\le\|t-y\|_{\infty}\,\forall t,y\,\}.$$ Obviously, the different Lipschitz constants: $\beta_1$ in our case and $1$ for the class $F^{r-1,1}_d$ do not play a major role since they do not change the order of error bounds. One difference between the two classes is that the common domain of functions from $F_{r,d}$ is ${\mathbb R}^d$, whereas for the class $F^{r-1,1}_d$ the common domain is $[0,1]^d$. A second difference is that we have Gaussian integration whereas Novak considered uniform integration, $\int_{[0,1]^d}f(t)\,dt$. As we shall see below these two differences are not really essential. In the second step of the proof, we use Novak’s proof technique. From [@N88; @N01] we know that for any positive $\e_1$ there are functions $f_1,f_2,\dots f_n$, with $n=\Theta(\e_1^{-d/r})$, from the class $F^{r-1,1}_d$ such that they take non-negative values, have disjoint supports in $[0,1]^d$ and - $\int_{[0,1]^d}f_i(t)\,dt\,=\,\e_1^{1+d/r}\quad i=1,2,\dots,n$, - $\sum_{i=1}^n\a_if_i\,\in\,F^{r-1,1}_d\quad \forall\,|\a_i|\le 1$. We use the same functions $f_i$ for our Gaussian integration for the class $F_{r,d}$. Since the support of $f_i$ is in $[0,1]^d$ we can extend $f_i$ by zero to ${\mathbb R}^d$. The extended functions $f_i$ have exactly the same smoothness as required for the class $F_{r,d}$, and there exists a positive $\beta_2$ depending only on $d,r$ and $K_0,K_1,\dots, K_r$ but independent of $i$ such that $\beta_2f_i\in F_{r,d}$. Note that $$\int_{{\mathbb R}^d}f_i(t)\,\nu_d(dt)\,=\, \int_{[0,1]^d}\rho_d(t)\,f_i(t)\,dt,$$ where $$\rho_d(t)\,=\, \frac{1}{\prod_{j=1}^d(2\pi\l_j)^{1/2}} \exp\left(-\sum_{j=1}^dt_j^2/2\right).$$ Since all $\l_j$ are positive, the function $\rho_d$ has positive minimum and maximum over $[0,1]^d$. That is, there are positive $\beta_3$ and $\beta_4$ depending on $d$ and $\l_1,\l_2,\dots,\l_d$ such that $$\beta_3\,\le\,\rho_d(t)\,\le \beta_4\quad \forall\,t_j\in[0,1].$$ Therefore for the functions $g_i=\beta_2f_i\in F_{r,d}$ we have $$\label{ostatnia} \beta_2\beta_3\,\e^{1+d/r}\,\le\, I(g_i)\,=\,\int_{{\mathbb R}^d}g_i(t)\,\nu_d(dt)\,\le\,\beta_2\beta_4 \e_1^{1+d/r},$$ and $$\sum_{i=1}^n\a_ig_i\,\in\,F_{r,d}\quad \forall\,|\a_i|\le 1.$$ Since $$I\left(\sum_{i=1}^n\a_ig_i\right)\,=\,\sum_{i=1}^n\a_iI(g_i)$$ we reduce our problem to summation of $n=\Theta(\e_1^{-d/r})$ terms for arbitrary $|\a_i|\le 1$. Let $$y_i\,=\,\frac{I(g_i)}{\beta_2\beta_4\e_1^{1+d/r}}\,\a_i.$$ Observe that by varying $\a_i$ from $[-1,1]$, the $y_i$ can take any value from $\beta_3/\beta_4[-1,1]$ due to the left hand side of (\[ostatnia\]). We need to compute an $\e$-approximation to $I(\sum_{i=1}^n\a_ig_i)$. This is equivalent to computing an $\e_2$-approximation to $$\frac1n\,\sum_{i=1}^ny_i$$ with $\e_2=\e/(n\beta_2\beta_4\e_1^{1+d/r})=\Theta(\e/\e_1)$. The summation problem $n^{-1}\sum_{i=1}^ny_i$ for our $y_i$ is not easier than the summation problem $n^{-1}\sum_{i=1}^ny_i$ for all $|y_i|\le\beta_3/\beta_4$. We can now apply the lower bound of Nayak and Wu, see [@NW98], that states that the minimal number of quantum queries needed to compute an $\e_2$-approximation with probability $3/4$ for the summation problem $n^{-1}\sum_{i=1}^ny_i$ with $|y_i|\le \beta_3/\beta_4$ is bounded from below by $$C\,\min\left(n,\frac{\beta_4}{\beta_3\,\e_2}\right)\,=\,\Theta\left( \min(\e_1^{-d/r},\e_1/\e)\right),$$ with some absolute positive number $C$. Finally, we take $\e_1$ such that $\e_1^{-d/r}=\e_1/\e$, i.e., $\e_1=\e^{r/(d+r)}$, and conclude that the minimal number $\qq$ of quantum queries is at least of order $\e^{-(1-r/(d+r))}$. Since $d>r(1-\a)/\a$ implies that $\a>r/(d+r)$, we have $$\e^{1-\a}\,\qq\,=\,\Omega\left(\e^{-(\a-r/(d+r))}\right)\, \to\,\infty \quad\mbox{as}\ \e\to 0.$$ This completes the proof. ------------------------------------------------------------------------ Appendix ======== We prove Theorem 1. We begin with $r=1$. It is shown in [@WW96] that $d=d(\e,F_{1})\le d^*$ where $d^*$ is an integer satisfying $$\sum_{i=d^*+1}^\infty\l_i\,\le\,\e^2/(2K_1)^2.$$ For $\l_i\,=\,\Theta(i^{-k})$ with $k>1$, we get $$d^*\,=\,\Theta\left((K_1/\e)^{2/(k-1)}\right) \quad\mbox{as}\ \e\to0.$$ For $\l_i=ai^{-k}$, we get $$d^*\,=\,\bigg\lceil \left(\frac{a}{k-1}\right)^{1/(k-1)} \left(\frac{2K_1}{\e}\right)^{2/(k-1)}\bigg\rceil.$$ For the Wiener measure we have $$d^*\,=\, \bigg\lceil\left(\frac1{\pi^2}\, \frac{2K_1}{\e}\right)^2\,+\frac12\bigg\rceil.$$ This establishes upper bounds on $d(\e,F_1)$. To get a lower bound, take the function $g:{\mathbb R}\to {\mathbb R}$ defined by $g(x)=c_1|x|/(1+|x|)$ with $c_1=\min(K_0,K_1)$. We have $\sup_{x\in {\mathbb R}}|g(x)|=c_1\le K_0$, and $g$ satisfies the Lipschitz condition with the constant $c_1\le K_1$. For $x\in H$, define $x_d=\sum_{j=d+1}^{2d}\il x,\eta_j\ir_H\eta_j$ and $$f\left({\rm Im}^{-1}x\right)\,=\,g(\|x_d\|).$$ Then $f\in F_1$ and $f_d=0$. We have $I(f)-I(f_d)=I(f)$ and $$I(f)\,=\,\int_Hg(\|x_d\|)\,\nu(dx)\,\ge\, c_1\int_{\|x_d\|\le1}\frac{\|x_d\|}{1+\|x_d\|}\,\nu(dx)\,\ge\, \frac{c_1}2\int_{\|x_d\|\le1}\|x_d\|\nu(dx).$$ Since $\left(\sum_{j=d+1}^{2d}a_j^2\right)^{1/2}\ge d^{-1/2} \sum_{j=d+1}^{2d}|a_j|$ for any $a_j\in {\mathbb R}$, we get $$I(f)\,\ge\,\frac{c_1}{ 2d^{1/2}}\, \sum_{j=d+1}^{2d}{\rm Int}_j$$ where $${\rm Int}_j\,=\,\prod_{i=1}^d(2\pi\l_i)^{-1/2}\int_{\sum_{i=d+1}^{2d}x_i^2\le1} |x_j|\exp\left(-\sum_{i=d+1}^{2d}x_i^2/(2\l_i)\right)\,dx.$$ There exist two positive numbers $\a_1$ and $\a_2$ such that $\a_1i^{-k}\le\l_i\le\a_2i^{-k}$ for all $i$. By changing variables $t_{i-d}=x_i/(\l_i)^{1/2}$ and noting that $\a_2d^{-k}\ge\l_d\ge \l_i\ge\l_{2d}\ge\a_1(2d)^{-k}$ we conclude that $$\begin{aligned} {\rm Int}_j\,&=&\, \frac{\sqrt{\l_j}}{(2\pi)^{d/2}} \int_{\sum_{i=1}^{d}\l_it_i^2\le1} |t_{j-d}|\exp\left(-\sum_{i=1}^{d}t_i^2/2\right)\,dx,\\ {\rm Int}_j\,&\ge&\,\frac{\a_1^{1/2}}{(2d)^{k/2}}\left( \frac1{(2\pi)^{d/2}}\int_{{\mathbb R}^d}|t_1|\,e^{-\|t\|^2/2}\,dt\,-\, \frac1{(2\pi)^{d/2}}\int_{\|t\|>(d^k/\a_2)^{1/2}} |t_1|\,e^{-\|t\|^2/2}\,dt\right).\end{aligned}$$ Here, $t=[t_1,t_2,\dots,t_d]$ and $\|t\|=(\sum_{j=1}^dt_j^2)^{1/2}$. The first integral is just $\sqrt{2/\pi}$. We now show that the second integral goes to zero with $d$. Indeed, let $\nu_d$ be for the Gaussian measure on ${\mathbb R}^d$ with zero mean and the identity covariance operator, and let $\alpha=(d^k/\a_2)^{1/2}$. Then the second integral is $\int_{{\mathbb R}^d\setminus B_\alpha}|t_1|\nu_d(dt)$, where $B_\alpha$ denotes the ball of radius $\alpha$, and is not greater than $$\left(\int_{{\mathbb R}^d}t_1^2\nu_d(dt)\right)^{1/2}\left(1-\nu_d(B_\alpha) \right)^{1/2}.$$ The integral with the integrand $t_1^2$ is just one, and using Lemma 2.9.2 from [@TWW88] p. 469 we conclude that $$1-\nu_d(B_\alpha)\,\le\, 5 \exp\left(-\alpha^2/(2d)\right).$$ Since $\alpha^2/(2d)=d^{k-1}\a_2/2$ and $k>1$ then this ratio goes to infinity, and $1-\nu_d(B_d)$ goes to zero. This means that ${\rm Int}_j$ is at least of order $d^{-k/2}$, and $I(f)$ is at least of order $d^{-(k-1)/2}$. Hence to guarantee that $I(f)=I(f)-I(f_d)\le {\e\over 2}$ we must take $d$ of order $\e^{-2/(k-1)}$ which completes the proof for the case $r=1$. Assume now that $r\ge 2$. We first establish an upper bound on $d(\e,F_r)$. For $x=\sum_{j=1}^\infty t_j\eta_j\in H$, define $t=[t_1,t_2,\dots]\in {\mathbb R}^\infty$ and $t^d=[t_1,t_2,\dots,t_d]$. Then we can identify $f(t)$ with $f({\rm Im}^{-1}x)$ and $f(t^d)$ with $f_d(t)$. By Taylor’s theorem we have $$f(t)\,=\,f(t^d)+f^{\prime}(t^d)(t-t^d)+ \int_0^1\left( f^{\prime}\left(t^d+u(t-t^d)\right)-f^{\prime}(t^d)\right)(t-t^d)\,du.$$ Note that $t-t^d=[0,\dots,0,t_{d+1},t_{d+2},\dots]$ and since $f^{\prime}(t^d)$ is a linear form we have $$f^{\prime}(t^d)(t-t^d)\,=\,\sum_{j=1}^\infty a_jt_j,$$ where $a_j=a_j(t_1,t_2,\dots,t_d)$. The mean element of $\nu$ is zero, which implies that $I(f^{\prime}(t^d)(t-t^d))=0$. Hence $$I(f)-I(f_d)\,=\, I\left(\int_0^1\left(f^{\prime}\left(t^d+u(t-t^d)\right)- f^{\prime}(t^d)\right)(t-t^d)\,du\right).$$ For $r\ge2$, $f^{\prime}$ satisfies the Lipschitz condition and we get $$\begin{aligned} \left|I(f)-I_d(f)\right|\,&\le&\,K_2\,I\left(\|t-t^d\|^2\,\int_0^1u\,du \right)\\ &\le&\,\frac{K_2}2\,I\left(\sum_{j=d+1}^\infty\il x,\eta_j\ir^2_H\right)\,=\, \frac{K_2}2 \sum_{j=d+1}^\infty\l_j.\end{aligned}$$ For $\l_j=\Theta(j^{-k})$ we obtain $$|I(f)-I_d(f)|\,=\,O\left(d^{-(k-1)}\right)$$ and for $d=O(\e^{-1/(k-1)})$ we guarantee that $|I(f)-I(f_d)|\le {\e\over 2}$ for all $f\in F_r$. For $\l_j=aj^{-k}$, we have $$\sum_{j=d+1}^\infty\l_j\,\le\, a\,\int_d^\infty u^{-k}\,du\,=\, \frac{a}{2(k-1)}\ \frac1{d^{k-1}}.$$ In this case it is enough to take $$d\,=\,\bigg \lceil \left(\frac{aK_2}{k-1}\right)^{1/(k-1)}\, \left(\frac1{\e}\right)^{1/(k-1)}\bigg \rceil.$$ For the Wiener measure, $$\sum_{j=d+1}^\infty\l_j\,\le\, \frac1{\pi^2}\,\int_d^\infty \left(u-1/2\right)^{-2}\,du\,=\, \frac1{\pi^2(d-1/2)},$$ and $$d\,=\, \bigg \lceil \frac{K_2}{\pi^2\e}\,+\,\frac12\bigg \rceil.$$ This establishes upper bounds on $d(\e,F_r)$. To get a lower bound, consider the function $g(x)=c_rx^2/(1+x^2)= c_r(1-1/(1+x^2))$ for $x\in {\mathbb R}$, where $c_r$ is a positive number chosen such that $\max_{0\le i\le r-1}\sup_{x\in {\mathbb R}}|g^{(i)}(x)|/K_i\le1$, and such that $g^{(r-1)}$ satisfies the Lipschitz condition with the constant $K_r$. It is easy to see that such a positive number $c_r$ exists. Indeed, the $j$th derivatives of $1/(1+x^2)$ can be written as the ratio of two polynomials $p_j(x)/(1+x^2)^{j+1}$ with the degree of $p_j$ being at most $j$, and therefore all derivatives go to zero as $|x|$ goes to infinity. As for the case $r=1$, we take $x_d=\sum_{j=d+1}^{2d}\il x,\eta_j\ir_H\eta_j$, and $f({\rm Im}^{-1}x)=g(\|x_d\|)$. Then $f\in F_r, f_d=0$ and $I(f)-I(f_d)=I(f)$. Similarly as for $r=1$ we have $$I(f)\,\ge\,\frac{c_r\l_{2d}}2\,\int_{\|t\|\le \alpha}\|t\|^2\nu_d(dt),$$ where $\alpha=(d^k/\a_2)^{1/2}$. We now show that the last integral tends to $d$. Indeed, it can be written as $$\int_{{\mathbb R}^d}\|t\|^2\nu_d(dt) \,-\,\int_{\|t\|>\alpha}\|t\|^2\nu_d(dt).$$ The first integral is obviously $d$, and we show that the integral over the outside of the ball tends to zero. For large $d$, the norm of $t$ is also large, and we can estimate $\|t\|^2\le\exp(c_d\|t\|^2/2)$ for $c_d=2\a_2\ln(d^k/\a_2)/d^k$. Then $$\int_{\|t\|>\alpha}\|t\|^2\nu_d(dt)\,\le\, (1-c_d)^{d/2}\, \int_{\|t\|>\alpha}\nu_{d,c}(dt),$$ where $\nu_{d,c}$ is a Gaussian measure on ${\mathbb R}^d$ with mean zero and covariance operator $(1-c_d)^{-1}I$. Again using Lemma 2.9.2 from [@TWW88] we obtain $$\int_{\|t\|>\alpha}\|t\|^2\nu_d(dt)\,\le\, 5(1-c_d)^{d/2} \exp\left(-\alpha^2(1-c_d)/d\right).$$ Since $k>1$, the quantity $(1-c_d)^d$ tends to $1$, and since $\alpha^2/d=\Theta(d^{k-1})$ tends to infinity, the integral goes to zero as claimed. Hence, $I(f)$ is at least of order $d^{-(k-1)}$ and $d$ must be at least of order $\e^{-1/(k-1)}$ to guarantee $I(f)\le {\e\over 2}$. This completes the proof for $r\ge2$.     ------------------------------------------------------------------------ . Acknowledgment {#acknowledgment .unnumbered} ============== We are grateful for the excellent facilities of the Santa Fe Institute where some of our research was conducted. The lower bound theorem of Section 7 was obtained during a stay at Los Alamos National Laboratory. We are also grateful to S. Heinrich, E. Novak, A. Papageorgiou, G. W. Wasilkowski and A. G. Werschulz for valuable comments on our paper. [50]{} D. S. Abrams and C. P. Williams, Fast quantum algorithms for numerical integrals and stochastic processes, LANL preprint quant-ph/9908083. G. Brassard, P. Høyer, M. Mosca and A. Tapp, Quantum amplitude amplification and estimation, LANL preprint quant-ph/0005055. F. Curbera, Delayed curse of dimension for Gaussian integration, [*J. Complexity*]{}, 16, 474-506, 2000. L. Grover, A fast quantum mechanical algorithm for database search. [*Physical Review Letters*]{}, 79, 325-328, 1996. See also http://arXiv.org/abs/quant-ph/9706033. L. Grover, A framework for fast quantum mechanical algorithms. Proc. 30 Annual ACM Symp. on the Theory of Computing, ACM Press New York. See also LANL preprint quant-ph/9711043 and Physical Review Letters [**80**]{}, 4329–4332, 1998. S. Heinrich, Quantum summation with an application to integration, [*J. Complexity*]{}, 18, 1-50, 2002. See also http://arXiv.org/abs/quant-ph/0105116. S. Heinrich, Quantum integration in Sobolev spaces, submitted for publication, 2001. S. Heinrich and E. Novak, Optimal summation and integration by deterministic, randomized, and quantum algorithms, in the Proceedings of the $4$th International Conference on Monte Carlo and Quasi-Monte Carlo Methods, Hong Kong 2000, (to appear) eds. F. Hickernell and H. Niederreiter, Springer Verlag. See also http://arXiv.org/abs/quant-ph/0105114. S. Heinrich and E. Novak, On a problem in quantum summation, submitted for publication, 2001. M. Kwas and Y.Li, Worst case complexity of multivariate Feynman-Kac path integration, submitted for publication. M. Kwas and H. Woźniakowski, Sharp error bounds on quantum Boolean summation in various settings, in preparation. A. Nayak and F. Wu, The quantum query complexity of approximating the median and related statistics, STOC, May, 1999, 384-393. See also LANL preprint quant-ph/9804066. H. Niederreiter, Random number generation and Quasi-Monte Carlo methods, [*CBMS-NSF Reg.Conf.Series Appl.Math.*]{}, 63, [*SIAM*]{}, Philadelphia, 1992. E. Novak, Deterministic and stochastic error bounds in numerical analysis, Lecture Notes in Mathematics, 1349, [*Springer Verlag*]{}, Berlin, 1988. E. Novak, The real number model in numerical analysis, [*J. Complexity*]{}, 11, 57-73, 1995. E. Novak, Quantum complexity of integration, [*J. Complexity*]{}, 17, 2-16, 2001. See also LANL preprint quant-ph/0008124. E. Novak, I. H. Sloan and H. Woźniakowski. Tractability of approximation for weighted Korobov spaces on classical and quantum computers, http://arXiv.org/abs/quant-ph/0206023. L. Plaskota, G.W. Wasilkowski and H. Woźniakowski, A new algorithm and worst case complexity for Feynman-Kac path integration, [*J. Comp. Phys.*]{}, 164, 355-353, 2000. P. W. Shor, Algorithms for quantum computation: Discrete logarithms and factoring, Proceedings of the 35th Annual Symposium on Foundations of Computer Science, [*IEEE Computer Society Press*]{}, Los Alamitos, Ca., 121-131. See also http://arXiv.org/abs/quant-ph/9508027. P. W. Shor, Quantum computing, [*Documenta Mathematica*]{}, extra volume ICM 1998, I, 467-486. J. F. Traub, A continuous model of computation, [*Physics Today*]{}, May,1999, 39-43. J. F. Traub, G. W. Wasilkowski, and H. Woźniakowski, Information-based complexity, [*Academic Press*]{}, New York, 1988. J. F. Traub and A. G. Werschulz, Information and complexity, [*Cambridge University Press*]{}, Cambridge, UK, 1998. N. N. Vakhania, V. I. Tarieladze, and S. A. Chobanyan, Probability distributions on Banach spaces, [*Reidel, Dordrecht*]{}, 1987. G. W. Wasilkowski, and H. Woźniakowski, On tractability of path integration, [*J. of Math. Physics*]{}, 37(4), 2071-2088, 1996. [^1]: This research was supported in part by the National Science Foundation. Effort sponsored by the Defense Advanced Research Projects Agency (DARPA) and Air Force Research Laboratory, Air Force Materiel Command, USAF, under agreement number F30602-01-2-0523. The U.S, Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Defense Advanced Research Projects Agency (DARPA), the Air Force Research Laboratory, or the U.S. Government. [^2]: We follow a convention of complexity theory that if the complexity growth is faster than polynomial then we say it is exponential. [^3]: The theta notation means that there exist positive numbers $c_1$ and $c_2$ such that $c_1j^{-k}\le \l_j\le c_2j^{-k}$ for all $j=1,2,\dots$.

--- abstract: 'Incompleteness of Dirac quantization scheme leads to a redundant set of solutions of the Wheeler-DeWitt equation for the wavefunction in superspace of quantum cosmology. Selection of physically meaningful solutions matching with quantum initial data can be attained by a reduction of the theory to the sector of true physical degrees of freedom and their canonical quantization. The resulting physical wavefunction unitarily evolving in the time variable introduced within this reduction can then be raised to the level of the cosmological wavefunction in superspace of 3-metrics to form a needed subset of all solutions of the Wheeler-DeWitt equation. We apply this technique in several simple but nonlinear minisuperspace models and discuss both at classical and quantum level physical reduction in [*extrinsic*]{} time – the time variable determined in terms of extrinsic curvature (or momentum conjugated to the cosmological scale factor). Only this extrinsic time gauge can be consis! tently used in vicinity of turning points and bounces where the scale factor reaches extremum and cannot monotonically parameterize the evolution of the system. Since the 3-metric scale factor is canonically dual to extrinsic time variable, the transition from the physical wavefunction to the wavefunction in superspace represents a kind of the generalized Fourier transform. This transformation selects square integrable solutions of the Wheeler-DeWitt equation, which guarantee Hermiticity of canonical operators of the Dirac quantization scheme. This makes this scheme consistent – the property which is not guaranteed on general solutions of the Wheeler-DeWitt equation. Semiclassically this means that wavefunctions are represented by oscillating waves in classically allowed domains of superspace and exponentially fall off in classically forbidden (underbarrier) regions. This is explicitly demonstrated in flat FRW model with a scalar field having a constant negative potential and for the case of phantom scalar field with a positive potential. The FRW model of a scalar field with a vanishing potential does not lead to selection rules for solutions of the Wheeler-DeWitt equation, but this does not violate Hermiticity properties, because all these solutions are anyway of plane wave type and describe cosmological dynamics without turning points and bounces. In models with turning points the description of classically forbidden domains goes beyond original principles of unitary quantum reduction to the physical sector, because it includes complexification of the physical time variable or complex nature of the physical Hamiltonian. However, this does not alter the formalism of the Wheeler-DeWitt equation which continues describing underbarrier quantum dynamics in terms of [*real*]{} superspace variables.' author: - 'A.O. Barvinsky' - 'A.Yu. Kamenshchik' title: 'Selection rules for the Wheeler-DeWitt equation in quantum cosmology' --- Introduction ============ Needless to say that quantum cosmology is an inalienable tool in the studies of the very early quantum Universe. Inflationary cosmology [@inflation], whose observational status was essentially strengthened after the first Planck release [@Planck], successfully resolves many traditional problems of the Big Bang theory, but it is unable to resolve the issue of initial conditions for the cosmological evolution. This issue is an ultimate goal of quantum cosmology. On the other hand, it would be not a great exaggeration to say that quantum cosmology is one of the most discredited areas in modern theoretical physics. The Wheeler-DeWitt equation [@DeWitt], as an incarnation of this theory, is widely considered as a decorative tool that would never lead to original achievements in gravity theory. At best, it would be used as a justification of the semiclassical results obtained by other much simpler and down-to-earth methods of quantum field theory in curved spacetime. Though much of this criticism seems to be really true, the situation with this equation very much resembles the status of modern $S$-matrix theory vs Schrödinger equation and its stationary perturbation theory. Despite the fact that scattering amplitudes are much simpler to obtain by the covariant Feynman diagrammatic technique, everyone clearly understands that their machinery is based on the fundamental but manifestly noncovariant Schrödinger equation, and without it this technique becomes a set of ungrounded contrived rules. A similar situation holds for the Wheeler-DeWitt equation which underlies the physical dynamically independent content of the theory and its numerous applications. However, there is a big difference – the formalism of the Wheeler-DeWitt equation itself, without additional assumptions, does not form a closed physical theory. These assumptions concern two main ingredients of the theory – selection of physically meaningful solutions of the Wheeler-DeWitt equation and the construction of the physical inner product on their subspace, that could generate observable amplitudes and expectation values. Matter of fact is that this equation has many more solutions than those having clear physical setup. This setup is dictated by dynamically independent degrees of freedom, which are intricately hidden among the full set of gravitational variables. Various ways to disentangle these degrees of freedom and separate them from the gauge variables give rise to different quantization schemes. One scheme consists in disentangling them at the classical level and canonically quantizing in the resulting reduced phase space. Another scheme is the Dirac quantization [@Dirac], when the first class Dirac constraints are imposed as equations on the quantum states in the representation space in which all original phase space variables (both physical and gauge ones) satisfy canonical commutation relations. This scheme is not closed as a physical theory, because it does not have by itself a well-defined conserved positive definite inner product that would provide unitarity of the theory. Ho! wever, exactly this approach is usually employed in numerous applications of quantum cosmology. As a rule they do not go beyond achieving the solution of Wheeler-DeWitt equation and giving it some interpretation or fundamentally restricting the gravitational sector to the semiclassical domain [@Rubakov] and using Born-Oppenheimer approximation [@BO]. The most striking feature of this approach is a mismatch between the nature of the Wheeler-DeWitt equation and the number of its boundary (or initial) value data. As a second order differential equation (of the Klein-Gordon type in minisuperspace applications) it requires two data at the Cauchy surface – the value of the function and its normal (or time) derivative, but any local unitary quantum theory assumes the existence of only one initial data function – wavefunction of the initial state. This clearly demonstrates incompleteness of the Dirac quantization, and the goal of our work is to achieve its completion by formulating the selection rules for solutions of the Wheeler-DeWitt equation and demonstration of these rules in several simple minisuperspace models. In the minisuperspace context the redundancy in solutions of this equation directly manifests itself in positive and negative frequency solutions, existing due to the hyperbolic rather than parabolic nature of the Wheeler-DeWitt equation. Usual interpretation of these solutions as describing expanding and contracting cosmological models breaks down in the vicinity of the bounce – the point of maximal or minimal size of the Universe, where expansion goes over into contraction or vice versa. Consistent description of this situation is possible if we start with the reduced phase space quantization with an appropriately chosen time variable. This automatically leads to the missing selection rule in the set of solutions of the Wheeler-DeWitt equation, and as a by-product raises the issue of Hermiticity of canonically commuting operators in the definition of quantum Dirac constraint. The structure of the paper is as follows. In Sect. II we present unitarity approach to the Wheeler-DeWitt formalism [@BarvU] which allows one to raise the reduced phase space quantum state to the level of the wavefunction in the DeWitt superspace of quantum cosmology. This procedure, described in Sect.III, implies a special [*time-nonlocal*]{} transform from the physical wavefunction, satisfying a conventional Schrödinger equation, to the Wheeler-DeWitt wavefunction and leads to the selection rules of the above type. In the following sections we apply this transform in several minisuperspace models and discuss relevant operator Hermiticity and unitarity properties. In Sects. IV-V we consider the model with a negative constant potential. In Sect. VI we consider the model of a massless scalar field with a vanishing potential. Section VII is devoted to the model, based on the phantom scalar field with a positive constant potential. Section VIII is devoted to discussion an! d conclusions. Appendices A and B present useful formulae for unitary canonical transformations and integral representations for Bessel and modified Bessel functions. Unitarity approach to quantum cosmology ======================================= Gravity theory in the canonical formalism has the action which in condensed DeWitt notations [@DeWitt1] is of the following form $$\begin{aligned} S=\int dt\,\big\{p_i\dot{q}^i -N^\mu H_\mu(q,p)\big\}. \label{1.1} \end{aligned}$$ Here $q^i$ represent spatial metric coefficients and matter fields. Their conjugated momenta are denoted by $p_i$. The condensed index $i$ includes discrete labels and also [*spatial*]{} coordinates, and Einstein summation rule implies integration over the latter. The same concerns the indices $\mu$ of the Lagrange multipliers $N^\mu$ which are the ADM lapse and shift functions [@ADM]. In the formal treatment of the infinite dimensional configuration space as a finite-dimensional manifold (what we assume in this section) the range of $i$ is $i=1,...n$ and the range of $\mu$ is $\mu=1,...m$ with $n>m$. In asymptotically-flat models the integrand of (\[1.1\]) contains also the contribution of the ADM surface term Hamiltonian $-H_0(q,p)$, but below we consider only spatially closed or spatially flat minisuperspace models without this term. The variation of the Lagrange multipliers leads to the set of nondynamical equations – constraints $$\begin{aligned} H_\mu(q,p)=0, \label{1.2} \end{aligned}$$ which in gravity theory comprise the set of Hamiltonian and momentum constraints. The constraint functions $H_\mu(q,p)$ belong to the first class and satisfy the Poisson-bracket algebra $$\begin{aligned} &&\{H_\mu,H_\nu\}=U^\lambda_{\mu\nu}H_\lambda, \label{1.3} \end{aligned}$$ with the structure functions $U^\lambda_{\mu\nu}=U^\lambda_{\mu\nu}(q,p)$ which depend on phase-space variables of the theory. Dirac quantization of the theory (\[1.1\]) consists in promoting the initial phase-space variables and the constraint functions to the operator level $(q,p,H_\mu)\rightarrow (\hat{q},\hat{p},\hat{H}_\mu)$ and in selecting the physical states $|\,{\mbox{\boldmath$\varPsi$}}\big>$ in the representation space of $(\hat{q},\hat{p},\hat{H}_\mu)$ by the equation $$\begin{aligned} \hat{H}_\mu|\,{\mbox{\boldmath$\varPsi$}}\big>=0. \label{1.5} \end{aligned}$$ The operators $(\hat{q},\hat{p})$ satisfy the canonical commutation relations $[\hat{q}^k,\hat{p}_l]=i\hbar\delta^k_l$ and the quantum constraints $\hat{H}_\mu$ as operator functions of $(\hat{q},\hat{p})$ should satisfy the correspondence principle with the classical $c$-number constraints and be subject to the commutator algebra $$\begin{aligned} [\hat{H}_\mu,\hat{H}_\nu]= i\hbar\hat{U}^{\lambda}_{\mu\nu} \hat{H}_\lambda. \label{1.6} \end{aligned}$$ with some operator structure functions $\hat{U}^{\lambda}_{\mu\nu}$ standing to the left of operator constraints. This algebra generalizes (\[1.3\]) to the quantum level and serves as integrability conditions for equations (\[1.5\]). Classical reduction to the physical sector: coordinate gauge conditions ----------------------------------------------------------------------- The theory with the action (\[1.1\]) is invariant under the set of transformations of phase space variables $(q^i,p_i)$ and Lagrange multipliers [@BarvU] signifying diffeomorphism invariance of the original action in the Lagrangian form. Conventional approach to this situation implies that the equivalence class of variables belonging to the orbit of these transformations corresponds to one and the same physical state. The description of this state in terms of physical variables consists in singling out the unique representative of each such class and in treating the independent labels of this representative as physical variables. This can be attained by imposing on original phase space variables the gauge conditions $$\begin{aligned} \chi^\mu(q,p,t)=0 \label{5.03} \end{aligned}$$ which determine in the $2n$-dimensional phase space the $(2n-m)$-dimensional surface (remember that $n$ is the range of index $i$, while $m$ is that of $\mu$) having a unique intersection with the orbit of gauge transformations. At least locally, the latter condition means the invertibility of the Faddeev-Popov matrix [@FaddeevPopov] with the nonvanishing determinant $$\begin{aligned} J=\det J^\mu_\nu,\quad J^\mu_\nu= \{\chi^\mu,H_\nu\}. \label{5.2} \end{aligned}$$ Gauge conditions of the form (\[5.03\]) impose restrictions only on phase space variables and single out the physical sector locally in time – all variables in the canonical action expressed in terms of true physical degrees of freedom. This goes as follows. To begin with, the Lagrange multipliers (which are not fixed by equations of motion for the action (\[1.1\])) become uniquely fixed as functions of $(q^i,p_i)$. This directly follows from the conservation in time of gauge conditions, serving as the equation for lapse and shift functions $$\begin{aligned} &&\frac{d}{dt}\chi^\mu=\{\chi^\mu,H_\nu\}N^\nu +\frac{\partial\chi^\mu}{\partial t}=0,\\ &&N^\mu=-J^{-1\,\mu}_{\,\,\,\;\;\;\,\nu} \frac{\partial\chi^\nu}{\partial t}. \label{lapse0} \end{aligned}$$ For the reparameterization invariant systems with $H_0(q,p)=0$ in (\[1.1\]), the gauge conditions should explicitly depend on time in order to generate dynamics in the physical phase space [@BarvU; @BKr]. For gravitational systems with the Lagrangian multipliers playing the role of lapse and shift functions this is obvious – nonzero values of the latter exist only for $\partial\chi^\nu/\partial t\neq 0$. [^1] The parametrization of $(q^i,p_i)$ in terms of phase space variables of the physical sector $(\xi^A,\pi_A)$, $A=1,...n-m$, in its turn, follows from solving together the system of constraints (\[1.2\]) and gauge conditions (\[5.03\]), which determine the embedding of the $2(n-m)$-dimensional physical phase space into the space of $(q^i,p_i)$ $$\begin{aligned} &&q^i=q^i(\xi^A,\pi_A,t),\\ &&p_i=p_i(\xi^A,\pi_A,t). \end{aligned}$$ Internal coordinates of this embedding should satisfy the canonical transformation law for the symplectic form restricted to the physical subspace $$\begin{aligned} p_i dq^i= \pi_A d\xi^A -H_{\rm phys}(\xi^A,\pi_A,t)dt +dF(q^i,\xi^A,t), \label{sympform} \end{aligned}$$ so that $\xi^A$ and $\pi_A$ can be respectively identified with physical phase space coordinates and conjugated momenta, $H_{\rm phys}(\xi^A,\pi_A,t)$ considered as a physical Hamiltonian and $F(q^i,\xi^A,t)$ – the generating function of this canonical transformation. The simplest form of this reduction is for gauge conditions imposed only on phase space coordinates, $\chi^\mu(q,t)=0$. Such [*coordinate*]{} gauge conditions determine the embedding of the $(n-m)$-dimensional space $\Sigma$ of physical [*coordinates*]{} $\xi^A$ directly into the space of original coordinates $q^i$, $$\begin{aligned} \Sigma: q^i=e^i(\xi^A,t),\,\,\, \chi^\mu(e^i(\xi,t),t)\equiv 0. \label{5.4} \end{aligned}$$ Here $\xi^A$ are identified with the physical coordinates, and the conjugated momenta $\pi_A$ and the physical Hamiltonian reads $$\begin{aligned} &&\pi_A=p_i\frac{\partial e^i}{\partial\xi^A},\\ &&H_{\rm phys}(\xi^A,\pi_A,t)=-p_i(\xi^A,\pi_A,t)\frac{\partial e^i(\xi,t)}{\partial t}, \label{5.8a} \end{aligned}$$ As we see, the physical momenta are the projections of the original momenta $p_i$ to the vectors tangential to $\Sigma$, $e^i_A\equiv\partial e^i/\partial\xi^A$. In view of the contact nature of this transformation the generating function $F(q^i,\xi^A,t)$ in (\[sympform\]) is vanishing. The normal projections of $p_i$ should be found from the constraints (\[1.2\]), the local uniqueness of their solution being granted by the nondegeneracy of the Faddeev-Popov determinant. Together with (\[5.4\]) this solution yields all the original phase space variables $(q^i,p_i)$ as known functions of the physical degrees of freedom $(\xi^A,\pi_A)$. The original action (\[1.1\]) reduced to the physical sector (that is to the subspace of constraints and gauge conditions) acquires the usual canonical form with the physical Hamiltonian (\[5.8a\]). Quantum reduction ----------------- Canonical quantization in the physical sector (or reduced phase space quantization) consists in promoting $\xi^A,\pi_A,H_{\rm phys}(\xi^A,\pi_A,t)$ to the level of operators $\hat\xi^A,\hat\pi_A,\hat H_{\rm phys}$, subject to canonical commutation relations $[\hat\xi^A,\hat\pi_A]=i\hbar\delta^A_B$, and postulating the Schrödinger equation for the quantum state of the system in the Hilbert space of these operators $$\begin{aligned} i\hbar\frac\partial{\partial t}\varPsi_{\rm phys}(t,\xi)=\hat H_{\rm phys} \varPsi_{\rm phys}(t,\xi). \label{SchroedEq} \end{aligned}$$ Here $\varPsi_{\rm phys}(t,\xi)=\langle\xi|\varPsi_{\rm phys}(t)\rangle$ is the wave function of this state in the coordinate representation. The kernel of its unitary evolution can be represented by the path integral over trajectories in the reduced phase space, $$\begin{aligned} &&\varPsi_{\rm phys}(t_+,\xi_+) =\int d\xi_-\nonumber\\ &&\qquad\qquad\times K(t_+,\xi_+|t_-,\xi_-)\, \varPsi_{\rm phys}(t_-,\xi_-), \label{5.10a}\\ \nonumber\\ &&K(t_+,\xi_+|t_-,\xi_-) =\int\limits_{\xi(t_\pm)=\xi_\pm}D[\xi,\pi]\nonumber\\ &&\qquad\quad\times\exp\frac{i}{\hbar} \int\limits_{t_-}^{t_+}dt \big\{\pi_A\dot\xi^A -H_{\rm phys}(\xi,\pi,t)\big\}, \label{unitaryK} \end{aligned}$$ where $D[\xi,\pi]=\prod_t d\xi(t)\,d\pi(t)$ is the Liouville measure of integration over trajectories interpolating between the two points $\xi_\pm$ – the arguments of the evolution kernel. Profound success in quantization of gauge theories achieved in seventies and eighties of the last century [@FaddeevPopov; @DeWitt; @DeWitt1; @FV; @BFV] was based on the identical transformation in this path integral from the variables of the reduced phase space to the variables of the original action (\[1.1\]). This transformation brings us to the Faddeev-Popov canonical path integral for the two-point kernel in the space of original coordinates $q^i$ – DeWitt superspace of 3-matrics and matter fields [@BarvU] $$\begin{aligned} &&\!\!\!{\mbox{\boldmath$K$}}(q_+,q_-) =\int\limits_{q(t_\pm)=q_\pm} D[\,q,p\,]\,DN\,\nonumber\\ &&\!\!\!\times\prod\limits_{t_+>t>t_-} J_t\,\delta(\chi_t)\,\exp\frac{i}{\hbar} \int\limits_{t_-}^{t_+}dt \big\{p_i\dot q^i-N^\mu H_\mu\big\}, \label{K} \end{aligned}$$ where $D[q,p]=\prod_t \prod_i dq^i(t)\,dp_i(t)$ is the Liouville measure of integration over trajectories interpolating between the points $q_\pm$, $$\begin{aligned} DN=\prod\limits_{t_+\geq t\geq t_-} \prod_{\mu}\, dN^{\mu}(t) \end{aligned}$$ is the integration measure over lapse and shift functions including the integration over $N^{\mu}(t_\pm)$ at boundary points $t_\pm$ and $\prod_t J_t\,\delta(\chi_t)$ is the Faddeev-Popov gauge fixing factor $$J_t\,\delta(\chi_t)\equiv \det J^\mu_\nu\big(q(t),p(t)\big) \prod\limits_\alpha\delta\big(\chi^\alpha(q(t),t)\big),$$ which restricts the integration over $q(t)$ at any $t\neq t_\pm$ to the gauge condition surface (\[5.03\]). Integration over $N^\mu(t_\pm)$ has a very important consequence. It implies that $$\hat H_\mu\Big(q_+,\frac{\hbar}i\frac{\partial}{\partial q_+}\Big) {\mbox{\boldmath$K$}}(q_+,q_-) = 0, \label{WdW-lapse}$$ i.e. one finds that the two-point kernel ${\mbox{\boldmath$K$}}$ is a solution of the quantum Dirac constraint $\hat H_\mu {\mbox{\boldmath$K$}}=0$ – the Wheeler-DeWitt equation. One can show that due to the well known gauge independence properties of the Faddeev-Popov path integral this kernel is independent of the choice of $\chi^\mu$ (in the class of admissible gauges). Also in view of the time parametrization invariance it is independent of $t_\pm$.[^2] The role of $\mbox{\boldmath$K$}(q,q')$ is revealed by the observation that in the semiclassical approximation it can be related to the unitary evolution operator (\[unitaryK\]) [@GenSem; @BKr; @BarvU]. This relation derived in [@BarvU] by slicing the path integral and confirmed in the semiclassical approximation in [@GenSem; @BKr] reads $$\begin{aligned} &&K(t,\xi|t',\xi')\nonumber \\ &&\quad= \left.\left(\frac{\overrightarrow{\!\!J}}M\right)^{1/2}\!\! {\mbox{\boldmath$K$}}(q,q') \left(\frac{\overleftarrow{J'}}{M'}\right)^{1/2} \right|_{\,q=e(\xi,t),\, q'=e(\xi',t')}\nonumber \\ &&\quad+O(\hbar), \label{5.9} \end{aligned}$$ where the operator-valued factors $\overrightarrow{J}$ and $\overleftarrow{J'}$ coincide with the Faddeev-Popov determinants in which the c-number momentum argument is replaced by its operator representation[^3], $$\begin{aligned} \overrightarrow{\!\!J}=J\Big(q,\frac{\hbar}i \frac{\overrightarrow\partial}{\partial q}\Big),\quad \overleftarrow{J'}=J\Big(q,-\frac{\hbar}i \frac{\overleftarrow \partial}{\partial q'}\Big), \label{operatorJ} \end{aligned}$$ and $M=M(q)$ is a measure of integration over the $(n-m)$-dimensional physical subspace $\Sigma$ in $n$-dimensional $q$-space, satisfying $$\begin{aligned} d^{n-m}\xi=d^n q\,\delta(\chi(q)) M(q). \label{M} \end{aligned}$$ This implies that the kernel ${\mbox{\boldmath$K$}}(q,q')$ similarly to the Schrödinger propagator $K(t,\xi|t',\xi')$ can be regarded as a propagator of the Dirac wavefunction ${\mbox{\boldmath$\varPsi$}}(q)$ in $q$-space. The boundary value problem for this wavefunction can be written down as $$\begin{aligned} &&\hat H_\mu\Big(q,\frac{\hbar}i\frac{\partial}{\partial q}\Big)\, \mbox{\boldmath$\varPsi$}(q)=0, \label{5.11}\\ &&\varPsi_{\rm phys}(\xi,t)=\left. \left(\frac{\overrightarrow{\!\!J}} M\right)^{1/2}\, \mbox{\boldmath$\varPsi$}(q)\, \,\right|_{\,q=e(\xi,t)} \!\!\!\!+O(\hbar), \label{5.11a} \end{aligned}$$ where the relation (\[5.11a\]) serves as “initial" condition for ${\mbox{\boldmath$\varPsi$}}(q)$ specified on the $(n-m)$-dimensional surface $\Sigma$, and $m$-equations (\[5.11\]), $\mu=1,...m$, propagate this initial data onto entire $n$-dimensional superspace.[^4] This propagation from the initial surface $\Sigma'$ via the two-point kernel $\mbox{\boldmath$K$}(q,q')$ reads in the semiclassical approximation as $${\mbox{\boldmath$\varPsi$}}(q) =\int dq'\,{\mbox{\boldmath$K$}}(q,q')\, \delta(\chi(q',t'))\, \overrightarrow{J'}\, {\mbox{\boldmath$\varPsi$}}(q') +O(\hbar). \label{5.11b}$$ Relations (\[5.9\])-(\[5.11b\]) could have been exact beyond the semiclassical approximation if the first-class constraints were linear in momenta $p_i$. In this case the equations (\[5.11\]) would have specified the derivatives of $\mbox{\boldmath$\varPsi$}(q)$ along gauge directions and the Faddeev-Popov determinant in the coordinate gauge would be just a $q$-dependent measure factor $J=J(q)$ independent of $p_i$. In quantum cosmology it is impossible, because the Hamiltonian constraint is quadratic in momenta, and $J(q,p)$ is a nonlinear function of momenta of power coinciding with the total number of Hamiltonian constraints (which is of course infinite in full gravity theory and formally equals the number of spatial points $\infty^3$). [^5] Formal treatment of infinite number of degrees of freedom can be avoided in minisuperspace applications of quantum cosmology, when only one Hamiltonian constraint ($m=1$) survives in the finite dimensional phase space of the FRW metric and homogeneous matter fields. However, another problem still remains with the boundary value problem (\[5.11\])-(\[5.11a\]). The Wheeler-DeWitt equation (\[5.11\]) is at least quadratic in derivatives $\partial/i\partial q$ and requires two initial conditions on $\Sigma$ – the value of $\mbox{\boldmath$\varPsi$}(q)$ and its first order derivative, so that the number of solutions is at least doubled as compared to the reduced phase space dynamics. This “positive-negative" frequency doubling serves as a source of a rather speculative third quantization concept [@third], which represents the attempt to go beyond physical phase space reduction. We will, however, remain within physical reduction concept which consists in lifting the physical wavefunction $\varPsi_{\rm phys}(t,\xi)$ to the level of the wavefunction in superspace $\mbox{\boldmath$\varPsi$}(q)$. The latter of course satisfies the Wheeler-DeWitt boundary value problem (\[5.11\])-(\[5.11a\]) but incorporates only the physical wavefunction information. In other words, a formal boundary value problem (\[5.11\])-(\[5.11a\]) contains many more solutions than the physically relevant ones which are encoded in the Schrödinger equation (\[SchroedEq\]) of the reduced phase space quantization. Thus our task will be finding the selection rules for the solutions of the Wheeler-DeWitt equation appropriate for the physical setting in the reduced phase space. Basis of classical constraints and their operator realization ------------------------------------------------------------- Classical theory is of course invariant under the change of the basis of constraints with any invertible matrix $\varOmega^\nu_\mu=\varOmega^\nu_\mu(q,p)$, $$H_\mu\to H'_\mu= \varOmega^\nu_\mu H_\nu, \label{basischange}$$ and under canonical transformations of phase space variables $$\begin{aligned} &&(q,p)\to (\tilde q,\tilde p), \label{cantran0}\\ &&p_i\,dq^i-\tilde p_i\, d\tilde q^i=dF(q,\tilde q). \label{cantran5} \end{aligned}$$ Here $F(q,\tilde q)$ is a generating function of this canonical transform, relating old and new symplectic forms. One should expect that at the quantum level this invariance should hold as unitary equivalence of Dirac quantization schemes in different constraint bases and different canonical parameterizations. At least in the semiclassical approximation (which includes one-loop prefactors) this issue formally has affirmative resolution. Interestingly, it is associated with the problem of operator realization of quantum constraints $\hat H_\mu$ which should satisfy a closed commutator algebra (\[1.6\]). As shown in [@BarvU; @BKr; @geom] in this approximation the quantum Dirac constraints are given by the Weyl ordering of their classical expressions $$\hat H_\mu=N_{\rm W} H_\mu(\hat q,\hat p) +O(\hbar^2). \label{Weylorder}$$ Remarkably, this prescription holds in any basis of classical constraints. Under the basis change (\[basischange\]) the formalism undergoes unitary transformation to new wavefunctions ${\mbox{\boldmath$\varPsi$}}'(q)$ and new operators $\hat H'_\mu$, $\hat H'_\mu{\mbox{\boldmath$\varPsi$}}'(q)=0$, $$\begin{aligned} &&\!\!\!\!{\mbox{\boldmath$\varPsi$}}(q) \to{\mbox{\boldmath$\varPsi$}}'(q) =\big({\rm det}\, \hat\varOmega^\nu_\mu\big)^{-1/2}\, {\mbox{\boldmath$\varPsi$}}(q) +O(\hbar), \label{tildepsi}\\ &&\!\!\!\!\hat H_\mu\to\hat H'_\mu=\big({\rm det}\, \hat\varOmega^\alpha_\beta\big)^{-1/2}\, \hat\varOmega^\nu_\mu\,\hat H_\nu\,\big({\rm det}\, \hat\varOmega^\alpha_\beta\big)^{1/2}\nonumber\\ &&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad +O(\hbar^2). \label{HtoH} \end{aligned}$$ In other words, Dirac wavefunction ${\mbox{\boldmath$\varPsi$}}(q)$ is a density of minus one half weight in the space of gauge indices [@geom].[^6] Similarly, the classical canonical transformation (\[cantran0\]) induces the unitary transformation (see derivation in Appendix A for a generic quantum system), $$\begin{aligned} &&{\mbox{\boldmath$\varPsi$}}(q)\to \tilde{\mbox{\boldmath$\varPsi$}}(\tilde q),\quad \hat{\widetilde H}_\mu\tilde{\mbox{\boldmath$\varPsi$}}(\tilde q)=0,\\ &&{\mbox{\boldmath$\varPsi$}}(q)= \int d\tilde q\, \left|\,{\rm det}\,\frac1{2\pi\hbar} \frac{\partial^2F(q,\tilde q)}{\partial q^i\;\partial \tilde q^k}\,\right|^{1/2}\nonumber\\ &&\qquad\qquad\quad \times\exp\left(\frac{i}{\hbar}\,F(q,\tilde q)\right) \tilde{\mbox{\boldmath$\varPsi$}}(\tilde q) +O(\hbar), \label{Fourier00} \end{aligned}$$ which was checked in [@geom] at least for contact transformations, $\tilde q=\tilde q(q)$, under which (\[Fourier00\]) reduces to the transformation law of the weight 1/2 density $$\begin{aligned} &&{\mbox{\boldmath$\varPsi$}}(q)= \left|\,{\rm det}\, \frac{\partial\tilde q}{\partial q}\, \right|^{1/2} \tilde{\mbox{\boldmath$\varPsi$}} (\tilde q(q))+O(\hbar). \label{Fourier01} \end{aligned}$$ These unitary equivalence transformations will be important in what follows, because we will have to go to another convenient canonical parametrization of the theory and also pick up a special normalization of the Hamiltonian constraint. Only their consistent treatment will guarantee correct transition between the physical and Wheeler-DeWitt wavefunctions of the theory. Minisuperspace models: from the physical sector to the Wheeler-DeWitt wavefunction ================================================================================== The goals formulated in the end of Sect.II.B can be explicitly implemented in case of minisuperspace quantum cosmology with one Hamiltonian constraint. For the index $\mu$ taking in minisuperspace case only one value, $$H_\mu(q,p)\equiv H(q,p),\quad \chi^\mu(q,p,t)\equiv\chi(q,p,t),$$ the gauge condition can be rewritten as expressing explicitly $t$ as a function on the phase space of $q^i$ and $p_i$, $$\chi(q,p,t)=0\quad \Rightarrow\quad \chi(q,p,t) \equiv T(q,p)-t=0, \label{Time}$$ so that the Faddeev-Popov determinant and the relevant lapse function (\[5.2\])-(\[lapse0\]) read $$J=\{T,H\},\quad N=\frac1{J}. \label{JvN}$$ The critical point of the physical reduction is the nondegeneracy of $J$ over the entire phase space. Degeneration of $J$ to zero at certain points in phase space implies the breakdown of the physical reduction known in the context of Yang-Mills type gauge theories as Gribov copies problem [@Gribov]. This problem arises when the surface of gauge conditions is not transversal everywhere to the orbits of gauge transformations and does not pick up a single representative of each class of gauge equivalent configurations. In gravity theory, and specifically in minisuperspace quantum cosmology, this problem manifests itself in the fact that the time variable $T(q,p)$ is not a monotonically growing function along all possible (on shell and off shell) histories. If $J$ changes sign, then according to (\[JvN\]) $N$ changes sign too, and according to the geometrical interpretation of the lapse function the spacelike hypersurface of constant time $t$ with growing $t$ starts movin! g in spacetime back to the past. Therefore a meaningful physical reduction makes sense only in domains where $J$ does not change sign. In classical theory this problem is usually circumvented by such a choice of time which is monotonic at the classical history. The same applies to semiclassical quantization which probes only infinitesimal neighborhood of the latter. However, even in this simplified case the requirement of monotonic $T$ imposes serious restrictions on its choice as a function of phase space variables. The role of the coordinate $q$ in minisuperspace models is basically played by the cosmological scale factor $a=e^\alpha$, so that $$q^i,p_i=\alpha,p_\alpha;\;\xi,\pi,$$ where $\xi,\pi$ are actually the matter degrees of freedom other than $\alpha,p_\alpha$. This immediately forbids the choice of the coordinate gauge or time $T=T(\alpha)$ (intrinsic metric time) in problems with bouncing cosmology when the evolution of $a$ undergoes a bounce from some of its maximum or minimum values. More generally, the coordinate gauge conditions $\chi(q,t)$ are forbidden even semiclassically, because their $J= \{\chi,H\}$ vanishes on caustics in multidimensional superspace [@UFN] ($\dot a=0$ is the simplest degenerate case of the caustic in one-dimensional superspace). Using classical theory as a guide in the search for admissible gauges we will chose such $T(q,p)$ which monotonically grows along the classical trajectory, and in the bounce cosmology it immediately leads to $p$-dependent choice – the so-called extrinsic time $T(\alpha,p_\alpha)$ [@extrinsic]. This transition to momentum-dependent gauges implies that the quantum reduction (\[5.11a\]) no longer applies directly. What we need is, first, to make a canonical transformation of the type (\[cantran0\]), $$\begin{aligned} &&\alpha,p_\alpha\;\to\; T=T(\alpha,p_\alpha),\quad p_T=p_T(\alpha,p_\alpha),\nonumber \\ && p_\alpha d\alpha-p_T dT=dF(\alpha,T), \label{canonical} \end{aligned}$$ with the relevant generating function $F(\alpha,T)$. Second, we make classical and quantum reduction in terms of new phase space variables in the gauge (\[Time\]). In these variables the Hamiltonian constraint and the Wheeler-DeWitt equation correspondingly read as $$\begin{aligned} &&H\equiv H(T,p_T;\xi,\pi)=0, \label{HinT}\\ &&\hat H\Big(T,\frac\partial{i\partial T};\hat\xi,\hat\pi\Big)\, |\,\widetilde{\mbox{\boldmath$\varPsi$}}(T)\,\rangle=0. \end{aligned}$$ The ket vector notation here refers to the state in the Hilbert space of matter operators $(\hat\xi,\hat\pi)$ and tilde labels the Wheeler-DeWitt wavefunction in the representation of the variable $T$. Quantum reduction to the physical sector involves the Faddeev-Popov “determinant" and its operator realization $$\begin{aligned} &&J(T,p_T;\xi,\pi)= \frac{\partial H}{\partial p_T},\\ &&\overrightarrow{\!\!J}= J\Big(T,\frac\partial{i\partial T}; \hat\xi,\hat\pi\Big). \label{JinT} \end{aligned}$$ The resulting physical wavefunction $|\,\varPsi_{\rm phys}(t)\,\rangle$ satisfies the Schrödinger equation (\[SchroedEq\]). In view of minisuperspace nature of the system the embedding (\[5.4\]), $q=e(\xi,t)$, of the physical space into superspace of $q=(T,\xi)$ is in fact a one to one map between $q$ and the arguments $(t,\xi)$ of $|\,\varPsi_{\rm phys}(t)\,\rangle=\varPsi_{\rm phys}(t,\xi)$. Therefore Eq.(\[5.11a\]) can be reversed to raise the physical quantum state to the level of the Wheeler-DeWitt wavefunction in superspace of the $T$-variable. Because of $dq\,\delta(\chi(q,t))\equiv dT\,\delta(T-t)\, d\xi=d\xi$, cf. (\[M\]), the integration measure in (\[5.11a\]) $M=1$, and this equation can be rewritten as[^7] $$\begin{aligned} |\,\widetilde{\mbox{\boldmath$\varPsi$}}(T)\,\rangle= \frac1{\big(\overrightarrow{\!\!J}\,\big)^{1/2}}\, |\,\varPsi_{\rm phys}(T)\,\rangle +O(\hbar). \label{5.11c} \end{aligned}$$ The invertibility of $J$ semiclassically guarantees the invertibility of the corresponding operator coefficient acting on the $T$ argument of $|\,\varPsi_{\rm phys}(T)\,\rangle$. Similarly to (\[Fourier00\]) the canonical transformation (\[canonical\]) implies at the quantum level the unitary transformation to the $\alpha$-representation from that of $T$ $$\begin{aligned} |\,{\mbox{\boldmath$\varPsi$}}(\alpha)\,\rangle= \int_{-\infty}^{\infty} dT\,\langle\,\alpha\,|\,T\,\rangle\, |\,\widetilde{\mbox{\boldmath$\varPsi$}}(T)\, \rangle, \label{genFtrans1} \end{aligned}$$ where $$\langle\,\alpha\,|\,T\,\rangle= \left|\frac1{2\pi\hbar} \frac{\partial^2F(\alpha,T)}{\partial\alpha\;\partial T}\right|^{1/2}e^{\frac{i}\hbar\,F(\alpha,T)} +O(\hbar), \label{genFtrans2}$$ whence $$\begin{aligned} &&|\,{\mbox{\boldmath$\varPsi$}}(\alpha)\,\rangle= \int_{-\infty}^{\infty} dT\, \left|\frac1{2\pi\hbar} \frac{\partial^2F(\alpha,T)}{\partial\alpha\;\partial T}\right|^{1/2}e^{(i/\hbar)\,F(\alpha,T)}\nonumber\\ &&\qquad\qquad\quad\times \frac1{\big(\overrightarrow{J}\,\big)^{1/2}}\, |\,\varPsi_{\rm phys}(T)\, \rangle+O(\hbar). \label{Fourier0} \end{aligned}$$ Note that this relation is nonlocal in time – only the knowledge of the entire Schrödinger evolution of $|\,\varPsi_{\rm phys}(t)\,\rangle$ allows one to recover the Wheeler-DeWitt wavefunction in superspace of $q$. Below we demonstrate how this equation works in several simple but essentially nonlinear minisuperspace models. What will be important for us is whether the resulting wavefunction is either exponentially suppressed or infinitely oscillating at superspace boundaries $\alpha=-\infty$ and $\alpha=\infty$ (similarly for $T$ and $\hat p_T$ at the boundaries $T=\pm\infty$). This would guarantee Hermiticity of the phase space operators – necessary property of the Dirac quantization and Wheeler-DeWitt equation formalism. Simultaneously this would provide us with the selection rules for physically motivated solutions of the Wheeler-DeWitt equation, which remove one half of the full set of their positive and negative frequency solutions. This restores the balance between the number of these solutions and the number of physical data in the reduced phase space quantization – a single physical wavefunction $|\,\varPsi_{\rm phys}(t)\,\rangle$ at the initial Cauchy surface. Flat FRW model with a homogeneous scalar field ============================================== We consider a flat Friedmann universe with the metric $$ds^2 = N^2(t) dt^2 - e^{2\alpha(t)}d{\bf x}^2, \label{Fried}$$ and a spatially homogeneous scalar field $\phi(t)$. Here $N(t)$ is the lapse function and $e^{\alpha(t)}$ is the cosmological scale factor. The range of the minisuperspace variable $\alpha$ $$-\infty<\alpha<\infty \label{range}$$ covers all values of the scale factor from singularity to infinite expansion. With the normalization of the gravitational constant $1/16\pi G=3/4$ the action of this model $$S = \int dx \sqrt{-g}\left(-\frac{R}{16\pi G} + \frac12g^{\mu\nu}\phi_{,\mu}\phi_{,\nu} - V(\phi)\right). \label{action}$$ gives rise to the minisuperspace Lagrangian and the Hamiltonian constraint $$\begin{aligned} &&L = -9e^{3\alpha}\frac{\dot\alpha^2}{2N} +e^{3\alpha}\frac{\dot{\phi}^2}{2N} -Ne^{3\alpha}V(\phi), \label{Lagrange1}\\ &&H=e^{-3\alpha}\left(-\frac1{18}\,p_{\alpha}^2 + \frac12\,p_{\phi}^2\right) + V(\phi)\,e^{3\alpha}, \label{H0} \end{aligned}$$ in terms of the canonical momenta for $\alpha$ and $\phi$ $$p_{\alpha} = -9\,\frac{\dot{\alpha} e^{3\alpha}}{N},\quad p_{\phi} = \frac{\dot{\phi} e^{3\alpha}}{N}. \label{momentum2}$$ According to discussion in Sect.II.D there is a freedom in operator realization of this constraint associated with its overall normalization, $H\to H'=\varOmega\,H$ (choice of the constraint basis (\[basischange\])). This freedom allows us to simplify the operator realization of $H'$. Multiplication of (\[H0\]) with $\varOmega\equiv e^{3\alpha}$ converts the constraint into the form $$H'=-\frac1{18}\,p_{\alpha}^2 + \frac12\,p_{\phi}^2 + V(\phi)\,e^{6\alpha}. \label{super}$$ The advantage of this normalization is that the kinetic term of $H$ is independent of minisuperspace coordinates, so that the Weyl ordering in the operator realization (\[Weylorder\]) is trivial. In the coordinate representation it reduces to the replacement of momenta by partial derivatives $$p_{\alpha} = -i\frac{\partial}{\partial \alpha},\quad p_{\phi} = -i\frac{\partial}{\partial \phi}, \label{momentum4}$$ and the minisuperspace operator of the Wheeler-DeWitt equation for the cosmological wave function ${\mbox{\boldmath$\varPsi$}}(\alpha,\phi)$ $$\hat H'\,{\mbox{\boldmath$\varPsi$}} (\alpha,\phi)=0 \label{WdW}$$ takes the form $$\hat H'=\frac1{18} \frac{\partial^2}{\partial \alpha^2} - \frac12\frac{\partial^2}{\partial \phi^2}+ V(\phi)e^{6\alpha}. \label{WdW1}$$ The case of a negative constant potential ========================================= Variational equations for (\[Lagrange1\]) in the gauge $N =1$ (the corresponding cosmic time we will denote by $\tau$) read $$\begin{aligned} &&\frac92\,h^2 = \frac{\dot{\phi}^2}{2} + V(\phi), \label{Fried1}\\ &&\ddot{\phi} + 3h\dot{\phi} + \frac{dV}{d\phi} = 0, \label{KG} \end{aligned}$$ where $h$ is the Hubble parameter $$h \equiv \dot\alpha, \label{Hubble}$$ and unusual coefficient of $h^2$ in (\[Fried1\]) is in fact $3/8\pi G=9/2$ in the chosen normalization of the gravitational constant. These equations essentially simplify for a constant negative potential – negative cosmological constant, $$V(\phi) = -V_0, \label{poten}$$ when the scalar field $\phi$ becomes a cyclic variable with a conserved momentum $p_{\phi}$. Then the equation for $\dot\phi$ in terms of $p_{\phi}$ $$\dot{\phi} = e^{-3\alpha}p_{\phi}, \label{velocity3}$$ when substituted in the Friedmann equation (\[Fried1\]) yields $$\begin{aligned} \frac92\,\dot\alpha^2 = \frac12\,p_{\phi}^2\,e^{-6\alpha} - V_0. \label{Fried3} \end{aligned}$$ Integrating this equation one gets $$\begin{aligned} &&e^{\alpha(\tau)} = \left(|p_{\phi}|\, \frac{\sin \sqrt{2V_0}\tau} {\sqrt{2V_0}}\right)^{1/3}, \label{radius}\\ &&h(\tau) = \frac{\sqrt{2V_0}} 3\cot \sqrt{2V_0}\tau. \label{Hubble1} \end{aligned}$$ Thus, the Universe begins its evolution at $\tau = 0$ (Big Bang cosmological singularity), reaches the point of the maximal expansion at $\tau = \pi/2\sqrt{2V_0}$ when $h = 0$ and begins contracting to the Big Crunch singularity at $\tau = \pi/\sqrt{2V_0}$. During this evolution the Hubble parameter is monotonously decreasing from $+\infty$ to $-\infty$. Below we consider quantum cosmology of this model. Note that the range of cosmic time is finite, $0<\tau<\pi/\sqrt{2V_0}$. General solution of the Wheeler-DeWitt equation ----------------------------------------------- With a constant potential (\[poten\]) in view of cyclic nature of $\phi$ the wavefunction ${\mbox{\boldmath$\varPsi$}}(\alpha,\phi)$ is easily represented by its Fourier transform $$\begin{aligned} {\mbox{\boldmath$\varPsi$}}(\alpha,\phi)= \int_{-\infty}^{\infty} dp_\phi\,\varPsi(\alpha,p_\phi)\, e^{ip_\phi\phi}. \end{aligned}$$ The wavefunction in the momentum representation $\varPsi(\alpha,p_\phi)$ then satisfies the Wheeler-DeWitt equation $$\left(\frac19\,\frac{\partial^2}{\partial \alpha^2} + p_{\phi}^2-2V_0 e^{6\alpha}\right) \varPsi(\alpha,p_{\phi}) = 0. \label{WdW2}$$ This equation has a general solution [@Olver] $$\begin{aligned} &&\varPsi(\alpha,p_{\phi}) = \psi_1(p_{\phi})\,I_{i|p_{\phi}|} \big(\sqrt{2V_0}\,e^{3\alpha}\big)\nonumber \\ &&\qquad\qquad\quad+\psi_2(p_{\phi})\, K_{i|p_{\phi}|} \big(\sqrt{2V_0}\,e^{3\alpha}\big), \label{WdW-sol} \end{aligned}$$ where $I_{\nu}(x)$ and $K_{\nu}(x)$ are the modified Bessel functions of the first and the second kind respectively, whereas $\psi_1(p_{\phi})$ and $\psi_2(p_{\phi})$ are generic functions of the momentum $p_{\phi}$. The two branches of the generic solution (\[WdW-sol\]) are drastically different. Near the cosmological singularity, $\alpha\to-\infty$, in view of imaginary value of $\nu=i|p_\phi|$ they both represent plane waves $\sim e^{\pm 3\nu\alpha}=e^{\pm 3i|p_\phi|\alpha}$. However, for $x=e^{3\alpha}\to+\infty$ one of them is rapidly growing, $I_\nu(x)\propto e^x/\sqrt{x}=\exp(e^{3\alpha}-3\alpha/2)$, and another is exponentially decaying $K_\nu(x)\propto e^{-x}/\sqrt{x}$. Therefore, Hermiticity of canonical momenta $\hat p_\alpha=-i\partial/\partial\alpha$ and other operators with respect to the $L_2$ inner product on the range of $\alpha$ (\[range\]) is possible only for quantum states represented by the second branch of the solution (\[WdW-sol\]). These Hermiticity properties are very important in the Dirac quantization scheme and in even more general BRST/BFV quantization scheme [@BFV; @BarvU; @WhyBFV].[^8] Violation of these properties leads to inconsistency of the formalism. Therefore, consistency of Dirac quantization should serve as a selection rule which retains only the MacDonald function branch $K_{i|p_{\phi}|} \big(\sqrt{2V_0}\,e^{3\alpha}\big)$ of (\[WdW-sol\]). Below we show that in this particular model the same selection rule follows from quantization in the physical sector. Physical wavefunction --------------------- From the discussion of Sect.II.C (see Eqs. (\[Time\])-(\[JvN\])) we remember that the gauge condition in the reduction to the physical sector can be cast into the form $$\chi(T,t) = T-t, \label{gauge}$$ explicitly depending on time $t$, and $T=T(\alpha,p_\alpha)$ is a new canonical variable, depending on the old phase space variables. For consistency of physical reduction, such a variable should monotonously grow with time at least on classical solutions of the model. For models with the cosmological expansion followed by contraction, the variables depending only on the scale factor $a$ (or its logarithm $\alpha$) are not monotonous. Thus, $T(\alpha,p_\alpha)$ should involve the canonical momentum, and it is called extrinsic time [@extrinsic] (involving extrinsic curvature rather than intrinsic geometry of a spatial surface in spacetime). A possible choice is the following function $$T(\alpha,p_\alpha) = \frac1{3\sqrt2}\, p_{\alpha}e^{-3\alpha}, \label{choice}$$ which is proportional to the Hubble parameter[^9] and therefore classically has infinite range, $-\infty<T<\infty$ (cf. beginning of Sect.V). The conjugated momentum for this variable is $$p_{T} = -\sqrt2\, e^{3\alpha}. \label{momentum5}$$ so that the inverse transform from $(T,P_T)$ to $(\alpha,p_\alpha)$ reads $$\begin{aligned} &&e^{3\alpha} = -\frac{p_T}{\sqrt2}, \label{inv}\\ &&p_{\alpha} = -3p_T T. \label{inv1} \end{aligned}$$ The constraint (\[super\]) in terms of new variables $$H'(T,p_T,p_\phi)\equiv-\frac12\,p_T^2 (T^2 + V_0)+\frac12 \,p_{\phi}^2=0 \label{constraint1}$$ has a solution for $p_T$ $$p_T = -\frac{|p_{\phi}|}{\sqrt{T^2+V_0}},$$ where a particular sign of the square root is chosen in accordance with the geometrical meaning of the momentum $p_T$ (minus the 3-dimensional volume of the cosmological model). Thus, the physical Hamiltonian in the gauge (\[gauge\]) reads (cf. Eq. (\[5.8a\])) $$H_{\rm phys}(p_\phi,t) = -p_T\,\big|_{\,T=t} = \frac{|p_{\phi}|}{\sqrt{t^2+V_0}}. \label{Hamilton-phys}$$ The corresponding Schrödinger equation for the physical wave function $$i\,\frac{\partial \varPsi_{\rm phys}(t,p_\phi)}{\partial t} = H_{\rm phys}(p_{\phi},t)\, \varPsi_{\rm phys}(t,p_\phi). \label{Schrodinger}$$ immediately gives the time evolution $$\begin{aligned} &&\varPsi_{\rm phys}(\tau,p_\phi) = \psi_0(p_{\phi}) \exp\left(-i|p_\phi|\,{\rm arcsinh} \frac{t}{\sqrt{V_0}}\right)\nonumber\\ &&=\psi_0(p_{\phi}) \left(\frac{\sqrt{V_0}}{t+\sqrt{t^2+V_0}} \right)^{i|p_\phi|}, \label{Schrodinger1} \end{aligned}$$ where $\psi_0(p_{\phi})$ is an arbitrary function of the momentum $p_{\phi}$ – the initial data for (\[Schrodinger\]) at $t=0$. Note that the time $t$ in contrast to the cosmic time $\tau$ has infinite range $-\infty<t<\infty$, and Eq.(\[Schrodinger\]) propagates the data from $t=0$ both forward and backward in time. From the physical wave function to the solution of the Wheeler-DeWitt equation ------------------------------------------------------------------------------ To lift the physical wavefunction to the level of the Wheeler -DeWitt wavefunction according to (\[5.11c\]) we have to act with the inverse square root of the operator version of the Faddeev-Popov operator. In the gauge (\[gauge\]) it reads $$J=\{T, H'\}= -p_T(T^2 + V_0), \label{JP1}$$ so that semiclassically $$\overrightarrow{\!\!J}=-(T^2 + V_0)\,\frac\partial{i\,\partial T},$$ Thus, with the same one-loop precision (disregarding the operator ordering in the above equation) $$\tilde\varPsi(T,p_\phi) = \left(-\frac\partial{i\,\partial T}\right)^{-1/2} \frac{\varPsi_{\rm phys}(T,p_\phi) } {\sqrt{T^2+V_0}}+O(\hbar). \label{JP2}$$ Now by the generalized Fourier transform (\[genFtrans1\])-(\[genFtrans2\]) we have to go from the $T$-representation to $\alpha$-representation. The classical generating function $F(\alpha,T)$ of this transform equals $$\begin{aligned} &&F(\alpha,T)=\sqrt2\,e^{3\alpha}T,\\ &&p_\alpha d\alpha-p_T dT=dF(\alpha,T). \end{aligned}$$ Therefore the kernel of the unitary transformation (\[genFtrans2\]) reads $$\langle\,\alpha\,|\,T\,\rangle= \sqrt{\frac3{\pi\sqrt2}}\,e^{3\alpha/2+i\sqrt2\,e^{3\alpha}T} +O(\hbar),$$ and $$\begin{aligned} &&\varPsi(\alpha,p_{\phi})=\int\limits_{-\infty}^{\infty}dT\, \langle\,\alpha\,|\,T\,\rangle\, \tilde\varPsi(T,p_\phi)\nonumber\\ &&\qquad\quad=\sqrt{\frac3{\pi\sqrt2}}\; e^{3\alpha/2}\int\limits_{-\infty}^{\infty}dT\, e^{i\sqrt2\,e^{3\alpha}T}\nonumber\\ &&\qquad\quad\times\left(-\frac\partial{i\,\partial T}\right)^{-1/2} \frac{\varPsi_{\rm phys}(T,p_\phi) } {\sqrt{T^2+V_0}}+O(\hbar). \label{X} \end{aligned}$$ “Integrating by parts" the time derivative in the nonlocal operator $(-\partial/i\partial T)^{1/2}$ – that is acting by $(\partial/i\partial T)^{1/2}$ to the left on the exponential function of $T$ (which is justified by rapid oscillations of the integrand, cf. Eq.(\[Schrodinger1\]), and decrease of its amplitude at $T\to\pm\infty$) – we get $$\varPsi(\alpha,p_{\phi}) =\sqrt{\frac3{2\pi}}\, \int\limits_{-\infty}^{\infty}\frac{dT\,\varPsi_{\rm phys}(T,p_\phi)}{\sqrt{T^2+V_0}}\, e^{i\sqrt2\,e^{3\alpha}T}+O(\hbar). \label{XX}$$ Substituting (\[Schrodinger1\]) and using as a new integration variable $$x \equiv {\rm arcsinh} \frac{T}{\sqrt{V_0}}, \label{new-var}$$ we have $$\varPsi(\alpha,p_{\phi}) =\sqrt{\frac3{2\pi}}\, \psi_0(p_{\phi})\int\limits_{-\infty}^{\infty}dx\, e^{-i\,|p_{\phi}|\,x +i\,e^{3\alpha}\sqrt{2V_0}\,\sinh x}. \label{Fourier1}$$ Then, in virtue of the formula for the MacDonald function $$K_{\nu}(x)=\frac12\, e^{\frac12 \nu\pi i} \int\limits_{-\infty}^{\infty}dt\, e^{ix\sinh t -\nu t}, \label{Watson2}$$ which is valid for real positive $x$ and the order $\nu$ belonging to the interval $-1 < {\rm Re}(\nu) < 1$ (see Appendix B for derivation), the wavefunction eventually takes the form $$\varPsi(\alpha,p_{\phi}) =\sqrt{\frac{6}{\pi}}\,\psi_0(p_{\phi})\, e^{- \frac12\pi|p_{\phi}|} K_{i|p_\phi|} \big(\sqrt{2V_0}e^{3\alpha}\big). \label{M-K3}$$ This is just one branch of the general solution of the Wheeler-DeWitt equation (\[WdW-sol\]) with[^10] $$\begin{aligned} &&{\psi}_1(p_\phi) = 0, \label{restr}\\ &&{\psi}_2(p_\phi) = \sqrt{\frac{6}{\pi}}\,\psi_0(p_{\phi}) \exp\left(-\frac{\pi|p_{\phi}|}2\right). \label{restr1} \end{aligned}$$ Thus, we have only one independent function in the solution of the Wheeler-DeWitt equation. Remarkably, this is exactly the MacDonald branch of (\[WdW-sol\]) which guarantees Hermiticity of the momentum operator $\hat p_\alpha=\partial/i\partial\alpha$ and makes Dirac quantization scheme consistent. Cosmic time ----------- Physical reduction can also be done in the cosmic time gauge. To find the extrinsic time variable $\tilde T(q,p)$ that would generate cosmic time with $N=1$ in phase space, one should solve the differential equation for $\tilde T$ in partial derivatives, $\{\tilde T,H'\}=e^{3\alpha}= -p_T/\sqrt2$.[^11] This solution $\tilde T$ turns out to be related to $T$ by the contact transformation $$\tilde T(\alpha,p_\alpha) = \frac{1}{\sqrt{2V_0}}\,{\rm arccot}\left(-\frac{T(\alpha,p_\alpha)}{\sqrt{V_0}}\right), \label{T-n}$$ where $T(\alpha,p_\alpha)$ is defined by Eq.(\[choice\]). It is instructive to demonstrate that starting with the physical wavefunction $\tilde\varPsi_{\rm phys}(\tau,p_\phi)$ built in the gauge $\tilde\chi\equiv\tilde T-\tau=0$ one again comes to the solution of the Wheeler-DeWitt equation (\[WdW2\]). Repeating the procedure of reduced phase space quantization in this gauge it is easy to see the set of relations between quantization schemes with $T$ and $\tilde T$ time variables $$\begin{aligned} &&\tilde J\equiv\{\tilde T,H'\}=\frac{\partial\tilde T}{\partial T}\,J,\\ &&\overrightarrow{\!\!\tilde J}=\frac{\partial\tilde T}{\partial T}\,\overrightarrow{\!\!J},\\ &&\tilde\varPsi_{\rm phys}(\tilde T,p_\phi) =\varPsi_{\rm phys}(T(\tilde T),p_\phi). \end{aligned}$$ There is also the relation between the generating functions of canonical transformation from $T$ to $\alpha$ and from $\tilde T$ to $\alpha$, $\tilde F(\alpha,\tilde T)=F(\alpha, T(\tilde T))$, so that $$\frac{\partial^2\tilde F(\alpha,\tilde T)}{\partial\alpha\,\partial\tilde T}= \frac{\partial^2F(\alpha,T)}{\partial\alpha\,\partial T}\, \frac{\partial T}{\partial\tilde T}.$$ Using these relations in the tilde version of (\[Fourier0\]) we see that, after the change of integration variable from $\tilde T$ to $T(\tilde T)$ all factors of $\partial\tilde T/\partial T$ cancel out, and it yields the same Wheeler-DeWitt wavefunction. This confirms the anticipated property of the formalism that quantization schemes in different gauges give rise to one and the same Wheeler-DeWitt wavefunction. The case of a vanishing potential ================================= Qualitatively different situation in the above model takes place in the case of a vanishing potential. Its classical evolution (\[radius\]) in the limit $V_0\to 0$ becomes $$e^\alpha=(|p_{\phi}|\,\tau)^{1/3}, \quad 0<\tau<\infty, \label{radius1}$$ for the positive range of cosmic time and corresponds to cosmological expansion from singularity to infinite scale factor. With $V_0=0$ the relation (\[T-n\]) between cosmic time $\tau$ and time variable $t$ reads $$\tau=-\frac1{\sqrt2\, t}, \label{tau-t}$$ so that (\[radius1\]) maps onto $$e^\alpha=\left(-\frac{|p_{\phi}|}t\right)^{1/3}, \quad -\infty<t<0, \label{radius10}$$ on a negative range of $t$. The contracting stage of the cosmological evolution can be described by opposite ranges of $\tau$ and $t$ $$\begin{aligned} e^\alpha&=&(-|p_{\phi}|\,\tau)^{1/3}\nonumber\\ &=& \left(\frac{|p_{\phi}|}t\right)^{1/3},\; 0<t<\infty,\;-\infty<\tau<0. \label{radius11} \end{aligned}$$ Both expansion and contraction can be unified as consecutive stages of a single evolution by gluing together semi-axes of $\tau$ or $t$ $$e^\alpha=(-|p_{\phi}|\,|\tau|)^{1/3}= \left(\frac{|p_{\phi}|}{|t|}\right)^{1/3},\quad -\infty<\tau,t<\infty. \label{radius12}$$ With this identifications the transition through the point $\tau=0$ implies the “bounce" at the singularity $e^\alpha=0$, whereas a similar transition through $t=0$ can be interpreted as a “turning" point from expansion to contraction at infinite scale factor, $e^\alpha\to\infty$. This unification is not physically natural however, because this transition through $\tau=0$ and $t=0$ lacks continuity and violates matching physical data at these junction points. As we will see below, this is even more manifest within physical reduction both at classical and quantum levels. At the quantum level the general solution of the Wheeler-DeWitt equation (\[WdW2\]) for $V_0 = 0$ represents a pure plane wave $$\varPsi(\alpha,p_{\phi}) = \psi_1(p_{\phi})\,e^{3i|p_{\phi}|\alpha} + \psi_2(p_{\phi})\,e^{-3i|p_{\phi}|\alpha}.\label{WdWmassless}$$ Therefore, in contrast to (\[WdW-sol\]) Hermiticity of momentum operator does not impose restrictions on coefficient functions $\psi_{1,2}(p_\phi)$. However, physical reduction continues selecting only one independent branch of this general solution. By choosing the range of time variable $T\leq 0$ or $T\geq 0$ one can restrict dynamics in the physical sector entirely to expanding or contracting phases of the evolution, the way it happens on solutions of equations of motion in classical theory. From the geometric meaning of $p_T$ as a negative quantity (\[momentum5\]) it follows that it reads as a solution of the Hamiltonian constraint $p_T=-|p_\phi|/|T|$ for both signs of $T$, and the physical Hamiltonian $H_{\rm phys}=|p_\phi|/|t|$ at the quantum level gives $$\varPsi_{\rm phys}^\pm(t,p_\phi)= \psi(p_\phi)\,\left(\frac{t_0}t\right)^{\pm i|p_\phi|}$$ respectively for contracting $(+)$ and expanding $(-)$ cases, for which $t$ and $t_0$ are correspondingly positive and negative. Here $t_0$ is the initial data moment of time when $\varPsi_{\rm phys}^\pm(t_0,p_\phi)=\psi(p_\phi)$, and it has the same sign as $t$. Then integration in Eq.(\[XX\]) with $V_0=0$ respectively over positive and negative values of $T$ gives $$\begin{aligned} &&\varPsi^\pm(\alpha,p_{\phi}) =\sqrt{\frac3{2\pi}}\,\psi(p_\phi)\, \left(\sqrt2|\,t_0\,|\right)^{\pm i|p_\phi|}\nonumber\\ &&\qquad\qquad\qquad\!\times\, e^{\frac{\pi|p_\phi|}2}\varGamma\big(\mp i|p_\phi|\,\big)\, e^{\pm 3i\,|p_\phi|\alpha}, \end{aligned}$$ which are of course the two branches of (\[WdWmassless\]).[^12] Thus, the physical reduction separately for contracting and expanding cosmological models leads to selection of a relevant branch in the full Wheeler-DeWitt solution. How natural is the unification of these two branches into a single picture mentioned above? In physical reduction this unification is possible only by the price of violating the geometrical meaning of $p_T$ as a negative quantity (\[momentum5\]), because the requirement of analyticity demands a replacement $H_{\rm phys}=|p_\phi|/|t|\to H_{\rm phys}=|p_\phi|/t$. For the full time range, $-\infty<t<\infty$, this generates a physical wavefunction $$\varPsi_{\rm phys}(t,p_\phi)= \psi(p_\phi)\,\left(\frac{t_0}t\right)^{i|p_\phi|}.$$ It has a branching point at $t=0$ and requires a prescription for analytical continuation either from $t>0$ to $t<0$ (“contracting" wavefunction) or from $t<0$ to $t>0$ (“expanding" wavefunction). Moreover, physical reduction at $T=0$ also becomes inconsistent because the main ingredient of this reduction, the Faddeev-Popov factor $J=-p_TT^2$, gets degenerate at $T=0$, and again a special prescription is needed how to detour this point by the path in the complex plane of $T$. Provided this is done, one can apply Eq.(\[XX\]) with integration over a full range of $T$, $-\infty<T<\infty$, and acquire $$\varPsi(\alpha,p_{\phi})= \left\{\begin{array}{l} 0,\\ \\ \big(1-e^{-2\pi|p_\phi|}\big)\, \varPsi^+(\alpha,p_{\phi}), \end{array}\right. \label{betas}$$ where the first case corresponds to the analytic continuation of $\varPsi_{\rm phys}(t,p_\phi)$ from the positive values of $t$ to the upper shore of the branch cut at negative semi-axis of $t$, $\varPsi_{\rm phys}(-|t|,p_\phi)=\varPsi_{\rm phys}(e^{i\pi}|t|,p_\phi)$, and the second case corresponds to the lower shore of this cut. Similar expressions in terms of $\varPsi^-(\alpha,p_{\phi})$ can be obtained if we start with the physical “expanding" wavefunction analytic near the negative semi-axis of $t$ and continue it to the branch cut along the positive semi-axis. In both cases only one branch of the general solution (\[WdWmassless\]) $\varPsi^\pm(\alpha,p_{\phi})$ is generated and physical reduction leads to selection of Wheeler-DeWitt wavefunctions. Similar conclusions can be reached within cosmic time reduction with extrinsic time variable $\tilde T$. Now it is hard to say how meaningful is this unification of expanding and contracting stages. This is, of course, a certain extension of the quantization concept in physical sector. Within the latter this unification seems as contrived as it is in classical theory. Classical solutions having no turning points at large values of the scale factor and no bounces close to singularities imply that quantum dynamics is also entirely restricted to either expansion or contraction, and both of them are related by time inversion. Intrinsic time -------------- Absence of turning points in classical dynamics implies that physical reduction can be done in the intrinsic time – the situation when the time variable $T$ is chosen as a function of only 3-metric of the theory and does not involve its conjugated momenta. In minisuperspace context this means identifying $T$ with, say, $\alpha$ and imposing a simple gauge $$\chi = \alpha-t. \label{gauge-alpha}$$ Then the physical Hamiltonian equals $-p_{\alpha}$ and as a solution of the constraint in our simple model with a vanishing scalar potential $V_0=0$ reads $$H_{phys} = -p_{\alpha} =-3\varepsilon\,|p_{\phi}|. \label{Ham-double}$$ In this case we do not have any reason to disregard any of the $\varepsilon=\pm 1$ sign factors which correspond respectively to expansion and contraction. Initial value data at a given $t$, that is at a given spacelike surface with the scale factor $e^{3t}$, includes the discrete degree of freedom $\varepsilon$ indicating the direction of evolution. The physical wavefunction becomes a two-component vector $\varPsi_{\rm phys}^\varepsilon(t,p_\phi)$ whose components evolve with time differently under the action of the Hamiltonian (\[Ham-double\]). Lifting this state to the level of the Wheeler-DeWitt wavefunction implies the generalized Fourier transform, analogous to (\[X\]) but apparently including summation over discreet values of $\varepsilon$. In this simple model with $V_0=0$ it trivially leads to the superposition (\[WdWmassless\]) with absolutely independent functions $\psi_{1,2}(p_\phi)$. Note that in the case of the vanishing potential the Hermiticity condition discussed above does not impose restrictions on the general solution of the Wheeler-DeWitt equation. The two terms in the general solution correspond respectively to expanding and contracting universes. When we construct the physical wave function, based on the intrinsic time choice we again obtain two physical wave functions, which can be lifted to the level of two branches of the general solution of the Wheeler-DeWitt equation. However, if we consider instead the extrinsic time parameter, then the corresponding physical wave function, when lifted to the level of the solution of the Wheeler-DeWitt equation, contains only one independent solution of this equation. This is explained by the fact that introduction of the extrinsic time implies a unique evolution instead of two independent evolutions – contraction and expansion. Namely, we have expansion-infinity-contraction for the time parameter associated with the Hubble variable and contraction-singularity-expansion for the cosmic time parameter. Selection of one branch of the Wheeler-DeWitt equation solution via physical reduction with external time might look unnatural. Such a selection is not enforced by Hermiticity requirement, but rather arises as an artifact of unifying the expansion and contraction of the universe as stages of unique evolution, whereas physically in this model these stages are separated either by cosmological singularity or by a domain of asymptotically infinite size of the universe. Thus, intrinsic time treatment and intrinsic time setting of Cauchy problem seems more natural here, because it yields two-component physical wavefunctions, which give rise to two independent branches of the Wheeler-DeWitt wavefunction, describing two different types of evolution in quantum ensemble – expansion and contraction. Phantom scalar field with a positive constant potential ======================================================= In order to see that nontrivial selection of solutions of the Wheeler-DeWitt equation matches with Hermiticity requirements in Dirac quantization we consider another example – a phantom scalar field with a positive constant potential. This field has a negative kinetic term, $$S = \int dx \sqrt{-g}\left(-\frac{R}{16\pi G}- \frac12g^{\mu\nu}\phi_{,\mu}\phi_{,\nu} - V_0\right),V_0>0, \label{phatomaction}$$ and despite obvious violation of unitarity in scattering problems this model recently attracted a lot of interest in context of dark energy models [@phantom]. As we will see it also raises interesting issues of underbarrier semiclassical behavior and boundaries in cosmological minisuperspace. This happens because this model has a cosmological bounce at small values of the scale factor – the situation similar to known prescriptions for the cosmological wave function of the universe, based on the ideas of Euclidean quantum gravity and quantum tunneling [@HH; @Vil; @tun1; @tun-we]. The Friedmann equations (\[Fried1\])-(\[Fried3\]) with inverted kinetic term of the scalar field and inverted constant potential ($p_\phi^2\to-p_\phi^2$, $V_0\to-V_0$) have a solution which in the cosmic time $\tau$ and the Hubble time $t$ of Sect.V read $$\begin{aligned} &&e^{3\alpha} =|p_{\phi}|\, \frac{\cosh\sqrt{2V_0}\tau}{\sqrt{2V_0}} =\frac{|p_\phi|}{\sqrt{2(V_0-t^2)}},\label{phantom2}\\ &&h=\frac{\sqrt{2V_0}}3\,\tanh \left(\sqrt{2V_0}\tau\right) =\frac{\sqrt2}3\,t. \label{phantom1} \end{aligned}$$ Time variables of this cosmological evolution from the moment of infinite size to the bounce and back to the moment of infinite expansion run in the ranges $$-\infty<\tau<\infty, \quad -\sqrt{V_0}<t<\sqrt{V_0}. \label{t-phantom}$$ In contrast to the model of Sect.V here the range of Hubble time is compact. At the level of Dirac quantization the Wheeler-DeWitt equation here reads $$\left(\frac19\,\frac{\partial^2}{\partial \alpha^2}-p_{\phi}^2 +2V_0\,e^{6\alpha}\right) \varPsi(\alpha,p_{\phi}\phi)=0 \label{WdW2-phan}$$ and has a general solution $$\begin{aligned} &&\varPsi(\alpha,p_{\phi}) = \psi_1(p_{\phi})\,J_{|p_\phi|} \big(\sqrt{2V_0}e^{3\alpha}\big) \nonumber \\ &&\qquad\qquad\quad +\psi_2(p_{\phi})\,J_{-|p_\phi|} \big(\sqrt{2V_0}e^{3\alpha}\big), \label{WdW-sol-phan} \end{aligned}$$ where $J_{\nu}(x)$ are the Bessel functions. Their leading behavior at $x \to \infty$ $$J_{\nu} (x) \sim \sqrt{\frac{2}{\pi x}}\cos\left(x-\frac{\nu\pi}{2} -\frac{\pi}{4}\right), \label{Bessel2-ph}$$ and therefore Hermiticity requirement at $\alpha \rightarrow \infty$ does not impose any restriction on the branches of (\[WdW-sol-phan\]). However, at the cosmological singularity $\alpha \rightarrow -\infty$ the Bessel function of a negative order diverges as $x^{\nu}$, and the requirement of Hermiticity selects only the first term of (\[WdW-sol-phan\]). Let us see if this selection also works if we start with quantization in the physical sector. Repeating the steps of Sect.V in the gauge $\chi = T-\tau=0$, where the Hubble time variable is chosen similarly to (\[choice\]) (opposite sign is taken to match the start of contraction from infinity with the negative value $T=-\sqrt{V_0}$) $$T = -\frac1{3\sqrt2}\, p_{\alpha}e^{-3\alpha},\quad p_T = \sqrt2\, e^{3\alpha}. \label{bar1}$$ The Hamiltonian constraint, Faddeev-Popov factor and the physical Hamiltonian now read $$\begin{aligned} &&H'(T,p_T,p_\phi)=-\frac12\, p_T^2 (T^2-V_0) -\frac12 \,p_{\phi}^2, \label{bar4}\\ &&J = -p_T\,(V_0-T^2), \label{FP-ph}\\ &&H_{\rm phys} = -p_{\bar{T}}= -\frac{|p_{\phi}|}{\sqrt{V_0-t^2}}, \label{Ham-ph} \end{aligned}$$ and the relevant solution of the Schrödinger equation is $$\varPsi_{\rm phys}(t,p_{\phi}) = \psi_0(p_{\phi})\exp\left(i\,|p_{\phi}|\,{\rm arcsin}\frac{t}{\sqrt{V_0}} \right), \label{Schrod-ph}$$ where of course $\psi_0(p_{\phi})$ is the physical data at $t=0$. Similarly to (\[X\])-(\[XX\]) the transition from $\varPsi_{\rm phys}(t,p_{\phi})$ to $\varPsi(\alpha,p_{\phi})$ becomes $$\varPsi(\alpha,p_{\phi}) =\sqrt{\frac3{2\pi}}\, \int\limits_{-\sqrt{V_0}}^{\sqrt{V_0}}\frac{dT\,\varPsi_{\rm phys}(T,p_\phi)}{\sqrt{T^2-V_0}}\, e^{-i\sqrt2\,e^{3\alpha}T}+O(\hbar), \label{XXXX}$$ where we retain the integration range (\[t-phantom\]) where only the unitary evolution with a [*real*]{} ${\rm arcsin}(t/\sqrt{V_0})$ is possible. Introducing a new variable $$\theta \equiv {\rm arcsin}\frac{T}{\sqrt{V_0}}. \label{theta}$$ we get $$\begin{aligned} &&\varPsi(\alpha,p_{\phi}) =\sqrt{6\pi}\, \psi_{0}(p_{\phi})\,I(x,\nu), \label{psi} \\ &&I(x,\nu)\equiv\frac1{2\pi} \int\limits_{-\pi/2}^{\pi/2} d\theta\,e^{-ix\sin\theta +i\nu\theta}, \label{I}\\ &&x\equiv\sqrt{2V_0}\, e^{3\alpha},\quad \nu\equiv|p_{\phi}|. \label{xnu} \end{aligned}$$ Underbarrier domains and minisuperspace boundaries -------------------------------------------------- Now we have to remember that our formalism of quantum physical reduction is known only semiclassically – up to $O(\hbar)$ terms which extend beyond one-loop order (classical exponent and prefactor). So let us check consistency of the obtained result and, in particular, the integration range in (\[I\]) within this approximation. It corresponds to the limit when both $x$ and $\nu$ are large, $$x,\,\nu=O\left(\frac1\hbar\right)\to\infty. \label{classical}$$ Classically allowed domain is defined by the overbarrier range of these parameters (\[xnu\]) $$x>\nu, \label{overbarrier}$$ where the asymptotic expansion for the integral in (\[psi\]) is given by the contributions of its two real stationary phase points $\theta_\pm=\pm\,{\rm arccos}(\nu/x)$, $$I(x,\nu)=\sqrt{\frac2\pi} \frac{\sin\left(\sqrt{x^2-\nu^2}-\nu\,{\rm arccos}\,\frac\nu{x}-\frac\pi4\right)} {(x^2-\nu^2)^{1/4}}+O(\hbar). \label{sine}$$ This coincides with the asymptotic approximation of the Bessel function $J_{\nu}(x)$ of simultaneously large argument and order (“approximation by tangents" Eq. 8.453.1 of [@Gradstein]), so that in this domain of parameters the integral (\[I\]) is in fact the “one-loop" approximation of the Bessel function $$I(x,\nu)=J_{\nu}(x)+O\left(\frac1x,\frac1\nu\right).$$ The phase of sine in (\[sine\]) is of course the function $S(x,\nu)$ satisfying in terms of $\alpha$ and $p_{\phi}$ the Hamilton-Jacobi equation for (\[WdW2-phan\]) $$-\frac19\,\left(\frac{\partial S}{\partial\alpha}\right)^2 -p_{\phi}^2 +2V_0\,e^{6\alpha}=0.$$ However, in the underbarrier regime of the semiclassical approximation $$x<\nu, \label{underbarrier}$$ the integral is given asymptotically by contributions of the boundary points $\theta=\pm\pi/2$, because there are no stationary phase points between them. These contributions are $O(1/x,1/\nu)=O(\hbar)$ and go beyond the one-loop approximation. This means that the expression (\[psi\]) does not reproduce correct semiclassical limit because it lacks in the underbarrier regime leading classical and one-loop terms. On the other hand, underbarrier phenomena are usually described by the transition into a complex plane of time variable. This serves as a strong motivation to extend the range of the variable $\theta$ beyond $\pm\pi/2$ to the upper half of the complex plane, $$\theta=\pm\frac\pi2+i\rho,\quad 0\leq\rho<\infty.$$ This, in its turn, corresponds to the extension of the range of the physical time variable from the segment $[-\sqrt{V_0},\sqrt{V_0}]$ to the full real axis $$-\infty<T<\infty.$$ On the new regions with $|T|>\sqrt{V_0}$ the physical Hamiltonian (\[Ham-ph\]) is imaginary, and the physical wavefunction becomes exponentially decaying for $\rho\to\infty$, $$\varPsi_{\rm phys}\big(\pm\sqrt{V_0}\cosh\rho,p_{\phi}\big) = \psi_0(p_{\phi})\, e^{\pm i\,\frac{\pi|p_{\phi}|}2-|p_\phi|\rho},$$ at the classically forbidden intervals of the whole time range $t=\sqrt{V_0}\sin(\pm\pi/2+i\rho)=\pm\sqrt{V_0}\cosh\rho$. Thus, if we want to include in the Wheeler-DeWitt formalism description of underbarrier phenomena the generalized Fourier transform from $\varPsi_{\rm phys}$ to $\varPsi(\alpha,p_{\phi})$ defined by (\[psi\])-(\[I\]) should involve integration over the full real axis of $T$, or in terms of $\theta$ $$\begin{aligned} &&\varPsi(\alpha,p_{\phi}) =\sqrt{6\pi}\, \psi_{0}(p_{\phi})\,J(x,\nu), \label{psiJ} \\ &&J(x,\nu)=\frac1{2\pi} \int\limits_{-\pi/2+i\infty}^{\pi/2+i\infty} d\theta\,e^{-ix\sin\theta +i\nu\theta}. \label{J} \end{aligned}$$ Here the integration contour runs vertically down from $\theta = -\pi/2+i\infty$ to $\theta = -\pi/2$, then along the real axis to $\theta = \pi/2$ and eventually goes vertically up to $\theta = \pi/2 + i\infty$. In the underbarrier range (\[underbarrier\]) this integral has a saddle point $\theta_+=i\ln(\nu/x+\sqrt{\nu^2/x^2-1})$ which contributes the leading one-loop term $$J(x,\nu)=\frac{e^{\sqrt{\nu^2-x^2}}} {\sqrt{2\pi}\,(\nu^2-x^2)^{1/4}} \left(\frac{x}{\nu+\sqrt{\nu^2-x^2}}\right)^\nu+O(\hbar)$$ missing in $I(x,\nu)$. This term is again the asymptotic expression for the Bessel function of large argument and order with $x<\nu$. Moreover, as shown in Appendix B, the integral (\[J\]) is exactly the representation of the Bessel function $J_{\nu}(x)$ for real positive argument $x$ and order $\nu$ $$J(x,\nu)=J_{\nu}(x), \quad x>0,\quad \nu>0, \label{int-B4}$$ so that finally $$\varPsi(\alpha,p_{\phi}) = \sqrt{6\pi}\,\psi_{0}(p_{\phi})\,J_{|p_{\phi}|} \big(\sqrt{2V_0}\,e^{3\alpha}\big)+O(\hbar),\label{WdW-ph5}$$ which is one of the branches of the general solution of the Wheeler-DeWitt equation (\[WdW-sol-phan\]), selected also by Hermiticity requirement. Extension of integration range from (\[I\]) to (\[J\]) still might seem contrived because it implies violation of principles of physical reduction. This reduction starts entirely in classical terms and remains consistent unless the Faddeev-Popov factor (\[FP-ph\]) degenerates to zero and physical evolution violates unitarity, that is for the domain $|T|<\sqrt{V_0}$. However, this domain does not cover full minisuperspace of $T$ or $\alpha$, and the artificial boundary at $T=\pm\sqrt{V_0}$ would mean a nonzero boundary value of $\varPsi(\alpha,p_{\phi})$ at $\alpha\to -\infty$ (or $x=0)$, because the integral (\[I\]) has a finite limit $I(0,\nu)=O(1)$ rather than exponential falloff $x^\nu\sim e^{3\alpha}$. Therefore, no Hermiticity properties of momentum operators in reduced superspace $-\sqrt{V_0}<T<\sqrt{V_0}$ can be spoken of. [^13] This strongly suggests extension of minisuperspace of $T$ up to infinity. This extension embraces a classically forbidden domain by the price of adding non-unitary dynamics in the physical sector (complex time or imaginary physical Hamiltonian) but retaining real values of the minisuperspace variable $T$ and $\alpha$. Conclusions =========== Dirac quantization of constrained systems does not form a closed physical theory. Quantum Dirac constraints, which in gravity theory context form the set of Wheeler-DeWitt equations, have many more solutions than those corresponding to the physical setting of the problem. The way to select physically meaningful solutions matching with quantum initial value data may consist in reduced phase space quantization. Performing reduction to a physical sector results after quantization in the physical wavefunction that can be raised to the level of the wavefunction in superspace of 3-metric and matter fields. This superspace wavefunction forms a subset of solutions of Wheeler-DeWitt equations parameterized by the physical initial data. This program can explicitly be realized in spatially homogeneous (minisuperspace) setup with one Hamiltonian constraint, and this was demonstrated for three simple but essentially nonlinear models: flat FRW cosmology with a scalar field having a negative constant potential (or negative $\varLambda$-term), vanishing potential and, finally, a phantom scalar field with a positive constant potential. Quite remarkably, the resulting selection rules for solutions of the minisuperspace Wheeler-DeWitt equation leave us only with those of its wavefunctions which guarantee Hermiticity of canonical phase space operators of the theory. This property is an important ingredient of the Dirac quantization scheme, but it is not a priori guaranteed to be true – generic set of solutions of the Wheeler-DeWitt equation is not square integrable and violates Hermiticity of canonical momenta operators. Central point of physical reduction is the choice of temporal gauge condition, or the choice of time $T$ as a function of phase space variables, which allows one to disentangle physical degrees of freedom, their Hamiltonian and unitarily evolving quantum state – the physical wavefunction. Consistency of this gauge fixing procedure, or nondegeneracy of the corresponding Faddeev-Popov operator, is equivalent to the requirement that monotonically growing time variable should be in one-to-one correspondence with evolving state of the system. The hint for construction of $T$ comes from considering two different types of the classical cosmological evolution - with or without turning points, i.e. the points of maximal expansion or the points of minimal contraction (bounces). In the case, when the classical evolution represents only expansion or contraction, it is sufficient to use gauges which fix the so called intrinsic time parameter, i.e. a parameter, which depends on the cosmological scale factor and is independent of its conjugated momentum. Two different physical Hamiltonians, as solutions of the quadratic Hamiltonian constraint equation, and the corresponding two physical wavefunctions arise in this case. The latter additively enter their Wheeler-DeWitt counterpart in minisuperspace and give its two independent branches without any selection rule. This, however, does not contradict Hermiticity requirement, because both branches turn out to be square integrable and admit integration of the derivatives by parts – Hermiticity of canonical momenta. This happens in the scalar field model with a vanishing potential. Qualitatively different situation arises in models whose evolution includes turning points – for a constant negative potential and for a phantom field with a positive potential. In this case, the intrinsic time gauges are inadequate, because a single value of the cosmological scale factor labels two different states - those of expansion and contraction. Instead, one should use extrinsic time – the function of the Hubble parameter whose values at least classically are in one-to-one correspondence with consecutive moments of the cosmological evolution, including the turning points and bounces. We make physical reduction with this Hubble time $T$, find the physical state evolving in this time and then raise it to the level of the wavefunction in minisuperspace of the scale factor $e^\alpha$ by a kind of a generalized Fourier transform from $T$-variable to $\alpha$-variable. Critical point of this procedure is a nontrivial selection rule – the result selects one branch of t! he generic solution of the Wheeler-DeWitt equation, which is square integrable and satisfies Hermiticity requirement. Note that the above method can also be extended to the degenerate case – the minisuperspace FRW model without matter fields entirely driven by a cosmological constant, used in pioneering papers for the construction of the tunneling [@Vil; @tun1] and no-boundary [@HH] wavefunctions of the Universe. Zero number of physical degrees of freedom in this case does not prevent us from applying the above procedure of raising the physical wavefunction to the minisuperspace level. The resulting selection rule, however, turns out to be different from the outgoing waves version of the tunneling cosmological state [@Vil-bound]. Neither it is related to cosmological singularities or behavior of the wavefunction at infinity of the scale factor. Rather, it is connected with the behavior in classically forbidden domains in minisuperspace – the shadow regions behind the turning point. Namely, in such regions one of the branches of the general solution of the Wheeler-DeWitt equation is infinitely growing and, hence, should be discarded. In this respect our selection rule seems closer to the no-boundary proposal of Hartle and Hawking [@HH], but a detailed comparison would require introduction of a spatial curvature and will be considered in the future publication. As we saw, for a vanishing potential no selection rules are enforced. This is natural because in this case one does not have classically forbidden regions in minisuperspace. This sounds like classical-to-quantum correspondence in cosmology. The structure of a classical evolution (presence or absence of turning points) determines the correct class of gauge conditions and absence or presence of selection rules for the solution of the Wheeler-DeWitt equation.[^14] The formalism of quantum reduction to the physical sector [@BarvU; @BKr; @geom] is known only semiclassically (including tree-level exponential and one-loop prefactor). The obtained results hold also with the same precision. Moreover, conformity of reduced phase space quantization with the Dirac quantization was found in [@BarvU; @BKr; @geom] only in the classically allowed (overbarrier) semiclassical domain. It holds in the sense that unitary evolution in the physical sector was mapped onto semiclassical oscillating solutions of the Wheeler-DeWitt equation. The model of phantom scalar field shows that this mapping can be extended to the classically forbidden (underbarrier) domain with exponentially damped wavefunctions. This represents the extension beyond original principles of reduced phase space quantization, because in this domain it deals with non-unitary dynamics in the physical sector. In particular, this extension uses a complex physical Hamiltonian or complex physi! cal time and encounters degeneration of the Faddeev-Popov factor (\[FP-ph\]) to zero at the boundary between the classically allowed domain and the forbidden one, $|T|=\sqrt{V_0}$. Nevertheless this extended quantum reduction provides important Hermiticity properties in Dirac quantization and selection of $L_2$-integrable Dirac-Wheeler-DeWitt wavefunctions. Moreover, as it was discussed in [@UFN] the caustic in the congruence of classical histories in superspace separating its overbarrier domains from the underbarrier ones always leads to vanishing Faddeev-Popov factor. This maintains the spirit of Euclidean quantum gravity – the concept underlying the notion of the no-boundary wavefunction which describes both classically allowed and forbidden phases of the cosmological state by [*real*]{} superspace variables [@HH; @Vil; @tun1; @tun-we]. Our principal conclusion on conformity of physical reduction and Hermiticity selection rules was attained only in simple models. Consideration of more complicated cosmological models with a full set of (inhomogeneous) field-theoretical modes can pose additional problems. For example, the one-loop approximation raises the issue of correspondence between covariant calculations and those based on explicit reduction to physical degrees of freedom. This was intensively discussed in the cosmological context [@one-loop] and in the context of the vacuum energy calculation on the background of wormholes and gravastars [@Garattini]. This means that beyond minisuperspace approximation one should be more cautious with regard to physical reduction. However, semiclassical nature of the method which captures the effect of superspace boundaries and underbarrier domains gives a hope that our main conclusion might be a generic feature of Dirac quantization, and we hope to study this ! conjecture in the future. Acknowledgments {#acknowledgments .unnumbered} =============== A.B. is grateful to the Section of INFN of Bologna and to the Department of Physics and Astronomy of the University of Bologna for hospitality during his visits to Bologna in March of 2012 and in May of 2013. A.K. is grateful to A. Vilenkin for fruitful discussions and for hospitality during his visit to the Tufts Institute of Cosmology in September of 2013. The work of A.B. was partially supported by the RFBR grant No 14-01-00489. The work of A.K. was partially supported by the RFBR grant No 14-02-01173. Unitary canonical transformations ================================= Here we remind the formalism of the unitary transformation (\[Fourier00\]) corresponding to the classical canonical transformation (\[cantran0\])-(\[cantran5\]). The latter implies that old and new canonical variables are related by $$p_m=\frac{\partial F(q,\tilde q)}{\partial q^m},\quad \tilde p_m= -\frac{\partial F(q,\tilde q)} {\partial\tilde q^m}.$$ To guarantee Hermiticity of canonical operators at the quantum level this transformation should be specified by Weyl ordering in the right hand sides of these relations $$\hat p_m=N_W\,\frac{\partial F(\hat q,\hat{\tilde q})}{\partial \hat q^m},\quad \hat{\tilde p}_m= -N_W\frac{\partial F(\hat q,\hat{\tilde q})} {\partial\hat{\tilde q}^m}.$$ The kernel of unitary transformation $\langle\,q\,|\,\tilde q\,\rangle$ can be found in the first (one-loop) order of $\hbar$-expansion by the following sequence of relations. First we rewrite Weyl normal ordering in terms of $q\tilde q$-ordering, when all operators $\hat q$ stand to the left of the operators $\hat{\tilde q}$ $$\begin{aligned} &&\!\!\!\!\!\!N_W\,\frac{\partial F(\hat q,\hat{\tilde q})}{\partial \hat q^m}= N_{q\tilde q}\left(\frac{\partial F(\hat q,\hat{\tilde q})}{\partial \hat q^m}\right.\nonumber\\ &&\!\!\!\!\!\!\quad\quad\left.-\frac12\,[\,\hat q^n,\hat{\tilde q}^k\,] \frac{\partial^2}{\partial\hat q^n \partial\hat{\tilde q}^k}\, \frac{\partial F(\hat q,\hat{\tilde q})}{\partial \hat q^m}\right)+O(\hbar^2), \end{aligned}$$ where the commutator $[\,\hat q^n,\hat{\tilde q}^k\,]$ with the same precision is given by the Poisson bracket $$\begin{aligned} &&[\,\hat q^n,\hat{\tilde q}^k\,]=i\hbar\,\{\,\hat q^n,\hat{\tilde q}^k\}+O(\hbar^2)=i\hbar\,\frac{\partial\hat{\tilde q}^k}{\partial\hat p_n}+O(\hbar^2)\nonumber\\ &&\qquad\qquad= i\hbar\,\left(\frac{\partial^2F(\hat q,\hat{\tilde q})}{\partial \hat q^n\partial\hat{\tilde q}^k}\right)^{-1}+O(\hbar^2). \end{aligned}$$ Therefore, since $\langle\,q\,|\,N_{q\tilde q}f(\hat q,\hat{\tilde q})\,|\,\tilde q\,\rangle=f(q,\tilde q)\, \langle\,q\,|\,\tilde q\,\rangle$ for any function $f(\hat q,\hat{\tilde q})$ of noncommuting operators, we have $$\begin{aligned} \frac\hbar{i}\,\frac\partial{\partial q^m}\, \langle\,q\,|\,\tilde q\,\rangle&=& \langle\,q\,|\,\hat p_m\,|\,\tilde q\,\rangle \nonumber\\ &=&\frac{\partial}{\partial q^m} \left(F -\frac{i\hbar}2\,\ln\,{\rm det} \frac{\partial^2 F}{\partial q^n \partial\tilde q^k}\right)\langle\,q\,|\,\tilde q\,\rangle\nonumber\\ &&\quad+O(\hbar^2), \end{aligned}$$ whence $$\begin{aligned} \!\!\!\langle\,q\,|\,\tilde q\,\rangle= \left|\,{\rm det}\,\frac1{2\pi\hbar} \frac{\partial^2F(q,\tilde q)}{\partial q^i\;\partial \tilde q^k}\,\right|^{1/2} e^{\frac{i}{\hbar}\,F(q,\tilde q)} +O(\hbar), \label{A} \end{aligned}$$ where the normalization coefficient follows from unitarity requirement, $\int d\tilde q\,\langle\,q\,|\,\tilde q\,\rangle \langle\,\tilde q\,|\,q'\,\rangle=\delta(q-q')$. In fact, this expression represents the Pauli-Van Vleck-Morette equation for the kernel of unitary evolution from $\tilde q$ to $q$ when $F(q,\tilde q)$ is identified with the relevant Hamilton-Jacobi function. The case of contact canonical transformations (\[Fourier01\]), $q\to\tilde q=\tilde q(q)$, requires somewhat different consideration because its classical generating function relates old coordinates $q$ with new momenta $\tilde p$, $$dF(q,\tilde p)=p\,dq+\tilde q\,d\tilde p,$$ and equals $F(q,\tilde p)=\tilde p_m\tilde q^m(q)$. Therefore, the kernel of transformation from $\hat q$ to $\hat{\tilde p}$ – the generalized Fourier transform – reads according to (\[A\]) $$\begin{aligned} \!\!\!\langle\,q\,|\,\tilde p\,\rangle= \left|\,{\rm det}\,\frac1{2\pi\hbar} \frac{\partial\tilde q}{\partial q}\;\right|^{1/2} e^{\frac{i}{\hbar}\,\tilde p\,\tilde q(q)} +O(\hbar), \label{A1} \end{aligned}$$ and the coordinate representation kernel $$\begin{aligned} \!\!\!\langle\,q\,|\,\tilde q\,\rangle&=&\frac1{\sqrt{2\pi\hbar}}\int d\tilde p\, \langle\,q\,|\,\tilde p\,\rangle\, e^{-\frac{i}{\hbar}\,\tilde p\,\tilde q}\nonumber\\ &=&\left|\,{\rm det}\, \frac{\partial\tilde q}{\partial q}\;\right|^{1/2} \delta(\tilde q(q)-\tilde q), \end{aligned}$$ yields the transformation law (\[Fourier01\]). Integral representations for Bessel and modified Bessel functions ================================================================= Eq. (\[Watson2\]) is based on the integral representation for the MacDonald function $K_{\nu}(x)$ [@Watson]: $$K_{\nu}(x) = \frac12e^{-\frac12 \nu \pi i} \int_{-\infty}^{\infty} dt\, e^{-ix\sinh t -\nu t}, \label{Mac0}$$ which is valid for real positive argument $x$ and order $\nu$ belonging to the range $-1 < {\rm Re}\,\nu < 1$. Using the symmetry $K_{-\nu}(z) = K_{\nu}(z)$ and changing the sign of the integration parameter $t$ we immediately come to $$K_{\nu}(x)=\frac12 e^{\frac12 \nu\pi i} \int_{-\infty}^{\infty}dt\,e^{ix\sinh t -\nu t}, \label{Watson21}$$ which is just Eq. (\[Watson2\]). To derive the exact integral representation (\[J\]) for the Bessel function (\[int-B4\]) we start with Eq. 8.412.6 of [@Gradstein] $$J_{\nu}(z) = \frac{1}{2\pi}\int\limits_{-\pi +i\infty}^{\pi+i\infty}d\theta\, e^{-iz\sin\theta +i\nu\theta}, \ {\rm Re}\ z > 0. \label{Bes-con}$$ The contour of integration here starts at $\theta = -\pi+i\infty$, runs vertically down to $\theta = -\pi$, follows along the real axis to the point $\theta = \pi$ and then goes vertically up to $\theta = \pi + i\infty$. The integral (\[J\]) has the integration contour similar to that of (\[Bes-con\]) with the points $\theta = \pm\pi$ replaced respectively by $\theta =\pm\pi/2$. Besides, the parameters $x$ and $\nu$ in the integrand of (\[J\]) are both real and positive. We want to show that this integral also gives the Bessel function $J_{\nu}(x)$. Since $$\begin{aligned} &&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \int\limits_{-\pi/2 +i\infty}^{\pi/2+i\infty}d\theta\,e^{-ix\sin\theta +i\nu\theta}= \left(\int\limits_{-\pi +i\infty}^{\pi+i\infty}\right. \nonumber\\ &&\!\!\!\!\!\!\!\!\!\!\!\!\qquad\left.-\int\limits_{-\pi +i\infty}^{-\pi/2+i\infty}-\int\limits_{\pi/2 +i\infty}^{\pi+i\infty}\right)\,d\theta\,e^{-ix\sin\theta +i\nu\theta} \label{int-B1} \end{aligned}$$ this statement reduces to the fact that the last two integrals in the right-hand side of this equation vanish. To prove it consider the first of these two integrals. It coincides with the integral over the horizontal segment of $$\theta = \beta + i\Lambda, \label{interval-B}$$ with the real part of $\theta$ running between $-\pi$ and $-\pi/2$, $-\pi \leq \beta \leq -\pi/2$, and the constant imaginary part $\Lambda$ tending to infinity, $\Lambda \rightarrow \infty$. At this segment the exponential of the integrand $$e^{-ix\sin\theta +i\nu\theta}=e^{-ix\sin\beta\cosh\Lambda +x\cos\beta\sinh\Lambda+i\nu\beta-\nu\Lambda}. \label{int-B2}$$ is dominated by a large real part $x\cos\beta\sinh\Lambda-\nu\Lambda$, which is negative in view of $\cos\beta \leq 0$ and $\nu>0$. Therefore, in the limit $\Lambda \rightarrow \infty$ the integrand uniformly tends to zero, and the integral $\int_{-\pi +i\infty}^{-\pi/2+i\infty}e^{-ix\sin\theta +i\nu\theta}d\theta$ vanishes. The same is true for the second integral $\int_{\pi/2 +i\infty}^{\pi+i\infty}e^{-ix\sin\theta +i\nu\theta}d\theta$. Thus, for real positive $x$ and $\nu$ the integral (\[J\]) coincides with the integral representation of the Bessel function (\[Bes-con\]) for $z=x$. [99]{} A.A. Starobinsky, in H.J. De Vega and N. Sanchez (eds.), [*Current trends in field theory quantum gravity and strings*]{}, Lecture Notes in Physics [**246**]{}, (Springer Verlag, Heidelberg, 1986), pp. 107-126; A.D. Linde, [*Particle Physics and Inflationary Cosmology*]{} (Harwood Academic, New York, 1990). P. A. R. Ade [*et al.*]{} \[Planck Collaboration\], arXiv:1303.5082 \[astro-ph.CO\]. B.S. DeWitt, Phys. Rev. [**160**]{}, 1113 (1967). P.A.M. Dirac, [*Lectures on Quantum Mechanics*]{} (Yeshiva University Press, New York, 1964). V. Lapchinsky and V.A. Rubakov, Acta Phys. Polon. B 10 (1979) 1041; T. Banks, Nucl. Phys. B 249 (1985) 332. R. Brout, Found. Phys.  [**17**]{}, 603 (1987); R. Brout and G. Venturi, Phys. Rev. D [**39**]{}, 2436 (1989); C. Bertoni, F. Finelli and G. Venturi, Class. Quant. Grav.  [**13**]{}, 2375 (1996); F. Finelli, G. P. Vacca and G. Venturi, Phys. Rev. D [**58**]{}, 103514 (1998). A.O. Barvinsky, Phys. Rept.  [**230**]{}, 237 (1993). B.S. DeWitt, [*Dynamical Theory of Groups and Fields*]{} (Gordon and Breach, New York, 1965). R. Arnowitt, S. Deser and C. Misner, in : Gravitation: an Introduction to Current Research, Ed. L. Witten (Wiley, New York, 1962), p. 227 L.D. Faddeev and V.N. Popov, Phys.Lett. B [**25**]{}, 29 (1967). A.O. Barvinsky and V. Krykhtin, Class.Quantum Grav. [**10**]{}, 1957 (1993). E.S. Fradkin and G.A. Vilkovisky, Phys. Lett. B [**55**]{}, 224 (1977); CERN report TH-2332 (1977). I.A. Batalin and G.A. Vilkovisky, Phys. Lett. B [**69**]{}, 309 (1977); I.A. Batalin and E.S. Fradkin, Riv. Nuovo Cim. [**9**]{}, 1 (1986); Ann. Inst. Henri Poincare [**49**]{}, 145 (1988). A.O. Barvinsky, JHEP 1310 (2013) 051, arXiv:1308.3270. A.O. Barvinsky, Phys. Lett. [**B241**]{}, 201 (1990). R. Marnelius, Nucl. Phys. [**B395**]{}, 647 (1993); I.A. Batalin and R. Marnelius, Nucl. Phys. B [**442**]{}, 669 (1995). S. Hawking, Phys. Rev. D [**37**]{}, 904 (1988); S. Coleman, Nucl. Phys. B [**307**]{}, 867 (1988); S.B. Giddings and A. Strominger, Nucl. Phys. B [**321**]{}, 481 (1988); M. McGuigan, Phyus. Rev. D [**38**]{}, 3031 (1988); V.A. Rubakov, Phys. Lett. B [**214**]{}, 503 (1988). A.O.Barvinsky, [*Geometry of the Dirac Quantization of Constrained Systems*]{}, gr-qc/9612003. V.N. Gribov, Nucl. Phys. B [**139**]{}, 1 (1978). B.L. Altshuler and A.O. Barvinsky, Physics-Uspekhi [**39**]{}, 429 (1996). J.W. York, Jr. , Phys. Rev. Lett. [**28**]{}, 1092 (1972). F. W. J. Olver, [*Asymptotics and Special Functions*]{} (Academic Press, New York, 1974). R.R. Caldwell, Phys. Lett. B [**545**]{}, 23 (2002). J.B. Hartle and S.W. Hawking, Phys. Rev. D [**28**]{}, 2960 (1983); S.W. Hawking, Nucl. Phys. B [**239**]{}, 257 (1984). A. Vilenkin, Phys. Lett. B [**117**]{}, 25 (1982); Phys. Rev. D [**27**]{}, 2848 (1983); Phys. Rev. D [**30**]{}, 509 (1984). Ya.B. Zeldovich and A.A. Starobinsky, Sov. Astron. Lett. [**10**]{}, 135 (1984); V.A. Rubakov, Phys. Lett. B [**148**]{}, 280 (1984); A.D. Linde, Lett. Nuovo Cim. [**39**]{}, 401 (1984). A.O. Barvinsky and A.Yu. Kamenshchik, Phys. Rev. D [**50**]{}, 5093 (1994); A.O. Barvinsky, A.Yu. Kamenshchik and I.V. Mishakov, Nucl. Phys. B [**491**]{}, 387 (1997). I.S. Gradsteyn and I.M. Ryzhik, [*Tables of Integrals, Series, and Products, Sixth edition*]{} (Academic Press, San Diego, CA, 2000). A. Vilenkin, Phys. Rev. D [**33**]{}, 3560 (1986). A.Yu. Kamenshchik and S. Manti, Phys. Rev. D [**85**]{}, 123518 (2012). P. A. Griffin and D. A. Kosower, Phys. Lett. B [**233**]{}, 295 (1989); A. O. Barvinsky, A. Y. .Kamenshchik and I. P. Karmazin, Annals Phys.  [**219**]{}, 201 (1992); A. Y. .Kamenshchik and I. V. Mishakov, Int. J. Mod. Phys. A [**7**]{}, 3713 (1992); D.V. Vassilevich, Int. J. Mod. Phys. A [**8**]{}, 1637 (1993); G. Esposito, A. Y. .Kamenshchik, I. V. Mishakov and G. Pollifrone, Class. Quant. Grav.  [**11**]{}, 2939 (1994); I. G. Moss and S. J. Poletti, Phys. Lett. B [**333**]{}, 326 (1994); D. V. Vassilevich, J. Math. Phys.  [**36**]{}, 3174 (1995); G. Esposito, A. Y. .Kamenshchik and G. Pollifrone, Fundam. Theor. Phys.  [**85**]{}, 1 (1997). R. Garattini, JHEP [**1309**]{}, 052 (2013). G.N. Watson, [*A Treatise on Theory of Bessel Functions, Second Edition*]{} (Cambridge University Press, 1944). [^1]: The geometrical meaning of $N^\mu$ is the collection of normal and tangential projections of 4-velocity with which a spacelike hypersurface moves in the embedding spacetime, so that degeneration of $N^\mu$ to zero implies freezing this surface at a fixed position. Then it does not scan the spacetime and no physical dynamics is probed within this time-independent gauge conditions [@BarvU]. [^2]: The kernel (\[K\]) is a truncation of the BFV unitary evolution operator in the relativistic phase space to the zero ghost sector [@BFV; @BarvU], and these properties directly follow from this truncation [@WhyBFV]. [^3]: Operator ordering in (\[operatorJ\]) is immaterial, because it is responsible for terms of higher order in $\hbar$ which go beyond the semiclassical approximation including the (one-loop) prefactor. [^4]: Note that $\Sigma$ is not the hypersurface and its codimension is $m>1$, so that $m$ equations (\[5.11\]) to recover $\mbox{\boldmath$\varPsi$}(q)$ on the full $n$-dimensional superspace from the boundary data (\[5.11a\]). [^5]: The semiclassical operator measure $\delta(\chi)\,\overrightarrow{\!\!J}+O(\hbar)$ in the physical inner product of Dirac wavefunctions in Eq.(\[5.11b\]) can be promoted to the level of an exact concept by embedding the Dirac quantization into the BFV quantization [@BFV] in the extended phase space of all canonical pairs of “matter" variables $q^i,p_i;N^\mu,\pi_\mu$ and pairs of Grassmann ghost variables $C^\mu,\bar{\cal P}_\mu;\bar C_\mu,{\cal P}^\mu$. This has been done within the concept of quantization on the so-called inner product spaces in [@BatalinMarnelius; @WhyBFV]. [^6]: $O(\hbar^2)$ and $O(\hbar)$ in (\[Weylorder\]) and (\[tildepsi\]) signify the same one-loop approximation because the semiclassical expansion for a quantum state begins with $1/\hbar$-order. [^7]: Here and in what follows we normalize for brevity $\hbar$ to unity and only label the commutator terms disregarded in the semiclassical approximation as $O(\hbar)$ or $O(\hbar^2)$. [^8]: BRST quantization has as one of its basic ingredients, the (! unphysical) inner product of $L_2$-type in the bosonic sector of phase space and Berezin integration inner product in the sector of its Grassmann ghost variables. [^9]: Note that the extrinsic time introduced in minisuperspace by Eqs. (\[gauge\])-(\[choice\]) coincides un to a numerical factor with the York extrinsic time introduced in [@extrinsic] for an arbitrary manifold $\tau \equiv \frac23\gamma^{-1/2}\pi$, $\pi = \gamma^{ab}\pi_{ab}$, where $\gamma_{ab}$ is a spatial metric and $\pi^{ab}$ is its conjugated momenta. [^10]: Note that the normalization of $H$ chosen in (\[super\]) should be kept fixed throughout the calculation leading to (\[M-K3\]). In particular it leads to the concrete normalization of $J\sim p_T$ in (\[JP1\]), origin of $(-\partial/i\partial T)^{-1/2}$ in (\[X\]) and cancelation of $e^{3\alpha/2}$ in (\[XX\]). Without this the resulting wavefunction would not satisfy the Wheeler-DeWitt equation in the operator realization (\[WdW2\]). [^11]: The normalization of the constraint $H'=e^{3\alpha}H$ implies rescaling the lapse function – the Lagrangian multiplier for the constraint – $N'=e^{-3\alpha}N$, so that in view of (\[JvN\]) the cosmic time corresponds to $\{\tilde T,H'\}=1/N'=e^{3\alpha}$. [^12]: Note that normalizability of this wavefunction with respect to $L_2$ inner product in $p_\phi$-space is the same as that of $\psi(p_\phi)$, because $e^{\pi|p_\phi|/2}|\varGamma\big(\mp i|p_\phi|\,\big)|\sim\sqrt{2\pi/|p_\phi|}$, for $|p_\phi|\to\infty$. [^13]: Not to mention that with the integration limits $T=\pm\sqrt{V_0}$ “integration by parts" of $(\partial/i\partial T)^{-1/2}$ in the derivation of (\[XXXX\]) (cf. discussion of Eq.(\[X\])) is impossible without uncontrollable boundary terms. [^14]: Another example of classical-to-quantum correspondence in cosmology was recently discussed in context of soft cosmological singularities [@Manti].

--- abstract: 'We analyze two large datasets from technological networks with location and social data: user location records from an online location-based social networking service, and anonymized telecommunications data from a European cellphone operator, in order to investigate the differences between individual and group behavior with respect to physical location. We discover agreements between the two datasets: firstly, that individuals are more likely to meet with one friend at a place they have not visited before, but tend to meet at familiar locations when with a larger group. We also find that groups of individuals are more likely to meet at places that their other friends have visited, and that the type of a place strongly affects the propensity for groups to meet there. These differences between group and solo mobility has potential technological applications, for example, in venue recommendation in location-based social networks.' author: - Chloë Brown - Neal Lathia - Anastasios Noulas - Cecilia Mascolo - Vincent Blondel title: Group colocation behavior in technological social networks --- Introduction {#introduction .unnumbered} ============ In today’s technologically connected world, a large volume of data is available about the social ties between individuals and also about their location. The study of these spatially embedded social networks has been a recent topic of interest for researchers, who have largely studied cellphone datasets [@Onnela07:Structure; @Gonzalez08:Understanding; @Lambiotte08:Geographical; @Krings09:Urban; @Onnela11:Geographic] and online social networks [@Backstrom10:Find; @Chang11:Location; @Cheng11:Exploring; @Cho11:Friendship; @Cranshaw10:Bridging; @Scellato11:Socio]. Datasets of mobile phone calls with cell tower locations, and those from online social networks with real-time indication of their users’ whereabouts, available through such means as geo-tagged tweets in Twitter and through user ‘check-ins’ in location-based social networks (LBSNs) such as Foursquare, afford the opportunity to study the relationship between the geographic positions of users and their friends at particular points in time, rather than using static locations such as home addresses. This enables us to investigate the relationship between friendship and colocation: the characteristics of the places where people meet with their friends, and how this may differ from the mobility of a lone individual. To date, there exists very little research examining the characteristics of the mobility of groups of friends in these networks. Some first work includes that by Crandall *et al.* [@Crandall10:Inferring], who analyzed how colocation affected the probability that two users of the online photo-sharing service Flickr were friends, showing that even a small number of colocations between users is a strong predictor of a social link. A further study concerning this topic was done by Cranshaw *et al.* [@Cranshaw10:Bridging], who examined the locations where people were colocated, and developed a measure of the entropy of a place in terms of the variety of unique visitors to that place. They observed that colocations at low entropy places such as homes are more likely to be between friends than those at high entropy places such as a shopping mall or university, which suggests that there may indeed be differences between the places that people visit with their friends and those where they go alone, and thus differences between individual and group mobility patterns. Calabrese *et al.* [@Calabrese11:Interplay] studied directly the interplay between face-to-face interactions and mobile phone communication, finding that colocations appear indicative of coordination calls, which occur just before face-to-face meetings. This study aims to build on this initial research and study whether there are differences between individual mobility and mobility in social groups, as viewed through the lens of technological networks such as the mobile phone communication network and location-based online social networks. We study two datasets, one cellphone dataset from a European mobile operator, and one from Foursquare, a popular online location-based social networking service, and examine the behavior of colocated friends, compared to the general behavior seen in each dataset. Knowledge of differences between the places where people go by themselves and those they visit when with their friends could be useful to the designers of location-aware social technologies, for example, by providing venue recommendations in location-based social networks, or better location prediction for location-aware advertising or search results returned on mobile phones [@DeDomenico13:Interdependence]. Please note that in this paper we use the word ‘friend’ to indicate ‘people who communicate with one another using the technological service being analyzed. We do not place restrictions on tie strength to define the social network, although the effect of tie strength on the results remains a potential direction for interesting future study. Results {#results .unnumbered} ======= Behavior of friend pairs in Foursquare {#behavior-of-friend-pairs-in-foursquare .unnumbered} -------------------------------------- We first study the places visited by individuals and by pairs of friends as recorded by their check-ins on Foursquare. Foursquare is an online location-based social network (LBSN), where users of the service may connect to their friends, and indicate their location by ‘checking in’ using an application on their mobile phones. We analyze a set of more than 2 million check-ins made by over 100,000 Foursquare users in New York over a period of 10 months (see Methods). Each check-in consists of the ID of the user who made the check-in, the venue to which the check-in was made (a specific place, as opposed to only co-ordinates), and a timestamp. We use in our analysis the public online social network indicated by the users. ### Social check-ins {#social-check-ins .unnumbered} We define a *social check-in* to be one where a user can be assumed to have been colocated at a venue with one of their friends in the social network. Foursquare does not provide ‘check-out’ information, so we consider a pair of friends to have been colocated when they have checked in to the same venue within one hour of one another. We analyze the distance of the locations of check-ins from a user’s most frequently visited location. We define a user’s *top location* to be the Foursquare venue where they have previously checked in the greatest number of times, and compute the distance between this location and the venue where a check-in takes place. We do not consider a user’s first check-in in the dataset, so such a location is always defined. In the case that a user has more than one top location, we consider all of those locations, and compute both the mean distance from the check-in venue to those locations, and the minimum distance from the check-in venue to one of those locations. Figure \[fig:distance\_pairs\] shows the cumulative distribution function (CDF) of the distance of a social check-in venue from a user’s top locations, and for comparison, the distribution of the distance of the venue of all check-ins (not just social check-ins) to the closest of the top locations of the user making the check-in. The results show that social check-ins tend to be closer to a top location of one of the friend pair concerned than do check-ins in general. We use information from Foursquare about the categories of venues to analyze the types of places where users tend to go with their friends. Foursquare defines 10 broad categories for venues: Arts and Entertainment (e.g. theaters, music venues), College and University (e.g. schools, university buildings), Food (e.g. cafes, restaurants), Nightlife (e.g. bars, clubs), Outdoors and Recreation (e.g. parks, nature spots), Professional (e.g. workplaces), Residence (e.g. homes), Shop and Service (e.g. shops, hospitals, churches), and Travel and Transport (e.g. railway stations, airports). We compute the ratio of the proportion of social check-ins in each category to the proportion of all check-ins in that category. Formally, define: - $num\_checkins(c)$ to be the total number of check-ins to category $c$ in the dataset. - $num\_social\_checkins(c)$ to be the number of social check-ins to category $c$. - $cats$ to be the set of all categories defined by Foursquare. Then for each category $c$ in $cats$: $$proportion(c) = \frac{num\_checkins(c)}{\sum_{c' \in cats}num\_checkins(c')}$$ That is, the proportion of all check-ins that occur in category $c$, and $$social\_proportion(c) = \frac{num\_social\_checkins(c)}{\sum_{c' \in cats}num\_social\_checkins(c')}$$ That is, the proportion of all social check-ins that occur in category $c$. We then quantify the propensity of each category to include venues where social check-ins particularly take place by defining the colocation ratio: $$colocation\_ratio(c) = \frac{social\_proportion(c)}{proportion(c)}$$ That is, the ratio of the proportion of social check-ins taking place in that category to the proportion of all check-ins taking place in that category. If this value is 1, it means that the category in question is equally likely to host both solo and social check-ins. If the value is markedly less than 1, the category is less likely to host social check-ins than it is to host solo check-ins, so places in this category might be less good to recommend for visits by pairs of friends. If the value is much more than 1, the category is more likely to host social check-ins than it is to host solo check-ins. We compute $colocation\_ratio(c)$ for each category $c$. Figure \[fig:categories\] shows that more than 1.5 times the proportion of social check-ins are to venues in the Arts and Entertainment and Nightlife Spot categories than the proportion of check-ins in general that are in these categories. Meanwhile, less than 0.7 of the proportion of social check-ins are to Residence, Shop and Service, and Travel and Transport venues than the proportion of check-ins in general. ![**Cumulative distribution function (CDF) of the distance from Foursquare users’ top locations to a check-in venue.** Top locations are the locations where a user has checked in the most in the past. ‘Closer’ refers to the distance to the closest of a pair of users’ top locations in a colocation event, ‘mean’ refers to the mean distance to the users’ top locations, and ‘single’ is the function for all check-ins, not just ‘social’ check-ins that make up colocations between friends. Social check-ins tend to take place closer to a top location of one of a pair of colocated friends than do general check-ins to a top location of the checking-in user.[]{data-label="fig:distance_pairs"}](figs/distance_analysis_pairs.pdf){width="\columnwidth"} ![**Ratio of the proportion of ‘social’ check-ins (part of a colocation between friends) in each category to the proportion of all check-ins in that category, for each of the categories of venue defined by Foursquare.** Red bars show categories where social check-ins are under-represented (ratio $<0.75$), yellow bars those where social check-ins are approximately in the same proportion as solo check-ins (ratio $0.75 - 1.25$), and green bars show categories where social check-ins are over-represented (ratio $>1.25$). Social check-ins are particularly likely to take place at venues in the Arts and Nightlife categories, and particularly unlikely to take place at venues in the Residence, Shop, and Transport categories.[]{data-label="fig:categories"}](figs/categories.pdf){width="\columnwidth"} ![**Cumulative distribution function (CDF) of the number of previous check-ins by a Foursquare user to the check-in venue.** The figure shows the functions for ‘social’ check-ins (part of a colocation between friends) and for all check-ins. Social check-ins are more likely to take place at new venues than check-ins in general.[]{data-label="fig:historical_pairs"}](figs/historical_analysis_cdf_pairs.pdf){width="\columnwidth"} ![**Cumulative distribution function (CDF) of the number of Foursquare users’ friends who have previously checked in at the check-in venue in the dataset.** The figure shows the functions for ‘social’ check-ins (part of a colocation between friends) and for all check-ins. Social check-ins are more likely to take place at venues where at least one of the user’s friends has been than check-ins in general.[]{data-label="fig:friends_pairs"}](figs/social_analysis_cdf_pairs.pdf){width="\columnwidth"} We examine how likely users are to visit new places with their friends. Figure \[fig:historical\_pairs\] shows the cumulative distribution function (CDF) of the number of previous visits in the dataset by the checking-in user to the visited venue, for all check-ins and for social check-ins. Social check-ins are more likely to take place at new venues than check-ins in general; the probability that a social check-in takes place at a previously visited venue is about 0.25, compared to about 0.38 for any check-in. Finally, we investigate how many social check-ins take place at venues where members of the friends’ wider social circles have previously checked in. Figure \[fig:friends\_pairs\] shows the cumulative distribution function (CDF) of the number of a user’s friends who have previously checked in to the venue in question, for all check-ins and for social check-ins (note that we require previous check-ins by friends to be at least an hour before the check-in in question, to avoid counting the first social check-in in a pair constituting a colocation event). Social check-ins are more likely than general check-ins to take place at a venue where a user’s friends have been before. About 18% of all check-ins are to a place visited by at least one friend before, but about 43% of social check-ins being to such venues. At first glance, this may seem contradictory to the observation that social check-ins tend to take place at venues where the user has not been before, but in fact it is the case that pairs of friends tend to check in together at venues that are new to the pair in question, but not to their wider social circle. It tends to be not one of the pair that has checked in to the venue before, but other friends of one or both of the colocated friends. Behavior of friend groups in a cellphone network {#behavior-of-friend-groups-in-a-cellphone-network .unnumbered} ------------------------------------------------ We extend our analysis from the behavior of colocated pairs of friends to larger friendship groups. Due to data sparsity, we are not able to obtain meaningful results for larger groups from the Foursquare dataset. Instead, we use a large anonymized dataset of billing records for over one million mobile phone users in Portugal, covering twelve months in 2006 and 2007 (see Methods). We extract colocated friendship groups by first constructing the social network by representing users by nodes and placing edges between them whenever one has called the other. We consider as colocated people who make calls using the same cell tower within one hour of one another, in agreement with the definition of social check-ins in the Foursquare dataset. Within a temporal window of one hour, we consider as groups the connected components of the subgraph of the social network that contains only edges between people colocated during that hour. We examine the distance of the places where a person meets a group of their friends from the top location of that person, defined for the cellphone data as the location of the cell tower from which that user has called the most frequently in the past. Figure \[fig:distance\_groups\] shows the cumulative distribution function of the distance of the colocation of a cellphone user with a group of their friends from that userÕs top locations. The figure also shows the distribution of the distance of all calls from the caller’s top locations (not just social colocations). Similarly to social check-ins in Foursquare, group colocations are more likely to take place near to one of the group’s top locations than calls in general. We note that the higher values of the CDF at very small distances compared to Foursquare is due to the coarser-grained spatial resolution of the cellphone dataset; in Foursquare we have individual venues for check-ins, giving single-building accuracy, whereas in the cellphone dataset we are limited to the resolution of cell towers. Figure \[fig:historical\_groups\] shows the CDF of the number of times a member of a colocated group has previously been seen at the place of colocation. Contrary to the behavior of pairs in Foursquare, an individual is less likely to meet a group of their friends at a new place than at a place where they have been before. We investigate this phenomenon further by analyzing separately groups of different sizes: pairs, trios, quartets, and quintets. Figure \[fig:historical\_breakdown\] shows the CDF for groups of each size. The results are in agreement with the results from Foursquare: pairs are more likely to meet at new places than people are to call from new places in general, but an individual is likely to meet a larger group somewhere they have been before. Finally, Figure \[fig:social\_groups\] shows the CDF of the number of an individual’s friends, excluding the group with whom they are colocated, who have been at the colocation place before. Again in agreement with the Foursquare data, a user is likely to meet a group at places where their wider circle of friends have been previously, compared to somewhere where none of their friends have been. ![**Cumulative distribution function (CDF) of the distance from cellphone users’ top locations to a group colocation venue.** Top locations are the locations where a user has made or received a phone call the most in the past. ‘Closer’ refers to the distance to the closest of a user’s top locations in a colocation event, ‘mean’ refers to the mean distance to the user’s top locations, and ‘single’ is the function for all calls, not just colocations. Groups tend to congregate closer to their members’ top locations than people tend to go in general in relation to their top locations.[]{data-label="fig:distance_groups"}](figs/distance_analysis_groups.pdf){width="\columnwidth"} ![**Cumulative distribution function (CDF) of the number of previous calls made or received by a cellphone user at a location.** The figure shows the functions for group colocations and for all calls. Group colocations are less likely to take place at new places, contrary to what was seen for pairs in the Foursquare data.[]{data-label="fig:historical_groups"}](figs/historical_analysis_groups.pdf){width="\columnwidth"} ![**Cumulative distribution function (CDF) of the number of previous calls made or received by a cellphone user at a location, for colocated groups of various sizes.** Pairs are more likely to meet at new places, in agreement with the Foursquare data, but bigger groups than this tend to meet at familiar locations.[]{data-label="fig:historical_breakdown"}](figs/historical_analysis_breakdown.pdf){width="\columnwidth"} ![**Cumulative distribution function (CDF) of the number of cellphone users’ friends who have previously made a call from a location.** Group colocations are more likely to take place somewhere where at least one of the user’s friends has been than calls in general.[]{data-label="fig:social_groups"}](figs/social_analysis_groups.pdf){width="\columnwidth"} Discussion {#discussion .unnumbered} ========== In this study we examine the differences between individual mobility and group colocation in two large datasets, one from the online location-based social network Foursquare, and one from a year’s worth of CDRs from a European cellphone operator. We find that there are indeed differences; first of all that in both datasets, friends are likely to meet closer to one of the individuals’ familiar locations than people may go in general. This is in agreement with the finding by Calabrese *et al.* [@Calabrese11:Interplay] that as the distance between the homes of pairs of colocated users increases, colocations take place in an area closer to one of the homes of the pair. We also analyze the venue category information available from Foursquare, and show that meetings between friends are over-represented in some categories such as Entertainment and Nightlife, and under-represented in others, such as Shop and Transport. This gives some indication that people are more likely to visit some categories of places with their friends than others, which could have implications for place recommendation in online location-based social networks. In both datasets, it is also the case that people are more likely to be colocated with friends at places where their other friends not part of that group have been before. In analyzing the number of times people have been recorded at a colocation venue previously, we found differing behavior between pairs of friends (both in Foursquare and cellphone dataset) and larger groups: while an individual is more likely to travel to a new place with a single friend than they are on their own, the opposite behavior is seen where larger groups are concerned. One way to interpret this could be in terms of research that has shown that people are more likely to take risks when with peers [@Gardner05:Peer], possibly including visiting a new place. Being with a friend may increase confidence and willingness to explore. However, this must be balanced, in a larger group situation, with the fact that the larger a group, the more difficult it is for that group to coordinate [@Ingham74:Ringelmann]. This could mean that it is easier for the group to meet at a place familiar to all of its members, than to identify and agree on a new meeting place. As with any study of this kind, there are some limitations that must be considered when interpreting the findings; these results should not be used to draw general conclusions about social and mobility behavior. The analysis of the behavior of Foursquare users in this paper is only certain to be relevant to those Foursquare users in the data, who are by their nature as users of the service not representative of a general population, and similarly for the mobile phone users whose behavior is reflected by the telecoms data. In particular, the Foursquare dataset contains check-ins that users shared publicly during the data collection period, and those users may have been to places where they did not check in, or kept certain check-ins private. It must be borne in mind that the likelihood of sharing a check-in at a given place may not be independent of the place category; for example, users might be less likely to share check-ins at home or at a train station than check-ins at entertainment venues. It is also possible that some of the patterns seen in Foursquare check-ins do extend outside the specifics of the services, since building on Cho, Myers, and Leskovec’s observation of considerable similarity between the mobility patterns in some LBSN datasets and in mobile telephone network data [@Cho11:Friendship], we have shown similarities also between the mobility behavior of groups seen in the Foursquare dataset analyzed in this work and the dataset from the mobile network operator. Of course, mobile phone datasets also have their inherent biases, and in this case it could be that we underestimate colocation between users because they did not happen to call one another around the time of meeting (similarly to people not checking in on Foursquare) or because they were connected to different cell towers with overlapping coverage areas, or that false positive colocations are inferred because people called one another while connected to the same cell tower despite not meeting face-to-face. We observe that the similarities in behavior that we find between the Foursquare and telecoms datasets provides some evidence in support of the differences not being due to these effects. However, the most reliable applications of results from the analysis of data such as these are most probably in areas directly relating to the use of those services, whether the mobile phone network or online social networks. Taken in the context of the technological service concerned, that is, for applications *within* such services, such as venue recommendation for groups in LBSNs, the results are relevant and potentially useful. We argue that our findings have potential applications in location-aware technological services. For example, one important task in online location-based social networks is that of venue recommendation: suggesting to the logged-in user places where that user might like to go. Given that we have found differences between the places that people may visit with their friends and those where they may go on their own, it may be that services such as Foursquare could benefit from suggesting different recommendations to a user based on knowledge of whether they are with friends or alone. Similarly, these differences in individual and group behavior could be used to improve mobility prediction, with possible uses in personalized services such as location-aware advertising and search results returned on mobile phones [@DeDomenico13:Interdependence]. Methods {#methods .unnumbered} ======= The Foursquare dataset consists of check-ins made by Foursquare users between November 2010 and September 2011. The dataset is composed of information publicly available on the Internet that can therefore be downloaded and analyzed in such a way as we have here. We downloaded all of the check-ins that users posted as non-private on Twitter, and which are thus publicly available, through the official Twitter search API by searching for tweets containing ‘4sq’ and a link (to the check-in page). This resulted in a dataset estimated to account for 20-25% of Foursquare check-ins made during this period. We were also able to download venue information including the venue location (latitude and longitude) and category (College, Food, Professional, etc.) directly from the Foursquare venue database, which is also publicly accessible. We analyze the set of 2,315,350 check-ins made to 109,314 venues in New York, by 104,266 users. We use check-ins as the representation of user visits to places; check-ins are voluntary so they are an under-estimate of the actual set of places visited by a user. However, the comparison between check-ins made with friends and alone shows considerable differences which may be reflected in the general place visit behavior of users, since the mobility patterns apparent from LBSN check-ins show strong similarity to those seen in other kinds of mobility data including cellphone records [@Cho11:Friendship]. In addition we downloaded the publicly available social networks of the users, taking a pair to be friends when each follows the other on Twitter. This social network has 99,725 nodes and 821,948 edges. The telecoms dataset is a large anonymized set of billing records for over one million mobile phone users in Portugal, gathered over a twelve month period between 2006 and 2007. The dataset contains information about mobile phone calls, but does not contain text messages (SMS) or data usage (Internet). The dataset was obtained directly from the operator and full permission was granted for its use in our analysis. In order to preserve privacy, individual phone numbers were anonymized by the operator. Each user in the anonymized dataset is identified by a hashed ID. Each entry is a CDR (Call Detail Record), consisting of a timestamp, the IDs of the caller and the callee, the call duration, and the cell tower IDs of the caller and callee towers. The dataset also includes the latitude and longitude of the cell towers, which allows us to study the relationship between the social network constructed by placing edges between nodes representing people whenever one person has called another, and the physical location of the users. The social network has 1,954,188 nodes and 19,370,004 edges. Acknowledgements {#acknowledgements .unnumbered} ================ Chloë Brown is a recipient of the Google Europe Fellowship in Mobile Computing, and this research is supported in part by this Google Fellowship. We acknowledge the support of the Engineering and Physical Sciences Research Council through grants GALE (EP/K019392) and UBHAVE (EP/I032673/1). [10]{} Onnela JP, Saram[ä]{}ki J, Hyv[ö]{}nen J, Szab[ó]{} G, Lazer D, et al. (2007) Structure and tie strengths in mobile communication networks. Proceedings of the National Academy of Sciences 104: 7332–7336. Gonzalez MC, Hidalgo CA, Barabasi AL (2008) Understanding individual human mobility patterns. Nature 453: 779–782. Lambiotte R, Blondel VD, de Kerchove C, Huens E, Prieur C, et al. (2008) Geographical dispersal of mobile communication networks. Physica A: Statistical Mechanics and its Applications 387: 5317–5325. Krings G, Calabrese F, Ratti C, Blondel VD (2009) Urban gravity: a model for inter-city telecommunication flows. Journal of Statistical Mechanics: Theory and Experiment 2009: L07003. Onnela JP, Arbesman S, Gonz[á]{}lez MC, Barab[á]{}si AL, Christakis NA (2011) Geographic constraints on social network groups. PLoS one 6: e16939. Backstrom L, Sun E, Marlow C (2010) Find me if you can: improving geographical prediction with social and spatial proximity. In: Proceedings of the 19th international conference on World wide web. ACM, pp. 61–70. Chang J, Sun E (2011) Location 3: How users share and respond to location-based data on social networking sites. In: Proceedings of the Fifth International AAAI Conference on Weblogs and Social Media. pp. 74–80. Cheng Z, Caverlee J, Lee K, Sui DZ (2011) Exploring millions of footprints in location sharing services. ICWSM 2011: 81–88. Cho E, Myers SA, Leskovec J (2011) Friendship and mobility: user movement in location-based social networks. In: Proceedings of the 17th ACM SIGKDD international conference on Knowledge discovery and data mining. ACM, pp. 1082–1090. Cranshaw J, Toch E, Hong J, Kittur A, Sadeh N (2010) Bridging the gap between physical location and online social networks. In: Proceedings of the 12th ACM international conference on Ubiquitous computing. ACM, pp. 119–128. Scellato S, Noulas A, Lambiotte R, Mascolo C (2011) Socio-spatial properties of online location-based social networks. ICWSM 11: 329–336. Crandall DJ, Backstrom L, Cosley D, Suri S, Huttenlocher D, et al. (2010) Inferring social ties from geographic coincidences. Proceedings of the National Academy of Sciences 107: 22436–22441. Calabrese F, Smoreda Z, Blondel VD, Ratti C (2011) Interplay between telecommunications and face-to-face interactions: A study using mobile phone data. PloS one 6: e20814. de Domenico M, Lima A, Musolesi M (2013) Interdependence and predictability of human mobility and social interactions. Pervasive and Mobile Computing 9: 798–807. Gardner M, Steinberg L (2005) Peer influence on risk taking, risk preference, and risky decision making in adolescence and adulthood: an experimental study. Developmental psychology 41: 625. Ingham AG, Levinger G, Graves J, Peckham V (1974) [The Ringelmann effect: Studies of group size and group performance]{}. Journal of Experimental Social Psychology 10: 371–384.

--- abstract: 'The ER=EPR conjecture is proposed to resolve the black hole information paradox without introducing a firewall in the black hole. The conjecture implies that entanglement of the EPR pair, which is thought to be a quantum mechanical effect, can be captured by classical gravity through the ER bridge. Using Bell inequality as a sharp test of entanglement, we study a holographic model with an EPR pair at the boundary and an ER bridge in the bulk by using the Schwinger-Keldysh formalism. By revealing how Bell inequality is violated by gravity in the bulk, our study sheds light on the entanglement of the original ER=EPR, which, unlike holography, has both ER and EPR living in the same spacetime dimensions.' author: - 'Jiunn-Wei Chen' - Sichun Sun - 'Yun-Long Zhang' title: Holographic Bell Inequality --- [^1] Introduction ============ Bell inequality plays an important role in quantum physics [@Bell:1964kc]. Correlations in local classical theories are bounded by the Bell inequality, which can be violated by the presence of the non-local entanglement in quantum mechanics. The violation of Bell inequality in the entangled Einstein-Podolsky-Rosen (EPR) pair [@Einstein:1935rr] indicates that two particles have an “instant interaction”, in contrast to theories of hidden variables that preserve strict locality [@Bell:1964kc; @Einstein:1935rr; @CHSH; @Cirelson; @Hartle:1992as]. In addition to the Bell’s tests in laboratories, there are also discussions of detections in cosmological scales, to confirm that whether the inflated primordial fluctuations are quantum mechanical in nature [@Maldacena:2015bha; @Choudhury:2016cso; @Chen:2017cgw]. Recently, Maldacena and Susskind proposed the ER=EPR conjecture [@Maldacena:2013xja; @Susskind:2016jjb] which stated that the quantum entanglement of the EPR pair can be attributed to the non-traversable Einstein-Rosen (ER) bridge that connects the pair [@Einstein:1935tc]. This conjecture was proposed to resolve the black hole information paradox without resorting to a firewall [@Almheiri:2012rt] behind the black hole horizon. Interestingly, the ER=EPR conjecture implies that entanglement, which is thought to be a quantum mechanical effect, of the EPR pair, can be captured by a classical theory, at least when the EPR pair is very massive. In this case, it is classical gravity with an ER bridge, although more general EPR may require the ingredients of quantum gravity. To test this extraordinary claim, Bell inequality is a natural choice as it provides a sharp test of entanglement. In this work, instead of working with the original ER=EPR setup, we employ a concrete holographic model of the EPR pair proposed in Refs. [@Jensen:2013ora; @Xiao:2008nr] based on the Anti-de Sitter/Conformal Field Theory(AdS/CFT) correspondence [@Maldacena:1997re]. The two particles of the boundary EPR pair are connected by a string in the AdS background of the bulk with an ER bridge on the string worldsheet. Therefore, this is a holographic realization of ER=EPR. We demonstrate that the Bell inequality violated by the EPR pair living at the boundary can also be violated by the gravitational theory with an ER bridge living in the bulk. Although the holographic setting is different from the original ER=EPR conjecture in which both ER and EPR live in the same spacetime dimensions, our study does shed light on how entanglement can be captured by a classical theory. Bell Inequality =============== The essence of Bell inequality is captured in the CHSH correlation parametrizations [@CHSH] which is reviewed here briefly. The entangled states made of a pair of spin $1/2$ particles (the generalization to particles of higher spin is straight forward) are detected by two observers, Alice and Bob, respectively. The operators correspond to measuring the spin along various axes with outcomes of eigenvalues $\pm 1$. Performing the operations $A$ and $A'$ on the first particle at Alice’s location, and operations $B$ and $B'$ on the second particle at Bob’s location. With the Pauli matrices $\vec{\sigma}=({\sigma_x, \sigma_y, \sigma_z})$, and the unit vector $\vec{n}=(n_x,n_y,n_z)$ to indicate the spatial direction of the measurement, we have the following operators $$\begin{aligned} A_s&= \vec{n}_A\dt \vec{\sigma} ,~ A'_s= \vec{n}_{A'}\dt \vec{\sigma} ,\\ B_s&= \vec{n}_B\dt \vec{\sigma},~ B'_s= \vec{n}_{B'} \dt \vec{\sigma}.\end{aligned}$$ Then the [CHSH correlation formulation]{} is introduced as [@CHSH] $$\begin{aligned} \label{eCHSH} \langle C_s \rangle = \langle A_s B_s \rangle + \langle A_s B'_s \rangle + \langle A'_s B_s \rangle - \langle A'_s B'_s \rangle,\end{aligned}$$ which is a linear combination of crossed expectation values of the measurements. In a local theory with hidden variables, the formula is bounded by the Bell inequality $|\langle C_s \rangle | \leq 2$. While in quantum mechanics, this inequality can be violated, with a higher bound $|\langle C_s \rangle |\leq 2\sqrt{2}$ [@Cirelson] (see also [@Maldacena:2015bha]). For example, if we choose the entanglement state of a spin singlet $$\begin{aligned} \label{eEPR} |\psi_s \rangle &= \frac{1}{\sqrt{2}}\big( |\!\uparrow\rangle \otimes |\!\downarrow\rangle - |\!\downarrow\rangle \otimes |\!\uparrow\rangle \big),\end{aligned}$$ and take the measurements along the $(x,y)$ plane, i.e. $n_A=(\cos\theta_A, \sin\theta_A, 0)$ etc., it is straightforward to show $$\begin{aligned} \label{Cdef} G_{AB}^s \equiv \langle\psi_s | A_s B_s |\psi_s \rangle&= - \cos \theta_{A B}.\end{aligned}$$ Here $\cos\theta_{A B} = \vec{n}_A \dt \vec{n}_B$ depends on the relative angle of the measurements. And from we have $$\begin{aligned} \langle\psi_s|C_s |\psi_s \rangle& =-\cos \theta_{A B}- \cos \theta_{A B'} -\cos \theta_{A' B} + \cos \theta_{A' B'}.\end{aligned}$$ In particular, if we fix the direction of $A$ and $A'$, as well as the angle between $B$ and $B'$ as $\pi/2$ $$\begin{aligned} \label{Achoices} \theta_A=0,\quad\theta_{A'}=\frac{\pi}{2},\quad \theta_{B'}=\theta_B-\frac{\pi}{2},\end{aligned}$$ then we have the relation depended on the direction of $B,B'$, $$\begin{aligned} \langle\psi_s |C_s |\psi_s\rangle =- 2 \sqrt{2} \cos \big( \theta_B - \frac{\pi}{4}\big)\end{aligned}$$ For $0{<}\theta_B{<}\pi/2$, the Bell inequality $ |\langle C_s \rangle | \leq 2$ can be violated, and we reach the maximal violation at $\theta_B = \pi/4$, with an extra factor of $\sqrt{2}$. Bell’s Test in Field Theory =========================== Before we move to holography, it is instructive to discuss the Bell’s test in the context of the boundary quantum field theory. The test can be described by a process with the transition amplitude $$\begin{aligned} \mathcal{A}={\langle\,}A^\uparrow B^\uparrow |T[P_A P_B]|\psi_s{\rangle}, \end{aligned}$$ where a spin singlet state of Eq.(\[eEPR\]) is measured by the projection operators $$\begin{aligned} P_A=\frac{1+\vec{n}_A{\cdot}\vec{\sigma}}{2},\quad P_B=\frac{1+\vec{n}_B{\cdot}\vec{\sigma}}{2},\end{aligned}$$ to the final state $|A^{\uparrow}B^{\uparrow} {\rangle}$ which is both spin up in the $\vec{n}_A$ and $\vec{n}_B$ directions. Note that the operators are time ordered since one can either measure $P_A$ or $P_B$ first. Squaring the amplitude to get the probability $$\begin{aligned} \label{timeordered} \mathcal{P} \propto | \mathcal{A}|^2 &=\sum_X {\langle\,}\psi_s|T[P_A P_B]^{\dagger}| X{\rangle}{\langle\,}X|T[P_A P_B]|\psi_s{\rangle}\nn\\ &={\langle\,}\psi_s|T[P_A P_B]|\psi_s{\rangle}, \end{aligned}$$ where only $|A^{\uparrow}B^{\uparrow} {\rangle}$ contributes in the complete set of state $| X{\rangle}$ in the first line and we have used the commutativity of $P_A$ and $P_B$, as well as $P_A^2=P_A$ and $P_B^2=P_B$ for projectors. The above discussion shows that the Bell’s test is measuring a time ordered Green’s function which does not have to vanish when $P_A$ and $P_B$ are outside of each other’s light cone. A familiar example of this is Feynman propagator which is also a time ordered Green’s function. Typically, this dramatic property of the Green’s function does not show up in physical observable. It is very interesting that Bell’s test relates those Green’s functions to observables through Eq.(\[timeordered\]). In the following holographic model, it is also a time order Green’s function that we will compute. Holographic EPR and Bell inequality =================================== In this section, we employ the holographic model of Ref. [@Jensen:2013ora] to study the time ordered Green’s function of Eq.(\[timeordered\]) and the spin-spin correlation of the Bell’s test. The model proposed that an entangled color singlet quark anti-quark ($q$-$\bar{q}$) pair in $\cal{N}$=4 supersymmetric Young-Mills theory (SYM) can be described by an open string with both of its endpoints attached to the boundary of AdS$_5$. The string connecting the pair is dual to the color flux tube between the two quarks, with a $1/r$ Coulomb potential as required by the scale invariance of boundary theory. Note that there is no confinement in this theory, therefore the pair can separate arbitrarily far away from each other. Various studies of related models can also be found in [@Sonner:2013mba; @Jensen:2014bpa; @Jensen:2014lua; @Chernicoff:2013iga; @Karch:2013gsa; @Hirayama:2010xi; @Caceres:2010rm]. There are numerical solutions of the string shapes with different boundary behaviors [@Herzog:2006gh; @Chernicoff:2008sa; @Chesler:2008wd]. But it is more convenient for us to work with the analytic solution for an accelerating string treated in the probe limit such that the back reaction to the AdS geometry is neglected [@Xiao:2008nr]. In the analytic solution the open string is also accelerated on the Poincáre patch of the AdS$_5$ $$\begin{aligned} \label{metric} {\d}s^2=\frac{\R^2}{\w^2}\big[-{\d}t^2+{\d}\w^2 +({\d}x^2+{\d}y^2+{\d}z^2)\big],\end{aligned}$$ with the AdS radius $\R$ and extra dimension $\w$. The string solution in the $AdS_5$ bulk is given by $$\begin{aligned} \label{solution} \xi^2=t^2+b^2-\w^2. \end{aligned}$$ The quark and anti-quark live on the AdS boundary $\w=0$. They are accelerating along the $\pm \xi$ direction, respectively, with the solution $\xi=\pm\sqrt{t^2+b^2}$. Therefore, the two entangled particles are out of causal contact with each other the whole time. [*String fluctuations.—*]{} To consider the string fluctuations, we transform the solution to the co-moving spacetime $(\tt, \tu, x,y,\tz)$ of the accelerating quarks via $$\begin{aligned} \label{transformation} |\xi| &= b \sqrt{1- \tu} {e^{\tz}} \cosh \tt, \nn \\ t &=b \sqrt{1- \tu}{e^{\tz}} \sinh \tt, \nn \\ \w &=b \sqrt{ \tu} {e^{\tz}},\quad x=b\, \tx, \quad y=b\, \ty .\end{aligned}$$ These two frames, which cover the regions $\xi\geq{0}$ and $\xi\leq{0}$ separately, are accelerating frames with a constant acceleration $a=1/b$ along opposite directions of $\xi$. And only maps the upper part of the string ($0<w<b$) into the proper frames of the accelerating quarks with $0<\tu< 1$. Plug this transformation in the string solution , one finds the string configuration becomes $\tz=0$ for both frames of the quark and anti-quark. Under this transformation , the metric becomes $$\begin{aligned} \label{newmetric} {\d}s^{2}{=}{\frac{\R^{2}}{ {\tu}}}\Big[{-}f({\tu}){\d}\tt^{2}{+}\frac{1}{4{\tu} }\frac{\d{\tu}^{2}}{f({\tu})}{+}{e^{-2\tz}} {\left(}{\d}\tx^{2}{+}{\d}\ty^{2}{\right)}{+}{\d}{\tz}^{2} \Big],\end{aligned}$$ where $f(\tu)=1-{\tu}$. Furthermore, with respect to the time $\tau {=} b \tt $, the Hawking temperature $T_H=\frac{1}{2\pi b}$ matches with the Unruh temperature ${T_a}=\frac{a}{2\pi}$ [@Xiao:2008nr; @Caceres:2010rm] and we have set the reduced Planck constant $\hbar$ and Boltzmann constant $k_B$ to be unit. Therefore, there is an event horizon at $\tu=1$ associated with the quark and another event horizon also at $\tu=1$ associated with the anti-quark. As shown in Fig. \[fig:hbell\], the two horizons are connected by part of the string which can be seen as ER bridge.Hence it is suggested to be a holographic realization of the ER=EPR conjecture [@Maldacena:2013xja; @Susskind:2016jjb]. However, in the original conjecture, both ER and EPR live in the same spacetime dimensions, unlike the holographic model, EPR lives at the boundary while ER lives in the bulk. The spin measurement of the quarks can be carried out by the Stern-Gerlach type experiment which applied a magnetic field gradience to generate a force that acts on the spins of the quarks. This introduces fluctuations to the world lines of quarks which set the boundary conditions for the [worldsheet]{} of the string fluctuations. Let $(\tt ,{\tu})$ be the new worldsheet coordinates in the current frame, then the string fluctuation is $X^{\mu }(\tt ,{\tu})=\big( \tt,{\tu}, \td_i (\tt ,{\tu})\big)$, with $i=(\tx,\ty,\tz)$. When ${\td_i }\ll 1$, the Nambu-Goto action of string with tension ${T_s}$ becomes$$\begin{aligned} \label{action} S &\simeq -{T_s}\R^{2}\int \frac{d{\tt}d{\tu}}{2\tu^{ 3/2}}\left\{ 1+ \Big[2{\tu}f(\tu) {\td_{i}}^{\prime}{\td_{j}}^{\prime}-\frac{1}{2f(\tu)} \dot{\td}_{i} \dot{\td}_{j}\Big] h^{ij}\right\} ,\end{aligned}$$ where ${\td_i}^{\prime}\equiv \frac{\partial {\td_i} }{\partial \tu}$, $\dot{\td}_i \equiv \frac{\partial {\td_i} }{\partial \tt}$ and $h^{ij}=\text{diag}[1,e^{2\tz},e^{2\tz}]$. The equations of motion for the fluctuations on the string are $$\begin{aligned} \label{EOM} \partial _{{\tu}}\Big( \frac{2 f {\td_i }^{\prime }}{{\tu}^{1/2}}\Big) -\partial _{\tt}\Big( \frac{\dot{\td}_i}{2f{\tu}^{ 3/2}}\Big) =0. \end{aligned}$$ Focusing on the transverse fluctuations $i=\tx,\ty$, $$\begin{aligned} {\td_i }({\tt},{\tu}) =\int \frac{d\omega }{2\pi }e^{ -i\omega {\tt }} {\td_i }(\omega) Y_{\omega }( {\tu}) ,\end{aligned}$$where $\td_i(\omega) $ is the Fourier transform of fluctuation on the boundary. ![Schematic diagram for the quark($q$)-antiquark($\bar{q}$) EPR pair in holography at a fixed time $t>0$. From , the trajectory of left(right) [worldsheet]{} horizon is on $\w=b$, which depicts the intersect of the string with the world volume horizon seen by $q$($\bar{q}$) in its co-moving frame. The Bell inequality test is performed by spin measurements at Alice and Bob’s locations. The AdS [black brane]{} is present only in the finite temperature case, which we briefly discuss in the last section. []{data-label="fig:hbell"}](Hbell2.pdf "fig:")\ Constructing Bell’s inequality ============================== The retarded Green’s function of the quark under effective random force $\cO^i(\tau)$ can be defined as ${\i}G_{\mathcal{R}}^{ij}(\tau)=\theta(\tau)\langle [\cO^i(\tau),\cO^j(0)] \rangle$ [@Xiao:2008nr]. In the AdS/CFT correspondence, $\cO^i(\tau)$ is the operator conjugate to the fluctuations $ q_i(\tau)$, where the dimensionful quantities are $\tau=b\tt, q_i=b\td_i$. In the low frequency limit $\omega\to 0$, it can be obtained analytically as $$\begin{aligned} G_{\mathcal{R}}^{ij}(\omega) &=-\frac{2{T_s}\R^{2} }{b^{2} \tu^{1/2}} f( {\tu}) Y_{-\omega }( {\tu}) \partial _{{\tu}}Y_{\omega}( {\tu}) \delta^{ij}\big\vert _{{\tu}\rightarrow 0} \nn \\ &= -\frac{a^2 \sqrt{\lambda}}{2\pi } \i\omega \delta^{ij}+O(\omega^2), $$and we have used the fact that ${T_s}\R^2=\frac{\sqrt{\lambda}}{2\pi}$. What we need for the Bell’s test is the contour time ordered Schwinger-Keldysh (SK) Green’s function, $$\begin{aligned} \i G_{AB}^{ij}(\tau, x)=\langle \cO_A^i(\tau, x) \cO_B^j (0) \rangle,\end{aligned}$$ where $\cO_A^i$ and $\cO_B^j$ are separately defined on the causally disconnected left and right wedges of the Penrose diagram, corresponding to the boundaries of different patches of the AdS space. This off-diagonal SK propagator is examined in the Supplemental Material and found to be related to the holographic retarded Green’s function $$\begin{aligned} \label{GABomega} &G_{AB}^{ij} ( \omega ) =\frac{2\i e^{-\omega/(2{T_a})}}{1-e^{-\omega/{T_a}}} \textrm{Im}G_{\mR}^{ij}\left( \omega \right),\end{aligned}$$ similar to what was found in Refs. [@Son:2007vk; @Herzog:2002pc; @Son:2002sd] but with different settings. For fluctuations coming from two causally separated quarks of an EPR pair along $x,y$ directions, and in the low frequency limit $\omega\to 0$, $$\begin{aligned} \label{Green} &\i G_{AB}^{xx} = \i G_{AB}^{yy} = \frac{\sqrt{\lambda} a^3}{2\pi^2} , \quad \i G_{AB}^{xy}=\i G_{AB}^{yx}=0, $$ which indicates that the spatial correlator $G^{ij}_{AB} \propto \delta^{ij}$. The $\sqrt{\lambda}$ factor is consistent with the observation that the entanglement entropy of the entangled pair is of order $\sqrt{\lambda}$ [@Jensen:2013ora]. It is also interesting that this SK correlator does not vanish when the quarks are separated at long distance. This is consistent with the non-local nature of entanglement. However, the SK correlator vanishes when the acceleration $a$ becomes zero and the EPR pair is always infinitely far apart. Thus, we can only approach the zero acceleration limit after we identify the spin correlation with the normalized operators as in the following Eq.(\[operator\]) in which $a$ dependence cancels. To study the correlators, we normalize the operators such that only the dependence on the spin wave function remains: $$\begin{aligned} A_{\cO}=(\cos\theta_A\cO_A^x+\sin\theta_A\cO_A^y)/\langle \cO_A^x \cO_B^x \rangle^{1/2},\nn\\ B_{\cO}=(\cos\theta_B\cO_B^x+\sin\theta_B\cO_B^y)/\langle \cO_A^x \cO_B^x \rangle^{1/2},\label{operator}\end{aligned}$$ The mixed measurements for correlators in [CHSH correlation formulation]{} become $$\begin{aligned} \label{cAB} \langle A_{\cO}B_{\cO} \rangle= \cos(\theta_{A}-\theta_{B})\equiv \cos\theta_{A B}.\end{aligned}$$ Together with the similar normalization of the operators $A'_{\cO}$ and $B'_{\cO}$, the [CHSH correlation formulations]{} becomes $$\begin{aligned} \label{Cdef} \langle C_{\cO} \rangle &= \langle A_{\cO}B_{\cO} \rangle + \langle A_{\cO}B'_{\cO} \rangle + \langle A'_{\cO}B_{\cO} \rangle - \langle A'_{\cO}B'_{\cO} \rangle\nn\\ &= \cos\theta_{A B}+ \cos\theta_{A B'\!}+ \cos\theta_{A'\!B}- \cos\theta_{A'\!B'\!}. \end{aligned}$$ For example, when $\theta_{A B}=\theta_{A B'}=\theta_{A' B}=\pi/4$, and $\theta_{A'B'}=3\pi/4$, we can reach the maximum value $2\sqrt{2}$. In this derivation, we see the bulk string fluctuations, which come from classical gravity, reproduce the quantum entanglement of an EPR pair on the boundary. Technically, this result relies on only two ingredients. The first one is that the observable in Bell’s test is a time ordered Greens function as shown in Eq.( \[timeordered\]). And it is well known that the time ordered Green’s function does not have to vanish when the measurements $P_A$ and $P_B$ are outside of each other’s light cone. Mathematically, this is because the behaviors of the SK correlators outside the two horizons need to be correlated, otherwise the solution is not smooth insides the horizons. The second one is that the equation of motion of the classical string, Eq.(\[EOM\]), has no coupling between $\td_x$ and $\td_y$ such that Eq.(\[Green\]) follows. This can be obtained as long as the string does not experience a force to propagate the fluctuation in the $x$-direction to the $y$-direction which breaks parity in general. It seems once these two conditions are satisfied, it does not matter whether there is an ER bridge in the bulk. Hence it is conceivable Bell inequality can still be violated in a holographic model where the EPR pair does not accelerate, similar to how holographic entanglement entropy is computed in a static system [@Ryu:2006bv; @Numasawa:2016emc]. At this point, it is also curious whether experimental observables associated with time ordered Green’s functions in our spacetime dimensions, not just in the bulk of holography, can be found. If it is found, then entanglement can be described in a classical theory without holography. In view of the original ER=EPR conjecture, this seems not completely impossible. Conclusion and Discussions ========================== The ER=EPR conjecture is proposed to resolve the black hole information paradox without introducing a firewall in the black hole. The conjecture implies that entanglement of the EPR pair, which is thought to be a quantum mechanical effect, can be captured by classical gravity through the ER bridge. Using Bell inequality as a sharp test of entanglement, we study a holographic model with an EPR pair at the boundary and an ER bridge in the bulk. By revealing how Bell inequality is violated by classical gravity in the bulk, our study sheds light on the possible conditions needed for the entanglement of the original ER=EPR. Since the original ER=EPR has both ER and EPR living in the same spacetime dimensions, it is curious whether experimental observables associated with classical time ordered Green’s functions in our spacetime dimensions, not just in the bulk of holography, can be found For future work, it is interesting to consider the back reaction by the measurements and see whether the ER bridge is broken due to the energy injected by measurements. This might provide an opportunity to study the “wave function collapse" typically used to describe how measurements change the states. Another interesting direction is the decoherence of the EPR pair in the environment. If the environmental effect can be described by thermal fluctuations, then we can add a black hole to the bulk of our model. When the distance of the EPR pair increases with time, the ER bridge also approaches the black brane horizon and then enters the horizon [@Dominguez:2008vd]. We expect the ER bridge breaks after it enters the horizon which might shed light on the decoherence process in the boundary field theory. Acknowledgements ================= We are grateful to A. Karch for many valuable comments and discussions. We thank helpful comments from D. Berenstein, F. L. Lin, and R. X. Miao. J. W. Chen and S. Sun are supported by the MOST and NTU-CTS at Taiwan. J.W. Chen is also partially supported by MIT MISTI program and the Kenda Foundation. Y. L. Zhang is supported by APCTP and CQUeST. [102]{} J. S. Bell, “On the Einstein-Podolsky-Rosen paradox,” [Physics [**1**]{}, 195 (1964)](https://inspirehep.net/record/31657). A. Einstein, B. Podolsky and N. Rosen, “Can quantum mechanical description of physical reality be considered complete?,” [Phys. Rev.  [**47**]{}, 777 (1935)](https://doi.org/10.1103/PhysRev.47.777). J. F. Clauser, M.A. Horne, A. Shimony, R.A. Holt, “Proposed experiment to test local hidden-variable theories," [Phys.   Rev.  Lett. [**23**]{}, 880 (1969)](https://doi.org/10.1103/PhysRevLett.23.880). B. S. Cirelson, “Quantum generalizations of Bell’s inequality,” [Lett. Math. Phys. [**4**]{}, 93 (1980)](https://doi.org/10.1007/BF00417500). J. B. Hartle, “Space-time quantum mechanics and the quantum mechanics of space-time,” [gr-qc/9304006](http://arxiv.org/abs/gr-qc/9304006). J. Maldacena, “A model with cosmological Bell inequalities,” [Fortsch. Phys.  [**64**]{}, 10 (2016)](http://dx.doi.org/10.1002/prop.201500097) \[[arXiv:1508.01082 \[hep-th\]](http://arxiv.org/abs/1508.01082)\]. S. Choudhury, S. Panda and R. Singh, “Bell violation in the Sky,” [Eur. Phys. J. C [**77**]{}, no. 2, 60 (2017)](http://dx.doi.org/10.1140/epjc/s10052-016-4553-3) \[[arXiv:1607.00237 \[hep-th\]](http://arxiv.org/abs/1607.00237)\]. J. W. Chen, S. H. Dai, D. Maity, S. Sun and Y. L. Zhang, “Towards Searching for Entangled Photons in the CMB Sky,” [arXiv:1701.03437 \[quant-ph\]](http://arxiv.org/abs/1701.03437). J. Maldacena and L. Susskind, “Cool horizons for entangled black holes,” [Fortsch. Phys.  [**61**]{}, 781 (2013)](http://dx.doi.org/10.1002/prop.201300020) \[[arXiv:1306.0533 \[hep-th\]](http://arxiv.org/abs/1306.0533)\]. L. Susskind, “Copenhagen vs Everett, Teleportation, and ER=EPR,” [Fortsch. Phys.  [**64**]{}, no. 6-7, 551 (2016)](http://dx.doi.org/10.1002/prop.201600036) \[[arXiv:1604.02589 \[hep-th\]](http://arxiv.org/abs/1604.02589)\]. A. Einstein and N. Rosen, “The Particle Problem in the General Theory of Relativity,” [Phys. Rev.  [**48**]{}, 73 (1935)](http://dx.doi.org/10.1103/PhysRev.48.73). A. Almheiri, D. Marolf, J. Polchinski and J. Sully, “Black Holes: Complementarity or Firewalls?,” [JHEP [**1302**]{}, 062 (2013)](http://dx.doi.org/10.1007/JHEP02(2013)062) \[[arXiv:1207.3123 \[hep-th\]](http://arxiv.org/abs/1207.3123)\]. K. Jensen and A. Karch, “Holographic Dual of an Einstein-Podolsky-Rosen Pair has a Wormhole,” [Phys. Rev. Lett.  [**111**]{}, no. 21, 211602 (2013)](http://dx.doi.org/10.1103/PhysRevLett.111.211602) \[[arXiv:1307.1132 \[hep-th\]](http://arxiv.org/abs/1307.1132)\]. B. W. Xiao, “On the exact solution of the accelerating string in AdS(5) space,” [Phys. Lett. B [**665**]{}, 173 (2008)](http://dx.doi.org/10.1016/j.physletb.2008.06.017) \[[arXiv:0804.1343 \[hep-th\]](http://arxiv.org/abs/0804.1343)\]. J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” \[[Adv. Theor. Math. Phys.  [**2**]{}, 231 (1998)](http://dx.doi.org/10.1023/A:1026654312961/)\] \[[hep-th/9711200](http://arxiv.org/abs/hep-th/9711200)\]. J. Sonner, “Holographic Schwinger Effect and the Geometry of Entanglement,” [Phys. Rev. Lett.  [**111**]{}, no. 21, 211603 (2013)](http://dx.doi.org/10.1103/PhysRevLett.111.211603) \[\[[arXiv:1307.6850 \[hep-th\]](http://arxiv.org/abs/1307.6850)\]. K. Jensen, A. Karch and B. Robinson, “Holographic dual of a Hawking pair has a wormhole,” [Phys. Rev. D [**90**]{}, no. 6, 064019 (2014)](http://dx.doi.org/10.1103/PhysRevD.90.064019) \[[arXiv:1405.2065 \[hep-th\]](http://arxiv.org/abs/1405.4817)\]. K. Jensen and J. Sonner, “Wormholes and entanglement in holography,” [Int. J. Mod. Phys. D [**23**]{}, no. 12, 1442003 (2014)](http://dx.doi.org/10.1142/S0218271814420036) \[[arXiv:1405.4817 \[hep-th\]](http://arxiv.org/abs/1405.4817)\]. M. Chernicoff, A. G[ü]{}ijosa and J. F. Pedraza, “Holographic EPR Pairs, Wormholes and Radiation,” [JHEP [**1310**]{}, 211 (2013)](http://dx.doi.org/10.1007/JHEP10(2013)211) \[[arXiv:1308.3695 \[hep-th\]](http://arxiv.org/abs/1308.3695)\]. A. Karch and S. Sun, “Matrix Flavor Brane and Dual Wilson Line,” [Phys. Rev. D [**89**]{}, no. 6, 066008 (2014)](http://dx.doi.org/10.1103/PhysRevD.89.066008) \[[arXiv:1312.2694 \[hep-th\]](http://arxiv.org/abs/1312.2694)\]. T. Hirayama, P. W. Kao, S. Kawamoto and F. L. Lin, “Unruh effect and Holography,” [Nucl. Phys. B [**844**]{}, 1 (2011)](http://dx.doi.org/10.1016/j.nuclphysb.2010.10.018) \[[arXiv:1001.1289 \[hep-th\]](http://arxiv.org/abs/1001.1289)\]. E. Caceres, M. Chernicoff, A. Guijosa and J. F. Pedraza, “Quantum Fluctuations and the Unruh Effect in Strongly-Coupled Conformal Field Theories,” [JHEP [**1006**]{}, 078 (2010)](http://dx.doi.org/10.1007/JHEP06(2010)078) \[[arXiv:1003.5332 \[hep-th\]](http://arxiv.org/abs/1003.5332)\]. C. P. Herzog, A. Karch, P. Kovtun, C. Kozcaz and L. G. Yaffe, “Energy loss of a heavy quark moving through N=4 supersymmetric Yang-Mills plasma,” [JHEP [**0607**]{}, 013 (2006)](http://dx.doi.org/10.1088/1126-6708/2006/07/013) \[[hep-th/0605158](http://arxiv.org/abs/hep-th/0605158)\]. M. Chernicoff and A. Guijosa, “Acceleration, Energy Loss and Screening in Strongly-Coupled Gauge Theories,” [JHEP [**0806**]{}, 005 (2008)](http://dx.doi.org/10.1088/1126-6708/2008/06/005) \[[arXiv:0803.3070 \[hep-th\]](http://arxiv.org/abs/0803.3070)\]. P. M. Chesler, K. Jensen and A. Karch, “Jets in strongly-coupled N = 4 super Yang-Mills theory,” [Phys. Rev. D [**79**]{}, 025021 (2009)](http://dx.doi.org/10.1103/PhysRevD.79.025021) \[[arXiv:0804.3110 \[hep-th\]](http://arxiv.org/abs/0804.3110)\]. D. T. Son and A. O. Starinets, “Minkowski-space correlators in AdS/CFT correspondence: Recipe and applications,” [JHEP [**0209**]{}, 042 (2002)](http://dx.doi.org/10.1088/1126-6708/2002/09/042). \[[arXiv:hep-th/0205051](http://arxiv.org/abs/hep-th/0205051)\]. C. P. Herzog and D. T. Son, “Schwinger-Keldysh propagators from AdS/CFT correspondence,” [JHEP **0303**, 046 (2003)](http://dx.doi.org/10.1088/1126-6708/2003/03/046). \[[arXiv:hep-th/0212072](http://arxiv.org/abs/hep-th/0212072)\]. D. T. Son and A. O. Starinets, “Viscosity, Black Holes, and Quantum Field Theory,” [Ann. Rev. Nucl. Part. Sci.  [**57**]{}, 95 (2007)](http://dx.doi.org/10.1146/annurev.nucl.57.090506.123120) \[[arXiv:0704.0240 \[hep-th\]](http://arxiv.org/abs/0704.0240)\]. S. Ryu and T. Takayanagi, “Holographic derivation of entanglement entropy from AdS/CFT,” [Phys. Rev. Lett.  [**96**]{}, 181602 (2006)](http://dx.doi.org/10.1103/PhysRevLett.96.181602) \[[hep-th/0603001](http://arxiv.org/abs/hep-th/0603001)\]. T. Numasawa, N. Shiba, T. Takayanagi and K. Watanabe, “EPR Pairs, Local Projections and Quantum Teleportation in Holography,” [JHEP [**1608**]{}, 077 (2016)](http://dx.doi.org/10.1007/JHEP08(2016)077) \[[arXiv:1604.01772 \[hep-th\]](http://arxiv.org/abs/1604.01772)\]. F. Dominguez, C. Marquet, A. H. Mueller, B. Wu and B. W. Xiao, [Nucl. Phys. A [**811**]{}, 197 (2008)](http://dx.doi.org/10.1016/j.nuclphysa.2008.07.004) \[[arXiv:0803.3234 \[nucl-th\]](http://arxiv.org/abs/0803.3234)\]. J. Casalderrey-Solana and D. Teaney, “Transverse Momentum Broadening of a Fast Quark in a N=4 Yang Mills Plasma,” [JHEP [**0704**]{}, 039 (2007)](http://dx.doi.org/10.1088/1126-6708/2007/04/039) \[[hep-th/0701123](http://arxiv.org/abs/hep-th/0701123)\]. W. G. Unruh, “Notes on black hole evaporation,” [Phys. Rev. D [**14**]{}, 870 (1976)](http://dx.doi.org/10.1103/PhysRevD.14.870). Appendix: Holographic Schwinger-Keldysh correlator {#AppendixSK} ================================================== In this supplemental material, we give a detailed derivation of the relation between holographic Schwinger-Keldysh correlator and retarded Green function in equation . The derivation largely follows Ref. [@Herzog:2002pc; @CasalderreySolana:2007qw]. Let $\tx^\mu=(\tt ,{\tu},\tx,\ty,\tz)$ be the dimensionless coordinates in the co-moving frame, with the square of line element ${\d}s^2=\tg_{\mu\nu}{\d} \tx^\mu {\d} \tx^\nu$, where $$\begin{aligned} \label{newmetric1} \tg_{\mu\nu}= \frac{\R^{2}}{{\tu}} \text{diag}\big[{-}f({\tu}), \frac{1}{4{\tu} f({\tu})}, {e^{-2\tz}}, {e^{-2\tz}} , 1 \big] .\end{aligned}$$ And $\tx^a=(\tt ,{\tu})$ are coordinates on the static string worldsheet. Without loss of generality, we can consider the string fluctuation as $X^{\mu }(\tt ,{\tu})=\big( \tt,{\tu}, \td_{i} (\tt ,{\tu})\big)$, which lead to an induced metric on the string worldsheet $g_{ab}{=}(\partial_a X^\mu)(\partial_b X^\nu) \tg_{\mu\nu}$. The Nambu-Goto action of string with tension ${T_s}$ is $S{=-}T_s{\int} \d{\tt}\d{\tu}\sqrt{-\det{g_{ab}}} $. When the fluctuation ${\td_i }\ll 1$, the action becomes$$\begin{aligned} S{\simeq}{-}{T_s}\R^{2}\!\! {\int}\frac{\d{\tt}\d{\tu}}{2\tu^{ 3/2}}\left\{1{+} \Big[2{\tu}f(\tu) {\td_{i}}^{\prime}{\td_{j}}^{\prime}{-}\frac{1}{2f(\tu)} \dot{\td}_{i} \dot{\td}_{j}\Big] h^{ij}\right\} ,\end{aligned}$$ where ${\td_i}^{\prime}\equiv \frac{\partial {\td_i} }{\partial \tu}$, $\dot{\td}_i \equiv \frac{\partial {\td_i} }{\partial \tt}$ and $h^{ij}=\text{diag}[1,e^{2\tz},e^{2\tz}]$. The equations of motion for the fluctuations $\td_{i}$ on the string are $$\begin{aligned} \partial _{{\tu}}\Big( \frac{2 f {\td_{i} }^{\prime }}{{\tu}^{1/2}}\Big) -\partial _{\tt}\Big( \frac{\dot{\td}_{i}}{2f{\tu}^{ 3/2}}\Big) =0. \label{stringeom} \end{aligned}$$ Performing a Fourier transform, $$\begin{aligned} \label{fourier} {\td_{i}}({\tt}, {\tu}) =\int \frac{d\omega }{2\pi }e^{ -i\omega {\tt }} {\td_i }(\omega) Y_{\omega }( {\tu}), \end{aligned}$$where $\td_i(\omega) $ is defined as the Fourier transform of fluctuation on the boundary, after choosing the normalization $\lim_{\tu\to 0}Y_{\omega }( {\tu }) =1$. Then becomes $$\begin{aligned} \!\! Y''_{\omega}( {\tu})- \frac{f(\tu)-2\tu{f'(\tu)}}{2 \tu f(\tu)}Y'_{\omega}({\tu}) +\frac{\omega^2 Y_{\omega}( {\tu})}{2f(\tu)\tu^{3/2}}=0.\end{aligned}$$Requiring the in-falling boundary condition at the horizon, this equation is solved by $$\begin{aligned} Y_{\omega}( {\tu})=(1-\tu)^{-\i\omega/2} F_{\omega}({\tu}).\end{aligned}$$ The complex conjugate $$\begin{aligned} Y^*_{\omega}( {\tu})=Y_{-\omega}( {\tu})=(1-\tu)^{+\i\omega/2} F_{-\omega}({\tu}),\end{aligned}$$ is the other solution with the outgoing boundary condition at the horizon. We need to extend these solutions into the Kruskal plane of the metric , with new coordinates $U$ and $V$, which are initially defined in the right-quadrant $\{{U<0}, {V>0}\}$, with $$\begin{aligned} {U}&=- e^{-2\tt} e^{-2\tu^*},\\ {V}&=+ e^{+2\tt} e^{-2\tu^*}.\end{aligned}$$ And $\tu^*$ is placed outside the worldsheet horizon ${0\!<\!\tu\!<\! 1}$, $$\begin{aligned} \tu^* \equiv \int_0^{\sqrt{\tu}}\frac{ d\tw }{f(\tw^2)} =\frac{1}{2} \ln\frac{1- {\sqrt{\tu}}}{1+\sqrt{\tu}}, \end{aligned}$$ with ${f(\tw^2)} =1- \tw^2$. The full extension of the metric and string worldsheet in the Kruskal plane is shown in Fig. \[fig:UV\]. ![The Kruskal plane of the metric in terms of coordinates $U$ and $V$. The string worldsheet covers four quadrants, with the time-like boundary A(Red) at the left quadrant, and B(Green) at the right quadrant.[]{data-label="fig:UV"}](Hplane2.pdf "fig:")\ In the right-quadrant the two solutions near the horizon are $$\begin{aligned} \label{Bi} \td^B_- &=e^{-i{\omega} {\tt}}{{Y}_{\omega}(\tu)} \sim e^{-i(\omega/2) \ln(V)} ,\\ \label{Bo} \td^B_+&=e^{-i{\omega} {\tt}}{Y^*_{\omega}(\tu)} \sim e^{i({\omega}/{2})\ln(-{U})}.\end{aligned}$$ And in the left quadrant $\{{U>0}, {V<0}\}$, $$\begin{aligned} \td^A_- &=e^{-i{\omega} {\tt}}{{Y}_{\omega}(\tu)} \sim e^{-i(\omega/2) \ln(-V)} \label{Ai},\\ \td^A_+ &=e^{-i{\omega} {\tt}}{Y^*_{\omega}(\tu)} \sim e^{i(\omega/2) \ln(U)} \label{Ao}.\end{aligned}$$ Similar to the Herzog-Son’s prescription [@Herzog:2002pc; @Unruh:1976db], two linear combinations are analytic over the full Kruskal plane, $$\begin{aligned} \td_+(\omega)&=\td^B_+ + e^{+\pi \omega/2} \td^A_+ \, ,\\ \td_-(\omega)&=\td^B_-+ e^{ -\pi \omega/2} \td^A_- \, ,\end{aligned}$$ which can be used as two bases for the string fluctuations, $$\begin{aligned} \label{basis} \td_i(\tt,\tu)=\int \frac{d{\omega}}{2\pi} \left[a_i(\omega)\td_+(\omega)+ b_i(\omega)\td_-(\omega)\right].\end{aligned}$$ The coefficients $a_i({\omega}), b_i({\omega})$ can be determined by the two boundary values $\td_i^A(\omega) $ and $\td_i^B(\omega)$ of the solutions, $$\begin{aligned} \label{cAB} a_i(\omega)&=\no\big[-\td_i^A({\omega}) +e^{\pi\omega/2} \td_i^B({\omega}) \big], \\ b_i(\omega)&=\no\big[ e^{\pi{\omega}}\td_i^A({\omega})-e^{\pi\omega/2}\td_i^B({\omega})\big],\end{aligned}$$ with $\no={1}/{(e^{\pi\omega}-1)}$. The total boundary term of the Nambu-Goto action in terms of the string solution turns out to be $$\begin{aligned} \! S_{\partial} &= \frac{\sqrt{\lambda} f( {\tu}) }{{\pi} b^{2} \tu^{1/2}}\big(\!\int_A -\int_B\! \big)\frac{d{\omega}}{2\pi} \Big[ {\td}_i(-\omega , {\tu}) \partial _{{\tu}}{\td}_j({\omega}, {\tu}) \delta^{ij}\Big]\!, \!\! \end{aligned}$$ After considering , it becomes $$\begin{aligned} S_{\partial} &= \!-\! \frac{1}{2} \! \int \! \frac{d \omega}{2\pi} \Big\{ \big[{\td}^A_i(-\omega){\td}^B_j(\omega) + {\td}^B_i(-\omega){\td}^A_j(\omega) \big] \nn\\ &~ ~\qquad\qquad \times \sqrt{\no(1+\no)} \big[ G_{\mA}^{ij}(\omega) -G_{\mR}^{ij}(\omega) \big] \nn\\ &~~~+{\td}^A_i(-\omega){\td}^A_j(\omega) \big[(1+n)G_{\mR}^{ij}(\omega) -n G_{\mA}^{ij}(\omega)\big] \nn\\ &~~~+{\td}^B_i(-\omega){\td}^B_j(\omega) \big[ n G_{\mR}^{ij}(\omega) -(1+n) G_{\mA}^{ij}(\omega)\big] \Big\},\!\end{aligned}$$where the holographic retarded and advanced Green’s functions are defined as $$\begin{aligned} G_{\mR}^{ij}(\omega) &=-\frac{\sqrt{\lambda} f( {\tu}) }{{\pi} b^{2} \tu^{1/2}} Y_{-\omega }( {\tu}) \partial _{{\tu}}Y_{\omega}( {\tu}) \delta^{ij}\big\vert _{{\tu}\rightarrow 0} \,, \\ G_{\mA}^{ij}(\omega) &=-\frac{\sqrt{\lambda} f( {\tu}) }{{\pi} b^{2} \tu^{1/2}} Y_{\omega }( {\tu}) \partial _{{\tu}}Y_{-\omega}( {\tu}) \delta^{ij}\big\vert _{{\tu}\rightarrow 0} \, .\end{aligned}$$ Taking functional derivatives of $S_{\partial}$ with respect to ${\td}^B_i(\omega)$ and ${\td}^A_j(\omega)$ yields precisely the Schwinger-Keldysh correlators. This off-diagonal one is related to the retarded Green’s function as in the main text, $$\begin{aligned} &G_{AB}^{ij} ( \omega ) =\frac{2\i e^{-\omega/(2T_a)}}{1-e^{-\omega/T_a}} \textrm{Im}G_{\mR}^{ij}\left( \omega \right),\end{aligned}$$ with ${T_a}= \frac{1}{2\pi b}=\frac{a}{2\pi}$ after restoring the physical units. [^1]: $^{\dag,\ddag}$corresponding authors.

--- abstract: | We have analysed radial velocity measurements for known transiting exoplanets to study the empirical signature of tidal orbital evolution for close-in planets. Compared to standard eccentricity determination, our approach is modified to focus on the rejection of the null hypothesis of a circular orbit. We are using a MCMC analysis of radial velocity measurements and photometric constraints, including a component of correlated noise, as well as Bayesian model selection to check if the data justifies the additional complexity of an eccentric orbit. We find that among planets with non-zero eccentricity values quoted in the literature, there is no evidence for an eccentricity detection for the $7$ planets CoRoT-5b, WASP-5b, WASP-6b, WASP-10b, WASP-12b, WASP-17b, and WASP-18b. In contrast, we confirm the eccentricity of HAT-P-16b, $e=0.034\pm0.003$, the smallest eccentricity that is reliably measured so far for an exoplanet as well as that of WASP-14b, which is the planet at the shortest period ($P=2.24$ d), with a confirmed eccentricity, $e= 0.088\pm0.003$. As part of the study, we present new radial velocity data using the HARPS spectrograph for CoRoT-1, CoRoT-3, WASP-2, WASP-4, WASP-5 and WASP-7 as well as the SOPHIE spectrograph for HAT-P-4, HAT-P-7, TrES-2 and XO-2. We show that the dissipative effect of tides raised in the planet by the star and vice-versa explain all the eccentricity and spin-orbit alignment measurements available for transiting planets. We revisit the mass-period relation [@Mazeh2005; @Pont2011] and consider its relation to the stopping mechanism of orbital migration for hot Jupiters. In addition to CoRoT-2 and HD 189733 [@Pont2009b], we find evidence for excess rotation of the star in the systems CoRoT-18, HAT-P-20, WASP-19 and WASP-43. author: - | Nawal Husnoo$^1$, Frédéric Pont$^1$, Tsevi Mazeh$^2$, Daniel Fabrycky$^{3}$, Guillaume Hébrard$^{4,5}$,François Bouchy$^{4,5}$, Avi Shporer$^{6}$\ $^1$ School of Physics, University of Exeter, Exeter, EX4 4QL, UK\ $^2$ School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel\ $^3$ Harvard-Smithsonian Centre for Astrophysics, Garden Street, Cambridge, MA\ $^4$ Institut d’Astrophysique de Paris, UMR7095 CNRS, Université Pierre & Marie Curie, 98bis boulevard Arago, 75014 Paris, France\ $^5$ Observatoire de Haute-Provence, CNRS/OAMP, 04870 Saint-Michel-l’Observatoire, France\ $^{6}$ Las Cumbres Observatory Global Telescope network, 6740 Cortona Drive, suite 102, Goleta, CA 93117, USA\ bibliography: - 'husnoo.bib' title: 'Observational constraints on tidal effects using orbital eccentricities.[^1]' --- \[firstpage\] planetary systems Introduction ============ Most of the information we have about the formation, evolution and structure of exoplanets have come from the study of transiting planets. This is possible because the combination of radial velocity measurements with transit photometry can provide powerful constraints on the physical and orbital parameters of an exoplanet, such as the planetary mass, radius, orbital eccentricity, etc. A selection effect due to geometry means that most transiting planets with radial velocity confirmation are found on very short period orbits with $P\sim1-20$ days. The close-in planets with periods of a few days are expected to experience strong tidal effects [e.g. @Rasio1996], which should increase sharply with decreasing period and these orbits are thus expected to circularise on a timescale much smaller than the system age. A higher tendency for such circular orbits is indeed observed in the sample of transiting planets, as compared to those from radial velocity surveys. This has been interpreted as a signature for tidal circularization. The transition from eccentric orbits to circular orbits at short period has also been seen in binary star systems, e.g. @Mathieu1988 and @Mazeh2008a. Over the last few years, we have carried out a monitoring programme to obtain several radial velocity measurements of known transiting planetary systems with the intention of refining the orbital properties such as orbital eccentricity and spin-orbit alignment angle. We have used the SOPHIE spectrograph in the Northern hemisphere and the HARPS spectrograph in the Southern hemisphere [e.g. @Loeillet2008; @Hebrard2008; @Husnoo2011; @Pont2011 ESO Prog. 0812.C-0312]. One issue is the difficulty of measuring the orbital eccentricity of exoplanets for faint stars, especially for low-mass planets. While it is impossible to prove that an orbit is circular, with $e=0$ exactly, we can place an upper limit on the eccentricity of a given orbit (e.g. reject $e>0.1$ at the 95% confidence level). In fact, a number of eccentric orbits have been detected at short period, but follow-up observations using photometry or additional radial velocity measurements led to the conclusion that some of these eccentricities had originally been overestimated. For example, the WASP-10 system [@Christian2009] was revisited by @Maciejewski2011, who showed that the initially reported eccentricity ($e=0.059^{+0.014}_{-0.004}$) had been overestimated and was in fact compatible with zero. The orbital eccentricity of WASP-12b [@Hebb2009 $e=0.049\pm0.015$] is similarly compatible with zero [@Husnoo2011], and the original detection was possibly due to systematic effects (weather conditions, instrumental drifts, stellar spots or scattered sunlight). The eccentricity distribution at short period has a crucial importance for any theory of planetary formation and orbital evolution. Planets on orbits that are consistent with circular gather in a well-defined region of the mass-period plane, close to the minimum period for any given mass [@Pont2011]. We now show that there are no exceptions to this pattern, and revisit some apparent exceptions as reported in the literature. As an ensemble, the totality of transiting planets considered in this study are in agreement with classical tide theory, with orbital circularisation due to tides raised on the planet by the star and tides on the star raised by the planet, to varying degree depending on the position of the planet-star system in the mass-period plane. In this study, we consider new radial velocity measurements made with HARPS and SOPHIE, as well as measurements present in the literature. We use photometric constraints in the form of the orbital period $P$ and mid-transit time $T_{tr}$, both of which can be measured accurately using transit photometry, and we also consider constraints from the secondary eclipse where available. In fact, if we define the orbital phase $\phi=(t-T_0)/P$ to be zero at mid-transit time $T_0=T_{tr}$, a planet on a circular orbit would have a mid-occultation phase of $\phi=0.5$ (by symmetry). A planet that is on an eccentric orbit will, however, have a mid-occultation phase different from 0.5 (unless the orbital apsides are aligned along the line of sight). This allows us to place a constraint on the $e\cos\omega$ projection of the eccentricity, as given by @Winn2005a (slightly modified): $$e\cos\omega\simeq\frac{\pi}{2}\left(\phi_{\rm occ}-0.5\right),$$ to first order in $e$, where we now define $\phi_{\rm occ}$ to be the phase difference between the mid-transit time and the mid-occultation time, i.e. $\phi_{\rm occ} = ({T_{sec}-T_{tr}})/P$, where $T_{sec}$ is the time of the secondary eclipse following the transit time $T_{tr}$. The component $e\sin\omega$ is dependent on a ratio involving the durations of the occultation and transit [@Winn2005a], $$e\sin\omega\simeq\frac{T_{\rm tra}-T_{\rm occ}}{T_{\rm tra}+T_{\rm occ}},$$ to first order in $e$, where $T_{\rm tra}$ and $T_{\rm occ}$ are the transit and occultation durations respectively, although this constraint is weaker than the one on $e\cos\omega$. In addition to the reanalysis of radial velocity measurements with photometric constraints, we also introduce two modifications to the Markov Chain Monte Carlo (MCMC) process commonly used by teams analysing radial velocity data to work out the orbital parameters of transiting exoplanets. This involves a new treatment of the correlated noise present in most radial velocity datasets, as well as analysing the data in model selection mode to check if an eccentric orbit is indeed justified, given the additional complexity of the eccentric version of a Keplerian orbit. The present time is significant in the study of exoplanets, because a number of high quality measurements are now available for the three main observable effects of tides: circularisation, synchronisation and spin-orbit alignment. In Section \[sec:observations\], we describe our new radial velocity measurements obtained with SOPHIE and HARPS for 10 objects, as well as the measurements we collected from the literature for this study. We then describe the analysis we performed, in Section \[sec:analysis\]. In Section \[sec:results\], we describe the objects in the classes “eccentric”, “compatible with circular, $e<0.1$”, and “poorly constrained” and present the updated eccentricities, as shown in Table \[tab:results\]. In Section \[sec:discussion\], we consider our orbital eccentricity estimates in the light of tidal effects inside the planet due to the star and vice-versa. We find that our results are compatible with classical tidal theory, removing the need for perturbing stellar or planetary companions to excite non-negligible eccentricities in short period orbits. @Winn2010b presented a discussion of the available measurements of the projected spin-orbit alignment angles, and found that hot planet-hosting stars ($T_{\textrm{eff}}>6250$ K) had random obliquities whereas cooler stars ($T_{\textrm{eff}}<6250$ K) tended to have aligned rotations. These authors suggested that this dichotomy can be explained if all these stars harbouring a planetary system start off with a random obliquity following some dynamical interaction, but only cool stars with a significant convective layer are able to undergo tidal effects leading to alignment. We verify in Section \[sec:discussion\] that the strong exceptions WASP-8 and HD 80606 are indeed systems with weak tidal interactions, and that the observation that these two are misaligned, is not incompatible with tidal theory. In a number of cases, such as HD 189733 [@Henry2008], WASP-19 [@Hebb2010] and CoRoT-2 [@Lanza2009a], the rotational period of the star is known from photometric monitoring. Assuming the results of @Winn2010b are correct in the sense that the convective layer in G dwarfs would cause tidal dissipation that aligns the stellar equator with the planetary orbit, the negligible value of the projected spin-orbit angle $\lambda$ means that the obliquity is indeed zero, i.e. the stellar equators are aligned with the orbital planes. In this case, a measurement of the projected equatorial rotational velocity of the star ($v\sin i$) through Doppler broadening yields the rotational period of the star. This means that for G dwarfs at least, we have enough information to observe the effect of tidal interactions on the stellar rotation. We show in Section \[sec:discussion\] that in addition to CoRoT-2 and HD 189733 [@Pont2009b], we find evidence for excess rotation of the star in the systems CoRoT-18, HAT-P-20, WASP-19 and WASP-43. The preliminary results from this study were published in @Pont2011, and the exact numerical values of the eccentricities have been updated in this paper to reflect our new choice of radial velocity measurement correlation timescale $\tau=1.5$ d (see Section \[sec:tau\]), as opposed to $\tau=0.1$ d in @Pont2011. The overall results, i.e. the clear separation between orbits that are consistent with circular and eccentric orbits in the mass-period plane, does not change in this paper. The mass-period relation of @Mazeh2005 is still clearly present, with low-mass hot Jupiters on orbits that are consistent with circular clumping in a definite region of the mass-period plane, with heavier objects moving closer in, to shorter periods. This strongly suggests that tidal effects are involved in the stopping mechanism of these objects. A similar stopping mechanism can be seen at higher planetary masses, but destruction of the planet is not excluded in many cases. Other effects, such as spin-orbit alignment and stellar spin-up also point strongly towards the scenario of @Rasio1996 where the short period orbits of hot Jupiters are formed by dynamical scattering, which produces eccentric and misaligned orbits. This is followed by tidal dissipation which leads to circularisation at short period, spin-orbit alignment and synchronisation of the rotation of the host star. Observations {#sec:observations} ============ We include 73 measurements for 6 objects with the HARPS spectrograph (Tables \[tab:rv\_corot1\], \[tab:rv\_corot3\], \[tab:rv\_wasp2\], \[tab:rv\_wasp4\], \[tab:rv\_wasp5\] and \[tab:rv\_wasp7\] ) and 45 measurements for 4 objects with the SOPHIE spectrograph (Tables \[tab:rv\_hatp4\], \[tab:rv\_hatp7\], \[tab:rv\_tres2\] and \[tab:rv\_xo2\]). Both are bench-mounted, fibre-fed spectrograph built on the same design principles and their thermal environments are carefully controlled, to achieve precise radial-velocity measurements. The two instruments have participated in the detection and characterisation of numerous transiting exoplanets, notably from the WASP and CoRoT transit searches. The wavelength calibrated high-resolution spectra from the instruments are analysed using a cross correlation technique which compares them with a mask consisting of theoretical positions and widths of the stellar absorption lines at zero velocity [@Pepe2002]. We carried out a literature survey and collected radial velocity measurements for 54 transiting planets, as well as other relevant data such as the orbital periods and the time of mid-transit. For the cases of CoRoT-1, CoRoT-2 and GJ-436, we also used the secondary eclipse constraint on the eccentricity component $e\cos\omega$ from @Alonso2009, @Alonso2009a and @Deming2007, respectively. Given the rapid rate of announcement of new transiting exoplanets, we had to stop the clock somewhere, and we picked the 1st of July 2010. We selected only objects that had been reported in peer-reviewed journals or on the online preprint archive ArXiV.org. Moreover, we selected systems with well measured parameters (planetary radius $R_p$ and mass $M_p$ to within 10%) and excluded faint objects ($V>15$). At that time, 64 such systems were known. We reanalyse the existing radial velocity data for 54 transiting systems, providing additional radial velocity measurements for 10 systems described above, and include 10 systems without further reanalysis of orbital ephemeris. These systems are listed in Table \[tab:results\]. In Section \[add\_systems\], we include a further 16 systems, most of which had been discovered in the mean time. The planets involved in this study are listed on the webpage http://www.inscience.ch/transits/, where we also include the parameters $v\sin i$ (the projected rotation velocity of the host star), $P_{\rm rot}$ (the orbital period of the host star) and the projected spin-orbit angle $\lambda$ where available. ------------------ --------------------- --------------------- Time RV $\sigma_{\rm RV}$  \[BJD-2450000\] \[[km s$^{-1}$]{}\] \[[km s$^{-1}$]{}\] 4385.86631 23.4168 0.0190 4386.83809 23.6726 0.0139 4387.80863 23.3155 0.0149 4419.81749 23.5811 0.0123 4420.80300 23.3290 0.0118 4421.81461 23.6586 0.0113 4446.77797 23.3936 0.0145 4447.75517 23.4562 0.0130 4448.77217 23.6982 0.0126 4479.67146 23.3161 0.0123 4480.65370 23.6836 0.0140 4481.63818 23.5129 0.0173 4525.59523 23.6451 0.0116 4529.56406 23.3324 0.0127 4530.58002 23.5743 0.0105 4549.58179 23.5793 0.0280 4553.49391 23.3652 0.0124 4554.57636 23.6696 0.0157 4768.77120 23.7041 0.0092 4769.76601 23.4802 0.0104 4770.80872 23.3613 0.0108 4771.76514 23.6955 0.0102 4772.76824 23.4379 0.0109 4773.76896 23.3980 0.0095 ------------------ --------------------- --------------------- : HARPS radial velocity measurements for CoRoT-1 (errors include random component only).[]{data-label="tab:rv_corot1"} ------------------ --------------------- --------------------- Time RV $\sigma_{\rm RV}$  \[BJD-2450000\] \[[km s$^{-1}$]{}\] \[[km s$^{-1}$]{}\] 4768.52895 -56.5039 0.0036 4769.51404 -58.1970 0.0038 4770.51329 -56.1017 0.0046 4772.52655 -55.9171 0.0047 4773.52957 -58.2227 0.0042 ------------------ --------------------- --------------------- : HARPS radial velocity measurements for CoRoT-3 (errors include random component only).[]{data-label="tab:rv_corot3"} ------------------ --------------------- --------------------- Time RV $\sigma_{\rm RV}$  \[BJD-2450000\] \[[km s$^{-1}$]{}\] \[[km s$^{-1}$]{}\] 4766.56990 -27.8402 0.0033 4767.52666 -27.6797 0.0023 4768.56373 -27.7842 0.0018 4769.54823 -27.7343 0.0017 4770.54665 -27.7131 0.0026 4771.54501 -27.8099 0.0031 4772.56012 -27.6489 0.0027 4773.56432 -27.8568 0.0019 ------------------ --------------------- --------------------- : HARPS radial velocity measurements for WASP-2 (errors include random component only).[]{data-label="tab:rv_wasp2"} ------------------ --------------------- --------------------- Time RV $\sigma_{\rm RV}$  \[BJD-2450000\] \[[km s$^{-1}$]{}\] \[[km s$^{-1}$]{}\] 4762.60256 57.6687 0.0028 4763.62220 57.5637 0.0022 4764.58386 57.9085 0.0035 4765.59031 57.9871 0.0038 4768.60378 57.9109 0.0022 4769.58081 57.9784 0.0023 4769.71186 58.0331 0.0017 4770.58784 57.6591 0.0024 4770.72474 57.7930 0.0023 4771.57892 57.6311 0.0021 4771.68481 57.5752 0.0019 4772.59125 57.9518 0.0025 4773.59429 57.9811 0.0018 4773.70377 58.0346 0.0024 ------------------ --------------------- --------------------- : HARPS radial velocity measurements for WASP-4 (errors include random component only).[]{data-label="tab:rv_wasp4"} ------------------ --------------------- --------------------- Time RV $\sigma_{\rm RV}$  \[BJD-2450000\] \[[km s$^{-1}$]{}\] \[[km s$^{-1}$]{}\] 4768.63152 19.7967 0.0022 4768.73169 19.8696 0.0018 4769.62838 20.1047 0.0023 4770.62473 20.1231 0.0022 4770.76117 20.2255 0.0031 4771.60846 19.7737 0.0017 4771.71520 19.7446 0.0021 4772.63762 20.2588 0.0023 4772.73505 20.2071 0.0022 4773.62311 19.8540 0.0021 4773.73277 19.9582 0.0025 ------------------ --------------------- --------------------- : HARPS radial velocity measurements for WASP-5 (errors include random component only).[]{data-label="tab:rv_wasp5"} ------------------ --------------------- --------------------- Time RV $\sigma_{\rm RV}$  \[BJD-2450000\] \[[km s$^{-1}$]{}\] \[[km s$^{-1}$]{}\] 4762.53711 -29.4388 0.0024 4763.57798 -29.4994 0.0023 4764.50127 -29.5469 0.0032 4765.54456 -29.3948 0.0032 4767.54077 -29.3485 0.0031 4768.57924 -29.5636 0.0021 4769.64528 -29.5332 0.0018 4770.64161 -29.4468 0.0022 4771.62297 -29.3421 0.0019 4772.65474 -29.4212 0.0022 4773.63924 -29.5829 0.0020 ------------------ --------------------- --------------------- : HARPS radial velocity measurements for WASP-7 (errors include random component only).[]{data-label="tab:rv_wasp7"} ------------------ --------------------- --------------------- Time RV $\sigma_{\rm RV}$  \[BJD-2450000\] \[[km s$^{-1}$]{}\] \[[km s$^{-1}$]{}\] 5003.41234 -1.3253 0.0206 5005.47636 -1.3669 0.0122 5006.50185 -1.3244 0.0123 5007.42327 -1.4822 0.0129 5008.39084 -1.4143 0.0131 5009.38832 -1.3465 0.0132 5010.39331 -1.4711 0.0131 5011.42884 -1.4181 0.0129 5012.46735 -1.3504 0.0131 5013.45923 -1.4487 0.0128 5014.43881 -1.4106 0.0123 5015.48148 -1.3212 0.0124 5016.41444 -1.4666 0.0119 ------------------ --------------------- --------------------- : SOPHIE Radial velocity measurements for HAT-P-4 (uncertainties include random component only).[]{data-label="tab:rv_hatp4"} ------------------ --------------------- --------------------- Time RV $\sigma_{\rm RV}$  \[BJD-2450000\] \[[km s$^{-1}$]{}\] \[[km s$^{-1}$]{}\] 5002.48517 -10.2995 0.0100 5003.52118 -10.6910 0.0103 5004.59910 -10.2681 0.0137 5005.49926 -10.6377 0.0101 5006.55335 -10.2564 0.0101 5007.53107 -10.5975 0.0101 5008.47624 -10.4027 0.0106 5010.43095 -10.5681 0.0102 5011.52259 -10.3835 0.0102 5013.60648 -10.3090 0.0093 5014.57426 -10.6862 0.0101 5015.58518 -10.2680 0.0103 5016.54123 -10.6808 0.0084 ------------------ --------------------- --------------------- : SOPHIE radial velocity measurements for HAT-P-7 (uncertainties include random component only).[]{data-label="tab:rv_hatp7"} ------------------ --------------------- --------------------- Time RV $\sigma_{\rm RV}$  \[BJD-2450000\] \[[km s$^{-1}$]{}\] \[[km s$^{-1}$]{}\] 5005.57207 -0.4489 0.0107 5006.57091 -0.2231 0.0100 5007.57482 -0.2619 0.0105 5008.44948 -0.4742 0.0106 5010.44234 -0.4505 0.0107 5011.51233 -0.2499 0.0110 5013.59448 -0.4207 0.0114 5014.56301 -0.1604 0.0108 5015.57356 -0.5090 0.0107 5016.55442 -0.2197 0.0110 ------------------ --------------------- --------------------- : SOPHIE radial velocity measurements for TrES-2 (uncertainties include random component only).[]{data-label="tab:rv_tres2"} ------------------ --------------------- --------------------- Time RV $\sigma_{\rm RV}$  \[BJD-2450000\] \[[km s$^{-1}$]{}\] \[[km s$^{-1}$]{}\] 4878.41245 46.7905 0.0091 4879.38681 46.9667 0.0084 4886.39349 46.7748 0.0095 4887.44867 46.9583 0.0084 4888.47514 46.7722 0.0085 4889.40965 46.8778 0.0086 4890.46546 46.8994 0.0085 4893.41643 46.8202 0.0087 4894.44335 46.8073 0.0121 ------------------ --------------------- --------------------- : SOPHIE radial velocity measurements for XO-2 (uncertainties include random component only).[]{data-label="tab:rv_xo2"} Analysis {#sec:analysis} ======== We used the radial velocity data , as well as the constraints on the orbital period $P$ and mid-transit time $T_{tr}$ (and $e\cos\omega$ where available) from photometry as described in Section \[sec:observations\]. To calculate the median values of the derived parameters and their corresponding uncertainties, we marginalise over their joint probability distribution using a Markov Chain Monte Carlo analysis with the Metropolis-Hastings algorithm. This has been described in the past by @holman2006 and our implementation is described in [@Pont2009b]. We model the radial velocity using a Keplerian orbit and run the MCMC for 500,000 steps, the first 50,000 of which are then dropped to allow the MCMC to lose memory of the initial parameters. We verify that the autocorrelation length of each chain is much shorter than the chain length to that ensure the relevant region of parameter space is properly explored. Although this procedure is common practice in the community, we bring two changes. The first is a modification to the merit function that is used to work out the likelihood of a set of parameters given the data, to include the effects of correlated noise. This is described in Section \[sec:sigmar\]. The second modification we bring is that we consider not only the case of an orbital model with a free eccentricity $e$, but we also work out the likelihood for a circular orbit (i.e. with $e$ fixed at zero). We then compare the two models, by including a penalty for the additional complexity in the eccentric one (i.e. two additional degrees of freedom). We do this by using the Bayesian Information Criterion, as described in Section \[sec:modelselection\]. We report the median value in the chain for each parameter, as well as the central 68.4% confidence interval on the parameter. A circular orbit model for an orbit that is in fact eccentric would artificially make the uncertainties in the derived parameters smaller, so in the case of the systemic velocity $V_0$ and the semi-amplitude $K$, we report the median values from a circular orbit model, yet we include the confidence intervals derived from the eccentric model. The treatment of correlated noise {#sec:sigmar} --------------------------------- Correlated noise can be important in the analysis of transit light curves [@Pont2006], and we included this in the analysis of radial velocity measurements in the case of WASP-12 [see @Husnoo2011]. If we assume uncorrelated Gaussian noise when analysing data that is affected by correlated noise, we run the risk of overestimating the importance of a series of measurements that were obtained in quick succession, and this can have implications for example in estimating the orbital eccentricity. From @Sivia2006, the likelihood function for some data, given a model, is given by: $${\rm P}({\rm\bf D}|{\bf \theta},I)=\frac{{\rm exp}\left[-\frac{1}{2}({\bf F}-{\bf D})^{\rm T}{\rm\bf C}^{-1}({\bf F}-{\bf D})\right]}{\sqrt{(2\pi)^N{\rm det}({\rm\bf C})}}, \label{eqn:likelihood}$$ where $\bf D$ is the radial velocity time series data expressed as a vector, $\bf \theta$ is the vector of model parameters, $\bf F$ is the predicted values from the Keplerian model. In this case, $\chi^2$ is defined by $$\chi^2=({\bf F}-{\bf D})^{\rm T}{\rm\bf C}^{-1}({\bf F}-{\bf D}),$$  \ where $\bf C$ is the covariance matrix, which remains constant throughout the MCMC analysis for each system. In the case of independent measurements, the components of the covariance matrix $\bf C$ would be obtained using $$C_{k,k'}= \left\{ \begin{array}{l l} \sigma_k^2 & \quad \mbox{for $k=k'$}\\ 0 & \quad \mbox{otherwise,}\\ \end{array} \right.$$ whereas in the presence of some correlated noise, we modify this to include a squared exponential covariance kernel so that $$C_{k,k'}=\delta_{k,k'}\sigma_k^2+\sum\limits^M_{i=1}\sigma_i^2\exp-\frac{(t_k-t_{k'})^2}{2\tau_i^2} \label{covariancesum}$$ where $\sigma_k$ is the formal uncertainty on each measurement $k$ as obtained from the data reduction for that measurement, and the sum over $M$ terms having the form $\sigma_i^2\exp-\frac{(t_k-t_{k'})^2}{2\tau_i^2}$ allows us to include a number of stationary covariance functions to account for correlations in the noise, occuring over the timescales of hours to days. In practise, it can be tricky to estimate the values of $\tau_i$ and $\sigma_i$ for radial velocity datasets, especially where the number of measurements is few or the phase-coverage is incomplete. Given the small datasets, we elect to use a single time-invariant correlation term, setting $M=1$. The timescale is now called $\tau$, and the corresponding value of $\sigma_i$ is now called $\sigma_r$, where the subscript $r$ indicates “red noise” [@Pont2006]. A fully Bayesian analysis would require that we assign priors to these two parameters and then marginalise over them. In practise, the sparse sampling and small datasets for radial velocity observations mean that it is very difficult to perform Bayesian marginalisation over these two parameters, and the results would depend on the prior space chosen (eg: $\tau$, $\log \tau$, etc). We therefore use a single pair of parameters for $\tau$ and $\sigma_r$. Using a single term in the sum in Equation \[covariancesum\] makes the expression less flexible, but prior experience shows that correlations over the $\sim 1$ day timescale are particularly important for radial velocity datasets, especially for measurements taken in the same night [see for example, @Husnoo2011 for the case of WASP-12]. For datasets where the reduced $\chi^2$ for a given model (circular or eccentric) was larger than unity, we estimated $\sigma_r$ by repeating the MCMC analysis with different values of $\sigma_r$ until the the best-fit orbit resulted in a reduced $\chi^2$ of unity for some optimal value of $\sigma_r$. We discuss the estimation of $\tau$ in the next sub-section. Estimation of $\tau$ {#sec:tau} -------------------- There is a degeneracy between $\sigma_r$ and $\tau$ for the time sampling typical of our RV data: if we assume a long timescale compared to the interval of time between the measurements, we are asserting that we have a reason to believe that several measurements may have been systematically offset in the same direction. A measurement that occurs within that timescale but is offset to a very different extent from nearby measurements (e.g. if the correlation timescale $\tau$ has been overestimated) will require a larger value of $\sigma_r$ for the dataset as a whole to yield a reduced $\chi^2$ of unity. To estimate $\tau$, we looked at several datasets for each of the instruments HARPS, HIRES and SOPHIE. We repeated the analysis in Section \[sec:sigmar\] using values of $\tau$ in the range 0.1–5 d, to check for weather-related correlations. To see the effects of choosing between an eccentric orbit or a circular orbit on our estimation of $\tau$, we carried each analysis twice, by adjusting $\sigma_r$ (see Section \[sec:sigmar\]) to obtain a reduced $\chi^2$ of unity (within 0.5%) for each orbital model (circular and eccentric). We plotted the optimal values of $\sigma_r$ against $\tau$ for several objects using data obtained from different instruments separately, as shown in Figure \[fig:tausigma\]. For WASP-2, we used our new HARPS measurements (Table \[tab:rv\_wasp2\]) as well as SOPHIE measurements from @Cameron2007. For WASP-4 and WASP-5, we used our new HARPS measurements (Tables \[tab:rv\_wasp4\] and \[tab:rv\_wasp5\]), and for HAT-P-7, we used our new SOPHIE measurements (Table \[tab:rv\_hatp7\]) as well as HIRES measurements from @Winn2009b. We found that for those datasets and objects where the orbital elements were well-constrained the plot showed a gentle increase in $\sigma_r$ with $\tau$, for $\tau\leq 1.5$d, then increased much faster for these datasets at a timescale of $\tau> 1.5$ d. For objects that have been observed with multiple instruments, this characteristic timescale is independent, both of the instrument used or the assumption about the eccentricity (i.e. free eccentricity or $e$ fixed at zero), suggesting that the correlated noise is probably related to weather conditions. We therefore assumed a correlation timescale of $1.5$ d in the rest of this study, unless otherwise noted. This means that we are accounting for the red noise in the same-night measurements, and for measurements that are taken further apart in time, this procedure reduces to the more familiar “jitter” term. The value of $\sigma_r$ inferred at $\tau=1.5$ d in some cases varies by a few percent depending on the model chosen, i.e. eccentric or circular, and varies across datasets, as discussed later. We also investigated the effects of varying $\tau$ on our final results. For the same systems discussed above, we plotted the 95% upper limit on the eccentricity as obtained from each dataset separately. The results are shown in Figure \[fig:tau\_ecc\], where it is clear that the choice of $\tau$ has no effect on the final result for $\tau\geq1.5$ d. The only exception is WASP-2 (HARPS), where we only have 8 measurements and the phase coverage is not as complete as for the other objects (see Figure \[fig:wasp2ph\]). Similarly, the derived parameters $V_0$ and $K$ did not vary appreciably with $\tau$. Model selection {#sec:modelselection} --------------- Determining whether an orbit is consistent with circular (Model 1) or eccentric (Model 2), is an exercise in model selection. If we assume the prior probability of the circular and eccentric models are the same, we can use the Bayesian Information Criterion (BIC) [@Liddle2007] to decide between the two models. This is equivalent to working out the Bayes factor $P({\rm data}|{\rm Model_1})/P({\rm data}|{\rm Model_2})$, subject to the assumptions described below. The Bayes factor is the ratio of marginal likelihoods for each model, each of which is given from $$P({\rm data}|{\rm Model_j}) = \displaystyle\int_{\Theta_j} L(\Theta_j|{\rm data}) \times P(\Theta_j|{\rm Model_j}) d\Theta_j$$ where $\Theta_j$ represent the vector of parameters for each model $j$, $L(\Theta_j|{\rm data})$ is the likelihood and $P(\Theta_j|{\rm Model_j})$ is the joint posterior distribution of the parameters. As described in Section \[sec:analysis\] above and in [@Pont2009b], the MCMC process produces the joint posterior distribution for the parameters, and we also obtain a maximum likelihood $L_{\rm max}$, corresponding to the smallest value of $\chi^2$ (as given in Section \[sec:sigmar\]) for each model. We then use the Bayesian Information Criterion [@Liddle2007] as given by, $${\rm BIC}=-2\ln L_{\rm max} +k \ln N, \label{eqn:bic}$$ where $N$ is the number of measurements, $k$ is the number of parameters in the model used. This simplifies the expression for the marginal likelihood by performing the integration using Laplace’s method and assumes a flat prior. If we replace $L_{\rm max}$ with the expression given by ${\rm P}({\rm\bf D}|{\bf \theta},I)$ in equation \[eqn:likelihood\] above, $${\rm BIC}=\chi_{\rm min}^2 +k \ln N + \ln \left((2\pi)^N|{\bf C}|\right), \label{eqn:bic2}$$ where $\chi_{\rm min}^2$ is the minimum value of $\chi^2$ achieved by the model, $N$ is the number of measurements, $k$ is the number of parameters in the model, and $|{\bf C}|$ is the determinant of the correlation matrix given in Section \[sec:sigmar\] above. The radial velocity data for a Keplerian orbit involves 6 free parameters: the period $P$, a reference time such as the mid-transit time $T_{\rm tr}$, a semi-amplitude $K$, a mean velocity offset $V_0$, the argument of periastron $\omega$ and the eccentricity $e$. Following @Ford2006, we use the two projected compoments $e\cos\omega$ and $e\sin\omega$ instead of $e$ and $\omega$, to improve the efficiency of the MCMC exploration. In this study, we use the period $P$, mid-transit time $T_{\rm tr}$ and their corresponding uncertainties as [*a priori*]{} information. We thus count them as two additional measurements in the calculation of the BIC, while the number of free parameters in each model (circular or eccentric) is now decreased by two. In this case, a circular model would have 2 free parameters ($V_0$ and $K$), while an eccentric model would have 4 free parameters ($V_0$, $K$, $e$, and $\omega$). The term $k\ln N$ thus penalises a model with a larger number of parameters (for example, an eccentric orbit), and we seek the model with smallest BIC. For each object, we repeated the MCMC analysis using the optimal value for $\sigma_r$ for a circular orbit and an eccentric orbit separately, at $\tau=1.5$ d, unless otherwise noted. We call these two families. For each family, we performed a fit with a circular model and an eccentric model. In most cases, the two families agreed on a circular model (indicated by “C” in Table \[tab:results\]) or an eccentric model (indicated by “E” if $e>0.1$ in Table \[tab:results\]), indicating this with a smaller BIC$_c$ or a smaller BIC$_e$ respectively. If the two families favoured a circular (or eccentric) orbit, we give the parameters from the family using an optimal value of $\sigma_r$ for the circular (or eccentric) orbit. In a number of such cases, however, the upper limits on the orbital eccentricity were larger than $e=0.1$. We labelled these eccentricities as “poorly constrained” (indicated by “P” in Table \[tab:results\]). In a few cases, the small number of measurements or the quality of measurements (e.g. for faint targets, or low mass planets) meant the two families disagreed: the family using the optimal value of $\sigma_r$ for a circular orbit gave a smaller value of BIC$_c$, favouring the circular orbit and the family using the optimal value of $\sigma_r$ for an eccentric orbit gave a smaller value of BIC$_e$, favouring the eccentric orbit. We labelled these cases “poorly constrained” as well. Results {#sec:results} ======= The results of this study are shown in Table \[tab:results\]. We place constraints on the eccentricities of transiting planets for which enough data is available. We analysed radial velocity data for 54 systems. For 8 systems, we used our new radial velocity data (described in Section \[sec:observations\]) in addition to existing RVs from the literature. For the other 46 systems, we reanalysed existing RVs from the literature. In Section \[sec:noteccentric\], we describe the planets for which we do not consider the evidence for an orbital eccentricity compelling, despite previous evidence of a departure from circularity ($e>1\sigma$ from zero), followed by Section \[sec:circular\], where we describe the planets for which we consider the orbital eccentricity to be either so small as to be undetectable or compatible with zero. In Section \[sec:eccentric\] we describe planets that can be safely considered to be on eccentric orbits and finally, in Section \[sec:unknown\], we describe the planets for which we consider the orbital eccentricity to be poorly constrained (as described in Section \[sec:modelselection\]). In the following Sections we also include a discussion of the evidence for eccentricity for 26 other systems from the literature. Planets with orbits that no longer qualify as eccentric according to this study {#sec:noteccentric} ------------------------------------------------------------------------------- In a number of cases in the past, the derived eccentricity from an MCMC analysis deviated from zero by more than 1$\sigma$, for example CoRoT-5b, GJ436b, WASP-5b, WASP-6b, WASP-10b, WASP-12b, WASP-14b, WASP-17b and WASP-18b. In this Section, we discuss the cases of 7 planets, CoRoT-5b, WASP-6b, WASP-10b, WASP-12b, WASP-17b, WASP-18b and WASP-5b, that are shown to have orbital eccentricities that are compatible with zero.  \ [**CoRoT-5**]{}\ CoRoT-5b is a $0.46$ M$_j$ planet on a $4.03$ day orbit around a F9 star (V=14.0), first reported by [@Rauer2009]. Using $6$ SOPHIE measurements (one of which is during the spectroscopic transit, which we ignore in this study) and $13$ HARPS measurements, the authors derived a value of eccentricity $e=0.09^{+0.09}_{-0.04}$. In our study, we used the formal uncertainties quoted with the data without any additional noise treatment, since they resulted in a reduced $\chi^2$ less than unity for both an eccentric and a circular orbit. We imposed the prior information from photometry $P=4.0378962(19)$ and $T_{tr}=2454400.19885(2)$ from the [@Rauer2009] and obtained a value of $\chi^2_c=15.97$ for the circular orbit and a value of $\chi^2_e=13.50$ for the eccentric orbit ($e=0.086^{+0.086}_{-0.054}$, $e<0.26$). Using $N=20$, $k=3$ and $k=5$ for the circular (two datasets, each with one $V_0$ and a single $K$) and eccentric orbits respectively, we obtained BIC$_{c}=151.05$ and BIC$_{e}=154.57$. A smaller value of BIC$_{c}$ means the circular orbit cannot be excluded.  \ [**WASP-6**]{}\ WASP-6b is a $0.50$ M$_j$ planet on a $3.36$ day orbit around a G8 star (V=11.9), first reported by [@Gillon2009b]. Using $35$ CORALIE measurements and $44$ HARPS measurements (38 of which occur near or during a spectroscopic transit, which we ignore in this study), the authors derived a value of eccentricity $e=0.054^{+0.018}_{-0.015}$. In our study, we used the 35 CORALIE measurements and the $6$ HARPS measurements that were not taken in the single night where the spectroscopic transit was observed. We used $\sigma_r=0$ [m s$^{-1}$]{}  for CORALIE (the data produces a reduced $\chi^2=0.89$ when fitted with a circular orbit, indicating overfitting) but for HARPS we used $\tau=1.5$ d and $\sigma_r=4.15$ [m s$^{-1}$]{} to obtain a reduced $\chi^2$ of unity for the circular orbit. We obtained a value of $\chi^2_c=38.09$ for the circular orbit and a value of $\chi^2_e=33.58$ for the eccentric orbit ($e=0.041\pm0.019$, $e<0.075$). Using $N=43$ (41 RVs and two constraints from photometry), $k=3$ and $k=5$ for the circular (two datasets, each with one $V_0$ and a single $K$) and eccentric orbits respectively, we obtained BIC$_{c}=333.25$ and BIC$_{e}=336.27$. We repeated the calculations, using $\sigma_r=0$ for CORALIE (the data produces a reduced $\chi^2=0.85$ when fitted with an eccentric orbit, indicating overfitting) but for HARPS we used $\tau=1.5$ d and $\sigma_r=3.59$ [m s$^{-1}$]{} to obtain a reduced $\chi^2$ of unity for the eccentric orbit. We obtained a value of $\chi^2_c=39.20$ for the circular orbit and a value of $\chi^2_e=34.47$ for the eccentric orbit ($e=0.043\pm0.019$, $e<0.075$). Using $N=43$, $k=3$ and $k=5$ for the circular and eccentric orbits respectively, we obtained BIC$_{c}=333.60$ and BIC$_{e}=336.39$. We therefore find that the circular orbital solution cannot be excluded, but the possibility that $e>0.1$ is rejected.  \ [**WASP-10**]{}\ WASP-10b is a $2.96$ M$_j$ planet on a $3.09$ day orbit around a K5 star (V=12.7), first reported by [@Christian2009]. Using $7$ SOPHIE measurements and $7$ FIES measurements, the authors derived a value of eccentricity $e=0.059^{+0.014}_{-0.004}$. The FIES data yielded a reduced $\chi^2$ less than unity with both eccentric and circular orbits, indicating overfitting, so we set $\sigma_r=0$ [m s$^{-1}$]{}. For the SOPHIE data, used $\tau=1.5$ d, $\sigma_r=54.5$ [m s$^{-1}$]{} to obtain a reduced $\chi^2$ of unity for the circular orbit. We reanalysed all the radial velocity measurements, and applied the prior from photometry $P=3.0927636(200)$ and $T_{tr}=2454357.8581(4)$ from [@Christian2009]. We obtained a value of $\chi^2_c=13.49$ for the circular orbit and a value of $\chi^2_e=7.47$ for the eccentric orbit ($e=0.049\pm 0.022$, less significant than the original claim). Using $14$ measurements and two priors from photometry ($N=16$), $k=3$ and $k=5$ for the circular (two datasets, each with one $V_0$ and a single $K$) and eccentric orbits respectively, we obtained BIC$_{c}=151.50$ and BIC$_{e}=151.01$. This now appears to show only a marginal support for an eccentric orbit. We plotted the SOPHIE radial velocity data against time, as shown in Figure \[fig:wasp10\] and overplotted a circular orbit as well as an eccentric orbit. Due to the long time between the first two measurements and the last five, we plot them in separate panels, shown on the left and right respectively. It is clear that the first measurement is pulling the eccentricity upwards, and we suspect from experience that the long term drifts in the SOPHIE zero point in HE mode for faint targets could have affected the first two measurements. We therefore repeated our calculations using only the last five measurements from the SOPHIE dataset and the whole FIES dataset, and set $\sigma_r=45.5$ [m s$^{-1}$]{} for SOPHIE. This time, we obtained a value of $\chi^2_c=11.81$ for the circular orbit and a value of $\chi^2_e=7.64$ for the eccentric orbit ($e=0.043\pm 0.035$). Using $12$ measurements and two priors from photometry ($N=14$), $k=3$ and $k=5$ for the circular (two datasets, each with one $V_0$ and a single $K$) and eccentric orbits respectively, we obtained BIC$_{c}=128.71$ and BIC$_{e}=129.83$, this time favouring the circular orbit. We repeated this calculation, and set $\sigma_r=0$ [m s$^{-1}$]{} for both SOPHIE and FIES, as each dataset gave a reduced $\chi^2$ of less than unity for the eccentric orbit (SOPHIE reduced $\chi^2=0.64$, FIES reduced $\chi^2=0.45$). This time, we obtained a value of $\chi^2_c=19.30$ for the circular orbit and a value of $\chi^2_e=10.65$ for the eccentric orbit ($e=0.080\pm 0.055$). Using $12$ measurements and two priors from photometry ($N=14$), $k=3$ and $k=5$ for the circular (two datasets, each with one $V_0$ and a single $K$) and eccentric orbits respectively, we obtained BIC$_{c}=128.33$ and BIC$_{e}=124.97$, this time favouring the eccentric orbit once again. It is therefore unclear to us whether or not the orbital eccentricity is non-zero as claimed in [@Christian2009]. @Maciejewski2011b, used transit timing variation analysis and reanalysed the radial velocity data, to obtain an eccentricity that is indistinguishable from zero ($e=0.013\pm0.063$). They argued instead that the original detection of an eccentricity had been influenced by starspots. The difference between our value of eccentricity and that derived by @Maciejewski2011b is probably due to the fact that the latter used a two planet model, which can reduce the derived eccentricity further — sparse sampling of the radial velocity from a two planet system can lead to an overestimated eccentricity.  \ [**WASP- 12**]{}\ WASP- 12b is a $1.41$ M$_j$ planet on a $1.09$ day orbit around a F9 star (V=11.7), first reported by [@Hebb2009]. Using SOPHIE measurements, the original authors derived a value of eccentricity $e=0.049\pm0.015$. @Husnoo2011 used new SOPHIE radial velocity measurements, as well as the original transit photometry from @Hebb2009 and the secondary eclipse photometry from @Campo2011 to suggest that the eccentricity was in fact compatible with zero ($e= 0.017^{+0.015}_{-0.010}$).  \ [**WASP-17**]{}\ WASP-17b is a $0.50$ M$_j$ planet on a $3.74$ day orbit around a F6 star (V=11.6), first reported by [@Anderson2010]. Using $41$ CORALIE measurements (three of which are during the spectroscopic transit, which we ignore in this study) and $3$ HARPS measurements, the authors considered three cases: first imposing a prior on the mass $M_*$ of the host star, secondly imposing a main-sequence prior on the stellar parameters and thirdly with a circular orbit. They derived values of eccentricity $e=0.129^{+0.106}_{-0.068}$ and $e=0.237^{+0.068}_{-0.069}$ for the first two cases respectively. We set $\sigma_r=0$ for both HARPS and CORALIE since we obtained a reduced $\chi^2$ of slightly less than unity for both eccentric and circular orbits for either dataset alone, indicating overfitting. We obtained a value of $\chi^2_c=37.98$ for the circular orbit and a value of $\chi^2_e= 35.94$ for the eccentric orbit. Using $41$ measurements and two priors from photometry ($N=43$), $k=3$ and $k=5$ for the circular (two datasets, each with one $V_0$ and a single $K$) and eccentric orbits respectively, we obtained BIC$_{c}=399.31$ and BIC$_{e}=404.80$. We thus find that the circular orbit cannot be excluded, agreeing with the third case ($e=0$, fixed) considered in [@Anderson2010] and rejecting the two derived values of eccentricity in that paper.  \ [**WASP-18**]{}\ WASP-18b is a $10.3$ M$_j$ planet on a $0.94$ day orbit around a F6 star (V=9.3), first reported by [@Hellier2009a]. Using $9$ CORALIE measurements (we drop the third measurement in our final analysis, since it produces a 5-$\sigma$ residual that is not improved by an eccentric orbit, suggesting that it is a genuine outlier), the authors derived a value of eccentricity $e=0.0092\pm0.0028$. In our study, we set $\tau=1.5$ d and $\sigma_r=20.15$ [m s$^{-1}$]{} to obtain a reduced $\chi^2$ of unity for the circular orbit. We obtained a value of $\chi^2_c=8.17$ for the circular orbit and a value of $\chi^2_e=6.64$ for the eccentric orbit ($e=0.007\pm0.005$, $e<0.018$). Using $N=10$, $k=2$ and $k=4$ for the circular (one dataset, with one $V_0$ and a single $K$) and eccentric orbits respectively, we obtained BIC$_{c}=75.34$ and BIC$_{e}=78.41$. We repeated the calculations using $\sigma_r=22.5$ [m s$^{-1}$]{} to obtain a reduced $\chi^2$ of unity for the eccentric orbit. We obtained a value of $\chi^2_c=7.14$ for the circular orbit and a value of $\chi^2_e=6.00$ for the eccentric orbit ($e=0.008\pm0.005$, $e<0.019$). Using $N=10$, $k=2$ and $k=4$ for the circular (one dataset, with one $V_0$ and a single $K$) and eccentric orbits respectively, we obtained BIC$_{c}=75.50$ and BIC$_{e}=78.97$. We thus find that the circular orbit cannot be excluded, in contrast to [@Hellier2009a]. The possibility that $e>0.1$ is excluded.  \ [**WASP-5 (new HARPS data) \[sec:wasp5\]**]{}\ WASP-5b is a 1.6 M$_j$ planet on a $1.63$ day orbit around a G4 star (V=12.3), first reported by @Anderson2008. @Gillon2009 used z-band transit photometry from the VLT to refine the eccentricity to $e=0.038^{+0.026}_{-0.018}$, and the authors made a tentative claim for the detection of a small eccentricity. We analysed our 11 new HARPS measurements for WASP-5 and the 11 CORALIE RVs from @Anderson2008 using the photometric constraints on the orbital period $P=1.6284246(13)$ and mid-transit time $T_{tr}=2454375.624956(24)$ from @Southworth2009b. We use $\tau=1.5$ d, $\sigma_r=10.6$ [m s$^{-1}$]{} for HARPS and $\sigma_r=4.3$ [m s$^{-1}$]{} for CORALIE to obtain a value of reduced $\chi^2$ of unity for the circular orbit for each dataset separately. We ran the MCMC twice: the first time fitting for the systemic velocity $v_0$ and semi-amplitude $K$, and the second time adding two parameters $e\cos\omega$ and $e\sin\omega$ to allow for an eccentric orbit. The best fit result is shown in Figure \[fig:wasp5\_circ\_noline\]. The residuals for a circular orbit are plotted, and a signal is clearly present in the residuals. The value of $\chi^2$ for the circular orbit is $24.36$ and that for an eccentric orbit is $20.57$. This results in a value of BIC$_c=169.40$ for the circular orbit and BIC$_e=171.97$ for the eccentric orbit, given 22 measurements, 2 constraints from photometry and 3 and 5 free parameters respectively for each model. We repeated the above analysis using $\tau=1.5$ d, $\sigma_r=9.4$ [m s$^{-1}$]{} for the HARPS dataset to obtain a value of reduced $\chi^2$ of unity for the eccentric orbit and $\sigma_r=0$ [m s$^{-1}$]{} for CORALIE (which resulted in a reduced $\chi^2$ of 0.58). This time, we obtained a value of $\chi^2$ for the circular orbit is $27.35$ and that for an eccentric orbit is $23.00$. This leads to a value of BIC$_c=170.37$ for the circular orbit and BIC$_e=172.38$ for the eccentric orbit. Once again, the circular orbit is favoured. A keplerian model, circular or eccentric ($e=0.012\pm0.007$) does not account for the scatter in the data the HARPS dataset as shown in Figure \[fig:wasp5\_circ\_noline\]. We have therefore plotted the radial velocity measurements, the bisector span, the signal to noise at order 49, the contrast and full width at half maximum for the cross-correlation function against the same time axis. The trend in radial velocity residuals can be seen to be correlated with both the bisector span and the full width at half maximum of the cross correlation function. This suggests a line shape change that’s related to either weather effects or instrumental systematics. The timescale of this variation is compatible with both scenarios. The bisector inverse span is generally directly correlated with the residuals, which weighs against a scenario involving stellar activity, but this is not so clear for the first three measurements — the drift could be due to stellar activity or an additional planetary or stellar companion. We extended the model with a linear acceleration of the form $$v(t) = v_{\rm keplerian}(t) + \dot{\gamma}(t-t_0), \label{eqn:lintrend}$$ and fitted the HARPS data alone using $t_0=2454768$ (to allow the MCMC to explore values of $\dot{\gamma}$ more efficiently) and reran the MCMC twice: once for a circular orbit and once for an eccentric orbit. Firstly, we used $\sigma_r=10.6$ [m s$^{-1}$]{} for the HARPS dataset, and the linear trend for a circular orbit resulted in $\dot{\gamma}=-2.6\pm2.9$ [m s$^{-1}$]{} yr$^{-1}$ and that for an eccentric orbit is $\dot{\gamma}=-2.0\pm2.9$ [m s$^{-1}$]{} yr$^{-1}$. The best fit result is shown in Figure \[fig:wasp5\_circ\_line\] and the residuals for a circular orbit are plotted in the bottom panel. The value of $\chi^2$ for the circular orbit is $10.24$ and that for an eccentric orbit is $7.70$. This results in a value of BIC$_{c, lin}=71.91$ for the circular orbit and BIC$_{e, lin}=74.49$ for the eccentric orbit, given 11 (N=13) measurements, 2 constraints from photometry and 3 and 5 free parameters respectively for each model. We repeated these calculations using $\sigma_r=9.4$ [m s$^{-1}$]{} for the HARPS dataset, and the linear trend for a circular orbit resulted in $\dot{\gamma}=-3.7\pm1.3$ [m s$^{-1}$]{} yr$^{-1}$ and that for an eccentric orbit is $\dot{\gamma}=-3.3\pm1.3$ [m s$^{-1}$]{} yr$^{-1}$. The value of $\chi^2$ for the circular orbit is $15.46$ and that for an eccentric orbit is $13.50$. This leads to a value of BIC$_{c, lin}=69.72$ for the circular orbit and BIC$_{e, lin}=72.89$ for the eccentric orbit. The circular orbit is not excluded, and the possibility that $e>0.1$ is excluded. The results for both models, one including the linear trend but excluding the CORALIE data, and one including the CORALIE data but excluding the linear trend are shown in Table \[tab:wasp5\]. In both cases, we give results for the case where $\sigma_r$ is chosen to yield a reduced $\chi^2$ of unity for the circular orbit. We attempted to repeat this using both the CORALIE and HARPS datasets, but we were unable to obtain a fit with the MCMC, because of the long time scale between the two datasets. -------------------------------------------------- ------------------- ----------------------------- ---------------------------------- Parameter @Anderson2008 HARPS only, [*this work*]{} HARPS & CORALIE, [*this work*]{} (with linear trend) (no linear trend) Centre-of-mass velocity $V_0$ \[[m s$^{-1}$]{}\] 20010.5$\pm$3.4 20018$\pm$12 20009.9$\pm$7.4 (HARPS) Orbital eccentricity $e$ 0 (adopted) $0.013\pm0.008$ ($<0.029$) $0.012\pm 0.007$ ($<0.026$) Argument of periastron $\omega$ \[$^o$\] 0 (unconstrained) 0 (unconstrained) 0 (unconstrained) $e\cos\omega$ – 0.002$\pm$0.003 0.003$\pm$0.003 $e\sin\omega$ – 0.012$\pm$0.010 0.011$\pm$0.009 Velocity semi-amplitude K \[m$\,$s$^{-1}$\] 277.8$\pm$7.8 266.4$\pm$1.3 266.9$\pm$1.3 -------------------------------------------------- ------------------- ----------------------------- ---------------------------------- Planets on circular orbits {#sec:circular} -------------------------- We establish that 20 planets have orbital eccentricities compatible with zero and the 95% upper limits are smaller than $e_{95}=0.1$. In this Section, we describe the planets WASP-4b, HAT-P-7b, TrES-2 and WASP-2b, for which we introduce new RVs. We also establish that the 95% upper limits on the eccentricities of WASP-5b, WASP-12b and WASP-18b, which have been described in Section \[sec:noteccentric\] above. In addition, we give the 95% upper limits on the eccentricities of CoRoT-1b, CoRoT-3b, HAT-P-8b, WASP-3b, WASP-16b, WASP-19b, WASP-22b, WASP-26b and XO-5b in Table \[tab:results\]. We discuss the evidence for circular orbits for HAT-P-13b, HD189733b, HD209458b and Kepler-5b at the end of this section.  \ [**WASP-4 (new HARPS data) \[sec:wasp4\]**]{}\ WASP-4b is a 1.2 M$_j$ planet on a $1.34$ day orbit around a G7 star (V=12.5), first reported by @Wilson2008. We analysed our 14 new HARPS measurements and the 14 CORALIE measurements from @Wilson2008 for WASP-4 and used the photometric constraints on the orbital period $P=1.33823214(71)$ and mid-transit time $T_{tr}=2454697.797562(43)$ from @Winn2009a. We estimate $\tau=1.5$ d, $\sigma_r=11$ [m s$^{-1}$]{} for the HARPS dataset and $\sigma_r=4.5$ [m s$^{-1}$]{} for the CORALIE dataset to obtain a reduced $\chi^2$ of unity for a circular orbit for each dataset separately. We ran the MCMC twice: the first time fitting for the systemic velocity $v_0$ and semi-amplitude $K$ only, ie. a circular orbit ($k=2$), and the second time adding two parameters $e\cos\omega$ and $e\sin\omega$ to allow for an eccentric orbit ($k=4$). The best fit result is shown in Figure \[fig:wasp4\_circ\_noline\]. The residuals for a circular orbit are plotted, and a signal is clearly present in the residuals. The value of $\chi^2$ for the circular orbit is $27.13$ and that for an eccentric orbit is $24.32$. This leads to a value of BIC$_c=208.62$ for the circular orbit and BIC$_e=212.55$ for the eccentric orbit, given 14 measurements, 2 constraints from photometry and 2 and 4 free parameters respectively for each model. We repeated the calculations, estimating $\tau=1.5$ d, $\sigma_r=10.1$ [m s$^{-1}$]{} for the HARPS dataset and $\sigma_r=7.1$ [m s$^{-1}$]{} for the CORALIE dataset to obtain a reduced $\chi^2$ of unity for an eccentric orbit for each dataset separately. The value of $\chi^2$ for the circular orbit is $27.29$ and that for an eccentric orbit is $24.33$. This leads to a value of BIC$_c=208.72$ for the circular orbit and BIC$_e=212.51$. Note the trend that is apparent in the residuals in Figure \[fig:wasp4\_circ\_noline\]. We have therefore plotted the radial velocity measurements, the bisector span, the signal to noise at order 49, the contrast and full width at half maximum for the cross-correlation function against the same time axis. For most measurements, the trend in radial velocity residuals can be seen to be correlated with both the bisector span and the full width at half maximum of the cross correlation function. This suggests a line shape change that’s related to either stellar activity, weather effects or instrumental systematics. The timescale of this variation is compatible with all three scenarios. We repeated the calculations for the HARPS dataset alone, and added a linear component to the radial velocity model in the same way we did for WASP-5 in Section \[sec:noteccentric\] and we set $t_0=2454762$ (to allow the MCMC to explore values of $\dot{\gamma}$ more efficiently) and reran the MCMC twice: once for a circular orbit and once for an eccentric orbit. We set $\tau=1.5$ d and $\sigma_r=11$ [m s$^{-1}$]{} for the HARPS dataset. The best fit result is shown in Figure \[fig:wasp4\_circ\_line\]. The residuals for a circular orbit are plotted, and a signal is clearly present in the residuals. The linear trend for a circular orbit results in $\dot{\gamma}=1023\pm490$ [m s$^{-1}$]{} yr$^{-1}$ and that for an eccentric orbit is $\dot{\gamma}=919\pm500$ [m s$^{-1}$]{} yr$^{-1}$. The value of $\chi^2$ for the circular orbit is $9.83$ and that for an eccentric orbit is $7.51$. This leads to a value of BIC$_c=92.02$ for the circular orbit and BIC$_e=95.26$ for the eccentric orbit, given 14 measurements, 2 constraints from photometry and 3 and 5 free parameters respectively for each model. We repeated the calculations, setting $\tau=1.5$ d, $\sigma_r=10.05$ [m s$^{-1}$]{} for the HARPS dataset. The value of $\chi^2$ for the circular orbit is $10.30$ and that for an eccentric orbit is $8.07$. This leads to a value of BIC$_c=91.18$ for the circular orbit and BIC$_e=94.49$. In all cases, the circular orbit is not excluded. The results for both models, one including the linear trend but excluding the CORALIE data, and one including the CORALIE data but excluding the linear trend are shown in Table \[tab:wasp4\]. In both cases, we give results for the case where $\sigma_r$ is chosen to yield a reduced $\chi^2$ of unity for the circular orbit. We reject the possibility that $e>0.1$. -------------------------------------------------- ------------------- ----------------------------- ---------------------------------- Parameter @Wilson2008 HARPS only, [*this work*]{} HARPS & CORALIE, [*this work*]{} (with linear trend) (no linear trend) Centre-of-mass velocity $V_0$ \[[m s$^{-1}$]{}\] 57733$\pm$2 57773$\pm$10 57790.8$\pm$5.7 Orbital eccentricity $e$ 0 (adopted) $0.004\pm0.003$ ($<$0.011) $0.005\pm0.003$ ($<$0.011) Argument of periastron $\omega$ \[$^o$\] 0 (unconstrained) 0 (unconstrained) 0 (unconstrained) $e\cos\omega$ – 0.004$\pm$0.003 0.003$\pm$0.003 $e\sin\omega$ – $-$0.002$\pm$0.004 $-$0.004$\pm$0.004 Velocity semi-amplitude K \[m$\,$s$^{-1}$\] 240$\pm$10 233.1$\pm$2.1 233.7$\pm$2.0 -------------------------------------------------- ------------------- ----------------------------- ----------------------------------  \ [**HAT-P-7 (new SOPHIE data)**]{}\ HAT-P-7b is a $1.8$ M$_j$ planet on a $2.20$ day orbit around an F6 star (V=10.5), first reported by [@Pal2008]. We use 13 new SOPHIE radial velocity measurements and 16 out of the 17 HIRES measurements in @Winn2009b (we drop one in-transit measurement) to work out the orbital parameters of HAT-P-7b. We impose the period $P=2.204733(10)$ d as given from photometry in @Welsh2010 and mid-transit time $T_{tr}=2454731.67929(43)$ BJD as given from photometry in @Winn2009b. We set $\tau=1.5$ d, $\sigma_r=9.41$ [m s$^{-1}$]{} for HIRES and $\sigma_r=12.9$ [m s$^{-1}$]{} for SOPHIE to obtain a reduced $\chi^2$ of unity for the best-fit circular orbit for each dataset separately. We used $29$ measurements in all, and count the two constraints from photometry as two additional data points to obtain $N=31$, and used $k=4$ for the circular orbit (two $V_0$, one for each dataset, the semi-amplitude $K$ and a constant drift term $\dot{\gamma}$, since @Winn2009b found evidence for a distant companion in the system and we set $t_0=2454342$). We repeated this analysis with an eccentric orbit $k=6$ (4 degrees of freedom for the circular orbit with two datasets and a linear acceleration, and 2 additional degrees of freedom for the eccentricity, $e\cos\omega$ and $e\sin\omega$). The orbital parameters are given in Table \[tab:hatp7\], and the radial velocity dataset is plotted in Figure \[fig:hatp7sophie\], with residuals shown for a circular orbit. The Figure also shows models of a circular and an eccentric orbit (with $e=0.014$), but they are almost undistinguishable. For the circular orbit, we obtained $\chi^2=26.94$, and a value of BIC$_c=222.81$ and for the eccentric orbit, we obtained $\chi^2=23.98$ and a value of BIC$_e=226.72$. We repeated the calculations and set $\tau=1.5$ d, $\sigma_r=8.2$ [m s$^{-1}$]{} for HIRES and $\sigma_r=8.2$ [m s$^{-1}$]{} for SOPHIE to obtain a reduced $\chi^2$ of unity for the best-fit eccentric orbit. For the circular orbit, we obtained $\chi^2=35.65$ and a value of BIC$_c=224.14$ and for the eccentric orbit, we obtained $\chi^2=31.89$ and a value of BIC$_e=227.25$. We therefore find that the circular orbit cannot be excluded for HAT-P-7b. Further, we exclude the possibility that $e>0.1$. Parameter HIRES, @Winn2009b HIRES+SOPHIE, [*this work*]{} ------------------------------------------------------------------------ ---------------------- ------------------------------------------------------- Centre-of-mass velocity $V_0$ \[[m s$^{-1}$]{}\] $-$51.2$\pm$3.6 $-$49.96$\pm$6.0 (HIRES) and $-$10510$\pm$10 (SOPHIE) Orbital eccentricity $e$ $e_{99\%}<$0.039 0.014$\pm0.010$ ($e<0.038$) Argument of periastron $\omega$ \[$^o$\] – 0 (unconstrained) $e\cos\omega$ $-$0.0019$\pm$0.0077 $-$0.007$\pm$0.004 $e\sin\omega$ 0.0037$\pm$0.0124 $-$0.011$\pm$0.015 Velocity semi-amplitude $K$ \[m$\,$s$^{-1}$\] 211.8$\pm$2.6 213.8$\pm$1.2 Constant radial acceleration $\dot{\gamma}$ \[m$\,$s$^{-1}$yr$^{-1}$\] 21.5$\pm$2.6 21.1$\pm$4.2  \ [**TrES-2 (new SOPHIE data)**]{}\ TrES-2b is a $1.3$ M$_j$ planet on a $2.47$ day orbit around a G0 star (V=11.4), first reported by [@ODonovan2006]. We use 10 new SOPHIE radial velocity measurements and the 11 HIRES measurements in [@ODonovan2006] to work out the orbital parameters of TrES-2b. We impose the period $P=2.470614(1)$ d and mid-transit time $T_{tr}=2453957.63492(13)$ BJD as given from photometry in @Raetz2009. We set $\tau=1.5$ d and $\sigma_r=6.8$ [m s$^{-1}$]{} for SOPHIE to obtain a reduced $\chi^2$ of unity for the best-fit circular orbit (using the SOPHIE data alone), and set $\sigma_r=0$[m s$^{-1}$]{} for the HIRES data since a circular orbit for that dataset alone yields a reduced $\chi^2$ of 0.72, indicating over-fitting. We used $21$ measurements in all, and count the two constraints from photometry as two additional datapoints to obtain $N=23$, and used $k=3$ for the circular orbit (two $V_0$, one for each dataset, and the semi-amplitude $K$). We repeated this analysis with an eccentric orbit $k=5$ (three degrees of freedom for the circular orbit, and two additional degrees of freedom for the eccentricity, $e\cos\omega$ and $e\sin\omega$). The orbital parameters are given in Table \[tab:tres2\], and the radial velocity dataset is plotted in Figure \[fig:tres2\], with residuals shown for a circular orbit. The Figure also shows models of a circular and an eccentric orbit (with $e=0.023$), but they are almost undistinguishable. For the circular orbit, we obtained $\chi^2=18.00$, yielding a value of BIC$_c=160.30$ and for the eccentric orbit, we obtained $\chi^2=15.91$ and a value of BIC$_e=164.48$. We repeated the calculations and set $\sigma_r=8.45$ [m s$^{-1}$]{} for SOPHIE to obtain a reduced $\chi^2$ of unity for the best-fit circular orbit (using the SOPHIE data alone), while we set $\sigma_r=0$[m s$^{-1}$]{} for the HIRES data since an eccentric orbit for that dataset alone yields a reduced $\chi^2$ of 0.56, indicating over-fitting. For a circular orbit, we obtained $\chi^2=15.97$, resulting in a value of BIC$_c=159.38$ and for an eccentric orbit, we obtained $\chi^2=13.88$ and a value of BIC$_e=163.56$. We therefore find that the circular orbit cannot be excluded for TrES-2b. Furthermore, we exclude the possibility that $e>0.1$. Parameter HIRES, @ODonovan2006 HIRES, SOPHIE, [*this work*]{} -------------------------------------------------- ---------------------- ---------------------------------------------------- Centre-of-mass velocity $V_0$ \[[m s$^{-1}$]{}\] – $-$29.8$\pm$2.4 (HIRES), $-$315.5$\pm$5.0 (SOPHIE) Orbital eccentricity $e$ 0 (adopted) 0.023$\pm$0.014, $e<0.051$ Argument of periastron $\omega$ \[$^o$\] 0 (unconstrained) 0 (unconstrained) $e\cos\omega$ – 0.002$\pm$0.009 $e\sin\omega$ – $-$0.022$\pm$0.016 Velocity semi-amplitude K \[m$\,$s$^{-1}$\] 181.3$\pm$2.6 181.1$\pm$2.5  \ [**WASP-2 (new HARPS data)**]{}\ WASP-2b is a $0.85$ M$_j$ planet on a $2.15$ day orbit around a K1 star (V=12), first reported by [@Cameron2007]. We use 8 new HARPS radial velocity measurements and 7 of the original 9 SOPHIE measurements (we drop the first measurement, which has an uncertainty about 15 times larger than the rest, and the fifth, which shows a 3-$\sigma$ deviation at a phase close to the transit) in [@Cameron2007] to work out the orbital parameters of WASP-2b. We impose the period $P=2.15222144(39)$ d and mid-transit time $T_{tr}=2453991.51455(17)$ BJD as given from photometry in @Southworth2010. We used $15$ measurements in all, and count the two constraints from photometry as two additional datapoints ($N=17$) and used $k=3$ for the circular orbit (two $V_0$, one for each dataset, and the semi-amplitude $K$). We estimated the timescale of correlated noise for both the HARPS and SOPHIE data to be $\tau=1.5$ d, and we estimated $\sigma_r=10.4$ [m s$^{-1}$]{} for the SOPHIE data and $\sigma_r=6.45$ [m s$^{-1}$]{} for the HARPS data to obtain a reduced $\chi^2$ of unity for the circular orbit. We repeated this analysis with an eccentric orbit $k=5$ (3 degrees of freedom for the circular orbit, and 2 additional degrees of freedom for the eccentricity, $e\cos\omega$ and $e\sin\omega$). The orbital parameters are given in Table \[tab:wasp2\], and the radial velocity dataset is plotted in Figure \[fig:wasp2ph\], with residuals shown for a circular orbit. The Figure also shows models of a circular and an eccentric orbit (with $e=0.027$), but they are almost undistinguishable. For the circular orbit, we obtained $\chi^2=15.60$, giving a value of BIC$_c=115.08$ and for the eccentric orbit, we obtained $\chi^2=13.88$ giving a value of BIC$_e=119.02$. We repeated these calculations to obtain a reduced $\chi^2$ of unity for the eccentric orbit and estimated $\sigma_r=10.4$ [m s$^{-1}$]{} for the SOPHIE data (the SOPHIE dataset did not allow the MCMC to converge and yield a reduced $\chi^2$ of unity with an eccentric orbit) and $\sigma_r=7.05$ [m s$^{-1}$]{} for the HARPS data. For the circular orbit, we obtained $\chi^2=15.16$, and a value of BIC$_c=115.47$ and for the eccentric orbit, we obtained $\chi^2=13.49$ and a value of BIC$_e=119.47$. We therefore find that the circular orbit cannot be excluded for WASP-2. Furthermore, we exclude the possibility that $e>0.1$. Parameter SOPHIE, @Cameron2007 SOPHIE and HARPS, [*this work*]{} -------------------------------------------------- ---------------------- --------------------------------------------------------- Centre-of-mass velocity $V_0$ \[[m s$^{-1}$]{}\] $-$27863$\pm$7 $-$27862$\pm$7.4 (SOPHIE), $-$27739.81$\pm$4.1 (HARPS), Orbital eccentricity $e$ 0 (adopted) 0.027$\pm0.023$ ($<$0.072) Argument of periastron $\omega$ \[$^o$\] 0 (unconstrained) 0 (unconstrained) $e\cos\omega$ – $-$0.003$\pm$0.003 $e\sin\omega$ – $-$0.027$\pm$0.027 Velocity semi-amplitude K \[m$\,$s$^{-1}$\] 155$\pm$7 156.3$\pm$2.1  \ [**Other planets**]{}\ HD189733b and HD209458b are both on orbits that are compatible with a circular model: @Laughlin2005 reported the 95% limits on eccentricity for HD 209458b ($e<0.042$) and we estimate the upper limit for HD189733b from @Triaud2009 assuming a Gaussian probability distribution, $e<0.008$). In both cases, the eccentricity is strongly constrained by the timing of the secondary eclipse. No radial velocity data was found for Kepler-5 in the literature or online, but we include the results of @Kipping2011 in this study: Kepler-5b has an eccentricity of $e=0.034_{-0.018}^{+0.029}$, with a 95% upper limit of $e<0.086 $. We therefore classify Kepler-5b as having a circular orbit. We also omitted an analysis of the two-planet system HAT-P-13, choosing to estimate the 95% limits on the orbital eccentricity of HAT-P-13b from the literature ($e<0.022$) and classify this orbit as circular. Planets on eccentric orbits {#sec:eccentric} --------------------------- In contrast to Section \[sec:noteccentric\], in this Section, we confirm the eccentricities of 10 planets. We verify the eccentricities of CoRoT-9b, GJ-436b and HAT-P-2b as a test for our procedures and we also confirm the eccentricities of HAT-P-16b and WASP-14b, with the former being the planet on a short period orbit with the smallest confirmed eccentricity, and the latter being the planet with the shortest period orbit having a confirmed eccentricity. Finally we note the confirmed orbital eccentricities of CoRoT-10b, HAT-P-15b, HD17156b, HD80606b and XO-3b.  \ [**CoRoT-9**]{}\ CoRoT-9b is a $0.84$ M$_j$ planet on a $95.3$ day orbit around a G3 star (V=13.5), first reported by [@Deeg2010], who found an eccentricity of $e=0.11\pm0.04$. We used the $14$ HARPS measurements from [@Deeg2010], setting $\tau=1.5$ d and $\sigma_r=3.7$ [m s$^{-1}$]{} to obtain a value of reduced $\chi^2$ of unity for the circular orbit. We imposed the prior information from photometry $P=95.2738(14)$ and $T_{tr}=2454603.3447(1)$ from [@Deeg2010] and obtained a value of $\chi^2_{c}=14.05$ and $\chi^2_{e}=7.90$. Using $N=16$, $k_c=2$ and $k_e=4$, we obtain BIC$_c=106.17$ and BIC$_e=105.57$, which provides marginal support for an eccentric orbit at $e=0.111\pm0.046$, with the 95% limit at $e<0.20$. We repeated the caculations, setting $\sigma_r=0$ [m s$^{-1}$]{} since this results in a reduced $\chi^2$ of less than unity for the eccentric orbit. This time, we obtained a value of $\chi^2_{c}=16.65$ and $\chi^2_{e}=9.63$. Using $N=16$, $k_c=2$ and $k_e=4$, we obtain BIC$_c=105.66$ and BIC$_e=104.18$, which supports an eccentric orbit at $e=0.111\pm0.039$.  \ [**GJ-436**]{}\ GJ-436b is a $0.071$ M$_j$ planet on a $2.64$ day eccentric orbit around a M2.5 star (V=10.7), first reported by [@Butler2004]. @Deming2007 detected the secondary eclipse using Spitzer, placing a constraint on the secondary eclipse phase $\phi_{\rm occ}=0.587\pm0.005$. This translates into $e\cos\omega=0.1367\pm0.0012$, which we apply as a Bayesian prior in the calculation of our merit function. We used the $59$ HIRES measurements from [@Maness2007], setting $\tau=1.5$ d and $\sigma_r=5.5$ [m s$^{-1}$]{} to obtain a value of reduced $\chi^2$ of unity for the circular orbit. We imposed the prior information from photometry $P=2.64385(9)$ from @Maness2007and $T_{tr}=2454280.78149(16)$ from @Deming2007 and obtained a value of $\chi^2_{c}=59.74$ and $\chi^2_{e}=38.86$. Using $N=61$ (59 measurements and 2 priors from photometry) and $k_c=2$ for the circular orbit, we obtain BIC$_c=371.38$. Using $N=62$ (59 measurements and 3 priors from photometry) and $k_e=3$ ($V_0$, $K$, $e\sin\omega$) for the eccentric orbit, we obtain BIC$_e=354.67$, which supports an eccentric orbit at $e=0.157\pm0.024$, with the 95% limit at $e<0.21$. We repeated the caculations, setting $\sigma_r=3.95$ [m s$^{-1}$]{} to obtain a reduced $\chi^2$ of unity for the eccentric orbit. This time, we obtained a value of $\chi^2_{c}=88.20$ and $\chi^2_{e}=59.39$. This time, we obtain BIC$_c=372.67$ and BIC$_e=348.03$, which supports an eccentric orbit at $e=0.153\pm0.017$, which is in agreement with @Deming2007, who reported $e=0.150\pm0.012$.  \ [**HAT-P-16**]{}\ HAT-P-16b is a $4.19$ M$_j$ planet on a $2.78$ day orbit around a F8 star (V=10.7), first reported by [@Buchhave2010]. The original authors found an eccentricity of $e=0.036\pm0.004$. We re-analysed the 7 high resolution FIES measurements, 14 medium resolution FIES measurements and 6 HIRES measurements, with two priors from photometry on the period and mid-transit time. We set $\tau=1.5$ d for all instruments and set $\sigma_r=115$, 185, and 28 [m s$^{-1}$]{} respectively for the three instruments to obtain a reduced $\chi^2$ of unity for each individually. We then analysed them together using both a circular ($\chi^2=28.83$) and an eccentric orbit ($\chi^2=3.81$). Using $N=29$, $k_c=4$ and $k_e=6$, we obtain BIC$_c=314.74$ and BIC$_e=296.45$, which supports an eccentric orbit at $e=0.034\pm0.010$. Figure \[fig:hatp16\] (left) shows the data from [@Buchhave2010], with a circular orbit overplotted with a solid line and an eccentric orbit with the dotted line. The residuals are plotted for the circular solution and they show a clear periodic signal. We repeated the analysis, this time setting $\sigma_r=0$ (reduced $\chi^2=0.62$, indicating over-fitting), 16 (reduced $\chi^2=33$), and 4.7 [m s$^{-1}$]{} respectively and separately for the three datasets (i.e. aiming for a reduced $\chi^2$ of unity for each dataset individually, with an eccentric orbit). We then analysed them together using both a circular ($\chi^2=347.86$) and an eccentric orbit ($\chi^2=44.62$). Using $N=29$, $k_c=4$ and $k_e=6$, we obtain BIC$_c=541.64$ and BIC$_e=245.14$, which supports an eccentric orbit at $e=0.034\pm0.003$. We thus confirm the eccentricity of HAT-P-16b, which means this is the planet with the smallest eccentricity that is reliably measured. This is in part helped by the fact that HAT-P-16b is a very massive planet, making the radial velocity signal for an eccentric orbit very clear. Figure \[fig:hatp16\] (right) shows the data from [@Buchhave2010] again, with an eccentric orbit overplotted with the dotted line.  \ [**WASP-14**]{}\ WASP-14b is a $7.3$ M$_j$ planet on a $2.24$ day orbit around a F5 star (V=9.8), first reported by [@Joshi2009], who found an eccentricity of $e=0.091\pm0.003$. @Husnoo2011 confirmed the eccentricity of the orbit and updated the precise value to $e=0.088\pm0.003$. This makes WASP-14b the planet that is closest to its host star but still has an eccentric orbit, taking the place from WASP-12b.  \ [**CoRoT-10, HAT-P-2, HAT-P-15, HD17156, HD80606 and XO-3**]{}\ The orbits of the planets CoRoT-10b ($e=0.110\pm0.039$), HAT-P-2b ($e=0.517\pm0.003$), HAT-P-15b ($e=0.190\pm0.019$), HD17156b ($e=0.677\pm0.003$), HD80606b ($e=0.934\pm0.001$) and XO-3b ($e=0.287\pm0.005$) are clearly eccentric from existing literature [See for example @Bonomo2010; @Loeillet2008; @Kovacs2010; @Nutzman2010; @Hebrard2010a; @Hebrard2008 respectively]. Planets with orbits that have poorly constrained eccentricities {#sec:unknown} --------------------------------------------------------------- For 26 of the transiting planets that we attempted to place upper limits on their eccentricities, we obtained limits that were larger than 0.1. We considered these eccentricities to be poorly determined. We discuss the cases of HAT-P-4b, WASP-7, XO-2b and Kepler-4b below.  \ [**HAT-P-4 (new SOPHIE data)**]{}\ HAT-P-4b is a $0.68$ M$_j$ planet on a $3.06$ day orbit around an F star (V=11.2), first reported by [@Kovacs2007]. We use 13 new SOPHIE radial velocity measurements and the 9 HIRES measurements in [@Kovacs2007] to work out the orbital parameters of HAT-P-4b. We impose the period $P=3.056536(57)$ d and mid-transit time $T_{tr}=2454248.8716(6)$ BJD as given from photometry in [@Kovacs2007]. We set $\tau=1.5$ d and $\sigma_r=3.35$ [m s$^{-1}$]{} for SOPHIE and $\sigma_r=3.75$[m s$^{-1}$]{} for HIRES, to obtain a reduced $\chi^2$ of unity for each dataset separately for the best-fit circular orbit. We used $22$ measurements in all, and count the two constraints from photometry as two additional datapoints ($N$=24), and used $k=3$ for the circular orbit (two $V_0$, one for each dataset, and the semi-amplitude $K$). We repeated this analysis with an eccentric orbit $k=5$ (three degrees of freedom for the circular orbit, and two additional degrees of freedom for the eccentricity, $e\cos\omega$ and $e\sin\omega$). The orbital parameters are given in Table \[tab:hatp4\], and the radial velocity dataset is plotted in Figure \[fig:hatp4sophie\], with residuals shown for a circular orbit. The Figure also shows models of a circular and an eccentric orbit (with $e=0.064$). For the circular orbit, we obtained $\chi^2=22.05$, giving a value of BIC$_c=161.96$ and for the eccentric orbit, we obtained $\chi^2=16.77$ giving a value of BIC$_e=163.04$. We repeated these calculations by setting $\tau=1.5d$, $\sigma_r=1.81$ [m s$^{-1}$]{} for HIRES, and kept $\sigma_r=3.35$ [m s$^{-1}$]{} for SOPHIE, since we were unable to determine a value of $\sigma_r$ that would allow the MCMC chain to converge and lead to a $\chi^2$ of unity for an eccentric orbit. This time, we obtained $\chi^2=25.88$ for the circular orbit, giving a value of BIC$_c=161.96$ and for the eccentric orbit, we obtained $\chi^2=20.05$ giving a value of BIC$_e=162.49$. We find that the circular orbit cannot be excluded for HAT-P-4b, but because the eccentricity is $e=0.064\pm0.028$ with an upper limit of $e<0.11$, which is above 0.1, we classify HAT-P-4b as having a poorly constrained eccentricity. Parameter HIRES, @Kovacs2007 HIRES+SOPHIE, [*this work*]{} -------------------------------------------------- -------------------- ---------------------------------------------------- Centre-of-mass velocity $V_0$ \[[m s$^{-1}$]{}\] 12.1$\pm$0.9 $20.3\pm 2.6$ (HIRES), $-$1402.0 $\pm$4.0 (SOPHIE) Orbital eccentricity $e$ 0 (adopted) 0.064$\pm$0.028, $e<0.11$ Argument of periastron $\omega$ \[$^o$\] 0 (unconstrained) 0 (unconstrained) $e\cos\omega$ – $-$0.018$\pm0.012$ $e\sin\omega$ – $-$0.061$\pm$0.027 Velocity semi-amplitude K \[m$\,$s$^{-1}$\] 81.1$\pm$1.9 81.3$\pm$2.6  \ [**WASP-7 (new HARPS data)**]{}\ WASP-7b is a 1.0 M$_j$ planet on a $4.95$ day orbit around a F5 star (V=9.5), first reported by @Hellier2009. We analysed our 11 new HARPS measurements for WASP-7 as well as $11$ measurements from @Hellier2009 using CORALIE, and used the photometric constraints on the orbital period $P=4.954658(55)$ and mid-transit time $T_{tr}=2453985.0149(12)$ from the same paper. For both instruments, we set $\tau=1.5$ d and for CORALIE, we set $\sigma_r=28.3$ [m s$^{-1}$]{} while for HARPS, we set $\sigma_r=210$ [m s$^{-1}$]{} in order to get a value of reduced $\chi^2$ equal to unity for the circular orbit. We performed the MCMC analysis twice: the first time fitting for the systemic velocity $v_0$ and semi-amplitude $K$, and the second time adding two parameters $e\cos\omega$ and $e\sin\omega$ to allow for an eccentric orbit. The best-fit parameters are given in Table \[tab:wasp7\]). We plot the radial velocity data against time (Figure \[fig:wasp7\_circ\], left) and phase (Figure \[fig:wasp7\_circ\], right). When the residuals for a circular orbit are plotted, and a scatter of about $30$ [m s$^{-1}$]{} is clearly seen, which is much larger than the median uncertainties of $\sigma=2.21$ [m s$^{-1}$]{} on the radial velocity measurements. This is similar to that found by @Hellier2009 from their CORALIE data. An eccentric orbit does not reduce the scatter. The value of $\chi^2$ for the circular orbit is $22.37$ and that for an eccentric orbit is $18.11$. This leads to a value of BIC$_c=250.62$ and BIC$_e=252.72$, respectively, for 22 measurements, 2 constraints from photometry and 3 and 5 free parameters respectively (Keplerian orbits, but with two $V_0$ to account for a possible offset between the two instruments). This shows that the circular orbit is still preferred, and an eccentric orbit does not explain the scatter. We repeated this using $\sigma_r=33.8$ [m s$^{-1}$]{} for CORALIE while for HARPS, we set $\sigma_r=158.5$ [m s$^{-1}$]{} in order to get a value of reduced $\chi^2$ equal to unity for the eccentric orbit. We performed the MCMC analysis both for a circular and eccentric orbit. The value of $\chi^2$ for the circular orbit is $146.95$ and that for an eccentric orbit is $146.86$. This leads to a value of BIC$_c=367.47$ and BIC$_e=373.91$, respectively, for 22 measurements, 2 constraints from photometry and 3 and 5 free parameters respectively (Keplerian orbits, but with two $V_0$ to account for a possible offset between the two instruments). This shows that the circular orbit is still preferred, and an eccentric orbit does not explain the scatter. WASP-7 is an F5V star, with a temperature of $T_{\rm eff}=6400\pm100$ K. Despite the result of the original paper that WASP-7 is not chromospherically active above the 0.02 mag level, @Lagrange2009 found evidence for other F5V stars showing radial velocity variability with a scatter at this level, for example HD 111998, HD 197692 or HD 205289, with scatters of $40$ [m s$^{-1}$]{} $30$ [m s$^{-1}$]{} and $29$ [m s$^{-1}$]{} respectively. Our derived value of eccentricity is $e=0.103\pm0.061$, with the 95% upper limit is at $e<0.25$. We therefore classify the eccentricity of the orbit of WASP-7b as poorly constrained In Figure \[fig:wasp7\_circ\], we have also plotted the bisector span, the signal to noise at order 49, the contrast and full width at half maximum for the cross-correlation function against the same time axis. The large scatter in radial velocity residuals can be seen to be correlated with both the bisector span and the full width at half maximum of the cross correlation function. Parameter @Hellier2009 HARPS, [*this work*]{} -------------------------------------------------- -------------------- --------------------------- Centre-of-mass velocity $V_0$ \[[m s$^{-1}$]{}\] $-$29850.6$\pm$1.7 $-$29455$\pm$103 Orbital eccentricity $e$ 0 (adopted) $0.103\pm0.061$ ($<$0.25) Argument of periastron $\omega$ \[$^o$\] 0 (unconstrained) 0 (unconstrained) $e\cos\omega$ – 0.021$\pm$0.068 $e\sin\omega$ – 0.101$\pm$0.074 Velocity semi-amplitude K \[m$\,$s$^{-1}$\] 97$\pm$13 96$\pm$14  \ [**XO-2 (new SOPHIE data)**]{}\ XO-2 is a $0.6$ M$_j$ planet on a $2.62$ day orbit around a K0 star (V=11.2), first reported by [@Burke2007]. We use 9 new SOPHIE radial velocity measurements and the 10 HJS measurements in [@Burke2007] to work out the orbital parameters of XO-2. We impose the period $P=2.6158640(21)$ d and mid-transit time $T_{tr}=2454466.88467 (17)$ BJD as given from photometry in @Fernandez2009. We set $\tau=1.5$ d and $\sigma_r=5.3$ [m s$^{-1}$]{} for SOPHIE and $\sigma_r=0$ [m s$^{-1}$]{} for HJS (because the HJS data alone, with a circular orbit, yield a reduced $\chi^2$ of 0.78, indicating overfitting) to obtain a reduced $\chi^2$ of unity for the best-fit circular orbit. We used $19$ measurements in all, and count the two constraints from photometry as two additional datapoints ($N=21$), and used $k=3$ for the circular orbit (two $V_0$, one for each dataset, and the semi-amplitude $K$). We repeated this analysis with an eccentric orbit $k=5$ (two degrees of freedom for the circular orbit, and two additional degrees of freedom for the eccentricity, $e\cos\omega$ and $e\sin\omega$). The orbital parameters are given in Table \[tab:xo2\], and the radial velocity dataset is plotted in Figure \[fig:xo2\], with residuals shown for a circular orbit. The Figure also shows models of a circular and an eccentric orbit (with $e=0.064$). For the circular orbit, we obtained $\chi^2=19.65$, giving a value of BIC$_c=165.55$ and for the eccentric orbit, we obtained $\chi^2=17.57$ giving a value of BIC$_e=169.55$. We repeated the calculations using $\sigma_r=7.05$ [m s$^{-1}$]{} for SOPHIE and $\sigma_r=0$ [m s$^{-1}$]{} for HJS (because the HJS data alone, with an eccentric orbit, yield a reduced $\chi^2$ of 0.56, indicating overfitting) to obtain a reduced $\chi^2$ of unity for the best-fit eccentric orbit. For the circular orbit, we obtained $\chi^2=18.02$, giving a value of BIC$_c=165.21$ and for the eccentric orbit, we obtained $\chi^2=16.01$ giving a value of BIC$_e=169.29$. In both cases, i.e. using the optimal value of $\sigma_r$ for a circular orbit and using the optimal value of $\sigma_r$ for an eccentric orbit, a circular orbit is favoured. The 95% upper limit is $e<0.14$, which is above 0.1, so we classify the orbital eccentricity of XO-2 as poorly constrained. Parameter HJS, @Burke2007 HJS, SOPHIE, [*this work*]{} -------------------------------------------------- ------------------- ----------------------------------------------- Centre-of-mass velocity $V_0$ \[[m s$^{-1}$]{}\] – $-1.3\pm 6.3$ (HJS), 46860.1$\pm$4.1 (SOPHIE) Orbital eccentricity $e$ 0 (adopted) 0.064$\pm$0.041 ($e<0.14$) Argument of periastron $\omega$ \[$^o$\] 0 (unconstrained) 0 (unconstrained) $e\cos\omega$ – 0.007$\pm$0.017 $e\sin\omega$ – $-0.063$$\pm$0.047 Velocity semi-amplitude K \[m$\,$s$^{-1}$\] 85$\pm$8 98.0$\pm$4.0  \ [**Kepler-4**]{}\ Kepler-4b has a derived eccentricity of $e=0.25_{-0.12}^{+0.11}$, with a 95% upper limit of $e<0.43$ [@Kipping2011], so we classify it as “poorly constrained eccentricity”.  \ [**Other objects**]{}\ For the 8 objects CoRoT-6, HAT-P-1, HAT-P-3, HAT-P-6, HD149026, Kepler-6, WASP-10 and WASP-21, we found the BIC$_e$ for an eccentric orbit was smaller than the BIC$_c$ for a circular orbit if we assume a $\sigma_r$ that yields a reduced $\chi^2$ of unity for an eccentric orbit, whereas the BIC$_c$ for a circular orbit was smaller than the BIC$_e$ for an eccentric orbit if we assume a $\sigma_r$ that yields a reduced $\chi^2$ of unity for a circular orbit. This suggests that the current RV datasets do not constrain the orbit enough for us to detect a finite eccentricity. We have already discussed the case of WASP-10b in Section \[sec:noteccentric\] above. Additional planetary systems\[add\_systems\] -------------------------------------------- In addition to the 64 planets considered so far, we now include 3 additional planets on eccentric orbits, 11 planets on orbits where $e>0.1$ is excluded at the 95% level and two brown dwarves. The additional planets on eccentric orbits are HAT-P-17b [@Howard2010a $e=0.346\pm 0.007$], HAT-P-21b [@Bakos2010a $e=0.228\pm0.016$] and HAT-P-31b [@Kipping2011b $e=0.245\pm0.005$]. The additional planets on orbits that are consistent with circular are:\ CoRoT-18b [$e<0.08$ at 3-$\sigma$, @Hebrard2011a],\ HAT-P-20b [$e<0.023$, estimated from @Bakos2010a],\ HAT-P-22b [$e<0.031$, estimated from @Bakos2010a],\ HAT-P-25b [$e<0.068$, estimated from @Quinn2010],\ HAT-P-30b [$e<0.074$, estimated from @Johnson2011],\ WASP-23b [$e<0.062$ at 3-$\sigma$, @Triaud2011],\ WASP-34b [$e<0.058$, estimated from @Smalley2011],\ WASP-43b [$e<0.04$ at 3-$\sigma$, @Hellier2011],\ WASP-45b [$e<0.095$, @Anderson2011],\ WASP-46b [$e<0.065$, @Anderson2011] and\ $\tau$ Boötis b [$e<0.045$, estimated from @Butler2006]. The two brown dwarves are OGLE-TR-122b [@Pont2005a $e=0.205\pm0.008$] and OGLE-TR-123b [@Pont2005 $e=0$]. In addition to the above, we also consider the case of WASP-38 [@Barros2011], which has an eccentricity of $e=0.031\pm0.005$, indicating it is in the process of circularisation, just like WASP-14 and HAT-P-16. ----------- -------------------------------------- ------- ------- ------ ----------- ---- ------- ------- Name Eccentricity E (literature) CoRoT-1b – 0.006 0.012 $(<$ $0.042)$ C 1.06 0.14 CoRoT-2b – 0.036 0.033 $(<$ $0.10)$ P 3.14 0.17 CoRoT-3b $0.008^{+0.015}_{-0.005}$ 0.012 0.01 $(<$ $0.039)$ C 21.61 1.2 CoRoT-4b $0\pm0.1$ 0.27 0.15 $(<$ $0.48)$ P 0.659 0.079 CoRoT-5b $0.09^{+0.09}_{-0.04}$ 0.086 0.07 $(<$ $0.26)$ P 0.488 0.032 CoRoT-6b $<0.1$ 0.18 0.12 $(<$ $0.41)$ P 2.92 0.30 CoRoT-9b 0.11$\pm$0.04 0.11 0.039 $(<$ $0.20)$ E 0.839 0.070 CoRoT-10b 0.53$\pm$0.04 0.53 0.04 – E 2.75 0.16 GJ-436b 0.150$\pm$0.012 0.153 0.017 – E 0.069 0.006 GJ-1214b $<0.27$ (95%) 0.12 0.09 $(<$ $0.34)$ P 0.020 0.003 HAT-P-1b $<0.067$ (99%) 0.048 0.021 $(<$ $0.087)$ P 0.514 0.038 HAT-P-2b 0.517$\pm$0.003 0.517 0.003 – E 8.76 0.45 HAT-P-3b – 0.1 0.05 $(<$ $0.20)$ P 0.58 0.17 HAT-P-4b – 0.063 0.028 $(<$ $0.107)$ P 0.677 0.049 HAT-P-5b – 0.053 0.061 $(<$ $0.24)$ P 1.09 0.11 HAT-P-6b – 0.047 0.017 $(<$ $0.078)$ P 1.031 0.053 HAT-P-7b $<0.039$ (99%) 0.014 0.01 $(<$ $0.037)$ C 1.775 0.070 HAT-P-8b – 0.011 0.019 $(<$ $0.064)$ C 1.340 0.051 HAT-P-9b – 0.157 0.099 $(<$ $0.40)$ P 0.767 0.10 HAT-P-11b 0.198$\pm$0.046 0.28 0.32 $(<$ $0.80)$ P 0.055 0.022 HAT-P-12b – 0.071 0.053 $(<$ $0.22)$ P 0.187 0.033 HAT-P-13b 0.014$^{+0.005}_{-0.004}$ 0.014 0.005 $(<$ $0.022)$ C 0.855 0.046 HAT-P-14b 0.107$\pm$0.013 0.11 0.04 $(<$ $0.18)$ P 2.23 0.12 HAT-P-15b 0.190$\pm$0.019 0.19 0.019 – E 1.949 0.077 HAT-P-16b 0.036$\pm$0.004 0.034 0.003 $(<$ $0.039)$ ES 4.20 0.11 HD17156b 0.677$\pm$0.003 0.675 0.004 – E 3.223 0.087 HD80606b 0.934$\pm$0.001 0.933 0.001 – E 3.99 0.33 HD149026b – 0.121 0.053 $(<$ $0.21)$ P 0.354 0.031 HD189733b 0.004$^{+0.003}_{-0.002}$ 0.004 0.003 $(<$ $0.0080)$ C 1.139 0.035 HD209458b 0.014$\pm$0.009 0.014 0.009 $(<$ $0.042)$ C 0.677 0.033 Kepler-4b $0.25_{-0.12}^{+0.11}$ ($<0.43$) 0.25 0.12 $(<$ $0.43)$ P 0.077 0.028 Kepler-5b $0.034_{-0.018}^{+0.029}$ ($<0.086$) 0.034 0.029 $(<$ $0.086)$ C 2.120 0.079 Kepler-6b $0.056_{-0.028}^{+0.044}$ ($<0.13$) 0.057 0.026 $(<$ $0.12)$ P 0.659 0.038 Kepler-7b $0.102_{-0.047}^{+0.104}$ ($<0.31$) 0.065 0.045 $(<$ $0.19)$ P 0.439 0.044 Kepler-8b $0.35_{-0.11}^{+0.15}$ ($<0.59$) 0.011 0.24 $(<$ $0.39)$ P 0.57 0.11 TrES-1b – 0.019 0.054 $(<$ $0.21)$ P 0.757 0.061 TrES-2b – 0.023 0.014 $(<$ $0.051)$ C 1.195 0.063 TrES-3b – 0.066 0.048 $(<$ $0.16)$ P 1.86 0.12 TrES-4b – 0.21 0.21 $(<$ $0.66)$ P 0.93 0.17 WASP-1b – 0.19 0.22 $(<$ $0.65)$ P 0.89 0.15 WASP-2b – 0.027 0.023 $(<$ $0.072)$ C 0.852 0.080 WASP-3b – 0.009 0.013 $(<$ $0.048)$ C 1.99 0.13 WASP-4b – 0.005 0.003 $(<$ $0.011)$ C 1.205 0.044 WASP-5b 0.038$^{+0.026}_{-0.018}$ 0.012 0.007 $(<$ $0.026)$ C 1.571 0.063 WASP-6b 0.054$^{+0.018}_{-0.015}$ 0.041 0.019 $(<$ $0.075)$ C 0.480 0.038 WASP-7b – 0.074 0.063 $(<$ $0.23)$ P 1.07 0.16 WASP-10b 0.057$^{+0.014}_{-0.004}$ 0.052 0.031 $(<$ $0.11)$ P 3.15 0.12 WASP-11b – 0.091 0.054 $(<$ $0.21)$ P 0.470 0.035 WASP-12b $0.049\pm0.015$ 0.018 0.018 $(<$ $0.05)$ C 1.48 0.14 WASP-13b – 0.14 0.1 $(<$ $0.32)$ P 0.458 0.064 WASP-14b 0.091$\pm$0.004 0.088 0.003 $(<$ $0.090)$ ES 7.26 0.59 WASP-15b – 0.056 0.048 $(<$ $0.17)$ P 0.548 0.059 WASP-16b – 0.009 0.012 $(<$ $0.047)$ C 0.846 0.072 WASP-17b $0.129^{+0.106}_{-0.068}$ 0.121 0.093 $(<$ $0.32)$ P 0.487 0.062 WASP-18b $0.009\pm0.001$ 0.007 0.005 $(<$ $0.018)$ C 10.16 0.87 WASP-19b 0.02$\pm$0.01 0.011 0.013 $(<$ $0.047)$ C 1.15 0.10 WASP-21b – 0.048 0.024 $(<$ $0.11)$ P 0.308 0.018 WASP-22b $0.023\pm0.012$ 0.022 0.016 $(<$ $0.057)$ C 0.56 0.13 WASP-26b – 0.033 0.025 $(<$ $0.086)$ C 1.018 0.034 XO-1b – 0.042 0.088 $(<$ $0.30)$ P 0.911 0.088 XO-2b – 0.064 0.041 $(<$ $0.14)$ P 0.652 0.032 XO-3b 0.287$\pm$0.005 0.287 0.005 – E 11.81 0.53 XO-4b – 0.28 0.15 $(<$ $0.50)$ P 1.56 0.30 XO-5b – 0.01 0.01 $(<$ $0.036)$ C 1.065 0.036 ----------- -------------------------------------- ------- ------- ------ ----------- ---- ------- ------- Discussion {#sec:discussion} ==========  \ [**The Mass-Period plane**]{}\ We now discuss the results of the previous sections in the context of tidal evolution in hot Jupiters. Figure \[fig:mass\_period\] shows a plot of the mass ratio $M_p/M_s$ against orbital period for transiting planets with orbital period $P<20$ days. The empty symbols represent orbits that are consistent with circular, and the black symbols represent eccentric orbits, whereas grey symbols represent objects with small ($e<0.1$), but significant eccentricities. The circles represent the G dwarfs and the squares represent F dwarfs. It appears that the low mass hot Jupiters on orbits that are consistent with circular around G dwarfs migrate inwards until they stop at a minimum period for a given mass, conglomerating on the mass-period relation of @Mazeh2005. In this case, the heavier planets can move in further before they are stopped. Planets heavier than about 1.2 $M_j$ can migrate inwards and raise tides on the star, leading to a spin-up of the host star, and even synchronisation in some cases where enough angular momentum can be transferred from the orbital motion into the stellar rotation. In cases where the planetary angular momentum is insufficient, the process can lead to a run-away migration until the planet is destroyed inside the star. The Roche limit for a planet is defined by $R_p=0.462a_R(M_p/M_s)^{-3}$. If we write the stopping distance $a=\alpha a_R$, [@Ford2006a] argued that slow migration on quasi-circular orbits would result in a value of $\alpha=1$, with the only surviving planets being those that stop at their Roche limit. On the other hand, if the planets were brought in on an eccentric orbit (eg: dynamical interactions within a system or capture from interstellar space), and then circularised by tidal interaction, the value of $\alpha$ should be two. In Figure \[fig:mass\_period\], the dashed line shows this case, with $\alpha=2$. This does not appear to be a very good fit for the hot Jupiters that are on orbits consistent with circular. The dotted lines show the range $\alpha=2.5$–$4.5$. As mentionned in @Pont2011, this larger value of $\alpha$ could indicate the planets had larger radii at the time their orbits were circularised. Subsequent thermal evolution of the planets would have shrunk them [eg: @Baraffe2004], leaving them further out from their current Roche limits. . \[fig:mass\_period\]  \ [**Circularisation Timescales**]{}\ The process of tidal circularisation, spin-orbit alignment and synchronisation are expected to occur roughly in this order, and over a similar timescale. For close-in systems, this timescale is expected to be small compared to the lifetime of the system. @Hut1981 derived equations for the tidal evolution due to the equilibrium tide using the assumption of weak friction, and constant time-lag $\Delta t$. @Leconte2010 re-visited this model and showed that the orbital eccentricity evolves according to $$\begin{aligned} \frac{1}{e}\frac{{\rm d} e}{{\rm d} t} = 11 \frac{a}{GM_sM_p} & \{K_p\left[\Omega_e(e)x_p\frac{\omega_p}{n} - \frac{18}{11}N_e(e)\right] \\ & +K_s\left[\Omega_e(e)x_s\frac{\omega_s}{n} - \frac{18}{11}N_e(e)\right] \} \nonumber,\end{aligned}$$ where $\Omega_e(e)$ and $N_e(e)$ are functions of $e$ and approximately equal to unity for small $e$; $x_p$ and $x_s$ are the cosines of the angle between the orbital plane and the planet and stellar equators respectively. $\omega_p$ and $\omega_s$ are the angular frequencies of rotation of the planet and star, and the two terms $$K_p = \frac{3}{2}k_{2,p}\Delta t_p \left(\frac{GM_p^2}{R_p}\right)\left(\frac{M_s}{M_p}\right)^2\left(\frac{R_p}{a}\right)^6n^2$$ and $$K_s = \frac{3}{2}k_{2,s}\Delta t_s \left(\frac{GM_s^2}{R_s}\right)\left(\frac{M_p}{M_s}\right)^2\left(\frac{R_s}{a}\right)^6n^2$$ describe the effect of tides on the planet by the star, and vice-versa, respectively. $n$ is the mean orbital motion and the semi-major axis is denoted $a$. Under the assumption of a constant-time delay between the exciting tidal potential and the response of the equilibrium tide in the relevant body, $k_{2,p} \Delta t_p$ and $k_{2,s} \Delta t_s$ are constants where $k_{2}$ are the potential Love numbers of degree 2 and $\Delta t$ are the constant time lags in each of the two bodies. We now consider two limits, firstly the case where only the tides in the planet dominate, and then the case where only tides in the star dominate. When tides in the planet dominate, $K_s\sim 0$ so that we obtain a timescale $$\tau_p = -\left(\frac{1}{e}\frac{{\rm d}e}{{\rm d}t}\right)^{-1} = \frac{2}{21G}\frac{1}{k_{2,p}\Delta t_{p}}\frac{M_p}{M_s^2}\frac{a^8}{R_p^5}\label{eqn:planet_isochrones}$$ where we have assumed that $\Omega_e=N_e\approx 1$, i.e. the equation is valid to lowest order in $e$; $\omega_p/n \sim 1$, i.e. synchronisation of the planetary rotation with the orbit and $x_p\sim 1$, i.e. the planet’s equator coincides with the orbital plane. A similar equation can be written for tides in the star, even though $\omega_s/n$ is not typically unity. As long as $\omega_s/n<18/11$, for small $e$, the effect of tides in the star will lead to a decrease in orbital eccentricity. We can therefore write, $$\tau_s = -\left(\frac{1}{e}\frac{{\rm d}e}{{\rm d}t}\right)^{-1} = \frac{2}{21G}\frac{1}{k_{2,s}\Delta t_{s}}\frac{M_s}{M_p^2}\frac{a^8}{R_s^5}\label{eqn:star_isochrones}$$ We take some typical values of $k_{2,p}\Delta t_{p}\sim 0.01$ s and $k_{2,s}\Delta t_{s}\sim 1$ s, which would correspond to tidal quality factors [@Goldreich1966] of about $10^6$ and $10^4$ respectively, in the constant-$Q$ model (in contrast to the constant $\Delta t$ model that we consider here) for an orbital period of about 5 d. We expect planets that are further out to be only weak affected by tides, whereas close-in planets will experience strong tides. Some of these close-in planets will be heavy enough and close enough to exert their own influence on the star by raising stellar tides. This can be seen in Figure \[fig:timescale\_circ\], where we have plotted the timescale of circularisation assuming tides inside the star alone against the timescale of circularisation assuming tides in the planet alone. The open symbols represent orbits that are consistent with circular, and the black symbols represent eccentric orbits, whereas the grey symbols represent objects with small ($e<0.1$), but significant eccentricities. The dashed lines represent lines of constant circularisation timescale, at 1 Myr, 10 Myr, 100 Myr, 1 Gyr and 10 Gyr. For the G dwarfs, orbits that are consistent with circular and eccentric orbits are cleanly segregated by the 10 Gyr isochrone, with HAT-P-16b ($e=0.034\pm0.003$) caught in the process of circularisation. For the F dwarfs (open symbols), WASP-14b ($T_{\mathrm{eff}}=6475 \pm 100$ K) has a small eccentricity $e=0.008\pm0.003$ and XO-3b ($T_{\mathrm{eff}}=6429 \pm 100$ K) has an eccentricity of $0.287\pm0.005$, whereas CoRoT-3b ($T_{\mathrm{eff}}=6740\pm 140$ K) is on an orbit that is consistent with circular. This suggests that in the dissipation factor in hotter stars may vary in an unknown fashion, although the small eccentricity of WASP-14b and the moderately small eccentricity of XO-3, together with the short timescale for stellar tides indicate that tides in the star are clearly important even in these cases.  \ [**Hot Neptunes**]{}\ GJ-436b is a planet on an eccentric orbit ($e=0.153\pm0.017$) in a region of the mass-scale plane where tidal effects on the planet are expected to be significant. The planet is a hot Neptune so it is possible that the structure is different enough that the tidal quality factor $Q$ is very much higher, leading to a longer circularisation timescale. In this case, GJ-436b would simply not have had enough time to circularise its orbit. Another possibility that was initially suggested by @Maness2007, is that a second companion may be present in the system and is pumping up the eccentricity of GJ-436b by secular interactions. Further measurements with radial velocity [@Ribas2009] and photometry [@Ballard2010] appear to rule this possibility out.  \ [**Synchronisation**]{}\ Tidal dissipation leading to orbital circularisation can occur in either the planet, the star, or both, according to the timescale for each case. On the other hand, synchronisation of the host star rotation with the orbital motion would depend on tidal effects inside the star alone. This would occur on a similar timescale as circularisation in the case of dissipation in the star alone. Figure \[fig:timescale2\] shows the same axes as Figure \[fig:timescale\_circ\], but on the left panel, the red star symbols represent objects with excess stellar rotation. In the case of CoRoT-3b and $\tau$ Boötis b, the rotation of the host star has been synchronised with the orbital period. Pont (2009) also pointed out that HD 189733 and CoRoT-2b were rotating faster than expected from the isochrones of Strassmeier & Hall (1988), even if the stellar rotations were not synchronised. We can now confirm that four more objects are clearly in this regime: CoRoT-18, HAT-P-20, WASP-19 and WASP-43. The rotation periods of these stars and the expected rotation periods are shown in Table \[tab:excess\_rot\]. From Figure \[fig:timescale2\], we note that the estimated timescale for orbital circularisation due to tidal effects in the star alone is less than 5 Gyr for the objects WASP-19, WASP-43, CoRoT-2, CoRoT-18 and CoRoT-3. This means that tidal dissipation in the star could lead to the excess rotation well within the lifetime of these stars. On the other hand, the two objects $\tau$ Boötis b and HAT-P-20 have timescales $\tau_s\sim10$ Gyr, while HD 189733b has $\tau_s\sim80$ Gyr. Even in this case, it should be noted that the tidal dissipation strength would have to be stronger by a single order of magnitude for these objects to have been spun up by tidal dissipation inside the star. Given that the tidal time lag is uncertain by up to about two orders of magnitude, this does not sound implausible. In contrast, orbital circularisation in many of these cases may well have occured due to dissipation in the planet instead. Planets that are unable to spin-up their parent stars to synchronisation may be doomed to destruction. @Hellier2009 pointed out that the existence of WASP-18 at its current position in the mass-period plane suggests that either the tidal dissipation in the system is several orders of magnitude smaller than expected, or that the system is caught at a very special time while it is in the last $10^{-4}$ of the estimated lifetime of the system. The latter possibility sounds more plausible, considering the striking paucity of heavy planets at short period. Name Expected $P_{\rm rot}$ (d) ----------- ------- ---------------------------- ---- CoRoT-2 4.52 0.02 36 CoRoT-18 6.3 0.9 49 HAT-P-20 11.3 2.2 57 HD 189733 12.95 0.01 57 WASP-19 10.5 0.2 42 WASP-43 7.6 0.7 57 : Table showing the systems with excess rotation in the left panel of Figure \[fig:timescale2\]. $P_{\textrm{rot}}$ is the stellar rotation period today, and ‘Expected $P_{\textrm{rot}}$’ is the expected rotation period of the star as estimated from the rotation isochrones of Strassmeier & Hall (1988).[]{data-label="tab:excess_rot"}  \ [**Spin-orbit alignment**]{}\ The right panel of Figure \[fig:timescale2\] shows the same axes (timescales), but now the circles represent G stars and the squares represent F stars. The empty symbols represent aligned systems ($\lambda<30^{\circ}$), and the filled symbols represent misaligned systems ($\lambda>30^{\circ}$). In this case, the G dwarfs are aligned, except for CoRoT-1($T_{\mathrm{eff}}=5950\pm150$ K) and WASP-1 ($T_{\mathrm{eff}}=6110\pm245$ K) are actually hot stars, and WASP-8 is outside the region of strong tides in the star. CoRoT-1, WASP-1 and the F dwarfs, display a spread in terms of aligned and misaligned, even in cases of strong tides. @Winn2010b found a link between the presence of a convective core and spin-orbit alignment by tidal effects. Thus, exoplanets could migrate inwards by planet-planet scattering, giving rise to orbits with a range of eccentricities and spin-orbit angles. Planets in orbit around cooler stars ($T_{\rm eff}<6250$ K, where the stellar convective region is significant), can have their orbital angular momentum aligned with the stellar rotation, while planets in orbit around hot stars ($T_{\rm eff}>6250$ K, where the extent of the convective region is negligible) manage to keep their initial misalignment. Conclusion ========== We have recalculated estimates of orbital eccentricity for a population of known transiting planets and included a noise treatment to account for systematic effects in the data. As @Laughlin2005 showed using synthetic data, analysis of radial velocity data can result in a derived eccentricity at a few $\sigma$ level even in cases where the orbit is in fact consistent with circular. In a similar way, correlated noise in the instrument or atmosphere, stellar activity, or additional companions to the host star can cause a spurious eccentricity detection, the cases of WASP-12 and WASP-10 being two examples highlighted in this paper. Once these confusing effects are accounted for, a much clearer picture emerges, highlighting the importance of tidal interactions in close-in exoplanet systems. The present observations support a scenario where low mass hot Jupiters migrate inwards and circularise their orbits until they stop at a minimum period for a given mass, conglomerating on the mass-period relation of @Mazeh2005. The heavier planets are able to move further inwards before they stop. Planets heavier than about 1.2 $M_j$ can raise tides on the star as they migrate inwards, leading to a spin-up of the host star [@Pont2009a], and even spin-orbit synchronisation in some cases where enough angular momentum can be transferred from the orbital motion into the stellar rotation. This appears to be the case for CoRoT-3b, $\tau$ Boötis b, HD 189733, CoRoT-2b, CoRoT-18, HAT-P-20, WASP-19 and WASP-43, where the first two are synchronised, and the rest show clear evidence of excess rotational angular momentum in the star. If the planetary angular momentum is insufficient, the process can lead to a run-away migration and the planet is destroyed, as appears to be the case for WASP-18b [@Hellier2009a]. This is also supported by the lack of such heavy planets at short period. As suggested by [@Winn2010b], tidal effects in G dwarfs are also responsible for aligning the spin of the star with the orbit of the planet, whereas the same effect is much less effective in the case of the hotter F stars. Overall, therefore, the present data on close-in exoplanets support the case for a prominent role for tidal interactions between the planet and the host star in the orbital evolution of hot Jupiters. Acknowledgements {#acknowledgements .unnumbered} ---------------- We are grateful to the anonymous referee for the encouraging remarks and the enormous amount of detailed feedback and sound advice, which helped us to make this paper better. We thank the editor for his encouraging comments. FP is grateful for the STFC grant and Halliday fellowship ST/F011083/1. We also thank the SOPHIE Exoplanet Consortium for arranging a flexible observation schedule for our programme, and the whole OHP/SOPHIE team for support. NH thanks Gilles Chabrier for his clear explanations of the equilibrium tides model. [^1]: Based on observations made at the 1.93-m telescopes at Observatoire de Haute-Provence (CNRS), France with the SOPHIE spectrograph.

--- abstract: 'We present a detailed theoretical and numerical study discussing the application and optimization of phase estimation algorithms (PEAs) to diamond spin magnetometry. We compare standard Ramsey magnetometry, the non-adaptive PEA (NAPEA) and quantum PEA (QPEA) incorporating error-checking. Our results show that the NAPEA requires lower measurement fidelity, has better dynamic range, and greater consistency in sensitivity. We elucidate the importance of dynamic range to Ramsey magnetic imaging with diamond spins, and introduce the application of PEAs to time-dependent magnetometry.' author: - 'N. M. Nusran' - 'M. V. Gurudev Dutt' title: Optimizing phase estimation algorithms for diamond spin magnetometry --- Introduction {#sec:intro} ============ Quantum sensors in robust solid-state systems offer the possibility of combining the advantages of precision quantum metrology with nanotechnology. Quantum electrometers and magnetometers have been realized with superconducting qubits [@Bylander11], quantum dots [@Vamivakas11], and spins in diamond [@Maze08; @Taylor08; @Balasub08; @Dolde11]. These sensors could be used for fundamental studies of materials, spintronics and quantum computing as well as applications in medical and biological technologies. In particular, the electronic spin of the nitrogen-vacancy (NV) color center in diamond has become a prominent quantum sensor due to optical transitions that allow for preparation and measurement of the spin state, stable fluorescence even in small nanodiamonds[@Bradac10], long spin lifetimes[@Balasub09], biological compatibility[@CCFu07; @McGuinness11] as well as available quantum memory that can be encoded in proximal nuclear spins[@Jelezko04b; @Dutt07]. The essential idea of quantum probes is to detect a frequency shift $\delta \nu$ in the probe resonance caused by the external perturbation to be measured. The standard method to do this with maximum sensitivity is the Ramsey interferometry scheme, which measures the relative phase $\phi = \delta \nu \times t$ accumulated by the prepared superposition of two qubit states $({\ensuremath{|{0}\rangle}} + {\ensuremath{|{1}\rangle}})/ \sqrt{2}$. These states will evolve to $({\ensuremath{|{0}\rangle}} + e^{-i \phi} {\ensuremath{|{1}\rangle}})/ \sqrt{2}$, and subsequent measurement along one of the two states will yield a probability distribution $P(\phi) \sim \cos (\phi)$, allowing the frequency shift $\delta \nu$ to be measured. For NV centers the detuning $\delta \nu = \gamma_e B $ where $\gamma_e \approx 2 \pi (28)$ GHz/T is the NV gyromagnetic ratio and $B$ is the field to be measured. The phase (or field) sensitivity is obtained by assuming that the phase has been well localized between the values $(\phi-\pi/2, \phi+\pi/2)$, where $\phi$ is the actual quantum phase value, and in practice to much better than this by making a linear approximation to the sinusoidal distribution. Thus, prior knowledge of the “working point” of the quantum sensor is key to obtaining the high sensitivity that makes the sensors attractive. When the actual phase $\phi$ is allowed to take the full range of values, then the quantum phase ambiguity (i.e. the multi-valued nature of the inverse cosine function) results in much larger phase variance than predicted by the standard methods. To overcome the quantum phase ambiguity, we require an estimator $\phi_{est}$ that can achieve high precision (small phase variance) over the entire phase interval $(-\pi, \pi)$ without any prior information. In terms of field sensing, this translates to a high dynamic range for magnetometry, i.e. to increase the ratio of maximum field strength ($B_{max}$) to the minimum measurable field ($\delta B_{min}$) per unit of averaging time. This would be a typical situation in most applications of nanoscale magnetometry and imaging, where unknown samples are being probed. In fact, as we show in [Section \[sec:dynrng\]]{}, as the sensitivity increases, it will be increasingly difficult to image systems where there is more than one type of spin present, and errors in the NV position will result in significant errors in the mapping. Further, since Ramsey imaging results in only one contour of the field being mapped out in a given scan, the acquisition time is greatly reduced, and thus several images must be made to accurately reconstruct the position of the target spins. Recently, phase estimation algorithms (PEA) were implemented experimentally with both the electronic [@Nusran12] and nuclear spin qubits [@Waldherr12] in diamond to address and resolve the dynamic range problem. While we note that the theory for the nuclear spin qubit has been presented in Ref. [@Said11], our work supplements this by applying the theory to the electronic spin qubit which is more commonly used for magnetometry. Some of the questions that we address in this work, and have not been studied earlier, include: (i) the importance of control phases in the PEA, (ii) the dependence of sensitivity on the control phases, (iii) the dependence of sensitivity on the dynamic range, and (iv) the impact of measurement fidelity on the PEAs. We have also studied the application of the PEA to magnetometry of time-dependant fields, and demonstrate the usefulness in measuring both amplitude and phase of an oscillating (AC) magnetic field. In [Section \[sec:qmag\]]{}, we present a brief introduction and overview of the NV center, Ramsey interferometry, and the importance of dynamic range in magnetic imaging. [Section \[sec:pea\]]{} introduces the two types of PEAs that we have compared in this work. [Section \[sec:results\]]{} shows some of the important results we obtain through the simulations. This includes a discussion on the importance of control phases, weighting scheme, required measurement fidelity and the possibility of implementing PEA for phase-lockable AC magnetic field detection. Finally, [Section \[sec:conclusion\]]{} summarizes the conclusions. Diamond quantum magnetometry {#sec:qmag} ============================ The NV ground state consists of a $S = 1$ spin triplet, with a natural quantization axis provided by the defect symmetry axis between the substitutional nitrogen and adjacent vacancy that constitute the color center. In the absence of a magnetic field, the ground state $m_s = 0$ spin sublevel is split by $D = 2 \pi (2.87)$ GHz from the $m_s = \pm 1$ levels. By applying a small magnetic field, the magnetic dipole moment of the NV causes the states ${\ensuremath{|{\pm 1}\rangle}}$ to split. The optically excited state of the NV defect also has the triplet $S = 1$ configuration, oriented along the same quantization axis and with similar magnetic moment, but its room temperature zero-field splitting is only $2 \pi (1.42)$ GHz[@Fuchs08; @Neumann09]. In the experiments of Ref.[@Nusran12], a static magnetic field of 40 mT is applied along the NV axis by a permanent magnet. Level anti-crossing (LAC) occurs between the ${\ensuremath{|{m_S = -1}\rangle}}$ and ${\ensuremath{|{m_S = 0}\rangle}}$ sublevels in the excited state, which results in dynamic nuclear-spin polarization (DNP) of $^{14}N$ nuclear spin (I=1) associated with most NV experiments  [@Jacques09]. Microwave is delivered via a 20 micron diameter copper wire placed on the diamond sample, which is soldered into an impedance matched strip-line. The resonant microwave radiation for the ${\ensuremath{|{m_S = 0}\rangle}} \leftrightarrow {\ensuremath{|{m_S = -1}\rangle}}$ transition leads to coherent manipulation of the spin. Due to the wide separation of the electronic spin levels and the DNP mechanism, we can treat the system as a pseudo-spin $\sigma = 1/2$ qubit, and write the Hamiltonian in the rotating frame, with the rotating wave approximation as $H = \hbar \delta \nu \sigma_z + \hbar \Omega (\sigma_x \cos \Phi + \sigma_y \sin \Phi)$ where $\Omega$ is the Rabi frequency, and $\Phi$ is the control phase. Our theoretical simulations below assume typical numbers from the experiments of Ref. [@Nusran12], but we shall discuss the consequences of improved experimental efficiency where appropriate. For clarity, we shall use the notations ${\ensuremath{|{0}\rangle}} \equiv {\ensuremath{|{m_S = 0}\rangle}}$ and ${\ensuremath{|{1}\rangle}} \equiv {\ensuremath{|{m_S = -1}\rangle}}$ to describe the qubit basis states from here onwards. Standard measurement limited sensitivity {#sec:sql} ---------------------------------------- The standard model for phase measurements in quantum metrology is depicted in [Fig. \[fig:mzi\]]{}. The equivalence of Mach-Zehnder interferometry (MZI) depicted in [Fig. \[fig:mzi\]]{}(a), the quantum network model (QNM) in [Fig. \[fig:mzi\]]{}(b)[@Cleve98], and Ramsey interferometry (RI) [Fig. \[fig:mzi\]]{}(c) for phase estimation allows us to treat all three problems in a unified framework. Thus, it was pointed out by Yurke et al [@Yurke86] that the states of photons injected into the MZI can be rewritten through application of Schwinger double-ladder operators to represent spin states. They showed that the number phase uncertainty relation for photons could be derived from the angular momentum commutation relations. Similarly, in the quantum network model [@Cleve98], auxiliary qubits (or classical fields) are prepared in an eigenstate of the operator $U$ such that $U{\ensuremath{|{\phi}\rangle}} = e^{-i \phi} {\ensuremath{|{\phi}\rangle}}$. The controlled-U operations on the system results in the following sequence of transformations on the qubits, $${\ensuremath{|{+}\rangle}}{\ensuremath{|{\phi}\rangle}} \xrightarrow{c-U} \frac{{\ensuremath{|{0}\rangle}} + e^{-i \phi} {\ensuremath{|{1}\rangle}}}{\sqrt{2}} {\ensuremath{|{\phi}\rangle}} \xrightarrow{H} \left(\cos(\frac{\phi}{2}){\ensuremath{|{0}\rangle}} + i \sin(\frac{\phi}{2}) {\ensuremath{|{1}\rangle}} \right) e^{-i \phi/2} {\ensuremath{|{\phi}\rangle}}$$ The state of the auxiliary register, being an eigenstate of $U$, is not altered along the network, but the eigenvalue $e^{-i \phi}$ is “kicked-out” in front of the ${\ensuremath{|{1}\rangle}}$ component of the control qubit. This model has allowed for application of ideas from quantum information to understand the limits of quantum sensing (see Refs.[@Huelga97; @Giovannetti04; @Giovannetti06]). Quantum metrology shows that the key resource for phase estimation is the number of interactions of $n$ spins with the field prior to measurement ($n$ measurement passes). Classical and quantum strategies differ in the preparation of uncorrelated or entangled initial states, respectively. Parallel and serial strategies differ in whether after the initial preparation, all spins are treated identically in terms of evolution and measurement. Thus, a serial strategy can trade-off running time with number of spins to achieve the same field uncertainty. In either case, the limiting resources can be expressed in one variable: the total interaction time $T = n t$. While quantum strategies can in principle achieve the Heisenberg phase uncertainty $ \langle (\Delta \phi)^2 \rangle \propto 1/n^2 \propto 1/ T^2$, classical strategies (whether parallel or serial), however, can at best scale with the phase uncertainty $ \langle (\Delta \phi)^2 \rangle \propto 1/n \propto 1/ T$. This limit, known as the standard measurement sensitivity (SMS) arises from the combination of two causes: the probabilistic and discrete nature of quantum spin measurements, and the well-known central limit theorem for independent measurements [@Giovannetti04; @Giovannetti06]. However, in obtaining the phase from the number of spins found to be pointing up or down after a measurement, there is an ambiguity. Because of the sinusoidal dependence of the phase accumulated, the above expression for the SMS assumes the phase has already been localized to an interval $(-\pi/2, \pi/2)$ around the true value. But for unknown fields, the entangled states typically accumulate phase $\sim n$ times faster than a similar un-entangled state. Thus, the working point must be known much more precisely for such strategies to be successful, which may defeat the original purpose of accurate field estimation. Thus, such quantum entangled strategies are better suited for situations where there are only likely to be small changes from a previously well-known field. Consider the case of the classical strategy: we can write the interaction time $T = n T_2$, where $T_2$ is the decoherence time, and since $\delta \phi \sim 1/ \sqrt{n}$ we obtain the field uncertainty $ \delta B \sim \frac{1}{\gamma_e \sqrt{T_2 T}}$. This also implies that the field must be known to lie in the range $\vert B \vert \leq B_{max} = \frac{\pi}{2 \gamma_e T_2}$. Putting these together, we have that the dynamic range $DR = \frac{B_{max}}{\delta B} \sim \frac{\pi}{2} \sqrt{\frac{T}{T_2}}$. Thus the dynamic range will decrease as the coherence time increase. The above expressions for the SMS do not take into account the effects of decoherence, measurement imperfections and other types of noise in experiments. We use the density matrix approach to describe the state of the quantum system and take into account these effects. Any unitary interaction on a single spin is essentially a rotation in the Bloch sphere. If we assume that the Rabi frequency $\Omega \gg \delta \nu$, we can assume that the rotations are instantaneous, and neglect the effect of the free evolution during the time of the pulses. In numerical simulations, we could also include the effect of finite $\Omega$ and $\delta \nu$ easily. For instance, a simple Ramsey experiment could be simulated as follows: an initial density matrix $\rho_0 = {\ensuremath{|{0}\rangle}}{\ensuremath{\langle{0}|}}$ is first brought to $\rho_1 = R_y(\pi/2) \rho_0 R_y(\pi/2)^{\dagger}$, where $R_n(\theta) = \exp{(-i (\vec{\sigma} . \vec{n}) \theta /2)}$ is the rotation operator along the $\hat{n}$ direction. This is equivalent to the action of a $(\pi/2)_y$ pulse in the experiment. Letting the system to evolve freely under the external magnetic field leads to the state: $\rho_t = U(t) \rho_1 U(t)^{\dagger}$, where $U(t)=\exp{(- i \delta \nu \sigma_z t)}$ is the time evolution operator. The application of the final $(\pi/2)_{\Phi}$ pulse is achieved using a z-rotation followed by $R_y(\pi/2)$: $$\rho_f = R_y(\pi/2) R_z(\Phi) \rho_t R_z(\Phi)^{\dagger} R_y(\pi/2)^{\dagger}$$ The effect of decoherence is introduced by multiplying the off-diagonal elements with the decay factor $D(t,T_2^*)=\exp{(-(t/T_2^*)^2)}$, where $T_2^*$ is the dephasing time set in our simulations. The probability for the measurement of the state in the Ramsey experiment is then given by[@Said11], $$P(u_m | \phi) = \frac{1 +(-1)^{u_m} D(t, T_2^*) \cos (\phi - \Phi)}{2} \label{eq:probumphik}$$ where measurement bit $u_m = 1 (0)$ is applied to to state ${\ensuremath{|{0}\rangle}}({\ensuremath{|{1}\rangle}})$. The bit $u_m$ is determined by comparing the measured signal level with a pre-defined threshold value. Further, feedback rotations $\Phi$ are simply achieved by controlling the phase of the second $\pi/2$ microwave pulse. Repeating the experiment $n$ times, we obtain the fraction of spins $n_0 (n_{1})$ that actually point up or down, thereby inferring the probability, e.g. $P(u_m = 1 | \phi) = \frac{n_0}{n}$. The last step is to take the inverse of this equation and obtain $\phi$. Unfortunately, as pointed out earlier, the inverse cosine is multi-valued, and thus we have the quantum phase ambiguity which requires us to have prior knowledge about the phase and the working point for the Ramsey experiments. The SMS limit can be calculated for our Ramsey experiments with NV centers, using the definition $(\delta \phi)^2 = \frac{\langle (\delta S)^2 \rangle}{\lvert d S / d \phi \rvert^2}$ i.e., by assuming that signal to noise ratio $SNR = 1$. Because the phase error $\delta \phi = \gamma_e \delta B t$, we can also calculate the sensitivity $\eta = \delta B \sqrt{T}$. Here, $S = \langle Tr (M \rho) \rangle$ represents the signal from Ramsey experiments, and the variance of the signal, $\langle (\delta S)^2 \rangle = Tr (M^2 \rho) - (Tr( M \rho))^2 $. The optical measurement operator $M = a {\ensuremath{|{0}\rangle}}{\ensuremath{\langle{0}|}} + b {\ensuremath{|{1}\rangle}}{\ensuremath{\langle{1}|}}$, where $a, b$ are Poisson random variables with means $\kappa \alpha_0 $ ($ \kappa \alpha_1$) that represent our experimental counts when the qubit is in the ${\ensuremath{|{0}\rangle}}$ (${\ensuremath{|{1}\rangle}}$) state respectively. Here, $\alpha_0$ and $\alpha_1$ represent the photon counts per optical measurement shot and $\kappa$ is the number of times the measurement is repeated till the qubit state can be distinguished with sufficiently high fidelity, $f_d$. For instance, standard quantum discrimination protocols imply that $f_d >0.66$ is sufficient to distinguish unknown pure states from a random guess [@Massar95]. The value of $\kappa$ can be tuned in the simulations and experiments, but after fixing $\kappa$ for a given experiment, $N$ is then simply the statistical repetitions needed to find the system phase $\phi$, thus the number of resources $ n = N \cdot \kappa$. In the limit of single-shot readout $\kappa = 1$ on the electronic spin state, it is clear that quantum projection noise limits for $n$ and $N$ are equivalent, and otherwise they are proportional by a scale factor that depends on experimental efficiency. We can explicitly calculate the sensitivity (with $SNR = 1$) for Ramsey measurements for general working points. From definitions, it can be shown analytically that $$\eta^2 = \frac{\kappa_{th} }{\gamma_e^2 t D(t, T_2^*)^2 \sin^2(\phi - \Phi)} \label{eq:etaSMS}$$ with, $$\kappa_{th} = 1 + 2 \frac{\alpha_0 + \alpha_1}{(\alpha_0 - \alpha_1)^2} + \frac{2 D(t, T_2^*) \cos(\phi - \Phi)}{\alpha_0 - \alpha_1} - D(t, T_2^*)^2 \cos^2(\phi - \Phi)$$ Similar results have also been derived by Refs. [@Taylor08; @Meriles10]. This expression reduces to the ideal SMS $\eta_{ideal}^{SMS} = \frac{e^{0.5}}{\gamma \sqrt{T_2^*}}$ in the limit of perfect experimental efficiency ($\kappa_{th} =1$), and assuming that $\phi = 0, \Phi = \pi/2$. The importance of the working point can clearly be seen in this derivation since small changes in $\phi$ from the working point result in quadratic increase of $\eta$. The factor $\kappa $ may be thought of as a loss mechanism, i.e. when we repeat the experiment $N$ times, we only gain information from $1/\kappa$ of the runs during measurement and hence we must repeat the experiment $\kappa$ times to achieve the same sensitivity. For ideal (single-shot) measurements, which could potentially be realized through resonant excitation and increased collection efficiency[@Babinec10], the SMS is given by taking $\alpha_1 / \alpha_0 \rightarrow 0$ and $\alpha_0 \gg 1$, resulting in $\kappa = 1$. To verify our simulation method needed for the phase estimation algorithms, we first carried out Monte-Carlo simulation procedure for Ramsey fringes where we have the analytical results derived above for comparison. [Fig. \[fig:simmethod\]]{} demonstrates first the measurement fidelity for distinguishing ${\ensuremath{|{0}\rangle}}$ and ${\ensuremath{|{-1}\rangle}}$ states for two different sample sizes. The measurement fidelity is defined as $$f_d = \frac{{\ensuremath{\langle{0}|{\rho^{0} | 0}\rangle}} + {\ensuremath{\langle{1}|{\rho^{1}|1}\rangle}}}{2}$$ where $\rho^{0(1)}$ corresponds to the state initially prepared in ${\ensuremath{|{0}\rangle}}$(${\ensuremath{|{1}\rangle}}$). The average photon counts per optical measurement for the state ${\ensuremath{|{0}\rangle}}$ (${\ensuremath{|{1}\rangle}}$) $\alpha_0 = 0.010$ ($\alpha_1 =0.007$) are set throughout our simulations to correspond with the experiments of Ref.[@Nusran12]. The experimental sequence was run with initialization of the spin into the ${\ensuremath{|{0}\rangle}}$ state, followed either immediately by fluorescence measurement or by a $\pi$ pulse and then measurement. The experimental threshold for the bit measurement $u_m$ is usually chosen as the average of the means of the two histograms. Our results, shown in [Fig. \[fig:simmethod\]]{}(a), show that by tuning the number of samples $\kappa$, we can achieve very high measurement fidelity. Impact of dynamic range on magnetic imaging {#sec:dynrng} ------------------------------------------- In the standard approach of magnetic imaging, the contour height (Ramsey detuning $\delta \nu$) is set by estimating what the expected field would be at the NV spin, and calculating the corresponding detuning. The resonance condition will be met when the field from the target spins projected on the NV axis is within one linewidth $\delta B/\gamma_e$ of the Ramsey detuning, and the corresponding pixel in the image is shaded to represent a dip in the fluorescence level. Thus, only a single contour line of the magnetic field is revealed as a “resonance fringe” and a sensitivity limited by the intrinsic linewidth $1/T_2^*$ could be achieved. However, a quantitative map of the magnetic field is impossible in a single run due to the restriction of a single contour[@Haberle13]. In order to further illustrate the importance of dynamic range, we show in [Fig. \[fig:imaging\]]{} the contours obtained for Ramsey imaging with a single NV center placed at a distance of 10 nm above different types of spins that are separated by $10$ nm. The contours are calculated by using the expressions for magnetic fields from point dipoles with magnetic dipole moments for the corresponding nuclear spin species. Our simulations do not take into account any measurement imperfections such as fringe visibility, and assume that the decoherence time of the NV spin can be made sufficiently long enough to detect the various species of spins, e.g. $T_2 = 100$ ms. From the figures, it is clear that when spin species of different types are present in the sample, the contours get greatly distorted and makes it difficult to reconstruct the position of the spin. Using the same procedures, we find that an error in the NV spin position inside the diamond lattice will significantly affect both resolution and image reconstruction. First, the resolution of the image is set by the gradient of the field from the target spin $\nabla B$ and the line width $\delta B$, giving rise to a resolution $\Delta x_{res} = \delta B / \nabla B$. Here $\nabla B = \frac{3\mu_0 m}{4 \pi r^4}$, where $m$ is the magnetic dipole moment of target spin, $\mu_0$ is the permeability in vacuum. From the prior expression for the line width, this becomes $\Delta x_{res} \sim\ \frac{\kappa}{\gamma_e \nabla B \sqrt{T_2 T}}$. For a line width $\delta B \approx 30$ pT and a target proton spin, we get $\Delta x_{res} \sim 0.75$ nm, which agrees well with the contour plots in [Fig. \[fig:imaging\]]{}. Secondly, if the NV position has an error of $\delta r$, the working point will shift by $\nabla B \, \delta r$. When the shift is comparable to the field sensing limit of Ramsey measurements $B_{max}$ defined earlier, $\frac{\pi}{2 \gamma_e T_2}$, we will lose the sensitivity needed to reconstruct the position. Putting in the numbers used for our simulations, we obtain that position reconstruction will not be possible if $\delta r \sim \frac{2 \pi^2 r^4}{3 \gamma_e \mu_0 m T_2} \approx 3.7$ nm. In practice the error in working point should be a fraction e.g. 30 – 50% of the dynamic range $B_{max}$ since the linewidth will be broadened otherwise. Under normal growth or implantation conditions for near-surface NV centers, the typical uncertainty in NV position is $\sim 1-10$ nm[@Mamin13; @Staudacher13]. By increasing the dynamic range of the imaging technique, we can relax the requirement for knowing the NV position more accurately. Phase Estimation Methods {#sec:pea} ======================== The quantum network model for quantum metrology allows us to apply ideas from quantum information to resolve the problem of dynamic range. To see how this works, let us consider the quantum phase estimation algorithm (QPEA) that utilizes the inverse quantum Fourier transform of Shor’s algorithm. The QPEA ([Fig. \[fig:algo\]]{}(a)) requires $K$ number of unitaries $(U^p, p=2^0,2^1,...,2^{K-1})$ to be applied in order to obtain an estimation $$\phi_{est} = 2 \pi (\frac{u_1}{2^0} + \frac{u_2}{2^1} \ldots +\frac{u_K}{2^{K-1}})\equiv 2 \pi (0. u_1 u_2 \ldots u_K)_2$$ for the classical phase parameter $\phi$ , with $K$ bits of precision. When the phase is expressible exactly in binary notation (i.e. a fraction of a power of two), the QPEA gives an exact result for the phase estimator $\phi_{est}$[@Kitaev96; @Higgins07]. Each application of $U$ is controlled by a different qubit which is initially prepared in the state of ${\ensuremath{|{+}\rangle}}=({\ensuremath{|{0}\rangle}}+{\ensuremath{|{1}\rangle}})/\sqrt{2}$. The control introduces a phase shift $e^{ip\phi}$ on the ${\ensuremath{|{1}\rangle}}$ component. Measurement takes place in the $\sigma_x$ basis $(X)$ and the results control the additional phase shifts (control phases $\Phi$) on subsequent qubits. This basically enables performing the inverse quantum Fourier transforms without using two-bit or entangled gates [@Kitaev96; @Higgins07; @Griffiths96]. As shown by Ref.[@Giedke06], the QPEA does not achieve the SMS that we derived earlier for the Ramsey experiments. However, it solves the problem of needing prior information about the working point. The reason the QPEA does not reach the best sensitivity is due to the fact that for arbitrary phases, we can view the QPEA estimator $\phi_{est} = 2 \pi (0. u_1 u_2 ... u_K)_2$ as a truncation of an infinite bit string representing the true phase. However, in the QPEA, every control rotation $\Phi_k$ depends on the measurement results of all the bits to the right of the $u_k$ bit. Thus, even if all measurements are perfect, the probabilistic nature of quantum measurements implies that there will be a finite probability to make an error especially for the most significant bits. Although the probability of error is low, the corresponding error is large, and therefore the overall phase variance is increased above the quantum limit. It was noted by Ref.[@Giedke06] that by weighting (error-checking) the QPEA for the most significant bits and using fewer measurements on the least significant bits, this problem could be reduced but not completely overcome. A modified version of the QPEA was introduced by Berry and Wiseman in Ref.[@Berry00] which would work for all phases, not just those that were expressible as fractions of powers of two. This model required adaptive control of the phase similar to the QPEA depending on all previous measurements, but also increased the complexity of the calculations. Surprisingly, a simpler version of the Berry and Wiseman algorithm, referred to in Refs.[@Higgins09; @Said11] as non-adaptive phase estimation algorithm (NAPEA) ([Fig. \[fig:algo\]]{}(b)) was also found to give nearly as good results, especially at lower measurement fidelities. In the NAPEA, the number of measurements vary as a function of $k$: $M(K,k)= M_K + F(K-k)$ and the control phase simply cycles through a fixed set of values typically $\Phi = \{ \Phi_1, \Phi_2,\ldots \Phi_{M(K,k)}\}$ after each measurement. Exactly as for the Ramsey measurements, the conditional probability of the $u_m$ measurement is given by: $$P(u_m | \phi,k) = \frac{1 + (-1)^{u_m} D(t, T_2^*) \cos (\phi_k - \Phi)}{2}$$ Now, with the assumption of a uniform *a priori* probability distribution for the actual phase $\phi$, Bayes’ rule can be used to find the conditional probability for the phase given the next measurement result: $$P(\phi | \vec{u}_{m+1}) \propto P(u_{m+1} | \phi) P(\phi | \vec{u}_{m})$$ where $P(\phi | \vec{u}_{m})\equiv P_m(\phi) $ is the likelihood function after $u_1, u_2, \ldots u_m$ bit measurements, and gets updated after each measurement. In Refs.[@Higgins09; @Said11], the best estimator is again obtained through an integral over this distribution. However, in our work, we simply use the the maximum likelihood estimator ($\phi_{MLE}$) of the likelihood function[@Said11; @Higgins09; @Berry09]. In our work, we have chosen to compare the NAPEA with the standard QPEA for several reasons. Firstly, the adaptive algorithm of Refs.[@Said11; @Higgins09; @Berry09] is more difficult to implement experimentally, and in practice seems to offer only slight improvements over the NAPEA. Secondly, the QPEA is a standard PEA which has a simple feed-forward scheme based purely on the bit results. Unless otherwise stated, we set $t_{min}=20$ ns and $T_2^*=1200$ ns in our simulations. Although these values were chosen to be comparable with the typical conditions and limitations in our experimental apparatus, the results and conclusions are valid for any condition in general. The necessary steps involved in the simulation of the both types of PEA are enumerated below. Simulation of PEAs: ------------------- For both QPEA and NAPEA, the following steps are common: 1. Parameter initialization: $t_{min}, K, M_K, F, k=K, \Phi=0$ 2. Preparation of the initial superposition state: $\rho_0 = {\ensuremath{|{+}\rangle}} {\ensuremath{\langle{+}|}}$ where ${\ensuremath{|{+}\rangle}}=({\ensuremath{|{0}\rangle}}+{\ensuremath{|{1}\rangle}})/\sqrt{2}$. 3. The unitary phase operation ${U^2}^{k-1}$ on $\rho_0$: $\rho_k ={U^2}^{k-1} \rho_0 {(U^{\dagger})^2}^{k-1}$\ where $U = {\ensuremath{{\ensuremath{|{0}\rangle}} {\ensuremath{\langle{0}|}}}} + e^{-i \phi } {\ensuremath{{\ensuremath{|{1}\rangle}} {\ensuremath{\langle{1}|}}}}$ and $\phi=\gamma_e B_{ext} t_{min}$. 4. The feedback rotations $\Phi$ on $\rho_k$: $\rho_f = R \rho_k R^\dagger$, where $R = {\ensuremath{{\ensuremath{|{0}\rangle}} {\ensuremath{\langle{0}|}}}} + e^{i \Phi } {\ensuremath{{\ensuremath{|{1}\rangle}} {\ensuremath{\langle{1}|}}}}$. 5. POVM measurement to obtain the signal $S=Tr[M.\rho_f]$ where M is the imperfect measurement operator as described in the text: $M = a {\ensuremath{{\ensuremath{|{0}\rangle}} {\ensuremath{\langle{0}|}}}} + b {\ensuremath{{\ensuremath{|{1}\rangle}} {\ensuremath{\langle{1}|}}}}$. 6. Assignment of the bit $u_m$ (0 or 1) by comparison of $S$ with the threshold signal. **QPEA:** 1. Repeat steps 2-6 $M(K,k)$ number of times. 2. Update the controls: $$\Phi = \sum_{j>k} \frac{u_k}{2^{j-k}} \pi ~, ~~~ k=k-1$$ where $u_k$ is chosen by majority vote among $\{u_m\}$ for a given $k$. 3. Repeat steps 2-8 until $k=1$. **NAPEA:** 7. Update the control phase $\Phi$ from the list $\{ \Phi_1, \Phi_2,\ldots \Phi_{M(K,k)}\}$ 8. Repeat steps 2-7 $M(K,k)$ number of times. 9. Update $k=k-1$. 10. Repeat steps 2-9 until $k=1$. Results {#sec:results} ======= [Fig. \[fig:simPEA\]]{}(a) shows the final phase likelihood distribution as parameter K is varied. The secondary peaks in the final likelihood distribution occur due to the phase ambiguity of individual measurements. As more measurements are performed, these secondary peaks become further suppressed. Note that the figure is given in log scale in order to make the secondary peaks more visible. Recall in QPEA, the bit string itself represents a binary estimate for the unknown phase. However, in order to make a fair comparison between the QPEA and NAPEA, we use the Bayesian approach to analyze the QPEA results as well. The digitization in the phase estimate in QPEA is clearly observed in its phase likelihood distribution. A phase that is perfectly represented by the bit string can lead to a perfect estimate, provided sufficient measurement fidelity is available. [Fig. \[fig:simPEA\]]{}(b) shows the histogram of $\phi_{MLE}$ when each PEA is performed 100 times. While QPEA shows only two possible outcomes for $\phi_{MLE}$, the histogram of $\phi_{MLE}$ for NAPEA is approximately Gaussian around the system phase (blue line). Interestingly, the difference of the two $\phi_{MLE}$ outcomes in QPEA is equal to $2 \pi /2^K $ where $K=6$ in this simulation. The phase readout $\phi_{MLE}$ is converted to a field readout by the linear relationship: $B_{MLE} = \phi_{MLE} / \gamma_e t_{min}$ in [Fig. \[fig:simPEA\]]{}(c) and agrees well with the external magnetic field $B_{ext}$. Hence PEA can be useful for sensing unknown magnetic fields in contrast to the standard Ramsey approach in which the readout is sinusoidally dependent on the external field. Multiple control phases in NAPEA {#sec:multiple} -------------------------------- To understand the choice of control phases in the NAPEA, one could imagine a simple version of NAPEA without multiple control phases, i.e., $\Phi=\{0\}$. The final phase likelihood distribution in this case will be symmetric about the origin $\phi = 0$ (see [Fig. \[fig:compDQOV\]]{}(a) inset), because the likelihood distribution is a product of many even cosine functions. Introducing a second control phase i.e., $\Phi=\{0, \pi/2 \}$ breaks this symmetry and result in a unique answer for $\phi_{MLE}$([Fig. \[fig:compDQOV\]]{}(b)). However, introducing even more control phases can be useful for obtaining a consistent sensitivity throughout the full field range. Table \[controlphases\] summarizes the terms that will be used in this paper, in describing the different sets of control phases. Control phases $\{\Phi\}$ Term $\sigma_{(\delta \phi_{MLE})^2}$ (rad$^2$) -------------------------------------------------------------------------------------------------------------- ------ -------------------------------------------- $\{0, \frac{\pi}{2} \}$ DUAL 4.13$\times 10^{-3}$ $\{0, \frac{\pi}{2}, \pi, \frac{3 \pi}{2} \}$ QUAD 4.28$\times 10^{-3}$ $\{0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \pi, \frac{5\pi}{4}, \frac{3\pi}{2} , \frac{7\pi}{4} \}$ OCT 1.80$\times 10^{-3}$ $\{0, \frac{\pi}{{M(K,k)}}, \frac{2\pi}{{M(K,k)}}, \ldots \frac{[{M(K,k)}-1]\pi}{{M(K,k)}} \}$ VAR 0.71$\times 10^{-3}$ : List of control phases used in NAPEA and corresponding terms used in the paper. []{data-label="controlphases"} The variance of the phase readout $(\delta \phi_{MLE})^2$ with respect to the given quantum phase is plotted in [Fig. \[fig:compDQOV\]]{}(c). It is noteworthy that a QUAD set of control phases is no better than the DUAL set. While former case leads to X and Y basis measurements, latter case corresponds to X, Y, -X and -Y basis measurements. Therefore similarity in results of DUAL and QUAD sets could be explained as follows. Imagine a condition that resulted in a bit measurement $u_m$ in the X(Y) basis. The same condition would have resulted a bit measurement $1-u_m$ in the -X(-Y) basis which will eventually result in the same probability distributions. Because the DUAL and the QUAD sets implies measurement in $\{X,Y\}$ and $\{X,Y,-X,-Y\}$ basis respectively, they tend to the same final results, and are technically equivalent. As seen in [Fig. \[fig:compDQOV\]]{}(c), the DUAL and the QUAD cases have relatively worse phase variance at working points corresponding to $\phi \sim 0$ or $ \pm \pi /2$. This effect is significantly suppressed in the case of eight control phases, the OCT set. Using the variable set of control phases (VAR) leads to further improvement in consistency because of the rapid increment in the number of control phases according to the weighting scheme. However, the VAR set can be comparatively difficult to implement in practice. The consistency of the various sets of control phases are summarized in Table \[controlphases\] by calculating the standard deviation of the variance over the entire interval $(-\pi,\pi)$. Weighting scheme and the measurement fidelity --------------------------------------------- In this section, we explore the effect of weighting scheme and the measurement fidelity on the NAPEA and QPEA. [Fig. \[fig:peascaling\]]{} gives phase sensitivity scaling $(\delta \phi_{MLE})^2 N$ when $K$ is increased from 1 to 9. Here $N$ is equivalent to the number of unitary operations in Refs.[@Higgins09; @Said11] and can be calculated as below. $$\label{eq:T} N = \sum_{k=1}^K M(K,k) \, 2^{(k-1)} \, = [M_K (2^K -1) + F (2^K -K -1)]$$ [Fig. \[fig:peascaling\]]{}(a) shows the scaling of sensitivity with $N$ for five different choices of quantum phases $\phi= (\pi/9.789, \pi/7.789, \pi/5.789, \pi/3.789, \pi/1.789) $, while the inset figure gives the average behavior. Here onward, we present only the average result in the scaling plots for clarity. It is important to present the average behavior rather than the behavior for a particular quantum phase, because the sensitivity is not necessarily the same for all phases as shown in the previous section. From [Fig. \[fig:peascaling\]]{}(b) and (c), it is clear that although weighting can play a role in NAPEA, there also exist non-weighted choice of optimal results. However, the optimal non-weighted parameters are highly dependent on $f_d$. For instance, with $f_d = 98.8\%$, the optimum non-weighted parameters were found to be {$K, M_K, F$}={7,8,0} while the same parameters led to $\sim 10^3$ worse sensitivity when $f_d = 92.9\%$. On the other hand the weighted parameters {7,8,8} resulted in nearly same sensitivities in either case. In QPEA, the change in control phase $\Phi$ occurs with the change in $k$. Moreover, only a single bit measurement result $u_k$ is available for each $k$, unlike in the case of NAPEA where there exist $M(k,K)$ bit measurements. However, in order to make a fair comparison, we still perform the weighting scheme on QPEA as described in section \[sec:pea\] to obtain $M(k,K)$ bit measurements. We use majority voting of bit measurements for determination of the control phases. It turns out that the best results in QPEA are obtained only with extremely high measurement fidelity ($f_d > 99 \%$) and requires no weighting ($M_K=1, F=0$). Further, even after using Bayesian estimation, the sensitivity in QPEA is ultimately limited by the minimum bit error of the phase readout given by $\delta \phi_{est} = \pi/2^K$. Field sensitivity and PEA performance {#sec:cmpsensdyrg} ------------------------------------- The corresponding scaling of field sensitivity $\eta^2 = (\delta B)^2 T $ for some of the data in [Fig. \[fig:peascaling\]]{} is shown in [Fig. \[fig:fieldsensitivity\]]{}(a). Here, $(\delta B)^2 = \langle (B_{MLE}-B_{ext})^2 \rangle$ is the variance of field with respect to the external magnetic field and $T = N t_{min} \kappa$ is the total evolution time of the PEA. The best results from NAPEA was obtained with a fidelity 92.9% and an OCT set of control phases. QPEA’s best results requires extremely high fidelity 99.9%, and furthermore show a significant fluctuation in the sensitivity over the full field range. The statistics obtained here along with PEA parameters used are summarized in the Table \[bestresults\]. PEA Fidelity Parameters $\sigma_{(\delta \phi_{MLE})^2}$ (rad$^2$) $\eta^2_{avg}$ ($\mu T^2/Hz$) $T$ (s) ------------ -------------------------------- --------------------- -------------------------------------------- ------------------------------- --------- NAPEA(OCT) 92.9% ($\kappa =2\kappa_{th}$) $M_K=20, F=0, K=6$ 0.022$\times 10^{-3}$ $1.58 \pm 0.35$ 0.202 QPEA 99.9%($\kappa =10\kappa_{th}$) $M_K = 1, F=0, K=7$ 0.197$\times 10^{-3}$ $3.67 \pm 1.62 $ 0.102 : Summery of best results from QPEA and NAPEA []{data-label="bestresults"} While NAPEA demands relatively lesser measurement fidelity than in QPEA, the total estimation time is larger. However, as shown in above table and [Fig. \[fig:fieldsensitivity\]]{}(c), the sensitivity obtained from NAPEA is better and more consistent compared to QPEA. Although it is possible to enhance the dynamic range of PEA by reducing $t_{min}$ and thereby achieving higher $K$, no significant improvement in sensitivity was observed because it is ultimately limited by the SMS at the longest evolution time. On the other hand reaching the best SMS in Ramsey limits the dynamic range. [Fig. \[fig:drtc\]]{}(b) show the total time and the dynamic range $DR=B_{max}/\delta B$ as a function of $t_{min}$ for different choices of NAPEA parameters: $M_K$ and $F$. The parameter K is chosen such that the longest evolution time interval is always the same i.e., $2^{K-1} t_{min} \approx T_2^*$. Here, $\delta B$ is the minimum detectable field amplitude. The corresponding Ramsey DR obtained for an averaging time equal to that of NAPEA with $M_K=F=8$ is also shown. Clearly, NAPEA gives better DR for smaller $t_{min}$. By a suitable choice of NAPEA parameters we can reduce the time constant without significant compromise between the sensitivity and DR. For instance, when $t_{min}$=10 ns, a change in the NAPEA parameters from $M_K=F=8$ to $M_K=F=4$ will reduce the time constant by 50% though the reduction in sensitivity and DR is only $\sim 28\%$. In principle, $t_{min}$ could be lowered to any value in order to achieve a desired dynamic range. However in practice, this is limited by the finite pulse length and gives a lower-bound; $t_{min} > t_{\pi}$. On the other hand, strong qubit driving can invalidate the RWA due to the effect known as the Bloch-Siegert shift[@Tuorila10; @Fuchs09]. Here, the qubit resonance is shifted by a factor of $(1+\Omega^2/4 \omega_0^2)$ in the rotating frame of the driving field where $\Omega$ is the Rabi frequency and $\omega_0$ is the qubit resonance frequency in the Lab frame. However, the RWA can still be reasonably applicable upto a $\sim 1\%$ of a Bloch-Siegert shift[@Fuchs09] corresponding to $\Omega / \omega_0 \sim 1/5$. This suggests that, in our application where a background magnetic field of $\sim$ 500 G leads to a qubit frequency $\omega_0 \approx 1.4$ GHz associated with $m_s=0 \leftrightarrow m_s=-1$ transition, the Rabi driving could be made as strong as $\Omega \sim 300$ MHz resulting to lower bound of $t_{min} \sim 3.4$ ns. In case of driving the $m_s=0 \leftrightarrow m_s=+1$ transition under the same conditions, qubit frequency is $\omega_0 \approx 4.3$ GHz and corresponds to a lower bound of $t_{min} \sim 1.2$ ns. Extrapolation from [Fig. \[fig:drtc\]]{}(b) data gives the upper bound for the dynamic range in this case, $DR \sim 10^5$, which should be sufficient to simultaneously detect the fields from both electron and nuclear spins in a single magnetic field image. PEA for AC Magnetometry {#sec:acpea} ----------------------- The best sensitivity in DC magnetometry is limited by the dephasing time $T_2^*$ which is usually much less than the decoherence time $T_2$. Therefore, one could be interested in implementing the PEA for AC magnetometry in order to achieve improvement in the sensitivity: $\eta_{AC}\approx \eta_{DC} \sqrt{\frac{T_2^*}{T_2}}$. Here we show by simulations, how PEA could be applied for sensing AC magnetic fields, $b(t)=b_{ac} \cos(\omega t + \theta)$. Our approach can be used to sense an unknown field amplitude $b_{ac}$ as well as the phase $\theta$ of the field. Because our focus is only to describe the method of implementation, we consider the ideal scenario of 100% photon efficiency and neglect the effect of decoherence for simplicity. Performing PEA requires the ability to accumulate several phases: $\phi, 2\phi, 4\phi$ etc, where $\phi$ is the unknown quantum phase to be measured. In DC magnetometry, these phase accumulations are achieved by varying the free evolution time in Ramsey sequence. In order to achieve the required phase accumulations from an AC-field, we can have two types of echo-based pulse sequences referred to as type-I and type-Q ([Fig. \[fig:acfig\]]{}(a)) in this paper. Type-I sequence is maximally sensitive to magnetic fields with $\theta=0^0$ or $\theta=180^0$ whereas completely insensitive (i.e., gives zero phase accumulation) to $\theta=\pm 90^0$. Type-Q sequence on the other hand, is maximally sensitive to magnetic fields with $\theta=\pm 90^0$ and completely insensitive to $\theta=0^0$ or $\theta=180^0$ ([Fig. \[fig:acfig\]]{}(b)). Further, a magnetic field with an arbitrary phase $\theta$ could be expanded as: $$b(t)=b_{ac} \cos(\omega t + \theta) = b_{ac} \cos(\theta) \cos(\omega t)-b_{ac} \sin(\theta) \sin(\omega t)$$ Therefore PEA with type-I and type-Q sequences lead to readout $\phi_I \propto b_{ac} \cos(\theta) $ and $\phi_Q \propto b_{ac} \sin (\theta)$ respectively. Hence, the phase information of the unknown field could be extracted: $\theta_{est} = \tan^{-1}[ \phi_Q / \phi_I ]$ ([Fig. \[fig:acfig\]]{}(c)). Application of PEA for AC magnetometry has recently been demonstrated. For further details see Ref.[@Nusran13] and its online supplementary material. Conclusion {#sec:conclusion} ========== In conclusion we have made a detailed investigation of PEA approach for magnetic field sensing via Monte-Carlo simulations and compared with Ramsey magnetometry. The importance of dynamic range for magnetic imaging of unknown samples was also emphasized. The high dynamic range and the linear response to the field amplitude makes PEA useful for many practical applications. When it comes to NAPEA, DUAL and QUAD set of control phases give similar results and have relatively worse sensitivities at working points corresponding to $\phi \sim 0$ or $ \pm \pi /2$. This effect can be suppressed by introducing more control phases. In particular, the use of OCT case set of control phase lead to a significant improvement in uniformity of the sensitivity over the full field range. The weighting scheme can play a role in NAPEA but not in QPEA. Even for NAPEA, there is always a choice of non-weighted PEA parameters that can lead to optimum results, but the optimum parameters in general depend on the measurement fidelity. The best results in NAPEA are however guaranteed for measurement fidelity above $\sim 90\%$. QPEA shows a significant variation in the sensitivity across the full field range as a consequence of the binary error in the readout. Further, the best results in QPEA demands extremely high fidelity $\sim 99\%$. Because multiple measurements are not required, the total estimation time for QPEA is much less than in NAPEA. In any case, NAPEA seems to be superior to QPEA due to (a) better sensitivity, (b) consistency in sensitivity throughout the full field range, (c) comparatively less demanding measurement fidelity and (d) for its simplicity in experimental realization. Finally, we have shown that PEA can also be implemented for detection of unknown AC magnetic fields. Our method allows for the detection of both field amplitudes, and the phase of the field. Acknowledgements ================ This work was supported by the Department of Energy Award No. DE-SC0006638 for optimization of magnetometry techniques, key equipment, materials and effort by N.M.N and G.D.; NSF Award No. DMR-0847195 for development of experimental setup, NSF Award No. PHY-1005341 for development of fabrication techniques and collaborations. G.D. gratefully acknowledges support from the Alfred P. Sloan Foundation. [39]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [****,  ()](http://dx.doi.org/10.1038/nphys1994) [****,  ()](\doibase 10.1103/PhysRevLett.107.166802) @noop [****,  ()]{} @noop [****, ()]{} @noop [****, ()]{} [****,  ()](\doibase 10.1038/NPHYS1969) [****,  ()](\doibase doi:10.1038/nnano.2010.56) [****,  ()](http://dx.doi.org/10.1038/nmat2420) @noop [****,  ()]{} [****,  ()](http://dx.doi.org/10.1038/nnano.2011.64) @noop [****, ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} [****,  ()](\doibase 10.1103/PhysRevB.83.125410) [****,  ()](\doibase 10.1103/PhysRevLett.101.117601) [****,  ()](http://stacks.iop.org/1367-2630/11/i=1/a=013017) [****,  ()](\doibase 10.1103/PhysRevLett.102.057403) [****,  ()](\doibase 10.1098/rspa.1998.0164) [****,  ()](\doibase 10.1103/PhysRevA.33.4033) [****,  ()](\doibase 10.1103/PhysRevLett.79.3865) [****,  ()](\doibase 10.1126/science.1104149) [****,  ()](\doibase 10.1103/PhysRevLett.96.010401) [****,  ()](\doibase 10.1103/PhysRevLett.74.1259) [****,  ()](\doibase http://dx.doi.org/10.1063/1.3483676) [****,  ()](\doibase doi:10.1038/nnano.2010.6) [****,  ()](\doibase 10.1103/PhysRevLett.111.170801) @noop [****,  ()]{} @noop [****,  ()]{} @noop [**** ()]{} @noop [****,  ()]{} [****, ()](\doibase 10.1103/PhysRevLett.76.3228) [****,  ()](\doibase 10.1103/PhysRevA.74.032316) [****,  ()](\doibase 10.1103/PhysRevLett.85.5098) [****, ()](http://stacks.iop.org/1367-2630/11/i=7/a=073023) [****,  ()](\doibase 10.1103/PhysRevA.80.052114) [****,  ()](\doibase 10.1103/PhysRevLett.105.257003) [****,  ()](\doibase 10.1126/science.1181193) [****, ()](\doibase 10.1103/PhysRevB.88.220410) \[1\][![[]{data-label="fig:mzi"}](fig1.pdf "fig:"){width="5in"}]{} \[1\][![[]{data-label="fig:simmethod"}](fig4.pdf "fig:"){width="5in"}]{} \[1\][![[]{data-label="fig:imaging"}](fig2.png "fig:"){width="5in"}]{} \[1\][![[]{data-label="fig:algo"}](fig3.pdf "fig:"){width="5in"}]{} \[1\][![ []{data-label="fig:simPEA"}](fig5.pdf "fig:"){width="5in"}]{} \[1\][![ []{data-label="fig:compDQOV"}](fig6.pdf "fig:"){width="5in"}]{} \[1\][![ []{data-label="fig:peascaling"}](fig7.pdf "fig:"){width="6in"}]{} \[1\][![[]{data-label="fig:fieldsensitivity"}](fig8.pdf "fig:"){width="6.5in"}]{} \[1\][![[]{data-label="fig:drtc"}](fig9.pdf "fig:"){width="5in"}]{} \[1\][![[]{data-label="fig:acfig"}](fig10.png "fig:"){width="5in"}]{}

--- abstract: 'At present, there is considerable interest in using atomic fermions in optical lattices to emulate the mathematical models that have been used to study strongly correlated electronic systems. Some of these models, such as the two dimensional fermion Hubbard model, are notoriously difficult to solve, and their key properties remain controversial despite decades of studies. It is hoped that the emulation experiments will shed light on some of these long standing problems. A successful emulation, however, requires reaching temperatures as low as $10^{-12}$K and beyond, with entropy per particle far lower than what can be achieved today. Achieving such low entropy states is an essential step and a grand challenge of the whole emulation enterprise. In this paper, we point out a method to literally squeeze the entropy out from a Fermi gas into a surrounding Bose-Einstein condensed gas (BEC), which acts as a heat reservoir. This method allows one to reduce the entropy per particle of a lattice Fermi gas to a few percent of the lowest value obtainable today.' author: - 'Tin-Lun Ho and Qi Zhou' title: Squeezing Out the Entropy of Fermions in Optical Lattices --- Currently, many laboratories are trying to realize the anti-ferromagnetic (AF) phase of the 3D Hubbard model using ultra-cold fermions in optical lattices[@Cho]. Although the AF phase is well known in condensed matter, its realization will be a major step for the emulation program since it requires overcoming the serious challenge mentioned above which is common to all cold atom emulations[@HoZhou]. To understand the origin of the problem, recall that the strongly correlated states of lattice fermions emerge in the lowest Bloch band of the optical lattice. To put all the fermions in the lowest band, a sufficiently deep optical lattice is required. In such deep lattices, many known methods of cooling fail. For example, standard evaporative cooling does not work because the magnetic repulsive potential used in the evaporation process is not strong enough to overcome the deep lattice. As a result, all current experiments on lattice quantum gases resort to the conventional cooling scheme, first cooling the quantum gas in a harmonic trap (without optical lattice) to the lowest entropy state possible, and then turning on the lattice adiabatically[@adiabatic][@Kohl]. The hope is that one could reach the state of interest when the lattice depth is sufficiently high. To realize these strongly correlated states with the current scheme, it is necessary that the entropy of the gas prior to switching on the lattice be less than that of the strongly correlated state one wishes to achieve. For 3D fermion Hubbard models, recent studies[@AG] show that it is possible to reach the antiferromagnetic (AF) phase slightly below the Neel temperature $T_{N}$ with the conventional cooling method. A similar calculation by Tremblay et.al.[@Tremblay] for the 2D Hubbard model, however, showed that the conventional scheme cannot reach even the pseudo-gap regime, which exists at a higher temperature than the anticipated superconducting phase. Although these calculations are for homogenous systems, they apply to confining traps as long as the majority of the sample is a Mott insulator with one fermion per site. These studies show that even under optimal conditions, one can at most reach the AF phase close to the magnetic ordering, but not low enough temperatures to study ground state properties. The problem of the conventional method is that it can only cool atoms before they are loaded onto an optical lattice. There is no way to reduce the entropy further once the lattice is switched on. Thus, the lowest entropy attainable today in an optical lattice plus harmonic trap is just the lowest entropy achievable in a harmonic trap alone. In the case of Fermi gases, it is $S/N = \pi^2(T/T_{F})$[@Castin], where $T_{F}$ is the Fermi temperature in the trap. For $T/T_{F}= 0.05$, which is very much the limit today, we have $S/N \sim 0.5$. Any strongly correlated states with lower entropy are unreachable by this method. It is therefore important to find ways to reduce the entropy of the system significantly below this value. (A) Our entropy removal scheme: =============================== The purpose of this paper is to present a scheme to produce a lattice Fermi gas with about one fermion per site with entropy per particle much lower than what can be achieved today. Our method is based on the principle of entropy [*redistribution*]{} and the removal of entropy by [*isothermal compression*]{}. It consists of the following steps: ${\bf (I)}$ We immerse the lattice fermions in a BEC which acts as a heat reservoir. The fermions are confined in a harmonic trap and a strong optical lattice. The bosons sees a much weaker trapping lattice than the one that traps for the fermions. They are confined in a loose trap and cover the entire fermion system. The traps for bosons and fermions are species specific[@specific] so that they can be varied separately. (See below for discussions of these potentials.) [**(II)**]{} We compress the fermion harmonic trap adiabatically to turn the fermions at the center into a band insulator, which has two fermions per site and essentially zero entropy. During this process, a substantial amount of the original fermion entropy is pushed into the bosons, while the entire system has little temperature change because of the large heat capacity of the BEC compared to the lattice fermions. Hence, even though the process is adiabatic for the entire Bose-Fermi system, it is essentially isothermal as far as the fermions are concerned. [**(III)**]{} After pushing out the fermion entropy into the BEC, we remove it by evaporating away the bosons all at once, leaving the remaining fermions to equilibrate. Since the band insulator is incompressible, only its density and entropy near the surface are affected during this process. This in turn severely limits entropy re-generation during the equilibration process. As a result, the entropy of the re-thermalized band insulator has a similar ultra-low value as before boson evaporation. [**(IV)**]{} We open up the the fermion harmonic trap adiabatically to lower the density of the lattice fermions. In this way, one can produce a Mott insulator or other states with fractional filling with the same ultra-low entropy. Before proceeding, we return to discuss the construction of the trapping potentials mentioned in $({\bf I})$. To have an optical lattice that confines the fermions tightly and the bosons loosely, the energy difference between the ground state ($S$-state) and the excited state ($P$-state) of fermions must be smaller than that of the bosons, $\Delta E_{f} < \Delta E_{b}$. In this way, one can choose a laser with frequency $(\omega)$ red detuned with respect to both excitation energies, ($\hbar \omega <\Delta E_{f}, \Delta E_{b}$), such that the detuning for the fermions is smaller than that for the bosons, $\Delta E_{b}-\hbar \omega > \Delta E_{f} - \hbar \omega$. For $^{40}$K fermions, its $4S$ to $4P$ transition has a wavelength 740$nm$, where as the wavelength of the $2S$ to $2P$ transition of $^{7}$Li boson is 671$nm$, which satisfies the aforementioned condition. Moreover, the difference between these two excitation energies are large enough so that the detuning $\Delta E_{b}-\hbar \omega$ and $ \Delta E_{f} - \hbar \omega$ sufficiently large to suppress heating due to spontaneous emission. Condition $({\bf I)}$ can therefore be satisfied. Another scheme that makes use of the hyperfine structure of the $P$ state of $^{87}$Rb was pointed out in ref.[@specific]. By tuning the laser frequency to 790.01$nm$, which is between the two hyperfine states $P_{3/2}$ and $P_{1/2}$ of $^{87}$Rb, Rb bosons sees no lattice because the red detuned lattice due to $P_{3/2}$ is cancelled by the blue detuned lattice due to $P_{1/2}$. On the other hand, such laser will generate an attractive potential for $^{6}$Li and $^{40}$K, since its wavelength is longer than those of the $S$-$P$ transitions of $^{6}$Li and $^{40}$K respectively. To illustrate our scheme, we shall first discuss the basic properties of lattice fermions and the important process of entropy-redistribution. (B) Number density and entropy distributions of lattice fermions: ================================================================= A Fermi gas in the lowest band of a (3D) optical lattice is described by the Hubbard model $$\hat{K} = \hat{J} + \hat{W} - \mu \hat{N}, \label{Hubbard}$$ where $\hat{J} = - J \sum_{\bf \langle R, R'\rangle, \sigma} a^{\dagger}_{\bf R, \sigma} a^{}_{\bf R', \sigma} $ describes hopping of fermions with spin $\sigma$, ($\sigma = \uparrow, \downarrow$) between neighbouring sites ${\bf R}$ and ${\bf R'}$, $J$ is the tunneling integral, $\hat{W} = U\sum_{\bf R}^{}n^{}_{\bf R, \uparrow} n^{}_{\bf R, \downarrow}$ describes the on-site repulsion $(U)$ between spin up and spin down fermions; $a^{\dagger}_{\bf R, \sigma}$ and $n^{}_{\bf R, \sigma} = a^{\dagger}_{\bf R, \sigma} a^{}_{\bf R, \sigma} $ are the creation and number operators of a fermion with spin $\sigma$ at site ${\bf R}$; $\hat{N} = \sum_{\bf R, \sigma}n^{}_{\bf R, \sigma}$ is the total fermion number, and $\mu$ is the chemical potential. The possible states of the fermions are: band insulator $(BI)$, Mott insulator $(MI)$, and “conducting” state $(C)$, corresponding to two, one and a non-integer number of fermions per site, respectively. The Mott insulator will develop AF order at the Neel temperature $T_{N}\sim J^2/U$, while the conducting state is expected to have a superfluid ground state. The strongly correlated regime emerges when $$U>>J >> J^2/U,$$ with $J^2/U$ being the smallest energy scale. At present, experiments operate in the temperature range $$U>T>J. \label{T-regime}$$ For simplicity, we set Boltzmann’s constant $k_{B}=1$. Our goal is to [*perform operations in this high temperature regime so that the fermions will lose a substantial amount of entropy*]{}. Within the temperature range described by eq.(\[T-regime\]), $\hat{J}$ can be treated as a perturbation and $\hat{K}$ is site-diagonal to zeroth order in $J$. The number occupation at site ${\bf R}$ is then $$n^{}_{\bf R}(T, \mu) = \sum_{\sigma} \langle \hat{n}^{}_{\bf R, \sigma}\rangle = \frac{ 2e^{\mu/T} + 2 e^{(2\mu - U)/T} }{ 1 + 2e^{\mu/T} + e^{(2\mu - U)/T} } . \label{nr}$$ The entropy per site is $s^{}_{\bf R} = \partial ( T{\rm ln}Z_{\bf R}^{})/\partial T $, where $Z_{\bf R}^{}(T,\mu) = {\rm Tr} e^{-\hat{K}_{\bf R}/T} = 1 + 2e^{\mu/T} + e^{(2\mu - U)/T} $ is the partition function at site ${\bf R}$. Explicitly, we have $$s_{\bf R}^{}(T, \mu) = {\rm ln}Z_{\bf R}^{}(T,\mu) + (E_{\bf R}^{}(T, \mu) -\mu n_{\bf R}^{}(T, \mu))/T \label{sr}$$ and $E_{\bf R}^{}(T, \mu) = U\langle \hat{n}_{\bf R, \uparrow}^{} \hat{n}_{\bf R, \downarrow}^{}\rangle = e^{(2\mu - U)/T}/Z_{\bf R}^{}(T,\mu)$. In a harmonic trap $V_{\omega}^{}({\bf R}) = M\omega^2 R^2/2$, the density and entropy distributions in the temperature range $U>T>J$ can be calculated from eq.(\[nr\]) and eq.(\[sr\]) using local density approximation (LDA) by replacing $\mu$ with $\mu({\bf R}) \equiv \mu - V_{\omega}^{}({\bf R})$. A typical distribution is shown in Figure 1. The chemical potential $\mu$ and temperature $T$ in eq.(\[nr\]) and (\[sr\]) are determined from the number and entropy constraints, $$N = \sum_{\bf R} n_{\bf R}^{}(T, \mu - V^{}_{\omega}({\bf R})), \,\,\,\,\,\, S = \sum_{\bf R} s_{\bf R}^{}(T, \mu - V^{}_{\omega}({\bf R})), \label{SNtotal}$$ where $N$ is the total number of fermions and $S$ is the total entropy produced in the convention cooling scheme before the lattice is turned on. Figure 1 shows the following phases: \[$ (BI) : n_{\bf R}^{} \rightarrow 2$, for $U< \mu({\bf R})$\]; \[$(MI) : n_{\bf R}^{} \rightarrow 1$ for $0<\mu({\bf R}) < U$\]; \[([*Vacuum*]{}) : $n_{\bf R}^{} \rightarrow 0$ for $ \mu({\bf R}) <0$\]. There are also two “conducting” phases $(C1)$ and $(C2)$ with non-integer number of fermions per site. They are $(C1): 1< n_{\bf R}^{} < 2$, with $\mu({\bf R}) \sim U$, and $ (C2): 0< n_{\bf R}^{} < 1$ with $\mu({\bf R}) \sim 0$. We shall denote the centres of the conducting regions $(C1)$ and $(C2)$ as $R_{1}^{}$ and $R_{2}^{}$ . They are determined by $$\mu - \frac{1}{2}M\omega^2 R^{2}_{1}=U \,\,\,\,\, {\rm and} \,\,\,\,\, \mu - \frac{1}{2}M\omega^2 R^{2}_{2}=0. \label{R1R2}$$ ![ Density distribution $n_{\textbf R}$ (red) and entropy distribution $s_{\textbf R}$ (blue) of a Fermi gas in the lowest band of an optical lattice and in the presence of an harmonic trap. $BI$ and $MI$ denote band insulator and Mott insulator, respectively. The regions labeled $C1$ and $C2$ are the “conducting" regions where the fermions have large number fluctuations and are mobile. []{data-label="fig1"}](newfig1.eps){width=".5\textwidth"} The $(BI)$ region has essentially zero entropy per particle, $s_{\bf R} \sim 0$, since the pair $(\uparrow\downarrow)$ is the only possible configuration at each lattice site. At temperatures $U> T> T^{}_{N}$, $(MI)$ has spin entropy $s_{\bf R}\sim {\rm ln}2$, since both $\uparrow$ and $\downarrow$, are equally probable. At the same temperature range, $(C1)$ has $s_{\bf R}\sim {\rm ln}3$, since the doublet $(\uparrow\downarrow)$ and the single spin states, $\uparrow$ and $\downarrow$ are all equally probable. At even higher temperatures, $s_{\bf R}$ of both $(MI)$ and $(C1)$ can rise as high as ${\rm ln}4$, as the probability of having an empty site increases from zero. Similar situation occurs in $(C2)$. While the total entropy and other properties of the system can be calculated in the temperature range $U>T>J$ using eqs.(\[nr\]), (\[sr\]), and (\[SNtotal\]), it is useful to understand them using simple estimates. The total entropy of the system is $$S = 4\pi R^{2}_{1} \Delta R_{1} \frac{\overline{s}_{C}}{d^3} + 4\pi R^{2}_{2} \Delta R_{2} \frac{\overline{s}_{C}}{d^3} + \frac{4\pi}{3} (R^{3}_{2} - R^{3}_{1}) \frac{\overline{s}_{M}}{d^3},$$ where $d$ is the lattice spacing, and $\Delta R_{1}$ and $\Delta R_{2}$ are the widths of the $(C1)$ and $(C2)$ regions. They are given by $\Delta \mu(R_{1})\sim T$ and $\Delta \mu(R_{2})\sim T$, or $$\Delta R_1 = T/(M\omega^2 R_1) \,\,\,\,\, \Delta R_2 = T/(M\omega^2 R_2).$$ The quantities $\overline{s}_{C}$ and $\overline{s}_{M}$ are the average entropy per site in $(C)$ and $(MI)$ regions: ${\rm ln 3} <\overline{s}_{C}< {\rm ln}4$, and ${\rm ln} 2< \overline{s}_{M} < {\rm ln 4} $. From eq.(\[R1R2\]), we also have $$R_{2}^2 - R^{2}_{1} = 2U/(M\omega^2),$$ showing that the $(MI)$ region shrinks as $\omega$ increases. (C) Entropy localization and reduction: ======================================= From Figure 1, we see that if we compress the trap [*adiabatically*]{} (by increasing $\omega$), the band insulator will grow at the expense of other phases. As a result, the Mott regions and the “conducting" regions, and hence all the entropy, are pushed to the surface, as shown in Figures 2a and 2b. Since all the entropy in the bulk is squeezed into the surface layer, the entropy density at the surface must rise above its former value to keep the total entropy constant, which means the temperature the system will increase, as shown in Figure 2. The fact that adiabatic compression causes heating is a consequence of thermodynamics[@HoZhou]. As compression proceeds, it reaches the point where the width of the $(MI)$ region becomes so thin that it is of the order of a few lattice spacings. (See Fig.2a and 2b). This occurs at $R_{2}-R_{1} << R_{1}\sim R_{2} \sim R$, where $R$ is the radius of the $(BI)$ region related to the total fermion number $N$ by $N=2(4\pi/3)(R/d)^3$ . In this case, we have $$\frac{S}{N} = \frac{3}{2} \left( \frac{8\pi}{3N}\right)^{2/3}\left( \frac{2T\overline{s}_{C} + U \overline{s}_{M}}{M\omega^2 d^2}\right). \label{SoverN}$$ Eq.(\[SoverN\]) shows that $T$ rises as $\omega^2$ during an adiabatic compression. If, however, we are able to keep the temperature constant during the compression, the entropy density at the surface will remain at its initial value, (see Figure 2c, 2d). As a result, the total entropy will be reduced by the same amount as the reduction of entropic volume (i.e. the region that contains entropy), which is substantial as the latter changes from a 3D volume to a surface layer. This can also be seen in eq.(\[SoverN\]), where $S/N$ drops as $\omega^{-2}$ in an isothermal compression. As isothermal compression continues, the widths of the $(C)$ and $(MI)$ regions will shrink down to a lattice spacing $d$, and the entropic surface region will eventually be reduced to a surface of thickness $d$. At this point, LDA breaks down. Due to thermal fluctuation, the thickness of this surface layer will remain approximately one lattice spacing, regardless of further increase of the trapping potential. As long as $T$ is above the spin ordering temperature $T_{N}^{}$, the entropy per site is still ${\rm ln}2$. This limits the lowest entropy attainable in isothermal compression for $T>J^2/U$ to $S^{\ast}=(4\pi R^2 d)(\overline{s}/d^3)$, or $$\frac{S^{\ast}}{N} = \frac{3}{2} \left( \frac{8\pi}{3N}\right)^{1/3}\overline{s}, \label{SNlimit}$$ where $\overline{s}\sim {\rm ln}2$. For $N = 5\times 10^6$ $(3\times 10^7)$, $S^{\ast}/N$ is 0.012 (0.007) which is 2$\%$ (1$\%$) of the lowest value attainable today. (D) Entropy transfer between fermions and BEC: ============================================== To keep the fermions at roughly constant temperature, they must be in contact with a heat reservoir. A BEC is an ideal medium for this purpose, for it has a much higher heat capacity than the fermions and can therefore keep the temperature roughly constant. In this section, we shall provide explicit calculations to demonstrate this fact. Our results show that the transfer of entropy between fermions and bosons is accurately described by the simple isothermal compression model in Section ${\bf (C)}$. We shall consider a mixture of BEC and fermions. The BEC is contained in a harmonic trap $V^{}_{B}({\bf R}) = \frac{1}{2}M^{}_{B} \omega_{B}^{2} {\bf R}^2$ with frequency $\omega_{B}$, where $M_{B}$ is the boson mass, and another trap $V^{}_{F}({\bf R}) = \frac{1}{2}M^{}_{F} \omega_{F}^{2} {\bf R}^2$ is used for the fermions. The fermions are also confined in an optical lattice. We shall assume these potentials are species specific as described in summary ${\bf (I)}$ in Section $({\bf A)}$, so that $V^{}_{F}$ can be varied independently of $V^{}_{B}$. This is important because the compression of fermions should not lead to a substantial compression of the BEC. Otherwise the temperature of the BEC will rise, making it less efficient in absorbing the entropy of the fermions. To study the entropy transfer between fermions and bosons, we first consider a homogenous Bose-Fermi system. In the grand canonical ensemble, the hamiltonian is $$\hat{K}=\hat{K}_{F}(\mu_{F})+\hat{K}_{B}(\mu_{B}) + \hat{H}_{BF},$$ where $\hat{K}_{F}$ is the Hubbard hamiltonian eq.(\[Hubbard\]) with $\mu$ now denoted as $\mu_{F}$. $\hat{K}_{B}$ is the hamiltonian for bosons with chemical potential $\mu_{B}$, $$K_{B} = \int \left( \frac{\hbar^2}{2M_{B} } \nabla \hat{\phi}^{\dagger} \cdot \nabla \hat{\phi} + \frac{g_{BB}}{2} \hat{\phi}^{\dagger} \hat{\phi}^{\dagger} \hat{\phi}^{} \hat{\phi}^{}-\mu_B\hat{\phi}^{\dagger} \hat{\phi}\right),$$ where $ \hat{\phi}^{\dagger} $ is the creation operator for bosons and $g_{BB} = 4\pi\hbar^2 a_{BB}/M_{B}$. $\hat{H}_{BF}$ is the boson-fermion interaction, $$H^{}_{BF} = \frac{g^{}_{BF}}{d^3} \sum_{{\bf R}} \hat{n}^{}_{F, {\bf R}} \hat{n}^{}_{B, {\bf R}},$$ with coupling constant $g_{BF}^{}$, $d$ is the lattice spacing, and $\hat{n}^{}_{F, {\bf R}}$ ($\hat{n}^{}_{B, {\bf R}}$) is the number of fermions (bosons) in a unit cell centered at site ${\bf R}$. ![Figure (2a) shows the number density $n_{\textbf R}$ of $N=2 \times 10^{6}$ $^{40}$K fermions in an optical lattice with lattice height $V_{o}=15E_{R}$ and $U=0.5E_{R}$ at harmonic trap frequencies $\nu_{F}=\omega_{F}/2\pi=$ 100Hz (blue), 150Hz (purple) and 600Hz (red), calculated from eqs.(\[nr\])-(\[SNtotal\]). The entropy per particle is fixed at $S/N=0.5$. The corresponding entropy distributions $s_{\textbf R}$ are shown in Figure (2b), with temperatures 27nK (blue), 88nK (purple), and 1880nK (red), respectively. As the trap frequency is increased from 100Hz (blue) to 150Hz (purple), Figs.(2a) and (2b) show that the Mott phase melts away and the entropy density at the center of the conducting layer increases, leading to a rise in temperature. This rise is very rapid. It scales as $\omega^2$ as seen in eq.(\[SoverN\]). Figures (2c) and (2d) show the number density $n_{\textbf R}$ and entropy density of $s_{\textbf R}$ for the same parameters as Figure (2a) and (2b), but now with the temperature fixed ($T=50$nK) instead of entropy per particle. As the harmonic trap is compressed from 100Hz (blue) to 150Hz (purple) and then 600Hz (red), $S/N$ decreases from 0.732 (blue) to 0.333 (purple) and finally, 0.021 (red), reaching a value which is only 4$\%$ of the lowest value attainable today, and has not yet reached the limit shown in eq.(\[SNlimit\]). []{data-label="fig2"}](newfig2.eps){width=".5\textwidth"} To calculate the properties the system, we make the following approximations : \(i) a mean field decomposition of $H_{BF}$, replacing it by $H_{BF}^{M} = g_{BF}\sum^{}_{{\bf R}} ( \hat{n}^{}_{F, {\bf R}} n^{}_{B} + n^{}_{F} \hat{n}^{}_{B, {\bf R}} - n^{}_{F} n^{}_{B} )$, where $n_{F}^{}=\langle \hat{n}^{}_{F, {\bf R}}\rangle$; $n_{B}^{}=\langle \hat{n}^{}_{B, {\bf R}}\rangle$; \(ii) treating the hopping term $J$ as a perturbation of $\hat{K}^{}_{F}$ as in Section [**(B)**]{}. This is justified in the temperature regime $U>T>J$; \(iii) applying the Hartree-Fock approximation for thermodynamics of the bosons[@Stringari]. With approximation (i), $\hat{K}$ becomes $$\hat{K}'= \hat{K}^{}_{F}(\mu^{}_{F} - g^{}_{BF} n_{B}) + \hat{K}^{}_{B}(\mu^{}_{B} - g^{}_{BF} n^{}_{F} ) - g^{}_{BF} n^{}_{B} n^{}_{F}. \label{K'}$$ The pressure $P(T, \mu_{F}, \mu_{B})= \Omega^{-1}T{\rm ln} {\rm Tr} e^{-K'/T}$ is then $$\begin{aligned} P(T, \mu_{B}, \mu_{F}) & = P^{}_{F}(T, \mu^{}_{F} - g^{}_{BF} n^{}_{B}) \hspace{0.9in} \nonumber \\ &+ P^{}_{B}(T, \mu^{}_{B} - g^{}_{BF} n^{}_{F}) + g_{BF}^{}n_{F}^{}n_{B}^{}, \end{aligned}$$ where $P^{}_{F,B}(T, \mu^{}_{F,B})= \Omega^{-1} T{\rm ln}e^{-K^{}_{F,B}(\mu^{}_{F,B})/T}$, and $\Omega$ is the volume of the system. The fermion and boson densities are given by $$n_{F} = \frac{\partial P}{\partial \mu_{F}}, \,\,\,\,\,\,\, n_{B} = \frac{\partial P}{\partial \mu_{B}}.$$ Within approximation (ii), the fermion density is readily given by eq.(\[nr\]) as $$n_{F} = n^{}_{F}(T, \mu^{}_{F} - g^{}_{BF} n^{}_{B}). \label{nF}$$ To apply approximation (iii), we follow the procedure in reference [@Stringari] to calculate the boson density for [*both*]{} the normal and superfluid parts of the boson cloud, which is of the form $$n_{B} = n^{}_{B}(T, \mu^{}_{B} - g^{}_{BF} n^{}_{F}) \label{nB}$$ according to eq.(\[K’\]). (The presence of the superfluid will depend on the value $ \mu^{}_{B} - g^{}_{BF} n^{}_{F}$). Eqs.(\[nF\]) to (\[nB\]) form a complete set of equations that determine ($n^{}_{B}$, $n^{}_{F}$) self consistently as a function of $(T, \mu_{B}^{}, \mu_{F}^{})$. All these have to be evaluated numerically. These solutions then allow one to evaluate the entropy density, which is $s(T, \mu_{F}^{}, \mu_{B}^{}) = \frac{\partial P}{\partial T}$, or $$s = \frac{\partial P}{\partial T} = s_{F}^{} + s^{}_{B} + s^{}_{BF},$$ where $s_{F}^{}, s_{B}^{}, s_{BF}^{}$ are defined as fermion, boson, and “interaction" entropy density respectively, $$s_{F}^{} \equiv \frac{\partial P^{}_{F}}{\partial T}, \,\,\,\,\, s_{B}^{} \equiv \frac{\partial P^{}_{B}}{\partial T}, \,\,\,\,\, s_{BF}^{} \equiv g_{BF}^{} \frac{ \partial (n_{B}^{} n_{F}^{}) }{\partial T},$$ which will also be functions $(T, \mu_{B}^{}, \mu_{F}^{})$. To find the density and entropy distribution in the presence of the boson and fermion traps, we can use LDA to replace $\mu^{}_{F,B}$ by $\mu^{}_{B,F}({\bf R}) = \mu_{B,F}^{} - V_{B,F}^{}$ . ![Density distribution $n_{\bf R}^{}$ and entropy distribution $s_{\bf R}^{}$ of bosons (blue) and fermions (red) in a mixture of BEC and lattice Fermi gas. The total number of fermions $N_F$ (Bosons $N_B$) is $5\times10^6$ ($5\times10^7$). The trapping frequency of bosons is fixed at $\nu_B=10 Hz$. The number density of bosons and fermions at fermion harmonic trap frequencies $100Hz$, $200Hz$, and $600Hz$ are shown in Figs.(3a)-(3c). The corresponding entropy densities are shown in Figs. (3d) to (3f). As the frequencies $\nu_F$ of the fermion harmonic trap increases, the edge of the band insulator sharpens and the entropy distribution becomes more and more narrowed, yet the temperature of the system changes very little as the fermion trap is tightened, from $60nK$ to $64.2nK$, and to $65.6nK$. As a result, the height of the entropy distribution remains roughly constant. The narrowing of the entropy distribution, however, leads to a rapid drop in the entropy per particle, from 0.691 to 0.184, to 0.016, as shown in Figs.(3d)-(3f). []{data-label="fig4"}](newfig4.eps){width=".5\textwidth"} In our numerical calculations, we consider $5\times 10^{6}$ of $^{40}$K fermions in a lattice immersed in a BEC of $^{87}$Rb with $N^{}_{B}=5\times 10^7$ bosons. The frequency of the fermion trap is initially set at $\nu^{}_{F}= \omega^{}/(2\pi)$ = 100Hz. The bosons are confined in a loose trap with frequency $\nu^{}_{B}= \omega^{}_{B}/(2\pi) = 10$Hz, so that it covers the entire fermion system. We take $a_{BB}^{}=5.45$ nm, and $g_{BF}^{}= g^{}_{BB}/2$. The lattice height is $V_{o}=15E_{R}$ and the fermions have a Hubbard interaction $U=1.5E_{R}$, where $E_{R}$ is the recoil energy. To demonstrate the transfer of entropy from the fermions to the BEC, we consider an initial state with initial temperature 60nK, corresponding to the entropy per particle $S_{F}/N_{F}=0.691$ for the fermions. For these parameters, most of the bulk is already a band insulator, similar to that in Figure (2a) and the entropy density is accumulated at the surface. (See Fig.(3a)). When the fermion trap frequency $\nu^{}_{F}$ increases from 100Hz to 600Hz, we see from Figs. (3a)-(3c) that the edge of the band insulator becomes sharper, while the entropy density becomes concentrated at the surface (Fig. (3d)-(3f)). Our calculation also show that the fermion entropy $S^{}_{F} \equiv \int s^{}_{f}$ decreases whereas the boson entropy $S_{B}^{}=\int s_{B}^{}$ increases, while $S_{BF }\equiv \int s_{BF}^{}$ remains much smaller than $S_{F}$ and $S_{B}$ during the compression, and the entropy of the entire Bose-Fermi mixture is a constant. Due to the large heat capacity of the bosons, the overall temperature $T$ rises only moderately, from 60nK to 65.6 nK. This shows that the simple isothermal compression model for the lattice fermions in Section $({\bf C})$ is a reasonable approximation of the entropy transfer process between the bosons and the fermions. At $\nu^{}_{F}$=600Hz, the fermion entropy per particle is $S^{}_{F}/N^{}_{F}\sim 0.016$, which is about $3\%$ of the best estimate achievable with conventional methods. (E): Equilibration of the lattice fermions after boson evaporation: =================================================================== After transferring the entropy of the fermions to the BEC as shown in Fig.(3c) at the temperature $65.6\,{\rm nK}(\equiv T^{(i)})$ and at trap frequency $\nu^{}_{F} = 600$Hz, we evaporate all the bosons suddenly so as to obtain a pure fermion system. The lattice Fermi gas will then relax to its equilibrium state corresponding to the new trap frequency $\omega^{}_{F}= 2\pi \nu^{}_{F}$. In this process, entropy will be generated and temperature will rise. However, because the band insulator is incompressible, and because the trap has a larger value of $\omega^{}_{F}$, entropy generation is limited. We find (in the calculation below) that the increases in entropy and temperature are about a few percent of their values prior to evaporation. In other words, the entropy of the band insulator after it reaches equilibrium is essentially the same as that before boson evaporation, i.e. $S_{F}^{}/N_{F}\sim 0.02$. This will be the entropy inherited by the Mott insulator that emerges from the band insulator as one decompresses the trap. The calculation for the above processes is as follows. Immediately after evaporation, the lattice Fermi gas has total energy $$\begin{aligned} E^{(i)}= & \sum_{\bf R} \epsilon (T^{(i)}, \mu_{F}^{} - V_{\omega}({\bf R}) - g^{}_{BF} n_{B}({\bf R}) ) \nonumber \\ & + \sum_{\bf R}V_{\omega}^{}({\bf R}) n_{F}^{}({\bf R}), \end{aligned}$$ where $\epsilon(T, \mu) \equiv U \langle \hat{n}_{{\bf R}, \uparrow} \hat{n}_{{\bf R}, \downarrow}\rangle$ is the internal energy per site of a [*homogeneous*]{} lattice fermion with hamiltonian $\hat{K}_{F}^{}$ at the temperature range $U>T>J$, and $$\epsilon (T, \mu)= \frac{Ue^{(2\mu-U)/T}} {1 + e^{\mu/T} + e^{(2\mu-U)/T}}.$$ $n_{B,F}^{}({\bf R})$ are boson and fermion densities prior to boson evaporation, which were calculated from eqs.(\[nF\]) and (\[nB\]). When the system finally relaxes to equilibrium, it will have a different temperature $T^{(f)}$ and chemical potential $\mu_{F}^{(f)}$. The energy of the final system is $$\begin{aligned} E^{(f)}= & \sum_{\bf R} \epsilon (T^{(f)}, \mu_{F}^{(f)} - V_{\omega}({\bf R}) ) \nonumber \\ & + \sum_{\bf R}V_{\omega}^{}({\bf R}) n_{F}^{} (T^{(f)}, \mu_{F}^{(f)} - V_{\omega}({\bf R}). \end{aligned}$$ Since energy is conserved in this process, we have $E^{(f)}= E^{(i)}$. In addition, the total number of particles is given by $N_{F} = \sum_{\bf R} n_{F}^{} (T^{(f)}, \mu_{F}^{(f)} - V_{\omega}({\bf R}) )$. These two relations uniquely determine the final temperature $T^{(f)}$ and chemical potential $\mu_{F}^{}$ of the band insulator, from which one can calculate the final entropy. With the parameters we mentioned before, we find that the entropy increase in this process is negligible, only about a few percent of the value before evaporation; hence our conclusions summarized in Section $({\bf A})$. Final Remarks: ============== We have introduced a method that allows extraction of a substantial fraction of the entropy of a Fermi gas in an optical lattice after a strong lattice is switched on. Moreover, the extraction process is conducted at the temperature regime $T>J$, much higher than the Neel temperature $T_{N}\sim J^2/U$. [*While our method makes explicit use of the band insulator, it is applicable to any system which has an equilibrium phase with a large gap.* ]{} The idea is to use the gapful phase to push away all the entropy in the bulk into a surrounding Bose-Einstein condensed gas. It should also be stressed that although the final stage of our method involves evaporating away the Bose-Einstein condensed gas, it is different from the usual sympathetic cooling both in purpose and in function. First, our evaporation must be preceded by the localization of fermion entropy in order to be effective. Second, the purpose of removing the bosons is not to decrease the energy of the system (as in the usual sympathetic cooling), but to remove the entropy it absorbed from the fermions. A natural question is whether one can efficiently reduce the entropy of a Mott phase which is in direct contact with a BEC by evaporating on the latter. The answer is negative, as shown by our calculations. The reason is that the interactions between bosons and fermions typically have very weak spin dependence. The removal of bosons therefore has little effect on the spin entropy of the Mott phase, which will remain close to its value before boson evaporation, ${\rm ln}2$ per particle. Finally, we point out that our scheme assumes that it is possible to expand the trap adiabatically to turn a band insulator into a Mott insulator. This implies that the relaxation time for particle re-distribution is sufficiently fast. Since mass transport occurs at the interface between different phases, (i.e. regions $(C1)$ and $(C2)$), it will take place within the time scale $\hbar/J$. It is therefore helpful to use lattices with sufficiently large tunneling, (say, around $10-15E_{R}$), and to reach the large U limit by increasing the interaction using a Feshbach resonance[@Chin]. This work is supported by NSF Grants DMR0705989, PHY05555576, and by DARPA under the Army Research Office Grant No. W911NF-07-1-0464. We thank Randy Hulet for discussions on creating different potentials for bosons and fermions, and Ed Taylor for a careful reading of the manuscript. [99]{} Cho A (2008), CONDENSED-MATTER PHYSICS: The Mad Dash to Make Light Crystals, *Science* 320: 312. Ho TL, Zhou Q (2007), Intrinsic Heating and Cooling in Adiabatic Processes for Bosons in Optical Lattices, *Phys Rev Lett* 99: 120404. Greiner M, Mandel O, Esslinger T, Hänsch TW, Bloch I (2002), Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms, *Nature* 415: 39-44. Köhl, M, Moritz H, Stöferle T, Günter K, Esslinger T (2005), Fermionic Atoms in a Three Dimensional Optical Lattice: Observing Fermi Surfaces, Dynamics, and Interactions, *Phys Rev Lett* 94: 080403. Werner F, Parcollet O, Georges A, Hassan SR (2005), Interaction-Induced Adiabatic Cooling and Antiferromagnetism of Cold Fermions in Optical Lattices, *Phys Rev Lett* 95: 056401. (2007) Daré A.-M., Raymond L, Albinet G, Tremblay A.-M. S *Phys Rev B* 76: 064402. Carr LD, Shlyapnikov GV, Castin Y (2004) *Phys Rev Lett* 92: 150404. LeBlanc L, Thywissen J (2007), Species-specific optical lattices, *Phys Rev A* 75: 053612. Pitaevskii L, Stringari S (2003) Bose-Einstein Condensation(Oxford University Press, Oxford), chapter 13. In the case of bosons, Cheng Chin’s group at Chicago has recently succeeded in producing a Mott phase of Cesium bosons using a Feshbach resonance in a lattice with relatively large tunneling, which has shortened considerably the equilibration time.

--- abstract: 'We introduce a dual logarithmic residue map for hypersurface singularities and use it to answer a question of Kyoji Saito. Our result extends a theorem of Lê and Saito by an algebraic characterization of hypersurfaces that are normal crossing in codimension one. For free divisors, we relate the latter condition to other natural conditions involving the Jacobian ideal and the normalization. This leads to an algebraic characterization of normal crossing divisors. As a side result, we describe all free divisors with Gorenstein singular locus.' address: - | M. Granger\ Université d’Angers, Département de Mathématiques\ LAREMA, CNRS UMR n^o^6093\ 2 Bd Lavoisier\ 49045 Angers\ France - | M. Schulze\ Department of Mathematics\ University of Kaiserslautern\ 67663 Kaiserslautern\ Germany author: - Michel Granger - Mathias Schulze bibliography: - 'dlr.bib' title: Normal crossing properties of complex hypersurfaces via logarithmic residues --- [^1] Introduction ============ Let $S$ be a complex manifold and $D$ be a (reduced) hypersurface $D$, referred to as a *divisor* in the sequel. In the landmark paper [@Sai80], Kyoji Saito introduced the sheaves of ${{\mathscr}{O}}_S$-modules of logarithmic differential forms and logarithmic vector fields on $S$ along $D$. Logarithmic vector fields are tangent to $D$ at any smooth point of $D$; logarithmic differential forms have simple poles and form a complex under the usual differential. Saito’s clean algebraically flavored definition encodes deep geometric, topological, and representation theoretic information on the singularities that is yet only partly understood. The precise target for his theory was the Gauß-Manin connection on the base $S$ of the semiuniversal deformation of isolated hypersurface singularities, a logarithmic connection along the discriminant $D$. Saito developed mainly three aspects of his logarithmic theory in loc. cit.: Free divisors, logarithmic stratifications, and logarithmic residues. Many fascinating developments grew out of Saito’s paper, a few of which we highlight in the following brief overview. A divisor is called free if the sheaf of logarithmic vector fields, or its dual, the sheaf of logarithmic $1$-forms, is a vector bundle; in particular, normal crossing divisors are free. Not surprisingly, discriminants of isolated hypersurface singularities are free divisors (see [@Sai80 (3.19)]). Similar results were shown for isolated complete intersection singularities (see [@Loo84 §6]) and space curve singularities (see [@vST95]). Both the reflection arrangements and discriminants associated with finite unitary reflection groups are free divisors (see [@Ter80b]). More recent examples include discriminants in certain prehomogeneous vector spaces (see [@GMS11]) whose study led to new constructions such as a chain rule for free divisors (see [@BC12 §4]). Free divisors can be seen as the extreme case opposite to isolated singularities: Unless smooth, free divisors have Cohen–Macaulay singular loci of codimension $1$. The freeness property is closely related to the complement of the divisor being a $K(\pi,1)$-space (see [@Sai80 (1.12)], [@Del72]), although these two properties are not equivalent (see [@ER95]). Even in special cases, such as that of hyperplane arrangements, freeness is not fully understood yet. For instance, Terao’s conjecture on the combinatorial nature of freeness for arrangements is one of the central open problems in arrangement theory. Saito’s second topic, the so-called logarithmic stratification of $S$, consists of immersed integral manifolds of logarithmic vector fields along $D$. Contrary to what the terminology suggests, the resulting decomposition of $S$ is not locally finite, in general. Saito attached the term holonomic to this additional feature: a point in $S$ is holonomic if a neighborhood meets only finitely many logarithmic strata. Along any logarithmic stratum, the pair $(D,S)$ is analytically trivial which turns holonomicity into a property of logarithmic strata. The logarithmic vector fields are tangent to the strata of the canonical Whitney stratification; the largest codimension up to which all Whitney strata are (neccessarily holonomic) logarithmic strata is called the holonomic codimension (see [@DM91 p. 221]). Holonomic free divisors were later called Koszul–free divisors. In case of a normal crossing divisor, the complex of logarithmic differential forms computes the cohomology of the complement of $D$ in $S$, an ingredient of Deligne’s mixed Hodge structure (see [@Del71]). The natural question, for which free divisors the same holds true for the complex of logarithmic differential forms is referred to as the logarithmic comparison theorem, or, for short, by the LCT (see [@Tor07] for a survey). For free divisors, this property turned out to be related to homogeneity properties of the singularities. Indeed, an explicit class of hypersurfaces for which the LCT holds true is that of (weakly) locally quasihomogeneous divisors (see [@CNM96; @Nar08]). Moreover, it is conjectured that the LCT implies strong Euler homogeneity of $D$, which has been proved only for Koszul free divisors, and in dimension $\dim S\le3$ (see [@CMNC02; @GS06]). For strongly Koszul–free divisors $D$ (see [@GS10 Def. 7.1]), the logarithmic comparison theorem holds true exactly if $D$ is strongly Euler homogeneous and $-1$ is the minimal integer root of all local $b$-functions (see [@CN05 Cor. 4.3] and [@Tor04 Cor. 1.8]). For isolated quasihomogeneous singularities, the LCT is equivalent to the vanishing of certain graded parts of the Milnor algebra (cee [@HM98]); there are related Hodge-theoretic properties in the non-quasihomogeneous case ([@Sch10]). The study of the LCT lead to a variant of $D$-module theory over the ring of logarithmic differential operators along a free divisor (see [@CU02; @CN05; @Nar08]). A key player in this context is the $D$-module $M^{\log D}$ defined by the ideal of logarithmic vector fields, considered as differential operators of order one (see [@CU02]); Saito-holonomicity of $D$ implies holonomicity of $M^{\log D}$ in the $D$-module-sense; but the converse is false. Much less attention has been devoted to Saito’s logarithmic residues, the main topic in this paper. It was Poincaré who first defined a residual $1$-form of a rational differential $2$-form on ${\mathds{C}}^2$ (see [@Poi87]). Later, the concept was generalized by de Rham and Leray to residues of closed meromorphic $p$-forms with simple poles along a smooth divisor $D$; these residues are holomorphic $(p-1)$-form on $D$ (see [@Ler59]). The construction of Deligne’s mixed Hodge structure uses, again holomorphic, residues of logarithmic differential forms along normal crossing divisors (see [@Del71]). Notably, in Saito’s generalization to arbitrary singular divisors $D$, the residue of a logarithmic $p$-form becomes a *meromorphic* $(p-1)$-form on $D$, or on its normalization ${\widetilde}D$. Using work of Barlet [@Bar78], Aleksandrov linked Saito’s construction to Grothendieck duality theory: The image of Saito’s logarithmic residue map is the module of regular differential forms on $D$ (see [@Ale88 §4, Thm.] and [@Bar78]). With Tsikh, he suggested a generalization for complete intersection singularities based on multilogarithmic differential forms depending on a choice of generators of the defining ideal (see [@AT01]). Recently, he approached the natural problem of describing the mixed Hodge structure on the complement of an LCT divisor in terms of logarithmic differential forms (see [@Ale12]). In Dolgachev’s work, one finds a different sheaf of logarithmic differential forms which is a vector bundle exactly for normal crossing divisors and whose reflexive hull is Saito’s sheaf of logarithmic differential forms (see [@Dol07]). Although his approach to logarithmic residues using adjoint ideals has a similar flavor to ours, he does not reach the conclusion of our main Theorem \[10\] (see Remark \[49b\]). While most constructions in Saito’s logarithmic theory and its generalizations have a dual counterpart, a notion of a dual logarithmic residue associated to a vector field was not known to the authors. The main motivation and fundamental result of this article is the construction of a dual logarithmic residue (see Section \[13\]). This turned out to have surprising applications including a proof of a conjecture of Saito, that was open for more than 30 years. Saito’s conjecture is concerned with comparing logarithmic residues of $1$-forms, that is, certain meromorphic functions on ${\widetilde}D$, with holomorphic functions on ${\widetilde}D$. The latter can also be considered as a weakly holomorphic function on $D$, that is, functions on the complement of the singular locus $Z$ of $D$, locally bounded near points of $Z$. While any such weakly holomorphic function is the residue of some logarithmic $1$-form, the image of the residue map can contain functions which are not weakly holomorphic. The algebraic condition of equality was related by Saito to a geometric and a topological property as follows (see [@Sai80 (2.13)]). \[28\] For a divisor $D$ in a complex manifold $S$, consider the following conditions: (A) \[28a\] The local fundamental groups of the complement $S\backslash D$ are Abelian. (B) \[28b\] In codimension $1$, that is, outside of an analytic subset of codimension at least $2$ in $D$, $D$ is normal crossing. (C) \[28c\] The residue of any logarithmic $1$-form along $D$ is a weakly holomorphic function on $D$. Then the implications $\Rightarrow$ $\Rightarrow$ hold true. Saito asked whether the the converse implications in Theorem \[28\] hold true. The first one was later established by Lê and Saito [@LS84]; it generalizes the Zariski conjecture for complex plane projective nodal curves proved by Fulton and Deligne (see [@Ful80; @Del81]). The implication $\Leftarrow$ in Theorem \[28\] holds true. Our duality of logarithmic residues turns out to translate condition in Theorem \[28\] into the more familiar equality of the Jacobian ideal and the conductor ideal. A result of Ragni Piene [@Pie79] proves that such an equality forces $D$ to have only smooth components if it has a smooth normalization. This is a technical key point which leads to a proof of the missing implication in Theorem \[28\]. \[10\] The implication $\Leftarrow$ in Theorem \[28\] holds true: If the residue of any logarithmic $1$-form along $D$ is a weakly holomorphic function on $D$ then $D$ is normal crossing in codimension $1$. \[49b\] Saito [@Sai80 (2.11)] proved Theorem \[10\] for plane curves. If $D$ has holonomic codimension at least $1$ (as defined above), this yields the general case by analytic triviality along logarithmic strata (see [@Sai80 §3]). Under this latter hypothesis, Theorem \[10\] follows also from a result of Dolgachev (see [@Dol07 Cor. 2.2]). However, for example, the equation $xy(x+y)(x+yz)=0$ defines a well-known free divisor with holonomic codimension $0$. The preceding results and underlying techniques serve to address two natural questions: The algebraic characterization of condition  through Theorem \[10\] raises the question about the algebraic characterizations of normal crossing divisors. Eleonore Faber was working on this question at the same time as the results presented here were developed. She considered freeness as a first approximation for being normal crossing and noticed that normal crossing divisors satisfy an extraordinary condition: The ideal of partial derivatives of a defining equation is radical. She proved the following converse implications (see [@Fab11; @Fab12]). Consider the following condition: (D) \[28e\] At any point $p\in D$, there is a local defining equation $h$ for $D$ such that the ideal ${{\mathscr}{J}}_h$ of partial derivatives is radical. (E) \[28f\] $D$ is normal crossing. Then the following holds: If $D$ is free then condition  decends to all irreducible components of $D$. Conditions  and are equivalent if $D$ is locally a plane curve or a hyperplane arrangement, or if its singular locus is Gorenstein. Motivated by Faber’s problem we prove the following \[16\] Extend the list of conditions in Theorem \[28\] as follows: (F) \[28d\] The Jacobian ideal ${{\mathscr}{J}}_D$ of $D$ is radical. (G) \[28g\] The Jacobian ideal ${{\mathscr}{J}}_D$ of $D$ equals the conductor ideal ${{\mathscr}{C}}_D$ of the normalization $\tilde D$. Then condition  implies condition . If $D$ is a free divisor then conditions , and are equivalent. \[9\] Note that ${{\mathscr}{J}}_h$ is an ${{\mathscr}{O}}_S$-ideal sheaf depending on a choice of local defining equation whereas its image ${{\mathscr}{J}}_D$ in ${{\mathscr}{O}}_D$ is intrinsic to $D$. In particular, condition  implies condition . We obtain the following algebraic characterization of normal crossing divisors. \[38\] For a free divisor with smooth normalization, any one of the conditions , , , , or implies condition . The implication $\Rightarrow$ in Theorem \[38\] improves Theorem A in [@Fab12] (see Remark \[9\]), which is proved using [@Pie79] like in the proof of our main result. Proposition C in [@Fab12] is the implication $\Rightarrow$ in Theorem \[38\], for the proof of which Faber uses our arguments. As remarked above, free divisors are characterized by their singular loci being (empty or) maximal Cohen–Macaulay. It is natural to ask when the singular locus of a divisor is Gorenstein. This question is answered by the following \[40\] A divisor $D$ has Gorenstein singular locus $Z$ of codimension $1$ if and only if $D$ is locally the product of a quasihomogeneous plane curve and a smooth space. In particular, $D$ is locally quasihomogeneous and $Z$ is locally a complete intersection. Theorem \[40\] complements a result of Kunz–Waldi [@KW84 Satz 2] saying that a Gorenstein algebroid curve has Gorenstein singular locus if and only if it is quasihomogeneous. Logarithmic modules and fractional ideals {#30} ========================================= In this section, we review Saito’s logarithmic modules, the relation of freeness and Cohen–Macaulayness of the Jacobian ideal, and the duality of maximal Cohen–Macaulay fractional ideals. We switch to a local setup for the remainder of the article. Let $D$ be a reduced effective divisor defined by ${{\mathscr}{I}}_D={{\mathscr}{O}}_S\cdot h$ in the smooth complex analytic space germ $S=({\mathds{C}}^n,0)$. Denote by $h\colon S\to T=({\mathds{C}},0)$ a function germ generating the ideal ${{\mathscr}{I}}_D={{\mathscr}{O}}_S\cdot h$ of $D$. We abbreviate by $$\Theta_S:=\operatorname{Der}_{\mathds{C}}({{\mathscr}{O}}_S)=\operatorname{Hom}_{{{\mathscr}{O}}_S}(\Omega^1_S,{{\mathscr}{O}}_S)$$ the ${{\mathscr}{O}}_S$-module of vector fields on $S$. Recall Saito’s definition [@Sai80 §1] of the ${{\mathscr}{O}}_S$-modules of logarithmic differential forms and of logarithmic vector fields. \[33\] $$\begin{aligned} \Omega^p(\log D)&:=\{\omega\in\Omega^p_S(D)\mid d\omega\in\Omega_S^{p+1}(D)\}\\ \operatorname{Der}(-\log D)&:=\{\delta\in\Theta_S\mid dh(\delta)\in{{\mathscr}{I}}_D\}\end{aligned}$$ These modules are stalks of analogously defined coherent sheaves of ${{\mathscr}{O}}_S$-modules (see [@Sai80 (1.3),(1.5)]). It is obvious that each of these sheaves ${{\mathscr}{L}}$ is torsion free and normal, and hence reflexive (see [@Har80 Prop. 1.6]). More precisely, $\Omega^1(\log D)$ and $\operatorname{Der}(-\log D)$ are mutually ${{\mathscr}{O}}_S$-dual (see [@Sai80 (1.6)]). Normality of a sheaf ${{\mathscr}{L}}$ means that ${{\mathscr}{L}}=i_*i^*{{\mathscr}{L}}$ where $i\colon S\setminus Z{\hookrightarrow}S$ denotes the inclusion of the complement of the singular locus of $D$. In case of ${{\mathscr}{L}}=\operatorname{Der}(-\log D)$, this means that $\delta\in\operatorname{Der}(-\log D)$ if and only if $\delta$ is tangent to $D$ at all smooth points. In addition, $\Omega^\bullet(\log D)$ is an exterior algebra over ${{\mathscr}{O}}_S$ closed under exterior differentiation and $\operatorname{Der}(-\log D)$ is closed under the Lie bracket. A divisor $D$ is called free if $\operatorname{Der}(-\log D)$ is a free ${{\mathscr}{O}}_S$-module. The definition of $\operatorname{Der}(-\log D)$ can be rephrased as a short exact sequence of ${{\mathscr}{O}}_S$-modules $$\label{1} {\SelectTips{cm}{}\xymatrix}{ 0&{{\mathscr}{J}}_D\ar[l]&\Theta_S\ar[l]_-{dh}&\operatorname{Der}(-\log D)\ar[l]&0\ar[l] }$$ where the Jacobian ideal ${{\mathscr}{J}}_D$ of $D$ is defined as the Fitting ideal $${{\mathscr}{J}}_D:={{\mathscr}{F}}^{n-1}_{{{\mathscr}{O}}_D}(\Omega_D^1)={{\left\langle\frac{{\partial}h}{{\partial}x_1},\dots,\frac{{\partial}h}{{\partial}x_n}\right\rangle}}\subset{{{\mathscr}{O}}_D}.$$ Note that ${{\mathscr}{J}}_D$ is an ideal in ${{\mathscr}{O}}_D$ and pulls back to ${{\left\langleh,\frac{{\partial}h}{{\partial}x_1},\dots,\frac{{\partial}h}{{\partial}x_n}\right\rangle}}$ in ${{\mathscr}{O}}_S$. We shall consider the singular locus $Z$ of $D$ equipped with the structure defined by ${{\mathscr}{J}}_D$, that is, $$\label{21} {{\mathscr}{O}}_Z:={{\mathscr}{O}}_D/{{\mathscr}{J}}_D.$$ Note that $Z$ might be non-reduced. There is the following intrinsic characterization of free divisors in terms of their singular locus (see [@Ale88 §1 Thm.] or [@Ter80a Prop. 2.4]). \[19\] The following are equivalent: 1. \[19a\] $D$ is a free divisor. 2. \[19b\] ${{\mathscr}{J}}_D$ is a maximal Cohen–Macaulay ${{\mathscr}{O}}_D$-module. 3. \[19c\] $D$ is smooth or $Z$ is Cohen–Macaulay of codimension $1$. If $dh(\Theta_S)$ does not minimally generate ${{\mathscr}{J}}_D$, then $D\cong D'\times({\mathds{C}}^k,0)$, $k>0$, by the triviality lemma [@Sai80 (3.5)]. By replacing $D$ by $D'$, we may therefore assume that is a minimal resolution of ${{\mathscr}{J}}_D$ as ${{\mathscr}{O}}_S$-module. Thus, the equivalence of and is due to the Auslander–Buchsbaum formula. By Lemma \[25\] below, ${{\mathscr}{J}}_D$ has height at least $1$ and the implication  $\Leftrightarrow$ is proved in [@HK71 Satz 4.13]. \[27\] Any $D$ is free in codimension $1$. By Theorem \[19\], the non-free locus of $D$ is contained in $Z$ and equals $$\{z\in Z\mid\operatorname{depth}{{\mathscr}{O}}_{Z,z}<n-2\}\subset D.$$ By Scheja [@Sch64 Satz 5], this is an analytic set of codimension at least $2$ in $D$. We denote by $Q(-)$ the total quotient ring. Then ${{\mathscr}{M}}_D:=Q({{\mathscr}{O}}_D)$ is the ring of meromorphic functions on $D$. A fractional ideal (on $D$) is a finite ${{\mathscr}{O}}_D$-submodule of ${{\mathscr}{M}}_D$ which contains a non-zero divisor. \[25\] ${{\mathscr}{J}}_D$ is a fractional ideal. By assumption $D$ is reduced, so ${{\mathscr}{O}}_D$ satisfies Serre’s condition $R_0$. This means that $Z\subset D$ has codimension at least $1$. In other words, ${{\mathscr}{J}}_D\not\subseteq{\mathfrak{p}}$ for all ${\mathfrak{p}}\in\operatorname{Ass}({{\mathscr}{O}}_D)$ where the latter denotes the set of (minimal) associated primes of ${{\mathscr}{O}}_D$. By prime avoidance, ${{\mathscr}{J}}_D\not\subseteq\bigcup_{{\mathfrak{p}}\in\operatorname{Ass}{{\mathscr}{O}}_D}{\mathfrak{p}}$. But the latter is the set of zero divisors of ${{\mathscr}{O}}_D$ and the claim follows. \[18\] The ${{\mathscr}{O}}_D$-dual of any fractional ideal ${{\mathscr}{I}}$ is again a fractional ideal ${{\mathscr}{I}}^\vee=\{f\in{{\mathscr}{M}}_D\mid f\cdot{{\mathscr}{I}}\subseteq{{\mathscr}{O}}_D\}$. The duality functor $$-^\vee=\operatorname{Hom}_{{{\mathscr}{O}}_D}(-,{{\mathscr}{O}}_D)$$ reverses inclusions. It is an involution on the class of maximal Cohen–Macaulay fractional ideals. See [@JS90a Prop. (1.7)]. For lack of reference, we prove the following \[39\] Let $A$ be a Gorenstein ring whose normalization $B$ is a finite $A$-module. Then $B$ is a reflexive $A$-module. We apply a standard characterization of reflexive modules (see [@BH93 Prop. 1.4.1.(b)]). To this end let ${\mathfrak{p}}\in\operatorname{Spec}A$ and let ${\mathfrak{q}}\in\operatorname{Spec}B$ with ${\mathfrak{q}}\cap A={\mathfrak{p}}$. By incomparability for integral extensions, ${\mathfrak{q}}$ induces a maximal ideal in $B_{\mathfrak{p}}$ and a minimal prime ideal in $B_{\mathfrak{p}}/{\mathfrak{p}}B_{\mathfrak{p}}$. In particular, $B_{\mathfrak{p}}$ has finitely many maximal ideals, the localization at one of them being $B_{\mathfrak{q}}$. By the Chinese remainder theorem, $B_{\mathfrak{q}}/{\mathfrak{p}}B_{\mathfrak{q}}$ is a direct factor of $B_{\mathfrak{p}}/{\mathfrak{p}}B_{\mathfrak{p}}$, and hence $B_{\mathfrak{q}}$ is a finite $A_{\mathfrak{p}}$-module by Nakayama’s lemma. By incomparability and going up for integral extensions, $\dim B_{\mathfrak{q}}=\dim A_{\mathfrak{p}}$. By hypothesis, $B$ satisfies Serre’s conditions $R_1$ and $S_2$. After these preparations, two cases need to be considered (see loc. cit.): If $\operatorname{depth}A_{\mathfrak{p}}\ge2$ then $\dim B_{\mathfrak{q}}=\dim A_{\mathfrak{p}}\ge 2$. Hence, $\operatorname{depth}_{A_{\mathfrak{p}}}B_{\mathfrak{q}}=\operatorname{depth}_{B_{\mathfrak{q}}} B_{\mathfrak{q}}\ge 2$ by $S_2$ for $B$ and invariance of depth under finite extensions (see [@Ser65 IV-18, Prop. 12]). But then also $\operatorname{depth}_{A_{\mathfrak{p}}}B_{\mathfrak{p}}\ge2$. If $\operatorname{depth}A_{\mathfrak{p}}\le1$ then also $\dim A_{\mathfrak{p}}\le 1$ by $S_2$ for $A$. Thus, $\dim B_{\mathfrak{q}}\le 1$ and $B_{\mathfrak{q}}$ is regular by $R_1$ for $B$. In particular, $B_{\mathfrak{q}}$, and hence $B_{\mathfrak{p}}$, is a maximal Cohen–Macaulay ring. By invariance of Cohen–Macaulayness under finite extensions (see [@Ser65 IV-18, Prop. 11])[^2], $B_{\mathfrak{p}}$ is then also a maximal Cohen–Macaulay $A_{\mathfrak{p}}$-module. As $A_{\mathfrak{p}}$ is Gorenstein, this implies that $B_{\mathfrak{p}}$ is reflexive. Logarithmic residues and duality {#13} ================================ In this section, we develop the dual picture of Saito’s residue map and apply it to find inclusion relations of certain natural fractional ideals and their duals. Let $\pi\colon{\widetilde}D\to D$ denote the normalization of $D$. Then ${{\mathscr}{M}}_D={{\mathscr}{M}}_{{\widetilde}D}:=Q({{\mathscr}{O}}_{{\widetilde}D})$ and ${{\mathscr}{O}}_{{\widetilde}D}$ is the ring of weakly holomorphic functions on $D$ (see [@JP00 Exc. 4.4.16.(3), Thm. 4.4.15]). Let $${\SelectTips{cm}{}\xymatrix}{ \Omega^p(\log D)\ar[r]^-{\rho^p_D}&\Omega_D^{p-1}\otimes_{{{\mathscr}{O}}_D}{{\mathscr}{M}}_D }$$ be Saito’s residue map [@Sai80 §2] which is defined as follows: By [@Sai80 (1.1)], any $\omega\in\Omega^p(\log D)$ can be written as $$\label{4} \omega=\frac{dh}{h}\wedge\frac{\xi}{g}+\frac{\eta}{g},$$ for some $\xi\in\Omega_S^{p-1}$, $\eta\in\Omega^p_S$, and $g\in{{\mathscr}{O}}_S$, which restricts to a non-zero divisor in ${{\mathscr}{O}}_D$. Then $$\label{34} \rho_D^p(\omega):=\frac{\xi}{g}\vert_D$$ is well defined by [@Sai80 (2.4)]. We shall abbreviate $\rho_D:=\rho_D^1$ and denote its image by $${{\mathscr}{R}}_D:=\rho_D(\Omega^1(\log D)).$$ Using this notation, condition  in Theorem \[28\], that the residue of any $\omega\in\Omega^1(\log D)$ is weakly holomorphic, can be written as ${{\mathscr}{O}}_{{\widetilde}D}={{\mathscr}{R}}_D$. The following result of Saito [@Sai80 (2.9)] can be considered as a kind of approximation of our main result Theorem \[10\]; in fact, it shall be used in its proof. Combined with his freeness criterion [@Sai80 (1.8)] it yields Faber’s characterization of normal crossing divisors in Proposition B of [@Fab12]. \[60\] Let $D_i=\{h_i=0\}$, $i=1,\dots,k$, denote irreducible components of $D$. Then the following conditions are equivalent: 1. \[60a\] $\Omega^1(\log D)=\sum_{i=1}^k{{\mathscr}{O}}_S\frac{dh_i}{h_i}+\Omega^1_S$. 2. \[60b\] $\Omega^1(\log D)$ is generated by closed forms. 3. \[60d\] The $D_i$ are normal and intersect transversally in codimension $1$. 4. \[60c\] ${{\mathscr}{R}}_D=\bigoplus_{i=1}^k{{\mathscr}{O}}_{D_i}$. The Whitney umbrella (which is not a free divisor) satisfies the conditions in Theorem \[28\], but not those in Theorem \[60\]. The implication  $\Leftarrow$ in Theorem \[60\] will be used in the proof of Theorem \[10\] after a reduction to nearby germs with smooth irreducible components. Its proof essentially relies on parts and of the following \[48\] \[48a\] Let $D=\{xy=0\}$ be a normal crossing of $2$ irreducible components. Then $\frac{dx}{x}\in\Omega^1(\log D)$ and $$(x+y)\frac{dx}{x}=y\frac{d(xy)}{xy}+dx-dy$$ shows that $$\rho_D\left(\frac{dx}{x}\right)=\frac{y}{x+y}\Big\vert_D.$$ On the components $D_1=\{x=0\}$ and $D_2=\{y=0\}$ of the normalization ${\widetilde}D=D_1\coprod D_2$, this function equals $1$ and $0$ respectively and is therefore not in ${{\mathscr}{O}}_D$. By symmetry, this yields $${{\mathscr}{R}}_D={{\mathscr}{O}}_{{\widetilde}D}={{\mathscr}{O}}_{D_1}\times{{\mathscr}{O}}_{D_2}={{\mathscr}{J}}_D^\vee$$ since ${{\mathscr}{J}}_D={{\left\langlex,y\right\rangle}}_{{{\mathscr}{O}}_D}$ is the maximal ideal in ${{\mathscr}{O}}_D={\mathds{C}}\{x,y\}/{{\left\langlexy\right\rangle}}$. This observation will be generalized in Proposition \[22\]. \[48b\] Conversely, assume that $D_1=\{h_1=x=0\}$ and $D_2=\{h_2=x+y^m=0\}$ are two smooth irreducible components of $D$. Consider the logarithmic $1$-form $$\omega=\frac{ydx-mxdy}{x(x+y^m)}=y^{1-m}\left(\frac{dh_1}{h_1}-\frac{dh_2}{h_2}\right)\in\Omega^1(\log(D_1+D_2))\subset\Omega^1(\log D).$$ Its residue $\rho_D(\omega)\vert_{D_1}=y^{1-m}\vert_{D_1}$ has a pole along $D_1\cap D_2$ unless $m=1$. Thus, if ${{\mathscr}{O}}_{{\widetilde}D}={{\mathscr}{R}}_D$ then $D_1$ and $D_2$ must intersect transversally. \[48c\] Assume that $D$ contains $D'=D_1\cup D_2\cup D_3$ with $D_1=\{x=0\}$, $D_2=\{y=0\}$, and $D_3=\{x-y=0\}$. Consider the logarithmic $1$-form $$\omega=\frac{1}{x-y}\left(\frac{dx}{x}-\frac{dy}{y}\right)\in\Omega^1(\log D')\subset\Omega^1(\log D).$$ Its residue $\rho_D(\omega)\vert_{D_1}=-\frac{1}{y}\vert_{D_1}$ has a pole along $D_1\cap D_2\cap D_3$ and hence ${{\mathscr}{O}}_{{\widetilde}D}\subsetneq{{\mathscr}{R}}_D$. After these preparations, we shall now approach the construction of the dual logarithmic residue. By definition, there is a short exact residue sequence $$\label{2} {\SelectTips{cm}{}\xymatrix}{ 0\ar[r]&\Omega^1_S\ar[r]&\Omega^1(\log D)\ar[r]^-{\rho_D}&{{\mathscr}{R}}_D\ar[r]&0. }$$ Applying $\operatorname{Hom}_{{{\mathscr}{O}}_S}(-,{{\mathscr}{O}}_S)$ to gives an exact sequence $$\label{44} {\SelectTips{cm}{}\xymatrix}@C-1em{ 0&\operatorname{Ext}^1_{{{\mathscr}{O}}_S}(\Omega^1(\log D),{{\mathscr}{O}}_S)\ar[l]&\operatorname{Ext}^1_{{{\mathscr}{O}}_S}({{\mathscr}{R}}_D,{{\mathscr}{O}}_S)\ar[l]&\Theta_S\ar[l]&\operatorname{Der}(-\log D)\ar[l]&0\ar[l]. }$$ The right end of this sequence extends to the short exact sequence . For the hypersurface ring ${{\mathscr}{O}}_D$, the change of rings spectral sequence $$\label{41} E^{p,q}_2=\operatorname{Ext}_{{{\mathscr}{O}}_D}^p(-,\operatorname{Ext}^q_{{{\mathscr}{O}}_S}({{\mathscr}{O}}_D,{{\mathscr}{O}}_S))\underset{p}\Rightarrow\operatorname{Ext}_{{{\mathscr}{O}}_S}^{p+q}(-,{{\mathscr}{O}}_S)$$ degenerates because $E^{p,q}_2=0$ if $q\neq 1$ and hence $$\label{43} \operatorname{Ext}_{{{\mathscr}{O}}_S}^1(-,{{\mathscr}{O}}_S)\cong E^{0,1}_2\cong\operatorname{Hom}_{{{\mathscr}{O}}_D}(-,{{\mathscr}{O}}_D)\cong-^\vee.$$ Therefore, the second term in the sequence  is ${{\mathscr}{R}}_D^\vee$. This motivates the following key technical result of this paper, describing the dual of Saito’s logarithmic residue. \[22\] There is an exact sequence $$\label{36} {\SelectTips{cm}{}\xymatrix}{ 0&\operatorname{Ext}^1_{{{\mathscr}{O}}_S}(\Omega^1(\log D),{{\mathscr}{O}}_S)\ar[l]&{{\mathscr}{R}}_D^\vee\ar[l]&\Theta_S\ar[l]_-{\sigma_D}&\operatorname{Der}(-\log D)\ar[l]&0\ar[l] }$$ such that $\sigma_D(\delta)(\rho_D(\omega))=dh(\delta)\cdot\rho_D(\omega)$. In particular, $\sigma_D(\Theta_S)={{\mathscr}{J}}_D$ as fractional ideals. Moreover, ${{\mathscr}{J}}_D^\vee={{\mathscr}{R}}_D$ as fractional ideals. The spectral sequence applied to ${{\mathscr}{R}}_D$ is associated with $$\operatorname{RHom}_{{{\mathscr}{O}}_S}(\Omega_S^1{\hookrightarrow}\Omega^1(\log D),h\colon{{\mathscr}{O}}_S\to{{\mathscr}{O}}_S).$$ Expanding the double complex $\operatorname{Hom}_{{{\mathscr}{O}}_S}(\Omega_S^1{\hookrightarrow}\Omega^1(\log D),h\colon{{\mathscr}{O}}_S\to{{\mathscr}{O}}_S)$, we obtain the following diagram of long exact sequences: $$\label{3}\small {\SelectTips{cm}{}\xymatrix}@C-2em{ &0\ar[d]&0\ar[d]\\ \operatorname{Ext}_{{{\mathscr}{O}}_S}^1({{\mathscr}{R}}_D,{{\mathscr}{O}}_S)\ar[d]^-{0}&\operatorname{Hom}_{{{\mathscr}{O}}_S}(\Omega^1_S,{{\mathscr}{O}}_S)\ar[l]\ar[d]^-{h}&\operatorname{Hom}_{{{\mathscr}{O}}_S}(\Omega^1(\log D),{{\mathscr}{O}}_S)\ar[l]\ar[d]^-{h}&0\ar[l]\\ \operatorname{Ext}_{{{\mathscr}{O}}_S}^1({{\mathscr}{R}}_D,{{\mathscr}{O}}_S)&\operatorname{Hom}_{{{\mathscr}{O}}_S}(\Omega^1_S,{{\mathscr}{O}}_S)\ar[l]\ar[d]&\operatorname{Hom}_{{{\mathscr}{O}}_S}(\Omega^1(\log D),{{\mathscr}{O}}_S)\ar[l]\ar[d]&0\ar[l]\ar[d]\\ &\operatorname{Hom}_{{{\mathscr}{O}}_S}(\Omega^1_S,{{\mathscr}{O}}_D)\ar[d]&\operatorname{Hom}_{{{\mathscr}{O}}_S}(\Omega^1(\log D),{{\mathscr}{O}}_D)\ar[l]\ar[d]&{{\mathscr}{R}}_D^\vee\ar[l]_-{\rho_D^\vee}\ar[d]^-\alpha&0\ar[l]\\ &0&\operatorname{Ext}^1_{{{\mathscr}{O}}_S}(\Omega^1(\log D),{{\mathscr}{O}}_S)\ar[l]&\operatorname{Ext}^1_{{{\mathscr}{O}}_S}({{\mathscr}{R}}_D,{{\mathscr}{O}}_S)\ar[l]\ar[d]\\ &&&0 }$$ We can define a homomorphism $\sigma_D$ from the upper left $\operatorname{Hom}_{{{\mathscr}{O}}_S}(\Omega^1_S,{{\mathscr}{O}}_S)$ to the lower right ${{\mathscr}{R}}_D^\vee$ by a diagram chasing process and we find that $\delta\in\Theta_S=\operatorname{Hom}_{{{\mathscr}{O}}_S}(\Omega^1_S,{{\mathscr}{O}}_S)$ maps to $$\sigma_D(\delta)={{\left\langleh\delta,\rho_D^{-1}(-)\right\rangle}}\vert_D\in{{\mathscr}{R}}_D^\vee$$ and that is exact. By comparison with the spectral sequence, one can check that $\alpha$ is the change of rings isomorphism applied to ${{\mathscr}{R}}_D$, and that $\alpha\circ\sigma_D$ coincides with the connecting homomorphism of the top row of the diagram, which is the same as the one in . Let $\rho_D(\omega)\in{{\mathscr}{R}}_D$ where $\omega\in\Omega^1(\log D)$. Following the definition of $\rho_D$ in , we write $\omega$ in the form . Then we compute $$\begin{aligned} \nonumber\sigma_D(\delta)(\rho_D(\omega))&=\\\label{5} {{\left\langleh\delta,\omega\right\rangle}}\vert_D&= dh(\delta)\cdot\frac{\xi}{g}\vert_D+h\cdot\frac{{{\left\langle\delta,\eta\right\rangle}}}{g}\vert_D= dh(\delta)\cdot\rho_D(\omega)\end{aligned}$$ which proves the first two claims. For the last claim, we consider the diagram dual to : $$\small {\SelectTips{cm}{}\xymatrix}@C-2em{ &&0\ar[d]&0\ar[d]\\ &0\ar[r]&\operatorname{Hom}_{{{\mathscr}{O}}_S}(\Theta_S,{{\mathscr}{O}}_S)\ar[r]\ar[d]^-{h}&\operatorname{Hom}_{{{\mathscr}{O}}_S}(\operatorname{Der}(-\log D),{{\mathscr}{O}}_S)\ar[r]\ar[d]^-{h}&\operatorname{Ext}_{{{\mathscr}{O}}_S}^1({{\mathscr}{J}}_D,{{\mathscr}{O}}_S)\ar[d]^-{0}\\ &0\ar[r]&\operatorname{Hom}_{{{\mathscr}{O}}_S}(\Theta_S,{{\mathscr}{O}}_S)\ar[r]\ar[d]&\operatorname{Hom}_{{{\mathscr}{O}}_S}(\operatorname{Der}(-\log D),{{\mathscr}{O}}_S)\ar[r]\ar[d]&\operatorname{Ext}_{{{\mathscr}{O}}_S}^1({{\mathscr}{J}}_D,{{\mathscr}{O}}_S)\\ 0\ar[r]&{{\mathscr}{J}}_D^\vee\ar[d]^\beta\ar[r]^-{dh^\vee}&\operatorname{Hom}_{{{\mathscr}{O}}_S}(\Theta_S,{{\mathscr}{O}}_D)\ar[r]\ar[d]&\operatorname{Hom}_{{{\mathscr}{O}}_S}(\operatorname{Der}(-\log D),{{\mathscr}{O}}_D)\\ &\operatorname{Ext}_{{{\mathscr}{O}}_S}^1({{\mathscr}{J}}_D,{{\mathscr}{O}}_S)\ar[r]&0 }$$ As before, we construct a homomorphism $\rho'_D$ from the upper right $\operatorname{Hom}_{{{\mathscr}{O}}_S}(\operatorname{Der}(-\log D),{{\mathscr}{O}}_S)$ to the lower left ${{\mathscr}{J}}_D^\vee$ such that $\beta\circ\rho'_D$ coincides with the connecting homomorphism of the top row of the diagram, where $\beta$ is the change of rings isomorphism applied to ${{\mathscr}{J}}_D$. By the diagram, $\omega\in\Omega^1(\log D)=\operatorname{Hom}_{{{\mathscr}{O}}_S}(\operatorname{Der}(-\log D),{{\mathscr}{O}}_S)$ maps to $$\rho'_D(\omega)={{\left\langleh\omega,dh^{-1}(-)\right\rangle}}\vert_D\in {{\mathscr}{J}}_D^\vee$$ which gives a short exact sequence $$\label{6} {\SelectTips{cm}{}\xymatrix}{ 0\ar[r]&\Omega^1_S\ar[r]&\Omega^1(\log D)\ar[r]^-{\rho'_D}&{{\mathscr}{J}}_D^\vee\ar[r]&0 }$$ similar to the sequence . Using and , we compute $$\rho'_D(\omega)(\delta(h))= \rho'_D(\omega)(dh(\delta))= {{\left\langleh\omega,\delta\right\rangle}}\vert_D= \rho_D(\omega)\cdot dh(\delta)= \rho_D(\omega)\cdot \delta(h)$$ for any $\delta(h)\in {{\mathscr}{J}}_D$ where $\delta\in\Theta_S$. Hence, $\rho'_D=\rho_D$ and the last claim follows using and . \[37\] There is a chain of fractional ideals $${{\mathscr}{J}}_D\subseteq{{\mathscr}{R}}_D^\vee\subseteq{{\mathscr}{C}}_D\subseteq{{\mathscr}{O}}_D\subseteq{{\mathscr}{O}}_{{\widetilde}D}\subseteq{{\mathscr}{R}}_D$$ in ${{\mathscr}{M}}_D$ where ${{\mathscr}{C}}_D={{\mathscr}{O}}_{{\widetilde}D}^\vee$ is the conductor ideal of $\pi$. In particular, ${{\mathscr}{J}}_D\subseteq{{\mathscr}{C}}_D$. By Lemma \[25\], ${{\mathscr}{J}}_D$ is a fractional ideal contained in ${{\mathscr}{R}}_D^\vee$ by Proposition \[22\]. By [@Sai80 (2.7),(2.8)], ${{\mathscr}{R}}_D$ is a finite ${{\mathscr}{O}}_D$-module containing ${{\mathscr}{O}}_{{\widetilde}D}$ and hence a fractional ideal. The remaining inclusions and fractional ideals are then obtained using Proposition \[18\]. \[24\] If $D$ is free then ${{\mathscr}{J}}_D={{\mathscr}{R}}_D^\vee$ as fractional ideals. If $D$ is free then the Ext-module in the exact sequence disappears and $\sigma_D$ becomes surjective. Then the claim is part of the statement of Proposition \[22\]. By Corollary \[37\], the inclusion ${{\mathscr}{O}}_{{\widetilde}D}\subset R_D$ always holds. For a free divisor $D$, the case of equality is translated into more familiar terms by the following \[42\] If $D$ is free then ${{\mathscr}{R}}_D={{\mathscr}{O}}_{{\widetilde}D}$ is equivalent to ${{\mathscr}{J}}_D={{\mathscr}{C}}_D$. By the preceding Corollary \[24\] and the last statement of Proposition \[22\], the freeness of $D$ implies that ${{\mathscr}{R}}_D$ and ${{\mathscr}{J}}_D$ are mutually ${{\mathscr}{O}}_D$-dual. By Lemma \[39\] and the definition of the conductor, the same holds true for ${{\mathscr}{O}}_{{\widetilde}D}$ and ${{\mathscr}{C}}_D$. The claim follows. Algebraic normal crossing conditions ==================================== In this section, we prove our main Theorem \[10\] settling the missing implication in Theorem \[28\]. We begin with some general preparations. \[35\] Any map $\phi\colon Y\to X$ of analytic germs with $\Omega^1_{Y/X}=0$ is an immersion. The map $\phi$ can be embedded in a map $\Phi$ of smooth analytic germs: $${\SelectTips{cm}{}\xymatrix}{ Y\ar@{^(->}[r]\ar[d]^-\phi&T\ar[d]^-\Phi\\ X\ar@{^(->}[r]&S. }$$ Setting $\Phi_i=x_i\circ\Phi$ and $\phi_i=\Phi_i+{{\mathscr}{I}}_Y$ for coordinates $x_1,\dots,x_n$ on $S$ and ${{\mathscr}{I}}_Y$ the defining ideal of $Y$ in $T$, we can write $\Phi=(\Phi_1,\dots,\Phi_n)$ and $\phi=(\phi_1,\dots,\phi_n)$ and hence $$\label{26} \Omega^1_{Y/X}=\frac{\Omega^1_Y}{\sum_{i=1}^n{{\mathscr}{O}}_Yd\phi_i}=\frac{\Omega^1_T}{{{\mathscr}{O}}_Td{{\mathscr}{I}}_Y+\sum_{i=1}^n{{\mathscr}{O}}_Td\Phi_i}.$$ We may choose $T$ of minimal dimension so that ${{\mathscr}{I}}_Y\subseteq{\mathfrak{m}}_T^2$ and hence $d{{\mathscr}{I}}_Y\subseteq{\mathfrak{m}}_T\Omega^1_T$. Now and the hypothesis $\Omega^1_{Y/X}=0$ show that $\Omega^1_T=\sum_{i=1}^n{{\mathscr}{O}}_Td\Phi_i+{\mathfrak{m}}_T\Omega^1_T$ which implies that $\Omega^1_T=\sum_{i=1}^n{{\mathscr}{O}}_Td\Phi_i$ by Nakayama’s Lemma. But then $\Phi$ and hence $\phi$ is a closed embedding as claimed. \[7\] If ${{\mathscr}{J}}_D={{\mathscr}{C}}_D$ and ${\widetilde}D$ is smooth then $D$ has smooth irreducible components. By definition, the ramification ideal of $\pi$ is the Fitting ideal $${{\mathscr}{R}}_\pi:={{\mathscr}{F}}^0_{{{\mathscr}{O}}_{{\widetilde}D}}(\Omega^1_{{\widetilde}D/D}).$$ As a special case of a result of Ragnie Piene [@Pie79 Cor. 1, Prop. 1] (see also [@OZ87 Cor. 2.7]), $${{\mathscr}{C}}_D{{\mathscr}{R}}_\pi={{\mathscr}{J}}_D{{\mathscr}{O}}_{{\widetilde}D}$$ By hypothesis, this becomes $${{\mathscr}{C}}_D{{\mathscr}{R}}_\pi={{\mathscr}{C}}_D$$ since ${{\mathscr}{C}}_D$ is an ideal in both ${{\mathscr}{O}}_D$ and ${{\mathscr}{O}}_{{\widetilde}D}$. By Nakayama’s lemma, it follows that that ${{\mathscr}{R}}_\pi={{\mathscr}{O}}_{{\widetilde}D}$ and hence that $\Omega^1_{{\widetilde}D/D}=0$. Since ${\widetilde}D$ is normal, irreducible and connected components coincide. By localization to a connected component ${\widetilde}D_i$ of ${\widetilde}D$ and base change to $D_i=\pi({\widetilde}D_i)$ (see [@Har77 Ch. II, Prop. 8.2A]), we obtain $\Omega^1_{{\widetilde}D_i/D_i}=0$. Then the normalization ${\widetilde}D_i\to D_i$ is an immersion by Lemma \[35\] and hence $D_i={\widetilde}D_i$ is smooth. We are now ready to prove our main results. In codimension $1$, $D$ is free by Corollary \[27\] and hence ${{\mathscr}{J}}_D={{\mathscr}{C}}_D$ by Corollary \[42\] and our hypothesis. Moreover, ${\widetilde}D$ is smooth in codimension $1$ by normality. By our language convention, this means that there is an analytic subset $A\subset D$ of codimension at least $2$ such that, for $p\in D\setminus A$, ${{\mathscr}{J}}_{D,p}={{\mathscr}{C}}_{D,p}$ and ${\widetilde}D$ is smooth above $p$. From Lemma \[7\] we conclude that the local irreducible components $D_i$ of the germ $(D,p)$ are smooth. The hypothesis ${{\mathscr}{R}}_D={{\mathscr}{O}}_{{\widetilde}D}$ at $p$ then reduces to the equality ${{\mathscr}{R}}_{D,p}=\bigoplus{{\mathscr}{O}}_{D_i}$. Thus, the implication $\Leftarrow$ in Theorem \[60\] yields the claim. In order to prove that implies , we may assume that $Z$ is smooth and reduce to the case of a plane curve as in the proof of Theorem \[40\]. Then the Mather–Yau theorem [@MY82] applies (see [@Fab12 Prop. 9] for details). Now assume that $D$ is free and normal crossing in codimension $1$. By the first assumption and Theorem \[19\], $Z$ is Cohen–Macaulay of codimension $1$ and, in particular, satisfies Serre’s condition $S_1$. By the second assumption, $Z$ also satisfies Serre’s condition $R_0$. Then $Z$ is reduced, and hence ${{\mathscr}{J}}_D$ is radical, by Serre’s reducedness criterion. This proves that implies for free $D$. The last equivalence then follows from Theorems \[28\] and \[10\] and Corollary \[42\]. By Theorems \[28\], \[10\], and \[16\], we may assume that ${{\mathscr}{J}}_D={{\mathscr}{C}}_D$. Then Lemma \[7\] shows that the irreducible components $D_i=\{h_i=0\}$, $i=1,\dots,m$, of $D$ are smooth, and hence normal. It follows that $${{\mathscr}{R}}_D={{\mathscr}{O}}_{{\widetilde}D}=\bigoplus_{i=1}^m{{\mathscr}{O}}_{{\widetilde}D_i}=\bigoplus_{i=1}^m{{\mathscr}{O}}_{D_i}.$$ By the implication $\Leftarrow$ in Theorem \[60\], this is equivalent to $$\label{51} \Omega^1(\log D)=\sum_{i=1}^m{{\mathscr}{O}}_S\frac{dh_i}{h_i}+\Omega^1_S.$$ On the other hand, Saito’s criterion [@Sai80 (1.8) i)] for freeness of $D$ reads $$\label{52} \bigwedge^n\Omega^1(\log D)=\Omega^n_S(D).$$ Combining and , it follows immediately that $D$ is normal crossing (see also [@Fab12 Prop. B]): As ${{\mathscr}{O}}_S$-module and modulo $\Omega_S^n$, the left hand side of is, due to , generated by expressions $$\begin{gathered} \label{53} \frac{d h_{i_1}\wedge\dots\wedge d h_{i_k}\wedge dx_{j_1}\wedge\dots\wedge dx_{j_{n-k}}}{h_{i_1}\cdots h_{i_k}},\\ \nonumber1\le i_1<\cdots<i_k\le m, \quad 1\le j_1<\cdots<j_{n-k}\le n\ge k\end{gathered}$$ whereas the right hand side of is generated by $$\frac{d x_1\wedge\dots\wedge d x_n}{h_1\cdots h_m}.$$ In order for an instance of to attain the denominator of this latter expression, it must satisfy $k=m$, and, in particular, $m\le n$. Further, comparing numerators, $d h_1\wedge\dots\wedge d h_m\wedge dx_{j_1}\wedge\dots\wedge dx_{j_{n-m}}$ must be a unit multiple of $d x_1\wedge\dots\wedge d x_n$. In other words, choosing $i_1,\dots,i_m$ such that $\{i_1,\dots,i_m,j_1,\dots,j_{n-m}\}=\{1,\dots,n\}$, $$\frac{{\partial}(h_1,\dots,h_m)}{{\partial}(x_{i_1},\dots,x_{i_m})}\in{{\mathscr}{O}}_S^*.$$ By the implicit function theorem, $h_1,\dots,h_m,x_{j_1},\dots,x_{j_{n-m}}$ is then a coordinate system and hence $D$ is a normal crossing divisor as claimed. Gorenstein singular locus ========================= In this section, we describe the canonical module of the singular locus $Z$ in terms of the module ${{\mathscr}{R}}_D$ of logarithmic residues. Then, we prove Theorem \[40\]. The complex of logarithmic differential forms along $D$ relative to the map $h\colon S\to T:=({\mathds{C}},0)$ defining $D$ is defined as $\Omega^\bullet(\log h):=\Omega^\bullet(\log D)/\frac{dh}h\wedge\Omega^{\bullet-1}(\log D)$ (see [@GMS09 §22] or [@DS12 Def. 2.7]). \[20\] The ${{\mathscr}{O}}_Z$-module ${{\mathscr}{R}}_h:={{\mathscr}{R}}_D/{{\mathscr}{O}}_D$ fits into an exact square $${\SelectTips{cm}{}\xymatrix}{ &0\ar[d]&0\ar[d]&0\ar[d]\\ 0\ar[r]&{{\mathscr}{O}}_S\ar[r]^-h\ar[d]^-{dh}&{{\mathscr}{O}}_S\ar[r]\ar[d]^-{\frac{dh}{h}}&{{\mathscr}{O}}_D\ar[r]\ar[d]&0\\ 0\ar[r]&\Omega^1_S\ar[r]\ar[d]&\Omega^1(\log D)\ar[r]^-{\rho_D}\ar[d]&{{\mathscr}{R}}_D\ar[r]\ar[d]&0\\ 0\ar[r]&\Omega^1_{S/T}\ar[r]\ar[d]&\Omega^1(\log h)\ar[r]\ar[d]&{{\mathscr}{R}}_h\ar[r]\ar[d]&0\\ &0&0&0 }.$$ If $D$ is free then $Z$ is Cohen–Macaulay with canonical module $\omega_Z={{\mathscr}{R}}_h$; in particular, $Z$ is Gorenstein if and only if ${{\mathscr}{R}}_D$ is generated by $1$ and one additional generator. We set $\omega_\emptyset:=0$ in case $D$ is smooth and assume $Z\ne\emptyset$ in the following. The exact square arises from the residue sequence using the Snake Lemma. Dualizing the short exact sequence $${\SelectTips{cm}{}\xymatrix}{ 0\ar[r]&{{\mathscr}{J}}_D\ar[r]&{{\mathscr}{O}}_D\ar[r]&{{\mathscr}{O}}_Z\ar[r]&0, }$$ one computes $$\label{29} \operatorname{Ext}_{{{\mathscr}{O}}_D}^1({{\mathscr}{O}}_Z,{{\mathscr}{O}}_D)={{\mathscr}{J}}_D^\vee/{{\mathscr}{O}}_D={{\mathscr}{R}}_D/{{\mathscr}{O}}_D={{\mathscr}{R}}_h.$$ which shows that ${{\mathscr}{R}}_h$ is an ${{\mathscr}{O}}_Z$-module. If $D$ is free then, by Theorem \[19\], $Z$ is Cohen–Macaulay of codimension $1$ and $\omega_Z={{\mathscr}{R}}_h$ by . Part  of the following proposition can also be found in Faber’s thesis (see [@Fab11 Prop. 1.29]). \[56\] Let $D$ be a free divisor. Then the following statements hold true: \[56e\] $\frac{dh}h$ is part of an ${{\mathscr}{O}}_S$-basis of $\Omega^1(\log D)$ if and only if $D$ is Euler homogeneous. \[56d\] $\frac{dh}h$ is part of an ${{\mathscr}{O}}_S$-basis of $\Omega^1(\log D)$ if and only if $1$ is part of a minimal set of ${{\mathscr}{O}}_D$-generators of ${{\mathscr}{R}}_D$. \[56g\] ${{\mathscr}{R}}_D$ is a cyclic ${{\mathscr}{O}}_D$-module if and only if $D$ is smooth. This is immediate from the existence of a dual basis and the fact that $\chi\in\operatorname{Der}(-\log D)$ is an Euler vector field exactly if ${{\left\langle\chi,\frac{dh}h\right\rangle}}=\frac{\chi(h)}h=1$. Then $\chi$ can be chosen as a member of some basis. We may assume that $D\cong D'\times S'$ with $S'=({\mathds{C}}^r,0)$ implies $r=0$. Indeed, a basis of $\Omega^1(\log D)$ is the union of bases of $\Omega^1(\log D')$ and $\Omega_{S'}^1$. This assumption is equivalent to $\operatorname{Der}(-\log D)\subseteq{\mathfrak{m}}_S\Theta_S$, where ${\mathfrak{m}}_S$ denotes the maximal ideal of ${{\mathscr}{O}}_S$. Dually, this means that no basis element of $\Omega^1(\log D)$ can lie in $\Omega_S^1$. Consider a basis $\omega_1,\dots,\omega_n$ of $\Omega^1(\log D)$ with $\omega_1=\frac{dh}h$ and set $\rho_i:=\rho_D(\omega_i)$ for $i=1,\dots,n$. If $\rho_1=1$ is not a member of some minimal set of generators of ${{\mathscr}{R}}_D$ then $\rho_1=\sum_{i=2}^na_i\rho_i$ with $a_i\in{\mathfrak{m}}_S$. Thus, the form $\omega_1':=\omega_1-\sum _{i=2}^na_i\omega_i$ can serve as a replacement for $\omega_1$ in the basis. But, by construction, $\rho_D(\omega'_1)=0$ which implies that $\omega'_1\in\Omega_S$ by . This contradicts our assumption on $D$. Conversely, suppose that $\frac{dh}h$ is not a member of any ${{\mathscr}{O}}_S$-basis $\omega_1,\dots,\omega_n$ of $\Omega^1(\log D)$. Then, by Nakayama’s lemma, there are $a_i\in{\mathfrak{m}}_S$ such that $\frac{dh}h=\sum_{i=1}^na_i\omega_i$. Applying $\rho_D$, this gives $1=\rho_D(\frac{dh}h)=\sum_{i=1}^na_i\rho_i\in{\mathfrak{m}}_D{{\mathscr}{R}}_D$. Again by Nakayama’s lemma, this means that $1$ is not a member of any minimal set of ${{\mathscr}{O}}_D$-generators of ${{\mathscr}{R}}_D$. If ${{\mathscr}{R}}_D\cong{{\mathscr}{O}}_D$ then also ${{\mathscr}{J}}_D\cong{{\mathscr}{O}}_D$ by Corollary \[24\] then $D$ must be smooth by Lipman’s criterion [@Lip69]. Alternatively, it follows that ${{\mathscr}{J}}_h+{{\mathscr}{O}}_S\cdot h={{\mathscr}{O}}_S$ and hence also ${{\mathscr}{J}}_h={{\mathscr}{O}}_S$ (see Remark \[9\]); so $D$ is smooth by the Jacobian criterion. As observed by Faber [@Fab12 Rmk. 54], condition  of Proposition \[56\] is satisfied given ${{\mathscr}{R}}_D={{\mathscr}{O}}_{{\widetilde}D}$ and one obtains Any free divisor $D$ with ${{\mathscr}{R}}_D={{\mathscr}{O}}_{{\widetilde}D}$ is Euler homogeneous. Assume that $Z$ is Gorenstein of codimension $1$ in $D$. The preimage ${{\mathscr}{J}}'_D$ of ${{\mathscr}{J}}_D$ in ${{\mathscr}{O}}_S$ is then a Gorenstein ideal of height $2$. As such, it is a complete intersection ideal by a theorem of Serre (see [@Eis95 Cor. 21.20]), and hence generated by two of the generators $h,{\partial}_1(h),\dots,{\partial}_n(h)$. Then the triviality lemma [@Sai80 (3.5)] shows that $D$ is either smooth or $D\cong C\times({\mathds{C}}^{n-2},0)$ and $C\subset({\mathds{C}}^2,0)$ a plane curve. Then also $C$ has Gorenstein singular locus and is hence quasihomogeneous by [@KW84]. Alternatively, the last implication follows from Propositions \[20\] and \[56\] using that quasihomogeneity of $C$ follows from Euler homogeneity of $C$ by Saito’s quasihomogeneity criterion [@Sai71] for isolated singularities. Finally, $D$ is quasihomogeneous and $Z$ is a complete intersection. The converse is trivial. Acknowledgements {#acknowledgements .unnumbered} ================ We are grateful to Eleonore Faber and David Mond for helpful discussions and comments. The foundation for this paper was laid during a “Research in Pairs” stay at the “Mathematisches Forschungsinstitut Oberwolfach” in summer 2011. [^1]: The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement n^o^ PCIG12-GA-2012-334355. [^2]: The proof in loc. cit. applies verbatim to the case where $B$ is semilocal.

--- abstract: 'It was recently argued that $f(T)$ gravity could inherit “remnant symmetry” from the full Lorentz group, despite the fact that the theory is not locally Lorentz invariant. Confusion has arisen regarding the implication of this result for the previous works, which established that $f(T)$ gravity is pathological due to superluminal propagation, local acausality, and non-unique time evolution. We clarify that the existence of the “remnant group” does not rid the theory of these various problems, but instead strongly supports it.' author: - Pisin Chen - Keisuke Izumi - 'James M. Nester' - Yen Chin Ong title: 'Remnant Symmetry, Propagation and Evolution in $f(T)$ Gravity' --- Introduction: $f(T)$ Gravity and Remnant Symmetry ================================================= General Relativity \[GR\] is a geometric theory of gravity formulated on a Lorentzian manifold equipped with the Levi-Civita connection. This connection is torsion-less and metric compatible – the gravitational field is completely described in terms of the Riemann curvature tensor. However, given a smooth manifold one can equip it with other connections. If one chooses to use the Weitzenböck connection, then the geometry is *flat* – the connection being curvature-free \[but still metric compatible\]. The gravitational field is now completely described in terms of the torsion tensor. Surprisingly GR can be recast into “teleparallel equivalent of GR” \[TEGR, or GR$_{\|}$\] which employs the Weitzenböck connection, a subject which has a large set of literature \[see, e.g., [@Moller:1961jj; @Cho:1975dh; @Hehl:1994ue; @Itin:1999wi; @deAndrade:2000kr; @BtelepHam; @Bgauge; @Hehl:2012pi; @Pereira.book]\]. The dynamical variable of TEGR, as well as its $f(T)$ extension [@eric], is the frame field \[vierbein\] $\{{e}_a(x)\}$, or equivalently its corresponding co-frame field $\{e^a (x) \}$. The vierbein is related to the coordinate vector fields $\{\partial_\mu\}$ by $e_a(x) = e_a^{~\mu} (x) \partial_\mu$, and similarly $e^a(x) =e^a_{~\mu} (x) dx^\mu$. The vierbein ${{e}_a(x)}$ forms an orthonormal basis for the tangent space $T_xM$ at each point $x$ of a given spacetime manifold $(M,g)$. The metric tensor $g$ is related to the vierbein field by $$g_{\mu\nu}(x)=\eta_{ab}\, e^a_{~\mu} (x)\, e^b_{~\nu} (x).$$ The Weitzenböck connection is defined by $$\overset{\mathbf{w}}{\nabla}_X Y :=(XY^a){e}_a,$$ where $Y=Y^a {e}_a$. This means that we declare the vierbein field to be *teleparallel*, i.e., covariantly constant: $\overset{\mathbf{w}}\nabla_X {e}_a = 0$. Equivalently, the connection coefficients are $$\label{Weitzenb} {\Gamma}^\lambda{}_{ \nu\mu} =e^\lambda_{~a}\: \partial_\mu e^a_{~\nu}.$$ It is then straightforward to show that this connection is curvature-less but the torsion tensor is nonzero in general. Under a linear transformation of the bases $\{e_a(x)\}$ of the tangent vector field $$e_a(x) \to e'_a(x) = L_a^{~b}(x) e_b(x), ~~\text{det}(L_a^{~b}) \neq 0,$$ the connection 1-form $$\Gamma^b{}_a(x) = \langle \theta^b, \overset{\mathbf{w}}{\nabla} e_a \rangle = {\Gamma}^b{}_{\mu a} dx^\mu,$$ transforms as $$\Gamma'^a{}_{\mu b} = (L^{-1})^a_{~d}\Gamma^d{}_{\mu c}L^c_{~b} + (L^{-1})^a_{~c}L^c_{~b,\mu},$$ where the comma in the subscript denotes the usual partial differentiation. We would like to emphasize that there is in fact a difference between “parallelizable” and “teleparallel”. A parallelizable manifold $M$ means that there exists a global frame field on $M$ \[that is, the frame bundle $FM$ has a global section\]. For example, $S^3$ is parallelizable but $S^2$ is not. Whether a manifold is parallelizable or not depends on the topology but *not* on the connection. In 4-dimensions, the necessary and sufficient condition for parallelizability is the vanishing of the second Stiefel-Whitney characteristic class. Teleparallel geometry means one has a *connection* which is flat everywhere, i.e., has vanishing curvature. \[A manifold with a teleparallel connection is always parallelizable.\] One then defines the contortion tensor, which is the difference between the Weitzenböck and Levi-Civita connections. In component form, it reads $$K^{\mu\nu}_{\:\:\:\:\rho}=-\frac{1}{2}\Big(T^{\mu\nu}_{ \:\:\:\:\rho} -T^{\nu\mu}_{\:\:\:\:\rho}-T_{\rho}^{\:\:\:\:\mu\nu}\Big).$$ For convenience, one usually also defines the tensor $$S_\rho^{\:\:\:\mu\nu}=\frac{1}{2}\Big(K^{\mu\nu}_{\:\:\:\:\rho} +\delta^\mu_\rho \:T^{\alpha\nu}_{\:\:\:\:\alpha}-\delta^\nu_\rho\: T^{\alpha\mu}_{\:\:\:\:\alpha}\Big).$$ In TEGR, the Teleparallel Lagrangian consists of the so-called “torsion scalar” $$T := S_\rho^{\:\:\:\mu\nu}\:T^\rho_{\:\:\:\mu\nu}.$$ It turns out that the “torsion scalar” only differs from the Ricci scalar \[obtained from the usual Levi-Civita connection\] by a boundary term: $T = -R + \text{div}(\cdot)$, and so it encodes all the dynamics of GR. One could then promote $T$ to a function $f(T)$, similar to how GR is generalized to $f(R)$ gravity. For a general $f$, this would lead to a dynamical gravity theory that would have second order field equations \[whereas $f(R)$ gravity gives higher order equations\], with some kind of non-linear dynamics that differs from GR but nevertheless reduces to GR in a certain limit. The hope was that this could potentially explain the acceleration of the universe [@eric]. It is well-known that, unlike TEGR, generalized theories such as $f(T)$ are *not* locally Lorentz invariant [@barrow]. Of course, at a purely mathematical level, a given manifold that can be parallelized admits infinitely many choices of vierbein. However, in a general teleparallel theory, such as $f(T)$ gravity, there exists a *preferred frame* compatible with the field equations. \[This is the difference between *kinematics* and *dynamics*; the latter is determined by a Lagrangian.\] While TEGR has local Lorentz symmetry, and like GR has just 2 dynamical degrees of freedom, for all non-trivial $f$’s it was thought that $f(T)$ theory would be a preferred frame theory with 5 degrees of freedom and no local Lorentz symmetry. The main message in a recent work by Ferraro and Fiorini [@1412.3424] is that this common belief is in fact much more subtle. They argued that, depending on the spacetime manifold, $f(T)$ gravity may “inherit” some “remnant symmetry” from the full \[orthochronous\] Lorentz group, and therefore there could exist more than one such preferred frame – even infinitely many. More precisely, Ferraro and Fiorini discovered a remarkable yet simple result that $f(T)$ gravity is only invariant under Lorentz transformations of the vierbein satisfying $$\label{1} d(\epsilon_{abcd} e^a \wedge e^b \wedge \eta^{de} L^c_{~f} (L^{-1})^f_{~e,\mu} dx^\mu)=0.$$ In other words, while this is fulfilled by a global Lorentz transformations, there are *local* Lorentz transformations that could also satisfy Eq.(\[1\]). The set of those local Lorentz transformations that satisfy Eq.(\[1\]), given a frame $e_a (x)$ \[or equivalently the co-frame $e^a(x)$\] that solves the field equations of $f(T)$ gravity, is denoted by $\mathcal{A}(e^a)$, and dubbed the “remnant group” \[which can be, but is not necessarily, a group\]. We will refer to the additional symmetry embodied in the remnant group \[in addition to the global Lorentz transformation\] as the “remnant symmetry”. Ferraro and Fiorini then asserted that the existence of the remnant group seems to be not consistent with[^1] the results obtained by us in [@1303.0993] and [@1309.6461], in which we showed that $f(T)$ gravity is generically problematic – it allows superluminal propagation and *local* acausality \[that is, temporal ordering is not well-defined even in an infinitesimal neighborhood\]. The theory also suffers from non-unique time evolution, i.e., Cauchy problem is ill-defined. That is, given a full set of the Cauchy data, one cannot predict what will happen in the future with certainty. Note that all these problems arise at the classical level. Ferraro and Fiorini did not explain the reason they think their results contradict ours. In this work we wish to clarify that the existence of the remnant group does not, in fact, contradict our previous works, but instead strongly supports it. Comments on Propagation and Evolution in $f(T)$ Gravity ======================================================= A good way to understand the number of dynamical degrees of freedom is from the Hamiltonian perspective. Following Dirac’s procedure we find primary constraints, introduce them into the Hamiltonian with Lagrange multipliers, determine the multipliers if possible and find any additional secondary constraints. The constraints are divided into two classes: *first class* are associated with gauge freedom, and *second class* related to non-dynamical variables. Here we are concerned with teleparallel theories. The primary dynamic variable is the orthonormal frame $e^a{}_\mu$. Its conjugate momentum is $P_a{}^\mu$. For the Hamiltonian analysis of teleparallel theories, see e.g., [@CCN; @telham; @BtelepHam; @Bgauge]. The Lagrangian *never* contains the time derivative of $e^a{}_0$, consequently we always have the primary constraints $P_a{}^0$. Preservation of these constraints lead to a set of 4 secondary constraints referred to as the Hamiltonian and momentum constraints. These 8 constraints are all first class, geometrically they generate spacetime diffeomorphisms. We need not consider them any further. For the special subclass of theories of the form $f(T)$ there are 6 more primary constraints, $P^{[\mu\nu]}\simeq0$ associated with the Lorentz sector. To keep the discussion simple and focus on the essentials, let us assume that they do not give rise to any additional constraints. The class of these 6 constraints is determined by the rank of the associated Poisson bracket matrix. If the rank is 0, then all constraints are first class, and they generate local Lorentz transformations — this is the TEGR special case. If the rank is maximal, i.e. 6, then all constraints are second class. Naturally, one would like to ask if there are other possibilities. Since the second class constraints come in pairs, we might imagine we could have rank 4 \[i.e., 2 extra first class and 4 second class constraints\] or rank 2 \[i.e., 4 extra first class and 2 second class constraints\]. The latter case seems to be actually not possible. These “extra” first class constraints each generates a one parameter subgroup of the Lorentz group. If we have 2 such generators their commutator is also a generator. Thus we would have an additional unwanted constraint, unless it vanished identically. Therefore, by counting, a local symmetry group which is an Abelian subgroup of the Lorentz group is sensible. It seems that there are no suitable 4 parameter subgroups of the Lorentz group. Thus the possible scenarios seem to be (i) all the 6 constraints are second class, (ii) 2 commuting first class constraints $+$ 4 second class constraints, (iii) all constraints are first class. The standard formula for counting degrees of freedom is \#() =&\[2() - 2()\ &- [\#]{}()\]/2, where $\#$ denotes “the number of”. This gives for the possibilities (i,ii,iii) above as, respectively, - $[2(16)-2(8)-6]/2 = 5$; - $[2(16)-2(10)-4]/2 = 4$; - $[2(16)-2(14)-0]/2 = 2$. The possibility (iii) corresponds to TEGR, while (i) is the generic case for $f(T)$ gravity, as confirmed by the detailed work of Miao Li et al. [@li], which was based on the Hamiltonian analysis of [@maluf]. In other words, $f(T)$ gravity generically propagates 5 degrees of freedom, i.e., there are 3 additional degrees of freedom compared to standard GR. These extra degrees of freedom are very non-linear in nature. For example, they do not show up in the linear perturbation of flat Friedmann-Lemaître-Robertson-Walker \[FLRW\] cosmological background [@1212.5774]. What about the option (ii) then? Case (ii) is “exotic”; it is some kind of geometry that is intermediate between a metric and preferred frame theory. Our Hamiltonian analysis identifies this possibility, which as we shall see below, *is precisely one of the types that was found by Ferraro and Fiorini*. This is the reason why the word “generically” is crucial, for in [@1303.0993] we found out that the number of physical degrees of freedom and the classes of Dirac constraints, can and do change depending on the values of the fields. That is, they are *expected to be different* on different background geometries[^2]. This is in fact, *in agreement* with the findings of Ferraro and Fiorini [@1412.3424] that different geometries give rise to a different number of “admissible frames”. We will further elaborate on this later. However, *precisely* because of the possibility that field configurations can change the number of degrees of freedom as well as the constraint structure of the theory, we expect anomalous propagation such as superluminal shock waves to arise. This is explained in detail already in [@1303.0993] in which we employed the well-understood PDE method of characteristics, pioneered by Cauchy and Kovalevskaya. \[See also Section 2 of [@1410.2289] and the references therein for further explanation regarding this method\]. One has to be careful in distinguishing between the symmetry of a *theory* and the symmetry of a *particular solution*. For example, consider a complex scalar field $\varphi$ with a simple potential: $$V = V_1(\varphi \varphi^*) + (1-\varphi \varphi^*) V_2 (\varphi).$$ Clearly only the $V_1$ term in the potential has local U(1) symmetry because it is a function of $\varphi \varphi^*$, i.e., the \[square of the\] absolute value of $\varphi$. On the other hand, $V_2$ term does not because it depends on the explicit $\varphi$ configuration. However, if the absolute value of $\varphi$ is unity, i.e. if $\varphi \varphi^*=1$, then the value of the potential $V$ is invariant under U(1) transformation. Thus, for some specific field configurations $V$ has U(1) symmetry, *but this is not the symmetry of the theory* for generic values of $\varphi$! The specific values that “restores” U(1) symmetry to $V$ has physical effect, it is the signal that the mode related to U(1) direction has become massless. Similarly in $f(T)$ gravity one expects that if the fields evolve such that some extra symmetries emerge, there would be physical effects that accompany the changes in the number and type of constraints \[much of the effort in [@1303.0993] and [@1309.6461] is spent on showing that the superluminal propagations are not simply due to gauge choice\]. To be more specific, one can consider a kinetic term for $\varphi$ of the form: $$(1-\varphi \varphi^*) \partial_\mu \varphi \partial^\mu \varphi^*.$$ This kinetic term does not have *local* $U(1)$ symmetry because the derivative is not covariant. Generically this term is well-defined, but when the field configurations approach the value such that $\varphi \varphi^*=1$, the dynamical term vanishes. Under the local $U(1)$ transformation, $\varphi \varphi^*=1$ still holds and the dynamical term remains naught. This is indeed a “remnant symmetry". However, the differential equations \[the equations of motion\] then behave very differently depending of the values of the fields. This is the situation faced by $f(T)$ gravity. In fact, the results of [@1412.3424] actually *support* our results. As we recall, generically $f(T)$ gravity has 5 degrees of freedom. However, for almost any $f$ it is very likely that there would exist solutions where the Poisson bracket matrix has less than the generic rank. For any such solution one or more of the generically second class constraints will now be first class. These first class constraints will generate some local Lorentz transformations. Thus what we expect to see is that for each solution there is some subgroup of the Lorentz group which acts as a local symmetry gauge group. In other words, our analysis involving rank changes already implicitly implied the same result that Ferraro and Fiorini now discovered, using a different, more straightforward analysis. The point is that the set $\mathcal{A}(e^a)$ \[which is generally not a group\] encodes all information about the change in rank of the Poisson bracket matrix – the size of this “group" \[at each spacetime point\] reflects the number of normally second class constraints, which have become first class for a particular frame, i.e., the change in the rank of the Poisson bracket matrix. For a given function $f(T)$, if $\mathcal{A}(e^a)$ is empty for every solution to the equations, then the propagation has no problems. For a given $f$, and for all solution frames, if this set is an Abelian group of size *independent of spacetime point*, then we may have good propagating modes. However, if this set is an Abelian group with size varying from point to point, then we have acausal propagation – which as we argued in [@1303.0993], arise as a consequences of the change in the number of degrees of freedom and the constraint structure of the theory. Most crucially, it should be noted that in [@1309.6461], we constructed an *explicit* example in which $f(T)$ gravity and its Brans-Dicke-generalization with scalar field suffers from non-unique evolution – starting with a perfectly homogeneous and isotropic, flat FLRW universe, anisotropy can suddenly emerge. By non-unique evolution, we do *not* simply mean the following: Given a spacetime geometry, a chosen tetrad field can evolve into another choice of tetrad, which corresponds to the same metric tensor \[if there are multitudes of “admissible frames” this would not be a problem notwithstanding the discussion above, *if physical observables only couple to the metric*\]. Instead we mean a stronger statement: even a *geometry*, described by the metric tensor, can change drastically under the evolution, and such change cannot be predicted from initial data alone. This problem cannot be evaded even if there are more than one “admissible frames” corresponding to a *fixed* geometry. Discussion ========== In this work we discuss why the existence of remnant symmetry as shown by Ferraro and Fiorini does not contradict our previous works, which established the existence of superluminal propagation, local acausality, and non-unique time evolution in $f(T)$ gravity \[and its Brans-Dicke generalization\]. In fact, these problems are closely related to the remnant symmetry – while our Hamiltonian analysis agrees with the results of Ferraro and Fiorini, the same analysis also shows that there are serious dynamical difficulties, especially if one approaches a point where the rank of the Poisson bracket matrix changes. We now conclude this work with some additional comments. In relation to the Hamiltonian analysis, the usual understanding is that generically teleparallel Lagrangians have no local frame gauge freedom. Beyond the relations that follow from diffeomorphism invariance \[which are connected with energy momentum\] the equations satisfy no other differential identities. According to Noether’s Second Theorem, a local gauge freedom means a differential identity. In the Hamiltonian formulation of teleparallel theories, generically the only first class constraints are the Hamiltonian and momentum constraints. If there are no other first class constraints, then there is no other local gauge symmetries. It would be very interesting to see how “remnant symmetries” fit into this scheme more explicitly. We now remark on the comment in [@1412.3424] on the possibility of constructing local inertial frames in $f(T)$ gravity, and the hope that Zeeman’s theorem on $\Bbb{R}^{3,1}$[@zeeman] would thus ensure local causality. In the context of Zeeman’s theorem, the invariance group $G$ of the Minkowski spacetime \[the orthochronous Lorentz group, the translation group and the dilatation group\] induces the light cone structure of $\Bbb{R}^{3,1}$. This structure provides some causality relation $C$ which allows the definition of causality group $G_c$. In 4 dimensions, it turns out that $G=G_c$. Indeed one could generalize the notion of Riemann normal coordinates to other geometries defined by different connections, in particular the Weitzenböck connection. This was accomplished in [@nester]. However, propagation is *not* defined at a point; it involves dynamical modes moving from a spacetime point $x$ to another \[although this distance could be in a $\varepsilon$-neighborhood of $x$\], and this is problematic in $f(T)$ gravity if the “size” of $\mathcal{A}(e^a)$ \[and thus the remnant symmetry\] changes from point to point. Furthermore, even local causality is certainly not avoided. To see this, one should consider also the equation that governs the propagation, derived in [@1303.0993]: $$\label{chareqn} \left[f_T M_a^{~\mu\nu}{}_b^{~\alpha\beta}+ 2f_{TT}S_a^{~\mu\nu}S_b^{~\alpha\beta}\right]k_\mu k_\alpha \bar{e}^b_{~\beta} = 0;$$ in which $$M_a^{~\mu\nu}{}_b^{~\alpha\beta}:=\dfrac{\partial S_a^{~\mu\nu}}{\partial T^b_{~\alpha\beta}}.\\$$ Here the notation $\bar{e}^b_{~\beta}$ represents the change of the frame in a certain direction, instead of the value of the frame. $k^\mu$ denotes the normal to a characteristic hypersurface which in this case could be timelike \[signaling a tachyonic propagation\]. Here, the first term in the square bracket is the healthy propagation; it is the only term presents in the case of TEGR, in which case $f(T)=T \Rightarrow f_{TT} \equiv 0$. This describes the normal light cones in \[TE\]GR. Differential equations are of course *local* in nature. Thus we see that even locally the characteristic cones in a generic $f(T)$ theory is *not* the same as TEGR, in particular the cones depend also on the second term \[in the square bracket\], which generically differs from one spacetime point to another. Lastly we would like to make a side remark that in [@1412.3424], two classic works by Hayashi and Shirafuji were mentioned [@HS1; @HS2]. In particular, it was pointed out that [@HS1] considered “restricted local invariance of this sort” in the context of “New General Relativity” \[NGR\] proposed in [@HS2]. Indeed Hayashi and Shirafuji considered the possibility of a new type of geometry somewhere in between teleparallel and Riemannian one. It would have what we have here called “remnant symmetry”, a preferred frame determined up to a certain dynamically determined subgroup of the Lorentz group. It is worth mentioning that there has since been some criticisms by Kopczyński and further developments that came out of that [@K; @N; @CCN; @CNY]. \[The work by Chen, Nester and Yo [@CNY] was the first to call attention to the effect of non-linear constraints and the relationship of changes in the rank of the Poisson matrix with tachyonic characteristics\]. Kopczyński’s objection was in regard to one of the complications that occurs when one has some “local symmetry" for the gravitational Lagrangian for a certain subclass of solutions. Briefly, if the LHS of the field equation has a local symmetry, then the RHS, i.e., the material energy momentum tensor must have the same symmetry. This could impose “unphysical" limitations on the matter sector. For the NGR theory, it was found that spin-1/2 Dirac field does not give rise to any problem, but for a hypothetical spin-3/2 field there is inconsistency \[unless if one subscribes to non-minimal coupling to save the theory\]. For $f(T)$ gravity this may not be a serious problem, as one could simply take the source energy-momentum tensor to be completely locally Lorentz invariant, as it is in GR. However this would mean that there is no material source for the extra degrees of freedom. Despite the pathologies of $f(T)$ gravity, the existence of remnant symmetry is indeed interesting and may provide further insights into the structure of teleparallel gravities, and perhaps modified gravity theories in general. The authors would like to thank Huan-Hsin Tseng for discussion. K.I. is supported by Taiwan National Science Council under Project No. NSC101-2811-M-002-103. [99]{} C. Moller, “Further Remarks on the Localization of the Energy in the General Theory of Relativity”, Annals Phys.  [**12**]{} (1961) 118. Y. M. Cho, “Einstein Lagrangian as the Translational Yang-Mills Lagrangian”, Phys. Rev. D [**14**]{} (1976) 2521. F. W. Hehl, J. D. McCrea, E. W. Mielke, Y. Ne’eman, “Metric Affine Gauge Theory of Gravity: Field Equations, Noether Identities, World Spinors, and Breaking of Dilation Invariance”, Phys. Rept.  [**258**]{} (1995) 1, [\[gr-qc/9402012\]](http://arxiv.org/abs/gr-qc/9402012). Y. Itin, “Coframe Teleparallel Models of Gravity: Exact Solutions”, Int. J. Mod. Phys. D [**10**]{} (2001) 547, [\[gr-qc/9912013\]](http://arxiv.org/abs/gr-qc/9912013). V. C. de Andrade, L. C. T. Guillen, J. G. Pereira, “Gravitational Energy Momentum Density in Teleparallel Gravity”, Phys. Rev. Lett.  [**84**]{} (2000) 4533, [\[gr-qc/0003100\]](http://arxiv.org/abs/gr-qc/0003100). M. Blagojević, M. Vasilić, “Gauge Symmetries of the Teleparallel Theory of Gravity”, Class. Quant. Grav **17** (2000) 3785, [\[arXiv:hep-th/0006080\]](http://arxiv.org/abs/hep-th/0006080). M. Blagojević, *Gravitation and Gauge Symmetries*, IoP Publishing, Bristol, 2002. F. W. Hehl, “Gauge Theory of Gravity and Spacetime”, [\[1204.3672 \[gr-qc\]\]](http://arxiv.org/abs/1204.3672). R. Aldrovandi, J. G. Pereira, [*[Teleparallel Gravity: An Introduction]{}*]{}, Springer, Dordrecht, 2013. E. V. Linder, “Einstein’s Other Gravity and the Acceleration of the Universe”, Phys. Rev. D **81** (2010) 127301, [\[arXiv:1005.3039 \[astro-ph.CO\]\]](http://arxiv.org/abs/1005.3039). B-J. Li, T. P. Sotiriou, J. D. Barrow, “f(T) Gravity and Local Lorentz Invariance”, Phys. Rev. D **83** (2011) 064035, [\[arXiv:1010.1041 \[gr-qc\]\]](http://arxiv.org/abs/1010.1041). R. Ferraro, F. Fiorini, “The Remnant Group of Local Lorentz Transformations in f(T) Theories”, [\[arXiv:1412.3424 \[gr-qc\]\]](http://arxiv.org/abs/1412.3424v2). Y. C. Ong, K. Izumi, J. M. Nester, P. Chen, “Problems with Propagation and Time Evolution in f(T) Gravity”, Phys. Rev. D **88** (2013) 024019, [\[arXiv:1303.0993 \[gr-qc\]\]](http://arxiv.org/abs/1303.0993). K. Izumi, J-A. Gu, Y. C. Ong, “Acausality and Nonunique Evolution in Generalized Teleparallel Gravity”, Phys. Rev. D **89** (2014) 084025, [\[arXiv:1309.6461 \[gr-qc\]\]](http://arxiv.org/abs/1309.6461). W-H. Cheng, D-C. Chern, J. M. Nester, “Canonical Analysis of the One-Parameter Teleparallel Theory”, Phys. Rev. D **38** (1988) 2656. J. M. Nester, “Positive Energy via the Teleparallel Hamiltonian”, Int. J. Mod. Phys. A [**4**]{} (1989) 1755. M. Li, R-X. Miao, Y-G. Miao, “Degrees of Freedom of $f(T)$ Gravity”, JHEP **1107** (2011) 108, [\[arXiv:1105.5934 \[hep-th\]\]](http://arxiv.org/abs/1105.5934v4). J. W. Maluf , J. F. da Rocha-Neto, “Hamiltonian Formulation of General Relativity in the Teleparallel Geometry”, Phys. Rev D **64** (2001) 084014, [\[arXiv: gr-qc/0002059\]](http://xxx.tau.ac.il/abs/gr-qc/0002059v2). K. Izumi, Y. C. Ong, “Cosmological Perturbation in f(T) Gravity Revisited”, JCAP **06** (2013) 029, [\[arXiv:1212.5774 \[gr-qc\]\]](http://arxiv.org/abs/1212.5774). S. Deser, K. Izumi, Y. C. Ong, A. Waldron, “Problems of Massive Gravities”, Mod. Phys. Lett. A 30 (2015) 1540006, [\[arXiv:1410.2289 \[hep-th\]\]](http://arxiv.org/abs/1410.2289). E. C. Zeeman, “Causality Implies the Lorentz Group”, J. Math. Phys. **5** (1964) 490. J. M. Nester, “Normal Frames for General Connections”, Ann. Phys. (Berlin) **19** (2010) 45. K. Hayashi, T. Shirafuji, “Addendum to ‘New General Relativity’ ”, Phys. Rev. D **24** (1982) 3312. K. Hayashi, T. Shirafuji, “New General Relativity”, Phys. Rev. D **19** (1979) 3524. W. Kopczyński, “Problems with Metric-Teleparallel Theories of Gravitation”, J. Phys. A: Math. Gen. **15** (1982) 493. J. M. Nester, “Is There Really a Problem with the Teleparallel Theory?”, Class. Quant. Grav. **5** (1988) 1003. H. Chen, J. M. Nester, H.J. Yo, “Acausal PGT Modes and the Nonlinear Constraint Effect”, Acta Phys. Polon. B **29** (1998) 961. [^1]: Indeed, they used a stronger expression “seems to discredit”. [^2]: Note that the determinant of the Poisson bracket matrix is a polynomial in the variables and their derivatives. Generically it has real roots. The rank cannot be constant if it admits generic solutions \[which include, but is not limited to, Minkowski and FLRW\].

--- abstract: 'We analyze the light curves of 413 radio sources at submillimeter wavelengths using data from the Submillimeter Array calibrator database. The database includes more than 20,000 observations at 1.3 and 0.8 mm that span 13 years. We model the light curves as a damped random walk and determine a characteristic time scale $\tau$ at which the variability amplitude saturates. For the vast majority of sources, primarily blazars and BL Lac objects, we find only lower limits on $\tau$. For two nearby low luminosity active galactic nuclei, M81 and M87, however, we measure $\tau=1.6^{+3.0}_{-0.9}$ days and $\tau=45^{+61}_{-24}$ days, respectively ($2\sigma$ errors). Including the previously measured $\tau=0.33\pm 0.16$ days for Sgr A\*, we show an approximately linear correlation between $\tau$ and black hole mass for these nearby LLAGN. Other LLAGN with spectra that peak in the submm are expected to follow this correlation. These characteristic time scales are comparable to the minimum time scale for emission processes close to an event horizon, and suggest that the underlying physics may be independent of black hole mass, accretion rate, and jet luminosity.' author: - 'Geoffrey C. Bower, Jason Dexter, Sera Markoff, Mark A. Gurwell, Ramprasad Rao, Ian McHardy' title: 'A Black Hole Mass-Variability Time Scale Correlation at Submillimeter Wavelengths' --- Introduction ============ In recent years there has been much work focused on understanding black hole accretion and its similarities across the mass scale, from stellar mass black holes in X-ray binaries (BHBs) to active galactic nuclei (AGN). Long term variability studies have found evidence for a mass-dependence in timing features that holds from BHB to AGN [@2006Natur.444..730M; @2009ApJ...698..895K; @2010ApJ...721.1014M]. A linear correlation can be understood if the emission originates in a region that is the same size for all systems in units of Schwarzschild radii, where $R_S=2GM/c^2$. However, the timescale is also inversely proportional to the Eddington accretion fraction, suggesting that the pertinent size of the emission zone is also regulated by the total system power. Similarly, studies of broadband spectra have found a strong correlation between radio and X-ray luminosity and black hole mass (the “Fundamental Plane of Black Hole Accretion”, or FP, see e.g., ), that have confirmed earlier theoretical frameworks that synchrotron spectral features scale predictably with black hole mass and accretion power. Combining these concepts, observations at a fixed frequency of black holes of the same mass with different accretion power should not yield similar timescales, because the frequency “selects” out different sized emission regions in both cases. The sub-millimeter (submm) band seems to be selecting out regions of event-horizon scale in two sources of drastically different mass and accretion rate: Sgr A\* and M87 [@2008Natur.455...78D; @2012Sci...338..355D; @doelemanetal2012]. Both sources belong to the class of nearby, low-luminosity AGN (LLAGN; @1999ApJ...516..672H) that fall on the FP. Variability provides an independent test of the size scales. @2014MNRAS.442.2797D used submm light curves to demonstrate that [Sgr A\*]{} follows a damped-random walk (DRW) variability pattern, with a characteristic time scale $\tau= 8_{-4}^{+3}$ h at 230 GHz at 95 per cent confidence, with consistent results at higher frequencies. This time scale is an order of magnitude larger than the period of the last stable orbit for a non-rotating black hole, which is most easily understood as resulting from accretion processes on a scale of a few to $\sim 10 R_S$. We present a variability study of AGN in the submm, including the LLAGN M81 and M87 as well as the 411 other radio sources included in the SMA Calibrator database, which span more than a decade in duration (§\[sec:data\]). This sample was previously analyzed by @2010arXiv1001.0806S, but with a much smaller number of objects and total span. The full sample of light curves are well described by a noise process, which we model with a damped random walk in order to quantitatively measure characteristic variability time scales (§\[sec:analysis\]). We show that the LLAGN follow a correlation between black hole mass and time scale, while the higher luminosity AGN have much longer characteristic time scales with no clear black hole mass dependence. Data Sample \[sec:data\] ======================== The SMA calibrator database [@2007ASPC..375..234G] includes observations of 412 sources from June 2002 to January 2015. A total of 23254 flux densities are recorded with 19111 of those flux densities obtained in the 1.3mm band. 410 light curves have $>10$ flux densities, 141 light curves have $>30$ flux densities, and 48 light curves have $>100$ flux densities. Observations were obtained at frequencies from 200 to 406 GHz with 70% obtained between 220 and 235 GHz. 165 observations (147 at 1.3mm) of M87 are included in the calibrator database spanning from January 2003 to January 2015. Data reduction of sources followed standard SMA techniques that set flux densities on absolute flux density scales determined by solar system objects with typical accuracy of 5 to 10%. Sources are selected for the SMA calibrator database on the basis of their suitability as phase reference sources for mm/submm interferometric observations, primarily reflected in their compact size, bright radio flux density, and declination above -50$^\circ$. Flux densities in the data base range from 26 mJy to 52 Jy with a median value of 1.2 Jy. 336 sources are matched to flat spectrum radio sources in the CRATES catalog [@2007ApJS..171...61H]. Non-matches to CRATES include steep spectrum calibrators such as 3C 286 and sources in the Galactic plane such as J1700-261 that may have been missed by the single dish surveys that form the basis of CRATES. The CRATES sample includes primarily blazars and BL Lac sources; 289 of the sources are found in the CGRABS catalog of $\gamma$-ray blazar candidates [@2008ApJS..175...97H]. We also included SMA monitoring observations of M81 in our analysis, which has been the target of a significant campaign (McHardy et al. 2015). This campaign included 86 observations between September 2009 and March 2012, of which 42 observations were obtained at frequencies higher than 300 GHz. We also include PdBI observations of M81 obtained in 2005 at 1.3 mm with a time resolution of $\sim 1$ hour . ![ Light curve for M87 at 1.3 mm from the SMA. The light curves show the characteristic of short-term variability coupled with long-term stability that is the signature of the damped random walk. \[fig:m87\] ](m87_lc-eps-converted-to.pdf){width="\textwidth"} Time Scale Analysis \[sec:analysis\] ==================================== The light curve for M87 is shown in Figure \[fig:m87\]. The light curve for M81 is shown in McHardy et al. (2015) and shares similar characteristics of short-term variability and long-term stability. These properties are visible in the structure functions (Fig.\[fig:sfun\]), defined for a light curve $s(t)$ with $N$ data points as $$S^2 (\Delta t) = {1 \over N} \Sigma\left( s(t) - s(t+\Delta t) \right)^2.$$ Both structure functions show increases in activity on short time scales followed by a plateau, similar to those of [Sgr A\*]{}[@2014MNRAS.442.2797D] but markedly different than those of typical calibrators in the database. For instance, the structure functions for 3C 84 and 279 rise smoothly without any apparent saturation time scale for $\Delta t < 10 \rm yr$. Structure functions are a useful guide for qualitative comparison; however, they functions are unreliable for quantitative analysis [@2010MNRAS.404..931E]. They can show artifacts from aliasing (e.g. peaks at $\sim 180$ days associated with the anual cycle), as well as spurious saturation time scales even with regularly sampled data. Fitting of structure functions for saturation time scales is also challenging since their values and errors at different time scales are correlated. Damped random walk model ------------------------ In order to quantitatively measure characteristic time scales, we model the entire set of submm light curves as the result of a stochastic damped random walk (DRW) process. The DRW is a simple 3-parameter model (mean, standard deviation of long-timescales, and transition time) that is well-suited to parametrize the most important properties of noise light curves [@2009ApJ...698..895K]. The DRW consists of red-noise on time scales less than $\tau$ and white-noise on longer time scales. The resultant structure function for the DRW is $$S^2(\tau) = S^2_{\rm \infty} \left(1-e^{-|\Delta t|/\tau}\right),$$ where $S^2_{\infty}$ is the power in the light curve on long time scales $\Delta t \gg \tau$. The characteristic time scale $\tau$ determines the de-correlation time on which the variability transitions from red noise on short timescales to white noise on long time scales. This model has been shown to successfully describe the optical light curves of quasars [@2009ApJ...698..895K; @2010ApJ...721.1014M], as well as the submm light curve of Sgr A\* [@2014MNRAS.442.2797D]. We convert the likelihood of the observed light curves arising from a given set of DRW parameters into the posterior probability of the set of model parameters using a Bayesian approach. @2009ApJ...698..895K and @2014MNRAS.442.2797D used a Metropolis-Hastings algorithm to sample the likelihood over the model parameter space. We use this approach to measure the model parameters in cases where the upper limit to $\tau$ is smaller than the light curve duration $T$. However, this is not the case for the vast majority of the SMA calibrator light curves. When no upper limit can be found, the Metropolis-Hastings algorithm fails, as it preferentially samples the long tail of the probability distribution rather than its peak. To avoid this issue, we sample the probability distribution over a regular $64^3$ grid in the parameters with uniform priors for the mean ($\mu$), standard deviation ($\log \sigma$), and $\log \tau$. Since for the vast majority of sources we are interested in lower limits on $\tau$, we choose the parameter ranges to ensure that the lower end and peak of the probability distribution are well captured, at the expense of the long tails to large $\tau$ (much longer than the duration of the light curve). From the probability distributions in $\tau$, we consider good limits to be cases where $\tau > \Delta t$ and $\tau < T$ at $99.7\%$ confidence, where $\Delta t$ is the minimum separation between two measured points in the light curve and $T$ is the total light curve duration. In all cases we report $2\sigma$ limits because of the non-Gaussian nature of the distributions that can include long tails. Although less efficient than Monte Carlo, this direct grid method recovers consistent parameter estimates for the M81 and M87 light curves. If the true light curve is not described by the DRW, as seen in Kepler quasar light curves [@2015MNRAS.451.4328K], the resulting $\tau$ estimate will be biased. For light curves that can successfully be fit by the model ($\chi_\nu^2 \sim 1$), such as the SMA data used here, the bias leads to $< 1\sigma$ changes in our estimates of $\tau$. Results ------- We use the DRW model to infer $\tau$ values from the full set of 664 230 and 345 GHz light curves from the 413 sources. A histogram of the resulting $95\%$ confidence lower and upper limits on $\tau$ is shown in Figure \[fig:hists\]. In total, we find 276 light curves with lower limits ($40\%$) and 16 with upper limits ($2\%$). The histograms are further broken down by sources with $> 30$ data points and those with available black hole mass estimates in the literature (see below). For the measured lower limits, the sub-samples are consistent with the full sample. Many of the measured upper limits are false positives in light curves with few data points caused by not sampling a large enough range in $\tau$. After those are removed, 5 detections of upper limits remain, all of which have $\chi_\nu^2$ close to unity, similar to Sgr A\*. In Table \[tab:detections\], we summarize the detections. The light curve residuals after model subtraction are normally distributed for M81 and M87, as expected for an accurate model, and similar to what was seen for Sgr A\* [@2014MNRAS.442.2797D]. The results of this time scale analysis are in stark contrast for the available LLAGN light curves (Sgr A\*, M81, M87) and those of ordinary AGN / blazars (the vast majority of the light curves). For the LLAGN, 3/3 sources have well measured values of $\tau$. For the rest of the sources, $1\%$ (4/664) have reliable measurements of $\tau$. It is therefore potentially possible to select LLAGN based on submm properties alone, with a low false positive rate and a high detection efficiency for well sampled light curves. There is no systematic difference in the number or cadence of measured flux densities that can explain this result: it is related to an intrinsic difference in the submm variability properties of the different source classes. ![ Structure functions for M87, M81, 3C 279, and 3C 84 from 1.3 mm SMA data. The structure function suggests that M87 and M81 have different variability characteristics from the two high power radio sources. Quantitative analysis with the DRW model confirms the differences. \[fig:sfun\] ](sma_1mm_sfun_short-eps-converted-to.pdf){width="\textwidth"} --------------------------------------------------------------- ---------------------------------------------------------------   --------------------------------------------------------------- --------------------------------------------------------------- [lrrrrrl]{} 0433+053 & 106 & 63 & 299 & 0.88 & $2.6 \times 10^7$ & @2002ApJ...579..530W\ 0721+713 & 75 & 56 & 155 & 0.83 & $5.2 \times 10^6$ & Fundamental Plane\ 1104+382 & 40 & 16 & 154 & 0.81 & $1.9 \times 10^8$ &@2002ApJ...579..530W\ 1751+096 & 106 & 50 & 331 & 1.04 & $2.2 \times 10^7$ & Fundamental Plane\ M87 & 45 & 21 & 106 & 0.87 & $6.2 \times 10^9$ & @2013ARAA..51..511K\ M81 & 1.6 & 0.7 & 4.6 & 1.14 & $6.5 \times 10^7$ & @2009MNRAS.394..660C\ Sgr A\* & 0.33 & 0.17 & 0.5 & 1.05 & $4.4 \times 10^6$ & @2010RvMP...82.3121G\ Black Hole Mass Estimates ------------------------- We compiled black hole masses from the literature. The three nearby LLAGN have the most accurately determined masses, often from multiple dynamical methods. [Sgr A\*]{} has a mass of $4.4 \times 10^6 M_{\sun}$ [@2010RvMP...82.3121G]. M87 has a mass of $3.5^{+0.9}_{-0.7} \times 10^9\, M_\sun$ from gas dynamic measurements [@2013ApJ...770...86W] and a mass of $6.6 \pm 0.4 \times 10^9\, M_\sun$ from stellar dynamic measurements [@2011ApJ...729..119G]. Following @2013ARAA..51..511K, we adopt a value of $6.2 \times 10^9\, M_\sun$. Similarly for M81, stellar and gas dynamical measurements exist [@2000AAS...197.9203B; @2003AJ....125.1226D] and we follow @2013ARAA..51..511K to adopt a value of $6.5 \times 10^7\, M_\sun$. For the remainder of the sample, we rely on published black hole mass estimates from various techniques. These include broad line region spectroscopy , black hole mass-bulge luminosity correlation [@2002ApJ...579..530W], and the FP between radio-X-ray luminosity [@2003MNRAS.345.1057M; @2012MNRAS.419..267P]. For the case of , which does not provide mass estimates, we use their measurements of the Mg II transition and the methodology of @2011ApJS..194...45S to estimate masses. FP estimates of the black hole mass are derived using ROSAT 0.2 - 2.0 keV X-ray fluxes and the 1.3mm flux density that is debeamed assuming a Doppler boosting factor of 7. We identify 194 black hole mass measurements for 148 sources. For sources with multiple mass estimates, we rely on the most recent published estimate. The accuracy of the mass estimates range from 0.16 dex for Mg II transitions to $>0.4$ dex for continuum luminosity [@2011ApJS..194...45S]. ![ Black hole mass versus DRW time scale for the SMA calibrator source sample. LLAGN Sgr A\*, M81, and M87 are shown as filled in circles. Other AGNs from the SMA calibrator database with black hole masses are shown as open circles for detections and up arrows for lower limits on $\tau$. The time scales associated with AGN/blazars are significantly longer at the same black hole mass than those of the LLAGN. The inferred time scales for LLAGN variability increase with black hole mass, and are consistent with the predicted linear relationship from General Relativity. The solid line gives the period for the innermost stable circular orbit (ISCO); the dashed line gives the infall time for a disk with radius $5 R_S$, a viscosity $\alpha=1$, and a height $H/R=1$. \[fig:taumbh\] ](tau_mbh_isco_5rs.pdf) Discussion \[sec:conclusions\] ============================== The $\tau$-$M_{\rm BH}$ relationship for the sub-sample of SMA calibrator sources with black hole mass estimates is shown in Figure \[fig:taumbh\]. The difference in submm variability properties between regular and low-luminosity AGN is immediately apparent: the vast majority of AGN only yield lower limits on $\tau$, while the LLAGN have measured time scales that lie well below many of these at comparable black hole mass. On the other hand, the measured $\tau$ values for the 3 LLAGN with good measurements of $\tau$ do show a significant trend of increasing $\tau$ with increasing black hole mass. Formally, the best power-law fit is $\tau \approx 0.3 {\rm\, day\,} (M_{\rm BH}/4 \times 10^6 M_\odot)^{0.7 \pm 0.1}$ (with a coefficient of determination $R^2=0.996$). We consider the observed trend consistent with a linear relationship, however, given the small number of sources and the large uncertainties in $\tau$ and $M_{\rm BH}$. If the M87 mass were taken from gas dynamical measurements, for instance, then a linear law would be consistent with the data. The reliability of the correlation rests on only three LLAGN. This is a minimally small number of objects for a correlation, but these are the only LLAGN for which $\tau$ can currently be determined. We are performing observational campaigns to increase the number of LLAGN with measured $\tau$. A prediction of general relativity is that time scales around black holes should scale as $\tau \sim R^{3/2} M_{\rm BH}$, where $R$ is the radius in units of the event horizon size. This time scale reaches a minimum for a given $M_{\rm BH}$ when $R$ is comparable to the size of the event horizon, and in that case one might expect to find a linear relationship between $\tau$ and $M_{\rm BH}$. This is clearly not borne out by the full sample: there is a large scatter in measured $\tau$ values for regular AGN, and they do not fall on a linear track with the LLAGN. The discovery of such a relationship for the LLAGN suggests that we are measuring such minimum time scales from emission close to an event horizon. This interpretation is consistent with our understanding of these sources inferred from their spectral energy distributions. For all three LLAGN, the spectra peak in the millimeter or submillimeter regime [@2015ApJ...802...69B; @2008ApJ...681..905M; @2013EPJWC..6108008D]. Synchrotron self-absorption leads to a large photosphere which shrinks until the spectral peak, where we can see down to the event horizon. At longer wavelengths, we expect longer timescales because of the larger scales of the photosphere. This larger photosphere is seen as a growing intrinsic image size with wavelength in VLBI observations of Sgr A\* [@2006ApJ...648L.127B] and M87 [e.g., @2011Natur.477..185H]. The same effect shows up in the variability time scales for Sgr A\* and M81 and likely for M87. For Sgr A\*, @2006ApJ...641..302M found a characteristic time scale at wavelengths from 0.7 to 20 cm that ranged from 6 days to hundreds of days, increasing with wavelength. Additionally, we performed a DRW analysis of the 2cm M81 light curve [@1999AJ....118..843H] and found a lower limit to $\tau$ of tens of days, much larger than the measured value at 1.3mm. In addition, for the case of Sgr A\*, the NIR variability time scale matches the submm value, consistent with the hypothesis of emission at both wavelengths emerging from the same region near the event horizon [@2012ApJS..203...18W]. Thus, one should expect a similar relationship for other LLAGN with spectra peaking in the submm. The blazars and high power radio sources presumably do not follow a linear correlation because these sources are still optically thick at this wavelength. X-ray binaries in the low/hard state may also show the same correlation in the optically thin regime, although this may be difficult to observe due to the ${\stackrel{\scriptstyle <}{\scriptstyle \sim}}1$ sec time scales implied by the relationship and the rapidly evolving dynamics of these sytems [@2014MNRAS.439.1390R]. A similar relationship was found in the X-ray [@2006Natur.444..730M] for black holes ranging from stellar to supermassive. This relationship, however, relied on a scaling factor based on the accretion rate, which we do not require here. This scaling factor could be interpreted as compensating between different values of $R$ in the time scale corresponding to the X-ray emission region. In the LLAGN, we instead appear to have reached the minimum variability time scale corresponding to emission from event horizon scales around black holes. The scaling relationship between black hole mass and variability time scale for LLAGN is an important insight for Event Horizon Telescope observations of these sources [@2013arXiv1309.3519F]. These imaging observations will have resolutions as good as a few Schwarzschild radii and have the potential to probe fundamental gravitational physics [e.g., @2014ApJ...788..120L; @2014ApJ...784....7B; @2014MNRAS.445.3370P; @2015MNRAS.446.1973R]. The $\tau-M_{BH}$ correlation shows that the compact sizes measured for these sources are actually colocated with the black hole. In addition, many of the fundamental parameters such as spin are degenerate with source physics parameters including the accretion rate, the jet inclination angle, the electron temperature distribution, the magnetic field strength and orientation, and the optical depth. Given the vastly different scales, environments, and physical properties of these three LLAGN systems, the existence of this correlation demonstrates a surprising coherence among these sources. These, and potentially other LLAGN with spectra peaking in the mm/submm can be treated as a unified class. Broadband spectra and light curve monitoring of other nearby LLAGN in the submm will add to the number of sources that fall into this class. Sources that do not follow this correlation, such as the majority of the SMA calibrators, may follow the relation at a higher frequency where the emission becomes optically thin. The Submillimeter Array is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics and is funded by the Smithsonian Institution and the Academia Sinica. This research has made use of the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. [46]{} natexlab\#1[\#1]{} , R. D. & [Konigl]{}, A. 1979, , 232, 34 , G. A., [Wilson]{}, A. S., [Heckman]{}, T. M., [Magorrian]{}, J., [Gebhardt]{}, K., [Richstone]{}, D. O., [Peterson]{}, B. M., & [Green]{}, R. F. 2000, in Bulletin of the American Astronomical Society, Vol. 32, American Astronomical Society Meeting Abstracts, 1566 , G. C., [Goss]{}, W. M., [Falcke]{}, H., [Backer]{}, D. C., & [Lithwick]{}, Y. 2006, , 648, L127 , G. C., [Markoff]{}, S., [Dexter]{}, J., [Gurwell]{}, M. A., [Moran]{}, J. M., [Brunthaler]{}, A., [Falcke]{}, H., [Fragile]{}, P. C., [Maitra]{}, D., [Marrone]{}, D., [Peck]{}, A., [Rushton]{}, A., & [Wright]{}, M. C. H. 2015, , 802, 69 , A. E., [Johannsen]{}, T., [Loeb]{}, A., & [Psaltis]{}, D. 2014, , 784, 7 , M., [Neumayer]{}, N., [Reunanen]{}, J., [van der Werf]{}, P. P., [de Zeeuw]{}, P. T., & [Rix]{}, H.-W. 2009, , 394, 660 , N., [Ford]{}, H., [Tsvetanov]{}, Z., & [Jacoby]{}, G. 2003, , 125, 1226 , J., [Kelly]{}, B., [Bower]{}, G. C., [Marrone]{}, D. P., [Stone]{}, J., & [Plambeck]{}, R. 2014, , 442, 2797 , S. S., [Fish]{}, V. L., [Schenck]{}, D. E., [Beaudoin]{}, C., [Blundell]{}, R., [Bower]{}, G. C., [Broderick]{}, A. E., [Chamberlin]{}, R., [Freund]{}, R., [Friberg]{}, P., [Gurwell]{}, M. A., [Ho]{}, P. T. P., [Honma]{}, M., [Inoue]{}, M., [Krichbaum]{}, T. P., [Lamb]{}, J., [Loeb]{}, A., [Lonsdale]{}, C., [Marrone]{}, D. P., [Moran]{}, J. M., [Oyama]{}, T., [Plambeck]{}, R., [Peimiani]{}, R. A., [Rogers]{}, A. E. E., [Smythe]{}, D. L., [Soohoo]{}, J., [Strittmatter]{}, P., [Tilanus]{}, R. P. J., [Titus]{}, M., [Weintroub]{}, J., [Wright]{}, M., [Young]{}, K. H., & [Ziurys]{}, L. M. 2012, Science, 338, 355 , S. S., [Fish]{}, V. L., [Schenck]{}, D. E., [Beaudoin]{}, C., [Blundell]{}, R., [Bower]{}, G. C., [Broderick]{}, A. E., [Chamberlin]{}, R., [Freund]{}, R., [Friberg]{}, P., [Gurwell]{}, M. A., [Ho]{}, P. T. P., [Honma]{}, M., [Inoue]{}, M., [Krichbaum]{}, T. P., [Lamb]{}, J., [Loeb]{}, A., [Lonsdale]{}, C., [Marrone]{}, D. P., [Moran]{}, J. M., [Oyama]{}, T., [Plambeck]{}, R., [Primiani]{}, R. A., [Rogers]{}, A. E. E., [Smythe]{}, D. L., [SooHoo]{}, J., [Strittmatter]{}, P., [Tilanus]{}, R. P. J., [Titus]{}, M., [Weintroub]{}, J., [Wright]{}, M., [Young]{}, K. H., & [Ziurys]{}, L. M. 2012, Science, 338, 355 , S. S., [Weintroub]{}, J., [Rogers]{}, A. E. E., [Plambeck]{}, R., [Freund]{}, R., [Tilanus]{}, R. P. J., [Friberg]{}, P., [Ziurys]{}, L. M., [Moran]{}, J. M., [Corey]{}, B., [Young]{}, K. H., [Smythe]{}, D. L., [Titus]{}, M., [Marrone]{}, D. P., [Cappallo]{}, R. J., [Bock]{}, D., [Bower]{}, G. C., [Chamberlin]{}, R., [Davis]{}, G. R., [Krichbaum]{}, T. P., [Lamb]{}, J., [Maness]{}, H., [Niell]{}, A. E., [Roy]{}, A., [Strittmatter]{}, P., [Werthimer]{}, D., [Whitney]{}, A. R., & [Woody]{}, D. 2008, , 455, 78 , A., [Hada]{}, K., [Nagai]{}, H., [Kino]{}, M., [Honma]{}, M., [Akiyama]{}, K., [Oyama]{}, T., & [Kono]{}, Y. 2013, in European Physical Journal Web of Conferences, Vol. 61, European Physical Journal Web of Conferences, 8008 , D., [McHardy]{}, I. M., & [Uttley]{}, P. 2010, , 404, 931 , H. & [Biermann]{}, P. L. 1995, , 293, 665 , H., [K[" o]{}rding]{}, E., & [Markoff]{}, S. 2004, , 414, 895 , V., [Alef]{}, W., [Anderson]{}, J., [Asada]{}, K., [Baudry]{}, A., [Broderick]{}, A., [Carilli]{}, C., [Colomer]{}, F., [Conway]{}, J., [Dexter]{}, J., [Doeleman]{}, S., [Eatough]{}, R., [Falcke]{}, H., [Frey]{}, S., [Gab[á]{}nyi]{}, K., [G[á]{}lvan-Madrid]{}, R., [Gammie]{}, C., [Giroletti]{}, M., [Goddi]{}, C., [G[ó]{}mez]{}, J. L., [Hada]{}, K., [Hecht]{}, M., [Honma]{}, M., [Humphreys]{}, E., [Impellizzeri]{}, V., [Johannsen]{}, T., [Jorstad]{}, S., [Kino]{}, M., [K[ö]{}rding]{}, E., [Kramer]{}, M., [Krichbaum]{}, T., [Kudryavtseva]{}, N., [Laing]{}, R., [Lazio]{}, J., [Loeb]{}, A., [Lu]{}, R.-S., [Maccarone]{}, T., [Marscher]{}, A., [Mart’[i]{}-Vidal]{}, I., [Martins]{}, C., [Matthews]{}, L., [Menten]{}, K., [Miller]{}, J., [Miller-Jones]{}, J., [Mirabel]{}, F., [Muller]{}, S., [Nagai]{}, H., [Nagar]{}, N., [Nakamura]{}, M., [Paragi]{}, Z., [Pradel]{}, N., [Psaltis]{}, D., [Ransom]{}, S., [Rodriguez]{}, L., [Rottmann]{}, H., [Rushton]{}, A., [Shen]{}, Z.-Q., [Smith]{}, D., [Stappers]{}, B., [Takahashi]{}, R., [Tarchi]{}, A., [Tilanus]{}, R., [Verbiest]{}, J., [Vlemmings]{}, W., [Walker]{}, R. C., [Wardle]{}, J., [Wiik]{}, K., [Zackrisson]{}, E., & [Zensus]{}, J. A. 2013, ArXiv e-prints , K., [Adams]{}, J., [Richstone]{}, D., [Lauer]{}, T. R., [Faber]{}, S. M., [G[ü]{}ltekin]{}, K., [Murphy]{}, J., & [Tremaine]{}, S. 2011, , 729, 119 , R., [Eisenhauer]{}, F., & [Gillessen]{}, S. 2010, Reviews of Modern Physics, 82, 3121 , M. A., [Peck]{}, A. B., [Hostler]{}, S. R., [Darrah]{}, M. R., & [Katz]{}, C. A. 2007, in Astronomical Society of the Pacific Conference Series, Vol. 375, From Z-Machines to ALMA: (Sub)Millimeter Spectroscopy of Galaxies, ed. A. J. [Baker]{}, J. [Glenn]{}, A. I. [Harris]{}, J. G. [Mangum]{}, & M. S. [Yun]{}, 234 , K., [Doi]{}, A., [Kino]{}, M., [Nagai]{}, H., [Hagiwara]{}, Y., & [Kawaguchi]{}, N. 2011, , 477, 185 , S. E., [Romani]{}, R. W., [Cotter]{}, G., [Michelson]{}, P. F., [Schlafly]{}, E. F., [Readhead]{}, A. C. S., [Giommi]{}, P., [Chaty]{}, S., [Grenier]{}, I. A., & [Weintraub]{}, L. C. 2008, , 175, 97 , S. E., [Romani]{}, R. W., [Taylor]{}, G. B., [Sadler]{}, E. M., [Ricci]{}, R., [Murphy]{}, T., [Ulvestad]{}, J. S., & [Winn]{}, J. N. 2007, , 171, 61 , L. C. 1999, , 516, 672 , L. C., [van Dyk]{}, S. D., [Pooley]{}, G. G., [Sramek]{}, R. A., & [Weiler]{}, K. W. 1999, , 118, 843 , V. P., [Vogeley]{}, M. S., & [Richards]{}, G. T. 2015, , 451, 4328 , B. C. & [Bechtold]{}, J. 2007, , 168, 1 , B. C., [Bechtold]{}, J., & [Siemiginowska]{}, A. 2009, , 698, 895 , J. & [Ho]{}, L. C. 2013, , 51, 511 , R.-S., [Broderick]{}, A. E., [Baron]{}, F., [Monnier]{}, J. D., [Fish]{}, V. L., [Doeleman]{}, S. S., & [Pankratius]{}, V. 2014, , 788, 120 , C. L., [Ivezi[ć]{}]{}, [Ž]{}., [Kochanek]{}, C. S., [Koz[ł]{}owski]{}, S., [Kelly]{}, B., [Bullock]{}, E., [Kimball]{}, A., [Sesar]{}, B., [Westman]{}, D., [Brooks]{}, K., [Gibson]{}, R., [Becker]{}, A. C., & [de Vries]{}, W. H. 2010, , 721, 1014 , J.-P. & [Bower]{}, G. C. 2006, , 641, 302 , S., [Nowak]{}, M., [Young]{}, A., [Marshall]{}, H. L., [Canizares]{}, C. R., [Peck]{}, A., [Krips]{}, M., [Petitpas]{}, G., [Sch[ö]{}del]{}, R., [Bower]{}, G. C., [Chandra]{}, P., [Ray]{}, A., [Muno]{}, M., [Gallagher]{}, S., [Hornstein]{}, S., & [Cheung]{}, C. C. 2008, , 681, 905 , I. M., [Koerding]{}, E., [Knigge]{}, C., [Uttley]{}, P., & [Fender]{}, R. P. 2006, , 444, 730 , A., [Heinz]{}, S., & [di Matteo]{}, T. 2003, , 345, 1057 , I., [Petitjean]{}, P., [Aubourg]{}, [É]{}., [Ross]{}, N. P., [Myers]{}, A. D., [Streblyanska]{}, A., [Bailey]{}, S., [Hall]{}, P. B., [Strauss]{}, M. A., [Anderson]{}, S. F., [Bizyaev]{}, D., [Borde]{}, A., [Brinkmann]{}, J., [Bovy]{}, J., [Brandt]{}, W. N., [Brewington]{}, H., [Brownstein]{}, J. R., [Cook]{}, B. A., [Ebelke]{}, G., [Fan]{}, X., [Filiz Ak]{}, N., [Finley]{}, H., [Font-Ribera]{}, A., [Ge]{}, J., [Hamann]{}, F., [Ho]{}, S., [Jiang]{}, L., [Kinemuchi]{}, K., [Malanushenko]{}, E., [Malanushenko]{}, V., [Marchante]{}, M., [McGreer]{}, I. D., [McMahon]{}, R. G., [Miralda-Escud[é]{}]{}, J., [Muna]{}, D., [Noterdaeme]{}, P., [Oravetz]{}, D., [Palanque-Delabrouille]{}, N., [Pan]{}, K., [Perez-Fournon]{}, I., [Pieri]{}, M., [Riffel]{}, R., [Schlegel]{}, D. J., [Schneider]{}, D. P., [Simmons]{}, A., [Viel]{}, M., [Weaver]{}, B. A., [Wood-Vasey]{}, W. M., [Y[è]{}che]{}, C., & [York]{}, D. G. 2014, , 563, A54 , U.-L. & [Broderick]{}, A. E. 2014, , 445, 3370 , R. M., [Markoff]{}, S., [Kelly]{}, B. C., [K[ö]{}rding]{}, E., & [Anderson]{}, S. F. 2012, , 419, 267 , A. & [Dexter]{}, J. 2015, , 446, 1973 , T. D., [Soria]{}, R., [Miller-Jones]{}, J. C. A., [Curran]{}, P. A., [Markoff]{}, S., [Russell]{}, D. M., & [Sivakoff]{}, G. R. 2014, , 439, 1390 , R., [Krips]{}, M., [Markoff]{}, S., [Neri]{}, R., & [Eckart]{}, A. 2007, , 463, 551 , J., [Vanden Berk]{}, D. E., [Schneider]{}, D. P., & [Hall]{}, P. B. 2008, , 135, 928 , Y., [Richards]{}, G. T., [Strauss]{}, M. A., [Hall]{}, P. B., [Schneider]{}, D. P., [Snedden]{}, S., [Bizyaev]{}, D., [Brewington]{}, H., [Malanushenko]{}, V., [Malanushenko]{}, E., [Oravetz]{}, D., [Pan]{}, K., & [Simmons]{}, A. 2011, , 194, 45 , A. L., [Siemiginowska]{}, A., [Gurwell]{}, M. A., & [Kelly]{}, B. C. 2010, ArXiv e-prints , J. L., [Barth]{}, A. J., [Ho]{}, L. C., & [Sarzi]{}, M. 2013, , 770, 86 , G., [Eckart]{}, A., [Bremer]{}, M., [Zamaninasab]{}, M., [Shahzamanian]{}, B., [Valencia-S.]{}, M., [Sch[ö]{}del]{}, R., [Karas]{}, V., [Lenzen]{}, R., [Marchili]{}, N., [Sabha]{}, N., [Garcia-Marin]{}, M., [Buchholz]{}, R. M., [Kunneriath]{}, D., & [Straubmeier]{}, C. 2012, , 203, 18 , J.-H. & [Urry]{}, C. M. 2002, , 579, 530

--- author: - | T. G. Birthwright, E. W. N. Glover, V. V. Khoze and P. Marquard.\ Department of Physics, University of Durham, Durham DH1 3LE, U.K.\ E-mail: title: 'Multi-Gluon Collinear Limits from MHV diagrams' --- Introduction {#sec:intro} ============ The interpretation of $\Neqfour$ supersymmetric Yang-Mills theory and QCD as a topological string propagating in twistor space [@Witten1], has inspired a new and powerful framework for computing tree-level and one-loop scattering amplitudes in Yang-Mills gauge theory. Notably, two distinct formalisms have been developed for calculations of scattering amplitudes in gauge theory – the ‘MHV rules’ of Cachazo, Svrček and Witten (CSW) [@CSW1], and the ‘BCF recursion relations’ of Britto, Cachazo, Feng and Witten [@BCF4; @BCFW]. In this paper, we wish to exploit these formalisms to examine the singularity structure of tree-level amplitudes when many gluons are simultaneously collinear. Understanding the infrared singular behaviour of multi-parton amplitudes is a prerequisite for computing infrared-finite cross sections at fixed order in perturbation theory. In general, when one or more final state particles are either soft or collinear, the amplitudes factorise. The first factor in this product is an amplitude depending on the remaining hard partons in the process (including any hard partons constructed from an ensemble of unresolved partons). The second factor contains all of the singularities due to the unresolved particles. One of the best known examples of this type of factorisation is the limit of tree amplitudes when two particles are collinear. This factorisation is universal and can be generalised to any number of loops [@Kosower:allorderfact]. Both frameworks, the MHV rules and the BCF recursion relations, are remarkably powerful in deriving analytic expressions for massless multi-particle tree-level amplitudes. At the same time, for the specific purpose of deriving general multi-collinear limits, we find the MHV rules approach to be particularly convenient. A useful feature of the MHV rules is that it is not required to set reference spinors $\eta_\alpha$ and $\eta_{\dot\alpha}$ to specific values dictated by kinematics or other reasons. In this way, on-shell (gauge-invariant) amplitudes are derived for arbitrary $\eta$’s, i.e. without fixing the gauge. By starting from the appropriate colour ordered amplitude and taking the collinear limit, the full amplitude factorises into an MHV vertex multiplied by a multi-collinear splitting function that depends on the helicities of the collinear gluons. Because the MHV vertex is a single factor, the collinear splitting functions have a similar structure to MHV amplitudes. Furthermore, the gauge or $\eta$-dependence of the splitting function drops out. One of the main points of our approach is that, in order to derive all required splitting functions we do not need to know the full amplitude. Out of the full set of MHV-diagrams contributing to the full amplitude, only a subset will contribute to the multi-collinear limit. This subset includes only those MHV-diagrams which contain an internal propagator which goes on-shell in the multi-collinear limit. In other words, the IR singularities in the MHV approach arise entirely from internal propagators going on-shell. This observation is specific to the MHV rules method and does not apply to the BCF recursive approach. We will see in Section [**\[sec:4.2.3\]**]{} that in the BCF picture collinear splitting functions generically receive contributions from the full set of allowed BCF diagrams[^1]. In view of this, we will employ the MHV rules of [@CSW1] for setting up the formalism and for derivations of general multi-collinear amplitudes. At the same time, various specific examples of multi-collinear splitting amplitudes derived in this paper will also be checked in Section [**\[sec:4.2\]**]{} using the BCF recursion relations [@BCF4]. The basic building blocks of the MHV rules approach [@CSW1] are the colour-ordered $n$-point vertices which are connected by scalar propagators. These MHV vertices are off-shell continuations of the maximally helicity-violating (MHV) $n$-gluon scattering amplitudes of Parke and Taylor [@ParkeTaylor; @BG]. They contain precisely two negative helicity gluons. Written in terms of spinor inner products [@SpinorHelicity], they are composed entirely of the holomorphic products $\spa{i}.{j}$ of the right-handed (undotted) spinors, rather than their anti-holomorphic partners $\spb{i}.{j}$, A\_n(1\^+,…,p\^-,…,q\^-,…,n\^+) = , \[MHV\] where we introduce the common notation $\spa{p_i}.{p_j}=\spa{i}.{j}$ and $\spb{p_i}.{p_j}=\spb{i}.{j}$. By connecting MHV vertices, amplitudes involving more negative helicity gluons can be built up. The MHV rules for gluons [@CSW1] have been extended to amplitudes with fermions [@GK]. New compact results for tree-level gauge-theory results for non-MHV amplitudes involving arbitrary numbers of gluons [@Zhu; @KosowerNMHV; @BBK], and fermions [@GK; @GGK; @Wu1; @Wu2] have been derived. They have been applied to processes involving external Higgs bosons [@DGK; @BGK] and electroweak bosons [@BFKM]. MHV rules for tree amplitudes have further been recast in the form of recursive relations  [@BBK; @BFKM; @BGK] which facilitate calculations of higher order non-MHV amplitudes in terms of the known lower-order results. In many cases new classes of tree amplitudes were derived, and in all cases, numerical agreement with previously known amplitudes has been found. MHV rules have also been shown to work at the loop-level for supersymmetric theories. Building on the earlier work of Bern et al [@BDDK1; @BDDK2], there has been enormous progress in computing cut-constructible multi-leg loop amplitudes in $\Neqfour$ [@CSW2; @BST; @CSW3; @BBKR; @Cachazo; @BCF1; @BDDK7; @BCF2; @BCF3; @BDKNMHV] and $\Neqone$ [@QuigleyRozali; @BBST1; @BBDD; @BBDP1] supersymmetric gauge theories. Encouraging progress has also been made for non-supersymmetric loop amplitudes [@BBST2; @BBDP2; @BDKrec]. Remarkably, the expressions obtained for the infrared singular parts of $\Neqfour$ one-loop amplitudes (which are known to be proportional to tree-level results) were found to produce even more compact expressions for gluonic tree amplitudes [@BDKNMHV; @RSV]. This observation led to the BCF recursion relations [@BCF4; @BCFW] discussed earlier as well as extremely compact six-parton amplitudes [@LW1; @LW2] and expressions for MHV and NMHV graviton amplitudes [@BBST3; @CS]. The factorisation properties of amplitudes in the infrared play several roles in developing higher order perturbative predictions for observable quantities. First, a detailed knowledge of the structure of unresolved emission enables phase space integrations to be organised such that the infrared singularities due to soft or collinear emission can be analytically extracted [@Giele:1992vf; @Frixione:1996ms; @Catani:1997vz]. Second, they enable large logarithmic corrections to be identified and resummed. Third, the collinear limit plays a crucial role in the unitarity-based method for loop calculations [@BDDK1; @BDDK2; @Bern:1996db; @Bern:1996je]. In general, to compute a cross section at N$^n$LO, one requires detailed knowledge of the infrared factorisation functions describing the unresolved configurations for $n$-particles at tree-level, $(n-1)$-particles at one-loop etc. The universal behaviour in the double collinear limit is well known at tree-level (see for example Refs. [@Altarelli:1977zs; @Bassetto:1983ik]), one-loop [@Bern:1995ix; @BDDK1; @Bern:split1gluon; @Kosower:split1; @Bern:split1QCD; @Catani:2000pi] and at two-loops [@Bern:2lsplit; @Badger:2lsplit]. Similarly, the triple collinear limit has been studied at tree-level [@Gehrmann-DeRidder:dblunres; @Campbell:dblunres; @Catani:NNLOcollfact; @Catani:IRtreeNNLO] and, in the case of distinct quarks, at one-loop [@Catani:2003]. Finally, the tree-level quadruple gluon collinear limit is derived in Ref. [@delduca]. Our paper is organised as follows. In Section [**\[sec:mhv\]**]{}, we briefly review the spinor helicity and colour ordered formalism that underpins the MHV rules. Section [**\[sec:limit\]**]{} describes the procedure for taking the collinear limit and deriving the splitting functions. We write down a general collinear factorization formula, which is valid for specific numbers of negative helicity gluons and an arbitrary number of positive helicity gluons and demonstrate that the gauge dependence explicitly cancels. We find it useful to classify our results according to the difference between the number of negative helicity gluons before taking the collinear limit, and the number after. We call this difference $\Delta M$. We provide formulae describing an arbitrary number of gluons for $\Delta M \leq 2$ in Section [**\[sec:gen\]**]{}. Specific explicit results for the collinear limits of up to six gluons are given in Sec. [**\[sec:expl\]**]{}. We have numerically checked that our results agree with the results available in the literature for three and four collinear gluons [@delduca]. Our findings are summarized in Sec. [**\[sec:concl\]**]{}. Colour ordered amplitudes in the spinor helicity formalism {#sec:mhv} ========================================================== Tree-level multi-gluon amplitudes can be decomposed into colour-ordered partial amplitudes as $${\cal A}_n(\{p_i,\lambda_i,a_i\}) = i g^{n-2} \sum_{\sigma \in S_n/Z_n} {\rm Tr}(T^{a_{\sigma(1)}}\cdots T^{a_{\sigma(n)}})\, A_n(\sigma(1^{\lambda_1},\ldots,n^{\lambda_n}))\,. \label{TreeColourDecomposition}$$ Here $S_n/Z_n$ is the group of non-cyclic permutations on $n$ symbols, and $j^{\lambda_j}$ labels the momentum $p_j$ and helicity $\lambda_j$ of the $j^{\rm th}$ gluon, which carries the adjoint representation index $a_i$. The $T^{a_i}$ are fundamental representation SU$(N_c)$ colour matrices, normalized so that ${\rm Tr}(T^a T^b) = \delta^{ab}$. The strong coupling constant is $\alpha_s=g^2/(4\pi)$. The MHV rules method of Ref. [@CSW1] is used to evaluate only the purely kinematic amplitudes $A_n.$ Full amplitudes are then determined uniquely from the kinematic part $A_n$, and the known expressions for the colour traces. In the spinor helicity formalism [@SpinorHelicity; @ParkeTaylor; @BG] an on-shell momentum of a massless particle, $p_\mu p^\mu=0,$ is represented as p\_[a a]{}  p\_\^\_[a a]{} = \_a\_[a]{}  , where $\lambda_a$ and $\tilde\lambda_{\dot a}$ are two commuting spinors of positive and negative chirality. Spinor inner products are defined by[^2] ,’=  \_[ab]{}\^a’\^b  , = -\_[ab]{} \^[a]{}’\^[b]{}  , and a scalar product of two null vectors, $p_{a\dot a}=\lambda_a \tilde\lambda_{\dot a}$ and $q_{a\dot a}=\lambda'_a\tilde\lambda'_{\dot a}$, becomes \[scprod\] p\_q\^= - ,’ . The MHV rules of Ref. [@CSW1] were developed for calculating purely gluonic amplitudes at tree level. In this approach all non-MHV $n$-gluon amplitudes (including $\overline{\rm MHV}$) are expressed as sums of tree diagrams in an effective scalar perturbation theory. The vertices in this theory are the MHV amplitudes of Eq. (\[MHV\]) continued off-shell as described below, and connected by scalar propagators $1/q^2$. When one leg of an MHV vertex is connected by a propagator to a leg of another MHV vertex, both legs become internal to the diagram and have to be continued off-shell. Off-shell continuation is defined as follows [@CSW1]: we pick an arbitrary reference spinor $\eta^{\dot a}$ and define $\lambda_a$ for any internal line carrying momentum $q_{a\dot a}$ by \[ofsh\] \_a=q\_[aa]{}\^[a]{} . External lines in a diagram remain on-shell, and for them $\lambda$ is defined in the usual way. For the off-shell lines, the same reference spinor $\eta$ is used in all diagrams contributing to a given amplitude. The multiple collinear limit {#sec:limit} ============================ To find the splitting functions we work with the colour stripped amplitudes $A_n$. For these colour ordered amplitudes, it is known that when the collinear particles are not adjacent there is no collinear divergence [@delduca]. Therefore, without loss of generality, we can take particles $1 \dots n$ collinear. The multiple collinear limit is approached when the momenta $p_1, \dots, p_n$ become parallel. This implies that all the particle subenergies $s_{ij}=(p_i+p_j)^2$, with $i,j=1,\dots,n$, are simultaneously small. We thus introduce a pair of light-like momenta $P^\nu$ and $\xi^\nu$ ($P^2=0, \xi^2=0$), and we write $$(p_1 + \dots + p_n)^\nu = P^\nu + \frac{s_{1,n} \; \xi^\nu}{2 \, \xi \cdot P} \;, \quad s_{i,j} = (p_i + \dots + p_j)^2 \;,$$ where $s_{1,n}$ is the total invariant mass of the system of collinear partons. In the collinear limit, the vector $P^\nu$ denotes the collinear direction, and the individual collinear momenta are $p_i^\nu \to z_i P^\nu$. Here the longitudinal-momentum fractions $z_i$ are given by $$z_i = \frac{\xi \cdot p_i}{\xi \cdot P}$$ and fulfil the constraint $\sum_{i=1}^m z_i =1$. To be definite, in the rest of the paper we work in the time-like region so that ($s_{ij} > 0, \; 1>z_i > 0$). ![Factorisation of an $N$-point colour ordered amplitude with gluons $p_1,\ldots,p_n$ collinear into splitting function for $P \to 1, \ldots, n$ multiplied by an $(N-n+1)$-point amplitude.[]{data-label="fig:limit"}](limit.eps){width="12cm"} As illustrated in Fig. \[fig:limit\], in the multi-collinear limit an $N$-gluon colour ordered tree amplitude factorises and can be written as \[eq:factorise\] A\_N(1\^[\_1]{},…,N\^[\_N]{}) && [([1]{}\^[\_[1]{}]{},…,n\^[\_n]{} P\^)]{} A\_[N-n+1]{}((n+1)\^[\_[n+1]{}]{},…,N\^[\_N]{},P\^).\ This labelling of the splitting amplitude ${\mathrm{split}(1^{\lambda_1},\ldots,n^{\lambda_n}\to P^{\lambda})}$ differs from the usual definition because we use the momentum and helicity that participates in the resultant amplitude $P^\lambda$ rather than $-P^{-\lambda}$. With this choice, it is easier to see how the helicity is conserved in the splitting, i.e. helicity $\lambda^1,\ldots,\lambda^n$ is replaced by $\lambda$. There are two different types of collinear limit [@CSW1], those that conserve the number of negative helicity gluons between the initial state and the final collinear state, and those that do not. Only the limits of the type ${\mathrm{split}(1^+,\ldots,n^+\to P^{+})}$ and ${\mathrm{split}(1^-,2^+,\ldots,n^+\to P^{-})}$ can contribute to the negative helicity conserving case, and these collinear splitting functions are straightforward to derive directly from the simple MHV vertex. All other limits belong to the second class which do not conserve the number of negative helicity gluons, and therefore we classify our results according to the difference between the number of negative helicity gluons before taking the collinear limit, and the number after, $\Delta M$. We find that $\Delta M$ corresponds to the order of MHV diagram needed to find a particular collinear limit, as follows, M=0 : &&1\^+,2\^+,3\^+, …,n\^+ P\^+\ &&1\^-,2\^+,3\^+, …,n\^+ P\^-\ & &\ M=1 : &&1\^-,2\^+,3\^+, …,n\^+ P\^+\ &&1\^-,2\^-,3\^+, …,n\^+ P\^- \[mmp\]\ & &\ M=2 : &&1\^-,2\^-,3\^+, …,n\^+ P\^+\ &&1\^-,2\^-,3\^-, …,n\^+ P\^-\ and so on for all $\Delta M>2$ cases. The splitting functions are derived by examining the general form of MHV diagrams, which consist of MHV vertices and scalar propagators. The general form of the $n$-particle collinear splitting functions is given by \[dmzero\] M =0:[  ]{} && ,\ \[dmnzero\] M 0:[  ]{} && , such that $(v-1)+(n-v)=n-1$, where $v$ is the number of vertices, and thus $v-1$ is the number of scalar propagators. From it follows that for an MHV-diagram to contribute to $ \Delta M \neq 0$ collinear limits, it is required to contain anti-holomorphic spinor products $\spb{i}.{j}$ of collinear momenta. However, because on-shell MHV vertices are entirely holomorphic, within the MHV rules there are only two potential sources of the anti-holomorphic spinor products. One source is scalar propagators $1/s_{ij}=1/\spa{i}.{j} \spb{j}.{i}$ which inter-connect MHV vertices. The second source is the off-shell continuation of the corresponding connected legs in the MHV vertices. Each off-shell continued leg of momentum $P$ gives rise to a factor $\langle i P \rangle \propto \langle i | P | \eta]$ which gives rise to anti-holomorphic factors of $[j \eta]$. When the reference spinors $\eta_{\dot\alpha}$ are kept general, and specifically, not set to be equal to one of the momenta in the collinear set, the $\eta$-dependence must cancel and the off-shell continuation cannot give rise to an overall factor of $\spb{i}.{j}$. Therefore, the only source of singular anti-holomorphic factors are MHV-diagrams that contain an internal propagator of momentum $q_{i+1,j}=p_{i+1}+\ldots + p_j$ which is a sum of external momenta from the collinear set such that $q^2=s_{i+1,j} \rightarrow 0$. Hence, we conclude that only a subset of MHV-diagrams contributes to multi-collinear limits of tree amplitudes. The subset is determined by requiring that all $v-1$ internal propagators are on-shell in the multi-collinear limit. This is a powerful constraint on the types of the contributing diagrams and it simplifies the calculation[^3]. We exploit the universal nature of the splitting function by choosing to start with an amplitude with $(n+3)$ external legs, i.e. setting $N=n+3$ in Eq. (\[eq:factorise\]). The helicities of the gluons are adjusted so that the remnant ‘hard’ four point MHV amplitude $A_4(P^\lambda, (n+1)^+, (n+2)^{-\lambda}, (n+3)^-)$ is given by\ && with $X=P$ for $ \lambda=-$ and $X=n+2$ for $\lambda=+$. To read off the collinear limits from the MHV rules, we use the limiting expressions for the spinor products: $\spa{a}.{q}$, $\spa{b}.{q}$ and $\spa{b}.{a}$. Here $a$ is a particle from the collinear set, $b$ is a particle which is not in the collinear set, and $q$ is the sum of the collinear momenta from $i+1$ to $j$. Hence, using .[q]{}=\_[l=i+1]{}\^[j]{}.[l]{} .  , .[q]{}=\_[l=i+1]{}\^[j]{}.[l]{} .  , and the expressions for spinors from the collinear set, |l= |P , |l\] = |P\]  , |a= |P , |a\] = |P\]  , we have, .[q]{}&& .\_[l=i+1]{}\^[j]{}.[l]{} ..[j]{}.[a]{} \[aqqq\]\ .[q]{}&& ..[P]{}\_[l=i+1]{}\^[j]{} z\_l \[Xq\]\ .[a]{}&& .[P]{} .\[Xa\] Here we introduced the definition $$\label{eq:7} \Del{i}.{j}.{a} = \sum_{l=i+1}^{j}\spa{a}.{l}\sqrt{z_l} .$$ Equations  and contain a factor $\spb{P}.{\eta}$ which, however, will always cancel in expressions for relevant splitting functions. As such we can read off the collinear limits of the amplitudes from the MHV-rules expressions by replacing terms on the left hand side of equations , and with the expressions on the right hand side of those equations, and further dropping the $\spb{P}.{\eta}$ factors. Certain terms in the sums that arise in MHV rules need special attention. These are the boundary terms involving either $\spa{0}.{1}$ or $\spa{n}.{n+1}$, and for these we have, && -, \[jn\]\ && .\[inp3\] We now present our results for the splitting functions. Results {#sec:results} ======= In this section we give the results for the multiple collinear limit of gluons. First we give the general results for an arbitrary number of gluons with $\Delta M \leq 2$. Afterwards we give explicit results for up to four collinear gluons for all independent helicity combinations, together with some specific examples for five and six collinear gluons. General results {#sec:gen} --------------- In this section we present the general results for the cases where the number of gluons with negative helicity changes by at most $\Delta M = 2$, and those related by parity where the number of gluons with positive helicity changes at most by the same amount. With the help of parity these general splitting amplitudes are sufficient to obtain the explicit expressions for all helicity combinations of up to six gluons. We will often use a more compact notation for the splitting amplitude. We denote the splitting amplitude for $n$ collinear gluons, of which $r$ have negative helicity, by: $$\begin{aligned} {\mathrm{split}(1^+,\ldots , m_1^-, \ldots, m_2^-, \ldots,m_r^-,\ldots , n^+\to P^{\pm})}={\mathrm{Split}^{(n)}}_{\pm}(m_1,\ldots,m_r).\end{aligned}$$ ### $\Delta M = 0$ {#sec:4.1.1} This is the simplest case which is read directly off the single MHV vertex. The denominator of an $N$-point MHV amplitude is factorised as follows (in the limit of collinear $p_1, \ldots, p_n$): &&.[1]{}.[2]{}….[n+1]{}….[N]{}\ &&= ( \_[l=1]{}\^[n-1]{} .[l+1]{}) (.[P]{} .[n+1]{}….[N]{} ) where the first factor contributes to the splitting function, and the second one is the denominator of the remaining hard MHV amplitude. Hence, the splitting function is $$\begin{aligned} {\mathrm{split}(1^+ ,\ldots, n^+ \to P^{+})}= \frac{1}{\sqrt{z_1 z_n} \prod_{l=1}^{n-1} \spa{l,}.{l+1} } \ ,\end{aligned}$$ and so by parity $$\begin{aligned} {\mathrm{split}(1^- ,\ldots , n^- \to P^{-})}= \frac{(-1)^{n-1}}{\sqrt{z_1 z_n} \prod_{l=1}^{n-1} \spb{l,}.{l+1} } \ .\end{aligned}$$ Similarly, $$\begin{aligned} {\mathrm{split}(1^+,\ldots,m_1^-, \ldots, n^+ \to P^{-})}= \frac{z_{m_1}^{2}}{\sqrt{z_1 z_n} \prod_{l=1}^{n-1} \spa{l,}.{l+1} } \ ,\end{aligned}$$ and $$\begin{aligned} {\mathrm{split}(1^-,\ldots,m_1^+, \ldots, n^- \to P^{+})}= \frac{(-1)^{n-1}z_{m_1}^{2}}{\sqrt{z_1 z_n} \prod_{l=1}^{n-1} \spb{l,}.{l+1} } \ .\end{aligned}$$ ### $\Delta M = 1$ {#sec:4.1.2} ![MHV diagrams contributing to ${\mathrm{Split}^{(n)}}_+(m_1)$. Negative helicity gluons are indicated by solid lines, while arbitrary numbers of positive helicity gluons emitted from each vertex are shown as dotted arcs.[]{data-label="fig:m1"}](m1.eps){width="5cm"} This is the next-to-MHV (NMHV) case, and in the collinear limit we need to take into account only a subset of MHV diagrams. In fact, there is only a single MHV diagram (or more precisely a single class of MHV diagrams) which can contribute to ${\mathrm{Split}^{(n)}}_+(m_1)$. It is shown in Fig. \[fig:m1\].[^4] In the limit where gluons $1, \ldots, n$ become collinear. The left vertex in Fig. \[fig:m1\] produces a ‘hard’ MHV amplitude while the right vertex generates the splitting function. We need to sum over $i$ and $j$ in Fig. \[fig:m1\] in such a way that only diagrams with a singular propagator are selected in the collinear limit. This puts a constraint $j\le n$ where $n$ is the number of collinear gluons. The resulting splitting function reads, $$\begin{aligned} \label{mnpp} {\mathrm{Split}^{(n)}}_+(m_1) &= \frac{1}{\sqrt{z_1 z_n} \prod_{l=1}^{n-1} \spa{l,}.{l+1} }\bigg( \sum_{i=0}^{m_1-1} \sum_{j=m_1}^{n} \frac{\Del{i}.{j}.{m_1}^4}{D(i,j,q_{i+1,j})} \bigg)\ ,\end{aligned}$$ where we define \[D\] D(i,j,q)= .[j]{}.[i]{}.[j]{}.[i+1]{}.[j]{}.[j]{}.[j]{}.[j+1]{} . ![MHV diagrams contributing to ${\mathrm{Split}^{(n)}}_-(m_1,m_2)$.[]{data-label="fig:m1m2"}](m1m2.eps){width="12cm"} Similarly, there are three (classes of) MHV-diagrams contributing to ${\mathrm{Split}^{(n)}}_-(m_1,m_2)$. They are shown in Fig. \[fig:m1m2\] and lead to a splitting function which reads $$\begin{aligned} \label{mmnpm} {\mathrm{Split}^{(n)}}_{-}(m_1,m_2)&= \frac{1}{\sqrt{z_1 z_n} \prod_{l=1}^{n-1} \spa{l,}.{l+1} }\Bigg( \sum_{i=0}^{m_1-1} \sum_{j=m_1}^{m_2-1} \frac{z_{m_2}^{2}\Del{i}.{j}.{m_1}^4}{D(i,j,q_{i+1,j})} \nonumber\\ &+\sum_{i=m_1}^{m_2-1} \sum_{j=m_2}^{n} \frac{z_{m_1}^{2}\Del{i}.{j}.{m_2}^4}{D(i,j,q_{i+1,j})}\nonumber\\ & +\sum_{i=0}^{m_1-1} \sum_{j=m_2}^{n} \frac{\spa{m_1}.{m_2}^4 }{D(i,j,q_{i+1,j})} \Big( \sum_{{l=i+1}}^{{j}}z_l \Big)^4\Bigg)\ .\end{aligned}$$ The remaining splitting amplitudes of the form $${\mathrm{split}(1^-, \ldots, m_1^+, \ldots, m_2^+, \ldots,m_r^+,\ldots, n^- \to P^{\pm})}$$ are obtained by parity transformation through the usual replacement $\spa{l,}.{k} \leftrightarrow -\spb{l,}.{k}$. ### $\Delta M = 2$ {#sec:4.1.3} ![MHV diagrams contributing to ${\mathrm{Split}^{(n)}}_+(m_1,m_2)$.[]{data-label="fig:m1m2plus"}](m1m2plus.eps){width="12cm"} The collinear limits with $\Delta M = 2$ are derived from next-to-next-to-MHV (NNMHV) diagrams. There are four (classes of) MHV-diagrams contributing to ${\mathrm{Split}^{(n)}}_{+}(m_1,m_2)$ which are shown in Fig. \[fig:m1m2plus\]. The corresponding splitting function is, $$\begin{aligned} \label{mmnpp} {\mathrm{Split}^{(n)}}_{+}&(m_1,m_2)= \frac{1}{\sqrt{z_1 z_n} \prod_{l=1}^{n-1} \spa{l,}.{l+1}}\nonumber\\& \Bigg( \sum_{i=0}^{m_1 -1}\sum_{j=m_2}^{n}\sum_{k=m_1}^{m_2-1}\sum_{r=m_2}^{j} \frac{\Del{i}.{j}.{m_1}^4\Del{k}.{r}.{m_2}^4}{{\mathrm{DD}}(i,j,q_{i+1,j};k,r,q_{k+1,r})}\nonumber\\ &+ \sum_{i=0}^{m_1 -1}\sum_{j=m_1}^{k}\sum_{k=m_1}^{m_2-1}\sum_{r=m_2}^{n} \frac{\Del{i}.{j}.{m_1}^4\Del{k}.{r}.{m_2}^4}{{\mathrm{DD}}(i,j,q_{i+1,j};k,r,q_{k+1,r})}\nonumber\\ & +\sum_{i=0}^{k}\sum_{j=m_2}^{n}\sum_{k=0}^{m_1-1}\sum_{r=m_1}^{m_2-1} \frac{\Del{i}.{j}.{m_2}^4 \Del{k}.{r}.{m_1}^4}{{\mathrm{DD}}(i,j,q_{i+1,j};k,r,q_{k+1,r})}\nonumber\\ &+\sum_{i=0}^{k}\sum_{j=m_2}^{n}\sum_{k=0}^{m_1-1}\sum_{r=m_2}^{j} \frac{\spa{m_1}.{m_2}^4 \Dell{i}.{j}.{k}.{r}^4} {{\mathrm{DD}}(i,j,q_{i+1,j};k,r,q_{k+1,r})} \Bigg)\end{aligned}$$ where $\Del{i}.{j}.{k}$ is given in Eq. (\[eq:7\]) and we introduce $$\label{eq:8} \Dell{i}.{j}.{k}.{r} = \sum_{u=i+1}^{j}\sum_{v=k+1}^{r}\spa{u}.{v}\sqrt{z_u z_v} \ .$$ The ‘effective propagator’ ${\mathrm{DD}}$ is defined by $${\mathrm{DD}}(i,j,q_1;k,r,q_2)=\chi(i,k,q_1,q_2) \chi(r,j,q_2,q_1) \chi(j,k,q_1,q_2) D(i,j,q_1)D(k,r,q_2)$$ in terms of $D$ defined previously in Eq. , and $\chi$ given by $$\label{chi} \chi(i,k,q_1,q_2)= \left \{\begin{array} {ll} 1 & i \neq k \\ \frac{\Dellq{1}.{2} \spa{i,}.{ i+1}}{ \Delq{ q_1}.{i+1} \Delq{q_2}.{i} } & i = k \end{array} \right. .$$ ![MHV topologies contributing to ${\mathrm{Split}^{(n)}}_-(m_1,m_2,m_3)$. The negative helicity gluons $m_1$, $m_2$ and $m_3$ are distributed in a cyclic way around each diagram. The remaining leg is the negative helicity gluon that remains after the collinear limit is taken.[]{data-label="fig:m1m2m3"}](m1m2m3.eps){width="12cm"} Finally there are 16 classes of MHV-diagrams contributing to ${\mathrm{Split}^{(n)}}_{-}(m_1,m_2,m_3)$, coming from the 5 topologies shown in Fig. \[fig:m1m2m3\] and their cyclic permutations. The individual contributions are given by $$\begin{aligned} \label{mmmnpm} {\mathrm{Split}^{(n)}}_{-}(m_1,m_2,m_3)=\frac{1}{\sqrt{z_1 z_n} \prod_{l=1}^{n-1} \spa{l,}.{l+1}}\sum_{i=1}^{16} A^{(i)}(m_1,m_2,m_3)\end{aligned}$$ where $$\begin{aligned} A^{(1)}(m_1,m_2,m_3)&=&\sum_{i=m_1}^{m_2-1}\sum_{j=m_3}^{n}\sum_{k=m_2}^{m_3-1}\sum_{r=m_3}^{j} z_{m_1}^2 \frac{\Del{i}.{j}.{m_2}^4 \Del{k}.{r}.{m_3}^4}{{\mathrm{DD}}(i,j,q_{i+1,j};k,r,q_{k+1,r})} \\ A^{(2)}(m_1,m_2,m_3)&=&\sum_{i=m_1}^{k}\sum_{j=m_3}^{n}\sum_{k=m_1}^{m_2-1}\sum_{r=m_2}^{m_3-1} z_{m_1}^2 \frac{\Del{i}.{j}.{m_3}^4 \Del{k}.{r}.{m_2}^4}{{\mathrm{DD}}(i,j,q_{i+1,j};k,r,q_{k+1,r})} \\ A^{(3)}(m_1,m_2,m_3)&=&\sum_{i=0}^{m_1-1}\sum_{j=m_2}^{m_3-1}\sum_{k=m_1}^{m_2-1}\sum_{r=m_2}^{j} z_{m_3}^2 \frac{\Del{i}.{j}.{m_1}^4 \Del{k}.{r}.{m_2}^4}{{\mathrm{DD}}(i,j,q_{i+1,j};k,r,q_{k+1,r})} \\ A^{(4)}(m_1,m_2,m_3)&=&\sum_{i=0}^{k}\sum_{j=m_2}^{m_3-1}\sum_{k=0}^{m_1-1}\sum_{r=m_1}^{m_2-1} z_{m_3}^2 \frac{\Del{i}.{j}.{m_2}^4 \Del{k}.{r}.{m_1}^4}{{\mathrm{DD}}(i,j,q_{i+1,j};k,r,q_{k+1,r})} \\ A^{(5)}(m_1,m_2,m_3)&=&\sum_{i=m_1}^{k}\sum_{j=m_3}^{n}\sum_{k=m_1}^{m_2-1}\sum_{r=m_3}^{j} z_{m_1}^2 \frac{\spa{m_2}.{m_3}^4 \Dell{i}.{j}.{k}.{r}^4}{{\mathrm{DD}}(i,j,q_{i+1,j};k,r,q_{k+1,r})} \\ A^{(6)}(m_1,m_2,m_3)&=&\sum_{i=0}^{k}\sum_{j=m_2}^{m_3-1}\sum_{k=0}^{m_1-1}\sum_{r=m_2}^{j} z_{m_3}^2 \frac{\spa{m_1}.{m_2}^4 \Dell{i}.{j}.{k}.{r}^4}{{\mathrm{DD}}(i,j,q_{i+1,j};k,r,q_{k+1,r})} \\ A^{(7)}(m_1,m_2,m_3)&=&\sum_{i=0}^{m_1-1}\sum_{j=m_3}^{n}\sum_{k=m_2}^{m_3-1}\sum_{r=m_3}^{j} \Big( \sum_{{l=i+1}}^{{j}}z_l \Big)^4 \frac{\spa{m_1}.{m_2}^4 \Del{k}.{r}.{m_3}^4}{{\mathrm{DD}}(i,j,q_{i+1,j};k,r,q_{k+1,r})} \\ A^{(8)}(m_1,m_2,m_3)&=&\sum_{i=0}^{m_1-1}\sum_{j=m_3}^{n}\sum_{k=m_1}^{m_2-1}\sum_{r=m_3}^{j} \Big( \sum_{{l=i+1}}^{{j}}z_l \Big)^4 \frac{\spa{m_2}.{m_3}^4 \Del{k}.{r}.{m_1}^4}{{\mathrm{DD}}(i,j,q_{i+1,j};k,r,q_{k+1,r})} \\ A^{(9)}(m_1,m_2,m_3)&=&\sum_{i=0}^{k}\sum_{j=m_3}^{n}\sum_{k=0}^{m_1-1}\sum_{r=m_2}^{m_3-1} \Big( \sum_{{l=i+1}}^{{j}}z_l \Big)^4 \frac{\spa{m_1}.{m_2}^4 \Del{k}.{r}.{m_3}^4}{{\mathrm{DD}}(i,j,q_{i+1,j};k,r,q_{k+1,r})} \\ A^{(10)}(m_1,m_2,m_3)&=&\sum_{i=0}^{m_1-1}\sum_{j=m_3}^{n}\sum_{k=m_1}^{m_2-1}\sum_{r=m_2}^{m_3-1} \Big( \sum_{{l=i+1}}^{{j}}z_l \Big)^4 \frac{\spa{m_1}.{m_3}^4 \Del{k}.{r}.{m_2}^4}{{\mathrm{DD}}(i,j,q_{i+1,j};k,r,q_{k+1,r})} \\ A^{(11)}(m_1,m_2,m_3)&=&\sum_{i=0}^{k}\sum_{j=m_3}^{n}\sum_{k=0}^{m_1-1}\sum_{r=m_1}^{m_2-1} \Big( \sum_{{l=i+1}}^{{j}}z_l \Big)^4 \frac{\spa{m_2}.{m_3}^4 \Del{k}.{r}.{m_1}^4}{{\mathrm{DD}}(i,j,q_{i+1,j};k,r,q_{k+1,r})} \\ A^{(12)}(m_1,m_2,m_3)&=&\sum_{i=m_1}^{m_2-1}\sum_{j=m_3}^{n}\sum_{k=m_2}^{r}\sum_{r=m_2}^{m_3-1} z_{m_1}^2 \frac{\Del{i}.{k}.{m_2}^4 \Del{r}.{j}.{m_3}^4}{{\mathrm{DD}}(i,k,q_{i+1,k};r,j,q_{r+1,j})} \\ A^{(13)}(m_1,m_2,m_3)&=&\sum_{i=0}^{m_1-1}\sum_{j=m_2}^{m_3-1}\sum_{k=m_1}^{r}\sum_{r=m_1}^{m_2-1} z_{m_3}^2 \frac{\Del{i}.{k}.{m_1}^4 \Del{r}.{j}.{m_2}^4}{{\mathrm{DD}}(i,k,q_{i+1,k};r,j,q_{r+1,j})} \\ A^{(14)}(m_1,m_2,m_3)&=&\sum_{i=0}^{m_1-1}\sum_{j=m_3}^{n}\sum_{k=m_2}^{r}\sum_{r=m_2}^{m_3-1} \Big( \sum_{{l=i+1}}^{{k}}z_l \Big)^4 \frac{\spa{m_1}.{m_2}^4 \Del{r}.{j}.{m_3}^4}{{\mathrm{DD}}(i,k,q_{i+1,k};r,j,q_{r+1,j})} \\ A^{(15)}(m_1,m_2,m_3)&=&\sum_{i=0}^{m_1-1}\sum_{j=m_3}^{n}\sum_{k=m_1}^{r}\sum_{r=m_1}^{m_2-1} \Big( \sum_{{l=r+1}}^{{j}}z_l \Big)^4 \frac{\spa{m_2}.{m_3}^4 \Del{i}.{k}.{m_1}^4}{{\mathrm{DD}}(i,k,q_{i+1,k};r,j,q_{r+1,j})} \\ A^{(16)}(m_1,m_2,m_3)&=&\sum_{i=0}^{m_1-1}\sum_{j=m_3}^{n}\sum_{k=m_1}^{m_2-1}\sum_{r=m_2}^{m_3-1} z_{m_2}^2 \frac{\Del{i}.{k}.{m_1}^4 \Del{r}.{j}.{m_3}^4}{{\mathrm{DD}}(i,k,q_{i+1,k};r,j,q_{r+1,j})}.\end{aligned}$$ Specific results for $n<7$. {#sec:4.2} --------------------------- In this section we present compact expressions for splitting amplitudes with up to six collinear gluons. These results are obtained directly from the general expressions given in Section \[sec:gen\]. First we note that splitting amplitudes satisfy reflection symmetry, $$\begin{aligned} {\mathrm{split}(1^{\lambda_1},\ldots ,n^{\lambda_n}\to P^{\pm})} = (-1) ^ {n+1} {\mathrm{split}(n^{\lambda_n},\ldots , 1^{\lambda_1}\to P^{\pm})}\end{aligned}$$ and the dual Ward identity, see e.g. [@delduca], $$\begin{aligned} &{\mathrm{split}(1^{\lambda_1},2^{\lambda_2},\ldots ,n^{\lambda_n}\to P^{\pm})} + {\mathrm{split}(2^{\lambda_2},1^{\lambda_1},\ldots ,n^{\lambda_n}\to P^{\pm})} +\cdots \nonumber \\ +\,\,&{\mathrm{split}(2^{\lambda_2},\ldots ,1^{\lambda_1},n^{\lambda_n}\to P^{\pm})} + {\mathrm{split}(2^{\lambda_2},\ldots ,n^{\lambda_n},1^{\lambda_1}\to P^{\pm})} = 0.\end{aligned}$$ These relations reduce the number of independent splitting amplitudes significantly. ### $n=2$ {#sec:4.2.1} For two collinear gluons there are two independent splitting amplitudes with $\Delta M = 0$. All others can be obtained by parity and reflection. Setting $z_1 = z$ and $z_2 = (1-z)$, we find $$\begin{aligned} {\mathrm{split}(1^+, 2^+\to P^{+})} &=& \, \frac{1}{\sqrt{z(1-z)}\, \langle 1 2\rangle}\, ,\\ {\mathrm{split}(1^-, 2^+\to P^{-})} &=& \,\frac{z^2}{\sqrt{z(1-z)}\, \langle 1 2\rangle}\, .\end{aligned}$$ As expected, the splitting amplitudes have a single pole proportional to $\spa{1}.{2}^{-1}$ . Note that in the soft limit $z \to 0$, we see that helicity conservation ensures that ${\mathrm{split}(1^-, 2^+\to P^{-})} \to 0$. ### $n=3$ result from MHV rules {#sec:4.2.2} For three collinear gluons there are three independent splitting amplitudes with $\Delta M=0$. They all follow directly from a single MHV vertex and are given by $$\begin{aligned} \label{eq:2} {\mathrm{split}(1^+,2^+,3^+\to P^{+})} &=& \frac{1}{\sqrt{z_1z_3}\spa{1}.{2}\spa{2}.{3}}\ ,\\ {\mathrm{split}(1^-,2^+,3^+\to P^{-})} &=& \frac{z_1^2}{\sqrt{z_1z_3}\spa{1}.{2}\spa{2}.{3}}\ ,\\ {\mathrm{split}(1^+,2^-,3^+\to P^{-})} &=& \frac{z_2^2}{\sqrt{z_1z_3}\spa{1}.{2}\spa{2}.{3}}\ .\end{aligned}$$ Parity and the reflection symmetry, ${\mathrm{split}(1^+,2^+,3^-\to P^{-})} = {\mathrm{split}(3^-,2^+,1^+\to P^{-})},$ give the rest. When $\Delta M = 1$, there are three amplitudes, \[eq:triple1\] [(1\^-, 2\^+, 3\^+P\^[+]{})]{} &=&\ &+&  ,\ \ \[eq:triple3\] [(1\^+, 2\^-, 3\^+P\^[+]{})]{} &=& -[(2\^-, 1\^+, 3\^+P\^[+]{})]{}-[(1\^+, 3\^+, 2\^-P\^[+]{})]{}\ &=&\ &+&\ &+&  ,\ \ [(1\^+, 2\^+, 3\^-P\^[+]{})]{} &=& [(3\^- ,2\^+, 1\^+P\^[+]{})]{}  .\ In addition to singular terms like $\spa{1}.{2}$, we see that the splitting functions contain mixed terms like $s_{1,3}$. The net singularity is schematically of the form $[~]\langle~\rangle$. Note that ${\mathrm{split}(1^-, 2^+, 3^+\to P^{+})}$ contains poles in $s_{1,2}$ and the triple invariant $s_{1,3} = s_{123}$ but not in $s_{2,3}$. This is because there is no MHV rule graph with a three-point vertex involving two positive helicity gluons. Expressions for these splitting functions are given in Eq. (5.52) of Ref. [@delduca]. The results given here are more compact and have a rather different analytic form. After adjusting the normalisation of the colour matrices, the splitting functions of Eqs. – numerically agree with those of Ref. [@delduca]. ### $n=3$ result from the BCF recursion relation {#sec:4.2.3} We now want to rederive the above results using the BCF recursion relation of [@BCF4]. In doing this we will (a) draw some useful comparisons between the ‘BCF recursion’ and the ‘MHV rules’ formalisms from the perspective of collinear amplitudes; and (b) test our expressions, such as Eq.  for ${\mathrm{split}(1^-, 2^+, 3^+\to P^{+})}$. ![BCF diagrams contributing to $A(1^-,2^+,3^+,4^+,5^-,6^-)$.[]{data-label="fig:bcf"}](BCF.eps){height="4cm"} We start with the six-point amplitude $A(1^-,2^+,3^+,4^+,5^-,6^-)$, and calculate it via the BCF recursive approach. We ultimately want to take the collinear limit $1\,||\,2\,||\,3 \rightarrow P^{+}$, so it will be convenient to choose the ‘marked’ gluons (required for the BCF recursive set-up) to be from this collinear set. Hence, we will mark the $\hat{1}^-$ and $\hat{2}^+$ gluons. There are only two BCF diagrams which contribute to the full amplitude, and they are shown in Fig. \[fig:bcf\]. We now note that in this particular collinear limit, only the second of these diagrams contains an on-shell propagator, $1/s_{23}$. Nevertheless, in distinction with the MHV rules approach which we have adopted previously, both BCF diagrams need to be taken into account in the collinear limit. The full amplitude reads $$A(1^-,2^+,3^+,4^+,5^-,6^-) = \frac{1}{{\ensuremath{\langle3 |1+2 |6]}}}\left( \frac{{\ensuremath{\langle5 |6+1 |2]}}^3}{{\ensuremath{[6 \,1]}}{\ensuremath{[1 \,2]}}{\ensuremath{\langle3 \,4\rangle}}{\ensuremath{\langle4 \,5\rangle}}s_{3,5}} + \frac{{\ensuremath{\langle1 |2+3 |4]}}^3}{{\ensuremath{[4 \,5]}}{\ensuremath{[5 \,6]}}{\ensuremath{\langle1 \,2\rangle}}{\ensuremath{\langle2 \,3\rangle}}s_{1,3}} \right) \ , \label{eq:1check}$$ where the two terms on the right hand side correspond to the two BCF diagrams above (cf. Eq. (2.9) of Ref. [@BCF4]). In the $1\,||\,2\,||\,3 \rightarrow P^{+} $ collinear limit, the first term becomes . This term factors into a contribution to the splitting amplitudes multiplied by a four-point MHV vertex. In contrast, in the collinear limit the second term factors onto the $\overline{{\rm MHV}}$ type diagram, written in terms of the anti-holomorphic spinor products, . For the special case of four-point amplitudes, the $\overline{{\rm MHV}}$ and MHV amplitudes coincide and we find an identical result to Eq. . Likewise, to test our expression for ${\mathrm{split}(1^+, 2^-, 3^+\to P^{+})}$ we start from Eq. (3.4) in [@BCF4]; $$\begin{aligned} A(1^+,2^-,3^+,4^-,5^+,6^-)&=& \frac{{\ensuremath{[1 \,3]}}^4 {\ensuremath{\langle4 \,6\rangle}}^4 }{{\ensuremath{[1 \,2]}}{\ensuremath{[2 \,3]}}{\ensuremath{\langle4 \,5\rangle}}{\ensuremath{\langle5 \,6\rangle}} s_{1,3}{\ensuremath{\langle6 |1+2 |3]}}{\ensuremath{\langle4 |2+3 |1]}}}\nonumber\\ &+&\frac{{\ensuremath{\langle2 \,6\rangle}}^4 {\ensuremath{[3 \,5]}}^4 }{{\ensuremath{\langle6 \,1\rangle}}{\ensuremath{\langle1 \,2\rangle}}{\ensuremath{[3 \,4]}}{\ensuremath{[4 \,5]}} s_{3,5}{\ensuremath{\langle6 |4+5 |3]}}{\ensuremath{\langle2 |3+4 |5]}}}\nonumber\\ &+&\frac{{\ensuremath{[1 \,5]}}^4 {\ensuremath{\langle2 \,4\rangle}}^4 }{{\ensuremath{\langle2 \,3\rangle}}{\ensuremath{\langle3 \,4\rangle}}{\ensuremath{[5 \,6]}}{\ensuremath{[6 \,1]}} s_{2,4}{\ensuremath{\langle4 |2+3 |1]}}{\ensuremath{\langle2 |3+4 |5]}}}. \label{eq:3check}\end{aligned}$$ Taking the collinear limit $1\,||\,2\,||\,3 \rightarrow P^{+} $, we find that $$\begin{aligned} \label{eq:4check} \mbox{Split}(1^+,2^-,3^+\to P^+) &=&\frac{z_2^2 z_3^2 {\ensuremath{[1 \,2]}}}{\sqrt{z_1 z_2 z_3} s_{1,2} (z_1+z_2)\left({\ensuremath{[1 \,3]}}\sqrt{z_1}+{\ensuremath{[2 \,3]}}\sqrt{z_2}\right)}\nonumber\\ &+& \frac{{\ensuremath{[1 \,3]}}^4}{s_{1,3}{\ensuremath{[1 \,2]}}{\ensuremath{[2 \,3]}}\left({\ensuremath{[1 \,3]}}\sqrt{z_1} +{\ensuremath{[2 \,3]}}\sqrt{z_2}\right)\left({\ensuremath{[1 \,2]}}\sqrt{z_2}+{\ensuremath{[1 \,3]}}\sqrt{z_3}\right)}\nonumber\\ &+&\frac{z_1^2 z_2^2 {\ensuremath{[2 \,3]}}}{\sqrt{z_1 z_2 z_3}s_{2,3}(z_2+z_3) \left({\ensuremath{[1 \,2]}}\sqrt{z_2}+{\ensuremath{[1 \,3]}}\sqrt{z_3}\right)}.\end{aligned}$$ This result has the same kinematic-invariant pole structure as Eq. , but otherwise is not obviously equivalent to Eq. . Note that Eq.  contains terms like $\left({\ensuremath{[1 \,2]}}\sqrt{z_2}+{\ensuremath{[1 \,3]}}\sqrt{z_3}\right)$ (rather than $\left({\ensuremath{\langle1 \,2\rangle}}\sqrt{z_2}+{\ensuremath{\langle1 \,3\rangle}}\sqrt{z_3}\right)$). Despite appearances, a more careful (e.g. numerical) comparison shows that these two results, Eqs.  and , are in fact the same. ### $n=4$ {#sec:4.2.4} For $n=4$, there are five collinear limits coming directly from MHV amplitudes where the number of gluons with negative helicity doesn’t change, $\Delta M=0$, $$\begin{aligned} {\mathrm{split}(1^+ ,2^+, 3^+, 4^+\to P^{+})}&=& \frac{1}{\sqrt{z_1z_4}\spa{1}.{2} \spa{2}.{3} \spa{3}.{4}},\\ {\mathrm{split}(1^- ,2^+, 3^+, 4^+\to P^{-})}&=& \frac{z_1^2}{\sqrt{z_1z_4}\spa{1}.{2} \spa{2}.{3} \spa{3}.{4}},\\ {\mathrm{split}(1^+ ,2^-, 3^+, 4^+\to P^{-})}&=& \frac{z_2^2}{\sqrt{z_1z_4}\spa{1}.{2} \spa{2}.{3} \spa{3}.{4}}.\end{aligned}$$ The remaining two are obtained by reflection symmetry, $$\begin{aligned} {\mathrm{split}(1^+, 2^+, 3^-, 4^+\to P^{-})}&=&-{\mathrm{split}(4^+ ,3^-, 2^+ ,1^+\to P^{-})}, \\ {\mathrm{split}(1^+ ,2^+ ,3^+ ,4^-\to P^{-})}&=&-{\mathrm{split}(4^- ,3^+ ,2^+, 1^+\to P^{-})}.\end{aligned}$$ When $\Delta M = 1$, there are ten splitting amplitudes however only three are imdependent, $$\begin{aligned} \lefteqn{{\mathrm{split}(1^- ,2^+, 3^+, 4^+\to P^{+})}=\mathcal{B}_1(1,2,3,4)} \nonumber \\ &=&{ -\frac { z_2^{3/2}\spa{1}.{2}} {\sqrt {z_1z_{{4}}}\spa{3}.{4}s_{{1,2}} \left( z_{{1}}+z_{{2}} \right) \Del{0}.{2}.{3}}}\nonumber\\&& +{\frac { \Del{0}.{3}.{1} ^{3}} {\sqrt {z_{{4}}}\spa{1}.{2}\spa{2}.{3}s_{{1,3}} \left( z_{{1}}+z_{{2}}+z_{{3}} \right) \Del{0}.{3}.{3}\Del{0}.{3}.{4}}}\nonumber\\&& -{\frac { \Del{0}.{4}.{1} ^{3}}{\spa{ 1}.{2}\spa{2}.{3}\spa{3}.{4}s_{{1,4}}\Del{0}.{4}.{4}}}, \label{eq:spb1}\end{aligned}$$ $$\begin{aligned} \lefteqn{ {\mathrm{split}(1^-,2^-,3^+,4^+\to P^{-})}=\mathcal{B}_2(1,2,3,4) }\nonumber \\ &=& -{\frac {z_1^{3/2} z_3^{3/2} \spa{2}.{3}}{ \sqrt {z_2z_{{4}}}s_{{2,3}}\Del{1}.{3}.{1}\Del{1}.{ 3}.{4}}}\nonumber\\&&-{\frac {{z_{{1}}}^{3/2} \Del{1}.{4}.{2} ^{3} }{\spa{2}.{3}\spa{3}.{4}s_{{2,4}} \Del{1}.{4}.{1}\Del{1}.{4}.{4} \left( 1-z_{{1}} \right) }}\nonumber\\&&-{\frac { \spa{1}.{2} \left( z_{{1}}+z_{{2}} \right) ^{3}} {\sqrt {z_1z_2z_{{4}}}\spa{3}.{4}s_{{1,2}}\Del{0}.{2}.{3}}}\nonumber\\ &&+{\frac { \spa {1}.{2} ^{3} \left( 1-z_{{4}} \right) ^{3}}{\sqrt {z_{{4}}}\spa{2}.{3}s_{{1,3}}\Del{0}.{3}.{1} \Del{0}.{3}.{3}\Del{0}.{3}.{4}}}\nonumber\\&&-{\frac { \spa{1}.{2} ^ {3}}{s_{{1,4}}\Del{0}.{4}.{1}\Del{0}.{4}.{4}\spa{2}.{3}\spa{3}.{4}}}, \label{eq:spb2}\end{aligned}$$ and, $$\begin{aligned} \lefteqn{{\mathrm{split}(1^-,2^+,3^-,4^+\to P^{-})}=\mathcal{B}_3(1,2,3,4)} \nonumber \\ &=&-{\frac {z_2^{3/2}{z_{{3}}}^{2} \spa{1}.{2}}{\sqrt{z_{{1}}z_{{4}}}\spa{3}.{4}s_{{1,2}} \left( z_{{1 }}+z_{{2}} \right) \Del{0}.{2}.{3}}}\nonumber\\&&-{\frac {z_1^{3/2}z_2^{3/2} \spa{2}.{3}}{\sqrt {z_{{3}}z_{{4}}}s_{{2,3}}\Del{1}.{3}.{1}\Del{1}.{3}.{4}}}\nonumber\\&&-{\frac {{ z_{{1}}}^{3/2} \Del{1}.{4}.{3} ^{4}}{\spa{2}.{3}\spa{3}.{4}s_{{2,4}}\Del{1}.{4}.{1} \Del{1}.{4}.{2}\Del{1}.{4}.{4} \left( 1-z_1 \right) }}\nonumber\\&&-{\frac {{z_{{1}}}^{3/2} \Del{2}.{4}.{3} ^{3}}{\spa{1}.{2}\spa{3}.{4}s_{{3,4}}\Del{2}.{4 }.{2}\Del{2}.{4}.{4} \left( z_{{3}}+z_{{4}} \right) }}\nonumber\\&&+{\frac { \spa{1}.{3} ^{4} \left( z_{{1}}+z_{{2}}+z_{{3}} \right) ^{3}}{\sqrt {z_{{4}}}\spa{1}.{2}\spa{2} .{3}s_{{1,3}}\Del{0}.{3}.{1}\Del{0}.{3}.{3}\Del{0}.{3}.{4}}}\nonumber\\&&-{\frac { \spa{1}.{3} ^{4}}{\spa{1}.{2}\spa{2}.{3}\spa{3}.{4}s_{{1,4}} \Del{0}.{4}.{4}\Del{0}.{4}.{1}}}, \label{eq:spb3}\end{aligned}$$ where $\Del{i}.{j}.{k}$ is given in Eq. . The seven remaining $\Delta M=1$ splitting functions can be obtained by using the dual ward identity, $$\begin{aligned} {\mathrm{split}(1^+ ,2^- , 3^+ ,4^+\to P^{+})}&=& -\mathcal{B}_1(2,1,3,4)-\mathcal{B}_1(2,3,1,4)-\mathcal{B}_1(2,3,4,1),\nonumber \\ {\mathrm{split}(1^+, 2^+, 3^-, 4^+\to P^{+})}&=& \phantom{-} \mathcal{B}_1(3,4,2,1)+\mathcal{B}_1(3,2,4,1)+\mathcal{B}_1(3,2,1,4),\nonumber \\ {\mathrm{split}(1^-, 2^+, 3^+,4^-\to P^{-})}&=& \phantom{-}\mathcal{B}_3(4,3,1,2) + \mathcal{B}_2(4,1,3,2) + \mathcal{B}_2(1,4,3,2) ,\nonumber \\ {\mathrm{split}(1^+, 2^-, 3^-, 4^+\to P^{-})}&=& - \mathcal{B}_3(2,1,3,4) - \mathcal{B}_2(2,3,1,4) - \mathcal{B}_2(2,3,4,1),\nonumber \\\end{aligned}$$ or reflection symmetry, $$\begin{aligned} {\mathrm{split}(1^+ ,2^+ ,3^+ ,4^-\to P^{+})}&=& - {\mathrm{split}(4^- ,3^+, 2^+, 1^+\to P^{-})},\nonumber \\ {\mathrm{split}(1^+,2^-, 3^+,4^-\to P^{-})}&=&- {\mathrm{split}(4^- ,3^+, 2^-, 1^+\to P^{-})},\nonumber \\ {\mathrm{split}(1^+,2^+ ,3^-,4^-\to P^{-})}&=&- {\mathrm{split}(4^-,3^- ,2^+,1^+\to P^{-})}.\end{aligned}$$ Finally splitting functions with $\Delta M = 2,~3$ are related to those given above by the parity transformation. Inspection of Eqs. , and reveals that each term is inversely proportional to a single invariant, in keeping with its MHV rules origins. For this type of collinear limit, there are potentially six invariants, the double invariants $s_{1,2}, s_{2,3}, s_{3,4}$, the triple invariants $s_{1,3}, s_{2,4}$ and $s_{1,4}$. Some poles are absent because the MHV rules forbid that type of contribution. For example, in ${\mathrm{split}(1^- ,2^+, 3^+, 4^+\to P^{+})}$, there are no contributions with poles in $s_{2,3}$, $s_{3,4}$ or $s_{2,4}$ precisely because these poles correspond to forbidden MHV diagrams. Expressions for the four gluon splitting functions are given in Ref. [@delduca]. The results given here are more compact and have a rather different analytic form. After adjusting the normalisation of the colour matrices, the splitting functions of Eqs. – numerically agree with those of Ref. [@delduca]. ### $n=5$ {#sec:4.2.5} In total there are 64 different splitting amplitudes, but only eleven are independent. The rest can be obtained with the help of parity, reflection and dual ward identities. The three simplest independent collinear limits can be obtained using only MHV rules, $$\begin{aligned} {\mathrm{split}(1^+ ,2^+, 3^+, 4^+, 5^+\to P^{+})}&=& \frac{1}{\sqrt{z_1z_5}\spa{1}.{2} \spa{2}.{3} \spa{3}.{4} \spa{4}.{5}},\\ {\mathrm{split}(1^-, 2^+, 3^+, 4^+, 5^+\to P^{-})} &=& \frac{z_1^2}{\sqrt{z_1z_5}\spa{1}.{2} \spa{2}.{3} \spa{3}.{4} \spa{4}.{5}}, \\ {\mathrm{split}(1^+, 2^-, 3^+, 4^+, 5^+\to P^{-})}&=& \frac{z_2^2}{\sqrt{z_1z_5}\spa{1}.{2} \spa{2}.{3} \spa{3}.{4} \spa{4}.{5}}.\end{aligned}$$ The amplitudes with $\Delta M = 1$ require the application of Eqs. (\[mnpp\]) and (\[mmnpm\]). There are 5 independent amplitudes in this class of splitting amplitudes, but we give here only two examples, one for each of the cases $-\to +$ and $-- \to -$, $$\begin{aligned} &{\mathrm{split}(1^-, 2^+, 3^+, 4^+, 5^+\to P^{+})}= \nonumber\\ &= {\frac { \left( \Del{0}.{2}.{1} \right) ^{3}\sqrt {z_{{1}}}}{\sqrt {z_ {{1}}z_{{5}}}\spa{1}.{2}\spa{3}.{4}\spa{4}.{5}s_{{1,2}} \left( z_{{1}} +z_{{2}} \right) \Del{0}.{2}.{2}\Del{0}.{2}.{3}}}\nonumber\\ &+{\frac { \left( \Del {0}.{3}.{1} \right) ^{3}\sqrt {z_{{1}}}}{\sqrt {z_{{1}}z_{{5}}}\spa{1} .{2}\spa{2}.{3}\spa{4}.{5}s_{{1,3}} \left( z_{{1}}+z_{{2}}+z_{{3}} \right) \Del{0}.{3}.{3}\Del{0}.{3}.{4}}}\nonumber\\ &+{\frac { \left( \Del{0}.{4}. {1} \right) ^{3}\sqrt {z_{{1}}}}{\sqrt {z_{{1}}z_{{5}}}\spa{1}.{2}\spa {2}.{3}\spa{3}.{4}s_{{1,4}} \left( z_{{1}}+z_{{2}}+z_{{3}}+z_{{4}} \right) \Del{0}.{4}.{4}\Del{0}.{4}.{5}}}\nonumber\\ &-{\frac { \left( \Del{0}.{5}. {1} \right) ^{3}}{\spa{1}.{2}\spa{2}.{3}\spa{3}.{4}\spa{4}.{5}s_{{1,5} }\Del{0}.{5}.{5}}},\end{aligned}$$ and, $$\begin{aligned} &{\mathrm{split}(1^-, 2^-, 3^+, 4^+, 5^+\to P^{-})}= \nonumber\\ & {\frac {{z_{{1}}}^{2} \left( \Del{1}.{3}.{2} \right) ^{3}}{\sqrt {z_{{ 1}}z_{{5}}}\spa{2}.{3}\spa{4}.{5}s_{{2,3}}\Del{1}.{3}.{1}\Del{1}.{3}.{ 3}\Del{1}.{3}.{4}}}\nonumber\\&+{\frac {{z_{{1}}}^{2} \left( \Del{1}.{4}.{2} \right) ^{3}}{\sqrt {z_{{1}}z_{{5}}}\spa{2}.{3}\spa{3}.{4}s_{{2,4}} \Del{1}.{4}.{1}\Del{1}.{4}.{4}\Del{1}.{4}.{5}}}\nonumber\\&-{\frac {{z_{{1}}}^{2} \left( \Del{1}.{5}.{2} \right) ^{3}\sqrt {z_{{5}}}}{\sqrt {z_{{1}}z_{ {5}}}\spa{2}.{3}\spa{3}.{4}\spa{4}.{5}s_{{2,5}}\Del{1}.{5}.{1}\Del{1}. {5}.{5} \left( z_{{2}}+z_{{3}}+z_{{4}}+z_{{5}} \right) }}\nonumber\\&+{\frac { \left( \spa{1}.{2} \right) ^{3} \left( z_{{1}}+z_{{2}} \right) ^{3} \sqrt {z_{{1}}}}{\sqrt {z_{{1}}z_{{5}}}\spa{3}.{4}\spa{4}.{5}s_{{1,2}} \Del{0}.{2}.{1}\Del{0}.{2}.{2}\Del{0}.{2}.{3}}}\nonumber\\&+{\frac { \left( \spa{1 }.{2} \right) ^{3} \left( z_{{1}}+z_{{2}}+z_{{3}} \right) ^{3}\sqrt {z _{{1}}}}{\sqrt {z_{{1}}z_{{5}}}\spa{2}.{3}\spa{4}.{5}s_{{1,3}}\Del{0}. {3}.{1}\Del{0}.{3}.{3}\Del{0}.{3}.{4}}}\nonumber\\&+{\frac { \left( \spa{1}.{2} \right) ^{3} \left( z_{{1}}+z_{{2}}+z_{{3}}+z_{{4}} \right) ^{3} \sqrt {z_{{1}}}}{\sqrt {z_{{1}}z_{{5}}}\spa{2}.{3}\spa{3}.{4}s_{{1,4}} \Del{0}.{4}.{1}\Del{0}.{4}.{4}\Del{0}.{4}.{5}}}\nonumber\\&-{\frac { \left( \spa{1 }.{2} \right) ^{3}}{\spa{2}.{3}\spa{3}.{4}\spa{4}.{5}s_{{1,5}}\Del{0}. {5}.{1}\Del{0}.{5}.{5}}}.\end{aligned}$$ The most complicated amplitudes are those with $\Delta M = 2 $ and require the use of Eq. (\[mmnpp\]). There are three independent splitting functions, but here we only give one example. $$\begin{aligned} &{\mathrm{split}(1^-, 2^-, 3^+, 4^+, 5^+\to P^{+})}= \nonumber\\& -{\frac { \left( \Del{0}.{3}.{1} \right) ^{3} \left( \Del{1}.{3}.{2} \right) ^{3}}{\sqrt {z_{{5}}}\spa{2}.{3}\spa{4}.{5}\Dell{0}.{3}.{1}.{ 3}s_{{1,3}} \left( z_{{1}}+z_{{2}}+z_{{3}} \right) \Del{0}.{3}.{4}s_{{ 2,3}}\Del{1}.{3}.{1}\Del{1}.{3}.{3}}}\nonumber\\&+{\frac { \left( \Del{0}.{4}.{1} \right) ^{3} \left( \Del{1}.{3}.{2} \right) ^{3}}{\sqrt {z_{{5}}}\spa {2}.{3}s_{{1,4}} \left( 1-z_5 \right) \Del{0 }.{4}.{4}\Del{0}.{4}.{5}s_{{2,3}}\Del{1}.{3}.{1}\Del{1}.{3}.{3}\Del{1} .{3}.{4}}}\nonumber\\&-{\frac { \left( \Del{0}.{4}.{1} \right) ^{3} \left( \Del{1} .{4}.{2} \right) ^{3}}{\sqrt {z_{{5}}}\spa{2}.{3}\spa{3}.{4}\Dell{0}.{ 4}.{1}.{4}s_{{1,4}} \left( 1 -z_5 \right) \Del{0}.{4}.{5}s_{{2,4}}\Del{1}.{4}.{1}\Del{1}.{4}.{4}}}\nonumber\\&-{\frac { \left( \Del{0}.{5}.{1} \right) ^{3} \left( \Del{1}.{3}.{2} \right) ^{ 3}}{\Del{1}.{3}.{4}\spa{2}.{3}\spa{4}.{5}s_{{1,5}}\Del{0}.{5}.{5}\Del{ 1}.{3}.{1}\Del{1}.{3}.{3}s_{{2,3}}}}\nonumber\\&-{\frac { \left( \Del{1}.{4}.{2} \right) ^{3} \left( \Del{0}.{5}.{1} \right) ^{3}}{s_{{2,4}}\Del{1}.{4 }.{1}\Del{1}.{4}.{4}\Del{1}.{4}.{5}\spa{2}.{3}\spa{3}.{4}s_{{1,5}}\Del {0}.{5}.{5}}}\nonumber\\&+{\frac { \left( \Del{1}.{5}.{2} \right) ^{3} \left( \Del {0}.{5}.{1} \right) ^{3}}{\Dell{0}.{5}.{1}.{5}\Del{1}.{5}.{1}\Del{1}.{ 5}.{5}s_{{2,5}}\spa{2}.{3}\spa{3}.{4}\spa{4}.{5}s_{{1,5}}}}\nonumber\\&+{\frac { \left( \spa{1}.{2} \right) ^{3} \left( \Dell{0}.{3}.{0}.{2} \right) ^ {3}}{\sqrt {z_{{5}}}\spa{4}.{5}s_{{1,3}} \left( z_{{1}}+z_{{2}}+z_{{3} } \right) \Del{0}.{3}.{3}\Del{0}.{3}.{4}s_{{1,2}}\Del{0}.{2}.{1}\Del{0 }.{2}.{2}\Del{0}.{2}.{3}}}\nonumber\\&+{\frac { \left( \spa{1}.{2} \right) ^{3} \left( \Dell{0}.{4}.{0}.{2} \right) ^{3}}{\sqrt {z_{{5}}}\spa{3}.{4}s _{{1,4}} \left( 1 - z_5 \right) \Del{0}.{4}.{4 }\Del{0}.{4}.{5}s_{{1,2}}\Del{0}.{2}.{1}\Del{0}.{2}.{2}\Del{0}.{2}.{3} }}\nonumber\\&+{\frac { \left( \spa{1}.{2} \right) ^{3} \left( \Dell{0}.{4}.{0}.{3 } \right) ^{3}}{\sqrt {z_{{5}}}\spa{2}.{3}s_{{1,4}} \left( 1-z_5 \right) \Del{0}.{4}.{4}\Del{0}.{4}.{5}s_{{1,3}} \Del{0}.{3}.{1}\Del{0}.{3}.{3}\Del{0}.{3}.{4}}}\nonumber\\&-{\frac { \left( \spa{1 }.{2} \right) ^{3} \left( \Dell{0}.{5}.{0}.{2} \right) ^{3}}{\spa{3}.{ 4}\spa{4}.{5}s_{{1,5}}\Del{0}.{5}.{5}s_{{1,2}}\Del{0}.{2}.{1}\Del{0}.{ 2}.{2}\Del{0}.{2}.{3}}}\nonumber\\&-{\frac { \left( \Dell{0}.{5}.{0}.{3} \right) ^ {3} \left( \spa{1}.{2} \right) ^{3}}{\Del{0}.{3}.{4}s_{{1,3}}\Del{0}.{ 3}.{1}\spa{2}.{3}\spa{4}.{5}s_{{1,5}}\Del{0}.{5}.{5}\Del{0}.{3}.{3}}}\nonumber\\&- {\frac { \left( \Dell{0}.{5}.{0}.{4} \right) ^{3} \left( \spa{1}.{2} \right) ^{3}}{\Del{0}.{4}.{5}\Del{0}.{4}.{1}\spa{2}.{3}\spa{3}.{4}s_{ {1,5}}\Del{0}.{5}.{5}s_{{1,4}}\Del{0}.{4}.{4}}}.\end{aligned}$$ ### $n=6$ {#sec:4.2.6} Finally, for six collinear gluons there are $2^7 = 128$ different splitting amplitudes, which can be expressed by 23 independent ones. To find all independent amplitudes we have to use Eq. (\[mmmnpm\]) for the first time. Due to the length of the results we give here only two examples obtained with the help of Eqs. (\[mnpp\]) and (\[mmnpm\]), $$\begin{aligned} &{\mathrm{split}(1^-, 2^+, 3^+, 4^+, 5^+, 6^+\to P^{+})}=\nonumber\\& {\frac { \left( \Del{0}.{2}.{1} \right) ^{3}\sqrt {z_{{1}}}}{\sqrt {z_ {{1}}z_{{6}}}\spa{1}.{2}\spa{3}.{4}\spa{4}.{5}\spa{5}.{6}s_{{1,2}} \left( z_{{1}}+z_{{2}} \right) \Del{0}.{2}.{2}\Del{0}.{2}.{3}}}\nonumber\\&+{ \frac { \left( \Del{0}.{3}.{1} \right) ^{3}\sqrt {z_{{1}}}}{\sqrt {z_{ {1}}z_{{6}}}\spa{1}.{2}\spa{2}.{3}\spa{4}.{5}\spa{5}.{6}s_{{1,3}} \left( z_{{1}}+z_{{2}}+z_{{3}} \right) \Del{0}.{3}.{3}\Del{0}.{3}.{4} }}\nonumber\\&+{\frac { \left( \Del{0}.{4}.{1} \right) ^{3}\sqrt {z_{{1}}}}{\sqrt {z_{{1}}z_{{6}}}\spa{1}.{2}\spa{2}.{3}\spa{3}.{4}\spa{5}.{6}s_{{1,4}} \left( z_{{1}}+z_{{2}}+z_{{3}}+z_{{4}} \right) \Del{0}.{4}.{4}\Del{0} .{4}.{5}}}\nonumber\\&+{\frac { \left( \Del{0}.{5}.{1} \right) ^{3}\sqrt {z_{{1}}} }{\sqrt {z_{{1}}z_{{6}}}\spa{1}.{2}\spa{2}.{3}\spa{3}.{4}\spa{4}.{5}s_ {{1,5}} \left( z_{{1}}+z_{{2}}+z_{{3}}+z_{{4}}+z_{{5}} \right) \Del{0} .{5}.{5}\Del{0}.{5}.{6}}}\nonumber\\&-{\frac { \left( \Del{0}.{6}.{1} \right) ^{3} }{\spa{1}.{2}\spa{2}.{3}\spa{3}.{4}\spa{4}.{5}\spa{5}.{6}s_{{1,6}}\Del {0}.{6}.{6}}},\\\nonumber\\ &{\mathrm{split}(1^-, 2^-, 3^+, 4^+, 5^+, 6^+\to P^{-})}=\nonumber\\& {\frac {{z_{{1}}}^{2} \left( \Del{1}.{3}.{2} \right) ^{3}}{\sqrt {z_{{ 1}}z_{{6}}}\spa{2}.{3}\spa{4}.{5}\spa{5}.{6}s_{{2,3}}\Del{1}.{3}.{1} \Del{1}.{3}.{3}\Del{1}.{3}.{4}}}\nonumber\\&+{\frac {{z_{{1}}}^{2} \left( \Del{1}. {4}.{2} \right) ^{3}}{\sqrt {z_{{1}}z_{{6}}}\spa{2}.{3}\spa{3}.{4}\spa {5}.{6}s_{{2,4}}\Del{1}.{4}.{1}\Del{1}.{4}.{4}\Del{1}.{4}.{5}}}\nonumber\\&+{ \frac {{z_{{1}}}^{2} \left( \Del{1}.{5}.{2} \right) ^{3}}{\sqrt {z_{{1 }}z_{{6}}}\spa{2}.{3}\spa{3}.{4}\spa{4}.{5}s_{{2,5}}\Del{1}.{5}.{1} \Del{1}.{5}.{5}\Del{1}.{5}.{6}}}\nonumber\\&-{\frac {{z_{{1}}}^{2} \left( \Del{1}. {6}.{2} \right) ^{3}\sqrt {z_{{6}}}}{\sqrt {z_{{1}}z_{{6}}}\spa{2}.{3} \spa{3}.{4}\spa{4}.{5}\spa{5}.{6}s_{{2,6}}\Del{1}.{6}.{1}\Del{1}.{6}.{ 6} \left( z_{{2}}+z_{{3}}+z_{{4}}+z_{{5}}+z_{{6}} \right) }}\nonumber\\&+{\frac { \left( \spa{1}.{2} \right) ^{3} \left( z_{{1}}+z_{{2}} \right) ^{3} \sqrt {z_{{1}}}}{\sqrt {z_{{1}}z_{{6}}}\spa{3}.{4}\spa{4}.{5}\spa{5}.{ 6}s_{{1,2}}\Del{0}.{2}.{1}\Del{0}.{2}.{2}\Del{0}.{2}.{3}}}\nonumber\\&+{\frac { \left( \spa{1}.{2} \right) ^{3} \left( z_{{1}}+z_{{2}}+z_{{3}} \right) ^{3}\sqrt {z_{{1}}}}{\sqrt {z_{{1}}z_{{6}}}\spa{2}.{3}\spa{4} .{5}\spa{5}.{6}s_{{1,3}}\Del{0}.{3}.{1}\Del{0}.{3}.{3}\Del{0}.{3}.{4}} }\nonumber\\&+{\frac { \left( \spa{1}.{2} \right) ^{3} \left( z_{{1}}+z_{{2}}+z_{{ 3}}+z_{{4}} \right) ^{3}\sqrt {z_{{1}}}}{\sqrt {z_{{1}}z_{{6}}}\spa{2} .{3}\spa{3}.{4}\spa{5}.{6}s_{{1,4}}\Del{0}.{4}.{1}\Del{0}.{4}.{4}\Del{0 }.{4}.{5}}}\nonumber\\&+{\frac { \left( \spa{1}.{2} \right) ^{3} \left( z_{{1}}+z_ {{2}}+z_{{3}}+z_{{4}}+z_{{5}} \right) ^{3}\sqrt {z_{{1}}}}{\sqrt {z_{{ 1}}z_{{6}}}\spa{2}.{3}\spa{3}.{4}\spa{4}.{5}s_{{1,5}}\Del{0}.{5}.{1} \Del{0}.{5}.{5}\Del{0}.{5}.{6}}}\nonumber\\&-{\frac { \left( \spa{1}.{2} \right) ^ {3}}{\spa{2}.{3}\spa{3}.{4}\spa{4}.{5}\spa{5}.{6}s_{{1,6}}\Del{0}.{6}. {1}\Del{0}.{6}.{6}}}.\end{aligned}$$ \[sec:expl\] Conclusion {#sec:concl} ========== In this paper we have considered the collinear limit of multi-gluon QCD amplitudes at tree level. We have used the new MHV rules for constructing colour ordered amplitudes from MHV vertices together with the general collinear factorization formula to derive timelike splitting functions that are valid for specific numbers of negative helicity gluons with an arbitrary number of positive helicity gluons (or vice versa). In this limit, the full amplitude factorises into an MHV vertex multiplied by a multi-collinear splitting function that depends on the helicities of the collinear gluons. These splitting functions are derived directly using MHV rules. Out of the full set of MHV-diagrams contributing to the full amplitude, only the subset of MHV-diagrams which contain an internal propagator which goes on-shell in the multi-collinear limit contribute. We find that the splitting functions can be characterised by $\Delta M$, the difference between the number of negative helicity gluons before taking the collinear limit, and the number after. $\Delta M+1$ also coincides with the number of MHV vertices involved in the splitting functions. Our main results are splitting functions for arbitrary numbers of gluons where $\Delta M =0,1,2$. Splitting functions where the difference in the number of positive helicity gluons $\Delta P = 0,1,2$ are obtained by the parity transformation. These general results are sufficient to describe [*all*]{} collinear limits with up to six gluons. We have given explicit results for up to four collinear gluons for all independent helicity combinations, which numerically agree with the results of Ref. [@delduca], together with new results for five and six collinear gluons. This method could be applied to higher numbers of negative helicity gluons, and via the MHV-rules for quark vertices, to the collinear limits of quarks and gluons. We anticipate that the results presented here will be useful in developing higher order perturbative predictions for observable quantities, such as jet cross sections at the LHC. Acknowledgements ================ EWNG and VVK acknowledge the support of PPARC through Senior Fellowships and TGB acknowledges the award of a PPARC studentship. [99]{} E. Witten, [*Perturbative gauge theory as a string theory in twistor space*]{}, [*Commun. Math. Phys.*]{} [**252**]{} (1004) 189 \[arXiv:hep-th/0312171\]. F. Cachazo, P. Svrček and E. Witten, [*MHV vertices and tree amplitudes in gauge theory*]{}, [*JHEP*]{} [**09**]{} (2004) 006 \[arXiv:hep-th/0403047\]. R. Britto, F. Cachazo and B. Feng, [*New recursion relations for tree amplitudes of gluons*]{}, arXiv:hep-th/0412308. R. Britto, F. Cachazo, B. Feng and E. Witten, [*Direct proof of tree-level recursion relation in Yang-Mills theory*]{}, arXiv:hep-th/0501052. D. A. Kosower, [*All-order collinear behavior in gauge theories*]{}, [ *Nucl. Phys.*]{} [**B552**]{} (1999) 319 \[hep-ph/9901201\]. S.J. Parke and T.R. Taylor, [*An amplitude for $n$-gluon scattering*]{}, [*Phys. Rev. Lett.*]{}  [**56**]{}, 2459 (1986). F. A. Berends and W. T. Giele, [*Recursive Calculations For Processes With N Gluons*]{}, Nucl. Phys.  [**B306**]{} (1988) 759. F.A. Berends, R. Kleiss, P. De Causmaecker, R. Gastmans and T.T. Wu, [*Single Bremsstrahlung processes in gauge theories*]{}, Phys. Lett. B [**103**]{} (1981) 124;\ P. De Causmaecker, R. Gastmans, W. Troost and T.T. Wu, [*Multiple Bremsstrahlung in gauge theories at high energies. 1. General formalism for quantum electrodynamics*]{}, Nucl. Phys. B [**206**]{} (1982) 53;\ Z. Xu, D.-H. Zhang, L. Chang, Tsinghua University preprint TUTP–84/3 (1984), unpublished;\ R. Kleiss and W.J. Stirling, [*Spinor techniques for calculating $p\bar{p} \to W^{\pm} Z^0\,+$ jets*]{}, Nucl. Phys. B [**262**]{} (1985) 235;\ J.F. Gunion and Z. Kunszt, [*Improved analytic techniques for tree graph calculations and the $ggq\bar{q}\ell\bar\ell$ subprocess*]{}, Phys. Lett. B [**161**]{} (1985) 333;\ Z. Xu, D.-H. Zhang and L. Chang, [*Helicity amplitudes for multiple Bremsstrahlung in massless nonabelian gauge theories*]{}, Nucl. Phys. B [**291**]{}, 392 (1987);\ M.L. Mangano and S.J. Parke, [*Multiparton amplitudes in gauge theories*]{}, Phys. Rept.  [**200**]{}, 301 (1991). G. Georgiou and V. V. Khoze, [*Tree amplitudes in gauge theory as scalar [MHV]{} diagrams*]{}, [*JHEP*]{} [**05**]{} (2004) 070 \[arXiv:hep-th/0404072\]. C.-J. Zhu, [*The googly amplitudes in gauge theory*]{}, [*JHEP*]{} [**04**]{} (2004) 032 \[arXiv:hep-th/0403115\]. D. A. Kosower, [*Next-to-maximal helicity violating amplitudes in gauge theory*]{}, [*Phys. Rev.*]{} [**D71**]{} (2005) 045007 \[arXiv:hep-th/0406175\]. I. Bena, Z. Bern and D. A. Kosower, [*Twistor-space recursive formulation of gauge theory amplitudes*]{}, [*Phys. Rev.*]{} [**D71**]{} (2005) 045008 \[arXiv:hep-th/0406133\]. G. Georgiou, E. W. N. Glover and V. V. Khoze, [*Non-[MHV]{} tree amplitudes in gauge theory*]{}, [*JHEP*]{} [**07**]{} (2004) 048 \[arXiv:hep-th/0407027\]. J.-B. Wu and C.-J. Zhu, [*[MHV]{} vertices and scattering amplitudes in gauge theory*]{}, [*JHEP*]{} [**07**]{} (2004) 032 \[arXiv:hep-th/0406085\]. J.-B. Wu and C.-J. Zhu, [*[MHV]{} vertices and fermionic scattering amplitudes in gauge theory with quarks and gluinos*]{}, [*JHEP*]{} [**09**]{} (2004) 063 \[arXiv:hep-th/0406146\]. L. J. Dixon, E. W. N. Glover and V. V. Khoze, [*[MHV]{} rules for higgs plus multi-gluon amplitudes*]{}, [*JHEP*]{} [**12**]{} (2004) 015 \[arXiv:hep-th/0411092\]. S. D. Badger, E. W. N. Glover and V. V. Khoze, [*[MHV]{} rules for higgs plus multi-parton amplitudes*]{}, arXiv:hep-th/0412275. Z. Bern, D. Forde, D. A. Kosower and P. Mastrolia, [*Twistor-inspired construction of electroweak vector boson currents*]{}, arXiv:hep-ph/0412167. Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, [*One loop n point gauge theory amplitudes, unitarity and collinear limits*]{}, [*Nucl. Phys.*]{} [**B425**]{} (1994) 217–260 \[arXiv:hep-ph/9403226\]. Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, [*Fusing gauge theory tree amplitudes into loop amplitudes*]{}, [*Nucl. Phys.*]{} [**B435**]{} (1995) 59–101 \[arXiv:hep-ph/9409265\]. F. Cachazo, P. Svrcek and E. Witten, [*Twistor space structure of one-loop amplitudes in gauge theory*]{}, [*JHEP*]{} [**10**]{} (2004) 074 \[arXiv:hep-th/0406177\]. A. Brandhuber, B. Spence and G. Travaglini, [*One-loop gauge theory amplitudes in [N]{} = 4 super yang-mills from [MHV]{} vertices*]{}, [*Nucl. Phys.*]{} [**B706**]{} (2005) 150 \[arXiv:hep-th/0407214\]. F. Cachazo, P. Svrcek and E. Witten, [*Gauge theory amplitudes in twistor space and holomorphic anomaly*]{}, [*JHEP*]{} [**10**]{} (2004) 077 \[arXiv:hep-th/0409245\]. I. Bena, Z. Bern, D. A. Kosower and R. Roiban, [*Loops in twistor space*]{}, arXiv:hep-th/0410054. F. Cachazo, [*Holomorphic anomaly of unitarity cuts and one-loop gauge theory amplitudes*]{}, arXiv:hep-th/0410077. R. Britto, F. Cachazo and B. Feng, [*Computing one-loop amplitudes from the holomorphic anomaly of unitarity cuts*]{}, [*Phys. Rev.*]{} [**D71**]{} (2005) 025012 \[arXiv:hep-th/0410179\]. Z. Bern, V. Del Duca, L. J. Dixon and D. A. Kosower, [*All non-maximally-helicity-violating one-loop seven-gluon amplitudes in [N]{} = 4 super-yang-mills theory*]{}, [*Phys. Rev.*]{} [**D71**]{} (2005) 045006 \[arXiv:hep-th/0410224\]. R. Britto, F. Cachazo and B. Feng, [*Coplanarity in twistor space of [N]{} = 4 next-to-[MHV]{} one-loop amplitude coefficients*]{}, arXiv:hep-th/0411107. R. Britto, F. Cachazo and B. Feng, [*Generalized unitarity and one-loop amplitudes in [N]{} = 4 super-yang-mills*]{}, arXiv:hep-th/0412103. Z. Bern, L. J. Dixon and D. A. Kosower, [*All next-to-maximally-helicity-violating one-loop gluon amplitudes in [N]{}=4 super-yang-mills theory*]{}, arXiv:hep-th/0412210. C. Quigley and M. Rozali, [*One-loop [MHV]{} amplitudes in supersymmetric gauge theories*]{}, [*JHEP*]{} [**01**]{} (2005) 053 \[arXiv:hep-th/0410278\]. S. J. Bidder, N. E. J. Bjerrum-Bohr, L. J. Dixon and D. C. Dunbar, [*[N]{} = 1 supersymmetric one-loop amplitudes and the holomorphic anomaly of unitarity cuts*]{}, [*Phys. Lett.*]{} [**B606**]{} (2005) 189 \[arXiv:hep-th/0410296\]. J. Bedford, A. Brandhuber, B. Spence and G. Travaglini, [*A twistor approach to one-loop amplitudes in [N]{} = 1 supersymmetric yang-mills theory*]{}, arXiv:hep-th/0410280. S. J. Bidder, N. E. J. Bjerrum-Bohr, D. C. Dunbar and W. B. Perkins, [*Twistor space structure of the box coefficients of [N]{} = 1 one-loop amplitudes*]{}, arXiv:hep-th/0412023. J. Bedford, A. Brandhuber, B. Spence and G. Travaglini, [*Non-supersymmetric loop amplitudes and [MHV]{} vertices*]{}, [*Nucl. Phys.*]{} [**B706**]{} (2005) 100 \[arXiv:hep-th/0412108\]. S. J. Bidder, N. E. J. Bjerrum-Bohr, D. C. Dunbar and W. B. Perkins, [*One-loop gluon scattering amplitudes in theories with N [$<$]{} 4 supersymmetries*]{}, arXiv:hep-th/0502028. Z. Bern, L. J. Dixon and D. A. Kosower, [*On-shell recurrence relations for one-loop QCD amplitudes*]{}, arXiv:hep-th/0501240. R. Roiban, M. Spradlin and A. Volovich, [*Dissolving N = 4 loop amplitudes into QCD tree amplitudes*]{}, arXiv:hep-th/0412265. M. X. Luo and C. K. Wen, [*Recursion relations for tree amplitudes in super gauge theories*]{}, arXiv:hep-th/0501121. M. X. Luo and C. K. Wen, [*Compact formulas for all tree amplitudes of six partons*]{}, arXiv:hep-th/0502009. J. Bedford, A. Brandhuber, B. Spence and G. Travaglini, [*A recursion relation for gravity amplitudes*]{}, arXiv:hep-th/0502146. F. Cachazo and P. Svrcek, [*Tree level recursion relations in general relativity*]{}, arXiv:hep-th/0502160. Z. Bern, N. E. J. Bjerrum-Bohr and D. C. Dunbar, [*Inherited twistor-space structure of gravity loop amplitudes*]{}, arXiv:hep-th/0501137. D. A. Kosower, [*Multiple singular emission in gauge theories*]{}, [*Phys. Rev.*]{} [**D67**]{} (2003) 116003 \[arXiv:hep-ph/0212097\]. W. T. Giele and E. W. N. Glover, [*Higher order corrections to jet cross-sections in $e^+ e^-$ annihilation*]{}, [*Phys. Rev.*]{} [**D46**]{} (1992) 1980–2010. S. Frixione, Z. Kunszt, and A. Signer, [*Three-jet cross sections to next-to-leading order*]{}, [*Nucl. Phys.*]{} [**B467**]{} (1996) 399–442, \[hep-ph/9512328\]. S. Catani and M. H. Seymour, [*A general algorithm for calculating jet cross sections in NLO QCD*]{}, [*Nucl. Phys.*]{} [**B485**]{} (1997) 291–419, \[ hep-ph/9605323\]. Z. Bern and A. G. Morgan, [*Massive loop amplitudes from unitarity*]{}, [*Nucl. Phys.*]{} [**B467**]{} (1996) 479–509, \[hep-ph/9511336\]. Z. Bern, L. J. Dixon, and D. A. Kosower, [*Progress in one-loop QCD computations*]{}, [*Ann. Rev. Nucl. Part. Sci.*]{} [**46**]{} (1996) 109 \[hep-ph/9602280\]. G. Altarelli and G. Parisi, [*Asymptotic freedom in parton language*]{}, [*Nucl. Phys.*]{} [**B126**]{} (1977) 298. A. Bassetto, M. Ciafaloni, and G. Marchesini, [*Jet structure and infrared sensitive quantities in perturbative [QCD]{}*]{}, [*Phys. Rept.*]{} [**100**]{} (1983) 201 Z. Bern and G. Chalmers, [*Factorization in one loop gauge theory*]{}, [*Nucl. Phys.*]{} [**B447**]{} (1995) 465 \[hep-ph/9503236\]. Z. Bern, V. Del Duca, and C. R. Schmidt, [*The infrared behavior of one-loop gluon amplitudes at next-to-next-to-leading order*]{}, [*Phys. Lett.*]{} [**B445**]{} (1998) 168 \[hep-ph/9810409\]. D. A. Kosower and P. Uwer, [*One-loop splitting amplitudes in gauge theory*]{}, [*Nucl. Phys.*]{} [**B563**]{} (1999) 477 \[hep-ph/9903515\]. Z. Bern, V. Del Duca, W. B. Kilgore, and C. R. Schmidt, [*The infrared behavior of one-loop [QCD]{} amplitudes at next-to-next-to-leading order*]{}, [*Phys. Rev.*]{} [**D60**]{} (1999) 116001, \[hep-ph/9903516\]. S. Catani and M. Grazzini, [*The soft-gluon current at one-loop order*]{}, [*Nucl. Phys.*]{} [**B591**]{} (2000) 435 \[hep-ph/0007142\]. Z. Bern, L. J. Dixon, and D. A. Kosower, [*Two-loop $g\rightarrow gg$ splitting amplitudes in [QCD]{}*]{}, [*JHEP*]{} [**08**]{} (2004) 012 \[hep-ph/0404293\]. S.D. Badger and E.W.N, Glover, [*Two-loop splitting functions in [QCD]{}*]{}, [*JHEP*]{} [**07**]{} (2004) 040 \[hep-ph/0405236\]. A. Gehrmann-De Ridder and E. W. N. Glover, [*“A complete ${\cal O}(\alpha \alpha_s)$ calculation of the photon + 1jet rate in $e^+ e^-$ annihilation*]{},” [*Nucl. Phys.*]{} [**B517**]{} (1998) 269 \[arXiv:hep-ph/9707224\]. J. M. Campbell and E. W. N. Glover, [*Double unresolved approximations to multiparton scattering amplitudes*]{}, [*Nucl. Phys.*]{} [**B527**]{} (1998) 264 \[hep-ph/9710255\]. S. Catani and M. Grazzini, [*Collinear factorization and splitting functions for next-to-next-to-leading order [QCD]{} calculations*]{}, [*Phys. Lett.*]{} [**B446**]{} (1999) 143 \[hep-ph/9810389\]. S. Catani and M. Grazzini, [*Infrared factorization of tree level [QCD]{} amplitudes at the next-to-next-to-leading order and beyond*]{}, [*Nucl. Phys.*]{} [**B570**]{} (2000) 287 \[hep-ph/9908523\]. S. Catani, D. de Florian and G. Rodrigo, [*The triple collinear limit of one-loop amplitudes*]{}, [*Phys. Lett.*]{} [**B586**]{} (2004) 323 \[hep-ph/0312067\]. V. Del Duca, A. Frizzo and F. Maltoni, [*Factorization of tree QCD amplitudes in the high-energy limit and in the collinear limit*]{}, Nucl. Phys. [**B568**]{}, 211 (2000) \[arXiv:hep-ph/9909464\]. L.J. Dixon, [*Calculating scattering amplitudes efficiently*]{}, hep-ph/9601359. [^1]: This is because the required IR poles in the BCF approach arise not only from propagators going on-shell, but also from the constituent BCF vertices. [^2]: Our conventions for spinor helicities follow [@Witten1; @CSW1], except that $[ij] = - [ij]_{CSW}$ as in ref. [@LDTASI]. [^3]: Note that this selection rule would not apply to neither gauge-fixed MHV rules (where $\eta$’s are fixed to be equal to kinematic variables from the collinear set), nor to the BCF rules which mix holomorphic MHV vertices with anti-holomorphic $\overline{\rm MHV}$ ones. [^4]: MHV diagrams where hard negative helicity gluons are emitted from more than one vertex do not give rise to on-shell propagators and do not contribute in the singular limit.

--- abstract: 'Pulsar “standard model”, that considers a pulsar as a rotating magnetized conducting sphere surrounded by plasma, is generalized to the case of oscillating star. We developed an algorithm for calculation of the Goldreich-Julian charge density for this case. We consider distortion of the accelerating zone in the polar cap of pulsar by neutron star oscillations. It is shown that for oscillation modes with high harmonic numbers $(l,m)$ changes in the Goldreich-Julian charge density caused by pulsations of neutron star could lead to significant altering of an accelerating electric field in the polar cap of pulsar. In the moderately optimistic scenario, that assumes excitation of the neutron star oscillations by glitches, it could be possible to detect altering of the pulsar radioemission due to modulation of the accelerating field.' author: - 'A. N. Timokhin' bibliography: - 'ns\_oscill.bib' date: 'Received: date / Accepted: date' subtitle: 'or could we see oscillations of the neutron star after the glitch in pulsar?' title: Impact of neutron star oscillations on the accelerating electric field in the polar cap of pulsar --- Introduction {#sec:introduction} ============ Neutron stars (NS) are probably the most dense objects in the Universe. There are extreme physical conditions inside NS, i.e. the magnetic field is close to the quantum limit, the pressure is of the order of the nuclear one and the typical radius of a NS is only about 2-3 times larger than its gravitational radius. Knowledge of properties of matter under such extreme conditions if very important for fundamental physics. It is impossible to reconstruct such physical circumstances in terrestrial laboratories, therefore, study of the NS’s internal structure would give an unique opportunity for experimental verification of several fundamental physical theories. To study interiors of a celestial body one have to perform some kind of seismological study, by comparing observed frequencies of eigenmodes with frequencies inferred from theoretical considerations. Eigenfrequencies of oscillations in the crust of NS as well as in its interiors were calculated in several papers [see e.g. @McDermott/1988; @Chugunov2006 and references there]. However, for seismological study there should exist both i) a mechanism for excitation of oscillations, ii) a mechanism modulating radiation of the celestial object. There are two types of known NSs: member of binary systems and isolated ones. The former radiate due to accretion of the matter from the companion. For these stars there are many possibilities to excite oscillations, for example by instabilities in the accretion flow. However, in this case it would be difficult to distinguish whether a particular feature in the power spectrum of the object is due to oscillations of the NS or it is caused by some processes in the accretion disc/column. Because of this ambiguity we think that the study of isolated NS should be more promising in regard of the seismology. The vast majority of known isolated neutron stars are radiopulsars. The glitch (sudden change of the rotational period) is probably the only possible mechanism for excitation of oscillations for isolated pulsars. Radiation of radiopulsars is produced mostly in the magnetosphere. In order to judge whether oscillations of the NS could produce detectable changes in pulsar radiation, the impact of the oscillations on the magnetosphere must be considered. There is a widely accepted model of radiopulsar as a highly magnetized NS surrounded by non-neutral plasma [@GJ]. Although, there is still no self-consistent theory of radiopulsars, there is a general agreement regarding basic picture for the processes in the magnetosphere. Oscillations of the star can generate electric field as it happens in the case of rotation. Generalization of the formalism developed for rotating NS to the case of oscillating star should help to obtain the desired information. The first attempt to generalize Goldreich-Julian (GJ) formalism to the case of oscillating NS was made in @TBS2000. It was developed a general algorithm for calculation of the GJ charge density in the near zone of an oscillating NS. Using this algorithm GJ charge density and electromagnetic energy losses were calculated for the case of toroidal oscillations of the NS. Here we apply this formalism to the case of spheroidal oscillation modes, representing wide class of stellar modes (r-,g-,p- modes). We consider also impact of stellar oscillations on the acceleration mechanism in the polar cap of pulsar and discuss the possibility of observation on the NS oscillations. Main Results ============ General formalism {#sec:general-formalism} ----------------- Let us start by considering the case of a non-rotating oscillating NS. Motion of the conducting NS surface in the strong magnetic field of the star generates electric field as in the case of rotation. Only oscillation modes with non-vanishing velocity ${\ensuremath{\vec{V}^\textrm{\tiny{}osc}}}$ at the surface will disturb the magnetosphere. For the same reason as it is in the pulsar “standard model”, the electric field in the magnetosphere of an oscillating star should be perpendicular to the magnetic field. Otherwise charged particles will be accelerated by a longitudinal (parallel to [$\vec{B}$]{}) electric field and their radiation will give rise to electron-positron cascades producing enough particles to screen the accelerating electric field [@Sturrock71]. As in the case of rotating stars we will define the Goldreich-Julian electric field ${\ensuremath{\vec{E}_\textrm{\tiny{}GJ}}}$ as the field which is everywhere perpendicular to the magnetic field of the star, ${\ensuremath{\vec{E}_\textrm{\tiny{}GJ}}}\perp{\ensuremath{\vec{B}}}$, and the GJ charge density as a charge density, which supports this field $$\label{eq:Rgj} {{\ensuremath{\rho_\textrm{\tiny{}GJ}}}}\equiv \frac{1}{4\pi} \nabla\cdot{\ensuremath{\vec{E}_\textrm{\tiny{}GJ}}}\:.$$ For simplicity we consider only a zone near the NS, at the distances $r\ll{}2\pi c/\omega$, where $\omega$ is the frequency of NS oscillations. For many global oscillation modes [see e.g. @McDermott/1988] the polar cap accelerating zone is well withing this distance, therefore, we can study the changes of the accelerating electric field in the polar cap caused by oscillation. In the near zone all physical quantities change harmonically with time, i.e. the time dependence enters only through the term $e^{-\mathrm{i}\omega t}$. We make an additional assumption, that changes of the magnetic field induced by currents in the NS crust are much larger than the distortion caused by currents flowing in the magnetosphere. This assumption is considered as a first order approximation according to the small parameter $\left(\xi/{{\ensuremath{R_\textrm{\tiny{}NS}}}}\right)$, where $\xi$ is the amplitude of oscillation and [[$R_\textrm{\tiny{}NS}$]{}]{}– the NS radius. In other words, outside of the NS $${\nabla\times}{\ensuremath{\vec{B}}}= 0\: \label{rot_B_eq_0}$$ in the first order in $\left(\xi/{{\ensuremath{R_\textrm{\tiny{}NS}}}}\right)$. This assumption can be rewritten in terms of a condition on the current density in the magnetosphere as $$j \ll {{\ensuremath{\rho_\textrm{\tiny{}GJ}}}}\; c\; \left(\frac{c}{\omega r}\right) \equiv {\ensuremath{j_\textrm{\tiny{}GJ}}} \left(\frac{c}{\omega r}\right)\:. \label{eq:j_ll_B_Rgj}$$ Condition (\[eq:j\_ll\_B\_Rgj\]) implies that the current density in the near zone of the magnetosphere is less that the GJ current density connected with oscillations multiplied by a large factor $c/(\omega r)$. So, if the current density in the magnetosphere is of the order of the GJ current density, assumption (\[rot\_B\_eq\_0\]) is valid. Under these assumption it is possible to solve the problem analytically in general case, i.e. to develop an algorithm for finding an analytical solution for the GJ charge density for arbitrary configuration of the magnetic field and arbitrary velocity field on the NS surface. Under assumption (\[rot\_B\_eq\_0\]) the magnetic field can be expressed through a scalar function $P$ as $${\ensuremath{\vec{B}}}= {\nabla}{\times}{\nabla}{\times}(P {\ensuremath{\vec{e_r}}})\: . \label{eq:B_P}$$ The GJ electric field depends also on a scalar potential ${{\ensuremath{\Psi_\textrm{\tiny{}GJ}}}}$ through the relation $${\ensuremath{\vec{E}_\textrm{\tiny{}GJ}}} = - \frac{1}{c} {\nabla}{\times}({\partial}_t P {\ensuremath{\vec{e_r}}}) - {\nabla}{{\ensuremath{\Psi_\textrm{\tiny{}GJ}}}}\: . \label{eq:E_P_Psi}$$ The GJ charge density is then expressed as $$\label{eq:Rgj_Psi} {{\ensuremath{\rho_\textrm{\tiny{}GJ}}}}= - \frac{1}{4\pi} {\Delta}{{\ensuremath{\Psi_\textrm{\tiny{}GJ}}}}\: .$$ An equation for ${{\ensuremath{\Psi_\textrm{\tiny{}GJ}}}}$ is $$\begin{aligned} \label{eq:EquationGeneral} {\Delta}_{\Omega} P \; {\partial}_r{{\ensuremath{\Psi_\textrm{\tiny{}GJ}}}}- {\partial}_r {\partial}_\theta P \; {\partial}_\theta {{\ensuremath{\Psi_\textrm{\tiny{}GJ}}}}- \frac{1}{\sin^2\theta}\: {\partial}_r {\partial}_\phi P \; {\partial}_\phi {{\ensuremath{\Psi_\textrm{\tiny{}GJ}}}}\nonumber \\ + \frac{1}{c \sin\theta} \left( {\partial}_r {\partial}_\phi P \; {\partial}_\theta {\partial}_t P - {\partial}_r {\partial}_\theta P \; {\partial}_\phi {\partial}_t P \right) = 0\:,\end{aligned}$$ where ${\Delta}_{\Omega}$ is an angular part of the Laplace operator. This is the first order linear partial differential equation for the GJ electric potential [[$\Psi_\textrm{\tiny{}GJ}$]{}]{}. As the equation for [[$\Psi_\textrm{\tiny{}GJ}$]{}]{} is linear, each oscillation mode can be treated separately. This equation is valid for arbitrary configuration of the magnetic field and for any amplitude of the surface oscillation, provided that condition (\[rot\_B\_eq\_0\]) is satisfied. Dependence on oscillation mode appears in the boundary conditions and also through the time derivative ${\partial}_t P$. For different oscillation modes both the equation and boundary conditions for [[$\Psi_\textrm{\tiny{}GJ}$]{}]{} are different. Derivation of eq. (\[eq:EquationGeneral\]), boundary condition for functions ${{\ensuremath{\Psi_\textrm{\tiny{}GJ}}}}$ as well as more detailed discussion of used approximations can be found in @TBS2000. {width="\columnwidth"} {width="\columnwidth"} {width=".95\columnwidth"} {width=".95\columnwidth"} {width="0.95\columnwidth"} {width="0.95\columnwidth"} {width="0.95\columnwidth"} {width="0.95\columnwidth"} Goldreich Julian charge density {#sec:goldr-juli-charge} ------------------------------- In @TBS2000 equation (\[eq:EquationGeneral\]) was solved for the case of small-amplitude toroidal oscillations and dipolar configuration of unperturbed magnetic field. Solutions had been obtained with a code written in computer algebra language MATHEMATICA. Now we have developed a new version of this code, which allows to obtain analytical solutions of equation (\[eq:EquationGeneral\]) for a more complicated case of spheroidal modes. Any vector field on a sphere can be represented as a composition of toroidal (${\nabla\cdot}{\ensuremath{\vec{V}^\textrm{\tiny{}osc}}} = 0$) and spheroidal (${\nabla\times}{\ensuremath{\vec{V}^\textrm{\tiny{}osc}}} = 0$) vector fields [@Unno1979]. So, now we are able to calculate GJ electric field and charge density for arbitrary oscillations of a NS with dipole magnetic field. Similar to the case of toroidal oscillations the small current approximation turned to be valid for a half of all oscillation modes. For oscillation modes with velocity field, which is symmetric relative to the equatorial plane (see an example of such mode in Fig. \[fig:shape\] (left)), solution of eq. (\[eq:EquationGeneral\]) is smooth everywhere (see Fig. \[fig:Pgj\] (left)). For modes with antisymmetrical velocity field (an example of such mode is shown in Fig. \[fig:shape\] (right)), [[$\Psi_\textrm{\tiny{}GJ}$]{}]{} is discontinuous at the equatorial plane (see Fig.\[fig:Pgj\] (right)). There is the following reason for such behavior. Dipolar magnetic field is antisymmetric relative to the equatorial plane. Antisymmetric motion of the field line footpoints give rise to a twisted configuration of the magnetic field, which cannot be curl-free. So, for such modes a strong current will flow along closed magnetic field lines, $j\gg{\ensuremath{j_\textrm{\tiny{}GJ}}}$. However, there is no physical reason why a smooth solution for GJ electric field can not exists also for such modes. An argument supporting this hypothesis is a solution for twisted force-free magnetic field found by @Wolfson1995. In his solution, which corresponds to the toroidal mode $(2,0)$, the configuration of force-free twisted magnetic field is supported by a strong current flowing along magnetic field lines. For the modes with smooth solutions our approximation should be valid. On the other hand, a strong electric current will flow only along closed magnetic field lines. The current density along open magnetic field lines should be close to ${\ensuremath{j_\textrm{\tiny{}GJ}}}$ and condition for the small current approximation (eq. (\[eq:j\_ll\_B\_Rgj\])), will be satisfied in the open field line domain. Then, [[$\Psi_\textrm{\tiny{}GJ}$]{}]{} in the polar cap could be obtained by solving eq. (\[eq:EquationGeneral\]). [[$\Psi_\textrm{\tiny{}GJ}$]{}]{} in the polar cap will differ from the solutions obtained here, because the boundary conditions should be set at the polar cap boundaries and not on the whole surface on the NS. The main properties of [[$\Psi_\textrm{\tiny{}GJ}$]{}]{} regarding its qualitative dependence on the coordinates, used in our discussion, would be, however, similar to the properties obtained from our solutions. As expected, the GJ charge density distribution follows the distribution of the velocity field (see e.g. Figs. \[fig:Rgj\_map\], \[fig:Rgj\_surf\], \[fig:Rgj\_\_44\_4\]). With increasing of the harmonic numbers, [[$\rho_\textrm{\tiny{}GJ}$]{}]{} falls more rapidly with the distance, what is also expected for multipolar solutions. A remarkable property of GJ charge density distribution near oscillating star is that the local maxima of [[$\rho_\textrm{\tiny{}GJ}$]{}]{} increases with increasing of both $l$ and $m$ (see Figs. \[fig:Rgj\_increase\_l\], \[fig:Rgj\_increase\_m\]). The reasons for this are as follows. The electric field induced by oscillations is of the order $$\label{eq:Egj_estimate} {{\ensuremath{E_\textrm{\tiny{}GJ}}}}\sim \frac{{{\ensuremath{V^\textrm{\tiny{}osc}}}}}{c} B\:.$$ The charge density supporting this electric field is of the order of $E/\Delta{}x$, where $\Delta{}x$ is a characteristic distance of electric field variation. For a mode with harmonic number $l$ this size is of the order of ${{\ensuremath{R_\textrm{\tiny{}NS}}}}/l$. Hence, for [[$\rho_\textrm{\tiny{}GJ}$]{}]{} we have an estimate $$\label{eq:Rgj_estimate} {{\ensuremath{\rho_\textrm{\tiny{}GJ}}}}\sim l\:\frac{E}{4\pi{{\ensuremath{R_\textrm{\tiny{}NS}}}}} \sim l\:\frac{{{\ensuremath{V^\textrm{\tiny{}osc}}}}}{c}\:\frac{B}{4\pi{{\ensuremath{R_\textrm{\tiny{}NS}}}}}\:.$$ So, for the same amplitude of velocity of oscillation the amplitude of variation of the GJ charge density is larger for higher harmonics. Equation (\[eq:Rgj\_estimate\]) is in a good agreement with the exact results shown in Figs. \[fig:Rgj\_increase\_l\], \[fig:Rgj\_increase\_m\]. ![View of the polar cap of pulsar from the top. Goldreich-Julian charge density for oscillation mode $(44,4)$ is shown by the color map (positive values in red, negative – in blue). The quantity changes harmonically with time. The polar cap boundary for a pulsar with period 3 ms is shown by the dashed line. []{data-label="fig:Rgj__44_4"}](fig5.eps){width=".95\columnwidth"} Particle acceleration in the polar cap of pulsar {#sec:part-accel-polar} ------------------------------------------------ The GJ charge density induced by NS oscillations influences particle acceleration mechanism in the polar cap of pulsar. As we will show, oscillations will have the strongest impact on the accelerating electric field in the polar cap of pulsar for models with free particle escape from the NS surface [@Arons/Scharlemann/78; @Muslimov/Tsygan92]. The accelerating electric field in pulsars arises due to deviation of the charge density of the plasma from the local GJ charge density. For pulsar models with Space Charge Limited Flow (SCLF) the charge density of the flow $\rho$ at the NS surface is equal to the local value of the GJ charge density, $\rho({{\ensuremath{R_\textrm{\tiny{}NS}}}})={{\ensuremath{\rho_\textrm{\tiny{}GJ}}}}({{\ensuremath{R_\textrm{\tiny{}NS}}}})$. Magnetic field lines diverge and the charge density of the flow decreases with increasing of the distance from the star. However, the local GJ charge density decreases in a different way and at some distance $r$ from the NS ${{\ensuremath{\rho_\textrm{\tiny{}GJ}}}}(r)\neq\rho(r)$. The discrepancy between these charge densities gives rise to a longitudinal electric field [see @Arons/Scharlemann/78; @Muslimov/Tsygan92]. The GJ charge density for oscillation modes with large $l$, $m$ falls very rapidly with the distance. Hence, the charge density of a charge-separated flow for oscillating NS will exceed the local GJ charge density at some distance from the star. This produces a *decelerating* electric field. Therefore, in the case of rotating and oscillating NS, if ${\ensuremath{{{\ensuremath{\rho_\textrm{\tiny{}GJ}}}}^\textrm{\tiny{}osc}}}<{\ensuremath{{{\ensuremath{\rho_\textrm{\tiny{}GJ}}}}^\textrm{\tiny{}rot}}}$, the effective accelerating electric field will be reduced periodically due to superposition of accelerating and decelerating electric fields. The oscillational GJ charge density ${\ensuremath{{{\ensuremath{\rho_\textrm{\tiny{}GJ}}}}^\textrm{\tiny{}osc}}}$ for modes with large $l$, $m$ decreases practically to zero already at 2-3 NS radii and the whole oscillational GJ charge density contributes to the decelerating electric field. While in the case of rotation only $\sim{}15\%$ of the GJ charge density contributes to the accelerating electric field for SCLF in the polar cap [@Muslimov/Tsygan92]. The most important factor increasing modification of the accelerating electric field is that the amplitude of ${\ensuremath{{{\ensuremath{\rho_\textrm{\tiny{}GJ}}}}^\textrm{\tiny{}osc}}}$ increases with increasing of the harmonic numbers of the mode. In Fig. \[fig:R\_m\_Rgj\] we show the difference between the charge density of SCLF and the local GJ charge density for dipolar magnetic field as a function of the distance for rotating and oscillating stars. In order to demonstrate the importance of the discussed effects we consider a rather grotesque case when the linear velocity of rotation is equal to the maximum velocity of oscillations. It is evident from this plot, that for large enough $l$ and $m$ the decelerating electric field caused by stellar oscillations could be of comparable strength with the accelerating electric field even if ${{\ensuremath{V^\textrm{\tiny{}osc}}}}\ll{{\ensuremath{V^\textrm{\tiny{}rot}}}}$. Let us estimate the harmonic number of the oscillation mode where decelerating electric field would have a given impact on the accelerating electric field induced by NS rotation. The decelerating electric will be $\kappa$ times less that the rotational accelerating electric field, $$\kappa \equiv \frac{ {\ensuremath{E^\textrm{\tiny{}osc}}}_\mathrm{dec} }{ {\ensuremath{E^\textrm{\tiny{}rot}}}_\mathrm{acc} } \sim \frac{ {\ensuremath{{{\ensuremath{\rho_\textrm{\tiny{}GJ}}}}^\textrm{\tiny{}osc}}} }{ 0.15{\ensuremath{{{\ensuremath{\rho_\textrm{\tiny{}GJ}}}}^\textrm{\tiny{}rot}}} }\:, \label{eq:kappa}$$ if $$\label{eq:l_Eacc_Edec_V} l \sim 0.15\:\kappa\: \frac{{{\ensuremath{V^\textrm{\tiny{}rot}}}}}{{{\ensuremath{V^\textrm{\tiny{}osc}}}}}\:,$$ here [[$V^\textrm{\tiny{}rot}$]{}]{} is the linear velocity of NS rotation at the equator. For example for $l\sim{}100$, from this equation we can see, that for canceling of the accelerating electric field it is sufficient that the velocity amplitude is only $\sim{}10^{-3}$ of rotational velocity at the NS equator. For pulsar operating in @Ruderman/Sutherland75 regime the impact of stellar oscillations on the accelerating field in the polar cap will be reduced by the factor of $\sim{}10$. In this model the accelerating electric field is generated in a vacuum gap, so the whole rotational GJ charge density contribute to the accelerating electric field and in this case $\kappa\sim{\ensuremath{{{\ensuremath{\rho_\textrm{\tiny{}GJ}}}}^\textrm{\tiny{}osc}}}/{\ensuremath{{{\ensuremath{\rho_\textrm{\tiny{}GJ}}}}^\textrm{\tiny{}rot}}}$. ![Dependency of the Goldreich-Julian charge density on the harmonic number $l$. Distribution of [[$\rho_\textrm{\tiny{}GJ}$]{}]{} on the NS surface in the polar cap of pulsar is shown for oscillation modes (from left to right) $(28,4)$, $(44,4)$, $(64,4)$ in polar coordinates $(\left|{{\ensuremath{\rho_\textrm{\tiny{}GJ}}}}^{lm}({{\ensuremath{R_\textrm{\tiny{}NS}}}},\theta,0)\right|, \theta)$. Positive values of charge density are shown by the solid line and negative ones by the dashed line. The red line shows the angle at which the last closed field line intersect the NS surface for a pulsar with period 3 ms. Circular segments correspond to the values of [[$\rho_\textrm{\tiny{}GJ}$]{}]{} in normalized units shown at the left side of each plot. The same normalization for [[$\rho_\textrm{\tiny{}GJ}$]{}]{} is used as in Fig. \[fig:Rgj\_surf\].[]{data-label="fig:Rgj_increase_l"}](fig6_1.eps "fig:")   ![Dependency of the Goldreich-Julian charge density on the harmonic number $l$. Distribution of [[$\rho_\textrm{\tiny{}GJ}$]{}]{} on the NS surface in the polar cap of pulsar is shown for oscillation modes (from left to right) $(28,4)$, $(44,4)$, $(64,4)$ in polar coordinates $(\left|{{\ensuremath{\rho_\textrm{\tiny{}GJ}}}}^{lm}({{\ensuremath{R_\textrm{\tiny{}NS}}}},\theta,0)\right|, \theta)$. Positive values of charge density are shown by the solid line and negative ones by the dashed line. The red line shows the angle at which the last closed field line intersect the NS surface for a pulsar with period 3 ms. Circular segments correspond to the values of [[$\rho_\textrm{\tiny{}GJ}$]{}]{} in normalized units shown at the left side of each plot. The same normalization for [[$\rho_\textrm{\tiny{}GJ}$]{}]{} is used as in Fig. \[fig:Rgj\_surf\].[]{data-label="fig:Rgj_increase_l"}](fig6_2.eps "fig:")   ![Dependency of the Goldreich-Julian charge density on the harmonic number $l$. Distribution of [[$\rho_\textrm{\tiny{}GJ}$]{}]{} on the NS surface in the polar cap of pulsar is shown for oscillation modes (from left to right) $(28,4)$, $(44,4)$, $(64,4)$ in polar coordinates $(\left|{{\ensuremath{\rho_\textrm{\tiny{}GJ}}}}^{lm}({{\ensuremath{R_\textrm{\tiny{}NS}}}},\theta,0)\right|, \theta)$. Positive values of charge density are shown by the solid line and negative ones by the dashed line. The red line shows the angle at which the last closed field line intersect the NS surface for a pulsar with period 3 ms. Circular segments correspond to the values of [[$\rho_\textrm{\tiny{}GJ}$]{}]{} in normalized units shown at the left side of each plot. The same normalization for [[$\rho_\textrm{\tiny{}GJ}$]{}]{} is used as in Fig. \[fig:Rgj\_surf\].[]{data-label="fig:Rgj_increase_l"}](fig7_1.eps "fig:") Discussion {#sec:discussion} ========== ![Dependency of the Goldreich-Julian charge density on the harmonic number $m$. Distribution of [[$\rho_\textrm{\tiny{}GJ}$]{}]{} on the NS surface in the polar cap of pulsar is shown for spheroidal oscillation modes (from left to right) $(64,4)$, $(64,8)$, $(64,14)$ in polar coordinates $(\left|{{\ensuremath{\rho_\textrm{\tiny{}GJ}}}}^{lm}({{\ensuremath{R_\textrm{\tiny{}NS}}}},\theta,0)\right|, \theta)$. Meaning of the lines is the same as in Fig.\[fig:Rgj\_increase\_l\].[]{data-label="fig:Rgj_increase_m"}](fig7_1.eps "fig:")   ![Dependency of the Goldreich-Julian charge density on the harmonic number $m$. Distribution of [[$\rho_\textrm{\tiny{}GJ}$]{}]{} on the NS surface in the polar cap of pulsar is shown for spheroidal oscillation modes (from left to right) $(64,4)$, $(64,8)$, $(64,14)$ in polar coordinates $(\left|{{\ensuremath{\rho_\textrm{\tiny{}GJ}}}}^{lm}({{\ensuremath{R_\textrm{\tiny{}NS}}}},\theta,0)\right|, \theta)$. Meaning of the lines is the same as in Fig.\[fig:Rgj\_increase\_l\].[]{data-label="fig:Rgj_increase_m"}](fig7_2.eps "fig:")   ![Dependency of the Goldreich-Julian charge density on the harmonic number $m$. Distribution of [[$\rho_\textrm{\tiny{}GJ}$]{}]{} on the NS surface in the polar cap of pulsar is shown for spheroidal oscillation modes (from left to right) $(64,4)$, $(64,8)$, $(64,14)$ in polar coordinates $(\left|{{\ensuremath{\rho_\textrm{\tiny{}GJ}}}}^{lm}({{\ensuremath{R_\textrm{\tiny{}NS}}}},\theta,0)\right|, \theta)$. Meaning of the lines is the same as in Fig.\[fig:Rgj\_increase\_l\].[]{data-label="fig:Rgj_increase_m"}](fig7_3.eps "fig:") ![$\Delta\rho\equiv\rho-{{\ensuremath{\rho_\textrm{\tiny{}GJ}}}}$ – difference between the charge density of a space charge limited flow and the local Goldreich-Julian charge density along magnetic field lines in the polar cap of pulsar. $\Delta\rho$ is shown (in arbitrary units) by solid lines for spheroidal modes with different ($l,m$), from top to bottom: (64,14), (54,14),(64,2),(54,2). The same relation for an aligned rotator is shown by the dashed line. Negative values of $\Delta\rho$ give rise to an accelerating electric field, negative ones – to a decelerating field.[]{data-label="fig:R_m_Rgj"}](fig8.eps){width=".9\columnwidth"} We have shown, that oscillations of the NS can induce changes in the accelerating electric field, which are more stronger than a naive estimation $({{\ensuremath{V^\textrm{\tiny{}osc}}}}/c)B$. Indeed, for high harmonics the induced electric field will be $\sim{}l$ times stronger. In order to make definitive predictions about observable parameters is necessary to study the polar cap acceleration zone more detailed. An accelerating electric potential and the height of the pair formation front should be calculated. However, we can do simple estimations using equation (\[eq:l\_Eacc\_Edec\_V\]). Let us estimate the harmonic number of the mode which can cancel the accelerating electric field, assuming the mode is excited by a glitch. As we pointed in Sec. \[sec:general-formalism\] only modes with non-zero amplitude on the NS surface can produce changes in the magnetosphere. The distribution of oscillational motion plays crucial role. If oscillations are trapped in the NS crust a rather small energy will be required to pump the oscillation amplitude to the level high enough for strong disturbance of the accelerating electric field. If a fraction $\epsilon$ of the NS mass ${\ensuremath{M_\textrm{\tiny{}NS}}}$ is involved in the oscillations the amplitude of the oscillational velocity is of the order $$\label{eq:Vosc_MNS} {{\ensuremath{V^\textrm{\tiny{}osc}}}}\sim \sqrt{ \frac{2{\ensuremath{W^\textrm{\tiny{}osc}}}}{\epsilon {\ensuremath{M_\textrm{\tiny{}NS}}}} }\:,$$ where ${\ensuremath{W^\textrm{\tiny{}osc}}}$ is the total energy of the mode. The energy transferred during the glitch of the amplitude $\Delta\Omega$ is $$\label{eq:Wglitch} W^\mathrm{glitch} = i\;{\ensuremath{I_\textrm{\tiny{}NS}}} \Omega \Delta \Omega = i\;{\ensuremath{I_\textrm{\tiny{}NS}}} \left(\frac{2\pi}{P}\right)^2 \frac{\Delta\Omega}{\Omega} \:,$$ where $i$ is the fraction of the total momentum of inertia of the NS ${\ensuremath{I_\textrm{\tiny{}NS}}}$ coupled to the crust. Let us assume that some fraction $\eta$ of this energy goes into excitation of oscillations. Using eqs. (\[eq:Wglitch\]), (\[eq:Vosc\_MNS\]), (\[eq:l\_Eacc\_Edec\_V\]) we get conditions for harmonic number of modes which would periodically cancel the accelerating electric field, as $$\label{eq:l_blok_final_SCLF} {\bf l} > 300 \; {\eta_\%^{-1/2}} \; {\epsilon^{1/2}}\; {i^{-1/2}}\; \left( \frac{\Delta\Omega}{\Omega} \right)_6^{-1/2} \;,$$ for SCLF model, and for @Ruderman/Sutherland75 model: $$\label{eq:l_blok_final_RS} {\bf l} > 2000 \; {\eta_\%^{-1/2}} \; {\epsilon^{1/2}}\; {i^{-1/2}}\; \left( \frac{\Delta\Omega}{\Omega} \right)_6^{-1/2} \;.$$ Here $\eta_\%$ is measured in per cents and the relative magnitude of the glitch $\left(\Delta\Omega/\Omega\right)_6$ is normalized to $10^{-6}$. It is widely accepted, that the origin of pulsar glitches is angular momentum transfer from the NS core to the crust. In the frame of this model the fraction of the energy which can go into excitation of NS oscillation $\eta$ is of the order of $\Delta\Omega/\Omega$, i.e. it is very small, of the order of $\sim{}10^{-6}$. We may speculate however, that excited oscillations are trapped in the NS crust, i.e. $\epsilon$ is also very small. Such global oscillation modes ($l\sim$ several hundreds) could induce substantial changes in the accelerating electric field. As we mentioned in Sec. \[sec:general-formalism\] all physical quantities in the solutions obtained here oscillate with the frequency of the star oscillations. The accelerating electric field close to the local geometrical maxima of the oscillational GJ charge density will be weakened periodically by the decelerating effect due to stellar pulsations. The field oscillation will influence the particle distribution in the open field line zone of the pulsar magnetosphere and it should produce some observable effects. Depending on oscillation mode and position of the line of sight a complicated pattern will appear periodically in individual pulse profiles. Although individual pulses are highly variable, the presence of periodical features should be possible to discover in the power spectra of pulsars, provided the oscillations are excited to a high enough level and observations have been made with hight temporal resolution. If one observes some feature, which appears just after the glitch, then decreases and disappears after some time, and it never appears in the normal pulsar emission, then one can undoubtedly attribute this feature to the NS oscillations. I wish to thank A. Alpar for discussion. This work was partially supported by RFBR grant 04-02-16720, and by the grants N.Sh.-5218.2006.2 and RNP-2.1.1.5940

--- abstract: | We investigate the role of Massey’s directed information in portfolio theory, data compression, and statistics with causality constraints. In particular, we show that directed information is an upper bound on the increment in growth rates of optimal portfolios in a stock market due to [causal]{} side information. This upper bound is tight for gambling in a horse race, which is an extreme case of stock markets. Directed information also characterizes the value of [causal]{} side information in instantaneous compression and quantifies the benefit of [causal]{} inference in joint compression of two stochastic processes. In hypothesis testing, directed information evaluates the best error exponent for testing whether a random process $Y$ [causally]{} influences another process $X$ or not. These results give a natural interpretation of directed information $I(Y^n \to X^n)$ as the amount of information that a random sequence $Y^n = (Y_1,Y_2,\ldots, Y_n)$ [causally]{} provides about another random sequence $X^n = (X_1,X_2,\ldots,X_n)$. A new measure, [*directed lautum information*]{}, is also introduced and interpreted in portfolio theory, data compression, and hypothesis testing. author: - 'Haim H. Permuter, Young-Han Kim, and Tsachy Weissman [^1]' title: 'Interpretations of Directed Information in Portfolio Theory, Data Compression, and Hypothesis Testing' --- Causal conditioning, causal side information, directed information, hypothesis testing, instantaneous compression, Kelly gambling, Lautum information, portfolio theory. Introduction ============ Mutual information $I(X;Y)$ between two random variables $X$ and $Y$arises as the canonical answer to a variety of questions in science and engineering. Most notably, Shannon [@Shannon48] showed that the capacity $C$, the maximal data rate for reliable communication, of a discrete memoryless channel $p(y|x)$ with input $X$ and output $Y$ is given by $$\label{eq:shannon} C = \max_{p(x)} I(X;Y).$$ Shannon’s channel coding theorem leads naturally to the operational interpretation of mutual information $I(X;Y) = H(X) - H(X|Y)$ as the amount of uncertainty about $X$ that can be reduced by observation $Y$, or equivalently, the amount of information that $Y$ can provide about $X$. Indeed, mutual information $I(X;Y)$ plays a central role in Shannon’s random coding argument, because the probability that independently drawn sequences $X^n = (X_1,X_2,\ldots, X_n)$ and $Y^n = (Y_1,Y_2,\ldots, Y_n)$ “look” as if they were drawn jointly decays exponentially with $I(X;Y)$ in the first order of the exponent. Shannon also proved a dual result [@Shannon60] that the rate distortion function $R(D)$, the minimum compression rate to describe a source $X$ by its reconstruction $\hat{X}$ within average distortion $D$, is given by $R(D) = \min_{p(\hat{x}|x)} I(X;\hat{X})$. In another duality result (the Lagrange duality this time) to , Gallager [@Gal77] proved the minimax redundancy theorem, connecting the redundancy of the universal lossless source code to the maximum mutual information (capacity) of the channel with conditional distribution that consists of the set of possible source distributions (cf. [@Ryabko79]). It has been shown that mutual information has also an important role in problems that are not necessarily related to describing sources or transferring information through channels. Perhaps the most lucrative of such examples is the use of mutual information in gambling. In 1956, Kelly [@Kelly56] showed that if a horse race outcome can be represented as an independent and identically distributed (i.i.d.) random variable $X$, and the gambler has some side information $Y$ relevant to the outcome of the race, then the mutual information $I(X;Y)$ captures the difference between growth rates of the optimal gambler’s wealth with and without side information $Y$. Thus, Kelly’s result gives an interpretation of mutual information $I(X;Y)$ as the financial value of side information $Y$ for gambling in the horse race $X$. In order to tackle problems arising in information systems with causally dependent components, Massey [@Massey90] introduced the notion of directed information, defined as $$\label{e_def_directed} I(X^n\to Y^n) := \sum_{i=1}^n I(X^i;Y_i|Y^{i-1}),$$ and showed that the normalized maximum directed information upper bounds the capacity of channels with feedback. Subsequently, it was shown that Massey’s directed information and its variants indeed characterize the capacity of feedback and two-way channels [@Kramer98; @Kramer03; @PermuterWeissmanGoldsmith09; @TatikondaMitter_IT09; @Kim07_feedback; @PermuterWeissmanChenMAC_IT09; @ShraderPemuter07ISIT] and the rate distortion function with feedforward [@Pradhan07Venkataramanan]. Note that directed information can be rewritten as $$\label{e_kim_identity} I(X^n \to Y^n) = \sum_{i=1}^n I(X_i; Y_i^n | X^{i-1}, Y^{i-1}),$$ each term of which corresponds to the achievable rate at time $i$ given side information $(X^{i-1}, Y^{i-1})$ (refer to [@Kim07_feedback] for the details). The main contribution of this paper is showing that directed information has a natural interpretation in portfolio theory, compression, and statistics when causality constraints exist. In stock market investment (Sec. \[s\_portfolio\_throey\]), directed information between the stock price $X$ and side information $Y$ is an upper bound on the increase in growth rates due to [*causal*]{} side information. This upper bound is tight when specialized to gambling in horse races. In data compression (Sec. \[s\_data\_compreession\]) we show that directed information characterizes the value of [causal]{} side information in instantaneous compression, and it quantifies the role of [causal]{} inference in joint compression of two stochastic processes. In hypothesis testing (Sec. \[s\_hypothesis\_testing\]) we show that directed information is the exponent of the minimum type II error probability when one is to decide if $Y_i$ has a [causal influence]{} on $X_i$ or not. Finally, we introduce the notion of directed Lautum[^2] information (Sec. \[s\_lautum\]), which is a causal extension of the notion of Lautum information introduced by Palomar and Verd[ú]{} [@Palomar_verdu08LautumInformation]. We briefly discuss its role in horse race gambling, data compression, and hypothesis testing. Preliminaries: Directed Information and Causal Conditioning {#sec: directed information} =========================================================== Throughout this paper, we use the [*causal conditioning*]{} notation $(\cdot||\cdot)$ developed by Kramer [@Kramer98]. We denote by $p(x^n||y^{n-d})$ the probability mass function of $X^n = (X_1,\ldots, X_n)$ *causally conditioned* on $Y^{n-d}$ for some integer $d\geq 0$, which is defined as $$\label{e_causal_cond_def} p(x^n||y^{n-d}) := \prod_{i=1}^{n} p(x_i|x^{i-1},y^{i-d}).$$ (By convention, if $i < d$, then $x^{i-d}$ is set to null.) In particular, we use extensively the cases $d=0,1$: $$\begin{aligned} p(x^n||y^{n}) &:= \prod_{i=1}^{n}p(x_i|x^{i-1},y^{i}),\\ p(x^n||y^{n-1}) &:= \prod_{i=1}^{n} p(x_i|x^{i-1},y^{i-1}).\end{aligned}$$ Using the chain rule, we can easily verify that $$\label{e_chain_rule} p(x^n,y^n)=p(x^n||y^{n})p(y^n||x^{n-1}).$$ The *causally conditional* entropy $H(X^n||Y^n)$ and $H(X^n||Y^{n-1})$ are defined respectively as $$\begin{aligned} H(X^n||Y^n) &:= E[\log p(X^n||Y^n)]=\sum_{i=1}^n H(X_i|X^{i-1},Y^i), \nonumber\\ H(X^n||Y^{n-1}) &:= E[\log p(X^n||Y^{n-1})]=\sum_{i=1}^n H(X_i|X^{i-1},Y^{i-1}). \label{e_causal_cond_entropy}\end{aligned}$$ Under this notation, directed information defined in (\[e\_def\_directed\]) can be rewritten as $$\label{e_def_dir_diff_entrop_alt} I(Y^n \to X^n)= H(X^n) - H(X^n||Y^n),$$ which hints, in a rough analogy to mutual information, a possible interpretation of directed information $I(Y^n \to X^n)$ as the amount of information that causally available side information $Y^n$ can provide about $X^n$. Note that the channel capacity results [@Massey90; @Kramer98; @Kramer03; @PermuterWeissmanGoldsmith09; @TatikondaMitter_IT09; @Kim07_feedback; @PermuterWeissmanChenMAC_IT09; @ShraderPemuter07ISIT] involve the term $I(X^n\to Y^n)$, which measures the amount of information transfer over the forward link from $X^n$ to $Y^n$. In gambling, however, the increase in growth rate is due to side information (the backward link), and therefore the expression $I(Y^n\to X^n)$ appears. Throughout the paper we also use the notation $I(Y^{n-1}\to X^n)$ which denotes the directed information from the vector $(\emptyset, Y^{n-1})$, i.e., the null symbol followed by $Y^{n-1}$, to the vector to $X^n$, that is,$$I(Y^{n-1} \to X^n) = \sum_{i=2}^n I(Y^{i-1}; X_i| X^{i-1}).$$ Using the causal conditioning notation, given in (\[e\_causal\_cond\_entropy\]), the directed information $I(Y^{n-1}\to X^n)$ can be written as $$\label{e_def_dir_diff_entrop2} I(Y^{n-1} \to X^n)= H(X^n) - H(X^n||Y^{n-1}).$$ Directed information (in both directions) and mutual information obey the following conservation law $$\label{e_conservation_law} I(X^n; Y^n) = I(X^n\to Y^n) + I(Y^{n-1}\to X^n),$$ which was shown by Massey and Massey[@Massey05]. The conservation law is a direct consequence of the chain rule (\[e\_chain\_rule\]), and we show later in Sec. \[s\_cost\_mismatch\] that it has a natural interpretation as a conservation of a mismatch cost in data compression. The causally conditional entropy rate of a random process $X$ given another random process $Y$ and the directed information rate from $X$ to $Y$ are defined respectively as $$\label{e_def_entropy_rate} \mathcal H(X||Y) := \lim_{n\to\infty} \frac{H(X^n||Y^n)}{n},$$ $$\label{e_def_directed_rate} \mathcal I(X \to Y) := \lim_{n\to\infty} \frac{I(X^n \to Y^n)}{n},$$ when these limits exist. In particular, when $(X,Y)$ is stationary ergodic, both quantities are well-defined, namely, the limits in (\[e\_def\_entropy\_rate\]) and (\[e\_def\_directed\_rate\]) exist [@Kramer98 Properties 3.5 and 3.6]. Portfolio theory {#s_portfolio_throey} ================ Here we show that directed information is an upper bound on the increment in growth rates of optimal portfolios in a stock market due to [causal]{} side information. We start by considering a special case where the market is a horse race gambling and show that the upper bound is tight. Then we consider a general stock market investment. Horse Race Gambling with Causal Side Information {#s_horse_race} ------------------------------------------------ Assume that there are $m$ racing horses and let $X_i$ denote the winning horse at time $i$, i.e., $X_i \in \mathcal{X} := \{1,2,\ldots,m\}$. At time $i$, the gambler has some side information which we denote as $Y_i$. We assume that the gambler invests all his/her capital in the horse race as a function of the previous horse race outcomes $X^{i-1}$ and side information $Y^i$ up to time $i$. Let $b(x_i|x^{i-1},y^{i})$ be the portion of wealth that the gambler bets on horse $x_i$ given $X^{i-1} = x^{i-1}$ and $Y^i = y^{i}$. Obviously, the gambling scheme should satisfy $b(x_i|x^{i-1},y^{i})\geq 0$ and $\sum_{x_i\in \mathcal{X}} b(x_i|x^{i-1},y^{i})=1$ for any history $(x^{i-1},y^i)$. Let $o(x_i|x^{i-1})$ denote the odds of a horse $x_i$ given the previous outcomes $x^{i-1}$, which is the amount of capital that the gambler gets for each unit capital that the gambler invested in the horse. We denote by $S(x^n||y^n)$ the gambler’s wealth after $n$ races with outcomes $x^n$ and causal side information $y^n$. Finally, $\frac{1}{n} W(X^n||Y^n)$ denotes the [*growth rate*]{} of wealth, where the growth $W(X^n||Y^n)$ is defined as the expectation over the logarithm (base 2) of the gambler wealth, i.e., $$\label{e_growth_rate_def} W(X^n||Y^n):= E[\log S(X^n||Y^n)].$$ Without loss of generality, we assume that the gambler’s initial wealth $S_0$ is 1. We assume that at any time $n$ the gambler invests all his/her capital and therefore we have $$S(X^n||Y^n)=b(X_{n}|X^{n-1},Y^{n})o(X_n|X^{n-1})S(X^{n-1}||Y^{n-1}).$$ This also implies that $$S(X^n||Y^n)=\prod_{i=1}^n b(X_{i}|X^{i-1},Y^{i})o(X_i|X^{i-1}).$$ The following theorem establishes the investment strategy for maximizing the growth. \[t\_gamble\_all\_money\] For any finite horizon $n$, the maximum growth is achieved when the gambler invests the money proportional to the causally conditional distribution of the horse race outcome, i.e., $$\label{e_b=p} b(x_i|x^{i-1},y^{i})=p(x_i|x^{i-1},y^{i})\quad \forall i\in\{1,...,n\}, x^i\in \mathcal X^i, y^i\in \mathcal Y^i,$$ and the maximum growth, denoted by $W^*(X^n||Y^n)$, is $$\label{e_growth_res} W^*(X^n||Y^n):=\max_{\{b(x_i|x^{i-1},y^{i-1})\}_{i=1}^n}W(X^n||Y^n)= E[\log o(X^n)]-H(X^n||Y^n).$$ Note that since $\{p(x_i|x^{i-1},y^{i})\}_{i=1}^n$ uniquely determines $p(x^n||y^{n})$, and since $\{b(x_i|x^{i-1},y^{i})\}_{i=1}^n$ uniquely determines $b(x^n||y^{n})$, then (\[e\_b=p\]) is equivalent to $$b(x^n||y^n)\equiv p(x^n||y^n).$$ Consider $$\begin{aligned} W^*(X^n||Y^n) &=\max_{b(x^{n}||y^n)} E[\log b(X^n||Y^n)o(X^n)]\nonumber \\ &= \max_{b(x^{n}||y^n)} E[\log b(X^n||Y^n)]+ E[\log o(X^n)]\nonumber \\ &=-H(X^n||Y^n)+ E[\log o(X^n)].\end{aligned}$$ Here the last equality is achieved by choosing $b(x^{n}||y^n)=p(x^{n}||y^n),$ and it is justified by the following upper bound: $$\begin{aligned} E[\log b(X^n||Y^n)] &=\sum_{x^n,y^n }p(x^n,y^n)\left[ \log p(x^n||y^n)+\log\frac{b(x^n||y^n)}{p(x^n||y^n)}\right] \nonumber \\ &=-H(X^n||Y^n)+\sum_{x^n,y^n }p(x^n,y^n)\log\frac{b(x^n||y^n)}{p(x^n||y^n)}\nonumber \\ &\stackrel{(a)}{\leq}-H(X^n||Y^n)+\log\sum_{x^n,y^n }p(x^n,y^n)\frac{b(x^n||y^n)}{p(x^n||y^n)}\nonumber \\ &\stackrel{(b)}{=}-H(X^n||Y^n)+\log\sum_{x^n,y^n }p(y^n||x^{n-1})b(x^n||y^n)\nonumber \\ &\stackrel{(c)}=-H(X^n||Y^n).\end{aligned}$$ where (a) follows from Jensen’s inequality, (b) from the chain rule, and (c) from the fact that $\sum_{x^n,y^n }p(y^n||x^{n-1})b(x^n||y^n)=1$. In case that the odds are fair, i.e., ${o}(X_i|X^{i-1})=1/m$, $$W^*(X^n||Y^n)= n\log m-H(X^n||Y^n),$$ and thus the sum of the growth rate and the entropy of the horse race process conditioned causally on the side information is constant, and one can see a duality between $H(X^n||Y^n)$ and $W^*(X^n||Y^n)$. Let us define $\Delta W(X^n||Y^n)$ as the increase in the growth due to causal side information, i.e., $$\Delta W(X^n||Y^n)=W^*(X^n||Y^n)-W^*(X^n).$$ \[c\_increase\_double\_rate\] The increase in growth rate due to the causal side information sequence $Y^n$ for a horse race sequence $X^n$ is $$\frac{1}{n}\Delta W(X^n||Y^n)=\frac{1}{n}I(Y^n\to X^n).$$ As a special case, if the horse race outcome and side information are pairwise i.i.d., then the (normalized) directed information $\frac{1}{n}I(Y^n\to X^n)$ becomes the single letter mutual information $I(X;Y)$, which coincides with Kelly’s result [@Kelly56]. From the definition of directed information (\[e\_def\_dir\_diff\_entrop\_alt\]) and Theorem \[t\_gamble\_all\_money\] we obtain $$\begin{aligned} W^*(X^n||Y^n)-W^*(X^n)&= -H(X^n||Y^n)+H(X^n)= I(Y^n\to X^n).\end{aligned}$$ \[ex1\] Consider the case in which two horses are racing, and the winning horse $X_i$ behaves as a Markov process as shown in Fig. \[f\_Markov\]. A horse that won will win again with probability $p$ and lose with probability $1-p$ ($0\leq p\leq 1$). At time zero, we assume that the two horses have equal probability of wining. The side information revealed to the gambler at time $i$ is $Y_i$, which is a noisy observation of the horse race outcome $X_i$. It has probability $1-q$ of being equal to $X_i$, and probability $q$ of being different from $X_i$. In other words, $Y_i=X_i+ V_i \mod 2$, where $V_i$ is a Bernoulli($q$) process. \[\]\[\]\[1\][Horse]{} \[\]\[\]\[1\][wins]{} \[\]\[\]\[1\][$1$]{} \[\]\[\]\[1\][$2$]{} \[\]\[\]\[1\][$p$]{} \[\]\[\]\[1\][$1\!-\!p$]{} \[\]\[\]\[1\][$Y^n$]{} \[\]\[\]\[1\] \[\]\[\]\[0.9\][$X\!=\!1$]{} \[\]\[\]\[1\][$$]{} \[\]\[\]\[1\][Horse 1 wins]{}\[\]\[\]\[0.9\][$X\!=\!2$]{} \[\]\[\]\[1\][Horse 2 wins]{}\[\]\[\]\[1\][1]{}\[\]\[\]\[1\][2]{}\[\]\[\]\[1\][$X$]{} \[\]\[\]\[1\][$Y$]{} \[\]\[\]\[1\] \[\]\[\]\[1\] \[\]\[\]\[1\][$q$]{} \[\]\[\]\[1\][$1-q$]{} ![The setting of Example 1. The winning horse $X_i$ is represented as a Markov process with two states. In state 1, horse number 1 wins, and in state 2, horse number 2 wins. The side information, $Y_i$, is a noisy observation of the winning horse, $X_i$. []{data-label="f_Markov"}](horse_race3.eps){width="6.6cm"} For this example, the increase in growth rate due to side information $\Delta W:=\frac{1}{n}\Delta W(X^n||Y^n)$ is $$\begin{aligned} \label{e_markov_qp} \Delta W&= h(p*q)-h(q),\end{aligned}$$ where $h(x):=-x\log x-(1-x)\log (1-x)$ is the binary entropy function, and $p*q=(1-p)q+(1-q)p$ denotes the parameter of a Bernoulli distribution that results from convolving two Bernoulli distributions with parameters $p$ and $q$. The increase $\Delta W$ in the growth rate can be readily derived using the identity in (\[e\_kim\_identity\]) as follows: $$\begin{aligned} \label{e_kim} \frac{1}{n} I(Y^n\to X^n) &\stackrel{(a)}{=} \frac{1}{n}\sum_{i=1}^n I(Y_i;X_i^n|X^{i-1},Y^{i-1}) \nonumber \\ &\stackrel{(b)}{=} H(Y_1|X_{0})-H(Y_1|X_1),\end{aligned}$$ where equality (a) is the identity from (\[e\_kim\_identity\]), which can be easily verified by the chain rule for mutual information [@Kim07_feedback eq. (9)], and (b) is due to the stationarity of the process. If the side information is known with some lookahead $k\geq 0$, meaning that at time $i$ the gambler knows $Y^{i+k}$, then the increase in growth rate is given by $$\begin{aligned} \Delta W&= \lim_{n\to \infty} \frac{1}{n} I(Y^{n+k}\to X^n) \nonumber \\ &\stackrel{}{=} H(Y_{k+1}|Y^{k},X_{0})-H(Y_1|X_1),\end{aligned}$$ where the last equality is due to the same arguments as (\[e\_kim\]). If the entire side information sequence $(Y_1,Y_2,\ldots)$ is known to the gambler ahead of time, then since the sequence $H(Y_{k+1}|Y^{k-1},X_{0})$ converges to the entropy rate of the process, we obtain mutual information [@Kelly56] instead of directed information, i.e., $$\begin{aligned} \Delta W&= \lim_{n\to \infty} \frac{1}{n} I(Y^{n};X^n)\nonumber\\ &=\lim_{n\to \infty} \frac{H(Y^n)}{n} -H(Y_1|X_1).\end{aligned}$$ Investment in a Stock Market with Causal Side Information --------------------------------------------------------- We use notation similar to the one in [@CovThom06 ch. 16]. A stock market at time $i$ is represented by a vector ${\bf X_i}=(X_{i1},X_{i2},\ldots,X_{im})$, where $m$ is the number of stocks, and the [*price relative $X_{ik}$*]{} is the ratio of the price of stock-$k$ at the end of day $i$ to the price of stock-$k$ at the beginning of day $i$. Note that gambling in a horse race is an extreme case of stock market investment—for horse races, the price relatives are all zero except one. We assume that at time $i$ there is side information $Y^i$ that is known to the investor. A [*portfolio*]{} is an allocation of wealth across the stocks. A nonanticipating or causal portfolio strategy with causal side information at time $i$ is denoted as ${\bf b}({\bf x}^{i-1},y^i)$, and it satisfies $\sum_{k=1}^m b_{k}({\bf x}^{i-1},y^i) =1$ and $b_{k}({\bf x}^{i-1},y^i)\geq 0$ for all possible $({\bf x}^{i-1},y^i)$. We define $S({\bf x}^n||y^n)$ to be the wealth at the end of day $n$ for a stock sequence ${\bf x}^n$ and causal side information $y^n$. We have $$S({\bf x}^n||y^n) = \left ({\bf b}^t({\bf x}^{n-1},y^n)\cdot {\bf x}_n\right ) S({\bf x}^{n-1}||y^{n-1}),$$ where ${\bf b}^t \cdot{\bf x}$ denotes inner product between the two (column) vectors ${\bf b}$ and ${\bf x}$. The goal is to maximize the growth $$W({\bf X}^n||Y^n):=E[\log S({\bf X}^n||Y^n) ].$$ The justifications for maximizing the growth rate is due to [@Algoet88 Theorem 5] that such a portfolio strategy will exceed the wealth of any other strategy to the first order in the exponent for almost every sequence of outcomes from the stock market, namely, if ${S^*({\bf X}^n||Y^n)}$ is the wealth corresponding to the growth rate optimal return, then $$\limsup_{n} \frac{1}{n} \log\left(\frac{S({\bf X}^n||Y^n)}{S^*({\bf X}^n||Y^n)}\right) \leq 0\quad\text{a.s.}$$ Let us define $$W({\bf X}_n|{\bf X}^{n-1},Y^n):=E[\log ({\bf b}^t({\bf X}^{n-1},Y^n) {\bf X}_n)].$$ From this definition follows the chain rule: $$W({\bf X}^n||Y^n)=\sum_{i=1}^n W({\bf X}_i|{\bf X}^{i-1},Y^i),$$ from which we obtain $$\begin{aligned} \label{e_maxWn_as_maxmize_each_one_market} \max_{\{{\bf b}({\bf x}^{i-1},y^i)\}_{i=1}^n}W({\bf X}^n||Y^n)&=\sum_{i=1}^n \max_{{\bf b}({\bf x}^{i-1},y^i)} W({\bf X}_i|X^{i-1},Y^{i})\nonumber \\ &=\sum_{i=1}^n \int_{{\bf x}^{i-1},y^i}f({\bf x}^{i-1},y^i)\max_{{\bf b}({\bf x}^{i-1},y^i)} W({\bf X}_i|{\bf x}^{i-1},y^{i}),\end{aligned}$$ where $f({\bf x}^{i-1},y^i)$ denotes the probability density function of $({\bf x}^{i-1},y^i)$. The maximization in (\[e\_maxWn\_as\_maxmize\_each\_one\_market\]) is equivalent to the maximization of the growth rate for the memoryless case where the cumulative distribution function of the stock-vector ${\bf X}$ is $P({\bf X}\leq {\bf x})=\Pr({\bf X}_i\leq {\bf x}|{\bf x}^{i-1},{ y}^i)$ and the portfolio ${\bf b} = {\bf b}({\bf x}^{i-1},y^i)$ is a function of $(x^{i-1}, y^i)$, i.e., $$\begin{aligned} \label{e_memoryless_opt} \text{maximize } & E[\log({\bf b}^t {\bf X})|X^{i-1} = x^{i-1}, Y^i = y^i]\nonumber \\ \text{subject to } & \sum_{i=1}^m b_k=1, \nonumber \\ & b_k\geq0, \forall k\in [1,2,\ldots,m].\end{aligned}$$ In order to upper bound the difference in growth rate due to [causal]{} side information we recall the following result which bounds the loss in growth rate incurred by optimizing the portfolio with respect to a wrong distribution $g({\bf x})$ rather than the true distribution $f({\bf x})$. \[t\_diff\_stock\] Let $f({\bf x})$ be the probability density function of a stock vector ${\bf X}$, i.e., ${\bf X}\sim f({\bf x})$. Let ${\bf b}_f$ be the growth rate portfolio corresponding to $f({\bf x})$, and let ${\bf b}_g$ be the growth rate portfolio corresponding to another density $g({\bf x})$. Then the increase in optimal growth rate $\Delta W$ by using ${\bf b}_f$ instead of ${\bf b}_g$ is bounded by $$\Delta W= E[\log ({\bf b}^t_f {\bf X})]-E[\log ({\bf b}^t_g {\bf X})]\leq D(f||g),$$ where $D(f||g):=\int f(x)\log \frac{f(x)}{g(x)} dx$ denotes the Kullback–Leibler divergence between the probability density functions $f$ and $g$. Using Theorem \[t\_diff\_stock\], we can upper bound the increase in growth rate due to causal side information by directed information as shown in the following theorem. \[t\_stock\_directed\_upper\_bound\] The increase in optimal growth rate for a stock market sequence ${\bf X}^n$ due to side information $Y^n$ is upper bounded by $$W^*({\bf X}^n||Y^n)- W^*({\bf X}^n) \leq I(Y^n\to {\bf X}^n),$$ where $W^*({\bf X}^n||Y^n)\triangleq \max_{\{{\bf b}({\bf X}^{i-1},Y^i)\}_{i=1}^n}W({\bf X}^n||Y^n)$ and $W^*({\bf X}^n):= \max_{\{{\bf b}({\bf X}^{i-1})\}_{i=1}^n}W({\bf X}^n)$. Consider $$\begin{aligned} \lefteqn{W^*({\bf X}^n||Y^n)- W^*({\bf X}^n)}\nonumber\\ &=\sum_{i=1}^n \int_{{\bf x}^{i-1},y^i}f({\bf x}^{i-1},y^i)\left[ \max_{{\bf b}({\bf x}^{i-1},y^i)} W({\bf X}_i|{\bf x}^{i-1},y^{i})-\max_{{\bf b_i}({\bf x}^{i-1})} W({\bf X}_i|{\bf x}^{i-1})\right] \nonumber \\ &\stackrel{(a)}{\leq}\sum_{i=1}^n \int_{{\bf x}^{i-1},y^i}f({\bf x}^{i-1},y^i)\left[ \int_{{\bf x}_i} f({\bf x}_i|{\bf x}^{i-1},y^i)\log \frac{f({\bf x}_i|{\bf x}^{i-1},y^i)}{f({\bf x}_i|{\bf x}^{i-1})} \right] \nonumber \\ &\stackrel{}{=}\sum_{i=1}^n E\left[ \log \frac{f({\bf X}_i|{\bf X}^{i-1},Y^i)}{f({\bf X}_i|{\bf X}^{i-1})} \right] \nonumber \\ &\stackrel{}{=}\sum_{i=1}^n h({\bf X}_i|{\bf X}^{i-1})-h({\bf X}_i|{\bf X}^{i-1},Y^i) \nonumber \\ &\stackrel{}{=}I(Y^n \to {\bf X}^n),\end{aligned}$$ where the inequality (a) is due to Theorem \[t\_diff\_stock\]. Note that the upper bound in Theorem \[t\_stock\_directed\_upper\_bound\] is tight for gambling in horse races (Corollary \[c\_increase\_double\_rate\]). Data Compression {#s_data_compreession} ================ In this section we investigate the role of directed information in data compression and find two interpretations: 1. directed information characterizes the value of causal side information in instantaneous compression, and 2. it also quantifies the role of causal inference in joint compression of two stochastic processes. Instantaneous Lossless Compression with Causal Side Information --------------------------------------------------------------- Let $X_1, X_2 \ldots$ be a source and $Y_1, Y_2, \ldots$ be side information about the source. The source is to be encoded losslessly by an instantaneous code with causally available side information, as depicted in Fig. \[f\_causal\_compression\]. More formally, an [ *instantaneous lossless source encoder with causal side information*]{} consists of a sequence of mappings $\{ M_i \}_{i \geq 1}$ such that each $M_i : \mathcal{X}^i \times \mathcal{Y}^i \mapsto \{0,1\}^*$ has the property that for every $x^{i-1}$ and $y^i$, $M_i (x^{i-1} \cdot , y^i)$ is an instantaneous (prefix) code. \[B\]\[\]\[1\][$X^i$]{} \[B\]\[\]\[1\][$\;\;\;\;\;\;\;\;\;\;\;M_i(X^i,Y^i)$]{} \[\]\[\]\[1\][Causal]{} \[\]\[\]\[1\][Encoder]{} \[\]\[\]\[1\][Decoder]{} \[\]\[\]\[1\][$\;Y^i$]{} \[B\]\[\]\[1\][$\;\;\;\;\;\;\hat X_i(M^i,Y^i)$]{} ![Instantaneous data compression with causal side information []{data-label="f_causal_compression"}](causal_compression.eps){width="8cm"} An instantaneous lossless source encoder with causal side information operates sequentially, emitting the concatenated bit stream $M_1 (X_1, Y_1) M_2(X^2, Y^2) \ldots$. The defining property that $M_i (x^{i-1} \cdot , y^i)$ is an instantaneous code for every $x^{i-1}$ and $y^i$ is a necessary and sufficient condition for the existence of a decoder that can losslessly recover $x^i$ based on $y^i$ and the bit stream $M_1 (x_1, y_1) M_2(x^2, y^2) \ldots$ as soon as it receives $M_1 (x_1, y_1) M_2(x^2, y^2) \ldots M_i(x^i, y^i)$ for all sequence pairs $(x_1,y_1), ( x_2, y_2), \ldots$, and all $i \geq 1$. Let $L(x^n|| y^n)$ denote the length of the concatenated string $M_1 (x_1, y_1) M_2(x^2, y^2) \ldots M_n(x^n, y^n)$. Then the following result is due to Kraft’s inequality and Huffman coding adapted to the case where causal side information is available. \[th: lossless insta coding with si\] Any instantaneous lossless source encoder with causal side information satisfies $$\label{eq: lossless source encoder converse} \frac{1}{n} E L(X^n|| Y^n) \geq \frac{1}{n} \sum_{i=1}^n H(X_i | X^{i-1}, Y^i) \ \ \ \ \ \forall n \geq 1.$$ There exists an instantaneous lossless source encoder with causal side information satisfying $$\label{eq: direct part for inst lossless causal} \frac{1}{n} E L(X^n|| Y^n) \leq \frac{1}{n} \sum_{i=1}^n r_i + H(X_i | X^{i-1}, Y^i) \ \ \ \ \ \forall i \geq 1,$$ where $r_i= \sum_{x^{i-1},y^{i}}p(x^{i-1},y^{i})\min(1, \max_{x_i}p(x_i|x^{i-1},y^{i-1})+0.086)$. The lower bound follows from Kraft’s inequality [@CovThom06 Theorem 5.3.1] and the upper bound follows from Huffman coding on the conditional probability $p(x_i|x^{i-1},y^{i})$. The redundancy term $r_i$ follows from Gallager’s redundancy bound [@Gallager78HuffmanCode], $\min(1, P_{i}+0.086)$, where $P_{i}$ is the probability of the most likely source letter at time $i$, averaged over side information sequence $(X^{i-1},Y^{i})$. Since the Huffman code achieves the entropy rate for dyadic probability, it follows that if the conditional probability $p(x_i|x^{i-1},y^{i-1})$ is dyadic, i.e., if each conditional probability equals to $2^{-k}$ for some integer $k$, then (\[eq: lossless source encoder converse\]) can be achieved with equality. Theorem \[th: lossless insta coding with si\], combined with the identity $\sum_{j=i}^n H(X_i | X^{i-1}, Y^i) = H(X^n) - I(Y^n \rightarrow X^n)$, implies that the compression rate saved in optimal sequential lossless compression due to the causal side information is upper bounded by $\frac{1}{n}I(Y^n\to X^n)-1$, and lower bounded by $\frac{1}{n}I(Y^n\to X^n)+1$. If all the probabilities are dyadic, then the compression rate saving is exactly the directed information rate $\frac{1}{n}I(Y^n\to X^n)$. This saving should be compared to $\frac{1}{n}I(X^n;Y^n)$, which is the saving in the absence of causality constraint. Cost of Mismatch in Data Compression {#s_cost_mismatch} ------------------------------------ Suppose we compress a pair of correlated sources $\{(X_i,Y_i)\}$ jointly with an optimal lossless variable length code (such as the Huffman code), and we denote by $E(L(X^n,Y^n))$ the average length of the code. Assume further that $Y_i$ is generated randomly by a forward link $p(y_i|y^{i-1},x^{i})$ as in a communication channel or a chemical reaction, and $X_i$ is generated by a backward link $p(x_i|y^{i-1},x^{i-1})$ such as in the case of an encoder or a controller with feedback. By the chain rule for causally conditional probabilities (\[e\_chain\_rule\]), any joint distribution can be modeled according to Fig. \[f\_compression\]. \[\]\[\]\[1\][$p(x_i|x^{i-1},y^{i-1})$]{} \[\]\[\]\[1\][Backward link]{} \[\]\[\]\[1\][$p(y_i|x^{i},y^{i-1})$]{} \[\]\[\]\[1\][$p(y_i|y^{i-1})$]{} \[\]\[\]\[1\][Forward link]{} \[\]\[\]\[1\][$X_i$]{} \[\]\[\]\[1\][$Y_i$]{} \[\]\[\]\[1\][$Y_{i-1}$]{} \[\]\[\]\[1\] \[\]\[\]\[1\] ![ [Compression of two correlated sources $\{X_i,Y_i\}_{i\geq 1 }$. Since any joint distribution can be decomposed as $p(x^n,y^n)=p(x^n||y^{n-1})p(y^n||x^n)$, each link embraces the existence of a forward or feedback channel (chemical reaction). We investigate the influence of the link knowledge on joint compression of $\{X_i,Y_i\}_{i\geq 1 }$]{}. \[f\_compression\]](joint_compression.eps){width="8cm"} Recall that the optimal variable-length lossless code, in which both links are taken into account, has the average length $$H(X^n,Y^n)\le E(L(X^n,Y^n)) < H(X^n,Y^n) + 1.$$ However, if the code is erroneously designed to be optimal for the case in which the forward link does not exist, namely, the code is designed for the joint distribution $p(y^n)p(x^n||y^{n-1})$, then the average code length (up to 1 bit) is $$\begin{aligned} E(L(X^n,Y^n))&=\sum_{x^n,y^n}p(x^n,y^n)\log \frac{1}{p(y^n)p(x^n||y^n-1)}\nonumber \\ &= \sum_{x^n,y^n}p(x^n,y^n)\log \frac{p(x^n,y^n)}{p(y^n)p(x^n||y^n-1)}+H(X^n,Y^n)\nonumber \\ &=\sum_{x^n,y^n}p(x^n,y^n)\log \frac{p(y^n||x^n)}{p(y^n)}+H(X^n,Y^n)\nonumber \\ &=I(X^n\to Y^n)+H(X^n,Y^n).\end{aligned}$$ Hence the redundancy (the gap from the minimum average code length) is $I(X^n\to Y^n)$. Similarly, if the backward link is ignored, then the average code length (up to 1 bit) is $$\begin{aligned} E(L(X^n,Y^n))&=\sum_{x^n,y^n}p(x^n,y^n)\log \frac{1}{p(y^n||x^n)p(x^n)}\nonumber \\ &= \sum_{x^n,y^n}p(x^n,y^n)\log \frac{p(x^n,y^n)}{p(y^n||x^n)p(x^n)}+H(X^n,Y^n)\nonumber \\ &=\sum_{x^n,y^n}p(x^n,y^n)\log \frac{p(x^n||y^{n-1})}{p(x^n)}+H(X^n,Y^n)\nonumber \\ &=I(Y^{n-1}\to X^n)+H(X^n,Y^n)\end{aligned}$$ Hence the redundancy for this case is $I(Y^{n-1}\to X^n)$. If both links are ignored, the redundancy is simply the mutual information $I(X^n; Y^n)$. This result quantifies the value of knowing causal influence between two processes when designing the optimal joint compression. Note that the redundancy due to ignoring both links is the sum of the redundancies from ignoring each link. This recovers the conservation law (\[e\_conservation\_law\]) operationally. Directed Information and Statistics: Hypothesis Testing {#s_hypothesis_testing} ======================================================= Consider a system with an input sequence $(X_1,X_2,\ldots,X_n)$ and output sequence $(Y_1,Y_2,\ldots,Y_n)$, where the input is generated by a stimulation mechanism or a controller, which observes the previous outputs, and the output may be generated either causally from the input according to $\{p(y_i|y^{i-1},x^{i})\}_{i=1}^n$ (the null hypothesis $H_0$) or independently from the input according to $\{p(y_i|y^{i-1})\}_{i=1}^n$ (the alternative hypothesis $H_1$). For instance, this setting occurs in communication or biological systems, where we wish to test whether the observed system output $Y^n$ is in response to one’s own stimulation input $X^n$ or to some other input that uses the same stimulation mechanism and therefore induces the same marginal distribution $p(y^n)$. The stimulation mechanism $p(x^n||y^{n-1})$, the output generator $p(y^n||x^n)$, and the sequences $X^n$ and $Y^n$ are assumed to be known. \[\]\[\]\[1\][$p(x_i|x^{i-1},y^{i-1})$]{} \[\]\[\]\[1\][Controller]{} \[\]\[\]\[1\][$p(y_i|x^{i},y^{i-1})$]{} \[\]\[\]\[1\][$p(y_i|y^{i-1})$]{} \[\]\[\]\[1\][Output generator]{} \[\]\[\]\[1\][$X_i$]{} \[\]\[\]\[1\][$Y_i$]{} \[\]\[\]\[1\][$Y_{i-1}$]{} \[\]\[\]\[1\][Hypothesis $H_0$:]{} \[\]\[\]\[1\][Hypothesis $H_1$]{} ![Hypothesis testing. $H_0$: The input sequence $(X_1,X_2\ldots,X_n)$ causally influences the output sequence $(Y_1,Y_2,\ldots,Y_n)$ through the causal conditioning distribution $p(y^n||x^n)$. $H_1$: The output sequence $(Y_1,Y_2,\ldots,Y_n)$ was not generated by the input sequence $(X_1,X_2,\ldots,X_n)$, but by another input from the same stimulation mechanism $p(x^n||y^{n-1})$. []{data-label="f_hypothesis"}](hypothesis_controller.eps){width="16cm"} An [*acceptance region*]{} $A$ is the set of all sequences $(x^n,y^n)$ for which we accept the null hypothesis $H_0$. The complement of $A$, denoted by $A^c$, is the rejection region, namely, the set of all sequences $(x^n,y^n)$ for which we reject the null hypothesis $H_0$ and accept the alternative hypothesis $H_1$. Let $$\label{e_error_def} \alpha:=\Pr(A^c|H_0),\quad \beta:=\Pr(A|H_1)$$ denote the probabilities of [*type I error*]{} and [*type II error*]{}, respectively. The following theorem interprets the directed information rate $\mathcal I(X\to Y)$ as the best error exponent of $\beta$ that can be achieved while $\alpha$ is less than some constant $\epsilon> 0$. \[t\_Chernoff-Stel\] Let $(X,Y) = \{X_i,Y_i\}_{i=1}^\infty$ be a stationary and ergodic random process. Let $A_n\subseteq {(\mathcal X\times \mathcal Y)}^n$ be an acceptance region, and let $\alpha_n$ and $\beta_n$ be the corresponding probabilities of type I and type II errors (\[e\_error\_def\]). For $0<\epsilon<\frac{1}{2}$, let $$\label{e_beta_n} \beta_n^{(\epsilon)}=\min_{A_n\subseteq {(\mathcal X\times \mathcal Y)}^n, \alpha_n<\epsilon} \beta_n.$$ Then $$\lim_{n\to\infty} -\frac{1}{n} \log \beta_n^{(\epsilon)}=\mathcal I(X\to Y),$$ where the directed information rate is the one induced by the joint distribution from $H_0$, i.e., $p(x^n||y^{n-1})p(y^n||x^n)$. Theorem \[t\_Chernoff-Stel\] is reminiscent of the achievability proof in the channel coding theorem. In the random coding achievability proof [@CovThom06 ch 7.7] we check whether the output $Y^n$ is resulting from a message (or equivalently from an input sequence $X^n$) and we would like to have the error exponent which is, according to Theorem \[t\_Chernoff-Stel\], $I(X^n\to Y^n)$ to be as large as possible so we can distinguish as many messages as possible. The proof of Theorem \[t\_Chernoff-Stel\] combines arguments from the Chernoff–Stein Lemma [@CovThom06 Theorem 11.8.3] with the Shannon–McMillan–Breiman Theorem for directed information [@Pradhan07Venkataramanan Lemma 3.1], which implies that for a jointly stationary ergodic random process $$\frac{1}{n}\log \frac{p(Y^n||X^n)}{P(Y^n)} \to \mathcal I( X\to Y) \text{ in probability.}$$ [*Achievability:*]{} Fix $\delta>0$ and let $A_n$ be $$\label{e_A_n} A_n=\left\{x^n,y^n:\left|\log \frac{p(y^n||x^n)}{p(y^n)}-\mathcal I( X\to Y)\right|<\delta \right\}$$ By the AEP for directed information [@Pradhan07Venkataramanan Lemma 3.1] we have that $\Pr (A_n|H_0)\to 1$ in probability; hence there exists $N(\epsilon)$ such that for all $n>N(\epsilon)$, $\alpha_n=\Pr (A_n^c|H_0)< \epsilon$. Furthermore, $$\begin{aligned} \beta_n&=\Pr(A_n|H_1)\nonumber \\ &=\sum_{x^n,y^n\in A_n} p(x^n||y^{n-1})p(y^n)\nonumber \\ &\stackrel{(a)}{\leq}\sum_{x^n,y^n\in A_n} p(x^n||y^{n-1})p(y^n||x^n)2^{-n(\mathcal I(X\to Y)-\delta)}\nonumber \\ &\stackrel{}{=}2^{-n(\mathcal I(X\to Y)-\delta)}\sum_{x^n,y^n\in A_n} p(x^n||y^{n-1})p(y^n||x^n)\nonumber \\ &\stackrel{(b)}{=}2^{-n(\mathcal I(X\to Y)-\delta)}(1-\alpha_n),\end{aligned}$$ where inequality (a) follows from the definition of $A_n$ and (b) from the definition of $\alpha_n$. We conclude that $$\label{e_Bn_limit} \lim \sup _{n\to\infty}\frac{1}{n}\log \beta_n\leq -\mathcal I(X\to Y)+\delta,$$ establishing the achievability since $\delta>0$ is arbitrary. [*Converse:*]{} Let $B_n\subseteq {(\mathcal X\times \mathcal Y)}^n$ such that $\Pr(B_n^c|H_0) < \epsilon < \frac{1}{2}$. Consider $$\begin{aligned} \Pr(B_n|H_1)&\geq \Pr(A_n\cap B_n|H_1) \nonumber \\ &= \sum_{(x^n,y^n) \in A_n\cap B_n} p(x^n||y^{n-1})p(y^n) \nonumber \\ &\geq \sum_{(x^n,y^n) \in A_n\cap B_n} p(x^n||y^{n-1})p(y^n||x^{n-1})2^{-n(\mathcal I(X\to Y)+\delta)} \nonumber \\ &= 2^{-n(\mathcal I(X\to Y)+\delta)}\Pr(A_n\cap B_n|H_0)\nonumber \\ &= 2^{-n(\mathcal I(X\to Y)+\delta)}(1-\Pr(A_n^c\cup B_n^c|H_0))\nonumber \\ &\geq 2^{-n(\mathcal I(X\to Y)+\delta)}(1-\Pr(A_n^c|H_0)-\Pr(B_n^c|H_0))%\nonumber \\\end{aligned}$$ Since $\Pr(A_n^c|H_0)\to 0$ and $\Pr(B_n^c|H_0)<\epsilon<\frac{1}{2}$, we obtain $$\lim \inf_{n\to\infty}\frac{1}{n}\log \beta_n\geq -(\mathcal I(X\to Y)+\delta).$$ Finally, since $\delta > 0$ is arbitrary, the proof of the converse is completed. Directed Lautum Information {#s_lautum} =========================== Recently, Palomar and Verd[ú]{} [@Palomar_verdu08LautumInformation] have defined the lautum information $L(X^n;Y^n)$ as $$L(X^n;Y^n):=\sum_{x^n,y^n}p(x^n)p(y^n)\log \frac{p(y^n)}{p(y^n|x^n)},$$ and showed that it has operational interpretations in statistics, compression, gambling, and portfolio theory, where the true distribution is $p(x^n)p(y^n)$ but mistakenly a joint distribution $p(x^n,y^n)$ is assumed. As in the definition of directed information wherein the role of regular conditioning is replaced by causal conditioning, we define two types of directed lautum information. The first type $$\begin{aligned} L_1(X^n\to Y^n) &:=\sum_{x^n,y^n}p(x^n)p(y^n)\log \frac{p(y^n)}{p(y^n||x^n)}\end{aligned}$$ and the second type $$\begin{aligned} L_2(X^n\to Y^n) &:=\sum_{x^n,y^n}p(x^n||y^{n-1})p(y^n)\log \frac{p(y^n)}{p(y^n||x^n)}.\end{aligned}$$ When $p(x^n||y^{n-1})=p(x^n)$ (no feedback), the two definitions coincide. We will see in this section that the first type of directed lautum information has operational meanings in scenarios where the true distribution is $p(x^n)p(y^n)$ and, mistakenly, a joint distribution of the form $p(x^n)p(y^n||x^n)$ is assumed. Similarly, the second type of directed information occurs when the true distribution is $p(x^n||y^{n-1})p(y^n)$, but a joint distribution of the form $p(x^n||y^{n-1})p(y^n||x^n)$ is assumed. We have the following conservation law for the first-type directed lautum information: For any discrete jointly distributed random vectors $X^n$ and $Y^n$ $$L(X^n;Y^n)=L_1(X^n\to Y^n)+L_1(Y^{n-1}\to X^n).$$ Consider $$\begin{aligned} \label{e_cons_lautum} L_1(X^n;Y^n)&\stackrel{(a)}{=} \sum_{x^n,y^n}p(x^n)p(y^n)\log \frac{p(y^n)p(x^n)}{p(y^n,x^n)}\nonumber\\ &\stackrel{(b)}{=}\sum_{x^n,y^n}p(x^n)p(y^n)\log \frac{p(y^n)p(x^n)}{p(y^n||x^n)p(x^n||y^{n-1})}\nonumber\\ &\stackrel{}{=}\sum_{x^n,y^n}p(x^n)p(y^n)\log \frac{p(y^n)}{p(y^n||x^n)}+\sum_{x^n,y^n}p(x^n)p(y^n)\log \frac{p(x^n)}{p(x^n||y^{n-1})}\nonumber\\ &\stackrel{}{=} L_1(X^n\to Y^n)+L_1(Y^{n-1}\to X^n),\end{aligned}$$ where (a) follows from the definition of lautum information and (b) follows from the chain rule $p(y^n,x^n)=p(y^n||x^n)p(x^n||y^{n-1})$. A direct consequence of the lemma is the following condition for the equality between two types of directed lautum information and regular lautum information. If $$L(X^n;Y^n)=L_1(X^n\to Y^n),$$ then $$\label{e_cond} p(x^n)=p(x^n||y^{n-1})\text{ for all }(x^n,y^n) \in \mathcal X^n\times \mathcal Y^n \text{ with } p(x^n,y^n)>0.$$ Conversely, if (\[e\_cond\]) holds, then $$L(X^n;Y^n)=L_1(X^n\to Y^n)=L_2(X^n\to Y^n).$$ The proof of the first part follows from the conservation law (\[e\_cons\_lautum\]) and the nonnegativity of Kullback–Leibler divergence [@CovThom06 Theorem 2.6.3] (i.e., $L_1(Y^{n-1}\to X^n)=0$ implies that $p(x^n)=p(x^n||y^{n-1})$). The second part follows from the definitions of regular and directed lautum information. The [*lautum information rate*]{} and [*directed lautum information rates*]{} are respectively defined as $$\begin{aligned} \mathcal L( X; Y)&:=\lim_{n\to \infty} \frac{1}{n}L( Y^n ; X^n),\label{e_lautum_rate} \\ \mathcal L_j(X \to Y)&:=\lim_{n\to \infty} \frac{1}{n}L_j( Y^n \to X^n)\quad \text{for } j=1,2\label{e_directed_lautum_rate},\end{aligned}$$ whenever the limits exist. The next lemma provides a technical condition for the existence of the limits. \[l\_suff\_lautum\_rate\] If the process $(X_n,Y_n)$ is stationary and Markov (i.e., $p(x_i,y_i|x^{i-1},y^{i-1})=p(x_i,y_i|x_{i-k}^{i-1},y_{i-k}^{i-1})$ for some finite $k$), then $\mathcal L(X; Y)$ and $\mathcal L_2(X \to Y)$ are well defined. Similarly, if the process $\{X^n,Y^n\}\sim p(x^n)p(y^n||x^n)$ is stationary and Markov, then $\mathcal L_1(X \to Y)$ is well defined. It is easy to see the sufficiency of the conditions for $\mathcal L_1( X \to \ Y)$ from the following identity: $$\begin{aligned} L_1(\mathcal X^n \to \mathcal Y^n) &= \sum_{x^n,y^n}p(x^n)p(y^n)\log \frac{p(x^n)p(y^n)}{p(x^n)p(y^n||x^n)}\nonumber \\ &=-H(X^n)-H(Y^n)- \sum_{x^n,y^n}p(x^n)p(y^n)\log {p(x^n)p(y^n||x^n)}.\end{aligned}$$ Since the process is stationary the limits $\lim_{n\to \infty }\frac{1}{n}H(X^n)$ and $\lim_{n\to \infty } \frac{1}{n}H(Y^n)$ exist. Furthermore, since $p(x^n,y^n)=p(x^n)p(y^n||x^n)$ is assumed to be stationary and Markov, the limit $\lim_{n\to \infty } \frac{1}{n}\sum_{x^n,y^n}p(x^n)p(y^n)\log {p(x^n)p(y^n||x^n)}$ exists. The sufficiency of the condition can be proved for $\mathcal L_2( X \to Y)$ and the lautum information rate using a similar argument. Adding causality constraints to the problems that were considered in [@Palomar_verdu08LautumInformation], we obtain the following results for horse race gambling and data compression. Horse Race Gambling with Mismatched Causal Side Information ----------------------------------------------------------- Consider the horse race setting in Section \[s\_horse\_race\] where the gambler has causal side information. The joint distribution of horse race outcomes $X^n$ and the side information $Y^n$ is given by $p(x^n)p(y^n||x^{n-1})$, namely, $X_i \to X^{i-1} \to Y^i$ form a Markov chain, and therefore the side information does not increase the growth rate. The gambler mistakenly assumes a joint distribution $p(x^n||y^n)p(y^n||x^{n-1})$, and therefore he/she uses a gambling scheme $b^*(x^n||y^n)=p(x^n||y^n)$. If the gambling scheme $b^*(x^n||y^n)=p(x^n||y^n)$ is applied to the horse race described above, then the penalty in the growth with respect to the gambling scheme $b^*(x^n)$ that uses no side information is $L_2(Y^n\to X^n)$. For the special case where the side information is independent of the horse race outcomes, the penalty is $L_1(Y^n\to X^n)$. The optimal growth rate where the joint distribution is $p(x^n)p(y^n||x^{n-1})$ is $W^*(X^n)= E[\log o(X^n)]-H(X^n)$. Let $E_{p(x^n)p(y^n||x^{n-1})}$ denotes the expectation with respect to the joint distribution $p(x^n)p(y^n||x^{n-1})$. The growth rate for the gambling strategy $b(x^n||y^n)=p(x^n||y^n)$ is $$\begin{aligned} W^*(X^n||Y^n)&\stackrel{}{=} E_{p(x^n)p(y^n||x^{n-1})}[\log b(X^n||Y^n)o(X^n)]\nonumber \\ &= E_{p(x^n)p(y^n||x^{n-1})}[\log p(X^n||Y^n)]+ E_{p(x^n)p(y^n||x^{n-1})}[\log o(X^n)];%\nonumber \\\end{aligned}$$ hence $W^*(X^n)-W^*(X^n||Y^n)=L_2(Y^n\to X^n)$. In the special case, where the side information is independent of the horse outcome, namely, $p(y^n||x^{n-1})=p(y^n)$, then $L_2(Y^n\to X^n)=L_1(Y^n\to X^n)$. This result can be readily extended to the general stock market, for which the penalty is [*upper bounded*]{} by $L_2(Y^n\to X^n)$. Compression with Joint Distribution Mismatch -------------------------------------------- In Section \[s\_data\_compreession\] we investigated the cost of ignoring forward and backward links when compressing a jointly $(X^n,Y^n)$ by an optimal lossless variable length code. Here we investigate the penalty of assuming forward and backward links incorrectly when neither exists. Let $X^n$ and $Y^n$ be independent sequences. Suppose we compress them with a scheme that would have been optimal under the incorrect assumption that the forward link $p(y^n||x^n)$ exists. The optimal lossless average variable length code under these assumptions satisfies (up to 1 bit per source symbol) $$\begin{aligned} E(L(X^n,Y^n))&=\sum_{x^n,y^n}p(x^n)p(y^n)\log \frac{1}{p(y^n||x^n)p(x^n)}\nonumber \\ &= \sum_{x^n,y^n}p(x^n)p(y^n)\log \frac{p(x^n)p(y^n)}{p(y^n||x^n)p(x^n)}+H(X^n)+H(Y^n)\nonumber \\ &= \sum_{x^n,y^n}p(x^n)p(y^n)\log \frac{p(y^n)}{p(y^n||x^n)}+H(X^n)+H(Y^n)\nonumber \\ &=L_1(X^{n}\to Y^n)+H(X^n)+H(Y^n).\end{aligned}$$ Hence the penalty is $L_1(X^{n}\to Y^n)$. Similarly, if we incorrectly assume that the backward link $p(x^{n}||y^{n-1})$ exists, then $$\begin{aligned} E(L(X^n,Y^n))&=\sum_{x^n,y^n}p(x^n)p(y^n)\log \frac{1}{p(x^n||y^{n-1})p(y^n)}\nonumber \\ &= \sum_{x^n,y^n}p(x^n)p(y^n)\log \frac{p(x^n)p(y^n)}{p(x^n||y^{n-1})p(y^n)}+H(X^n)+H(Y^n)\nonumber \\ &= \sum_{x^n,y^n}p(x^n)p(y^n)\log \frac{p(x^n)}{p(x^n||y^{n-1})}+H(X^n)+H(Y^n)\nonumber \\ &= L_1(Y^{n-1}\to X^{n})+H(X^n)+H(Y^n).\end{aligned}$$ Hence the penalty is $L_1(Y^{n-1}\to X^n)$. If both links are mistakenly assumed, the penalty[@Palomar_verdu08LautumInformation] is lautum information $L(X^n; Y^n)$. Note that the penalty due to wrongly assuming both links is the sum of the penalty from wrongly assuming each link. This is due to the conservation law (\[e\_cons\_lautum\]). Hypothesis Testing ------------------ We revisit the hypothesis testing problem in Section V, which is describe in Fig. \[f\_hypothesis\]. As a dual to Theorem 6, we characterize the minimum type I error exponent given the type II error probability: \[t\_Chernoff-Stel\_lautum\] Let $(X,Y) = \{X_i,Y_i\}_{i=1}^\infty$ be stationary, ergodic, and Markov of finite order such that $p(x^n,y^n) = 0$ implies $p(x^n||y^{n-1})p(y^n) = 0$. Let $A_n\subseteq {(\mathcal X\times \mathcal Y)}^n$ be an acceptance region, and let $\alpha_n$ and $\beta_n$ be the corresponding probabilities of type I and type II errors (\[e\_error\_def\]). For $0<\epsilon<\frac{1}{2}$, let $$\label{e_alpha_n} \alpha_n^{(\epsilon)}=\min_{A_n\subseteq {(\mathcal X\times \mathcal Y)}^n, \beta_n<\epsilon} \alpha_n.$$ Then $$\lim_{n\to\infty} -\frac{1}{n} \log \alpha_n^{(\epsilon)}=\mathcal L_2(X\to Y),$$ where the directed lautum information rate is the one induced by the joint distribution from $H_0$, i.e., $p(x^n||y^{n-1})p(y^n||x^n)$. The proof of Theorem \[t\_Chernoff-Stel\_lautum\] follows very similar steps as in Theorem \[t\_Chernoff-Stel\] upon letting $$A_n^c=\left\{x^n,y^n:\left|\log \frac{p(y^n)}{p(y^n||x^n)}-\mathcal L_2( X\to Y)\right|<\delta \right\},$$ analogously to (\[e\_A\_n\]), and upon using the Markov assumption for guaranteeing the AEP; hence we omit the details. Concluding Remarks ================== We have established the role of directed information in portfolio theory, data compression, and hypothesis testing. Put together with its key role in communications [@Kramer98; @Kramer03; @PermuterWeissmanGoldsmith09; @TatikondaMitter_IT09; @Kim07_feedback; @PermuterWeissmanChenMAC_IT09; @ShraderPemuter07ISIT; @Pradhan07Venkataramanan], and in estimation [@PermuterKimTsachy_ContinousDirectedConCom09], directed information is thus emerging as a key information theoretic entity in scenarios where causality and the arrow of time are crucial to the way a system operates. Among other things, these findings suggest that the estimation of directed information can be an effective diagnostic tool for inferring causal relationships and related properties in a wide array of problems. This direction is under current investigation [@Lei09directed_estimation]. In this paper we have seen that directed information is a fundamental measure in problems with causality constraints, in addition to feedback capacity and rate distortion with feed-forward. Directed information characterizes the increase in growth rate in a horse race gambling due to causal side information, and it upper bounds the growth rate in a stock market due to causal side information. It quantifies the value of [causal]{} side information in instantaneous compression and the benefit of [causal]{} inference in joint compression of two random processes. And it evaluates the best error exponent for testing whether a random process $Y$ [causally]{} influences another process $X$ or not. These results give a natural interpretation of directed information as a measure of causality between two random processes and it further motivates us to find additional mathematical properties of directed information and their applications in various fields. Acknowledgment {#acknowledgment .unnumbered} ============== The authors would like to thank Ioannis Kontoyiannis for helpful discussions. [10]{} C. E. Shannon. A mathematical theory of communication. , 27:379–423 and 623–656, 1948. C. E. Shannon. Coding theorems for a discrete source with fidelity criterion. In R. E. Machol, editor, [*Information and Decision Processes*]{}, pages 93–126. McGraw-Hill, 1960. R. G. Gallager. Source coding with side information and universal coding. unpublished manuscript, Sept. 1976. B. Y. Ryabko. Encoding a source with unknown but ordered probabilities. , pages 134–139, 1979. J. L. Kelly. A new interpretation of information rate. , 35:917–926, 1956. J. Massey. Causality, feedback and directed information. , pages 303–305, Nov. 1990. G. Kramer. . Ph.[D]{}. dissertation, Swiss Federal Institute of Technology (ETH) Zurich, 1998. G. Kramer. Capacity results for the discrete memoryless network. , IT-49:4–21, 2003. H. H. Permuter, T. Weissman, and A. J. Goldsmith. Finite state channels with time-invariant deterministic feedback. , 55(2):644–662, 2009. S. Tatikonda and S. Mitter. The capacity of channels with feedback. , 55:323–349, 2009. Y.-H. Kim. A coding theorem for a class of stationary channels with feedback. , 25:1488–1499, April, 2008. H. H. Permuter, T. Weissman, and J. Chen. Capacity region of the finite-state multiple access channel with and without feedback. , 55:2455–2477, 2009. B. Shrader and H. H. Permuter. On the compound finite state channel with feedback. In [*Proc. Internat. Symp. Inf. Theory*]{}, Nice, France, 2007. R. Venkataramanan and S. S. Pradhan. Source coding with feedforward: Rate-distortion theorems and error exponents for a general source. , IT-53:2154–2179, 2007. D. P. Palomar and S. Verd[ú]{}. Lautum information. , 54:964–975, 2008. J. Massey and P.C. Massey. Conservation of mutual and directed information. , pages 157–158, 2005. T. M. Cover and J. A. Thomas. . Wiley, New-York, 2nd edition, 2006. P. H Algoet and T. M. Cover. A sandwich proof of the [S]{}hannon-[McMillan]{}-[B]{}reiman theorem. , 16:899–909, 1988. A. R. Barron and T. M. Cover. A bound on the financial value of information. , IT-34:1097–1100, 1988. R. Gallager. Variations on a theme by huffman. , 24:668–674, 1978. H. H. Permuter, Y. H Kim, and T. Weissman. Directed information, causal estimation and communication in continuous tim. In [*Proc. Control over Communication Channels (ConCom)*]{}, Seoul, Korea., June, 2009. L. Zhao, T. Weissman, Y.-H Kim, and H. H. Permuter. Estimation of directed information. in preparation, Dec. 2009. [^1]: The work was supported by the NSF grant CCF-0729195 and BSF grant 2008402. Author’s emails: haimp@bgu.ac.il, yhk@ucsd.edu, and tsachy@stanford.edu. [^2]: [*Lautum* ]{}(“elegant" in Latin) is the reverse spelling of “mutual” as aptly coined in [@Palomar_verdu08LautumInformation].

--- abstract: 'In this paper we give a short, elementary proof of the following too extreme cases of the Leopoldt conjecture: the case when ${\mathbb{K}}/{\mathbb{Q}}$ is a solvable extension and the case when it is a totally real extension in which $p$ splits completely. The first proof uses Baker theory, the second class field theory. The methods used here are a sharpening of the ones presented at the SANT meeting in Göttingen, 2008 and exposed in [@Mi2], [@Mi1].' address: Mathematisches Institut der Universität Göttingen author: - Preda Mihăilescu bibliography: - 'leobak.bib' date: 'Version 1.0 ' title: Applications of Baker Theory to the Conjecture of Leopoldt --- *He will always carry on Some things are lost, some things are found, They will keep on speaking his name Some things are changed, some still the same.[^1]* To Alan Baker Introduction ============ Let ${\mathbb{K}}/{\mathbb{Q}}$ be a finite galois extension and $p$ be a rational prime. It was conjected by Leopoldt in [@Le] that the $p$ - adic regulator of ${\mathbb{K}}$ does not vanish. Some equivalent statements are explained below. The conjecture was proved for abelian extensions in 1967 by Brumer [@Br], using a local version of Baker’s linear forms in logarithms: the result is known as the Baker-Brumer theorem. A theorem proved by Ax in [@Ax] allows to relate the Leopoldt conjecture for abelian extensions to transcendency theory. In his paper, Ax mentions that he could expect his method to work also for non - abelian extensions. This was attempted by Emsalem and Kisilevski, who obtained in [@EK] results for some particular, non abelian extensions. The main result of this paper is \[main\] Let ${\mathbb{K}}/{\mathbb{Q}}$ be a galois extension and $p$ an odd prime. The Leopoldt conjecture holds for ${\mathbb{K}}$ and $p$ in the following cases: - The group ${\mbox{ Gal }}({\mathbb{K}}/{\mathbb{Q}})$ is solvable. - The extension ${\mathbb{K}}/{\mathbb{Q}}$ is totally real and splits $p$ completely. We state from [@Br] the central theorem on $p$ - adic forms in logarithms, which we shall use here: \[BB\] Let $\overline{{\mathbb{Q}}}_p$ be an algebraic closure of ${\mathbb{Q}}_p$ and ${\mathbb{U}}\subset \overline{{\mathbb{Q}}}_p$ be the units. Let $\alpha_1, \alpha_2, \ldots, \alpha_n$ be elements of ${\mathbb{U}}$ which are algebraic over ${\mathbb{Q}}$ and whose $p$ - adic logarithms exist and are independent over ${\mathbb{Q}}$. These logarithms are then independent over ${\mathbb{Q}}'$, the algebraic closure of ${\mathbb{Q}}$ in $\overline{{\mathbb{Q}}}_p$. Baker theory and Leopoldt’s conjecture ====================================== Let ${\mathbb{K}}/{\mathbb{Q}}$ be an arbitrary galois field with group $G$, let $p$ be a rational prime and $P =\{ \wp \subset {\mathcal{O}}({\mathbb{K}}) \ : \ ( p ) \subset \wp \}$ be the set of conjugate prime ideals above $p$ in ${\mathbb{K}}$. We shall prove in this section two important consequences of the Theorem \[BB\], one for absolute and one for relative galois extensions. The algebra ${\mathfrak{K}}({\mathbb{K}}) = {\mathbb{K}}\otimes_{{\mathbb{Q}}} {\mathbb{Q}}_p$ is the product of all completions of ${\mathbb{K}}$ at the places in $P$: $${\mathbb{K}}_p = \prod_{\wp \in P} {\mathbb{K}}_{\wp} .$$ The global field ${\mathbb{K}}$ is dense in ${\mathbb{K}}_{\wp}$ in the product topology and $G$ acts on this completion faithfully, so for any $x \in {\mathbb{K}}_p, x = \lim_n x_n, x_n \in {\mathbb{K}}$ and for all $g \in G$ we have $g ( x ) = \lim_n g ( x_n )$. The field ${\mathbb{K}}$ is dense in ${\mathfrak{K}}$ (under the product of the topologies of ${\mathbb{K}}_{\wp}$), the units $U \subset {\mathbb{K}}_p$ are products of the units in $U_{\wp} \subset {\mathbb{K}}_{\wp}$ and $E$ embeds diagonally to $\overline{E} \subset U$. We denote by $\iota_{\wp}$ the natural map ${\mathfrak{K}} \ra {\mathbb{K}}_{\wp}$ and identify $a \in {\mathbb{K}}$ with its image in ${\mathfrak{K}}$. The one - units $U^{(1)}$ are defined naturally as the product $U^{(1)} = \prod_{\wp \in P} U^{(1)}_{\wp}$, with $U^{(1)}_{\wp} = \{ x \in U_{\wp} : x - 1 \in {\mathcal{M}}_{\wp}\}$ and ${\mathcal{M}}_{\wp}$ the maximal ideal of $U_{\wp}$. Let ${\mathbb{M}}/{\mathbb{K}}$ be the product of all ${\mathbb{Z}}_p$ - extensions and $\Delta = {\mbox{ Gal }}({\mathbb{M}}/{\mathbb{K}})$. The global Artin symbol is an homomorphism $\varphi: U^{(1)} \ra \Delta$ of ${\mathbb{Z}}_p[ G ]$ - modules and there is an exact sequence $$\begin{aligned} \label{glart} 1 \ra \overline{E} \ra U^{(1)} \ra \Delta \ra 1.\end{aligned}$$ We let $U' = \{x \in U^{(1)} : {\mbox{\bf N}}_{{\mathbb{K}}/{\mathbb{Q}}}(x) = 1\}$; then $\overline{E} \subset U'$ by definition. Therefore $U^{(1)}/U' \cong U^{(1)} \cap {\mathbb{Q}}_p = U^{(1)}({\mathbb{Q}}_p)$ is a quotient which is fixed by $G$. It is known that $\td{U^{(1)}} \cong {\mathbb{Q}}_p[ G ]$, so $\Delta$ is a quasi-cyclic ${\mathbb{Z}}_p[ G ]$ (see also [@Mi]). Since ${\mathbb{K}}_{\infty}/{\mathbb{K}})$ is a ${\mathbb{Z}}_p$ - extension whose galois group is a quotient of $\Delta$, invariant under $G$; but $\Delta$ being ${\mathbb{Z}}_p[ G ]$ - cyclic, it follows that this quotient is unique up to quasi-isomorphism. We let $\Delta' = {\mbox{ Gal }}({\mathbb{M}}/{\mathbb{K}}_{\infty})$; we then have the additional exact sequence $$\begin{aligned} \label{glartinf} 1 \ra \overline{E} \ra U' \ra \Delta' \ra 1.\end{aligned}$$ We refer to [@Mi], §§2.1, 2.2 and 3.1 for more details on Minkowski units, idempotents of non commutative group rings and the associated annihilators, supports and components of ${\mathbb{Z}}_p[ G ]$ - modules. We also refer to §2.3 for the description of a choice of the base field ${\mathbb{K}}$, which contains the [$p^{\kappa} {\rm - th }$]{} roots of unity and has some pleasant properties, such as the fact that the $p$ - ranks of all $\Lambda$ - modules of finite rank are stationary, all ideals that capitulate have order bounded by $p^{\kappa}$ and $v_p(| G |) \leq \kappa$. In the same section we describe Weierstrass modules – which are ${\mathbb{Z}}_p$ - torsion free, infinite $\Lambda$ - modules of finite $p$ - rank – and prove the fundamental formula $${\hbox{ord} }(a_n) = p^{n+1+z(a)} \quad \forall n > 0,$$ which characterizes the orders of $a = (a_n)_{n \in {\mathbb{N}}} \in W \subset A$, when $W$ is Weierstrass. Here ${\mathbb{Z}}\owns z(a) \leq \kappa$ is a constant depending on $a$ but not on $a_n$. We use the notation $\varsigma(x) = x^{p^{\kappa}}$ for $x$ in an abelian group; the choice of $\varsigma$ is such that $\varsigma(A)$ is a Weierstrass module and for $a \in \underline{A}$, the finite $p$ - torsion part of $A$, we have $\varsigma(a) = 1$. We write ${\mathbb{H}}, \Omega$ for the maximal $p$ - abelian, unramified, respectively $p$ - ramified extensions of ${\mathbb{K}}_{\infty}$. If ${\mathbb{F}}/{\mathbb{K}}_{\infty}$ is any extensions and $F_0 = {\mbox{ Gal }}({\mathbb{F}}/{\mathbb{K}}_{\infty})^{\circ}$ is the ${\mathbb{Z}}_p$ - torsion of its galois group, we write $\overline{{\mathbb{F}}} = {\mathbb{F}}^{F_0}$: an extension which is either trivial or has a Weierstrass - module as galois group; this group may still be a free $\Lambda$ - module. The conjecture of Leopoldt says that $${{\mathbb{Z}}_p\hbox{-rk}}( \overline{E} ) = {{\mathbb{Z}}\hbox{-rk} }( E ) .$$ Let $\delta \in E$ be a Minkowski unit with $\delta \equiv 1 \bmod p^2$. Then the $p$ - adic logarithms of $\delta^g$ exist in all completions ${\mathbb{K}}_{\wp}$ and for all $g \in G$. If $A \subset {\mathbb{K}}_p$ is a multiplicative group, we write the action of $G$ exponentially, so $a^g = g(a)$. If $G$ is not commutative and $g, h \in G$ we have $$\begin{aligned} \label{CV} a^{g h} = \left(a^g\right)^h = h \circ g ( a ), \end{aligned}$$ and the definition of a contravariant multiplication $G \times G \ra G$ with $g \cdot h = h \circ g$ makes $A$ into a right ${\mathbb{Z}}_p[ G ]$ - module, and likewise for ${\mathbb{Z}}[ G ]$ - modules. In particular, $U, \overline{E}$ and are ${\mathbb{Z}}_p[ G ]$ - modules and Minkowski units generate submodules of maximal ${\mathbb{Z}}_p$ - rank: since ${\mathbb{K}}$ is dense in ${\mathbb{K}}_p$, it follows that $ {{\mathbb{Z}}_p\hbox{-rk}}( \overline{ E } ) = {{\mathbb{Z}}_p\hbox{-rk}}( \delta^{{\mathbb{Z}}_p[ G ]})$. With this structure we also define $$\begin{aligned} \delta^{\top} = \{ x \in {\mathbb{Z}}[ G ] \ : \ \delta^x = 1 \}, \quad \delta_p^{\top} = \{ x \in {\mathbb{Z}}_p[ G ] \ : \ \delta^x = 1 \}, \end{aligned}$$ the ${\mathbb{Z}}$ - and ${\mathbb{Z}}_p$ annihilators of $\delta$. Then Leopoldt’s conjecture is also equivalent to $$\begin{aligned} \label{bul} \delta_p^{\top} = \delta^{\top} \otimes_{{\mathbb{Z}}} {\mathbb{Z}}_p.\end{aligned}$$ In the context of this conjecture we are interested in ranks and not in torsion of modules over rings. It it thus a useful simplification to tensor these modules with fields, so we introduce the following \[tens\] Let $G$ be a finite group and $A, B$ a ${\mathbb{Z}}$, respectively a ${\mathbb{Z}}_p$ - module, which are torsion free. Let $a \in A, b \in B$. We denote $$\begin{aligned} \hat{A} & = & A \otimes_{{\mathbb{Z}}} {\mathbb{Q}}, \quad \hat{a} = a \otimes 1, \\ \td{B} & = & B \otimes_{{\mathbb{Z}}_p} {\mathbb{Q}}_p, \quad \td{b} = a \otimes 1, \end{aligned}$$ Note that ${{\mathbb{Z}}\hbox{-rk} }(A) = {{\mathbb{Q}}\hbox{-rk} }(\hat{A})$ and ${{\mathbb{Z}}_p\hbox{-rk}}(B) = {{\mathbb{Q}}_p\hbox{-rk} }(\td{B})$. We shall simply write ${{\rm rank } \ }(X)$ for the rank of a module when the ring of definition is clear (being one of ${\mathbb{Z}}, {\mathbb{Z}}_p$ or ${\mathbb{Q}}, {\mathbb{Q}}_p$.) For instance, $\td{E} = \overline{E} \otimes_{{\mathbb{Z}}_p} {\mathbb{Q}}_p$. The definition of $\hat{E}$ is not important for absolute extensions, but relevant in relative extensions ${\mathbb{L}}/{\mathbb{K}}$, when ${\mbox{\bf N}}_{{\mathbb{L}}/{\mathbb{K}}}(E({\mathbb{L}})) \subsetneq E({\mathbb{K}})$. We start with the case of an absolute extension ${\mathbb{K}}/{\mathbb{Q}}$, as introduced above. Let $r = r_1 + r_2 - 1 = {{\mathbb{Z}}\hbox{-rk} }(E)$ and $H = \{ g_1, g_2, \ldots, g_r \} \subset G \setminus \{ 1 \}$ be a maximal set of automorphisms, such that $\delta^{g_i}$ are ${\mathbb{Z}}$ - independent. In particular, there is a ${\mathbb{Z}}$ - linear map $e : {\mathbb{Z}}[ G ] \ra {\mathbb{Z}}[ H ]$ such that $$\begin{aligned} \label{rat} \delta^{\sigma} = \delta^{e(\sigma)}\end{aligned}$$ for each $\sigma \in G$. The map is the identity on $H$ and extends to $G$ due to the Minkowski property, which implies that $\delta^{{\mathbb{Z}}[ H ]} = \delta^{{\mathbb{Z}}[ G ]}$. We have the following consequence of Theorem \[BB\] \[baker-abs\] Let the notations be like above and ${\mathbb{Z}}' = {\mathbb{Q}}' \cap {\mathbb{Z}}_p$ be the integers in the algebraic closure ${\mathbb{Q}}' \subset {\mathbb{Q}}_p$ of ${\mathbb{Q}}$. Then $$\delta_p^{\top} \cap {\mathbb{Z}}'[ G ] = \delta^{\top}.$$ In particular, if $\delta_p^{\top} = \alpha {\mathbb{Z}}_p[ G ]$ with $\alpha \in {\mathbb{Z}}'[ G ]$, then Leopoldt’s conjecture holds for ${\mathbb{K}}$. Let $\wp \in P$ be fixed and $\delta_{\tau} = \iota_{\wp}(\delta^{\tau})$; then $\delta_{\tau} \in {\mathbb{Z}}'$. Since $\{ \delta^{\tau} : \tau \in H\}$ are ${\mathbb{Z}}$ - independent, $\{ \delta_{\tau} : \tau \in H\}$ are a fortiori ${\mathbb{Z}}$ - independent. Indeed, if $t \in {\mathbb{Z}}[ H ]$ was a linear dependence for $\delta_{\tau}$, such that $\iota_{\wp}(\delta^t) = 1$, then $d = \delta^t \in E$ verifies $\iota_{\wp}(d) = 1$. But in the diagonal embedding of $E$, a projection is $1$ if and only if the unit itself is $1$, thus $d = 1$: a contradiction of the independence of $\delta^{\tau}, \tau \in H$. Let $\theta_0 \in \delta_p^{\top} \cap {\mathbb{Z}}'[ G ]$; in view of [(\[rat\])]{}, $\theta = e(\theta_0) \in \delta_p^{\top} \cap {\mathbb{Z}}'[ H ]$ is also an annihilator. Let $\theta = \sum_{\tau \in H} c_{\tau} \tau, \ c_{\tau} \in {\mathbb{Z}}'$. We show that Theorem \[BB\] implies $\theta = 0$, so $\theta_0 \in e^{-1}(0) \subset {\mathbb{Z}}[ G ]$ for all $\theta_0 \in \delta_p^{\top} \cap {\mathbb{Z}}'[ G ]$, which is the claim. We have $\iota_{\wp}(\delta^{\theta}) = \prod_{\tau \in H} \delta_{\tau}^{c_{\tau}} = 1 \in {\mathbb{K}}_{\wp}, $ and taking the $p$ - adic logarithm we find the vanishing linear form in logarithms $$\sum_{\tau \in H} c_{\tau} \log_p(\delta_{\tau}) = 0.$$ Since $c_{\tau}, \delta_{\tau} \in {\mathbb{Z}}'$ and $\{ \delta_{\tau} : \tau \in H\}$ are ${\mathbb{Z}}$ - independent, the Theorem of Baker and Brumer implies that $\theta = 0$. Consequently, if $\delta_p^{\top} = \theta_0 {\mathbb{Z}}_p[ G ]$ and $\theta_0 \in {\mathbb{Z}}'[ G ]$, then the proof above shows that $\theta_0 \in {\mathbb{Z}}[ G ]$, which implies [(\[bul\])]{} and confirms Leopoldt’s conjecture. The following definition brings relative annihilators into focus. \[rel\] Let ${\mathbb{L}}\supset {\mathbb{K}}$ be an extension of number fields with the following properties: - ${\mathbb{L}}/{\mathbb{Q}}$ is a galois extension with group $G$ and $H = {\mbox{ Gal }}({\mathbb{L}}/{\mathbb{K}})$. - Let the *relative annihilator* of $e \in E({\mathbb{L}})$ be defined by $$\begin{aligned} \td{e}_{{\mathbb{L}}/{\mathbb{K}}}^{\top} & = & \{ x \in {\mathbb{Q}}_p[ H ] : \td{e}^x \in \td{E({\mathbb{K}})}\}, \\ e_{{\mathbb{L}}/{\mathbb{K}}}^{\top} & = & \td{e}_{{\mathbb{L}}/{\mathbb{K}}}^{\top} \cap {\mathbb{Z}}_p[ H ].\end{aligned}$$ Then for any global Minkowski unit $\delta \in E({\mathbb{L}})$ we have $$\td{\delta}_{{\mathbb{L}}/{\mathbb{K}}}^{\top} = {\mbox{\bf N}}_{{\mathbb{L}}/{\mathbb{K}}} \cdot {\mathbb{Q}}_p[ H ].$$ If points 1. and 2. hold for ${\mathbb{L}}/{\mathbb{K}}$, we say that ${\mathbb{L}}/{\mathbb{K}}$ is *relative Leopoldt extension, or rL - extension*. If in addition ${\mathbb{L}}$ is real, then the extension is a real relative Leopoldt, or RL. Bruno Anglès observed in connection with earlier attempts to use Baker theory, that these attempts suggest an approach using relative extensions, maybe even a *relative Leopoldt conjecture*, stating that if ${\mathbb{L}}/{\mathbb{K}}$ is RL and Leopoldt’s conjecture holds for ${\mathbb{K}}$, then it holds for ${\mathbb{L}}$. The use of relative annihilators leads to a proof of statement A. in Theorem \[main\]. However the relative conjecture encouters a severe obstruction due to the fact that Baker theory only allows statements on the relative annihilator in ${\mathbb{Q}}_p[ H ]$, but the relative conjecture requires annihilators in ${\mathbb{Q}}_p[ G ]$. We consider next the case of relative abelian extensions: \[baker-rel\] Abelian extensions ${\mathbb{L}}/{\mathbb{K}}$ with ${\mathbb{L}}/{\mathbb{Q}}$ galois are relative Leopoldt extensions. Furthermore, if ${\mathbb{L}}/{\mathbb{K}}$ is galois such that ${\mathbb{L}}_{\wp}/{\mathbb{K}}_{\wp}$ is abelian for all prime ideals $\wp \in P$, then ${\mathbb{L}}_{\wp}/{\mathbb{K}}_{\wp}$ is a local relative Leopoldt extension with respect to $\iota_{\wp}(e)$ for all Minkowski units $e \in E({\mathbb{L}})$. Let $H = {\mbox{ Gal }}({\mathbb{L}}/{\mathbb{K}})$ be abelian; the extension ${\mathbb{L}}/{\mathbb{K}}$ arises from a succession of cyclic extensions of prime degree, so it suffices to assume this case. Let $H = {\langle}\sigma {\rangle}$ with $| H | = [ {\mathbb{L}}: {\mathbb{K}}] = q$, for a prime $q$ which is not necessarily different from $p$. The group ${\mathbb{Q}}_p[ H ]$ decomposes as ${\mathbb{Q}}_p[ H ] = e_1 {\mathbb{Q}}_p[ H ] \oplus (1-e_1) {\mathbb{Q}}_p[ H ]$, where $e_1$ is the idempotent $\frac{N}{q}$, with $N ={\mbox{\bf N}}_{{\mathbb{L}}/{\mathbb{K}}}$ . Suppose thus that $\td{\delta}_{{\mathbb{L}}/{\mathbb{K}}}^{\top} = (a e_1 + b e_{\chi}) {\mathbb{Q}}_p[ H ]$, where $e_{\chi}$ is a (non trivial) sum of central idempotents for the augmentation part ${\mathbb{Q}}_p[ I_{{\mathbb{L}}/{\mathbb{K}}} ]$ and $a, b \in \{0, 1\}$. We shall show that $a = 1$ and $b = 0$. From the definition of $\td{\delta}_{{\mathbb{L}}/{\mathbb{K}}}^{\top}$ we have $$\td{\delta}^{a e_1 + b e_2} = N(\td{\delta})^a \cdot \td{\delta}^{b e_2} \in \td{E}({\mathbb{K}}).$$ Since $N(\td{\delta}) \in \td{E}({\mathbb{K}})$, we also have $d := \td{\delta}^{b e_2} \in \td{E}({\mathbb{K}})$. The group $H$ is cyclic and $e_2$ is in the augmentation, so $e_2 N = 0$. Taking the norm in the definition of $d$ and using the fact that $d^{\sigma} = d$ and thus $N(d) = d^q$, we find that $$\td{\delta}^{b e_2 N } = d^q = \td{\delta}^{b e_2 q } = 1.$$ But $e_2 q \in {\mathbb{Z}}_p[ H ]$ and thus $\delta^{b e_2 q} = 1$: starting from a relative relation we deduced an absolute annihilator of $\delta$ which is algebraic. We may apply the Lemma \[baker-abs\], concluding that $e_2 = 0$, since by hypothesis there is no rational dependence for $\delta$ in the augmentation. This completes the proof. As a consequence, we have \[relsolv\] Solvable extensions ${\mathbb{L}}/{\mathbb{K}}$ with ${\mathbb{L}}/{\mathbb{Q}}$ real and galois are RL - extensions. Since $H$ is solvable, there is a chain of intermediate extensions ${\mathbb{K}}_0 = {\mathbb{K}}\subset {\mathbb{K}}_1 \subset {\mathbb{K}}_2 \subset \ldots \subset {\mathbb{K}}_r = {\mathbb{L}}$ such that ${\mathbb{K}}_{i+1}/{\mathbb{K}}_i$ is abelian for $i = 0, 1, \ldots, r-1$ and ${\mathbb{L}}/{\mathbb{K}}_i$ is solvable for all $i$. The Lemma \[baker-rel\] holds for all ${\mathbb{K}}_{i+1}/{\mathbb{K}}_i$. Let $N_i = \sum_{\sigma \in {\hbox{\tiny Gal}}({\mathbb{K}}_{i+1}/{\mathbb{K}}_i)} \sigma$; then $N = N_0 \circ N_1 \circ \ldots \circ N_{r-1}$. The claim follows by induction and we illustrate this for the case $r = 2$, so ${\mbox{ Gal }}({\mathbb{L}}/{\mathbb{K}}_1) = H_1, {\mbox{ Gal }}({\mathbb{K}}_1/{\mathbb{K}}) = H_0$ and $H = H_0 \ltimes H_1$. Furthermore, ${\mathbb{Q}}_p[ H ] = {\mathbb{Q}}_p[ H_0 ] \ltimes{\mathbb{Q}}_p[ H_1 ]$ where the semidirect product $a_0 \ltimes a_1$, with $a_i \in {\mathbb{Q}}_p[ H_i ], i = 0, 1$ is defined term-wise; $N = N_0 \ltimes N_1$ follows from this definition. We know from the lemma that $\td{\delta}_{{\mathbb{L}}/{\mathbb{K}}_1}^{\top} = N_1 {\mathbb{Q}}_p[ H_1 ]$ and letting $\delta_1 = N_1(\delta) \in {\mathbb{K}}_1$, the same lemma yields $\td{\delta_1}_{{\mathbb{K}}_1/{\mathbb{K}}}^{\top} = N_0 {\mathbb{Q}}_p[ H_0 ]$. It follows that $\td{\delta}_{{\mathbb{L}}/{\mathbb{K}}}^{\top} \subset \td{\delta_1}_{{\mathbb{K}}_1/{\mathbb{K}}}^{\top} \ltimes N_1 {\mathbb{Q}}_p[ H_1 ] = N_0 {\mathbb{Q}}_p[ H_0 ] \ltimes N_1 {\mathbb{Q}}_p[ H_1 ] = N {\mathbb{Q}}_p[ H ]$. This way we may prove inductively that ${\mathbb{L}}/{\mathbb{K}}_i$ is RL for $i = r-2, r-3, \ldots, 0$. The last lemma leads to: \[A\] Point A. in Theorem \[main\] is true. Suppose that ${\mathbb{L}}/{\mathbb{Q}}$ is a real extension with solvable group $H$. If $\delta \in E({\mathbb{L}})$ is a Minkowski unit, then Lemma \[relsolv\] implies that its relative annihilator $\td{\delta}_{{\mathbb{L}}/{\mathbb{Q}}}^{\top} = {\mbox{\bf N}}_{{\mathbb{L}}/{\mathbb{Q}}} {\mathbb{Q}}_p[ H ]$. Since the base field is ${\mathbb{Q}}$, the relative annihilator is equal to the absolute one and it follows that ${{\mathbb{Z}}_p\hbox{-rk}}(\overline{E}) = {{\mathbb{Z}}\hbox{-rk} }(E)$ and Leopoldt’s conjecture is true. The restriction that ${\mathbb{L}}$ be real is not important. If ${\mathbb{L}}$ is complex, with solvable group, then $< \jmath > = {\mbox{ Gal }}({\mathbb{L}}/{\mathbb{L}}^+) \subset H$ is a normal subgroup, so ${\mathbb{K}}^+/{\mathbb{Q}}$ is galois, and Leopoldt’s conjecture holds in this case too. We are prepared to prove \[lmain\] Let ${\mathbb{K}}/{\mathbb{Q}}$ be a totally real extension with group $G$ and ${\mathbb{M}}/{\mathbb{K}}$ be the product of all ${\mathbb{Z}}_p$ extensions of ${\mathbb{K}}$, ${\mathbb{K}}_{\infty}$ the cyclotomic ${\mathbb{Z}}_p$ - extension of ${\mathbb{K}}$ and ${\mathbb{H}}/{\mathbb{K}}_{\infty}$ be the maximal $p$ - abelian $p$ - unramified extension. Then ${\mathbb{H}}\cap {\mathbb{M}}= {\mathbb{K}}_{\infty}$. We adopt a class field theoretic approach for our proof. Let ${\mathbb{K}}$ be like in the hypothesis and $({\mathbb{K}}_n)_{n \in {\mathbb{N}}}$ be the intermediate fields of its cyclotomic ${\mathbb{Z}}_p$ - extension. We assume that ${\mathbb{K}}$ is such that the Leopoldt defect ${\mathcal{D}}({\mathbb{K}}_n) = {\mathcal{D}}({\mathbb{K}})$ is constant for all $n \geq 0$ and let ${\mathbb{L}}= {\mathbb{K}}[\zeta_p]$. Firstly, we note that we can exclude the case that ${\mathbb{M}}/{\mathbb{K}}_{\infty}$ contains subextensions which split the primes above $p$: this follows by using the Iwasawa skew symmetric pairing and was proved in [@Mi], Theorem 3 of §3.3. It remains that ${\mathbb{M}}\subset \Omega_E$ and ${\mbox{ Gal }}({\mathbb{M}}/{\mathbb{K}}_{\infty}) \hookrightarrow {\mbox{\bf B}}$, where ${\mbox{\bf B}} \subset A$ is the submodule generated by classes containing ramified primes above $p$, as defined in [@Mi]. Let $E_n = E({\mathbb{K}}_n)$ and $U'_n = U'_n({\mathbb{K}}_n)$; we define $U'_{\infty} = \cup_n U'_n$ and $E_{\infty} = \cup_n E_n$. Then it is known that $U'_{\infty}/E_{\infty}$ is a torsion $\Lambda$ - module and thus, by choice of $\varsigma$, we obtain a Weierstrass module $\varsigma(U'_{\infty}/E_{\infty})$; since ${\mathbb{K}}$ is totally real, by reflection we see that $$\varsigma(U'_{\infty}/E_{\infty})^{\bullet} \hookrightarrow A^-({\mathbb{L}}).$$ Let $X''_n = \{ x \in X_n : N_{{\mathbb{K}}_n,{\mathbb{K}}}(x) = 1, n > \kappa\}$ for $X \in \{U'_n, E_n\}$. We let $W_n = \varsigma(U''_{n}/\overline{E}''_{n})$ and $W = \varsigma(U''_{\infty}/\overline{E}''_{\infty})$. Then $W$ is a Weierstrass module and we denote by $F$ its characteristic polynomial. Since the $T$ - part of $W$ is trivial, $T \nmid F(T)$. The Leopoldt defect is stationarity, so $\td{E}_n'' = (\td{U}_n'')^+$. Applying $F$ annihilates the diverging part in $W_n$ and we obtain: $$\begin{aligned} \label{mb} \left[ \left((U''_n)^+\right)^{F(T)} : (E''_n)^{F(T)} \right] < M, \end{aligned}$$ for a fixed upper bound $M$. In particular, since $(T, F(T)) = 1$, there is a fixed $m \geq \kappa$, depending on $F(T)$ and $| G |$, such that for all $n > 0$ we have $$\begin{aligned} \label{ucyc} ((U''_n)^+)^{p^m} \subset E''_n \cdot {U''_n}^{T}. \end{aligned}$$ Note that $E_n'' \cap {U_n''}^T = {E_n''}^T$; indeed, let $e \in E_n'' \cap {U_n''}^T$. By Hilbert 90 and the choice of $e$, there are a $w \in {\mathbb{K}}_n^{\times}$ and $\xi \in U_n''$ with $e = w^T = \xi^T$, and it follows that $w = w_0 \cdot \xi$ for some $w_0 \in U({\mathbb{K}})$. The ideal $(w)$ is ambig above ${\mathbb{K}}$; since $p^{\kappa}$ annihilates the class group $A_0$, we have $(w^{p^{\kappa}}) = (\pi \cdot \gamma)$, with $\gamma \in {\mathbb{K}}, (\gamma, p) = 1$ and $\pi$ a product of ideals above $p$. There is a unit $e_1 \in E_n$ such that $e_1 \pi \gamma = w^{p^{\kappa}} = (w_0 \xi)^{p^{\kappa}}$ and since $\xi$ is a local unit, it follows that $\pi | w_0$ or $\pi = 1$; moreover, $(\pi \gamma)^T = \gamma^T = 1$. It remains that $e^{p^{\kappa}} = w^{T p^{\kappa}} = e_1^T$, so $e \in E_n^T \cap E_n''$. Finally we show that we may choose $e_1 \in E_n''$. From $(\xi/e_1)^T = 1$ we have $e_1 = c \xi, c \in {\mathbb{K}}$ and $N_{n,0}(e_1) = c^{p^{n-\kappa}}$, so $c \in E({\mathbb{K}})$; the unit $e_2 := e_1/c \in E''_n$ verifies $e_2^T = e$, which confirms this claim. In particular, for sufficiently large $n$, we have $$\begin{aligned} \label{perk} \quad \quad {p\hbox{-rk} }\left(E''_n/\left({E''}_n^{p^n} \cdot ({U''_n}^T \cap E''_n) \right)\right) & = & {p\hbox{-rk} }\left(E''_n/{E''}_n^{(p^n, T)}\right) \\ & = & r_2 - 1.\nonumber\end{aligned}$$ Assume that ${\mbox{\bf B}}$ is infinite, and let $\alpha^{\top} \in {\mathbb{Q}}_p[ G ]$ be its canonic annihilator[^2]. We shall write ${{\mathbb{Z}}_p\hbox{-rk}}(\varsigma({\mbox{\bf B}})) = {\mathcal{D}}({\mathbb{K}})$: this is the case for the totally real extensions ${\mathbb{K}}$ which split the primes above $p$, as we show in the corollary below; note however that the claim of this proposition is more general and holds independently of this assumption. It does follow from the general case of Leopoldt’s conjecture too. We shall use [(\[ucyc\])]{} and [(\[perk\])]{} for proving that ${\mathcal{D}}({\mathbb{K}}) = 0$. The core observation is that, if ${\mathcal{D}}({\mathbb{K}}) > 0$, then there is a defect emerging in the ${\mathbb{Z}}$ - rank of $E''_n$, which raises a contradiction to [(\[perk\])]{}. For $\wp \in P$, we let $\wp_n \in A_n$ be the primes above $\wp$ and $a_n = [\wp_n] \in A_n$ be their classes, with diverging orders ${\hbox{ord} }(a_n) = p^{n+1+z(a)}$. If $\alpha_n$ approximates $| G | \alpha$ to the power $p^{n+\kappa+1}$, say, then there is a $\nu_n \in {\mathbb{K}}_n$ such that $(\nu_n) = \wp_n^{p^{\kappa} \cdot \alpha_n}$ and $\nu_n^T = e_0 \in E''_n$. Since $\nu_n^{| G | (1-\alpha_n)} \in ({\mathbb{K}}_n^{\times})^{p^n}$, it follows that the unit $e_0$ is annihilated in $E''_n/{E''}_n^{(T, p^n)}$ by $|G|(1-\alpha_n)$. We shall show that ${p\hbox{-rk} }\left(E''_n/{E''}_n^{T, p^n} \right) = r_2 - 1 - {\mathcal{D}}({\mathbb{K}})$, so [(\[perk\])]{} implies ${\mathcal{D}}({\mathbb{K}}) = 0$. Let $B \subset {\mbox{\bf B}}^{\top}$ be an irreducible elementary module with $\td{B}^{\top} = \beta \in {\mathbb{Q}}_p[ G ]$, an idempotent dividing $\alpha$, and let $\beta_n$ be rational approximants of $| G | \cdot \beta$. Suppose that there is a unit $e \in E''_n$ with $e^{\beta_n} \in \nu_n^{T {\mathbb{Z}}[ G ]}$; since $N_n(e) = 1$, it follows from [@Mi], Lemma 16, that $\varsigma(e) = \pi^T$ for some $p$ - unit $\pi$, so there is a $\theta \in {\mathbb{Z}}[ G ]$, with $(\pi) = \wp_n^{\theta}$. We may write $$| G | \theta \equiv c \alpha_n + b(1-\alpha_n) \ \bmod p^{n} {\mathbb{Z}}[ G ]; \ c, b \in p^{\kappa} {\mathbb{Z}}[ G ],$$ and claim that $b \equiv 0$ modulo a large power of $p$. Upon multiplication with $| G |(1-\alpha_n)$ we obtain a unit $e_1 = e^{| G |(1-\alpha_n)} = \pi_1^T$ with $(\pi_1) = \wp_n^{b| G |(1-\alpha_n) + O(p^{n})}$. The identity requires that the ideal $\wp_n^{b| G |(1-\alpha_n)}$ be principal and since $\alpha$ is the minimal annihilator of $a$, for large $n$, this implies $b \equiv 0 \bmod p^{n-(m+\kappa)}$, which was our claim on $b$. It follows that $\beta_n \in \alpha_n {\mathbb{Z}}[ G ] + p^{n-(m+\kappa)} {\mathbb{Z}}[ G ]$. But then, for $m' = 2(m+\kappa)$ and $n > m'$, the quotient $(E_n'')/((E_n'')^T \cdot (E_n'')^{p^{n-m'}})$ has $p$ - rank $r_2 - 1 - {\mathcal{D}}({\mathbb{K}})$ and [(\[perk\])]{} implies ${\mathcal{D}}({\mathbb{K}}) = 0$, which completes this proof. As a consequence, \[B\] Point B. in Theorem \[main\] is true. Assume that ${\mathbb{K}}/{\mathbb{Q}}$ is totally real and splits $p$ completely and let $\wp \in P$ be any prime above $p$. Then ${\mathbb{K}}_{\wp} = {\mathbb{Q}}_p$ and ${\mathbb{M}}_{\wp}/{\mathbb{K}}_{\wp}$ is in the product of the two ${\mathbb{Z}}_p$ - extensions of ${\mathbb{Q}}_p$: the unramified and the cyclotomic. Consequently ${\mathbb{M}}/{\mathbb{K}}_{\infty}$ must be unramified at $p$. This holds for all primes above $p$, so ${\mathbb{M}}/{\mathbb{K}}_{\infty}$ is totally unramified. However, by Theorem \[lmain\] we know that ${\mathbb{M}}\cap {\mathbb{H}}= {\mathbb{K}}_{\infty}$, so we must have ${\mathbb{M}}= {\mathbb{K}}_{\infty}$ and the Leopoldt conjecture holds in this case. The same argument was used by Greenberg in [@Gr] for showing that $\lambda$ may take arbitrarily large values in abelian extenions ${\mathbb{K}}/{\mathbb{Q}}$. It has been believed for a longer time that the two extreme cases treated by Theorem \[main\] are the easire one for Lepoldt’s conjecture, so this short proof only confirms this general belief. The general proof, given in [@Mi], requires deeper class field theory. The obstruction encountered, when trying to generalize the results of this paper is the following: let ${\mathbb{K}}/{\mathbb{Q}}$ have group $G$ and $\wp \in P$. The facts proven on relative annihilators imply quite easily that $\iota_{\wp}(\overline{E} \cdot {\mathbb{Q}}_p = {\mathbb{K}}_{\wp}$. Let $\Delta_{\wp} \subset \Delta = {\mbox{ Gal }}({\mathbb{M}}/{\mathbb{K}})$; Leopoldt’s conjecture would follow from Theorem \[lmain\], if we can prove that $$\Delta_{\wp} \cong U^{(1)}_{\wp}/\iota_{\wp}(\overline{E}),$$ which may appear as a ’reasonable’ localization of [(\[glart\])]{}. It needs however not be true and all we can say is the following: if $D \subset G$ is the decomposition group of $\wp$, $\td{\delta} = \td{\xi}^{\alpha}$ and $\alpha = \sum_{\tau \in C} c_{\tau} \tau$ with $c_{\tau} \in {\mathbb{Q}}_p[ D ]$ and $C = G/D$, then $\sum_{\tau \in C} c_{\tau} \in {\mbox{\bf N}}_{{\mathbb{M}}_{\wp}/{\mathbb{K}}_{\wp}}$. **Acknowledgments:** Much of the material presented here was completed after a two day visit of intensive work at the Laboratoire de Mathématique Nicolas Oresme of the University of Caen. I am most grateful to Bruno Anglés and David Vauclair for the helfpul and stimulating discussions which had an important contribution for clarifying the central ideas of these two papers. [^1]: From a Hymn of *Pretenders* [^2]: see [@Mi] §7.1 for a definition of canonic annihilators for cyclic modules over non - abelian group rings

--- abstract: 'Mysterious Fast Radio Bursts (FRB), still eluding a rational explanation, are astronomical radio flashes with durations of milliseconds. They are thought to be of an extragalactic origin, with luminosities orders of magnitude larger than any known short timescale radio transients. Numerous models have been proposed in order to explain these powerful and brief outbursts but none of them is commonly accepted, it is not clear which of these scenarios might account for real FRB. The crucial question that remains unanswered is: what makes FRB so exceptionally powerful and so exceptionally rare?! If the bursts are related with something happening with a star-scale object and its immediate neighborhood, why all detected FRB events take place in very distant galaxies and not in our own galaxy!? In this paper we argue that the non-linear phenomenon - self-trapping - which may provide efficient but rarely occurring beaming of radio emission towards an observer, coupled with another, also rare but powerful phenomenon providing the initial radio emission, may account for the ultra-rare appearance of FRB.' author: - | G. Machabeli,$^{1}$, A. Rogava$^{1}$[^1] and B. Tevdorashvili$^{1}$\ $^{1}$Centre for Theoretical Astrophysics, ITP, Ilia State University, Tbilisi 0162, Georgia\ title: 'Self-trapping as the possible beaming mechanism for FRB' --- \[firstpage\] Fast Radio Bursts - FRB Introduction ============ Fast Radio Bursts [@lor07; @tho13; @cha17] (hereafter referred as FRB) are spatially sporadic and temporarily intermittent radio emission outbursts of mysterious nature, happening throughout the universe, with duration of milliseconds. On the basis of a few credible observational arguments (e.g., observed dispersion measures greater than the maximum expected from the Galaxy, their spatial distribution mostly off the Galactic plane) it is strongly ascertained that FRB are most likely of extragalactic origin. This circumstance would necessarily imply that the radio luminosities of related astronomical sources are by several orders of magnitude larger than any previously detected millisecond-scale radio transient sources [@cor16]. Originally FRB were detected with large radio telescopes, with localization accuracy of the order of a few arcminutes. Evidently, localization efforts have been made and were related with the survey of simultaneous variability of the immediate neighborhood, adjacent area galaxies [@kea16] or possible presence of peculiar field stars [@loe14]. Until recently these systematic and repeated efforts failed to pinpoint their location, to lead to the detection of precise sources of FRB or,at least, their host galaxies with a satisfactory level of accuracy. However recently, by means of high-time-resolution radio interferometric observations, allowing direct imaging of the bursts [*per se*]{}, one of these source, FRB 121102, was localized with a sub-arcsecond accuracy [@cha17]. It appeared to be related to a persistent and faint radio source with non-thermal continuum spectrum. It also appears to have a very faint, 25-th magnitude, optical counterpart. Evidently FRB 121102 remains quite exceptional: so far it turns out to be the *only* known *repeating FRB* [@spi14; @spi16; @pet15; @sch16]. Even if FRB 121102 is unique member of the ’family’ of RRBs, still the repetitive nature of its bursts, makes less likely different kinds of ’catastrophic’ scenarios, happening with an astronomical object once in a lifetime. Another, very important and noteworthy aspect of FRB, is that when it happens no enhancement of the radiation emission in any other spectral range has ever been detected. Evidently, there are quite a number of different models of FRB. For instance, as early as in 2013 [@kas13], in order to explain four FRB reported in [@tho13], it was suggested that binary white dwarf mergers could lead to FRB. A birth of a quark star from a parent neutron star experiencing a quark nova was also suggested to be an explanation of FRBs [@sha16]. It was also suggested that FRB might be generated by ’cosmic bomb’ - a regular pulsar, otherwise unnoticeable at a cosmological distance, producing a FRB when its magnetosphere is suddenly “combed” by a nearby, strong plasma stream toward the anti-stream direction [@zha17]. It was also argued that a black hole absorbing a neutron star companion on the the battery phase of the binary, when the black hole interacts with the neutron star magnetic field could become a source of at least a subclass of FRB [@min15]. However, this mechanism is expected to produce electromagnetic radiation mainly in the high-energy (X-rays and/or gamma-rays) range, while FRB are observed only in the radio range. In another interesting model [@fal14] FRBs are surmised to represent final signals of a supermassive rotating neutron star: it is supposed that initially they are above the critical mass for non-rotating models, supported by their rapid rotation. But magnetic braking constantly reduces their spins, and at some moment of time these neutron stars start suddenly collapsing to a black hole, producing a FRB. A somewhat similar model was suggested in [@ful15]: neutron star collapsing as a result of ’sedimentation’ of dark matter (dark matter particles sinking to the center of a neutron star and becoming the same temperature as the star) within its core. Eventually, black hole is created at the center of the neutron star, with the collapse leading to the powerful radio outburst. Obviously special attention is focused on the repeating FRB 121102 source. Its persistent radio counterpart is believed to have number density of particles of the order of $N\sim 10^{52}$, energy about $E_{N} \sim10^{ 48} $erg, and its length-scale of the order of $R \sim10^{17} $cm. The FRB source is argued to be a nebula heated and expanded by an intermittent outflow from a peculiar magnetar a neutron star powered not by its rotational energy but by its magnetic energy [@bel17]. The peculiarity of the object is related with its very young age; it is supposed to liberate its energy frequently, in giant magnetic flares driven by accelerated ambipolar diffusion in the neutron star core. The flares would eventually feed the nebula and produce bright millisecond bursts. In [@viy17] yet another model for repeating FRBs was proposed, implying the existence of a variable and relativistic electron-positron beam, being boosted by an impulsive MHD mechanism, interacting with a plasma cloud at the center of a dwarf galaxy. According to this model, the interaction leads to the development of plasma turbulence and creates areas of high electrostatic field - cavitons - in the cloud. It is argued that as a result short-lived, bright coherent radiation bursts, FRB, are generated. Summarizing, we can cite a very recent review paper by J. I. Katz, where he says ’More than a decade after their discovery, astronomical Fast Radio Bursts remain enigmatic. They are known to occur at ’cosmological’ distances, implying large energy and radiated power, extraordinarily high brightness and coherent emission. Yet their source objects, the means by which energy is released and their radiation processes remain unknown’. [@kat18]. We believe that before trying to involve exotic phenomena for the explanation of FRB it is reasonable to try to explain FRB on the basis of traditional, well-established physical phenomenon. In this letter, we argue that a well-known nonlinear optical phenomenon - self trapping [@chi64; @chi65] - could serve as an alternative FRB model. The advantage of the proposed model is in its self-sustained and autonomous nature: it doesn’t require additional sources of energy and it naturally provides sufficiently high emission in a narrow spectral range without leading to a simultaneous radiation outburst in any other spectral ranges. It is also worthwhile to note that outbursts of similar nature are observed for certain objects in our galaxy, for instance, gamma-ray bursts for the Crab Nebula [@buh14]. Recently for the explanation of these powerful bursts the elements of nonlinear optics [@mac15] have been used. Before coming directly to the core of the problem and the contents of our model let us note that nonlinear optics is based on the fundamental principle of self-focusing of a powerful electromagnetic wave passing through a nonlinear medium. The self-focusing effect is related with the dependence of the medium dielectric permittivity on the wave intensity. A good example of a nonlinear medium is a liquid/plasma/gas which under the influence of a powerful electromagnetic wave develops coherent orientation of its molecules along the field. It leads, in its turn, to the anisotropy of the medium, increase of the electric field and the growth of the refraction index. In these circumstances the medium behaves as a focusing lens for incoming electromagnetic waves with transverse intensity gradient. The details of the mechanism are considered in the next section of the paper, while discussion of the model and its implications in the context of FRB phenomenon are given in the final section. Main Consideration ================== Let us consider a cloud of relativistic electron-proton plasma which sustains powerful electrostatic Langmuir waves (plasma oscillations) of electrons relative to heavy ions. Their wavelength can not be less than Debye radius, which for a relativistic plasma is: $$R_{D} = c {{\gamma_{p^{3/2}}}/{\omega_{p}}} ,$$ where $\omega_{p} = (4 \pi e^{2} n_{p}/m)^{1/2}$ is Langmuir (plasma) frequency of nonrelativistic plasma, while $\gamma_{p}$ and $n_{p}$ are Lorentz factor and number density of particles, respectively, and $c$ is the speed of light. The spectrum of Langmuir waves in relativistic plasma has the form $$\omega = \sqrt{{\omega_{p}^{2}}/{\gamma_{p}^{3}} + 3 {\bf k}^{2} c^{2}}$$ where ${\bf k}$ is the wavenumber vector of the electrostatic wave. The factor $3$ is related with the spatial isotropy of the medium. For a magnetized plasma the isotropy is violated and instead of the factor $3$ in (2) we have $1$. For Langmuir waves the Debye radius $R_{D}$ is the charge separation length-scale and is defined by the distance at which electron density fluctuation can be shifted on the plasma oscillation period time-scale due to the thermal motion of electrons. It leads to the polarization of the medium in the Debye volume caused by the grouping of charged particles. If one knows the value of the Debye radius and the average distance between the particles ${\langle d \rangle} = n_{p}^{1/3}$, then it is possible to estimate the number of dipoles ($N_{d} = (R_{D}/{\langle d \rangle})^{3}$) in the Debye volume. We assume that the plasma cloud contains a large number of Debye volumes but only in one of them the radiation is directed along the line of sight.Further, we suppose that the current of charged particles is continuous. Relativistic particles rapidly leave the volume but they are substituted by other, identical particles.Therefore the onset of the system with Langmuir waves can be considered quasi stationary. It allows us to use the dipole approximation. Let us now suppose that the Debye volume is filled with spatially aligned dipoles, oriented towards the line of sight. Obviously, it is an idealization because, in reality, in the Debye volume, only a part of the dipoles could be aligned with the line of sight. Furthermore, let us also assume that through the Debye volume an electromagnetic wave of radio frequency is passing. Let us assume that its energy is less than the energy of the electrostatic waves, but nevertheless it is powerful enough to cause the shifting of the multitude of dipoles. The polarization vector has the following form: $${\bf P} = e N_{D} {\bf r}$$ Even when the electric field ${\bf E}(t)$ of the incident wave is small it still manages to shift charged particles by a small displacement value ${\bf r}(t)$. The shifting, in its turn, causes the appearance of the restoring force ${\bf f}(t) = - \eta {\bf r}(t)$, where $\eta$ ia an analogue of the spring constant in Hooke’s law. However, when the field ${\bf E}(t)$ is not too small the displacement ${\bf r}(t)$ can be more considerable and the expression for the restoring force will contains also a second, nonlinear term: $${\bf f}(t) = - \eta {\bf r}(t) - q {\bf r}^{3}(t)$$ where $q$ is a constant coefficient. Its value does not have a decisive role in the framework of the present consideration. The value of the electron displacement ${\bf r}(t)$ can be determined from the equation of motion [@mac15]. In a relativistic case it has the following form: $$m \gamma^{3} {{d^{2} {\bf r}}\over{dt^{2}}} = - m \gamma \Gamma {{d {\bf r}}\over{dt}} - \eta {\bf r} - q {\bf r}^{3} + e {\bf E}$$ where $\gamma$ is corresponding Lorentz factor and where the first term on the right hand side of the equation describes dissipation, with $\Gamma$ being the damping rate. Taking into account the definition (3) of the polarization vector [**P**]{} from (4) we derive: $${{d^{2} {\bf P}}\over{dt^{2}}} + \Gamma {{d {\bf P}}\over{dt}} + \left({{\omega_0}\over{\gamma}} \right)^{2} {\bf P} + Q {\bf P}^{3} = {\left({{e^{2} N_D}\over{m \gamma^{3}}} \right)} {\bf E}$$ where $Q \equiv q/m e^{2} N_{p}^{3} \gamma^{3}$ and $\omega_0$, in this case, is the Langmuir oscillation frequency $\omega_{0} = \omega_{p}$. We have noted the incident wave’s electric field $\bf E$ is supposed to be large, but still much less than the intensity of the internal field within the cloud. In this case the nonlinear term in (6) can be considered to be small and we can solve the equation by means of the method of successive approximations. In particular, supposing $\bf P \equiv \bf P_{L} + \bf P_{NL}$, with $\bf P_{L} \gg \bf P_{NL}$, and neglecting the nonlinear term we obtain: $${{d^{2} {\bf P_{L}}}\over{dt^{2}}} + \Gamma {{d {\bf P_{L}}}\over{dt}} + \left({{\omega_{0}}\over{\gamma}} \right)^{2} {\bf P_{L}} = {\left({{e^{2} N_D}\over{m \gamma^{3}}} \right)} {\bf E}$$ If we further write down the electric field of the wave as $${\bf E}(t) = {\bf A} \cos (\omega t)$$ then the solution of (7) is found to be: $${\bf P_{L}}(t) = \left(\frac{e^{2} N_{D}}{m \gamma^{3} \sqrt{(\omega^{2} - \omega_{0}^{2})^{2} + 4 \Gamma^{2} \omega^{2}}}\right) {\bf A} \cos (\omega t + \Phi)$$ where $\tan(\Phi) = \Gamma \omega/(\omega^{2} - \omega_{0}^{2})$. Note that the Langmuir frequency $\omega \gg \omega_{0}$ is the frequency of radio emission. Therefore, the range of frequencies we consider is far from the resonance: $| \omega_{0}^{2} - \omega^{2} | \gg 4 \Gamma^{2}$ and the impact of the dissipation term can be neglected. The vector of polarization $\bf P $ is related to the electric field through the polarization of the medium $\mu$ in the following way: $\bf P = \mu \bf E$. This expression can be written as: $${\bf P}(t) = \mu(\omega) {\bf E}(t)$$ After this linear solution is found the equation in the nonlinear approximation has the following form: $$\frac{\partial^{2} {\bf P}_{NL}}{\partial t^{2}} + \omega_{0}^{2} {\bf P}_{NL} = - \left( \frac{q \mu^{3}(\omega)}{m \gamma^{3} e^{2} N_{D}^{2}} \right) {\bf E}^3(t)$$ Let us rewrite ${\bf E}^{3}(t)$ using trigonometric identity $cos^{3}(\omega t) = (1/4) [ 3 cos (\omega t) + cos (3 \omega t)]$. Then on the right hand side of (11) we have two terms, describing the contribution of the first and the third harmonics. Accordingly one can write: $${\bf P}(t) = \mu(\omega, A) {\bf E}(t)$$ with $\mu(\omega, A)$ defined by: $ \mu(\omega, A) = \mu(\omega){\left[1+ {{3 q \mu^{2}(\omega) A^{2}}\over{4m n^{2} e^{2} \omega^{2}}} \right]}. $ while $\mu(\omega)$ is determined from the solution of (7): $$\mu(\omega) = {{e^{2} N}\over{m \gamma^{3} \omega^{2}}}$$ Note that in the serial expansion of $\mu(\omega, A)$ only first non-vanishing terms are maintained. The dielectric permittivity of the medium is described by the tensor $\varepsilon_{ij}(\omega, A)$. The connection between $\varepsilon_{ij}(\omega, \b E)$ and $\mu_{ij}(\omega, \b E)$ tensors is given by: $$\varepsilon_{ij}(\omega, \b E) = \delta_{ij} + 4 \pi \mu_{ij}(\omega, A)$$ The induction vector $\bf D = \bf E + 4 \pi \bf P$, where $D_{i} = \varepsilon_{ij}(\omega, E) E_{j}$. Subsequently, taking into account(12) and (14), we write down Maxwell equation: $$\nabla \times {\bf B} - (1/c)\left[\varepsilon(\omega) + \frac{3 \pi q \mu^{3}(\omega)A^{2}}{mN^{2} e^{2} \omega^{2}} \right] \frac{\partial \bf E}{\partial t} = 0$$ From this equation it is evident that the influence of the nonlinear term is equivalent to the change of the dielectric permittivity or the refraction index of the medium. When an electromagnetic wave is propagating in this medium the refraction index is $H = c/v_{ph}$ and it depends on the wave frequency. Hence, the dispersion of the electromagnetic radiation depends on the refraction index. From $H^{2} = \varepsilon$ it turns out that the refraction index is equal to $H = H_{L} + H_{NL}$, where $H_{L}^{2} = \varepsilon(\omega)$, while $H_{NL} \approx H_{2} A^{2}$ and for $H_{2}$ we have: $$H_2 = 6 \pi q \mu^{3}(\omega)/m N^{2} e^{2} \omega^{2}$$ Therefore, if $H_{2}>0$ the refraction index in the cavern $H = H_{L} + H_{NL}$ turns out to be larger than the refraction index of the ambient beyond the Debye sphere [@mac15], which remains equal to $H=H_{L}$. Finally in the whole Debye volume let us separate rays directed to the observer. Due to the linear diffraction these rays has to diverge, feature angular diffusion across the the line of observation and before leaving the Debye volume they have to be confined within the cone with the opening angle $2 \theta_{D}$ where $$\theta_{D} \approx \frac{\lambda}{r_{D} H_{L}}$$ where $\lambda$ is the wavelength of the electromagnetic wave. However, when the rays leave the nonlinear medium and enter the ambient with the refraction index $H_{L}$, the rays experience nonlinear refraction. If the ray falls on the boundary between nonlinear, optically more dense medium and linear, optically less dense one and if the angle of incidence $\theta_{0} > \theta_{D}$ then all diffracted rays will undergo a total internal reflection. We are interested in the regime when $\theta_{0} \approx \theta_{D}$ when rays assemble in a parallel beam and the observer sees enhanced intensity of radiation [@mac15]. The limiting critical incidence angle for the total internal reflection is determined by the following equation: $$cos \theta_{0} = H_{L}/(H_{L} + H_{2} A^{2})$$ For the small value $\theta_{0}$ we find: $$\theta_{0}^{2} \approx 2 (H_{2}/H_{L}) A^{2}$$ Substituting (16) in (18) we find out that $$H_{2} \simeq 1/\omega^{8}$$ and if the condition is satisfied for certain frequencies in a given region and at a given moment of time for other values of frequencies it would not hold. The frequency dependence is very strongly nonlinear, which implies that self-trapping will work only for a very narrow frequency range. Discussion and conclusions ========================== In this paper we argue that the actual reason of FRB could be the self-trapping phenomenon. This nonlinear mechanism implies that to the radiation beam directed towards the observer additional rays are added which, in the absence of the self-trapping, would pass beyond the actual radiation pattern. As a result, the observer, while the beam is being self-trapped sees an enhanced intensity of radiation in a very narrow frequency range! This scenario is self-sustaining and fully autonomous because unlike many other mechanisms it does not require additional, external sources of energy. Besides, self-trapping depends on a quite large number of parameters, in particular, on the proper value of ratio of the wave amplitude to the amplitude of an incident electrostatic wave, on the temperature of the medium, direction of these waves relative to the line of sight. Any kind of, even a slight, deviation of any of these parameters from the “favorable” values may lead to the violation of the nonlinearity condition. That is why this is a very finely tuned, random and very rare phenomena. Its occurrence and the arrival of the self-trapped, self-focused enhanced beam to the observer has to be a totally random and extremely rare phenomenon. Initially we select a volume, which, at the moment when they pass through a randomly appearing nonlinear medium. contains waves directed towards the observer. It can be said that rays in this volume constitute a cylindrical beam with maximum energy concentrated at its center. The area of maximum intensity at the same time is optically thicker one [@akh68]. However the given volume, apart from the rays directed to the observer, contains also rays which propagate with some nonzero angle to the line of sight. Most of these rays, providing they pass only through a linear medium would not reach the observer. However, if these rays move from optically thicker area to optically less thick area they would be refracted towards the maximum energy area. Nonlinear area, selected by us, is significantly smaller than the plasma cloud in which the waves of the given frequency are generated. Therefore, it is reasonable to suppose that a significant number of the given frequency waves pass through the nonlinear region with propagation directions constituting small angles to the line of sight, which satisfy the condition of the total internal reflection at the boundary between the linear and nonlinear media. Evidently the coincidence of the channel axis with the line of sight has to be totally random. That is why FRB are happening rarely and on a totally random basis. Even a slight alteration of these parameters leads to the violation of the self-trapping condition and, therefore, disappearance of the wave intensity enhancement - disappearance of the burst. If our model is correct and relevant to actual FRB, the self-trapping condition may hold only for a few milliseconds. Hence, it is reasonable to expect that the probability of the coincidence between the line of sight and the direction of self-trapping has to be quite small. Therefore, what could be a serious drawback for a commonly occurring phenomena in this case ’works’ just the opposite way - it strengthens our confidence in believing that self-trapping could be the very reason of the appearance of this extremely rare and energetic phenomenon - fast radio burst or FRB. Additionally, self-trapping mechanism does not exclude other, physically plausible, repetitive or non-repetitive, catastrophic or non-catastrophic mechanism proposed for the FRB. Moreover, we believe that sel-trapping may be the very ’beaming’ mechanism which might be needed for interpreting FRB as narrowly beamed radio bursts [@kat17]. Any of those mechanisms giving credible explanation of rarity of FRB, coupled with self-trapping mechanism would imply the simultaneous occurrence of two, quite rare processes. Probably this is the very reason why FRB are not just rare, or very rare, but ultra-rare phenomenon, until now observed only from very distant, extragalactic sources. Acknowledgments {#acknowledgments .unnumbered} =============== This research was supported by Shota Rustaveli National Science Foundation of Georgia (SRNSFG) \[grant number FR/516/6-300/14\]. Andria Rogava wishes to thank for hospitality Centre for mathematical Plasma Astrophysics, KULeuven (Leuven, Belgium), where a part of this study was finalized. [99]{} Akhmanov, S. A., Sukhorukov, A. P., & Khokhlov, R. V., 1968, Sov. Phys. Uspekhi, 10, 609 Beloborodov A. M., 2017, 843, L26 Buhler, R. & Blandford, R., 2014, Rep. Prog. Phys., 77, 066901 Chatterjee, S. et al. 2017, Nature, 541, 58 Chiao, R. Y., Garmire, E., & Townes, C. H. 1964, , 13, 479 Chiao, R., Garmire, E., & Townes, C. 1965, , 14, 1056 Cordes, K. M. & Wasserman, I., 2016, , 457, 232 Falcke, H. & Rezzolla, L., 2014, , 562, A137 Fuller, J. & Ott, C. D., 2015, , 450, L71 Kashiyama, K., Ioka, K., & Meszaros P., 2013, , 776, L39 Katz, J. I., 2018, eprint:arXiv: 1804.09092 Katz, J. I., 2017, , 467, L96 Keane, E. F. et al. 2016, Nature, 530, 453 Loeb, A., Shvartzvald, Y. & Maoz, D., 2014, MNRAS, 439, L46 Lorimer, D. R., Bailes, M., McLaughlin, M.A., Narkevic, D. J., & Crawford, F. A., 2007, Science, 318, 777 Machabeli, G., Rogava, A., & Shapakidze, D., 2015, , 814, 38 Mingarelli, C., Levin, J., & Lazio, J., 2015, , 814, L20 Scholz, P. et al. 2016, , 833, 177 Shand, Z., et al. 2016, Res. Astron. Astrophys., 16, id.80 Spitler, L. G. et al. 2014, , 790, 101 Spitler, L. G. et al. 2016, Nature, 531, 202 Thornton, D. et al. 2013, Science, 341, 53 Viyero, F. L., et al., 2017, , 602, id. A64 Zhang, B., 2017, , 836, L32 \[lastpage\] [^1]: E-mail: andria.rogava@iliauni.edu.ge

--- abstract: 'In Online Sum-Radii Clustering, $n$ demand points arrive online and must be irrevocably assigned to a cluster upon arrival. The cost of each cluster is the sum of a fixed opening cost and its radius, and the objective is to minimize the total cost of the clusters opened by the algorithm. We show that the deterministic competitive ratio of Online Sum-Radii Clustering for general metric spaces is $\Theta(\log n)$, where the upper bound follows from a primal-dual algorithm and holds for general metric spaces, and the lower bound is valid for ternary Hierarchically Well-Separated Trees (HSTs) and for the Euclidean plane. Combined with the results of (Csirik et al., MFCS 2010), this result demonstrates that the deterministic competitive ratio of Online Sum-Radii Clustering changes abruptly, from constant to logarithmic, when we move from the line to the plane. We also show that Online Sum-Radii Clustering in metric spaces induced by HSTs is closely related to the Parking Permit problem introduced by (Meyerson, FOCS 2005). Exploiting the relation to Parking Permit, we obtain a lower bound of $\Omega(\log\log n)$ on the randomized competitive ratio of Online Sum-Radii Clustering in tree metrics. Moreover, we present a simple randomized $O(\log n)$-competitive algorithm, and a deterministic $O(\log\log n)$-competitive algorithm for the fractional version of the problem.' author: - Dimitris Fotakis - Paraschos Koutris bibliography: - 'clustering.bib' title: 'Online Sum-Radii Clustering[^1]' --- Algorithms, Competitive Analysis, Sum-$k$-Radii Clustering [^1]: This work was supported by the project AlgoNow, co-financed by the European Union (European Social Fund - ESF) and Greek national funds, through the Operational Program “Education and Lifelong Learning”, under the research funding program THALES. Part of this work was done while P. Koutris was with the School of Electrical and Computer Engineering, National Technical University of Athens, Greece. An extended abstract of this work appeared in the Proceedings of the 37th Symposium on Mathematical Foundations of Computer Science (MFCS 2012), Branislav Rovan, Vladimiro Sassone, and Peter Widmayer (Editors), Lecture Notes in Computer Science 7464, pp. 395-406, Springer, 2012.

--- author: - | Archil Gulisashvili $\cdot$\ Elias M. Stein title: Asymptotic Behavior of the Stock Price Distribution Density and Implied Volatility in Stochastic Volatility Models --- **Abstract We study the asymptotic behavior of distribution densities arising in stock price models with stochastic volatility. The main objects of our interest in the present paper are the density of time averages of the squared volatility process and the density of the stock price process in the Stein-Stein and the Heston model. We find explicit formulas for leading terms in asymptotic expansions of these densities and give error estimates. As an application of our results, sharp asymptotic formulas for the implied volatility in the Stein-Stein and the Heston model are obtained.\ \ **Keywords Stein-Stein model $\cdot$ Heston model $\cdot$ Mixing distribution density $\cdot$ Stock price $\cdot$ Bessel processes $\cdot$ Ornstein-Uhlenbeck processes $\cdot$ CIR processes $\cdot$ Asymptotic formulas $\cdot$ Implied volatility**** Introduction {#S:ashes} ============ In [@keyGS1] and [@keyGS2], we found sharp asymptotic formulas for the distribution density of the stock price process in the Hull-White model. These formulas were used in [@keyGS3] to characterize the asymptotic behavior of the implied volatility in the Hull-White model. The present paper is a continuation of [@keyGS1], [@keyGS2], and [@keyGS3]. It concerns the asymptotic behavior of various distribution densities arising in the Stein-Stein and the Heston model. The stochastic differential equations characterizing the stock price process $X_t$ and the volatility process $Y_t$ in the Stein-Stein model have the following form: $$\left\{\begin{array}{ll} dX_t=\mu X_tdt+\left|Y_t\right|X_tdW_t \\ dY_t=q\left(m-Y_t\right)dt+\sigma dZ_t, \end{array} \right. \label{E:SS}$$ where $W_t$ and $Z_t$ are standard Brownian motions. The initial condition for $X_t$ is denoted by $x_0$ and for $Y_t$ by $y_0$. The stochastic volatility model in (\[E:SS\]) was introduced and studied in [@keySS1]. In this model, the absolute value of an Ornstein-Uhlenbeck process plays the role of the volatility of the stock. We will assume that that $\mu\in\mathbb{R}$, $q> 0$, $m\ge 0$, and $\sigma> 0$. The Heston model was introduced in [@keyH]. In this model, the stock price process $X_t$ and the volatility process $Y_t$ satisfy the following system of stochastic differential equations: $$\left\{\begin{array}{ll} dX_t=\mu X_tdt+\sqrt{Y_t}X_tdW_t \\ dY_t=\left(a+bY_t\right)dt+c\sqrt{Y_t}dZ_t, \end{array} \right. \label{E:H}$$ where $W_t$ and $Z_t$ are standard Brownian motions. We will assume that $\mu\in\mathbb{R}$, $a\ge 0$, $b< 0$, and $c> 0$. The initial conditions for $X_t$ and $Y_t$ are denoted by $x_0$ and $y_0$, respectively. The volatility equation in (\[E:H\]) can be rewritten in the mean-reverting form. This gives $$\left\{\begin{array}{ll} dX_t=\mu X_tdt+\sqrt{Y_t}X_tdW_t \\ dY_t=r\left(m-Y_t\right)dt+c\sqrt{Y_t}dZ_t, \end{array} \right.$$ where $r=-b$ and $m=-\frac{a}{b}$. The volatility equation in (\[E:H\]) is uniquely solvable in the strong sense, and the solution $Y_t$ is a positive stochastic process. This process is called a Cox-Ingersoll-Ross process (a CIR-process). This process was studied in [@keyCIR]. Interesting results concerning the Heston model were obtained in [@keyDY]. It will be assumed throughout the present paper that the models described by (\[E:SS\]) and (\[E:H\]) are uncorrelated. This means that the Brownian motions $W_t$ and $Z_t$ driving the stock price equation and the volatility equation in (\[E:SS\]) and (\[E:H\]) are independent. In the analysis of the probability distribution of the stock price $X_t$, the mean-square averages of the volatility process over finite time intervals play an important role. For the Stein-Stein model, we set $$\alpha_t=\left\{\frac{1}{t}\int_0^tY_s^2ds\right\}^{\frac{1}{2}}. \label{E:mix1}$$ Here $Y_t$ satisfies the second equation in (\[E:SS\]). For the Heston model, we have $$\alpha_t=\left\{\frac{1}{t}\int_0^tY_sds\right\}^{\frac{1}{2}}, \label{E:mix2}$$ where $Y_t$ satisfies the second equation in (\[E:H\]). It will be shown below that for every $t> 0$, the probability distribution of the random variable $\alpha_t$ defined by (\[E:mix1\]) for the model in (\[E:SS\]) and by (\[E:mix2\]) for the model in (\[E:H\]) admits a distribution density $m_t$ (see Lemma \[L:disden\] and Remark \[R:mixis\] below). The function $m_t$ is called the mixing distribution density. The distribution density of the stock price $X_t$ will be denoted by $D_t$. The existence of the density $D_t$ follows from formula (\[E:stvoll1\]). In this paper, we obtain sharp asymptotic formulas for the distribution density $D_t$ in the case of the uncorrelated Stein-Stein and Heston models. Note that in [@keyDY] and [@keySS1], the behavior of $D_t$ was studied for the Heston model and the Stein-Stein model, respectively, using a rough logarithmic scale in the asymptotic formulas. Moreover, no error estimates were given in these papers. The results established in the present paper are considerably sharper. We find explicit formulas for leading terms in asymptotic expansions of $D_t$ in the Heston and the Stein-Stein model with error estimates. It would be interesting to obtain similar results for correlated models, since such models have more applications in finance. We hope that the methods employed in the present paper may be useful in the study of the correlated case. We will next quickly overview the structure of the present paper. In Section \[S:afsp\], the main results of the paper (theorems \[T:maino\], \[T:main\], \[T:ohh\], and \[T:oh\]) are formulated. They concern the asymptotic behavior of the mixing distribution density $m_t$ and the stock price distribution density $D_t$ in the Heston model and the Stein-Stein model. Section \[S:imvol\] is devoted to applications of our main results. In this section, we obtain sharp asymptotic formulas for the implied volatility in the Heston and the Stein-Stein model. In section \[S:CIRB\], we gather several known facts about CIR-processes and Bessel processes. We also formulate a theorem of Pitman and Yor concerning exponential functionals of Bessel processes. This theorem plays an important role in the present paper. In Section \[S:ilt\], we prove an asymptotic inversion theorem for the Laplace transform in a certain class of functions (see Theorem \[T:lti\]). This theorem is useful in the study of the asymptotic behavior of the mixing distribution density, since it is often easier to find explicit formulas for the Laplace transform of this density than to characterize the density itself. In sections \[S:firh2\] - \[S:qv\], we prove theorems \[T:maino\] - \[T:oh\] and describe the constants appearing in these theorems. Asymptotic formulas for distribution densities {#S:afsp} ============================================== The next four theorems are the main results of the present paper. The first two of them provide explicit formulas for the leading term in the asymptotic expansion of the distribution density $D_t$ of the stock price $X_t$ in the Stein-Stein model and the Heston model, while the other two concern the asymptotic behavior of the mixing distribution density $m_t$ in these models. \[T:maino\] Let $D_t$ be the stock price distribution density in model (\[E:SS\]) with $q\ge 0$, $m\ge 0$, and $\sigma> 0$. Then there exist positive constants $B_1$, $B_2$, and $B_3$ such that $$D_t\left(x_0e^{\mu t}x\right)=B_1x^{-B_3}e^{B_2\sqrt{\log x}}(\log x)^{-\frac{1}{2}} \left(1+O\left((\log x)^{-\frac{1}{4}}\right)\right),\quad x\rightarrow\infty. \label{E:dop1}$$ \[T:main\] Let $D_t$ be the stock price distribution density in model (\[E:H\]) with $a\ge 0$, $b\le 0$, and $c> 0$. Then there exist positive constants $A_1$, $A_2$, and $A_3$ such that $$D_t\left(x_0e^{\mu t}x\right)=A_1x^{-A_3}e^{A_2\sqrt{\log x}}(\log x)^{-\frac{3}{4}+\frac{a}{c^2}} \left(1+O\left((\log x)^{-\frac{1}{4}}\right)\right),\quad x\rightarrow\infty. \label{E:dopo}$$ \[T:ohh\] Suppose that $q\ge 0$, $m\ge 0$, and $\sigma> 0$ in model (\[E:SS\]). Then there exist positive constants $E$, $F$, and $G$ such that $$m_t(y)=Ee^{-Gy^2}e^{Fy} \left(1+O\left(y^{-\frac{1}{2}}\right)\right),\quad y\rightarrow\infty. \label{E:ohh1}$$ \[T:oh\] Suppose that $a\ge 0$, $b\le 0$, and $c> 0$ in model (\[E:H\]). Then there exist positive constants $A$, $B$, and $C$ such that $$m_t(y)=Ae^{-Cy^2}e^{By}y^{-\frac{1}{2}+\frac{2a}{c^2}} \left(1+O\left(y^{-\frac{1}{2}}\right)\right),\quad y\rightarrow\infty. \label{E:oh1}$$ It is known that for uncorrelated stochastic volatility models, the asymptotic behavior of the stock price distribution density $D_t$ near zero is determined by the behavior of $D_t$ near infinity. Indeed, we have $\displaystyle{D_t\left(x_0e^{\mu t}x^{-1}\right)=x^3D_t\left(x_0e^{\mu t}x\right)}$, $x> 0$. This equality can be derived from the formula $$D_t\left(x_0e^{\mu t}x\right)=\frac{1}{x_0e^{\mu t}\sqrt{2\pi t}}x^{-\frac{3}{2}}\int_0^{\infty}y^{-1}m_t(y) \exp\left\{-\left[\frac{1}{2ty^2}\log^2x+\frac{ty^2}{8}\right]\right\}dy \label{E:stvoll1}$$ (see, e.g., Section 4 in [@keyGS2]). It follows from Theorem \[T:main\] that the stock price distribution density $D_t(x)$ in the Heston model behaves at infinity roughly as the function $x^{-A_3}$ and at zero as the function $x^{A_3-3}$. For the Stein-Stein model, Theorem \[T:maino\] implies that $D_t(x)$ behaves at infinity roughly as the function $x^{-B_3}$ and at zero as the function $x^{B_3-3}$. In [@keySS1], the latter fact was established (in part, only heuristically) in the logarithmic scale (the same scale was used in [@keyDY]). More precisely, the following definition of the asymptotic equivalence was used in [@keyDY] and [@keySS1]: Two functions $F(x)$ and $G(x)$ are called asymptotically equivalent as $x\rightarrow\infty$ (or as $x\rightarrow 0$) if $\frac{\log F(x)}{\log G(x)}\rightarrow 1\quad\mbox{as}\quad x\rightarrow\infty\quad\mbox{(or $x\rightarrow 0$)}$. It is not hard to see that the constant $\gamma$ in [@keySS1], wich characterizes the decay of the stock price density $D_t$ near infinity (see formula (20) in [@keySS1]), coincides with the constant $B_3$ in (\[E:dop1\]) (see lemmas \[L:con1\] and \[L:con2\] for the description of the constant $B_3$). However, there is an error in formula (21) in [@keySS1]. It is stated that $D_t(x)$ behaves near zero as the function $x^{-1+\gamma}$. By Theorem \[T:maino\], the correct power function that characterizes the behavior of the density $D_t$ near zero is the function $x^{-3+\gamma}$. The values of the constants $A_3$ and $B_3$ appearing in theorems \[T:maino\] and \[T:main\] are given by the following formulas: $$A_3=A_3(t,b,c)=\frac{3}{2}+\frac{\sqrt{8C+t}}{2\sqrt{t}},\quad C=C(t,b,c) =\frac{t}{2c^2}\left(b^2+\frac{4}{t^2}r^2_{\frac{t|b|}{2}}\right),$$ and $$B_3=B_3(t,b,c)=\frac{3}{2}+\frac{\sqrt{8G+t}}{2\sqrt{t}},\quad G=G(t,q,\sigma) =\frac{t}{2\sigma^2}\left(q^2+\frac{1}{t^2}r^2_{qt}\right)$$ (see lemmas \[L:con1\], \[L:con2\], \[L:con3\], and \[L:con4\] below). Here $r_s$ denotes the smallest positive root of the entire function $z\mapsto z\,cos z+s\,sin z$. The zeroes of this function are studied in Section \[S:firh2\]. Note that $A_3$ does not depend on $a$, while $B_3$ does not depend on $m$. It is clear that $A_3> 2$ and $B_3> 2$. In [@keyGS1] and [@keyGS2], we studied the tail behavior of the stock price distribution density $D_t$ in the Hull-White model. It was established that the function $D_t(x)$ behaves at infinitiy like the function $x^{-2}$ on the power function scale. This is an extremely slow behavior. No uncorrelated stochastic volatility model has the function $D_t(x)$ decaying like $x^{-2+\epsilon}$, $\epsilon> 0$ (see [@keyGS2]). The tail of the stock price distribution in the Hull-White model is “fatter" than the corresponding tail in the Heston and the Stein-Stein model. Applications. Asymptotic behavior of the implied volatility {#S:imvol} =========================================================== The implied volatility in an option pricing model is the volatility in the Black-Scholes model such that the corresponding Black-Scholes price of the option is equal to its price in the model under consideration. In the present paper, we will only consider European call options, and the implied volatility will be studied as a function of the strike price $K$. Let us denote by $V_0$ the pricing function for the European call option in the Heston (or the Stein-Stein) model. Then the impled volatility $I(K)$ satisfies $C_{BS}(K,I(K))=V_0(K)$, where $C_{BS}$ stands for the Black-Scholes pricing function. In [@keyGS3], we studied the implied volatility in the Hull-White model. In the present paper, we characterize the asymptotic behavior of the implied volatility in the Stein-Stein and the Heston model. Our asymptotic formulas are sharper than the formulas which can be obtained form more general results due to Lee, Benaim, and Friz (see [@keyL], [@keyBF], [@keyBFL]). We will first consider the model in (\[E:H\]). It will be assumed below that the market price of volatility risk $\gamma$ is equal to zero (see, e.g., [@keyFPS] for the definition of $\gamma$). Then, under the corresponding martingale measure $\mathbb{P}^{*}$, the system of stochastic differential equations in (\[E:H\]) can be rewritten in the following form: $$\left\{\begin{array}{l} dX_t=rX_tdt+Y_tX_tdW_t^{*} \\ dY_t=\left(a+bY_t\right)dt+c\sqrt{Y_t}dZ_t^{*}, \end{array} \right. \label{E:sde2}$$ where $W_t^{*}$ and $Z_t^{*}$ are independent standard one-dimensional Brownian motions, and $r> 0$ is a constant interest rate. This means that under the measure $\mathbb{P}^{*}$ the new system also describes a Heston model. This is the reason why we assumed that $\gamma=0$. Let us consider a European call option associated with the stock price model in (\[E:sde2\]). The price of such an option at $t=0$ is given by the following formula: $$V_0(K)=\mathbb{E}^{*}\left[e^{-rT}\left(X_T-K\right)_{+}\right], \label{E:opt}$$ where $T$ is the expiration date and $K$ is the strike price. Put $D_T(x)=D_T\left(x;r,\nu,\xi,x_0,y_0\right)$. Then we have $$V_0(K)=e^{-rT}\int_K^{\infty}xD_T(x)dx-e^{-rT}K\int_K^{\infty} D_T(x)dx. \label{E:option}$$ The implied volatility is often considered as a function of the log-strike $k$, which in our case is related to the strike price $K$ by the formula $k=\log\frac{K}{x_0e^{rT}}$. In terms of $k$, the implied volatility is defined as follows: $\hat{I}(k)=I(K)$, $-\infty< k<\infty$, $0< K<\infty$. For uncorrelated stochastic volatlity models, the behavior of the implied volatility near zero is completely determined by how it behaves near infinity. More precisely, it is known that the implied volatility is symmetric in the following sense: $I\left(\left(x_0e^{rT}\right)^2K^{-1}\right)=I\left(K\right)$ for all $K> 0$ (see, e.g., [@keyGS2]). It is clear that the previous equality can be formulated in terms of the log-strike $k$ as follows: $\hat{I}(k)=\hat{I}(-k)$, $-\infty< k<\infty$. Important results concerning the behavior of the implied volatility $\hat{I}(k)$ in a general case were obtained by Lee (see [@keyL]). He characterized the asymptotic behavior of the implied volatlity for large strikes in terms of the moments of the stock price process. In [@keyBF], the asymptotic behavior of the implied volatility was linked to the tail behavior of the distribution of the stock price process (see also [@keyBFL]). The reader interested in more aspects of the asymptotic behavior of the implied volatility can consult Chapter 5 in [@keyFPS]. The following theorem will be established below: \[T:impik\] For the Heston model, $$\hat{I}(k)=\beta_1k^{\frac{1}{2}}+\beta_2+\beta_3\frac{\log k}{k^{\frac{1}{2}}}+O\left(\frac{\psi(k)}{k^{\frac{1}{2}}}\right) \label{E:impik1}$$ as $k\rightarrow\infty$, where $$\beta_1=\frac{\sqrt{2}}{\sqrt{T}}\left(\sqrt{A_3-1}-\sqrt{A_3-2}\right),$$ $$\beta_2=\frac{A_2}{\sqrt{2T}}\left(\frac{1}{\sqrt{A_3-2}}-\frac{1}{\sqrt{A_3-1}}\right),$$ $$\beta_3=\frac{1}{\sqrt{2T}}\left(\frac{1}{4}-\frac{a}{c^2}\right)\left(\frac{1}{\sqrt{A_3-1}}-\frac{1}{\sqrt{A_3-2}}\right),$$ and $\psi$ can be any positive increasing function on $(0,\infty)$ such that $\displaystyle{\lim_{k\rightarrow\infty}\psi(k)=\infty}$. For the Stein-Stein model, $$\hat{I}(k)=\gamma_1k^{\frac{1}{2}}+\gamma_2+O\left(\frac{\psi(k)}{k^{\frac{1}{2}}}\right) \label{E:impik2}$$ as $k\rightarrow\infty$, where $$\gamma_1=\frac{\sqrt{2}}{\sqrt{T}}\left(\sqrt{B_3-1}-\sqrt{B_3-2}\right),$$ $$\gamma_2=\frac{B_2}{\sqrt{2T}}\left(\frac{1}{\sqrt{B_3-2}}-\frac{1}{\sqrt{B_3-1}}\right),$$ and the function $\psi$ is as above. CIR processes and Bessel processes {#S:CIRB} ================================== The volatility in model (\[E:H\]) is described by the square root of an CIR-process. It is known that CIR-processes are related to squared Bessel processes. In the present section, we gather several results concerning Bessel processes and CIR-processes. Let $\delta\ge 0$ and $x\ge 0$, and consider the following stochastic differential equation: $dT_t=\delta dt+2\sqrt{T_t}dZ_t$, $T_0=x$ a.s. This equation has a unique nonnegative strong solution $T_t$, which is called the squared $\delta$-dimensional Bessel process started at $x$. The following notation is often used for the squared Bessel process: $T_t=BESQ^{\delta}_x(t)$. We refer the reader to [@keyBS; @keyCS; @keyD; @keyGY; @keyPY; @keyRY] for more information on Bessel processes. The next lemma links Bessel processes and CIR-processes (see, e.g., [@keyGY]). \[L:sviaz1\] Let $Y_t$ be a CIR process satisfying the equation $dY_t=\left(a+bY_t\right)dt+c\sqrt{Y_t}dZ_t$ with $Y_0=x$ $\mathbb{P}$-a.s., and put $T_t=BESQ^{\frac{4a}{c^2}}_x(t)$. Then $Y_t=e^{bt}T\left(\frac{c^2}{4b}\left(1-e^{-bt}\right)\right)$. \[R:vto\] If $b=0$, then we have $Y_t=BESQ_x^{\frac{4a}{c^2}}\left(\frac{c^2}{4}t\right)$. Pitman and Yor proved the following assertion (see [@keyPY; @keyRY]): \[T:kri\] Let $\lambda> 0$. Then $$\mathbb{E}\left[\exp\left\{-\frac{\lambda^2}{2}\int_0^tBESQ_x^{\delta}(u)du\right\}\right] =\left[\cosh(\lambda t)\right]^{-\frac{\delta}{2}}\exp\left\{-\frac{x\lambda}{2}\tanh(\lambda t) \right\}.$$ The next statement can be derived from Theorem \[T:kri\]. This statement can be found, e.g., in [@keyBS]. \[T:liu\] Let $a\ge 0$, $b< 0$, $c> 0$, and let $Y_t$ be a CIR-process in Lemma \[L:sviaz1\] such that $Y_0=y_0$ a.s. Then for every $\eta>\frac{1}{2}$, $$\begin{aligned} & \\ &=\exp\left\{-\frac{abt}{c^2}\right\}\left(\frac{2\eta} {2\eta\cosh(bt\eta)-\sinh(bt\eta)}\right)^{\frac{2a}{c^2}}\exp\left\{-\frac{by_0\left(4\eta^2-1\right)\sinh(bt\eta)} {2c^2\eta\cosh(bt\eta)-c^2\sinh(bt\eta)}\right\}.\end{aligned}$$ Theorem \[T:liu\] is equivalent to the following assertion: \[T:lapp\] Let $a\ge 0$, $b< 0$, $c> 0$, and let $Y_t$ be a CIR process in Lemma \[L:sviaz1\] such that $Y_0=y_0$ a.s. Then for every $\lambda> 0$, $$\begin{aligned} &\mathbb{E}_{y_0}\left[\exp\left\{-\lambda\int_0^tY_sds\right\}\right] \nonumber \\ &=\exp\left\{-\frac{abt}{c^2}\right\}\left(\frac{\sqrt{b^2+2c^2\lambda}} {\sqrt{b^2+2c^2\lambda}\cosh(\frac{1}{2}t\sqrt{b^2+2c^2\lambda}) -b\sinh(\frac{1}{2}t\sqrt{b^2+2c^2\lambda})}\right)^{\frac{2a}{c^2}} \nonumber \\ &\quad\exp\left\{-\frac{2y_0\lambda\sinh(\frac{1}{2}t\sqrt{b^2+2c^2\lambda})} {\sqrt{b^2+2c^2\lambda}\cosh(\frac{1}{2}t\sqrt{b^2+2c^2\lambda}) -b\sinh(\frac{1}{2}t\sqrt{b^2+2c^2\lambda})}\right\}. \label{E:lai2}\end{aligned}$$ Theorem \[T:lapp\] characterizes the Laplace transform of the probability distribution of the random variable $\int_0^tY_sds$. In the next section, we will study the asymptotic behavior of the inverse Laplace transform in a class of functions which look like the function on the right-hand side of (\[E:lai2\]). An asymptotic inverse of the Laplace transform {#S:ilt} ============================================== Let us assume that $M$ is a function on $(0,\infty)$ whose Laplace transform is given by the following formula: $$\int_0^{\infty}e^{-\lambda y}M(y)dy=I(\lambda),\quad\lambda> 0,$$ where $$I(\lambda)=\lambda^{\gamma_1}G_1(\lambda)^{\gamma_2}G_2(\lambda)e^{F(\lambda)}$$ with $\gamma_1\ge 0$ and $\gamma_2\ge 0$. We will next explain what restrictions are imposed on the functions $G_1$, $G_2$, and $F$. The function $G_2$ is analytic in the closed half-plane $\overline{\mathbb{C}}_{+}=\left\{\lambda:Re(\lambda)\ge 0\right\}$ and such that $G_2(0)\neq 0$. It is also assumed that the function $G_1$ is analytic in $\overline{\mathbb{C}}_{+}$ except for a simple pole at $\lambda=0$ with residue $1$, that is, $G_1(\lambda)=\frac{1}{\lambda}+\widetilde{G}(\lambda)$ where $\widetilde{G}$ is an analytic function in $\overline{\mathbb{C}}_{+}$. In addition, we suppose that $G_1$ is a nowhere vanishing function in $\mathbb{C}_{+}=\left\{\lambda:Re(\lambda)> 0\right\}$. Similarly, the function $F$ is analytic in $\overline{\mathbb{C}}_{+}$ and has a simple pole at $\lambda=0$ with residue $\alpha> 0$, that is, $F(\lambda)=\frac{\alpha}{\lambda}+\widetilde{F}(\lambda)$, where $\widetilde{F}$ is an analytic function in $\overline{\mathbb{C}}_{+}$. We also suppose that $$\left|G_1(\lambda)^{\gamma_2}G_2(\lambda)e^{F(\lambda)}\right|\le\exp\left\{-|\lambda|^{\delta}\right\} \label{E:lti2}$$ as $|\lambda|\rightarrow\infty$ in $\overline{\mathbb{C}}_{+}$, for some $\delta> 0$ (much less is needed about the decay of the function in (\[E:lti2\])). \[E:stch\]We assume that the function $G_1$ does not vanish in $\mathbb{C}_{+}$ in order to justify the existence of the power $G_1^{\gamma_2}$. Here we use the fact that for a nowhere vanishing analytic function $f$ in $\mathbb{C}_{+}$, there exists an analytic function $g$ on $\mathbb{C}_{+}$ such that $f(\lambda)=e^{g(\lambda)},\,\lambda\in\mathbb{C}_{+}$ (see Theorem 6.2 in [@keySS2]). The next result provides an asymptotic formula for the inverse Laplace transform of the function $I$. \[T:lti\] Suppose that the functions $G_1$, $G_2$, and $F$ are such as above. Then the following asymptotic formula holds: $$M(y)=\frac{1}{2\sqrt{\pi}}\alpha^{\frac{1}{4}+\frac{\gamma_1-\gamma_2}{2}}G_2(0)e^{\widetilde{F}(0)} y^{-\frac{3}{4}+\frac{\gamma_2-\gamma_1}{2}}e^{2\sqrt{\alpha}\sqrt{y}}\left(1+O\left(y^{-\frac{1}{4}}\right)\right) \label{E:lti3}$$ as $y\rightarrow\infty$. *Proof. Using the Laplace transform inversion formula, we see that for every $\varepsilon> 0$ we have\ $M(y)=\frac{1}{2\pi i}\int_{z=\varepsilon+ir}I(z)e^{yz}dz$. It follows that $M(\alpha y)=\frac{1}{2\pi i}\int_{z=\varepsilon+ir}I(z)e^{\alpha yz}dz$. By (\[E:lti2\]) and Cauchy’s formula, we can deform the contour of integration into a new contour $\eta$ consisting of the following three parts: the half-line $(-\infty i,-y^{-\frac{1}{2}}i]$, the half-circle $\Gamma$ in the right half-plane of radius $y^{-\frac{1}{2}}$ centered at $0$ (it is oriented counterclockwise), and finally the half-line $[y^{-\frac{1}{2}}i,\infty i)$. It follows that $$\begin{aligned} &M(\alpha y)=\frac{1}{2\pi i}\int_{\eta}I(z)e^{\alpha yz}dz =\frac{1}{2\pi}\int_{-\infty}^{-y^{-\frac{1}{2}}} (ir)^{\gamma_1}G_1(ir)^{\gamma_2}G_2(ir)e^{\widetilde{F}(ir)}e^{-\frac{i\alpha}{r}}e^{i\alpha yr}dr \nonumber \\ &\quad+\frac{1}{2\pi i}\int_{\Gamma}z^{\gamma_1}G_1(z)^{\gamma_2}G_2(z)e^{\widetilde{F}(z)}e^{\frac{\alpha}{z}} e^{\alpha yz}dz+\frac{1}{2\pi}\int_{y^{-\frac{1}{2}}}^{\infty}(ir)^{\gamma_1}G_1(ir)^{\gamma_2}G_2(ir)e^{\widetilde{F}(ir)} e^{-\frac{i\alpha}{r}}e^{i\alpha yr}dr \nonumber \\ &=I_1(y)+I_2(y)+I_3(y). \label{E:lti4}\end{aligned}$$* We will first estimate $I_2(y)$. This will give the main contribution to the asymptotics. By making a substitution $z=y^{-\frac{1}{2}}e^{i\theta}$, $-\frac{\pi}{2}\le\theta\le\frac{\pi}{2}$, we see that $$\begin{aligned} I_2(y)&=\frac{1}{2\pi}y^{-\frac{1+\gamma_1}{2}}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} e^{i\theta\gamma_1}G_1\left(y^{-\frac{1}{2}}e^{i\theta}\right)^{\gamma_2} G_2\left(y^{-\frac{1}{2}}e^{i\theta}\right) \\ &\quad\exp\left\{\widetilde{F}\left(y^{-\frac{1}{2}}e^{i\theta}\right)\right\} \exp\left\{\alpha\sqrt{y}e^{-i\theta}\right\} \exp\left\{\alpha\sqrt{y}e^{i\theta}\right\}e^{i\theta}d\theta.\end{aligned}$$ Next, taking into account the formula $\sqrt{y}\left(e^{i\theta}+e^{-i\theta}\right)=2\sqrt{y}\cos\theta=2\sqrt{y} +2\sqrt{y}(\cos\theta-1)$, we obtain $$\begin{aligned} I_2(y)&=\frac{1}{2\pi}y^{-\frac{1+\gamma_1}{2}}e^{2\alpha\sqrt{y}}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} e^{i\theta\left(1+\gamma_1\right)} G_1\left(y^{-\frac{1}{2}}e^{i\theta}\right)^{\gamma_2}G_2\left(y^{-\frac{1}{2}}e^{i\theta}\right) \nonumber \\ &\quad\exp\left\{\widetilde{F}\left(y^{-\frac{1}{2}}e^{i\theta}\right)\right\} \exp\left\{2\alpha\sqrt{y}(\cos\theta-1)\right\}d\theta. \label{E:I_2}\end{aligned}$$ It is easy to see that $$e^{i\theta\left(1+\gamma_1\right)}=1+O(\theta), \label{E:thet}$$ $$G_2\left(y^{-\frac{1}{2}}e^{i\theta}\right)-G_2(0)=O\left(y^{-\frac{1}{2}}\right),\quad \exp\left\{\widetilde{F}\left(y^{-\frac{1}{2}}e^{i\theta}\right)\right\}-e^{\widetilde{F}(0)} =O\left(y^{-\frac{1}{2}}\right) \label{E:tlast2}$$ on the contour $\Gamma$. Moreover, using (\[E:lti0\]) and the mean value theorem, we obtain $$\begin{aligned} G_1\left(y^{-\frac{1}{2}}e^{i\theta}\right)^{\gamma_2}-\left(\sqrt{y}e^{-i\theta}\right)^{\gamma_2} =\left[\sqrt{y}e^{-i\theta} +\widetilde{G}\left(y^{-\frac{1}{2}}e^{i\theta}\right)\right]^{\gamma_2}-\left(\sqrt{y}e^{-i\theta}\right)^{\gamma_2} =O\left(y^{\frac{\gamma_2-1}{2}}\right)\end{aligned}$$ on $\Gamma$. Therefore, $$G_1\left(y^{-\frac{1}{2}}e^{i\theta}\right)^{\gamma_2}-y^{\frac{\gamma_2}{2}}= O\left(y^{\frac{\gamma_2}{2}}|\theta|\right)+O\left(y^{\frac{\gamma_2-1}{2}}\right) \label{E:gt}$$ on $\Gamma$. It follows from (\[E:I\_2\]), (\[E:thet\]), (\[E:tlast2\]), and (\[E:gt\]) that $$\begin{aligned} I_2(y)&=\frac{1}{2\pi}G_2(0)e^{\widetilde{F}(0)} y^{-\frac{1+\gamma_1-\gamma_2}{2}}e^{2\alpha\sqrt{y}}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \exp\left\{2\alpha\sqrt{y}(\cos\theta-1)\right\} \nonumber \\ &\quad\left(1+O\left(y^{-\frac{1}{2}}\right)+O(|\theta|)\right)d\theta. \label{E:lappp}\end{aligned}$$ We will next employ Laplace’s method to estimate the integral appearing in (\[E:lappp\]). Consider the integral $\displaystyle{\int_a^be^{-s\Phi(x)}\psi(x)dx}$, where $\Phi\in C^{\infty}[a,b]$ and $\psi\in C^{\infty}[a,b]$ (much less is needed from the functions $\Phi$ and $\psi$), and assume that there is an $x_0\in(a,b)$ such that $\Phi^{\prime}\left(x_0\right)=0$, and $\Phi\left(x_0\right)> 0$ throughout $[a,b]$. Then the following assertion holds: \[T:Lapl\] Under the above assumptions, with $s> 0$ and $s\rightarrow\infty$, $$\int_a^be^{-s\Phi(x)}\psi(x)dx=e^{-s\Phi(x_0)}\left[\frac{A}{\sqrt{s}}+O\left(\frac{1}{s}\right)\right], \label{E:tlast5}$$ where $A=\sqrt{2\pi}\psi\left(x_0\right)\left(\Phi^{\prime\prime}\left(x_0\right)\right)^{-\frac{1}{2}}$. The proof of Theorem \[T:Lapl\] can be found, e.g., in [@keySS2]. Using (\[E:tlast5\]) with $a=-\frac{\pi}{2}$, $b=\frac{\pi}{2}$, $\Phi(x)=1-\cos x$, $\psi(x)=1$, $x_0=0$, and $s=2\alpha\sqrt{y}$, we see that $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}e^{2\alpha\sqrt{y}(\cos\theta-1)}d\theta=\frac{\sqrt{\pi}}{\sqrt{\alpha}} y^{-\frac{1}{4}}+O\left(y^{-\frac{1}{2}}\right),\quad y\rightarrow\infty. \label{E:tlast7}$$ Similarly $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}|\theta|e^{2\alpha\sqrt{y}(\cos\theta-1)}d\theta=O\left(y^{-\frac{1}{2}}\right), \quad y\rightarrow\infty. \label{E:tlast8}$$ Therefore, (\[E:lappp\]), (\[E:tlast7\]), and (\[E:tlast8\]) give $$I_2(y) =\frac{1}{2\sqrt{\pi\alpha}}G_2(0)e^{\widetilde{F}(0)} y^{-\frac{3}{4}+\frac{\gamma_2-\gamma_1}{2}}e^{2\alpha\sqrt{y}}\left(1+O\left(y^{-\frac{1}{4}}\right)\right), \quad y\rightarrow\infty. \label{E:tlast9}$$ Moreover, using (\[E:lti2\]) we obtain $$I_1(y)+I_3(y)=O\left(\exp\left\{-cy^{\delta}\right\}\right),\quad y\rightarrow\infty, \label{E:tlast10}$$ for some $c> 0$. It follows from (\[E:lti4\]), (\[E:tlast9\]), and (\[E:tlast10\]) that $$M\left(\alpha y\right)=\frac{1}{2\sqrt{\pi\alpha}}G_2(0)e^{\widetilde{F}(0)} y^{-\frac{3}{4}+\frac{\gamma_2-\gamma_1}{2}}e^{2\alpha\sqrt{y}}\left(1+O\left(y^{-\frac{1}{4}}\right)\right), \quad y\rightarrow\infty. \label{E:UU}$$ Now it is not hard to see that (\[E:UU\]) implies (\[E:lti3\]). This completes the proof of Theorem \[T:lti\]. Proof of Theorem \[T:oh\] {#S:firh2} ========================= We will first discuss the properties of the following complex function: $$\Phi_s(z)=z\,\cos z+s\,\sin z\quad,\quad z\in\mathbb{C}, \label{E:o}$$ where $s\ge -1$. The next lemma concerns the zeros of $\Phi_s$. A special case of this lemma was stated in [@keySS1] without proof and was used in [@keyDY] and [@keySS1]. \[L:seque\] For all $s\ge -1$, the function $\Phi_s$ has only real zeros. *Proof of Lemma \[L:seque\]. For every $n\ge 1$, put $P_n(z)=z\prod_{k=1}^n\left(1-\frac{z^2}{k^2}\right)$. The function $P_n$ is a polynomial of degree $2n+1$, all of whose roots ($z=k$, $k\in\mathbb{Z}$, $|k|\le n$) are real. Put $Q_n(z)=z^{-s+1}\frac{d}{dz}\left(z^sP_n(z)\right)=sP_n(z)+zP_n^{\prime}(z)$. Then $Q_n$ is a polynomial of degree $2n+1$ which vanishes at $z=0$. Also, by Rolle’s theorem, the function $\displaystyle{\frac{d}{dz}\left(z^sP_n(s)\right)}$ vanishes at points strictly between $k$ and $k+1$, $-n\le k< n$, since the function $z^sP_n(z)$ vanishes at those points. It follows from the previous considerations that $Q_n(z)$ has all its $2n+1$ roots that are real. But $\displaystyle{P_n(z)\rightarrow\frac{\sin\pi z}{\pi}}$ by the product formula. Hence, $\displaystyle{Q_n(z)\rightarrow\frac{s}{\pi}\sin\pi z+z\cos\pi z}$, and the desired conclusion that all the roots are real follows from the Rouché-Hurwitz theorem. The proof above implicitly used the condition $s> -1$ (for otherwise $z^sP_n(z)$ does not vanish at the origin). The result for $s=-1$ can be derived from that for $s> -1$ by a limiting argument.* This completes the proof of Lemma \[L:seque\]. In the present paper, we consider only the case where $s\ge 0$. It is clear that the function $\Phi_s$ is odd and satisfies $\Phi_s(0)=0$. \[D:seq\] For $s\ge 0$, the smallest positive zero of the function $\Phi_s$ will be denoted by $r_s$. The number $r_s$ plays an important role throughout the paper. It is not hard to see that $r_0=\frac{\pi}{2}$, and $r_s\uparrow\pi$ as $s\rightarrow\infty$. Moreover, the function $s\mapsto r_s$ is differentiable and increasing on $(0,\infty)$. Indeed, the value of $r_s$ for $0< s<\infty$ is equal to the first coordinate of the point in $\mathbb{R}^2$ where the segment $y=-s^{-1}x$, $\frac{\pi}{2}< x<\pi$, intersects the curve $y=\tan x$. In addition, we have $r_s=\phi^{-1}(s)$, $0< s<\infty$, where $\phi(u)=-u(\tan u)^{-1}$, $\frac{\pi}{2}< u<\pi$. It is also clear that $\sin(r_0)=1$, $\cos(r_0)=0$, and $\Phi^{\prime}_0\left(r_0\right)=-\frac{\pi}{2}$. Moreover, if $s> 0$, then $$\sin(r_s)> 0,\quad\cos(r_s)< 0,\quad\mbox{and}\quad\Phi^{\prime}_s\left(r_s\right)< 0. \label{E:urr2}$$ By Lemma \[L:seque\], the function $\rho_s$ defined by $$\rho_s(z)=z\cosh z+s\sinh z=-i\Phi_s(iz) \label{E:fro}$$ has only imaginary zeros. The next lemma concerns the mixing distribution densities. \[L:disden\] Suppose that $a\ge 0$, $b< 0$, and $c>0$ in the Heston model. Then the probability distribution of the random variable $\alpha_t$ in (\[E:mix2\]) admits a density $m_t$ for every $t> 0$. *Proof. We will first prove that the probability distribution of the random variable $\int_0^tY_sds$ admits a density $\overline{m}_t$. This will allow us to prove the existence of the mixing distribution $m_t$, since $m_t$ can be determined from the formula $$\overline{m}_t(y)=\frac{1}{2\sqrt{ty}}m_t\left(t^{-\frac{1}{2}}y^{\frac{1}{2}}\right). \label{E:hof}$$* It follows from Theorem \[T:lapp\] that $$\begin{aligned} &\mathbb{E}_{y_0}\left[\exp\left\{-\frac{\lambda}{2c^2}\int_0^tY_sds\right\}\right] =\exp\left\{-\frac{abt}{c^2}\right\}\left(\frac{\sqrt{b^2+\lambda}} {\sqrt{b^2+\lambda}\cosh(\frac{1}{2}t\sqrt{b^2+\lambda}) -b\sinh(\frac{1}{2}t\sqrt{b^2+\lambda})}\right)^{\frac{2a}{c^2}} \nonumber \\ &\quad\exp\left\{-\frac{y_0c^{-2}\lambda\sinh(\frac{1}{2}t\sqrt{b^2+\lambda})} {\sqrt{b^2+\lambda}\cosh(\frac{1}{2}t\sqrt{b^2+\lambda}) -b\sinh(\frac{1}{2}t\sqrt{b^2+\lambda})}\right\}. \label{E:laika}\end{aligned}$$ Denote by $\Psi_1$ and $\Psi_2$ the functions on the right-hand side and the left-hand side of formula (\[E:laika\]), respectively, and put $$u_{b,t}=-4t^{-2}r_{\frac{1}{2}t|b|}^2 \label{E:ver}$$ where $t>0$ and $b\le 0$. It is not hard to see that the function $\Psi_1$ can be continued analytically from the half-line to the half-plane $$\mathbb{C}_{b,t}=\left\{\lambda\in\mathbb{C}:\,Re(\lambda)>-b^2+u_{b,t}\right\}. \label{E:fof}$$ This can be shown using the properties of the zeros of the function $\rho_s$ defined by (\[E:fro\]) and the fact that $\lambda=0$ is not a singularity of the function $\Psi_1$. On the other hand, the function $\Psi_2$ is analytic in the half-plane $\mathbb{C}_{+}=\left\{\lambda\in\mathbb{C}:\,Re(\lambda)> 0\right\}$ (use formula (\[E:laika\]) with $\lambda> 0$ to prove the finiteness of the derivative of the function $\Psi_2$ in $\mathbb{C}_{+}$). It follows that formula (\[E:laika\]) holds for all $\lambda\in\mathbb{C}_{b,t}$. Since $i\xi\in \mathbb{C}_{b,t}$ for all $\xi\in\mathbb{R}^1$, equation (\[E:laika\]) implies the following formula: $$\int_0^{\infty}e^{-i\xi x}d\nu_t(x)=\Psi_1(i\xi),\quad\xi\in\mathbb{R}^1, \label{E:vera}$$ where $\nu_t$ stands for the probability distribution of the random variable $\frac{1}{c^2}\int_0^tY_sds$. It is not hard to see that $$\Psi_1(\lambda)-c_1\left(\frac{z}{\rho_s(z)}\right)^{\frac{2a}{c^2}}\exp\left\{-\frac{\left(c_2z^2-c_3\right)\sinh z} {\rho_s(z)}\right\}$$ where $z=\frac{1}{2}t\sqrt{b^2+\lambda}$, $s=-b$, $\rho_s$ is defined by (\[E:fro\]), and $c_1$, $c_2$, and $c_3$ are some positive constants. Next, we observe that for $z$ lying in a proper sector of the right-hand plane and $z\rightarrow\infty$, we have $$\left(c_2z^2-c_3\right)\frac{\sinh z}{\rho_s(z)}=c_2z+O(1)$$ and $$\frac{z}{\rho_s(z)}=2e^{-z}+O\left(|e^{-2z}|\right).$$ This implies that $$\left|\Psi_1(\lambda)\right|\le C_1\exp\left\{-C_2|\lambda|^{\frac{1}{2}}\right\} \label{E:tut}$$ for $\lambda\in\mathbb{C}_{b,t}$ and $\lambda\rightarrow\infty$. It follows from (\[E:tut\]) that the function $\xi\mapsto\left|\Psi_1(i\xi)\right|$ belongs to the space $L^2\left(\mathbb{R}^1\right)$. Taking into account (\[E:vera\]) and using Plancherel’s Theorem, we see that the measure $\nu_t$ is absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}^1$. Therefore, the density $\bar{m}_t$ exists. This implies the existence of the mixing density $m_t$, and the proof of Lemma \[L:disden\] is thus completed. We will next prove Theorem \[T:oh\]. The following lemma will be used in the proof. In this lemma, we use the notation introduced in (\[E:o\]) and (\[E:ver\]). \[L:hers\] Let $a\ge 0$, $b< 0$, $c> 0$, and let $Y_t$ be a CIR process satisfying the equation\ $dY_t=\left(a+bY_t\right)dt+c\sqrt{Y_t}dZ_t$ and such that $Y_0=y_0$ a.s. Then the following formula holds: $$\begin{aligned} &\int_0^{\infty}e^{-\lambda y}y^{-\frac{1}{2}}\exp\left\{\left(b^2-u_{b,t}\right)y\right\} m_t\left(\frac{\sqrt{2}c}{\sqrt{t}}\sqrt{y}\right)dy =\frac{\sqrt{2t}}{c}\exp\left\{-\frac{abt}{c^2}\right\} \left(\frac{it\sqrt{\lambda+u_{b,t}}}{2\Phi_{\frac{1}{2}t|b|}\left(i\frac{1}{2}t\sqrt{\lambda+u_{b,t}}\right)}\right)^{\frac{2a}{c^2}} \nonumber \\ &\quad \exp\left\{-\frac{ity_0\left(\lambda+u_{b,t}-b^2\right)\sinh\left(\frac{1}{2}t\sqrt{\lambda+u_{b,t}}\right)} {2c^2\Phi_{\frac{1}{2}t|b|}\left(i\frac{1}{2}t\sqrt{\lambda+u_{b,t}}\right)}\right\}, \label{E:laai}\end{aligned}$$ The functions on the both sides of (\[E:laai\]) are analytic in the right half-plane $\mathbb{C}_0=\left\{\lambda\in\mathbb{C}:\,Re(\lambda)>0\right\}$, and the equality in (\[E:laai\]) holds for all $\lambda\in\mathbb{C}_0$. *Proof. It was established above that formula (\[E:laika\]) holds for all $\lambda\in\mathbb{C}_{b,t}$, where $\mathbb{C}_{b,t}$ is the half-plane defined by (\[E:fof\]). It is not hard to see that Lemma \[L:hers\] follows from (\[E:laika\]), Lemma \[L:disden\], and (\[E:hof\]).* We will next compute the residue $\lambda_0$ of the function $$\begin{aligned} \Lambda(\lambda)&=\frac{\sqrt{\lambda+u_{b,t}}}{\sqrt{\lambda+u_{b,t}}\cosh\left(\frac{1}{2}t\sqrt{\lambda+u_{b,t}}\right) +|b|\sinh\left(\frac{1}{2}t\sqrt{\lambda+u_{b,t}}\right)} \nonumber \\ &=\frac{it\sqrt{\lambda+u_{b,t}}}{2\Phi_{\frac{1}{2}t|b|}\left(i\frac{1}{2}t\sqrt{\lambda+u_{b,t}}\right)} \label{E:lll}\end{aligned}$$ at $\lambda=0$. Since $$\Phi^{\prime}_{\frac{1}{2}t|b|}\left(r_{\frac{1}{2}t|b|}\right)=\left(1+\frac{1}{2}t|b|\right)\cos\left(r_{\frac{1}{2}t|b|}\right) -r_{\frac{1}{2}t|b|}\sin\left(r_{\frac{1}{2}t|b|}\right), \label{E:fofa}$$ it is not hard to see that $$\begin{aligned} &\lambda_0=\lim_{\lambda\rightarrow 0}\lambda\Lambda(\lambda)=-r_{\frac{1}{2}t|b|}\lim_{\lambda\rightarrow 0} \frac{\lambda}{\Phi_{\frac{1}{2}t|b|}\left(i\frac{1}{2}t\sqrt{\lambda+u_{b,t}}\right)} \nonumber \\ &=-8t^{-2}r^2_{\frac{1}{2}t|b|}\Phi^{\prime}_{\frac{1}{2}t|b|}\left(r_{\frac{1}{2}t|b|}\right)^{-1} =8t^{-2}r^2_{\frac{1}{2}t|b|}\left|\Phi^{\prime}_{\frac{1}{2}t|b|}\left(r_{\frac{1}{2}t|b|}\right)\right|^{-1} \nonumber \\ &=8t^{-2}r^2_{\frac{1}{2}t|b|}\left|\left(1+\frac{1}{2}t|b|\right)\cos\left(r_{\frac{1}{2}t|b|}\right) -r_{\frac{1}{2}t|b|}\sin\left(r_{\frac{1}{2}t|b|}\right)\right|^{-1}. \label{E:qqq}\end{aligned}$$ In the proof of (\[E:qqq\]), we used (\[E:ver\]) and (\[E:urr2\]). Our next goal is to apply Theorem \[T:lti\] to the Laplace transform in formula (\[E:laai\]). The numbers $\gamma_1$ and $\gamma_2$ and the functions $G_1$, $G_2$, and $F$ in Theorem \[T:lti\] are chosen as follows: $\gamma_1=0$, $\gamma_2=\frac{2a}{c^2}$, and $G_1(\lambda)=\frac{1}{\lambda_0}\Lambda(\lambda)$, where $\Lambda$ and $\lambda_0$ are defined by (\[E:lll\]) and (\[E:qqq\]), $G_2(\lambda)=\lambda_0^{\frac{2a}{c^2}}\frac{\sqrt{2t}}{c}\exp\left\{-\frac{abt}{c^2}\right\}$, and $$F(\lambda)=-\frac{ity_0\left(\lambda+u_{b,t}-b^2\right)\sinh\left(\frac{1}{2}t\sqrt{\lambda+u_{b,t}}\right)} {2c^2\Phi_{\frac{1}{2}t|b|}\left(i\frac{1}{2}t\sqrt{\lambda+u_{b,t}}\right)}.$$ Note that $G_1$ is a nowhere vanishing function in $\mathbb{C}_{+}$.It follows from (\[E:tut\]) that the decay condition in (\[E:lti2\]) is satisfied. The next lemma provides explicit formulas for the residue $\alpha$ of the function $F$ at $\lambda=0$ and for the number $\widetilde{F}(0)$. \[L:alfo\] The following formulas hold: $$\alpha=\frac{ty_0\eta_1}{2c^2|\rho_1|}\left(b^2+4t^{-2}r_{\frac{t|b|}{2}}^2\right) > 0. \label{E:lti0}$$ and $$\widetilde{F}(0)=\frac{ty_0}{2c^2\rho_1^2}\left[\left(\eta_1-\left(4t^{-2}r^2_{\frac{1}{2}t|b|}+b^2\right)\eta_2\right)\rho_1 +\left(4t^{-2}r^2_{\frac{1}{2}t|b|}+b^2\right)\rho_2\right]. \label{E:lti00}$$ The constants $\eta_1$, $\eta_2$, $\rho_1$, and $\rho_2$ in (\[E:lti0\]) and (\[E:lti00\]) are given by $$\eta_1=\sin\left(r_{\frac{1}{2}t|b|}\right),\quad\eta_2=-\frac{t^2\cos\left(r_{\frac{1}{2}t|b|}\right)}{8r_{\frac{1}{2}t|b|}}, \label{E:eta}$$ $$\rho_1=\frac{t^2}{8r_{\frac{1}{2}t|b|}}\left[\left(1+\frac{1}{2}t|b|\right)\cos\left(r_{\frac{1}{2}t|b|}\right) -r_{\frac{1}{2}t|b|}\sin\left(r_{\frac{1}{2}t|b|}\right)\right], \label{E:rho1}$$ and $$\rho_2=\frac{t^4}{128}\left[r^{-3}_{\frac{1}{2}t|b|}\left[\left(1+\frac{1}{2}t|b|\right)\cos\left(r_{\frac{1}{2}t|b|}\right) -r_{\frac{1}{2}t|b|}\sin\left(r_{\frac{1}{2}t|b|}\right)\right]+2r^{-2}_{\frac{1}{2}t|b|}\sin\left(r_{\frac{1}{2}t|b|} \right)\right]. \label{E:rho2}$$ *Proof. We will need the first two coefficients in the power series representation for the function $\lambda F(\lambda)$. It is not hard to see that $$\sinh\left(\frac{1}{2}t\sqrt{\lambda+u_{b,t}}\right)=i\eta_1+i\eta_2\lambda+\cdots \label{E:bbi2}$$ where $\eta_1$ and $\eta_2$ are defined by (\[E:eta\]). Next, using formula (\[E:fofa\]) and taking into account that $$\Phi^{\prime\prime}_{\frac{1}{2}t|b|}\left(r_{\frac{1}{2}t|b|}\right)=-\Phi_{\frac{1}{2}t|b|}\left(r_{\frac{1}{2}t|b|}\right) -2\sin\left(r_{\frac{1}{2}t|b|}\right)=-2\sin\left(r_{\frac{1}{2}t|b|}\right),$$ we obtain $$\Phi_{\frac{1}{2}t|b|}\left(i\frac{1}{2}t\sqrt{\lambda+u_{b,t}}\right) =\rho_1\lambda+\rho_2\lambda^2+\cdots \label{E:bbi1}$$ where $\rho_1$ and $\rho_2$ are defined by (\[E:rho1\]) and (\[E:rho2\]). Using (\[E:urr2\]), we see that $\eta_1> 0$ and $\eta_2> 0$. Moreover, (\[E:bbi2\]), (\[E:bbi1\]), and the definition of $F$ imply the following equality: $$\begin{aligned} \lambda F(\lambda)&=\frac{ty_0}{2c^2}\frac{\left(-4t^{-2}r^2_{\frac{1}{2}t|b|}-b^2+\lambda\right) \left(\eta_1+\eta_2 \lambda+\cdots\right)}{\rho_1+\rho_2\lambda+\cdots} \nonumber \\ &=\frac{ty_0}{2c^2}\frac{\left(-4t^{-2}r^2_{\frac{1}{2}t|b|}-b^2\right)\eta_1 +\left(\eta_1-\left(4t^{-2}r^2_{\frac{1}{2}t|b|}+b^2\right)\eta_2\right)\lambda+\cdots} {\rho_1+\rho_2\lambda+\cdots} \label{E:sco}\end{aligned}$$ where $\rho_1$, $\rho_2$, $\eta_1$, and $\eta_2$ are defined in (\[E:rho1\]), (\[E:rho2\]), and (\[E:eta\]).* Now, it is clear that Lemma \[L:alfo\] follows from (\[E:sco\]). We will next complete the proof of Theorem \[T:oh\] and compute the constants appearing in it. By applying Theorem \[T:lti\] to the Laplace transform in (\[E:laai\]), we see that $$\begin{aligned} &y^{-\frac{1}{2}}\exp\left\{\left(b^2-u_{b,t}\right)y\right\} m_t\left(\frac{\sqrt{2}c}{\sqrt{t}}\sqrt{y};a,b,c,y_0\right) \nonumber \\ &=\frac{1}{2\sqrt{\pi}}\alpha^{\frac{1}{4}-\frac{a}{c^2}} \lambda_0^{\frac{2a}{c^2}}\frac{\sqrt{2t}}{c}\exp\left\{-\frac{abt}{c^2}\right\}e^{\widetilde{F}(0)}y^{-\frac{3}{4}+\frac{a}{c^2}} e^{2\sqrt{\alpha}\sqrt{y}}\left(1+O\left(y^{-\frac{1}{4}}\right)\right) \label{E:obshch2}\end{aligned}$$ as $y\rightarrow\infty$. Next, replacing $y$ by $y^2\frac{t}{2c^2}$ in formula (\[E:obshch2\]), we obtain $$\begin{aligned} m_t\left(y;a,b,c,y_0\right)&=\frac{1}{2\sqrt{\pi}}\alpha^{\frac{1}{4}-\frac{a}{c^2}} \lambda_0^{\frac{2a}{c^2}}\frac{\sqrt{2t}}{c}\exp\left\{-\frac{abt}{c^2}\right\}e^{\widetilde{F}(0)} \left(\frac{t}{2c^2}\right)^{-\frac{1}{4}+\frac{a}{c^2}} \nonumber \\ &\quad y^{-\frac{1}{2}+\frac{2a}{c^2}} \exp\left\{\frac{\sqrt{2\alpha t}}{c}y\right\}\exp\left\{-\frac{t\left(b^2-u_{b,t}\right)}{2c^2}y^2\right\} \left(1+O\left(y^{-\frac{1}{2}}\right)\right) \label{E:obshch22}\end{aligned}$$ as $y\rightarrow\infty$. Now it is clear that (\[E:obshch22\]) implies Theorem \[T:oh\]. In addition, the following lemma describes the constants appearing in Theorem \[T:oh\]: \[L:con1\] The constants $A$, $B$, and $C$ in Theorem \[T:oh\] are given by $$A=\frac{1}{\sqrt{\pi}}\left(\frac{t}{2c^2}\right)^{\frac{1}{4}+\frac{a}{c^2}} \alpha^{\frac{1}{4}-\frac{a}{c^2}}\lambda_0^{\frac{2a}{c^2}}\exp\left\{-\frac{abt}{c^2}\right\}e^{\widetilde{F}(0)},$$ $$B=c^{-1}\sqrt{2\alpha t},\quad\mbox{and}\quad C=\left(2c^2\right)^{-1}t\left(b^2+4t^{-2}r_{\frac{t|b|}{2}}^2\right),$$ where the numbers $\lambda_0> 0$, $\alpha> 0$, and $\widetilde{F}(0)$ are defined in (\[E:qqq\]), (\[E:lti0\]), and (\[E:lti00\]), respectively. This completes the proof of Theorem \[T:oh\]. \[C:Hes\] The following formula holds: $$m_t\left(y,a,0,c,y_0\right)=Ae^{-Cy^2}e^{By}y^{-\frac{1}{2}+\frac{2a}{c^2}} \left(1+O\left(y^{-\frac{1}{2}}\right)\right)$$ as $y\rightarrow\infty$, where $A=\frac{2^{\frac{1}{4}+\frac{a}{c^2}}y_0^{\frac{1}{4}-\frac{a}{c^2}}}{c\sqrt{t}}\exp\left\{\frac{4y_0}{c^2t}\right\}$, $B=\frac{2\sqrt{2y_0}\pi}{c^2t}$, and $C=\frac{\pi^2}{2c^2t}$. Corollary \[C:Hes\] can be derived from Theorem \[T:oh\] and Lemma \[L:con1\] by using the equality $r_0=\frac{\pi}{2}$ and the fact that for $b=0$ we have $\displaystyle{\lambda_0=\frac{4\pi}{t^2}}$, $\displaystyle{\alpha=\frac{4y_0\pi^2}{c^2t^3}}$, and $\displaystyle{\widetilde{F}(0)=-\frac{3y_0}{c^2t}}$. Proof of Theorem \[T:main\] {#S:proofs} =========================== We will next formulate a theorem concerning the asymptotic behavior of certain integral operators. This result is a minor modification of Theorem 4.5 established in [@keyGS2]. \[T:into\] Let $A$, $\zeta$, and $b$ be positive Borel functions on $[0,\infty)$, and suppose that the following conditions hold: 1. The function $A$ is integrable over any finite sub-interval of $[0,\infty)$. 2. The function $b$ is bounded and $\displaystyle{\lim_{y\rightarrow\infty}b(y)=0}$. 3. There exist $y_1> 0$, $c> 0$, and $\gamma$ with $0<\gamma\le 1$ such that $\zeta$ and $b$ are differentiable on $[y_1,\infty)$, and in addition $\left|\zeta^{\prime}(y)\right|\le cy^{-\gamma}\zeta(y)$ and $\left|b^{\prime}(y)\right|\le cy^{-\gamma}b(y)$ for all $y\ge y_1$. 4. For every $a> 0$, there exists $y_a> 0$ such that $b(y)\zeta(y)\ge\exp\left\{-ay^4\right\}$, $y> y_a$. 5. There exists a real number $l$ such that $A(y)=e^{ly}\zeta(y)(1+O(b(y)))$ as $y\rightarrow\infty$. Then, for every fixed $k> 0$, $$\begin{aligned} &\int_0^{\infty}A(y)\exp\left\{-\left(\frac{w^2}{y^2}+k^2y^2\right)\right\}dy \\ &=\frac{\sqrt{\pi}}{2k} \exp\left\{\frac{l^2}{16k^2}\right\} \zeta\left(k^{-\frac{1}{2}}w^{\frac{1}{2}}\right)\exp\left\{lk^{-\frac{1}{2}}w^{\frac{1}{2}}\right\} e^{-2kw}\left[1+O\left(w^{-\frac{\gamma}{2}}\right)+O\left(b\left(k^{-\frac{1}{2}}w^{\frac{1}{2}}\right)\right)\right], \quad w\rightarrow\infty.\end{aligned}$$ The only difference between Theorem \[T:into\] formulated above and Theorem 4.5 in [@keyGS2] is that in Theorem \[T:into\] we do not assume the integrability of the function $\zeta$ near zero. It is not hard to see that Theorem \[T:into\] can be derived from Theorem 4.5 if we replace the function $\zeta(y)$ by the function $A(y)e^{-ly}$ near zero. Let $A(y)=y^{-1}m_t(y)e^{Cy^2}$, $k=\sqrt{C+\frac{t}{8}}$, $l=B$, $\zeta(y)=Ay^{-\frac{3}{2}+\frac{2a}{c^2}}$, and $b(y)=y^{-\frac{1}{2}}$, where the constants $A$, $B$, and $C$ are as in Theorem \[T:oh\]. Then, it is not hard to see that condition 2 in Theorem \[T:into\] follows from formula (\[E:oh1\]). In addition, it is clear that condition 3 with $\gamma=1$ holds. The next lemma shows that condition 1 in Theorem \[T:into\] also holds. \[L:dodo\] For every $s> 0$, $\displaystyle{\int_0^sy^{-1}m_t(y)dy<\infty}$. *Proof. The function on the right-hand side of (\[E:laai\]) is integrable with respect to $\lambda$ over the interval $(1,\infty)$. Suppose $h$ is any positive function on $[0,\infty)$ such that its Laplace transform has this property. Then $\int_0^sh(y)y^{-1}dy<\infty$ for all $s> 0$. It follows from this fact and (\[E:laai\]) that the function $y\mapsto y^{-\frac{3}{2}}m_t\left(\frac{\sqrt{2}c}{\sqrt{t}}\sqrt{y}\right)$ is integrable over any interval of the form $[0,s]$ with $s> 0$. This implies Lemma \[L:dodo\].* Next, applying Theorem \[T:into\] we see that $$\begin{aligned} &\int_0^{\infty}y^{-1}m_t(y)\exp\left\{-\left(\frac{z^2}{y^2}+\frac{ty^2}{8}\right)\right\}dy \nonumber \\ &=A\frac{\sqrt{\pi}}{2k}\exp\left\{\frac{B^2}{16k^2}\right\}k^{\frac{3}{4}-\frac{a}{c^2}}z^{-\frac{3}{4}+\frac{a}{c^2}} \exp\left\{Bk^{-\frac{1}{2}}\sqrt{z}\right\}e^{-2kz}\left(1+O\left(z^{-\frac{1}{4}}\right)\right) \label{E:dop2}\end{aligned}$$ as $z\rightarrow\infty$. Replacing $z$ by $\frac{\log x}{\sqrt{2t}}$ in formula (\[E:dop2\]) and taking into account formula (\[E:stvoll1\]) and the equality $\displaystyle{k=\frac{\sqrt{8C+t}}{2\sqrt{2}}}$, we obtain $$\begin{aligned} &D_t\left(x_0e^{\mu t}x\right)=\frac{A}{x_0e^{\mu t}}2^{-\frac{3}{4}+\frac{a}{c^2}}t^{-\frac{1}{8}-\frac{a}{2c^2}} (8C+t)^{-\frac{1}{8}-\frac{a}{2c^2}}\exp\left\{\frac{B^2}{2(8C+t)}\right\} \nonumber \\ &\quad(\log x)^{-\frac{3}{4}+\frac{a}{c^2}} \exp\left\{\frac{B\sqrt{2}}{t^{\frac{1}{4}}(8C+t)^{\frac{1}{4}}}\sqrt{\log x}\right\}x^{-\left(\frac{3}{2}+ \frac{\sqrt{8C+t}}{2\sqrt{t}}\right)} \left(1+O\left((\log x)^{-\frac{1}{4}}\right)\right) \label{E:dop3}\end{aligned}$$ as $x\rightarrow\infty$. Now it is clear that formula (\[E:dop3\]) implies Theorem \[T:main\]. Moreover, the following lemma holds: \[L:con2\] The constants $A_1$, $A_2$, and $A_3$ in Theorem \[T:main\] are given by $$A_1=\frac{A}{x_0e^{\mu t}}2^{-\frac{3}{4}+\frac{a}{c^2}}t^{-\frac{1}{8}-\frac{a}{2c^2}} (8C+t)^{-\frac{1}{8}-\frac{a}{2c^2}}\exp\left\{\frac{B^2}{2(8C+t)}\right\},$$ $A_2=\frac{B\sqrt{2}}{t^{\frac{1}{4}}(8C+t)^{\frac{1}{4}}}$, and $A_3=\frac{3}{2}+ \frac{\sqrt{8C+t}}{2\sqrt{t}}$, where $A$, $B$, and $C$ are defined in Lemma \[L:con1\]. Let $p\in\mathbb{R}$, and denote by $l_p$ the moment of order $p$ of the function $D_t$, that is,\ $l_p=\mathbb{E}\left[X_t^p\right]=\int_0^{\infty}x^pD_t(x)dx$. The next result was obtained in [@keyAP]. \[L:blup\] For $a\ge 0$, $b\le 0$, $c> 0$, and $p\in\mathbb{R}$, the following statement holds: $q_p<\infty$ if and only if $2-A_3< p< A_3-1$, where the constant $A_3$ is such as in Theorem \[T:main\]. For $b=0$, $q_p<\infty$ if and only if $$\frac{1}{2}-\frac{\sqrt{4\pi^2+c^2t^2}}{2ct}< p<\frac{1}{2}+\frac{\sqrt{4\pi^2+c^2t^2}}{2ct}.$$ It is not hard to see that Lemma \[L:blup\] and more precise integrability theorems for the distribution of the stock price follow from Theorem \[T:main\]. Proof of Theorem \[T:ohh\] {#S:progen} ========================== Let $Y_t$ be the volatility process in model (\[E:SS\]). Then $Y_t^2$ is a squared Ornstein-Uhlenbeck process. The Laplace transform of the law of the squared Ornstein-Uhlenbeck process was found by Wenocur [@keyW] and by Stein and Stein [@keySS1]. Another explicit expression for this Laplace transform is given in the next formula: $$\begin{aligned} &\mathbb{E}_{y_0}\left[\exp\left\{-\lambda\int_0^tY_s^2ds\right\}\right] =2\sqrt{t}e^{\frac{qt}{2}}\left(\frac{\sqrt{w}}{\sqrt{w}\cosh\left(t\sqrt{w}\right)+q\sinh\left(t\sqrt{w}\right)} \right)^{\frac{1}{2}} \nonumber \\ &\quad\exp\left\{-\frac{y_0^2\lambda\sinh\left(t\sqrt{w}\right)}{\sqrt{w}\cosh\left(t\sqrt{w}\right) +q\sinh\left(t\sqrt{w}\right)}\right\} \exp\left\{-\frac{2mqy_0\lambda\left(\cosh\left(t\sqrt{w}\right)-1\right)}{\sqrt{w} \left(\sqrt{w}\cosh\left(t\sqrt{w}\right)+q \sinh\left(t\sqrt{w}\right)\right)}\right\} \nonumber \\ &\quad\exp\left\{\frac{m^2q^2\lambda\left(\sinh\left(t\sqrt{w}\right)-t\sqrt{w}\cosh\left(t\sqrt{w}\right)\right)} {w\left(\sqrt{w}\cosh\left(t\sqrt{w}\right)+q \sinh\left(t\sqrt{w}\right)\right)}\right\} \nonumber \\ &\quad\exp\left\{\frac{m^2q^3\lambda \left(4\sinh^2\left(\frac{t\sqrt{w}}{2}\right)-t\sqrt{w} \sinh\left(t\sqrt{w}\right)\right)}{w^{\frac{3}{2}}\left(\sqrt{w}\cosh\left(t\sqrt{w}\right)+q \sinh\left(t\sqrt{w}\right)\right)}\right\} \label{E:ochdli}\end{aligned}$$ where $\lambda> 0$ and $w=q^2+2\sigma^2\lambda$. We will next sketch the proof of formula (\[E:ochdli\]). The first step is to replace the symbols $\delta$, $\theta$, $k$, $\sigma_0$, and $\lambda$ in formula (8) in [@keySS1] by $q$, $m$, $\sigma$, $y_0$, and $t\lambda$, respectively, and take into account the following relations between the notation in [@keySS1] and in the present paper: $A=-\frac{q}{\sigma^2}$, $B=\frac{mq}{\sigma^2}$, $C=-\frac{\lambda}{\sigma^2}$, $a=\frac{1}{\sigma^2}\sqrt{q^2+2\sigma^2\lambda}$, $b=\frac{q}{\sqrt{q^2+2\sigma^2\lambda}}$, and $ak^2t=t\sqrt{q^2+2\sigma^2\lambda}$. We also combine the terms $\frac{a-A}{2a^2}a^2k^2t$ and $-\frac{1}{2}\log\left\{\frac{1}{2}\left(\frac{A}{a}+1\right)+\frac{1}{2} \left(1-\frac{A}{a}\right)e^{2ak^2t}\right\}$ in the expression for $N$ in formula (7) in [@keySS1], and after somewhat long and tedious computations show that formula (8) in [@keySS1] and formula (\[E:ochdli\]) in the present paper are equivalent. \[R:mixis\]Lemma \[L:disden\] states that for the Heston model, there exists the mixing distribution density $m_t$ for every $t> 0$. The same statement holds for the Stein-Stein model. This can be established using formula (\[E:ochdli\]) and reasoning as in the proof of Lemma \[L:disden\]. It follows from Remark \[R:mixis\] that $$\mathbb{E}_{y_0}\left[\exp\left\{-\lambda\int_0^tY_s^2ds\right\}\right]= \int_0^{\infty}e^{-\lambda ty^2}m_t(y)dy=\frac{1}{2\sqrt{t}} \int_0^{\infty}e^{-\lambda y}y^{-\frac{1}{2}}m_t\left(t^{-\frac{1}{2}}y^{\frac{1}{2}}\right)dy. \label{E:otchi}$$ We will next obtain a sharp asymptotic formula for the mixing distribution density in the Stein-Stein model using formulas (\[E:ochdli\]), (\[E:otchi\]), and the methods developed in Section \[S:firh2\]. Recall that for $s\ge 0$, we denoted by $r_s$ the smallest strictly positive zero of the function $\Phi_s(z)=z\cos z+s\sin z$. It is clear that $z\cosh z+s\sinh z=-i\Phi_s(iz)$. For $q> 0$ and $t> 0$, put $v_{q,t}=-\frac{r^2_{qt}}{t^2}$. Now (\[E:ochdli\]) and (\[E:otchi\]) give $$\begin{aligned} &\int_0^{\infty}e^{-\lambda y}y^{-\frac{1}{2}}\exp\left\{\left(q^2-v_{q,t}\right)y\right\} m_t\left(\frac{\sqrt{2}\sigma\sqrt{y}}{\sqrt{t}}\right)dy \nonumber \\ &=\frac{\sqrt{2t}}{\sigma}\exp\left\{\frac{qt}{2}\right\} F_1(\lambda)\exp\left\{F_2(\lambda)+F_3(\lambda)+F_4(\lambda)+F_5(\lambda)\right\}, \label{E:expo1}\end{aligned}$$ where $$F_1(\lambda)=\left(\frac{it\sqrt{\lambda+v_{q,t}}}{\Phi_{qt}\left(it\sqrt{\lambda+v_{q,t}}\right)}\right)^{\frac{1}{2}}, \quad F_2(\lambda)=-\frac{iy_0^2t\left(\lambda+v_{q,t}-q^2\right)\sinh\left(t\sqrt{\lambda+v_{q,t}}\right)} {2\sigma^2\Phi_{qt}\left(it\sqrt{\lambda+v_{q,t}}\right)}, \label{E:expo2}$$ $$F_3(\lambda)=-\frac{imqy_0t\left(\lambda+v_{q,t}-q^2\right)\left[\cosh\left(t\sqrt{\lambda+v_{q,t}}\right)-1\right]} {\sigma^2\sqrt{\lambda+v_{q,t}}\Phi_{qt}\left(it\sqrt{\lambda+v_{q,t}}\right)}, \label{E:expo3}$$ $$F_4(\lambda)=\frac{im^2q^2t\left(\lambda+v_{q,t}-q^2\right)\left[\sinh\left(t\sqrt{\lambda+v_{q,t}}\right)-t \sqrt{\lambda+v_{q,t}}\cosh\left(t\sqrt{\lambda+v_{q,t}}\right)\right]}{2\sigma^2\left(\lambda+v_{q,t}\right) \Phi_{qt}\left(it\sqrt{\lambda+v_{q,t}}\right)}, \label{E:expo4}$$ and $$F_5(\lambda)=\frac{im^2q^3t\left(\lambda+v_{q,t}-q^2\right)\left[4\sinh^2\frac{t\sqrt{\lambda+v_{q,t}}}{2} -t\sqrt{\lambda+v_{q,t}}\sinh\left(t\sqrt{\lambda+v_{q,t}}\right)\right]}{2\sigma^2\left(\lambda+v_{q,t}\right) ^{\frac{3}{2}}\Phi_{qt}\left(it\sqrt{\lambda+v_{q,t}}\right)}. \label{E:expo5}$$ It is not hard to see that the functions $F_1$, $F_2$, $F_3$, $F_4$, and $F_5$ have removable singularities at $$\lambda=-v_{q,t}=\frac{r^2_{qt}}{t^2}.$$ In addition, these functions are analytic in $\mathbb{C}_{+}$. Let us denote by $\lambda_1$ the residue of the function $F_1$ at $\lambda=0$. It is not hard to see that $$\lambda_1=\frac{2r_{qt}^2}{t^2\left|\left(1+qt\right)\cos\left(r_{qt}\right)-r_{qt}\sin\left(r_{qt}\right)\right|}. \label{E:sim1}$$ Our next goal is to apply Theorem \[T:lti\] to (\[E:expo1\]). Put $$\gamma_1=0,\quad\gamma_2=\frac{1}{2},\quad G_1(\lambda)=\frac{it\sqrt{\lambda+v_{q,t}}} {\Phi_{qt}\left(it\sqrt{\lambda+v_{q,t}}\right)}, \quad G_2(\lambda)=\frac{\sqrt{2t}}{\sigma} e^{\frac{qt}{2}}\lambda_1^{\frac{1}{2}}, \label{E:data1}$$ and $$F(\lambda)=F_2(\lambda)+F_3(\lambda)+F_4(\lambda)+F_5(\lambda). \label{E:data2}$$ In the sequel, the symbols $\alpha_j$ and $\widetilde{F}_j(0)$ will stand for the numbers in the formulation of Theorem \[T:lti\] associated with the function $F_j$, $2\le j\le 5$. We will next compute these numbers. The following formula will be helpful in the computations: $$\frac{\left(v_{q,t}-q^2+\lambda\right)\lambda}{\Phi_{qt}\left(it\sqrt{\lambda+v_{q,t}}\right)} =\frac{\left(v_{q,t}-q^2\right)+\lambda}{\zeta_1+\zeta_2\lambda+\cdots}=\tau_1+\tau_2\lambda+\cdots \label{E:sim2}$$ where $$\zeta_1=\frac{t^2\left(\left(1+qt\right)\cos\left(r_{qt}\right)-r_{qt}\sin\left(r_{qt}\right)\right)}{2r_{qt}},$$ $$\zeta_2=\frac{t^2\left(\left(1+qt\right)\cos\left(r_{qt}\right)+r_{qt}\sin\left(r_{qt}\right)\right)}{8r_{qt}^3},$$ $$\tau_1=\frac{2\left(v_{q,t}-q^2\right)r_{qt}}{t^2\left(\left(1+qt\right)\cos\left(r_{qt}\right)-r_{qt}\sin\left(r_{qt}\right)\right)}> 0,$$ and $$\begin{aligned} &\tau_2=\frac{\zeta_1-\left(v_{q,t}-q^2\right)\zeta_2}{\zeta_1^2} \\ &=\frac{4r_{qt}^2 \left(\left(1+qt\right)\cos\left(r_{qt}\right)-r_{qt}\sin\left(r_{qt}\right)\right)-\left(v_{q,t}-q^2\right) \left(\left(1+qt\right)\cos\left(r_{qt}\right)+r_{qt}\sin\left(r_{qt}\right)\right)}{2r_{qt}t^2 \left(\left(1+qt\right)\cos\left(r_{qt}\right)-r_{qt}\sin\left(r_{qt}\right)\right)^2}.\end{aligned}$$ Here we use the facts that $v_{q,t}=-\frac{r_{qt}^2}{t^2}< 0$,$\cos\left(r_{qt}\right)< 0$,$\sin\left(r_{qt}\right)> 0$, $\Phi_{qt}\left(r_{qt}\right)=0$, $$\Phi_{qt}^{\prime}\left(r_{qt}\right)=(1+qt)\cos\left(r_{qt}\right)-r_{qt}\sin\left(r_{qt}\right),$$ and $$\Phi_{qt}^{\prime\prime}\left(r_{qt}\right)=-2\sin\left(r_{qt}\right)-\Phi_{qt}\left(r_{qt}\right)=-2\sin\left(r_{qt}\right).$$ We will next employ (\[E:expo2\])-(\[E:expo5\]) to find explicit formulas for the numbers $\alpha_j$ and $\widetilde{F}_j(0)$ for\ $2\le j\le 5$. It follows from (\[E:expo2\]), (\[E:sim2\]), and the formula $$\sinh\left(t\sqrt{\lambda+v_{q,t}}\right)=i\sin\left(r_{qt}\right)-i\frac{t^2}{2r_{qt}}\cos\left(r_{qt}\right)\lambda+\cdots$$ that $$\begin{aligned} \lambda F_2(\lambda)&=-\frac{iy_0^2t}{2\sigma^2}\left(\tau_1+\tau_2\lambda+\cdots\right) \left(i\sin\left(r_{qt}\right)-i\frac{t^2}{2r_{qt}}\cos\left(r_{qt}\right)\lambda+\cdots\right) \\ &=\frac{y_0^2t}{2\sigma^2}\tau_1\sin\left(r_{qt}\right)+\frac{y_0^2t}{2\sigma^2}\left(\tau_2\sin\left(r_{qt}\right) -\tau_1\frac{t^2}{2r_{qt}}\cos\left(r_{qt}\right)\right)\lambda+\cdots\end{aligned}$$ Therefore, $$\alpha_2=\frac{y_0^2t}{2\sigma^2}\tau_1\sin\left(r_{qt}\right)> 0 \label{E:vcr0}$$ and $$\widetilde{F}_2(0)= \frac{y_0^2t}{2\sigma^2}\tau_1\sin\left(r_{qt}\right)+\frac{y_0^2t}{2\sigma^2}\left(\tau_2\sin\left(r_{qt}\right) -\tau_1\frac{t^2}{2r_{qt}}\cos\left(r_{qt}\right)\right). \label{E:vcr1}$$ Moreover, (\[E:expo3\]), (\[E:sim2\]), and the fact that $$\frac{\cosh\left(t\sqrt{\lambda+v_{q,t}}\right)-1}{\sqrt{\lambda+v_{q,t}}}=i\frac{1-\cos\left(r_{qt}\right)}{r_{qt}} -i\frac{t^3\left(2r_{qt}\sin\left(r_{qt}\right)+\cos\left(r_{qt}\right)-1\right)}{2r_{qt}^3}\lambda+\cdots$$ give $$\begin{aligned} &\lambda F_3(\lambda)=-\frac{im^2q^2t}{2\sigma^2}\left(\tau_1+\tau_2\lambda+\cdots\right)\left(i\frac{1-\cos\left(r_{qt}\right)}{r_{qt}} -i\frac{t^3\left(2r_{qt}\sin\left(r_{qt}\right)+\cos\left(r_{qt}\right)-1\right)}{2r_{qt}^3}\lambda+\cdots\right) \\ &=\frac{m^2q^2t}{2\sigma^2}\tau_1\frac{1-\cos\left(r_{qt}\right)}{r_{qt}}+\frac{m^2q^2t}{2\sigma^2} \left[\tau_2\frac{1-\cos\left(r_{qt}\right)}{r_{qt}}-\tau_1 \frac{t^3\left(2r_{qt}\sin\left(r_{qt}\right)+\cos\left(r_{qt}\right)-1\right)}{2r_{qt}^3}\right]\lambda+\cdots\end{aligned}$$ Hence, $$\alpha_3=\frac{m^2q^2t}{2\sigma^2}\tau_1\frac{1-\cos\left(r_{qt}\right)}{r_{qt}}> 0 \label{E:vcr2}$$ and $$\widetilde{F}_3(0)=\frac{m^2q^2t}{2\sigma^2} \left[\tau_2\frac{1-\cos\left(r_{qt}\right)}{r_{qt}}-\tau_1 \frac{t^3\left(2r_{qt}\sin\left(r_{qt}\right)+\cos\left(r_{qt}\right)-1\right)}{2r_{qt}^3}\right]. \label{E:vcr3}$$ In addition, (\[E:expo4\]), (\[E:sim2\]), and the formulas $r_{qt}\cos\left(r_{qt}\right)+qt\sin\left(r_{qt}\right)=0$ and $$\begin{aligned} &\frac{\sinh\left(t\sqrt{\lambda+v_{q,t}}\right)-t\sqrt{\lambda+v_{q,t}} \cosh\left(t\sqrt{\lambda+v_{q,t}}\right)}{\lambda+v_{q,t}} \\ &=-i\frac{t^2(1+qt)\sin\left(r_{qt}\right)}{r^2_{qt}}-i\frac{t^4\sin\left(r_{qt}\right)(1-qt-r_{qt}^2)} {r_{qt}^4}\lambda+\cdots\end{aligned}$$ imply that $$\begin{aligned} &\lambda F_4(\lambda)=\frac{im^2q^2t}{2\sigma^2}\left(\tau_1+\tau_2\lambda+\cdots\right) \left(-i\frac{t^2(1+qt)\sin\left(r_{qt}\right)}{r^2_{qt}}-i\frac{t^4\sin\left(r_{qt}\right)\left(1-qt-r^2_{qt}\right)} {r_{qt}^4}\lambda+\cdots\right).\end{aligned}$$ It follows that $$\alpha_4=\tau_1\frac{m^2q^2t^3}{2\sigma^2}\frac{(1+qt)\sin\left(r_{qt}\right)}{r^2_{qt}}> 0 \label{E:vcr4}$$ and $$\widetilde{F}_4(0)=\frac{m^2q^2t}{2\sigma^2}\left(\tau_1\frac{t^4\sin\left(r_{qt}\right)\left(1-qt-r^2_{qt}\right)} {r_{qt}^4}+\tau_2\frac{t^2(1+qt)\sin\left(r_{qt}\right)}{r^2_{qt}}\right). \label{E:vcr5}$$ Finally, we have $$\begin{aligned} &\frac{4\sinh^2\frac{t\sqrt{\lambda+v_{q,t}}}{2} -t\sqrt{\lambda+v_{q,t}}\sinh\left(t\sqrt{\lambda+v_{q,t}}\right)}{\left(\lambda+v_{q,t}\right) ^{\frac{3}{2}}}=i\frac{t^3\left(r_{qt}\sin\left(r_{qt}\right)-4\sin^2\frac{r_{qt}}{2}\right)}{r_{qt}^3} \\ &\quad+\left[\frac{i3t^5}{2r_{qt}^5}\left(r_{qt}\sin\left(r_{qt}\right)-4\sin^2\frac{r_{qt}} {2}\right)+\frac{it^5}{2r_{qt}^3}\left(\frac{3}{r_{qt}}\sin\left(r_{qt}\right)-\cos\left(r_{qt}\right)\right) \right]\lambda +\cdots,\end{aligned}$$ and hence $$\lambda F_5(\lambda)=\frac{im^2q^3t}{2\sigma^2}\left(\tau_1+\tau_2\lambda+\cdots\right) \left[i\frac{t^3\left(r_{qt}\sin\left(r_{qt}\right)-4\sin^2\frac{r_{qt}}{2}\right)}{r_{qt}^3} +D\lambda +\cdots\right]$$ where $$D=\frac{i3t^5}{2r_{qt}^5}\left(r_{qt}\sin\left(r_{qt}\right)-4\sin^2\frac{r_{qt}} {2}\right)+\frac{it^5}{2r_{qt}^3}\left(\frac{3}{r_{qt}}\sin\left(r_{qt}\right)-\cos\left(r_{qt}\right)\right).$$ Using the previous equalities, we obtain $$\alpha_5=\frac{m^2q^3t^4}{2\sigma^2}\tau_1\frac{4\sin^2\frac{r_{qt}}{2}-r_{qt}\sin\left(r_{qt}\right)}{r_{qt}^3} \label{E:vcr6}$$ and $$\begin{aligned} &\widetilde{F}_5(0)=\frac{m^2q^3t^4}{2\sigma^2}\tau_2\frac{4\sin^2\frac{r_{qt}}{2} -r_{qt}\sin\left(r_{qt}\right)}{r_{qt}^3} \\ &\quad-\frac{m^2q^3t^4}{2\sigma^2}\tau_1\left[\frac{3t^5}{2r_{qt}^5}\left(r_{qt} \sin\left(r_{qt}\right)-4\sin^2\frac{r_{qt}} {2}\right)+\frac{t^5}{2r_{qt}^3}\left(\frac{3}{r_{qt}}\sin\left(r_{qt}\right)-\cos\left(r_{qt}\right)\right)\right]. \label{E:vcr7}\end{aligned}$$ Our next goal is to prove that $\alpha_5> 0$. Indeed, we have $$4\sin^2\frac{r_{qt}}{2}-r_{qt}\sin\left(r_{qt}\right)=2\sin\frac{r_{qt}}{2}\left(2\sin\frac{r_{qt}}{2} -r_{qt}\cos\frac{r_{qt}}{2} \right). \label{E:chto}$$ Since $\frac{\pi}{4}<\frac{r_{qt}}{2}<\frac{\pi}{2}$ and $x<\tan x$ for $\frac{\pi}{4}< x<\frac{\pi}{2}$, inequality $\alpha_5> 0$ follows from (\[E:vcr6\]) and (\[E:chto\]). We can now apply Theorem \[T:lti\] with the data given by (\[E:data1\]) and (\[E:data2\]). Recall that we denoted by $\alpha$ the positive number given by $$\alpha=\alpha_2+\alpha_3+\alpha_4+\alpha_5 \label{E:chi1}$$ where $\alpha_2$, $\alpha_3$, $\alpha_4$, and $\alpha_5$ are defined in (\[E:vcr0\]), (\[E:vcr2\]), (\[E:vcr4\]), and (\[E:vcr6\]), respectively. We also denoted by $F$ the function $F_2+F_3+F_4+F_5$ (see (\[E:data2\])), and put $$\widetilde{F}(0)=\widetilde{F}_2(0)+\widetilde{F}_3(0)+\widetilde{F}_4(0)+\widetilde{F}_5(0) \label{E:chi2}$$ where $\widetilde{F}_2(0)$, $\widetilde{F}_3(0)$, $\widetilde{F}_4(0)$, and $\widetilde{F}_5(0)$ are given by (\[E:vcr1\]), (\[E:vcr3\]), (\[E:vcr5\]), and (\[E:vcr7\]), respectively. By applying Theorem \[T:lti\] to (\[E:expo1\]), we see that $$y^{-\frac{1}{2}}\exp\left\{\left(q^2-v_{q,t}\right)y\right\} m_t\left(\frac{\sqrt{2}\sigma}{\sqrt{t}}\sqrt{y}\right) =\frac{\sqrt{t}}{\sqrt{2\pi}\sigma} e^{\frac{qt}{2}}\lambda_1^{\frac{1}{2}}e^{\widetilde{F}(0)}y^{-\frac{1}{2}} e^{2\sqrt{\alpha}\sqrt{y}}\left(1+O\left(y^{-\frac{1}{4}}\right)\right) \label{E:ou2}$$ as $y\rightarrow\infty$. Next, replacing $y$ by $y^2\frac{t}{2\sigma^2}$ in formula (\[E:ou2\]), we obtain $$\begin{aligned} m_t\left(y\right)&=\frac{\sqrt{t}}{\sqrt{2\pi}\sigma} e^{\frac{qt}{2}}\lambda_1^{\frac{1}{2}}e^{\widetilde{F}(0)} \exp\left\{\frac{\sqrt{2\alpha t}}{\sigma}y\right\}\exp\left\{-\frac{t\left(q^2-v_{q,t}\right)}{2\sigma^2}y^2\right\} \left(1+O\left(y^{-\frac{1}{2}}\right)\right) \label{E:ou3}\end{aligned}$$ as $y\rightarrow\infty$. Now, it is clear that formula (\[E:ou3\]) implies Theorem \[T:ohh\]. Moreover, the following lemma holds: \[L:con3\] The constants $E$, $F$, and $G$ in Theorem \[T:ohh\] are given by $$E=\frac{\sqrt{t}}{\sqrt{2\pi}\sigma} e^{\frac{qt}{2}}\lambda_1^{\frac{1}{2}}e^{\widetilde{F}(0)},\quad F=\sigma^{-1}\sqrt{2\alpha t},\quad\mbox{and}\quad G=\left(2\sigma^2\right)^{-1}t\left(q^2+t^{-2}r_{qt}^2\right),$$ where the numbers $\lambda_1> 0$, $\alpha> 0$, and $\widetilde{F}(0)$ are defined in (\[E:sim1\]), (\[E:chi1\]), and (\[E:chi2\]), respectively. This completes the proof of Theorem \[T:ohh\]. Proof of Theorem \[T:maino\] {#S:qv} ============================ We will first show that it is possible to apply Theorem \[T:into\] with $A(y)=y^{-1}m_t(y)e^{Gy^2}$, $k=\sqrt{G+\frac{t}{8}}$, $l=F$, $\zeta(y)=Ey^{-1}$, and $b(y)=y^{-\frac{1}{2}}$, where the constants $E$, $F$, and $G$ are such as in Lemma \[L:con3\]. It is not hard to see that condition 2 in Theorem \[T:into\] follows from formula (\[E:ohh1\]). In addition, it is clear that condition 3 with $\gamma=1$ holds. The validity of condition 1 in Theorem \[T:into\] can be shown by reasoning as in the proof of Lemma \[L:dodo\]. Here we use formula (\[E:expo1\]) instead of formula (\[E:laai\]). It follows from Theorem \[T:into\] that $$\begin{aligned} &\int_0^{\infty}y^{-1}m_t(y)\exp\left\{-\left(\frac{z^2}{y^2}+\frac{ty^2}{8}\right)\right\}dy \nonumber \\ &=E\frac{\sqrt{\pi}}{2k}\exp\left\{\frac{F^2}{16k^2}\right\}k^{\frac{1}{2}}z^{-\frac{1}{2}} \exp\left\{Fk^{-\frac{1}{2}}\sqrt{z}\right\}e^{-2kz}\left(1+O\left(z^{-\frac{1}{4}}\right)\right) \label{E:do2}\end{aligned}$$ as $z\rightarrow\infty$. Replacing $z$ by $\displaystyle{\frac{\log x}{\sqrt{2t}}}$ in formula (\[E:do2\]) and taking into account formula (\[E:stvoll1\]) and the equality $\displaystyle{k=\frac{\sqrt{8G+t}}{2\sqrt{2}}}$, we obtain $$\begin{aligned} &D_t\left(x_0e^{\mu t}x\right)=\frac{E}{x_0e^{\mu t}}2^{-\frac{1}{2}}t^{-\frac{1}{4}} (8G+t)^{-\frac{1}{4}}\exp\left\{\frac{F^2}{2(8G+t)}\right\} \nonumber \\ &\quad(\log x)^{-\frac{1}{2}} \exp\left\{\frac{F\sqrt{2}}{t^{\frac{1}{4}}(8G+t)^{\frac{1}{4}}}\sqrt{\log x}\right\}x^{-\left(\frac{3}{2}+ \frac{\sqrt{8G+t}}{2\sqrt{t}}\right)} \left(1+O\left((\log x)^{-\frac{1}{4}}\right)\right) \label{E:do3}\end{aligned}$$ as $x\rightarrow\infty$. Now, it is clear that formula (\[E:do3\]) implies Theorem \[T:maino\]. In addition, the following lemma holds: \[L:con4\] The constants $B_1$, $B_2$, and $B_3$ are given by $B_1=\frac{E}{x_0e^{\mu t}}2^{-\frac{1}{2}}t^{-\frac{1}{4}} (8G+t)^{-\frac{1}{4}}\exp\left\{\frac{F^2}{2(8G+t)}\right\}$, $B_2=\frac{F\sqrt{2}}{t^{\frac{1}{4}}(8G+t)^{\frac{1}{4}}}$, and $B_3=\frac{3}{2}+ \frac{\sqrt{8G+t}}{2\sqrt{t}}$, where the numbers $E$, $F$, and $G$ are defined in Lemma \[L:con3\]. It is an interesting fact that Theorem \[T:maino\] with $m=0$ is a special case of Theorem \[T:main\]. This will be explained below. Recall that in model (\[E:SS\]), the volatility process is the absolute value of an Ornstein-Uhlenbeck process. The following explicit representation is valid for the Ornstein-Uhlenbeck process $\widetilde{Y}_t$ satisfying the stochastic differential equation $d\widetilde{Y}_t=q\left(m-\widetilde{Y}_t\right)dt+\sigma dZ_t$: $$\widetilde{Y}_t\left(q,m,\sigma,y_0\right)=e^{-qt}y_0+\left(1-e^{-qt}\right)m+\sigma e^{-qt}\int_0^te^{qu}dZ_u$$ (see, e.g., [@keyN], Proposition 3.8). Therefore, $$\widetilde{Y}_t\left(q,m,\sigma,y_0\right)=\widetilde{Y}_t\left(q,0,\sigma,y_0+\left(e^{qt}-1\right)m\right). \label{E:ss2}$$ It is known that squared Ornstein-Uhlenbeck processes are related to CIR-processes. Indeed, it is not hard to see, using the Itô formula, that the squared Ornstein-Uhlenbeck process $T_t=\widetilde{Y}_t\left(q,0,\sigma,y_0\right)^2$ satisfies the following stochastic differential equation: $dT_t=\left(\sigma^2-2qT_tdt\right)+2\sigma\sqrt{T_t}dZ_t$. Therefore, the uniqueness implies that the process $\widetilde{Y}_t\left(q,0,\sigma,z_0\right)^2$ is indistinguishable from the CIR-process $Y_t\left(\sigma^2,-2q,2\sigma,z_0^2\right)$. It follows from (\[E:ss2\]) that $$\widetilde{Y}_t\left(q,m,\sigma,y_0\right)^2=Y_t\left(\sigma^2,-2q,2\sigma,\left(y_0+\left(e^{qt}-1\right)m\right)^2\right), \label{E:beou1}$$ and hence in the case where $m=0$, the mixing distribution densities corresponding to the processes on the both sides of (\[E:beou1\]) coincide. Therefore, all the results concerning the distribution density of the stock price process in model (\[E:H\]) can be reformulated for model (\[E:SS\]) with $m=0$. For instance, formula (\[E:lai2\]) becomes $$\begin{aligned} &\int_0^{\infty}\exp\left\{-\lambda y\right\}y^{-\frac{1}{2}} m_t\left(t^{-\frac{1}{2}}y^{\frac{1}{2}};q,0,\sigma,y_0\right)dy \\ &=2\sqrt{t}\exp\left\{\frac{qt}{2}\right\}\left(\frac{\sqrt{q^2+2\sigma^2\lambda}} {\sqrt{q^2+2\sigma^2\lambda}\cosh(t\sqrt{q^2+2\sigma^2\lambda}) +q\sinh(t\sqrt{q^2+2\sigma^2\lambda})}\right)^{\frac{1}{2}} \\ &\quad\exp\left\{-\frac{y_0^2\lambda\sinh(t\sqrt{q^2+2\sigma^2\lambda})} {\sqrt{q^2+2\sigma^2\lambda}\cosh(t\sqrt{q^2+2\sigma^2\lambda}) +q\sinh(t\sqrt{q^2+2\sigma^2\lambda})}\right\}.\end{aligned}$$ Summarizing what was said above, we see that Theorem \[T:maino\] with $m=0$ follows from Theorem \[T:main\] and formula (\[E:beou1\]). The next statement concerns the moment explosion problem for the Stein-Stein model. \[L:blupo\] Let $q\ge 0$, $m\ge 0$, $\sigma> 0$, and $p\in\mathbb{R}$. Then, the following statement is true for the moment $l_p$ of the stock price distribution density $D_t$ in model (\[E:SS\]): $$l_p<\infty\Longleftrightarrow\frac{1}{2}-\frac{\sqrt{8C+t}}{2\sqrt{t}}< p< \frac{1}{2}+\frac{\sqrt{8C+t}}{2\sqrt{t}},$$ where $\displaystyle{C=\frac{1}{2\sigma^2}\left(tq^2+t^{-1}r_{tq}^2\right)}$. It is not hard to see that Lemma \[L:blupo\] follows from Lemma \[L:blup\] and formula (\[E:beou1\]). Proof of Theorem \[T:impik\] ============================ Let $\bar{K}> 0$, and let $f$ and $g$ be positive functions on the interval $[\bar{K},\infty)$. We will write $f(x)\approx g(x)$, $x\rightarrow\infty$, if there exist constants $c_1> 0$, $c_2> 0$, and $K_0>\bar{K}$ such that the inequalities $c_1g(x)\le f(x)\le c_2g(x)$ hold for all $x> K_0$. The following notation will be used below: $\hat{V}_0(k)=V_0(K)$ (see (\[E:opt\]) for the definition of $V_0$). The next lemma was obtained in [@keyGS3]. \[L:genet\] Suppose that there exist positive increasing continuous functions $\psi$ and $\phi$ such that\ $\lim_{k\rightarrow\infty}\psi(k)=\lim_{k\rightarrow\infty}\phi(k)=\infty$ and $$\hat{V}_0(k)\approx\frac{\psi(k)}{\phi(k)}\exp\left\{-\frac{\phi(k)^2}{2}\right\} \label{E:gene30}$$ as $K\rightarrow\infty$. Then the following asymptotic formula holds: $$\hat{I}(k)=\frac{1}{\sqrt{T}}\left(\sqrt{2k+\phi(k)^2}-\phi(k)\right) +O\left(\frac{\psi(k)}{\phi(k)}\right)$$ as $K\rightarrow\infty$. For the model in (\[E:H\]), let $\psi$ be a function such as in the formulation of Lemma \[L:genet\]. Our next goal is to find a function $\phi$ for which formula (\[E:gene30\]) holds. It follows from (\[E:option\]) that $$\hat{V}_0(k)=e^{-rT}\left(x_0e^{rT}\right)^2\left[\int_{e^k}yD_T\left(x_0e^{rT}y\right)dy-e^k\int_{e^k}^{\infty} D_T\left(x_0e^{rT}y\right)dy\right]. \label{E:zza1}$$ Now it is not hard to see that (\[E:dopo\]) and (\[E:zza1\]) imply $$\hat{V}_0(k)\approx k^{-\frac{3}{4}+\frac{a}{c^2}}e^{A_2\sqrt{k}}e^{-\left(A_3-2\right)k},\quad k\rightarrow\infty. \label{E:zza2}$$ Put $$\phi(k)=\sqrt{\left(2A_3-4\right)k-2A_2\sqrt{k} +\left(\frac{1}{2}-\frac{2a}{c^2}\right)\log k-2\log\psi(k)}.$$ Then we have $\phi(k)\approx\sqrt{k}$, and (\[E:zza2\]) shows that condition (\[E:gene30\]) in Lemma \[L:genet\] holds. Applying this lemma and the mean value theorem, we see that $$\begin{aligned} \hat{I}(k)&=\frac{\sqrt{2}}{\sqrt{T}}\left[\sqrt{\left(A_3-1\right)k -A_2\sqrt{k}+\left(\frac{1}{4}-\frac{a}{c^2}\right)\log k}-\sqrt{\left(A_3-2\right)k-A_2\sqrt{k}+\left(\frac{1}{4} -\frac{a}{c^2}\right)\log k}\right] \\ &+O\left(\frac{\psi(k)}{\sqrt{k}}\right),\quad k\rightarrow\infty.\end{aligned}$$ Next using the fact that $\sqrt{1-h}=1-\frac{1}{2}h+O\left(h^2\right)$ as $h\downarrow 0$, we obtain (\[E:impik1\]). The proof of (\[E:impik2\]) is similar. Here we use (\[E:dop1\]) instead of (\[E:dopo\]). [11]{} Andersen, L. B. G., Piterbarg, V. V.: [Moment explosions in stochastic volatility models.]{} Finance Stoch. **11, 29-50 (2007) Benaim, S., Friz, P.: [Regular variation and smile asymptotics.]{} To appear in Math. Finance. Benaim, S., Friz, P., Lee, R.: [The Black-Scholes implied volatility at extreme strikes.]{} Preprint (2007) Borodin, A. N., Salminen, P.: [Handbook of Brownian Motion - Facts and Formulae.]{} Birkhäuser Verlag, Basel (1996) Carr, P., Schröder, M.: [Bessel processes, the integral of Geometric Brownian motion, and Asian options.]{} Theory Probab. Appl. **48, 400-425 (2004) Cox, J. C., Ingersoll, J. E., Ross, S. A.: [A theory of the term structure of interest rates.]{} Econometrica **53, 385-407 (1985) Dr$\rm\breve{a}$gulescu, A. A., Yakovenko, V. M.: [Probability distribution of returns in the Heston model with stochastic volatility.]{} Quantitative Finance **2, 443-453 (2002) Dufresne, D.: [Bessel processes and a functional of Brownian motion.]{} University of Melbourne ePrints Repository (2004) Fouque, J.-P., Papanicolaou, G., Sircar, K. R.: [Derivatives in Financial Markets with Stochastic Volatility.]{} Cambridge University Press, Cambridge (2000) Göing-Yaeschke, A., Yor, M.: [A survey and some generalizations of Bessel processes.]{} Bernoulli **9, 313-349 (2003) Gulisashvili, A., Stein, E. M.: [Asymptotic behavior of the distribution of the stock price in models with stochastic volatility: the Hull-White model.]{} C. R. Acad Sci. Paris, Ser. I **343, 519-523 (2006) Gulisashvili, A., Stein, E. M.: [Asymptotic behavior of distribution densities in models with stochastic volatility, I.]{} Submitted for publication. Gulisashvili, A., Stein, E. M.: [Implied volatility in the Hull-White model.]{} To be published in Mathematical Finance. Heston, S. L.: [A closed-form solution for options with stochastic volatility, with applications to bond and currency options.]{} Review of Financial Studies **6, 327-343 (1993) Lee, R.: [The moment formula for implied volatility at extreme strikes.]{} Math. Finance **14, 469-480 (2004) Nielsen, L. T.: [Pricing and Hedging of Derivative Securities.]{} Oxford University Press, Oxford (1999) Pitman, J., Yor, M.: [A decomposition of Bessel bridges.]{} Z. Wahrscheinlichkeitstheorie verw. Gebiete **59, 425-457 (1982) Revuz, D., Yor, M.: [Continuous Martingales and Brownian motion.]{} Springer-Verlag, Berlin (1991) Stein, E. M., Stein, J.: [Stock price distributions with stochastic volatility: An analytic approach.]{} Review of Financial Studies **4, 727-752 (1991) Stein, E. M., Shakarchi, R.: [Complex Analysis.]{} Princeton University Press, Princeton and Oxford (2003) Wenocur, M. L.: [Ornstein-Uhlenbeck process with quadratic killing,]{} Journal of Applied Probability **27, 707-712 (1990)**********************

--- abstract: | The [Fréchet]{} distance is a popular distance measure for curves. We study the problem of clustering time series under the [Fréchet]{} distance. In particular, we give $(1+{{\varepsilon}})$-approximation algorithms for variations of the following problem with parameters ${\ensuremath{k}}$ and ${\ensuremath{\ell}}$. Given $n$ univariate time series $P$, each of complexity at most $m$, we find ${\ensuremath{k}}$ time series, not necessarily from $P$, which we call *cluster centers* and which each have complexity at most ${\ensuremath{\ell}}$, such that (a) the maximum distance of an element of $P$ to its nearest cluster center or (b) the sum of these distances is minimized. Our algorithms have running time near-linear in the input size for constant ${{\varepsilon}}, {\ensuremath{k}}$ and ${\ensuremath{\ell}}$. To the best of our knowledge, our algorithms are the first clustering algorithms for the [Fréchet]{} distance which achieve an approximation factor of $(1+{{\varepsilon}})$ or better. Keywords: time series, longitudinal data, functional data, clustering, [Fréchet]{} distance, dynamic time warping, approximation algorithms. author: - 'Anne Driemel [^1]' - 'Amer Krivošija [^2]' - 'Christian Sohler [^3]' bibliography: - 'Trajectories.bib' title: 'Clustering time series under the [Fréchet]{}distance [^4] ' --- Introduction ============ Time series are sequences of discrete measurements of a continuous signal. Examples of data that are often represented as time series include stock market values, electrocardiograms (ECGs), temperature, the earth’s population, and the hourly requests of a webpage. In many applications, we would like to automatically analyze time series data from different sources: for example in industry 4.0 applications, where the performance of a machine is monitored by a set of sensors. An important tool to analyze (time series) data is clustering. The goal of clustering is to partition the input into groups of similar time series. Its purpose is to discover hidden structure in the input data and/or summarize the data by taking a representative of each cluster. Clustering is fundamental for performing tasks as diverse as data aggregation, similarity retrieval, anomaly detection, and monitoring of time series data. Clustering of time series is an active research topic [@ar-dcaa-2013; @Boecking2014129; @cmp-fcis-07; @Hsu2014358; @jp-fdc-13; @Li2013243; @liao05survey; @tc-sbc-12; @Petitjean2011678; @rm-fcbwm-06; @sf-mpfa-14; @hd-tsc-14; @wsh-cbc-06; @zt-MODIS-14; @qg-nadtw-12] however, most solutions lack a rigorous algorithmic analysis. Therefore, in this paper we study the problem of clustering time series from a theoretical point of view. Formally, a *time series* is a recording of a signal that changes over time. It consists of a series of paired values $(w_i,t_i)$, where $w_i$ is the $i$th measurement of the signal, and $t_i$ is the time at which the $i$th measurement was taken. A common approach is to treat time series data as point data in high-dimensional space. That is, a time series $S_j=(w_1,t_1),\dots,(w_m,t_m)$ of the input set is treated as the point $S^{*}_j=(w_1,\dots,w_m)$ in $m$-dimensional Euclidean space. Using this simple interpretation of the data, any clustering algorithm for points can be applied. Despite it being common practice (for a survey, see [@liao05survey]), it has many limitations. One major drawback is the requirement that all time series must have the same length and the sampling must be regular and synchronized. In particular, the latter requirement is often hard to achieve. In this paper, we follow a different approach that has also received much attention in the literature. In order to formulate the objective function of our clustering problem, we consider a distance measure that allows for irregular sampling and local shift and that only depends on the “shape” of the analyzed time series: the [Fréchet]{} distance. The [Fréchet]{} distance is defined for continuous functions $f:[0,1]\rightarrow {{\rm I\!\hspace{-0.025em} R}}$. Two functions $f,g$ have [Fréchet]{} distance at most $\delta$, if one can move simultaneously but at different and possibly changing positive speed on the domains of $f$ and $g$ from $0$ to $1$ such that at all times $|f(x)-g(y)|\le \delta$, where $x$ and $y$ are the current positions on the respective domains. The [Fréchet]{} distance is the infimum over the values $\delta$ that allow such a movement. For a formal definition, see [Section \[sec:problem\]]{}. The [Fréchet]{} distance between two monotone functions $f,g:[0,1] \rightarrow {{\rm I\!\hspace{-0.025em} R}}$ equals the maximum of $|f(0)-g(0)|$ and $|f(1)-g(1)|$. This also implies, that the [Fréchet]{} distance between two functions is completely determined by the sequence of local extrema ordered from $0$ to $1$. In order to consider the [Fréchet]{} distance of time series, we view a time series as a specification of the sequence of local extrema of a function. Once we have defined the distance measure we can also formulate the clustering problem we want to study. We will look at two different variants: $k$-center clustering and $k$-median clustering. Both methods are based on the idea of representing a cluster by a cluster center, which can be thought of as being a representative for the cluster. In $k$-center clustering, the objective is to find a set of $k$ time series (the cluster centers) such that the maximum [Fréchet]{} distance to the nearest cluster center is minimized over all input time series. In $k$-median clustering, the goal is to find a set of $k$ centers that minimizes the sum of distances to the nearest centers. However, this is not yet the problem formulation we consider. We would like to address another problem that often occurs with time series: noise. In many applications where we study time series data, the measurements are noisy. For example, physical measurements typically have measurement errors and when we want to determine trends in stock market data, the effects of short term trading is usually not of interest. Furthermore, a cluster center that minimizes the [Fréchet]{} distance to many curves may have a complexity (length of the time series) that equals the sum of the complexities of the time series in the cluster. This will typically lead to a vast overfitting of the data. We address this problem by limiting the complexity of the cluster center. Thus, our problem will be to find $k$ cluster centers each of complexity at most $\ell$ that minimize the $k$-center and $k$-median objective under the [Fréchet]{} distance. It seems that none of the existing approaches summarizes the input along the time-dimension in such a way. #### Our results To the best of our knowledge, clustering under the [Fréchet]{} distance has not been studied before. We develop the first $(1+{{\varepsilon}})$-approximation algorithm for the $k$-center and the $k$-median problem under [Fréchet]{} distance when the complexity of the centers is at most $\ell$. For constant ${{\varepsilon}}, k$ and $\ell$, the running time of our algorithm is $\widetilde O(nm)$, where $n$ is the number of input time series and $m$ their maximum complexity. (see [Theorem \[theo:k:l:center:main\]]{} and [Theorem \[theo:k:l:median:main\]]{}.) We also prove that clustering univariate curves is [**NP**]{}-hard and show that the doubling dimension of the [Fréchet]{} (pseudo)metric space is infinite for univariate curves. #### Challenge and ideas High-dimensional data pose a common challenge in many clustering applications. The challenge in clustering under the [Fréchet]{}distance is twofold: High dimensionality of the joint parametric space of the set of time series: For the seemingly simple task of computing the ([Fréchet]{}) *median* of a fixed group of time series, state-of-the-art algorithms run exponentially in the number of time series [@ahn2015middle; @hr-fdre-14], since the standard approach is to search the joint parametric space for a monotone path [@ag-cfdbt-95; @akw-mpcfd-10; @buchin2012four; @bbw-eapcm-09; @feldman2012gps; @GudVahr12; @hr-fdre-14; @mssz-fdsl-2011]. High dimensionality of the metric space: the doubling dimension of the [Fréchet]{}metric space is infinite, as we will show, even if we restrict the length of the time series. Existing ($1+{{\varepsilon}})$-approximation algorithms [@abs-cm-10; @kumar2010lineartime] with comparable running time are only known to work for special cases as the Euclidean $k$-median problem or (more generally) for metric spaces with bounded doubling dimension [@abs-cm-10]. These two challenges make the clustering task particularly difficult, even for short univariate time series. Our approach exploits the low dimensionality of the ambient space of the time series and the fact that we are looking for low-complexity cluster centers which best describe the input. We introduce the concept of *signatures* ([Definition \[def:signature\]]{}) which capture critical points of the input time series. We can show that each signature vertex of an input curve needs to be matched to a different vertex of its nearest cluster center ([Lemma \[lemma:nec:suff\]]{}). Furthermore, we use a technique akin to *shortcutting*, which has been used before in the context of partial curve matching [@bbw-eapcm-09; @dh-jydfd-13]. We show that any vertex of an optimal solution that is *not* matched to a signature vertex, can be omitted from the solution without changing its cost ([Theorem \[theo:remove:one\]]{}). These ingredients enable us to generate a constant-size set of candidate solutions. The ideas going into the individual parts can be summarized as follows. For the $k$-center problem we generate a candidate set based on the entire input and a decision parameter. If the candidate set turns out too large, then we conclude that there exists no solution for this parameter, since more vertices would be needed to “describe” the input. For the $k$-median problem we use the approach of random sampling used previously by Kumar [*et al.*]{} [@kumar2010lineartime] and Ackermann [*et al.*]{} [@abs-cm-10]. We show that one can generate a constant-size candidate set that contains a $(1+{{\varepsilon}})$-approximation to the $1$-median based on a sample of constant size. To achieve this, we observe that a vertex of the optimal solution that is matched to a signature vertex and which is unlikely to be induced by our sample, can be omitted without increasing the cost by a factor of more than $(1+{{\varepsilon}})$ ([Lemma \[lemma:omit:low:prob\]]{}). #### Related Work A distance measure that is closely related to the [Fréchet]{} distance is *Dynamic time warping (DTW)*. DTW has been popularized in the field of data mining [@dtswk-08; @mueller07dtw] and is known for its unchallenged universality. It is a discrete version of the [Fréchet]{}distance which measures the total sum of the distances taken at certain fixed points along the traversal. The process of traversing the curves with varying speeds is sometimes referred to as “time-warping” in this context. It has been successfully used in the automated classification of time series describing phenomena as diverse as surgical processes [@Forestier2012255], whale singing [@bmp-ackw-07], chromosomes [@Legrand2008215], fingerprints [@888711], electrocardiogram (ECG) frames [@1013101] and many others. While DTW was born in the 80’s from the use of dynamic programming with the purpose of aligning distorted speech signals, the [Fréchet]{} distance was conceived by Maurice [Fréchet]{} at the beginning of the 20th century in the context of the study of general metric spaces. The best known algorithms for computing either distance measure between two input time series have a worst-case running time which is roughly quadratic in the number of time stamps [@buchin2012four; @mueller07dtw]. Faster algorithms exist for both problems under certain input assumptions [@dhw-afd-12; @keogh2005exact]. Recently, Bringmann showed [@Bringmann14] that the [Fréchet]{} distance between two polygonal curves lying in the plane cannot be computed in strongly subquadratic time, unless the Strong Exponential Time Hypothesis (SETH) fails. This was extended by Bringmann and Mulzer to the 1-dimensional case [@bm-adfd-15]. Both distance measures consider only the ordering of the measurements and ignore the explicit time stamps. This makes them robust against local deformations. Both distance measures deal well with irregular sampling and have been used in combination with curve simplification techniques [@ahmw-nltcs-05; @dhw-afd-12; @keogh1999scaling]. However, while the [Fréchet]{}distance is inherently independent of the sampling of the curves, DTW does not work well when one of the two curves is sampled much less frequently (see [Figure \[fig:subsampling\]]{}). Since we are interested in finding cluster centers of low complexity, we therefore focus on the [Fréchet]{}distance. ![Illustration why a continuous distance measure performs better than a discrete distance measure if sampling rates differ substantially. Shown is the curve $\tau$ and the same curve subsampled at times $s_1,\dots,s_5$ (for better visibility, the subsampled curve $\tau'$ is translated by $L$). A discrete assignment between the vertices of the curves is shown in light blue. $\tau$ records a linear process with some error ${{\varepsilon}}$. A suitable distance measure should estimate the distance between $\tau$ and $\tau'$ in the interval $[0,2{{\varepsilon}}]$. This is true for the (normalized) DTW, if the rate of subsampling is high enough. Similarly it is true for the (continuous) [Fréchet]{}distance at any rate of subsampling. However, as the rate of subsampling decreases, DTW will estimate the distance increasingly larger. ](subsampling.pdf) [\[fig:subsampling\]]{} The problem of clustering points in general metric spaces has been extensively studied in many different settings. The problem is known to be computationally hard in most of its variants [@aloise2009nphard; @jain2002greedy; @megiddo1984geo; @Vaziranibook]. A number of polynomial-time constant-factor approximation algorithms are known  [@AryaGKMMP04; @CharikarG05; @cgts-kmedian-02; @CharikarL12; @c-kmc-09; @feder1988optimal; @Gonzalez1985; @HochbaumShmoys1985; @jain2002greedy; @ls-akm-13]. For the $k$-center problem the best such algorithm achieves a $2$-approximation [@Gonzalez1985; @HochbaumShmoys1985] and this is also the best lower bound for the approximation ratio of the polynomial-time algorithms unless $\textbf{P}={\textbf{NP}}$. This picture looks much different if one makes certain assumptions on the graph that defines the metric, as done in the work by Eisenstadt [*et al.*]{} [@emk-akcpg-14], for example. For the $k$-median problem in a general metric space a $(1+\sqrt{3}+{{\varepsilon}})$-approximation can be achieved [@ls-akm-13]. The best lower bound for the approximation ratio of polynomial time algorithm is $1+2/e\approx 1.736$ unless ${\textbf{NP}}\subseteq \text{DTIME}\left[ n^{O(\log\log n)}\right]$ [@jain2002greedy]. If the data points are from the Euclidean space ${{\rm I\!\hspace{-0.025em} R}}^d$, a number of different algorithms are known that compute a $(1+{{\varepsilon}})$-approximation to the $k$-median problem [@c-kmc-09; @FeldmanLangberg11; @HarPeledKushal07; @harpeledmazumdar2004; @KR99; @kumar2010lineartime]. Ackermann [*et al.*]{} [@abs-cm-10] show that under certain conditions a randomized $(1+{{\varepsilon}})$-approximation algorithm by Kumar [*et al.*]{} [@kumar2010lineartime] can be extended to general distance measures. In particular, they show that one can compute a $(1+{{\varepsilon}})$-approximation to the $k$-median problem if the distance metric is such that the $1$-median problem can be $(1+{{\varepsilon}})$-approximated using information only from a random sample of constant size. They show that this is the case for metrics with bounded doubling dimension. In our case, the doubling dimension is, however, unbounded. A different approach would be to use the technique of metric embeddings, which proved successful for clustering in Euclidean high-dimensional spaces. A metric embedding is a mapping between two metric spaces which preserves distances up to a certain distortion. Unlike the similar Hausdorff distance, for which non-trivial embeddings are known, it is not known if the [Fréchet]{}distance can be embedded into an $\ell_{p}$ space using finite dimension [@IndMat04]. Any finite metric space can be embedded into $\ell_\infty$, which is therefore considered as “the mother of all metrics”. In turn, any bounded set in ${{\rm I\!\hspace{-0.025em} R}}^d$ with $\ell_{\infty}$ can be embedded as curves with the [Fréchet]{}distance (see also [Section \[sec:nphard\]]{}). Recent work by Bartal [*et al.*]{} [@bartal2014impossible] suggests that, due to the infinite doubling dimension, a metric embedding of the [Fréchet]{}distance into an $\ell_p$-space needs to have at least super-constant distortion. On the positive side, Indyk showed how to embed the [Fréchet]{}distance into an inner product space for the purpose of designing approximate nearest-neighbor data structures [@i-approxnn-02]. However, this work focuses on a different version of the [Fréchet]{}distance, namely the *discrete* [Fréchet]{}distance, which is aimed at sequences instead of continuous curves. For any $t>1$, the resulting data structure is $O({\!\left({\log m \log \log n}\right)}^{t-1})$-approximate, uses space exponential in $tm^{1/t}$, and achieves $(m+\log n)^{O(1)}$ query time, where $m$ is the maximum length of a sequence and $n$ is the number of sequences stored in the data structure. The problem of clustering under the DTW distance has also been studied in the data mining community. In particular, substantial effort has been put into extending Lloyd’s algorithm [@lloyd82] to the case of DTW, an example is the work of Petitjean [*et al.*]{} [@Petitjean2011678]. In order to use Lloyds algorithm, one first has to be able to compute the mean of a set of time series. For DTW, this turns out to be a non-trivial problem and current solutions are not satisfactory. One of the problems is that the complexity of the mean is prone to become quite high, namely linear in the order of the total complexity of the input. This can lead to overfitting in the same way as discussed for the [Fréchet]{}distance. For a more extensive discussion we refer to [@ar-dcaa-2013] and references therein. Also in statistics, the problem of clustering longitudinal data, or functional data, is an active research topic and solutions based on wavelet decomposition and principal component analysis have been suggested [@cmp-fcis-07; @jp-fdc-13; @rm-fcbwm-06; @wsh-cbc-06]. Preliminaries ============= A *time series* is a series $(w_1,t_1),\ldots ,(w_m,t_m)$ of measurements $w_i$ of a signal taken at times $t_i$. We assume $0=t_1<t_2<\ldots <t_m=1$ and $m$ is finite. A time series may be viewed as a continuous function $\tau: [0,1] \rightarrow {{\rm I\!\hspace{-0.025em} R}}$ by linearly interpolating $w_1,\dots,w_m$ in order of $t_i$, $i=1,\ldots m$. We obtain a polygonal curve with *vertices* $w_1=\tau(t_1),\dots, w_m=\tau(t_m)$ and segments between $w_i$ and $w_{i+1}$ called *edges* $\overline{w_i w_{i+1}}=\{xw_i+(1-x)w_{i+1}|x\in [0,1]\}$. We will simply refer to $\tau$ as a *curve*. When defining such a curve, we may write “curve $\tau:[0,1]\rightarrow{{\rm I\!\hspace{-0.025em} R}}$ with $m$ vertices”, or “curve $\tau=w_1,\ldots, w_m$”. [^5] We say that such a curve $\tau$ has *complexity* $m$. We denote its set of vertices with ${\ensuremath{\mathcal{V}}}(\tau)$. For any $t_i<t_j \in [0,1]$, we denote the subcurve of $\tau$ starting at $\tau(t_i)$ and ending at $\tau(t_j)$ with $\tau[t_i,t_j]$. We define ${\ensuremath{\min( \tau[t^{-},t^{+}])}}=\min{\ensuremath{{\left\{ { \tau(t) ~|~ t \in [t^{-},t^{+}]} \right\}} }}$, and ${\ensuremath{\max( \tau[t^{-},t^{+}])}}=\max{\ensuremath{{\left\{ { \tau(t) ~|~ t \in [t^{-},t^{+}]} \right\}} }}$, to denote the minimum and maximum along a (sub)curve. Furthermore, we will use the following non-standard notation for an interval. For any $a,b\in {{\rm I\!\hspace{-0.025em} R}}$ we define ${\left\langle{a,b}\right\rangle}={\left[ {\min(a,b), \max(a,b)} \right]}$. We denote ${\ensuremath{[h]_{\delta}}}=[h-\delta, h+\delta]$. For any set $P$, we denote its cardinality with $|P|$. Let $\mathcal H$ denote the set of continuous and increasing functions $f:[0,1]\rightarrow[0,1]$ with the property that $f(0)=0$ and $f(1)=1$. For two given functions $\tau:[0,1]\rightarrow {{\rm I\!\hspace{-0.025em} R}}$ and $\pi:[0,1]\rightarrow {{\rm I\!\hspace{-0.025em} R}}$, their *[Fréchet]{}distance* is defined as $$\label{def:frechet} {\ensuremath{d_F{\!\left({\tau,\pi}\right)}}}=\inf_{f\in \mathcal H}\; \max_{t \in [0,1]} \| \tau(f(t))-\pi(t) \|,$$ The [Fréchet]{} distance between two time series is defined as the [Fréchet]{} distance of their corresponding continuous functions. [^6] Note that any $f \in \mathcal H$ induces a bijection between the two curves. We refer to the function $f$ that realizes the [Fréchet]{}distance as a *matching*. It may be that such a matching exists in the limit only. That is, for any ${{\varepsilon}}>0$, there exists a $f\in \mathcal H$ that matches each point on $\tau$ to a point on $\pi$ within distance ${\ensuremath{d_F{\!\left({\tau,\pi}\right)}}}+{{\varepsilon}}$. The [Fréchet]{} distance is a pseudo metric [@ag-cfdbt-95], i.e. it satisfies all properties of a metric space except that there may be two different functions that have distance $0$. If one considers the equivalence classes that are induced by functions of pairwise distance $0$ we can obtain a metric space $(\Delta,{\ensuremath{d_F}})$ defined by the [Fréchet]{}distance and the set $\Delta$ of all (equivalence classes of) univariate time series. We denote with $\Delta_m$ the set of all univariate time series of complexity at most $m$. We notice that only the ordering of the $w_i$ is relevant and that under [Fréchet]{}distance two curves can be thought of being identical, if they have the same sequence of local minima and maxima. Therefore, we can assume that a curve is induced by its sequence of local minima and maxima and we will use the term curve in the paper to describe the equivalence class of curves with pairwise [Fréchet]{}distance 0. [\[def:concatenation\]]{} Let two curves $\tau_1:[a_1,b_1]\rightarrow\mathbb{R}$, $0\leq a_1\leq b_1\leq 1$, and $\tau_2:[a_2,b_2]\rightarrow\mathbb{R}$, $0\leq a_2\leq b_2\leq 1$ be given, such that $\tau_1(b_1)=\tau_2(a_2)$. The concatenation of $\tau_1$ and $\tau_2$ is a curve $\tau$ defined as $\tau=\tau_1\oplus\tau_2:[0,1]\rightarrow\mathbb{R}$, such that $$\tau(t)=\left(\tau_1\oplus\tau_2\right)(t)=\begin{cases}\tau_1\left(a_1+\left(b_1-a_1+b_2-a_2\right)\cdot t\right) & \text{if } t\leq \frac{b_1-a_1}{b_1-a_1+b_2-a_2} \\ \tau_2\left(b_2-\left(b_1-a_1+b_2-a_2\right)\cdot \left(1-t\right)\right) & \text{if } t> \frac{b_1-a_1}{b_1-a_1+b_2-a_2} \end{cases}$$ We are going to use the following simple observations throughout the paper. [\[obs:fd:concat\]]{} Let two curves $\tau:[0,1]\rightarrow {{\rm I\!\hspace{-0.025em} R}}$ and $\pi:[0,1]\rightarrow {{\rm I\!\hspace{-0.025em} R}}$ be the concatenations of two subcurves each, $\tau=\tau_1\oplus \tau_2$ and $\pi=\pi_1\oplus \pi_2$, then it holds that $${\ensuremath{d_F{\!\left({\tau,\pi}\right)}}}\leq \max\lbrace {\ensuremath{d_F{\!\left({\tau_1,\pi_1}\right)}}}, {\ensuremath{d_F{\!\left({\tau_2,\pi_2}\right)}}}\rbrace$$ [\[obs:segments\]]{} Given two edges $\overline{a_1 a_2}$ and $\overline{b_1 b_2}$ with $a_1,a_2,b_1,b_2\in {{\rm I\!\hspace{-0.025em} R}}$, it holds that $${\ensuremath{d_F{\!\left({\overline{a_1 a_2},\overline{b_1 b_2}}\right)}}}=\max{\!\left({|a_1-b_1|,|a_2-b_2|}\right)}.$$ We make the following *general position* assumption on the input. For every input curve ${{\ensuremath{{{\ensuremath{\tau}}}_{}}}}$ we assume that no two vertices ${{\ensuremath{{{\ensuremath{\tau}}}_{}}}}$ have the same coordinates and any two differences between coordinates of two vertices of ${{\ensuremath{{{\ensuremath{\tau}}}_{}}}}$ are different. This assumption can easily be achieved by symbolic perturbation. Furthermore we assume that ${{\ensuremath{{{\ensuremath{\tau}}}_{}}}}$ has no edges of length zero and its vertices are an alternating sequence of minima and maxima, i.e. no vertex lies in the linear interpolation of its two neighboring vertices. Problem statement ----------------- [\[sec:problem\]]{} Given a set of $n$ time series $P=\lbrace{{\ensuremath{{{\ensuremath{\tau}}}_{1}}}},\ldots , {{\ensuremath{{{\ensuremath{\tau}}}_{n}}}}\rbrace \subseteq \Delta_m$ and parameters ${\ensuremath{k}},{\ensuremath{\ell}}\in {{\rm I\!\hspace{-0.025em} N}}$, we define a *$({\ensuremath{k}},{\ensuremath{\ell}})$-clustering* as a set of ${\ensuremath{k}}$ time series $C=\lbrace{{\ensuremath{{{\ensuremath{c}}}_{1}}}},\ldots , {{\ensuremath{{{\ensuremath{c}}}_{{\ensuremath{k}}}}}}\rbrace$ taken from $\Delta_{{\ensuremath{\ell}}}$ which minimize one of the following cost functions: $$\label{def:kcenter} \operatorname{cost}_{\infty}(P,C)=\max_{i=1,\ldots n}\; \min_{j=1,\ldots {\ensuremath{k}}} {\ensuremath{d_F{\!\left({{{\ensuremath{{{\ensuremath{\tau}}}_{i}}}},{{\ensuremath{{{\ensuremath{c}}}_{j}}}}}\right)}}}.$$ $$\label{def:kmedian} \operatorname{cost}_1(P,C)=\sum_{i=1,\ldots n}\; \min_{j=1,\ldots {\ensuremath{k}}} {\ensuremath{d_F{\!\left({{{\ensuremath{{{\ensuremath{\tau}}}_{i}}}},{{\ensuremath{{{\ensuremath{c}}}_{j}}}}}\right)}}}.$$ We refer to the clustering problem as *$({\ensuremath{k}},{\ensuremath{\ell}})$-center* (Equation \[def:kcenter\]) and *$({\ensuremath{k}},{\ensuremath{\ell}})$-median* (Equation \[def:kmedian\]), respectively. We denote the cost of the optimal solution as $$\operatorname{opt}^{(i)}_{{\ensuremath{k}},{\ensuremath{\ell}}}(P) = \min_{C \subset \Delta_{{\ensuremath{\ell}}}} \operatorname{cost}_{i}(P,C),$$ where the restrictions on $C$ are as described above and $i \in \{\infty,1\}$. Note that this corresponds to the classical definition of the $k$-median problem (resp. $k$-center problem) if $D$ is a distance measure on $\Delta_{\ell} \cup P$, defined as $D(p,q)=\infty$ for $p,q \in P$ and $D(p,q)={\ensuremath{d_F{\!\left({p,q}\right)}}}$ otherwise. Note that the new distance measure $D$ does not satisfy the triangle inequality and is therefore not a metric. On signatures of time series ============================ [\[sec:on:signatures\]]{} Before introducing our signatures, we first review similar notions traditionally used for the purpose of curve compression. A simplification of a curve is a curve which is lower in complexity (it has fewer vertices) than the original curve and which is similar to the original curve. This is captured by the following standard definitions. [\[def:min:error:simp\]]{} We call a curve $\pi$ a minimum-error $\ell$-simplification of ${{\ensuremath{{{\ensuremath{\tau}}}_{}}}}$ if for any curve $\pi'$ of at most $\ell$ vertices, it holds that ${\ensuremath{d_F{\!\left({\pi',\tau}\right)}}} \geq {\ensuremath{d_F{\!\left({\pi,\tau}\right)}}}$. [\[def:min:size:simp\]]{} We call a curve $\pi$ a minimum-size $\varepsilon$-simplification of ${{\ensuremath{{{\ensuremath{\tau}}}_{}}}}$ if ${\ensuremath{d_F{\!\left({\pi,{{\ensuremath{{{\ensuremath{\tau}}}_{}}}}}\right)}}}\leq \varepsilon$ and for any curve $\pi'$ such that ${\ensuremath{d_F{\!\left({\pi',{{\ensuremath{{{\ensuremath{\tau}}}_{}}}}}\right)}}}\leq \varepsilon$, it holds that the complexity of $\pi'$ is at least as much as the complexity of $\pi$. The simplification problem has been studied under different names for multidimensional curves and under various error measures, in domains, such as cartography [@dp-73; @ramer1972iterative], computational geometry [@godau1991natural], and pattern recognition [@pratt2002search]. Often, the simplified curve is restricted to vertices of the input curve and the endpoints are kept. However, in our clustering setting, we need to use the more general problem definitions stated above. Historically, the first minimal-size curve simplification algorithm was a heuristic algorithm independently suggested in the 1970’s by Ramer and Douglas and Peucker [@dp-73; @ramer1972iterative] and it remains popular in the area of geographic information science until today. It uses the Hausdorff error measure and has running time $O(n^2)$ (where $n$ denotes the complexity of the input curve), but does not offer a bound to the size of the simplified curve. Recently, worst-case and average-case lower bounds on the number of vertices obtained by this algorithm were proven by Daskalakis [*et al.*]{} [@daskalakis2010good]. Imai and Iri [@ii-pac-1988] solved both the minimum-error and minimum-size simplification problem under the Hausdorff distance by modeling it as a shortest path problem in directed acyclic graphs. Curve simplification using the [Fréchet]{}distance was first proposed by Godau [@godau1991natural]. The current state-of-the-art approximation algorithm for simplification under the [Fréchet]{}distance was suggested by Agarwal [*et al.*]{} [@ahmw-nltcs-05]. This algorithm computes a $2$-approximate minimal-size simplification in time $O(n\log n)$. The framework of Imai and Iri is also used for the streaming algorithm of Abam [*et al.*]{} [@abh-sals-10] under the [Fréchet]{}distance. Driemel and Har-Peled [@dh-jydfd-13] introduced the concept of a *vertex permutation* with the aim of preprocessing a curve for curve simplification. The idea is that any prefix of the permutation represents a bicriteria approximation to the minimal-error curve simplification. In [Section \[sec:computing:signatures\]]{} we will use this concept to develop improved algorithms in our setting where the curves are time series. For time series, a concept similar to simplification called *segmentation* has been extensively studied in the area of data mining [@bingham2006segmentation; @himberg01; @terzi2006efficient]. The standard approach for computing exact segmentations is to use dynamic programming which yields a running time of $O(n^2)$.  [\[fig:signature\]]{} We now proceed to introduce the concept of signatures ([Definition \[def:signature\]]{}). Our definition aligns with the work on computing important minima and maxima in the context of time series compression [@pratt2002search]. Intuitively, the signatures provide us with the “shape” of a time series at multiple scales. Signatures have a unique hierarchical structure (see [Lemma \[lemma:canonical:signature\]]{}) which we can exploit in order to achieve efficient clustering algorithms. Furthermore, the signatures of a curve approximate the respective simplifications under the [Fréchet]{}distance (see [Lemma \[lemma:apx:min:error:simp\]]{}). [Figure \[fig:signature\]]{} shows an example of a signature. We show several crucial properties of signatures in [Subsection \[sec:properties:signatures\]]{}. Signatures always exist and are easy to compute (see [Section \[sec:computing:signatures\]]{}). In particular, we show how to compute the $\delta$-signature of a curve in one pass in linear time ([Theorem \[theo:computing:delta:signature\]]{}), and how to preprocess a curve in near-linear time for fast queries of the signature of a certain size ([Theorem \[theo:compute:signatures\]]{}). [\[def:signature\]]{} We define the $\delta$-signature of any curve $\tau: [0,1]\rightarrow {{\rm I\!\hspace{-0.025em} R}}$ as follows. The signature is a curve $\sigma: [0,1] \rightarrow {{\rm I\!\hspace{-0.025em} R}}$ defined by a series of values $0=t_1 <\dots<t_{\ensuremath{\ell}}=1$ as the linear interpolation of $\tau(t_i)$ in the order of the index $i$, and with the following properties.\ For $1 \leq i \leq {\ensuremath{\ell}}-1$ the following conditions hold: (non-degeneracy) if $i \in [2,{\ensuremath{\ell}}-1]$ then $\tau(t_i) \notin \langle \tau(t_{i-1}), \tau(t_{i+1})\rangle $, (direction-preserving)\ if $\tau(t_i)<\tau(t_{i+1})$ for $t < t' \in [t_i,t_{i+1}]$: $\tau(t)-\tau(t') \leq 2\delta$, and\ if $\tau(t_i)>\tau(t_{i+1})$ for $t < t' \in [t_i,t_{i+1}]$: $\tau(t')-\tau(t) \leq 2\delta$, (minimum edge length)\ if $i \in [2,{\ensuremath{\ell}}-2]$ then $|\tau(t_{i+1})-\tau(t_i)| > 2\delta$, and\ if $i\in \{1,{\ensuremath{\ell}}-1\}$ then $|\tau(t_{i+1})-\tau(t_{i})| > \delta$, (range) for $t \in [t_i,t_{i+1}]$:\ if $i \in [2,{\ensuremath{\ell}}-2]$ then $\tau(t) \in \langle\tau(t_i), \tau(t_{i+1})\rangle$, and\ if $i=1$ and ${\ensuremath{\ell}}>2$ then $\tau(t) \in \langle \tau(t_i),\tau(t_{i+1})\rangle \cup \langle \tau(t_i)-\delta,\tau(t_i)+\delta\rangle$, and\ if $i={\ensuremath{\ell}}-1$ and ${\ensuremath{\ell}}>2$ then $\tau(t) \in \langle \tau(t_{i-1}),\tau(t_i)\rangle \cup \langle \tau(t_i)-\delta,\tau(t_i)+\delta\rangle$, and\ if $i=1$ and ${\ensuremath{\ell}}=2$ then $\tau(t) \in \langle \tau(t_{1}),\tau(t_2)\rangle \cup \langle \tau(t_1)-\delta,\tau(t_1)+\delta\rangle \cup \langle \tau(t_2)-\delta,\tau(t_2)+\delta\rangle.$ It follows from the properties (i) and (iv) of [Definition \[def:signature\]]{} that the parameters $t_i$ for $i\in [1,\ell]$ specify vertices of $\tau$. Furthermore, it follows that the vertex $\tau(t_i)$ is either a minimum or maximum on $\tau[t_{i-1}, t_{i+1}]$ for $i\in [2,\ell-1]$. For a signature $\sigma$ we will simply write *signature $\sigma:[0,1]\rightarrow{{\rm I\!\hspace{-0.025em} R}}$ with ${\ensuremath{\ell}}$ vertices* or *signature $\sigma=v_1,\ldots,v_{\ensuremath{\ell}}$*, instead of *signature $\sigma:[0,1]\rightarrow{{\rm I\!\hspace{-0.025em} R}}$, with vertices $v_1=\sigma(s_1),\ldots,v_{\ensuremath{\ell}}=\sigma(s_{\ensuremath{\ell}})$, where $0=s_1<\ldots<s_{\ensuremath{\ell}}=1$*. We assume that the parametrization of $\sigma$ is chosen such that $\sigma(s_j)=\tau(s_j)$, for any $j\in\lbrace 1,\ldots,{\ensuremath{\ell}}\rbrace$. We remark that while the above definition is somewhat cumbersome, the stated properties turn out to be the exact properties needed to prove [Theorem \[theo:remove:one\]]{}, which in turn enables the basic mechanics of our clustering algorithms. Useful properties of signatures ------------------------------- [\[sec:properties:signatures\]]{} [\[lemma:fd:signature\]]{} It holds for any $\delta$-signature $\sigma$ of $\tau$ that ${\ensuremath{d_F{\!\left({\sigma,\tau}\right)}}}\leq \delta$. Let $t_1<\dots<t_l \in [0,1]$ be the series of parameter values of vertices on $\tau$ that describe $\sigma$. We construct a greedy matching between each signature edge $e_i=\overline{\tau(t_i)\tau(t_{i+1})}$ and the corresponding subcurve $\widehat{\tau}=\tau[t_i,t_{i+1}]$ of $\tau$. Assume first for simplicity that it holds $\tau(t_i)<\tau(t_{i+1})$ (i.e. the traversal of the signature is directed upwards at the time) and none of its endpoints are endpoints of $\tau$. We process the vertices of the subcurve $\widehat{\tau}$ while keeping a current position $v$ on the edge $e$. The idea is to walk as far as possible on $\widehat{\tau}$ while walking as little as possible on $e_i$. We initialize $v=\tau(t_i)$ and match the first vertex of $\widehat{\tau}$ to $v$. When processing a vertex $w$, we update $v$ to $\max(v,w-\delta)$ and match $w$ to the current position $v$ on $e_i$. By the direction-preserving condition in [Definition \[def:signature\]]{} and by [Observation \[obs:segments\]]{} every subcurve of $\widehat{\tau}$ is matched to a subsegment of $e_i$ within [Fréchet]{}distance $\delta$. If for the edge $e_i$ it holds that $\tau(t_i)>\tau(t_{i+1})$ (traversal directed downwards) the construction can be done symmetrically by walking backwards on $\widehat{\tau}$ and $e_i$. If the first vertex of $\widehat{\tau}$ is an endpoint of $\tau$, we start the above construction with the first vertex that lies outside the range $[\tau(0)-\delta,\tau(0)+\delta]$. The skipped vertices can be matched to $\tau(0)$. As for the remaining case if the last vertex of $\widehat{\tau}$ is an endpoint of $\tau$, we can again walk backwards on $\widehat{\tau}$ and $e_i$ and the case is analogous to the above. [^7] [\[lemma:nec:suff\]]{} Let $\sigma=v_1,\dots,v_{\ensuremath{\ell}}$ be a $\delta$-signature of $\tau=w_1,\dots,w_m$. Let $r_i=[v_i-\delta,v_i+\delta]$, for $1\leq i \leq {\ensuremath{\ell}}$, be ranges centered at the vertices of $\sigma$ ordered along $\sigma$. It holds for any curve $\pi$ if ${\ensuremath{d_F{\!\left({\tau,\pi}\right)}}}\leq\delta$, then $\pi$ has a vertex in each range $r_i$, and such that these vertices appear on $\pi$ in the order of $i$. For any $i=\lbrace 3,\ldots {\ensuremath{\ell}}-2\rbrace$ the vertices $v_{i-1}, v_i$ and $v_{i+1}$ satisfy that $|v_i-v_{i-1}|> 2\delta$ and $|v_{i+1}-v_{i}|> 2\delta$. This implies that $r_{i-1}\cap r_i = \emptyset$, $r_{i}\cap r_{i+1} = \emptyset$. Let $\pi(p_i)$ be the point matched to $v_i$ under a matching that witnesses ${\ensuremath{d_F{\!\left({\tau,\pi}\right)}}}$ and $p_i\in [0,1]$, for all $1\leq i \leq \ell$. It holds that $p_1<p_2<\ldots <p_\ell$. Therefore the curve $\pi$ visits the ranges $r_{i-1}$, $r_i$ and $r_{i+1}$ in the order of the index $i$. Since $v_i \notin \langle v_{i-1}, v_{i+1}\rangle $ the curve $\pi$ must change direction (from increasing to decreasing or vice versa) between visiting $r_{i-1}$ and $r_{i+1}$. Furthermore, $\pi$ cannot go beyond $r_i$ between visiting $r_{i-1}$ and $r_{i+1}$, i.e. there is no point $x\in\pi\left[p_{i-1}, p_{i+1} \right]$ such that it holds that $x\notin r_i$ and there is an ordering $v_{i-1}<v_i<v_i+\delta<x$ or $v_{i-1}>v_i>v_i-\delta>x$. This follows from $v_i$ being a local extremum on $\tau$. Therefore, the change of the direction of $\pi$ takes place in a vertex in $r_i$. For $i=2$ we use a similar argument. Note that $\pi(0)$ has to be matched to $v_1$ by the definition of the [Fréchet]{}distance. As before, $\pi$ has to visit the ranges $r_2$ and $r_3$ in this order and it holds that $r_2 \cap r_3 = \emptyset$. Either the first vertex of $\pi$ already lies in $r_2$, or again $\pi$ has to change direction and therefore needs to have a vertex in $r_2$. The case $i={\ensuremath{\ell}}-1$ is symmetric. The fact that the points $\tau(0)$ and $\tau(1)$ have to be matched to $\pi(0)$ and $\pi(1)$ closes the proof. The following is a direct implication of [Lemma \[lemma:nec:suff\]]{} and the minimum-edge-length condition in [Definition \[def:signature\]]{}, since $\sigma$ is a $\delta$-signature and there has to be at least one vertex in each of the ranges centered in vertices which are not endpoints of $\tau$. [\[cor:nec:suff\]]{} Let $\sigma$ be a signature of $\tau$ with ${\ensuremath{\ell}}$ vertices and ${\ensuremath{d_F{\!\left({\sigma,\tau}\right)}}}\leq \delta$. Then any curve $\pi$ with ${\ensuremath{d_F{\!\left({\pi,\tau}\right)}}}\leq \delta$ needs to have at least ${\ensuremath{\ell}}-2$ vertices. [[\[theo:remove:one\]]{}]{} Let $\sigma=v_1,\dots,v_{\ensuremath{\ell}}$ be a $\delta$-signature of $\tau=w_1,\dots,w_m$. Let $r_j=[v_j-\delta,v_j+\delta]$ be ranges centered at the vertices of $\sigma$ ordered along $\sigma$, where $r_1=[v_1-4\delta,v_1+4\delta]$ and $r_{\ensuremath{\ell}}=[v_{\ensuremath{\ell}}-4\delta,v_{\ensuremath{\ell}}+4\delta]$. Let $\pi$ be a curve with ${\ensuremath{d_F{\!\left({\tau,\pi}\right)}}} \leq \delta$ and let $\pi'$ be a curve obtained by removing some vertex $u_i=\pi(p_i)$ from $\pi$ with $u_i \notin \bigcup_{1\leq j \leq {\ensuremath{\ell}}} r_j$. It holds that ${\ensuremath{d_F{\!\left({\tau,\pi'}\right)}}} \leq \delta$. In order to prove this theorem, we have the following lemma, which is a slight variation of the Theorem and it simplifies the case when the [Fréchet]{}distance is obtained in the limit. [\[lemma:remove:one\]]{} Let $\sigma=v_1,\dots,v_{\ensuremath{\ell}}$ be a $\delta$-signature of $\tau=w_1,\dots,w_m$. Let $r_j=[v_j-\delta,v_j+\delta]$ be ranges centered at the vertices of $\sigma$ ordered along $\sigma$, where $r_1=[v_1-4\delta,v_1+4\delta]$ and $r_{\ensuremath{\ell}}=[v_{\ensuremath{\ell}}-4\delta,v_{\ensuremath{\ell}}+4\delta]$. Let $\pi$ be a curve with ${\ensuremath{d_F{\!\left({\tau,\pi}\right)}}} <\delta$ and let $\pi'$ be a curve obtained by removing some vertex $u_i=\pi(p_i)$ from $\pi$ with $u_i \notin \bigcup_{1\leq j \leq {\ensuremath{\ell}}} r_j$. For any ${{\varepsilon}}>0$, it holds that ${\ensuremath{d_F{\!\left({\tau,\pi'}\right)}}} \leq \delta+{{\varepsilon}}$. We obtain the [Theorem \[theo:remove:one\]]{} from [Lemma \[lemma:remove:one\]]{} as follows. By the theorem statement, we are given $\tau,\pi$ and $\delta$, such that ${\ensuremath{d_F{\!\left({\tau,\pi}\right)}}} \leq \delta$. By the definition of the [Fréchet]{}distance it holds for any ${{\varepsilon}}> 0$ that ${\ensuremath{d_F{\!\left({\tau,\pi}\right)}}} < \delta+{{\varepsilon}}$. Let $\delta'=\delta+{{\varepsilon}}$ for some ${{\varepsilon}}>0$ small enough such that: the $\delta$-signature of $\tau$ is equal to the $\delta'$-signature of $\tau$ (see also [Lemma \[lemma:canonical:signature\]]{} for the existence of such a signature), and any vertex $u_i$ of $\pi$ satisfying the conditions in [Theorem \[theo:remove:one\]]{} also satisfies the conditions of [Lemma \[lemma:remove:one\]]{} for $\delta'$. Now we can apply [Lemma \[lemma:remove:one\]]{} using $\delta'$, implying that ${\ensuremath{d_F{\!\left({\tau,\pi'}\right)}}} \leq \delta+2{{\varepsilon}}$. Since this is implied for any ${{\varepsilon}}> 0$ small enough, we have $\lim_{{{\varepsilon}}\rightarrow 0} {\ensuremath{d_F{\!\left({\tau,\pi'}\right)}}} \leq \lim_{{{\varepsilon}}\rightarrow 0} \left( \delta+2{{\varepsilon}}\right) = \delta$. Let $f$ denote the witness matching from $\pi$ to $\tau$, that maps each point on $\pi$ to a point on $\tau$ within distance $\delta$.[^8] Intuitively, we removed $u_i$ and its incident edges from $\pi$ by replacing the incident edges with a new “edge” connecting the two subcurves which were disconnected by the edge removal. The obtained curve is called $\pi'$. We want to construct a matching $f'$ from $\pi'$ to $\tau$ based on $f$ to show that their [Fréchet]{}distance is at most $\delta$. Because of the continuity of the curves, we have to describe the “edge” connecting disconnected parts. Let $\pi(p_{i-1})$ and $\pi(p_{i+1})$ be the endpoints of the disconnected components. Let $\pi[p^{-},p^{+}]$ denote the subcurve by which $\pi$ and $\pi'$ differ. In particular, $p^{-}$ and $p^{+}$ are such that $\pi'$ can be written as a concatenation of a prefix and a suffix curve of $\pi$: $\pi'= \pi[0,p^{-}] \oplus \pi[p^{+},1]$ and $p_i$ is contained in the open interval $(p^{-},p^{+})$. Note that $\pi(p^{-})=\pi(p^{+})$. Furthermore, it is clear that $\pi[p^{-},p^{+}]$ consists of two edges with $u_i$ being the minimum or maximum connecting them. (Otherwise, if $u_i$ was neither a minimum nor a maximum on $\pi$, then $\pi[p^{-},p^{+}]$ is empty. In this case the claim holds trivially.) The new “edge” $\pi'[p_{i-1}, p_{i+1}]$ consists of three parts: the edge $\pi[p_{i-1}, p^-]$, the point $\pi[p^-]$ and the edge $\pi[p^+, p_{i+1}]$. This is illustrated by [Figure \[fig:vertexremovalexample\]]{}. ![The removal of the vertex $\pi(p_i)$ from $\pi$. The curve $\pi[p^-,p^+]$ is marked red](th37image.pdf "fig:")\ [\[fig:vertexremovalexample\]]{} In the construction of $f'$ we need to show that the subcurve $\tau[f(p^{-}),f(p^{+})]$, which was matched by $f^{-1}$ to the missing part, can be matched to some subcurve of $\pi'$, while respecting the monotonicity of the matching. The proof is a case analysis based on the structure of the two curves. In order to focus on the essential arguments, we first make some global assumptions stated below. The first two assumptions can be made without loss of generality. We also introduce some basic notation which is used throughout the rest of the proof. We assume that $\pi(p_i)$ is a local minimum on $\pi$ (otherwise we first mirror the curves $\tau$ and $\pi$ across the horizontal time axis to obtain this property without changing the [Fréchet]{}distance). Let $z_{\min}= \operatorname*{arg\,min}_{t \in [f(p^{-}),f(p^{+})]} \tau(t)$. Let $\tau[s_j,s_{j+1}]$ be the subcurve of $\tau$ bounded by two consecutive signature vertices, such that $z_{\min} \in [s_j,s_{j+1}]$. We assume that $\tau(s_j) < \tau(s_{j+1})$ (otherwise we first reparametrize the curves $\tau$ and $\pi$ with $\phi(t)=1-t$, i.e., reverse the direction of the time axis, to obtain this property without changing the [Fréchet]{}distance; note that this does not change the property of $\pi(p_i)$ being a local minimum). [\[ass:sign:edge:asc\]]{} We assume that neither $s_j=0$, nor $s_{j+1}=1$ (These are boundary cases which will be handled at the end of the proof). [\[ass:inner:sig:edge\]]{} By [Definition \[def:signature\]]{} we can assume that $\tau(s_{j+1}) - \tau(s_{j}) > 2\delta$,\[item:sig:length\] $\tau(s_{j}) = {\ensuremath{\min( \tau[s_{j-1},s_{j+1}])}}$,\[item:sig:sj:min\] $\tau(s_{j+1}) = {\ensuremath{\max( \tau[s_j,s_{j+2}])}}$,\[item:sig:sj1:max\] $\tau(t) \geq \tau(t')-2\delta$ for $s_{j}\leq t' < t \leq s_{j+1}$.\[item:sig:descent\] $\tau(s_{j+1}) - \tau(s_{j+2}) > 2\delta$,\[item:sig:length:2\] By the general position assumption the minimum $\tau(s_{j})$ and the maximum $\tau(s_{j+1})$ are unique on their respective subcurves. [\[prop:signature\]]{} Any two points matched by $f$ have distance at most $\delta$ from each other. In particular, for any two $0\leq p' < p \leq 1$, it holds that $\tau(f(p)) -\delta \leq \pi(p) \leq \tau(f(p)) + \delta$,\[item:frechet:dist\] ${\ensuremath{\min( \tau[f(p'),f(p)])}} - \delta \leq {\ensuremath{\min( \pi[p',p])}} \leq {\ensuremath{\min( \tau[f(p'),f(p)])}} + \delta$,\[item:frechet:min\] ${\ensuremath{\max( \tau[f(p'),f(p)])}} - \delta \leq {\ensuremath{\max( \pi[p',p])}} \leq {\ensuremath{\max( \tau[f(p'),f(p)])}} + \delta$.\[item:frechet:max\] [\[prop:frechet\]]{} Our proof is structured as case analysis. We consider first the case $\tau(z_{\min}) \geq \pi(p^{-}) - \delta$. This is illustrated by [Figure \[fig:shortcutting:case1\]]{}. $\tau(z_{\min}) \geq \pi(p^{-}) - \delta$ [\[case:trivial\]]{} ![Example of [Case \[case:trivial\]]{}. The broken part of the matching $f$ is indicated by fat lines.](shortcutting_case1234.pdf "fig:")\ [\[fig:shortcutting:case1\]]{} As a warm-up exercise we quickly check that the above case is indeed trivial. In this case, we would simply match $\pi(p^{-})$ to the subcurve $\tau[f(p^{-}),f(p^{+})]$ and the remaining subcurves $\pi[0,p^{-}]$ and $\pi[p^{+},1]$ can be matched as done by $f$.[^9] Indeed, $${\ensuremath{{\left\{ { \tau(t) ~|~ t \in [f(p^{-}),f(p^{+})]} \right\}} }} \subseteq {\ensuremath{[\pi(p^{-})]_{\delta}}},$$ since by the case distinction $${\ensuremath{\min( \tau[f(p^{-}),f(p^{+})])}} = \tau(z_{\min}) \geq \pi(p^{-}) - \delta$$ and by [Property \[prop:frechet\]]{}, $${\ensuremath{\max( \tau[f(p^{-}),f(p^{+})])}} \leq {\ensuremath{\max( \pi[p^{-},p^{+}])}} +\delta = \pi(p^{-}) +\delta.$$ We assume in the rest of the proof that $\tau(z_{\min}) < \pi(p^{-}) - \delta$ (non-trivial case). [\[ass:nontrivialcase\]]{} Intuitively, we want to extend the subcurves of the trivial case in order to fix the broken matching. The difficulty lies in finding suitable subcurves which cover the broken part $\tau[f(p^{-}),f(p^{+})]$ and whose [Fréchet]{}distance is at most $\delta$. Furthermore, the endpoints need to line up suitably such that we can re-use $f$ for the suffix and prefix curves. The next two claims settle the question, to which extent signature vertices can be included in the subcurve $\tau[f(p^{-}),f(p^{+})]$ for which we need to fix the broken matching. If $s_{j+1} \in [f(p^{-}),f(p^{+})]$ then ${\ensuremath{{\left\{ { \tau(t) ~|~ t \in [s_{j+1},f(p^{+})]} \right\}} }} \subseteq [\tau(s_{j+1})-2\delta, \tau(s_{j+1})]$. [\[claim:descent:after:sj1\]]{} We have to prove that $$\min(\tau[s_{j+1},f(p^+)])\geq \tau(s_{j+1})-2\delta \text{ and } \max(\tau[s_{j+1},f(p^+)])\leq \tau(s_{j+1}).$$ The subcurve $\pi[p^-, p^+]$ consists of two edges $\overline{\pi(p^-)\pi(p_i)}$ and $\overline{\pi(p_i)\pi(p^+)}$ and $\pi(p_i)$ is the minimum of the subcurve. For the lower bound we distinguish two cases: $p_i\leq f^{-1}(s_{j+1})\leq p^+$ and $p^-\leq f^{-1}(s_{j+1}) < p_i$. If $p_i\leq f^{-1}(s_{j+1})\leq p^+$, then by [Property \[prop:frechet\]]{} $$\min(\tau[s_{j+1},f(p^+)])\geq \min(\pi[f^{-1}(s_{j+1}), p^+])-\delta = \pi(f^{-1}(s_{j+1})) -\delta \geq \tau(s_{j+1})-2\delta.$$ If $p^-\leq f^{-1}(s_{j+1}) < p_i$, then since $z_{\min}<s_{j+1}$ it holds that $$\tau(z_{\min}) = \min(\tau[f(p^-),s_{j+1}]) \geq \min(\pi[p^-,f^{-1}(s_{j+1})]) - \delta = \pi(f^{-1}(s_{j+1})) - \delta \geq \tau(s_{j+1})-2\delta.$$ It follows that $$\min(\tau[s_{j+1},f(p^+)])\geq \min(\tau[f(p^-), f(p^+)]) =\tau(z_{\min}) \geq \tau(s_{j+1})-2\delta$$ as claimed. Furthermore, by [Property \[prop:signature\]]{}(\[item:sig:length:2\]) it follows that $$\min(\tau[s_{j+1},f(p^+)])\geq \tau(s_{j+1})-2\delta > \tau(s_{j+2}),$$ and therefore $s_{j+2}\notin [f(p^{-}), f(p^{+})]$. Now, [Property \[prop:signature\]]{}(\[item:sig:sj1:max\]) implies the upper bound: $ {\ensuremath{\max( \tau[s_{j+1},f(p^{+})])}} \leq \tau(s_{j+1}).$ $s_j \notin [f(p^{-}),f(p^{+})]$ [\[claim:no:minimum\]]{} For the sake of contradiction, assume the claim is false, i.e. $s_j \in [f(p^{-}),f(p^{+})]$. We have (by definition) $$z_{\min} \in [f(p^{-}),f(p^{+})] \cap [s_j,s_{j+1}]$$ Furthermore, by definition $\tau(z_{\min}) = {\ensuremath{\min( \tau[f(p^{-}),f(p^{+})])}}$, and by [Property \[prop:signature\]]{}(\[item:sig:sj:min\]), we have $\tau(s_j) = {\ensuremath{\min( \tau[s_{j-1},s_{j+1}])}}$. This would imply that $$\tau(z_{\min}) = \min {\left\{ {\tau(t)~|~ t \in [f(p^{-}),f(p^{+})] \cap [s_j,s_{j+1}]} \right\}} = \tau(s_j).$$ By the theorem statement $\pi(p_i) \notin {\ensuremath{[\tau(s_j)]_{\delta}}} = {\ensuremath{[\tau(z_{\min})]_{\delta}}}$. However, by [Property \[prop:frechet\]]{}, $$\pi(p_i) = {\ensuremath{\min( \pi[p^{-},p^{+}])}} \in {\ensuremath{[ {\ensuremath{\min( \tau[f(p^{-}),f(p^{+})])}}]_{\delta}}}={\ensuremath{[\tau(z_{\min})]_{\delta}}}.$$ We now introduce some more notation which will be used throughout the proof. $$\begin{aligned} t_{\min} &=& \operatorname*{arg\,min}_{t \in [f(p^{-}),s_{j+1}] } \tau(t)\\ x &=& \max\{p \in [0,p^{-}] ~|~ \pi(p) = \min(\tau(t_{\min}) + \delta, \tau(s_{j+1}) - \delta)\}\\ p_{\max}&=& \operatorname*{arg\,max}_{p \in [x, p^{-}] } \pi(p)\\ y &=& \min\{t \in [t_{\min}, 1] ~|~ \tau(t) = \pi(p_{\max}) - \delta\}\end{aligned}$$ In the next few claims we argue that these variables are well-defined. In particular, that $x$ and $y$ always exist in the non-trivial case ([Claim \[claim:x:exists\]]{} and [Claim \[claim:y:exists\]]{}). Clearly $t_{\min}$ is well-defined and by our initial assumptions we have $z_{\min} \leq t_{\min}$ (since $z_{\min} \in [s_j,s_{j+1}]$). We also derive some bounds along the way, which will be used throughout the later parts of the proof. It holds that $\min(\tau(t_{\min}) + \delta, \tau(s_{j+1})-\delta) \in \{\pi(p) ~|~ p \in [f^{-1}(s_j),p^{-}]\}$ ${\ensuremath{\min( \pi[x,p^{-}])}} \geq \min(\tau(t_{\min}) + \delta, \tau(s_{j+1})-\delta) = \pi(x)$ $\tau(s_j) < \tau(t_{\min})$ [\[claim:x:exists\]]{} [\[claim:x:high\]]{} [\[claim:sj:low\]]{} We first prove part (i) of the claim. We show that there exist two parameters $f^{-1}(s_j) \leq p_1 \leq p_2 \leq p^{-}$ such that $$\pi(p_1) \leq \min{\!\left({\tau(t_{\min})+\delta, \tau(s_{j+1})-\delta}\right)} \leq \pi(p_2).$$ Since the curve is continuous, this would imply the claim. Indeed, we can choose $p_1=f^{-1}(s_j)$ and $p_2=p^{-}$. If $s_{j+1}\geq f(p^{+})$ we have $$\pi(p_2)=\pi(p^{-}) \geq \tau(z_{\min}) + \delta \geq \tau(t_{\min}) + \delta, $$ (since we assume the non-trivial case). Otherwise, if $s_{j+1} < f(p^{+})$ and therefore $\tau(s_{j+1})={\ensuremath{\max( \tau[f(p^{-}),f(p^{+})])}}$, then by [Property \[prop:signature\]]{} and [Property \[prop:frechet\]]{}, $$\pi(p_2)=\pi(p^{-}) = {\ensuremath{\max( \pi[p^{-},p^{+}])}} \geq {\ensuremath{\max( \tau[f(p^{-}),f(p^{+})])}} - \delta = \tau(s_{j+1}) -\delta.$$ Thus, in both cases, it holds that $\pi(p_2) \geq \min{\!\left({\tau(t_{\min})+\delta, \tau(s_{j+1})-\delta}\right)}$. As for $p_1$, by [Claim \[claim:no:minimum\]]{} we have $0\leq s_j \leq f(p^{-}) \leq t_{\min} \leq s_{j+1}$ and by [Property \[prop:frechet\]]{} $$\pi(p_1) =\pi(f^{-1}(s_j)) \leq \tau(s_j) +\delta .$$ It follows by [Property \[prop:signature\]]{}(\[item:sig:sj:min\]) that $\pi(p_1) < \tau(t_{\min}) + \delta$ and by [Property \[prop:signature\]]{}(\[item:sig:length\]) that $\pi(p_1) < \tau(s_{j+1})-\delta.$ Now, part (ii) of the claim follows directly from the above, since $\pi(x)$ is defined as the last point along the prefix subcurve $\pi[0,p^{-}]$ with the specified value and since $\pi(x) \leq \pi(p^{-})$. Note that part (iii) we indirectly proved above. It holds that $ \pi(p_{\max}) - \delta \in \{\tau(t) ~|~ t \in [t_{\min},s_{j+1}]\}$ ${\ensuremath{\max( \tau[t_{\min},y])}} \leq \pi(p_{\max}) - \delta =\tau(y)$ $\pi(p_{\max}) < \tau(s_{j+1}) + \delta$ [\[claim:y:exists\]]{} [\[claim:sj1:high\]]{} To prove part (i) of the claim we show that there exist two parameters $t_{\min} \leq t_1 \leq t_2 \leq s_{j+1}$, such that $$\tau(t_1)\leq \pi(p_{\max})-\delta \leq \tau(t_2).$$ We choose $t_1=t_{\min}$ and $t_2=s_{j+1}$. Since we have the non-trivial case, we know $$\pi(p_{\max}) \geq \pi(p^{-}) \geq \tau(z_{\min})+\delta \geq \tau(t_{\min})+\delta = \tau(t_1)+\delta.$$ Now, for $t_2$, we know that $s_j \leq f(x) \leq f(p_{\max}) \leq f(p^{-}) \leq s_{j+1}$. By [Property \[prop:frechet\]]{} and by [Property \[prop:signature\]]{}(\[item:sig:sj1:max\]) $$\tau(t_2) = \tau(s_{j+1}) \geq \tau(f(p_{\max})) \geq \pi(p_{\max})-\delta.$$ Since the subcurve is continuous, there must be a parameter $t_1\leq t\leq t_2$ which satisfies the claim. Now, part (ii) of the claim also follows directly, since $\tau(y)$ is the first point along the suffix subcurve $\tau[t_{\min},1]$ with the specified value and since $\tau(y) \geq \tau(t_{\min})$. Note that part (iii) we just proved above. The following claim follows directly from [Claim \[claim:x:exists\]]{} and [Claim \[claim:y:exists\]]{}. $s_j \leq f(x) \leq y \leq s_{j+1}$ [\[claim:xy:inside\]]{} The following claim will be used throughout the proof. $\pi(p_{\max}) - 2\delta \leq \pi(x)$ [\[claim:not:so:bad\]]{} [\[claim:max:dist\]]{} We need to show that $$\pi(p_{\max}) - 2 \delta \leq \min(\tau(t_{\min}) + \delta, \tau(s_{j+1})-\delta)$$ [Claim \[claim:y:exists\]]{} immediately implies $\pi(p_{\max}) \leq \tau(s_{j+1}) + \delta$. On the other hand, by [Claim \[claim:xy:inside\]]{}, $$s_j \leq f(x) \leq f(p_{\max}) \leq f(p^{-}) \leq t_{\min} \leq s_{j+1}.$$ By [Property \[prop:frechet\]]{} and by [Property \[prop:signature\]]{}(\[item:sig:descent\]), $$\pi(p_{\max}) \leq \tau(f(p_{\max})) + \delta \leq \tau(t_{\min}) + 3\delta$$ The next two claims ([Claim \[claim:frechet:y\]]{} and [Claim \[claim:frechet:x\]]{}) show that our choice of $x$ and $y$ is suitable for fixing some part of the broken matching: the subcurve $\pi[x,p^{-}]$ can be matched entirely to $\tau(y)$ and the subcurve $\tau[f(x),y]$ can be matched entirely to $\pi(x)$. After that, it remains to match the subcurve $\pi[p^{+}, f^{-1}(y)]$. For this we have the case analysis that follows. ${\ensuremath{{\left\{ { \pi(p) ~|~ p \in [x,p^{-}]} \right\}} }} \subseteq [\pi(p_{\max}) -2\delta, \pi(p_{\max})] = {\ensuremath{[\tau(y)]_{\delta}}}.$ [\[claim:frechet:y\]]{} [\[claim:frechet:ya\]]{} By [Claim \[claim:x:high\]]{} and [Claim \[claim:not:so:bad\]]{}, $${\ensuremath{\min( \pi[x,p^{-}])}} \geq \min{\!\left({\tau(t_{\min})+\delta, \tau(s_{j+1})-\delta}\right)} \geq \pi(p_{\max})-2\delta.$$ On the other hand, by definition of $p_{\max}$, we have ${\ensuremath{\max( \pi[x,p^{-}])}} = \pi(p_{\max}).$ The last equality of the claim follows directly from the definition of $y$ and from [Claim \[claim:y:exists\]]{} ($y$ is well-defined). ${\ensuremath{{\left\{ { \tau(t) ~|~ t \in [f(x),y]} \right\}} }} \subseteq {\ensuremath{[\min{\!\left({\tau(t_{\min}) + \delta, \tau(s_{j+1})-\delta}\right)}]_{\delta}}} = {\ensuremath{[\pi(x)]_{\delta}}}.$ [\[claim:frechet:x\]]{} We first prove the lower bound on the minimum of the subcurve $\tau[f(x),y]$. By [Claim \[claim:x:high\]]{}, and by [Property \[prop:frechet\]]{}, we have $${\ensuremath{\min( \tau[f(x),f(p^{-})])}} \geq {\ensuremath{\min( \pi[x,p^{-}])}} - \delta \geq \min{\!\left({\tau(t_{\min}), \tau(s_{j+1})-2\delta}\right)}.$$ By definition, $\tau(t_{\min})$ is a minimum on $\tau[f(p^{-}), s_{j+1}]$, thus $${\ensuremath{\min( \tau[f(p^{-}),y])}} \geq \tau(t_{\min}) \geq \min{\!\left({\tau(t_{\min}), \tau(s_{j+1})-2\delta}\right)},$$ for $y \leq s_{j+1}$, which is ensured by [Claim \[claim:xy:inside\]]{}. We now prove the upper bound on the maximum of the subcurve $\tau[f(x),y]$. Since by [Claim \[claim:sj1:high\]]{} $s_j \leq f(x) \leq t_{\min} \leq s_{j+1}$ and since by [Property \[prop:signature\]]{}(\[item:sig:descent\]), $\tau[s_j, s_{j+1}]$ may not descend by more than $2\delta$, it follows that $${\ensuremath{\max( \tau[f(x),t_{\min}])}} \leq \tau(t_{\min})+2\delta.$$ By definitions of $x$ and $y$ and by [Claim \[claim:max:dist\]]{} $${\ensuremath{\max( \tau[t_{\min},y])}} \leq \pi(p_{\max})-\delta \leq \pi(x) +\delta \leq \tau(t_{\min}) + 2\delta.$$ By [Property \[prop:signature\]]{}(\[item:sig:sj1:max\]) we also have, $${\ensuremath{\max( \tau[f(x),y])}} \leq \tau(s_{j+1}),$$ for $y \leq s_{j+1}$, which is ensured by [Claim \[claim:xy:inside\]]{}. Together this implies the claim. The last equality of the claim follows directly from the definition of $x$ and from [Claim \[claim:x:exists\]]{} ($x$ is well-defined). Now we have established the basic setup for our proof. In the following, we describe the case analysis based on the structure of the two curves $\tau$ and $\pi$. Consider walking along the subcurve $\pi[p^{+},1]$. At the beginning of the subcurve, we have $\pi(p^{+}) \in [\pi(x), \pi(p_{\max})]$. One of the following events may happen during the walk: either we go above $\pi(p_{\max})$, or we go below $\pi(x)$, or we stay inside this interval. Let $q$ denote the time at which for the first time one of these events happens. Formally, we define the intersection function $g:{{\rm I\!\hspace{-0.025em} R}}\rightarrow [p^{+},1] \cup {\left\{ {p_{\infty}} \right\}}$, $$\begin{aligned} g(h) &=& \min (\{p \in [p^{+}, 1] ~|~ \pi(p)=h \} \cup \{p_{\infty}\}) \\ q &=& \min{\!\left({g(\pi(p_{\max})), g(\pi(x))}\right)}, $$ where $p_{\infty}>1$ is some fixed constant for the case that the suffix curve $\pi[p^{+},1]$ does not contain the value $h$. We distinguish the following main cases. In each of the cases, we devise a matching scheme to fix the broken matching. For each case, our construction ensures that the extended subcurves cover the subcurve $\tau[f(p^{-}),f(p^{+})]$ and that the subcurves line up with suitable prefix and suffix curves, such that we can always use $f$ for the parts of $\pi$ and $\tau$ not covered in the matching scheme. We need to prove in each case that the [Fréchet]{}distance between the specified subcurves is at most $\delta$. If this is the case, we call the matching scheme *valid*. We have to make further distinction between the case when $f(p^{+}) \leq y$ and the case $f(p^{+}) > y$. If $f(p^{+}) \leq y$ holds, the three aforementioned events are described by [Case \[case:level\]]{}, [Case \[case:upwards\]]{} and [Case \[case:downwards\]]{}. If it happens that $f(p^{+}) > y$, it becomes more complicated to repair the matching. This is discussed in [Case \[case:matroska\]]{}. $p^{+} \leq f^{-1}(y) \leq q$. [\[case:level\]]{} ![Example of [Case \[case:level\]]{}. The broken part of the matching $f$ is indicated by fat lines.](shortcutting_case1234.pdf "fig:")\ [\[fig:shortcutting:case2\]]{} [Case \[case:level\]]{} is the simplest case. We intend to use the following matching scheme: $$\begin{aligned} \pi(x) &\Leftrightarrow& \tau[f(x),y]\\ \pi[x,p^{-}] &\Leftrightarrow& \tau(y)\\ \pi[p^{+},f^{-1}(y)] &\Leftrightarrow& \tau(y)\end{aligned}$$ [Claim \[claim:frechet:x\]]{} implies that the [Fréchet]{}distance between $\tau[f(x),y]$ and $\pi(x)$ is at most $\delta$. [Claim \[claim:frechet:y\]]{} implies that the [Fréchet]{}distance between $\pi[x,p^{-}]$ and $\tau(y)$ is at most $\delta$. Finally, by our case distinction and by [Claim \[claim:not:so:bad\]]{} $${\ensuremath{{\left\{ { \pi(p) ~|~ p \in [p^{+},f^{-1}(y)]} \right\}} }} \subseteq [\pi(x), \pi(p_{\max})] \subseteq [\pi(p_{\max})-2\delta, \pi(p_{\max})] = {\ensuremath{[\tau(y)]_{\delta}}}.$$ Therefore, also the [Fréchet]{}distance between $\pi[p^{+},f^{-1}(y)]$ and $\tau(y)$ is at most $\delta$. $q < f^{-1}(y)$ and $q=g(\pi(p_{\max}))$ [\[case:upwards\]]{} ![Example of [Case \[case:upwards\]]{}. The broken part of the matching $f$ is indicated by fat lines.](shortcutting_case1234.pdf "fig:")\ [\[fig:shortcutting:case3\]]{} In [Case \[case:upwards\]]{}, let $y'=\max\{t\in [0,f(q)]~ |~ \tau(t)=\tau(y) \}$ and $z=\max\{p^{+},f^{-1}(y')\}$. We intend to use the following matching scheme: $$\begin{aligned} \pi(x) &\Leftrightarrow& \tau[f(x),y']\\ \pi[x,p^{-}] &\Leftrightarrow& \tau(y')\\ \pi[p^{+},z] &\Leftrightarrow& \tau(y')\\ \pi(z) &\Leftrightarrow& \tau[y',f(p^{+})]\end{aligned}$$ (Note that if $y' > f(p^{+})$, then the last line of the above matching scheme is simply dropped.) We first argue that $y'$ exists. To this end, we show that there exist two parameters $0\leq t_1 < t_2 \leq f(q)$, such that $$\tau(t_1) \leq \tau(y) = \pi(p_{\max})-\delta \leq \tau(t_2).$$ We choose $t_1=z_{\min}$ and $t_2=f(q)$. Note that $z_{\min} \leq f(p^{+}) \leq f(q)$. Now, by [Property \[prop:frechet\]]{}, $$\tau(t_2) = \tau(f(q)) \geq \pi(q)-\delta = \pi(p_{\max})-\delta.$$ Since we are assuming the non-trivial case, $$\pi(p_{\max}) \geq \pi(p^{-}) \geq \tau(z_{\min}) + \delta = \tau(t_1) + \delta.$$ Thus, since $\tau[0,f(q)]$ is continuous, $y'$ must exist and it holds that $f(p^{-}) \leq z_{\min} \leq y'$. It remains to prove that the matching scheme is valid. Since $y'\leq f(q) < y$, [Claim \[claim:frechet:x\]]{} implies that the [Fréchet]{}distance between $\tau[f(x),y']$ and $\pi(x)$ is at most $\delta$. [Claim \[claim:frechet:y\]]{} implies that the [Fréchet]{}distance between $\pi[x,p^{-}]$ and $\tau(y')$ is at most $\delta$. For the last two lines of the matching scheme we distinguish two cases. If $y' > f(p^{+})$, then $z=f^{-1}(y')$ and we need to prove that $${\ensuremath{{\left\{ { \pi(p) ~|~ p \in [p^{+},f^{-1}(y')]} \right\}} }} \subseteq {\ensuremath{[\tau(y')]_{\delta}}}.$$ By our case distinction and by [Claim \[claim:not:so:bad\]]{} $${\ensuremath{{\left\{ { \pi(p) ~|~ p \in [p^{+},q]} \right\}} }} \subseteq [\pi(x), \pi(p_{\max})] \subseteq [\pi(p_{\max})-2\delta, \pi(p_{\max})] = {\ensuremath{[\tau(y')]_{\delta}}}.$$ Since $f^{-1}(y') \leq q$, this implies the validity of the matching. Otherwise, if $y' \leq f(p^{+})$, then $z=p^{+}$ and we need to prove that $${\ensuremath{{\left\{ { \tau(t) ~|~ t \in [y',f(p^{+})]} \right\}} }} \subseteq {\ensuremath{[\pi(p^{+})]_{\delta}}}.$$ This can be derived as follows. On the one hand, by [Property \[prop:frechet\]]{}, since $y'\in [f(p^{-}),f(p^{+})]$ $${\ensuremath{\max( \tau[y',f(p^{+})])}} \leq {\ensuremath{\max( \pi[f^{-1}(y'),p^{+}])}}+\delta = {\ensuremath{\max( \pi[p^{-},p^{+}])}}+\delta = \pi(p^{+})+\delta.$$ On the other hand, by the definition of $y'$ and since $y' \leq f(p^{+}) \leq f(q)$ $${\ensuremath{\min( \tau[y',f(p^{+})])}} = \tau(y') = \pi(p_{\max})-\delta \geq \pi(p^{+}) -\delta.$$ $q < f^{-1}(y)$ and $q=g(\pi(x))$. [\[case:downwards\]]{} ![Example of [Case \[case:downwards\]]{}. The broken part of the matching $f$ is indicated by fat lines.](shortcutting_case1234.pdf "fig:")\ [\[fig:shortcutting:case4\]]{} In [Case \[case:downwards\]]{}, let $y''=\min\{ t \in [f(p_{\max}),1] ~|~ \tau(t) = \tau(y) \}$. We intend to use the following matching scheme: $$\begin{aligned} \pi(p_{\max}) &\Leftrightarrow& \tau[f(p_{\max}), y'']\\ \pi[p_{\max},p^{-}] &\Leftrightarrow& \tau(y'')\\ \pi[p^{+},q] &\Leftrightarrow& \tau(y'')\\ \pi(q) &\Leftrightarrow& \tau[y'',f(q)]\end{aligned}$$ Clearly, $y''$ exists in the non-trivial case, since $$\tau(f(p_{\max})) \geq \tau(y) \geq \tau(z_{\min}),$$ and $f(p_{\max}) \leq f(p^{-}) \leq z_{\min}$. We prove the validity of the matching scheme line by line. Note that by definition $\tau(y'') = \tau(y) = \pi(p_{\max})-\delta$. For the first matching: by the definition of $y''$ and by [Property \[prop:frechet\]]{}, $${\ensuremath{{\left\{ { \tau(t) ~|~ t \in [f(p_{\max}),y'']} \right\}} }} \subseteq {\ensuremath{[\pi(p_{\max})]_{\delta}}}.$$ The validity of the second matching follows from [Claim \[claim:frechet:y\]]{} since $p_{\max} \geq x$. For the third matching: By our case distinction and by [Claim \[claim:not:so:bad\]]{} $${\ensuremath{{\left\{ { \pi(t) ~|~ t \in [p^{+},q]} \right\}} }} \subseteq [\pi(x), \pi(p_{\max})] \subseteq [\pi(p_{\max})-2\delta, \pi(p_{\max})] ={\ensuremath{[\tau(y'')]_{\delta}}}.$$ As for the last matching, since $f(x) \leq f(p_{\max}) \leq y''$ and since by our case distinction $f(q) < y$, [Claim \[claim:frechet:x\]]{} implies $${\ensuremath{{\left\{ { \tau(t) ~|~ t \in [y'',f(q)]} \right\}} }} \subseteq {\ensuremath{[\pi(x)]_{\delta}}} = {\ensuremath{[\pi(q)]_{\delta}}}.$$ $f(p^{+}) > y$. [\[case:matroska\]]{} [Case \[case:matroska\]]{} seems to be the most difficult case to handle. However, we have already established a suitable set of tools in the previous cases. We devise an iterative matching scheme and prove an invariant ([Claim \[claim:frechet:xa\]]{}) to verify that the [Fréchet]{}distance of the subcurves is at most $\delta$. We first define $z_{\min}^{(1)}=z_{\min}$, $t_{\min}^{(1)}=t_{\min}$, $x^{(1)}=x$, and $y^{(1)}=y$. Now, for $a = 2, \dots$ let $$\begin{aligned} z_{\min}^{(a)} &=& \operatorname*{arg\,min}_{t\in [y^{(a-1)},f(p^{+})] } \tau(t)\\ t_{\min}^{(a)} &=& \operatorname*{arg\,min}_{t\in [y^{(a-1)},s_{j+1}]} \tau(t)\\ x^{(a)}&=& \min{\left\{ {p\in [x^{(a-1)},p_{\max}] ~|~ \pi(p) = \min{\!\left({\tau(t_{\min}^{(a)})+\delta, \tau(s_{j+1})-\delta}\right)} } \right\}} \\ y^{(a)}&=& \min \{t \in [t_{\min}^{(a)},s_{j+1}] ~|~ \tau(t) = \pi(p_{\max}) - \delta\},\end{aligned}$$ We describe the intended matching scheme, beginning with the following subcurves: $$\begin{aligned} \pi(x^{(1)}) &\Leftrightarrow& \tau[f(x^{(1)}),y^{(1)}]\\ \pi[x^{(a-1)},x^{(a)}] &\Leftrightarrow& \tau(y^{(a-1)})\\ \pi(x^{(a)}) &\Leftrightarrow& \tau[y^{(a-1)},y^{(a)}],\end{aligned}$$ where the last two matchings are repeated while incrementing $a$ (starting with $a=2$). After each iteration, we are left with the unmatched subcurves $\pi[x^{(a)}, p^{-}]$ and $\tau[y^{(a)}, f(p^{+})]$. We would like to complete the matching with the following scheme $$\begin{aligned} \pi[x^{(a)},p^{-}] &\Leftrightarrow& \tau(y^{(a)})\\ \pi(p^{+}) &\Leftrightarrow& \tau[y^{(a)},f(p^{+})]\end{aligned}$$ This is indeed possible if $$\pi(p^{-}) \leq \tau(z_{\min}^{(a+1)}) + \delta.$$ The above is the equivalent to the trivial case ([Case \[case:trivial\]]{}). We first prove correctness in this case ([Case \[case:matroska\]]{}(\[item:trivial\])). To this end, we extend [Claim \[claim:frechet:x\]]{} as follows. Note that this claim will also be used in the non-trivial cases that follow. ${\ensuremath{{\left\{ { \tau(t) ~|~ t \in [y^{(a-1)},y^{(a)}]} \right\}} }} \subseteq {\left[ {\tau(t_{\min}^{(a)}) ~,~ \min(\tau(t_{\min}^{(a)}) + 2\delta, \tau(s_{j+1}) )} \right]} \subseteq {\ensuremath{[\pi(x^{(a)})]_{\delta}}}.$ [\[claim:frechet:xa\]]{} Recall that $s_j \leq y \leq y^{(a-1)} \leq y^{(a)} \leq s_{j+1}$ by [Claim \[claim:xy:inside\]]{}. By definition of $t_{\min}^{(a)}$, $${\ensuremath{\min( \tau[y^{(a-1)},y^{(a)}])}} \geq {\ensuremath{\min( \tau[y^{(a-1)},s_{j+1}])}} \geq \tau(t_{\min}^{(a)}).$$ By [Property \[prop:signature\]]{}(\[item:sig:descent\]) and the definitions of $y^{(a)}$ and $t_{\min}^{(a)}$, we also have that $${\ensuremath{\max( \tau[y^{(a-1)},y^{(a)}])}} \leq \tau(t_{\min}^{(a)}) + 2\delta,$$ and by [Property \[prop:signature\]]{}(\[item:sig:sj1:max\]), we also have that $${\ensuremath{\max( \tau[y^{(a-1)},y^{(a)}])}} \leq \tau(s_{j+1}).$$ This proves the first part of the claim. For the second part we use the definition of $\pi(x^{(a)}) = \min(\tau(t_{\min}^{(a)})+\delta, \tau(s_{j+1})-\delta)$, which implies $$\begin{aligned} \tau(t_{\min}^{(a)}) &\geq & \pi(x^{(a)}) - \delta\\ \min(\tau(t_{\min}^{(a)}) + 2\delta, \tau(s_{j+1})) &=& \pi(x^{(a)}) + \delta.\end{aligned}$$ If for some value of $a$, it holds that $\pi(p^{-}) \leq \tau(z_{\min}^{(a+1)}) + \delta$ then the above matching scheme is valid. [\[claim:trivial:subcase\]]{} By [Claim \[claim:frechet:x\]]{}, [Claim \[claim:frechet:ya\]]{} and [Claim \[claim:frechet:xa\]]{} the iterative part of the matching scheme is valid. It remains to prove the validity of the last two matchings. By [Claim \[claim:frechet:y\]]{}, $${\ensuremath{{\left\{ { \pi(p) ~|~ p \in [x,p^{-}]} \right\}} }} \subseteq [\pi(p_{\max}) -2\delta, \pi(p_{\max})] = {\ensuremath{[\tau(y)]_{\delta}}} = {\ensuremath{[\tau(y^{(a)})]_{\delta}}}.$$ Since $x \leq x^{(a)} \leq p^{-}$, this implies that the [Fréchet]{}distance between $\pi[x^{(a)},p^{-}]$ and $\tau(y^{(a)})$ is at most $\delta$. As for the other matching, we have by our case distinction $$\pi(p^{-}) \leq \tau(z_{\min}^{(a+1)}) + \delta = {\ensuremath{\min( \tau[y^{(a)},f(p^{+})])}} + \delta,$$ while (by [Property \[prop:frechet\]]{}) the matching $f$ testifies that $${\ensuremath{\max( \tau[f(p^{-}),f(p^{+})])}} \leq {\ensuremath{\max( \pi[p^{-},p^{+}])}} + \delta = \pi(p^{-}) + \delta.$$ Since $f(p^{+}) \geq y^{(a)} \geq y \geq f(p^{-})$, the above implies $${\ensuremath{{\left\{ { \tau(t) ~|~ t \in [y^{(a)},f(p^{+})]} \right\}} }} \subseteq {\ensuremath{[\pi(p^{-})]_{\delta}}} = {\ensuremath{[\pi(p^{+})]_{\delta}}}.$$ Note that the proof holds both if $s_{j+1} < f(p^{+})$ or $s_{j+1} \geq f(p^{+})$. From now on, we will assume the non-trivial (sub)case. Our matching scheme is based on a stopping parameter ${\overline{a}}$, which (intuitively) depends on whether $f$ matched some point on the missing subcurve $\pi[p^{-},p^{+}]$ to a signature vertex $\tau(s_{j+1})$ of $\tau$. (Stopping parameter ${\overline{a}}$) If $s_{j+1} \geq f(p^{+})$ then let ${\overline{a}}$ be the minimal value of $a$ satisfying $f(p^{+}) \leq y^{(a)}.$ Otherwise, let ${\overline{a}}$ be the minimal value of $a$ such that $y^{(a)} = y^{(a+1)} \leq s_{j+1}.$ [\[def:a:stop\]]{} The stopping parameter ${\overline{a}}$ ([Definition \[def:a:stop\]]{}) is well-defined and the matching scheme is valid for $a\leq {\overline{a}}.$ We first argue that there must be a value of $a$ such that $t_{\min}^{(a+1)}=y^{(a)}=y^{(b)}$ for any $b>a$. Recall that by our initial assumptions, $z_{\min} \in [s_j,s_{j+1}]$ and thus $z_{\min} \leq t_{\min} \leq s_{j+1}$. As a consequence, [Claim \[claim:y:exists\]]{} testifies that in the non-trivial case, the point $\tau(y)$ exists and is well-defined. We defined $y^{(1)}=y$ and for $a>1$ we defined $$y^{(a)}= \min \{t \in [t_{\min}^{(a)},s_{j+1}] ~|~ \tau(t) = \pi(p_{\max}) - \delta\}.$$ Since by [Claim \[claim:sj1:high\]]{}, $\tau(s_{j+1}) \geq \pi(p_{\max})-\delta = \tau(y^{(a)})$, there must be a value of $a$ such that $${\ensuremath{\min( \tau[y^{(a)},s_{j+1}])}} \geq \tau(y^{(a)}).$$ Let this value of $a$ be denoted ${\widehat{a}}$. In this case, it follows by definition that $t_{\min}^{({\widehat{a}}+1)} = y^{({\widehat{a}})}$, which implies that $y^{({\widehat{a}}+1)}=y^{({\widehat{a}})}$ and $t_{\min}^{({\widehat{a}}+2)}=y^{({\widehat{a}}+1)}, \dots$. This has the effect that $y^{({\widehat{a}})}=y^{(b)}$ for any $b > {\widehat{a}}$. Now, if $s_{j+1} < f(p^{+})$, then the above analysis implies that ${\overline{a}}$ is well-defined. However, if $s_{j+1} \geq f(p^{+})$, we defined ${\overline{a}}$ to be the minimal value of $a$ such that $f(p^{+}) \leq y^{(a)}$. Now it might happen that $y^{({\widehat{a}})} \leq f(p^{+}) \leq s_{j+1}$. In this case, there exists no value of $a$ such that $f(p^{+}) \leq y^{(a)}$, thus $y^{({\overline{a}})}$ does not exist. We can reduce this case to the trivial case ([Case \[case:matroska\]]{}(\[item:trivial\])) as follows. By [Claim \[claim:sj1:high\]]{}, $\tau(s_{j+1}) \geq \pi(p_{\max})-\delta$ and by [Property \[prop:signature\]]{}(\[item:sig:sj1:max\]), $\tau(s_{j+1})$ must be a maximum on $\tau[s_{j}, s_{j+2}]$. Thus, by definition of $t_{\min}^{({\overline{a}})}$, we would have $\tau\left(z_{\min}^{({\overline{a}})}\right) = \tau\left(t_{\min}^{({\overline{a}})}\right) = \tau(y^{({\overline{a}}-1)}) = \pi(p_{\max})-\delta \geq \pi(p^-)-\delta$, which is [Case \[case:matroska\]]{}(i) (the trivial case). Thus, also in the non-trivial case, ${\overline{a}}$ is well-defined. The validity of the matching scheme for $a\leq {\overline{a}}$ follows from [Claim \[claim:frechet:x\]]{}, [Claim \[claim:frechet:ya\]]{} and [Claim \[claim:frechet:xa\]]{}. It follows that the iterative part of the matching scheme is valid for $a\leq {\overline{a}}.$ Now we are left with the unmatched subcurves $\pi[x^{({\overline{a}})}, p^{-}]$ and $\tau[y^{({\overline{a}})}, f(p^{+})]$ and we have to complete the matching scheme. In order to set up a case analysis with a similar structure as before, we define $$\begin{aligned} q'&=& \min{\!\left({g(\pi(p_{\max})), g(\tau(t_{\min}^{({\overline{a}}-1)})+\delta)}\right)}\\ q''&=& \min{\!\left({g(\pi(p_{\max})), g(\tau(s_{j+1})-\delta)}\right)}\end{aligned}$$ [|c|m[7.5cm]{}|m[5cm]{}|]{} case & definition & intended matching\ \[case:matroska\](\[item:trivial\]) & $\exists a: \pi(p^{-}) \leq \tau(z_{\min}^{(a+1)}) + \delta$ & $\pi[x^{(a)},p^{-}] \Leftrightarrow \tau(y^{(a)})$$\pi(p^{+}) \Leftrightarrow \tau[y^{(a)},f(p^{+})]$\ \[case:matroska\](\[item:level\]) & $p^{+} \leq f^{-1}(y^{({\overline{a}})}) \leq q'$ & $\pi[x^{({\overline{a}})},p^{-}] \Leftrightarrow \tau(y^{{\overline{a}}})$$\pi[ p^{+}, f^{-1}(y^{{\overline{a}}})] \Leftrightarrow \tau(y^{{\overline{a}}})$\ \[case:matroska\](\[item:upwards\]) & $p^+\leq q' < f^{-1}(y^{({\overline{a}})})$ and $q'= g(\pi(p_{\max}))$ & This case can be reducedto [Case \[case:matroska\]]{}(i)\ \[case:matroska\](\[item:downwards\]) & $p^+\leq q' < f^{-1}(y^{({\overline{a}})})$ and$ q' = g(\tau(t_{\min}^{{\overline{a}}-1})+\delta)$ & $\pi[x^{({\overline{a}}-1)},p^{-}] \Leftrightarrow \tau(y^{({\overline{a}}-1)})$$\pi[p^{+}, q'] \Leftrightarrow \tau(y^{({\overline{a}}-1)})$$\pi(q') \Leftrightarrow \tau[y^{({\overline{a}}-1)}, f(q')]$\ \[case:matroska\](\[item:upwards:sig\]) & $p^{+} > f^{-1}(y^{({\overline{a}})})$ and $q'' = g(\pi(p_{\max}))$For the matching scheme, let $x'=\min\{t\in [x^{({\overline{a}})},p_{\max}]~|~ \tau(t)=\tau(s_{j+1})-\delta \}$ $y'=\max\{t\in [0,f(q)]~|~ \tau(t)=\tau(y) \}$ $z=\max\{p^{+},f^{-1}(y')\}$ & $\pi[x^{({\overline{a}})},x'] \Leftrightarrow \tau(y^{({\overline{a}})})$$\pi(x') \Leftrightarrow \tau[y^{({\overline{a}})},y']$$\pi[x',p^{-}] \Leftrightarrow \tau(y')$$\pi[p^{+}, z] \Leftrightarrow \tau(y')$$\pi[z] \Leftrightarrow \tau[y',f(p^{+})]$\ \[case:matroska\](\[item:downwards:sig\]) & $p^{+} > f^{-1}(y^{({\overline{a}})})$ and $q'' = g(\tau(s_{j+1})-\delta)$ & $\pi[x^{({\overline{a}})},p^{-}] \Leftrightarrow \tau(y^{({\overline{a}})})$$\pi[p^{+},q''] \Leftrightarrow \tau(y^{({\overline{a}})})$$\pi(q'') \Leftrightarrow \tau[y^{({\overline{a}})},f(q'')]$\ [\[tab:matroska:subcases\]]{} The exact case distinction is specified in [Table \[tab:matroska:subcases\]]{}: trivial case (see [Claim \[claim:trivial:subcase\]]{}),\[item:trivial\] $\pi$ stays level, \[item:level\] $\pi$ tends upwards,\[item:upwards\] $\pi$ tends downwards,\[item:downwards\] unmatched signature vertex and $\pi$ tends upwards, \[item:upwards:sig\] unmatched signature vertex and $\pi$ tends downwards. \[item:downwards:sig\] It remains to prove that the case analysis is complete and to prove correctness in each of these subcases. The case distinction of subcases of [Case \[case:matroska\]]{} ([Table \[tab:matroska:subcases\]]{}) is complete. [\[claim:matroska:complete\]]{} We assume that we are not in the trivial case [Case \[case:matroska\]]{}(\[item:trivial\]). If $f(p^{+}) \leq y^{({\overline{a}})}$ (also $f(p^{+}) \leq y^{({\overline{a}})} \leq s_{j+1}$) we get one of [Case \[case:matroska\]]{}(\[item:level\])-(\[item:downwards\]). Otherwise we have $f(p^{+}) > y^{({\overline{a}})}$ (also $f(p^{+}) > s_{j+1} \geq y^{({\overline{a}})}$). In this case, we get one of [Case \[case:matroska\]]{}(\[item:upwards:sig\])-(\[item:downwards:sig\]). In the following, we argue that, indeed, if the subcurve of $\tau$ specified by the parameter interval $[f(p^{-}),f(p^{+})]$ contains the signature vertex at $s_{j+1}$, it must be that $$\tau(s_{j+1}) - \delta \in {\ensuremath{{\left\{ { \pi(p) ~|~ p \in [p^{+},1]} \right\}} }}$$ and thus, $q''\neq p_\infty$ and $\pi(q'')$ is be one of ${\left\{ {\pi(p_{\max}), \tau(s_{j+1})-\delta} \right\}}$. Assume that $s_{j+2} \neq 1$, i.e., the next signature vertex after $\tau(s_{j+1})$ is not the last signature vertex. In this case, by [Property \[prop:signature\]]{} and [Property \[prop:frechet\]]{}, we have $$\tau(s_{j+1}) \geq \tau(s_{j+2})+2\delta \geq \pi(f^{-1}(s_{j+2})) + \delta.$$ Since $\pi$ is continuous, this implies that there must exist a point $\pi(t)$ with $t \geq p^{+}$ and $\pi(t) \leq \tau(s_{j+1})-\delta$. Now, assume that $s_{j+2} = 1$. In this case, we have by the theorem statement that $\pi(p_i) \notin {\ensuremath{[\tau(s_{j+2})]_{4\delta}}}$. It must be that either $\tau(s_{j+1}) \geq \pi(p_i) > \tau(s_{j+2}) + 4\delta $ (in which case we can apply the above argument), or $\pi(p_i) < \tau(s_{j+2}) - 4\delta \leq \tau(s_{j+1}) - 5\delta$. The second case is not possible since by [Claim \[claim:max:dist\]]{} and by [Property \[prop:frechet\]]{} we have $$\begin{aligned} \pi(p_i) &\geq& \tau(f(p_i)) -\delta \geq \tau(t_{\min}) -\delta \geq \pi(p_{\max}) - 4\delta \\ && \geq \pi(p^{+})-4\delta \geq \pi(f^{-1}(s_{j+1}))-4\delta \geq \tau(s_{j+1}) - 5\delta.\end{aligned}$$ Here, $\pi(p^{+}) \geq \pi(f^{-1}(s_{j+1}))$ follows from $s_{j+1} \in [f(p^{-}), f(p^{+})]$ in [Case \[case:matroska\]]{}(\[item:upwards:sig\])-(\[item:downwards:sig\]) and the fact that $${\ensuremath{\max( \pi[p^{-},p^{+}])}} = \pi(p^{+}),$$ by our initial assumptions. By [Claim \[claim:frechet:y\]]{} the [Fréchet]{}distance between $\pi[x^{({\overline{a}})},p^{-}]$ and $\tau(y^{({\overline{a}})})$ is at most $\delta$. By our case distinction, $${\ensuremath{{\left\{ { \pi(p) ~|~ p \in [p^{+},f^{-1}(y^{({\overline{a}})})]} \right\}} }} \subseteq [\tau(t_{\min}^{({\overline{a}}-1)})+\delta, \pi(p_{\max})] \subseteq [\pi(p_{\max}) -2\delta, \pi(p_{\max})] = {\ensuremath{[\tau(y^{({\overline{a}})})]_{\delta}}},$$ since by [Claim \[claim:max:dist\]]{}, $$\tau(t_{\min}^{({\overline{a}}-1)})+\delta \geq \tau(t_{\min})+\delta \geq \pi(p_{\max}) -2\delta.$$ (Note that $t_{\min}^{({\overline{a}}-1)}$ always exists since $y^{(1)} \leq s_{j+1}$ by [Claim \[claim:xy:inside\]]{}.) This implies that also the second matching is valid. We can reduce this case to [Case \[case:matroska\]]{}(\[item:trivial\]) (the trivial case) as follows. By our case distinction, $f(p^{+}) \leq f(q') < y^{({\overline{a}})}$. Let $b$ be the maximal value of $a$ such that $f(q') \in [y^{(a)}, y^{({\overline{a}})}]$. By [Property \[prop:frechet\]]{} it must be that $\tau(f(q')) \geq \pi(p_{\max}) - \delta = \tau(y^{(b)})$. Thus $\min(\tau[y^{(b)},f(q')])\geq \pi(p_{\max})-\delta$. This holds since for any $a'$, $\tau$ goes upwards in $\tau(y^{(a')})$, then intersects $\pi(p_{\max})-\delta$ downwards and goes upwards again in $\tau(y^{(a'+1)})$. By our case distinction, $f(p^{+}) \in [y^{(b)}, f(q')]$. Thus, $$\tau(z_{\min}^{(b+1)}) = {\ensuremath{\min( \tau[y^{(b)},f(p^{+})])}} \geq {\ensuremath{\min( \tau[y^{(b)},f(q')])}} \geq \pi(p_{\max}) - \delta\geq \pi(p^-) - \delta.$$ In this case, we rollback the last two matchings of the iterative matching scheme and instead end with $a={\overline{a}}-1$. Thus, we are left with the unmatched subcurves $\pi[x^{({\overline{a}}-1)}, p^{-}]$ and $\tau[y^{({\overline{a}}-1)}, f(p^{+})]$. We complete the matching scheme as defined in [Table \[tab:matroska:subcases\]]{}. The validity of the first matching follows directly from [Claim \[claim:frechet:y\]]{}, since $x^{({\overline{a}}-1)} > x$. By the definition of $q'$ and our case distinction, $${\ensuremath{{\left\{ { \pi(p) ~|~ p \in [p^{+},q']} \right\}} }} \subseteq [\tau(t_{\min}^{({\overline{a}}-1)} +\delta, \pi(p_{\max}))] \subseteq {\ensuremath{[\tau(y^{({\overline{a}}-1)})]_{\delta}}}.$$ This proves validity of the second matching. [Claim \[claim:frechet:xa\]]{} implies the validity of the last matching. ![Examples of [Case \[case:matroska\]]{}(\[item:trivial\])-(\[item:downwards\]). The broken part of the matching $f$ is indicated by fat lines.](shortcutting_case5.pdf "fig:")\ ![Examples of [Case \[case:matroska\]]{}(\[item:trivial\])-(\[item:downwards\]). The broken part of the matching $f$ is indicated by fat lines.](shortcutting_case5.pdf "fig:")\ ![Examples of [Case \[case:matroska\]]{}(\[item:trivial\])-(\[item:downwards\]). The broken part of the matching $f$ is indicated by fat lines.](shortcutting_case5.pdf "fig:")\ ![Examples of [Case \[case:matroska\]]{}(\[item:trivial\])-(\[item:downwards\]). The broken part of the matching $f$ is indicated by fat lines.](shortcutting_case5.pdf "fig:")\ [\[fig:shortcutting:case5i\]]{} ![Examples of [Case \[case:matroska\]]{}(\[item:upwards:sig\])-(\[item:downwards:sig\]). The broken part of the matching $f$ is indicated by fat lines.](shortcutting_case5.pdf "fig:")\ ![Examples of [Case \[case:matroska\]]{}(\[item:upwards:sig\])-(\[item:downwards:sig\]). The broken part of the matching $f$ is indicated by fat lines.](shortcutting_case5.pdf "fig:")\ ![Examples of [Case \[case:matroska\]]{}(\[item:upwards:sig\])-(\[item:downwards:sig\]). The broken part of the matching $f$ is indicated by fat lines.](shortcutting_case5.pdf "fig:")\ ![Examples of [Case \[case:matroska\]]{}(\[item:upwards:sig\])-(\[item:downwards:sig\]). The broken part of the matching $f$ is indicated by fat lines.](shortcutting_case5.pdf "fig:")\ [\[fig:shortcutting:case5v\]]{} We have now handled [Case \[case:matroska\]]{}(\[item:trivial\])-(\[item:downwards\]). Examples of these cases are shown in [Figure \[fig:shortcutting:case5i\]]{}. We now move on to prove correctness of the remaining cases [Case \[case:matroska\]]{}(\[item:upwards:sig\]) and [Case \[case:matroska\]]{}(\[item:downwards:sig\]). Observe that in this case $q''=q=g(\pi(p_{\max}))$, as in [Case \[case:upwards\]]{}. Therefore, $y'$ and $z$ are the same as in [Case \[case:upwards\]]{} and must exist. We argue that $x'$ must also exist. Indeed, we can derive $\pi(x^{({\overline{a}})}) \leq \tau(s_{j+1})-\delta \leq \pi(p_{\max})$, as follows. Recall that by our case distinction $f(p^{-}) \leq s_{j+1} \leq f(p^{+})$. By [Property \[prop:frechet\]]{}, it follows that $$\tau(s_{j+1}) = {\ensuremath{\max( \tau[f(p^{-}),f(p^{+})])}} \leq {\ensuremath{\max( \pi[p^{-},p^{+}])}} +\delta \leq \pi(p_{\max}) +\delta.$$ Now we need to prove the validity of the matching scheme. The first line follows from [Claim \[claim:frechet:ya\]]{}. For the second line we need to prove that $${\ensuremath{{\left\{ { \tau(t) ~|~ t \in [y^{({\overline{a}})},y']} \right\}} }} \subseteq [\tau(s_{j+1})-2\delta, \tau(s_{j+1})] = {\ensuremath{[\pi(x')]_{\delta}}}.$$ The upper bound follows from [Property \[prop:signature\]]{}(\[item:sig:sj1:max\]). As for the lower bound, by the definition of the stopping parameter, $${\ensuremath{\min( \tau[y^{({\overline{a}})},s_{j+1}])}} = \tau(y^{({\overline{a}})})= \pi(p_{\max})-\delta \geq \tau(s_{j+1})-2\delta,$$ (as we just proved above). By [Claim \[claim:descent:after:sj1\]]{}, $${\ensuremath{\min( \tau[s_{j+1},f(p^{+})])}} \geq \tau(s_{j+1})-2\delta.$$ By our case distinction and by [Property \[prop:frechet\]]{}, $${\ensuremath{\min( \tau[f(p^{+}),f(q'')])}} \geq {\ensuremath{\min( \pi[p^{+},q''])}}-\delta \geq \tau(s_{j+1})-2\delta.$$ The validity of the third matching is implied by [Claim \[claim:frechet:y\]]{}. For the last two matchings we can apply the respective part of the proof of [Case \[case:upwards\]]{} verbatim. The validity of the first matching follows from [Claim \[claim:frechet:y\]]{} and since $x^{({\overline{a}})} \geq x$. By our case distinction, $${\ensuremath{{\left\{ { \pi(p) ~|~ p \in [p^{+},q'']} \right\}} }} \subseteq [\tau(s_{j+1})-\delta, \pi(p_{\max})] \subseteq {\ensuremath{[\tau(y^{({\overline{a}})}]_{\delta}}}).$$ Thus, also the second matching is valid. For the last matching we need to prove that $${\ensuremath{{\left\{ { \tau(t) ~|~ t \in [y^{({\overline{a}})},f(q'')]} \right\}} }} \subseteq [\tau(s_{j+1})-2\delta, \tau(s_{j+1})] = {\ensuremath{[\pi(q'')]_{\delta}}}.$$ Again, as in [Case \[case:matroska\]]{}(\[item:upwards:sig\]), it holds that $${\ensuremath{{\left\{ { \tau(t) ~|~ t \in [y^{({\overline{a}})},f(p^{+})]} \right\}} }} \subseteq [\tau(s_{j+1})-2\delta, \tau(s_{j+1})].$$ By our case distinction and by [Property \[prop:frechet\]]{} $${\ensuremath{\min( \tau[f(p^{+}),f(q'')])}} \geq {\ensuremath{\min( \pi[p^{+},q''])}} -\delta \geq \tau(s_{j+1})-2\delta$$ This also implies that $f(q'') < s_{j+2}$, by [Property \[prop:signature\]]{}(\[item:sig:length\]). Thus, by [Property \[prop:signature\]]{}(\[item:sig:sj1:max\]), we conclude $${\ensuremath{\max( \tau[f(p^{+}),f(q'')])}} \leq \tau(s_{j+1}).$$ Together this implies the validity of the last matching. We now proved correctness of the last two cases [Case \[case:matroska\]]{}(\[item:upwards:sig\]) and [Case \[case:matroska\]]{}(\[item:downwards:sig\]). Examples of these cases are shown in [Figure \[fig:shortcutting:case5v\]]{}. It remains to prove the boundary cases, which we have ruled out so far by [Assumption \[ass:inner:sig:edge\]]{}. There are three boundary cases: $s_j=0$ and $s_{j+1}=1$, $s_j=0$ and $s_{j+1}<1$, $s_j>0$ and $s_{j+1}=1$. To prove the claim in each of these cases, we can use the above proof verbatim with minor modifications. Note that in the proof, we used $s_j$ in its function as the minimum on the signature edge $\overline{s_js_{j+1}}$, resp., we used $s_{j+1}$ in its function as the maximum on this edge. Thus, let $$s_{\min}=\operatorname*{arg\,min}_{s \in [s_j,s_{j+1}]} \tau(s), ~~~ s_{\max}=\operatorname*{arg\,max}_{s \in [s_{j},s_{j+1}]} \tau(s).$$ In each of the cases (B1), (B2) and (B3), it holds that $f(p_i) \in [s_{\min},s_{\max}]$ and $\tau(s_{\max})-\tau(s_{\min}) \geq 4\delta$. [\[claim:pi:inside\]]{} By the theorem statement and by [Definition \[def:signature\]]{}, it holds that $$\pi(p_i) \notin {\ensuremath{[v_1]_{4\delta}}} \cup {\ensuremath{[v_{\ell}]_{4\delta}}} = {\ensuremath{[\tau(0)]_{4\delta}}} \cup {\ensuremath{[\tau(1)]_{4\delta}}}$$ i.e., the removed vertex $\pi(p_i)$ lies very far from the endpoints of the curve $\tau$. At the same time, by [Definition \[def:signature\]]{}, in case $s_j=0$, $$\tau(0) \geq \tau(s_{\min}) \geq \tau(0) - \delta$$ and, in case $s_{j+1}=1$, $$\tau(1) \leq \tau(s_{\max}) \leq \tau(1) + \delta.$$ By the direction-preserving property of [Definition \[def:signature\]]{} and by [Property \[prop:frechet\]]{}, this implies that $f(p_i) \in [s_{\min},s_{\max}]$. In the cases where $s_j=0$, this implies $$\tau(f(p_i)) \geq \tau(s_{\min}) \geq \tau(0) - \delta,$$ therefore, by the above, $\tau(f(p_i)) \geq \tau(0) + 4\delta \geq \tau(s_{\min})+4\delta$. Similarly in the cases, where $s_{j+1}=1$, we can derive that $\tau(f(p_i)) \leq \tau(1) - 4\delta \leq \tau(s_{\max})-4\delta$. In each of the cases (B1), (B2) and (B3), this implies the second part of the claim. We replace [Property \[prop:signature\]]{} with the following property. $\tau(s_{\max}) - \tau(s_{\min}) > 2\delta$, $\tau(s_{\min}) = {\ensuremath{\min( \tau[s_{j-1},s_{j+1}])}}$ (if $s_j=0$, then $\tau(s_{\min}) = {\ensuremath{\min( \tau[0,s_{j+1}])}}$), $\tau(s_{\max}) = {\ensuremath{\max( \tau[s_j,s_{j+2}])}}$ (if $s_{j+1}=1$, then $\tau(s_{\max}) = {\ensuremath{\max( \tau[s_j,1])}}$), $\tau(t) \geq \tau(t')-2\delta$ for $s_{\min}\leq t' < t \leq s_{\max}$, if $s_{j}=0$, then $\tau(s_{\max}) - \tau(s_{j+2}) > 2\delta$. [\[prop:signature:boundary\]]{} [Property \[prop:signature:boundary\]]{}(\[item:sig:sj:min\]), (\[item:sig:sj1:max\]), (\[item:sig:descent\]), and (\[item:sig:length:2\]) hold by [Definition \[def:signature\]]{}. [Property \[prop:signature:boundary\]]{}(\[item:sig:length\]) follows from [Claim \[claim:pi:inside\]]{}. Instead of [Claim \[claim:descent:after:sj1\]]{} we use the claim If $s_{\max} \in [f(p^{-}),f(p^{+})]$ then ${\ensuremath{{\left\{ { \tau(t) ~|~ t \in [s_{\max},f(p^{+})]} \right\}} }} \subseteq [\tau(s_{\max})-2\delta, \tau(s_{\max})]$. Instead of [Claim \[claim:no:minimum\]]{} we use the claim $s_{\min} \notin [f(p^{-}),f(p^{+})]$. Now, the theorem follows in the boundary cases (B1), (B2) and (B3), by replacing $s_j$ with $s_{\min}$ and replacing $s_{j+1}$ with $s_{\max}$. This closes the proof of [Lemma \[lemma:remove:one\]]{} $({\ensuremath{k}},{\ensuremath{\ell}})$-center =============================================== [\[alg:candidate:generator:center\]]{} For each ${{\ensuremath{{{\ensuremath{\tau}}}_{i}}}}$, let ${\ensuremath{\mathcal{V}}}_i$ be the vertex set of its $\alpha$-signature computed by [Algorithm \[alg:low:pass\]]{} Compute the union $U$ of the intervals $r=\left[w-4\alpha,w+4\alpha\right]$ for $w \in {\ensuremath{\mathcal{V}}}=\bigcup_{i=1}^{n} {\ensuremath{\mathcal{V}}}_i$ [\[lemma:candidate:generator:center\]]{} Given a set of curves $P=\{{{\ensuremath{{{\ensuremath{\tau}}}_{1}}}},\dots,{{\ensuremath{{{\ensuremath{\tau}}}_{n}}}}\} $ and parameters $\alpha,\beta > 0$, and ${\ensuremath{k}},{\ensuremath{\ell}}\in {{\rm I\!\hspace{-0.025em} N}}$, then [Algorithm \[alg:candidate:generator:center\]]{} generates a set of candidate solutions $\Gamma^{{\ensuremath{k}},{\ensuremath{\ell}}}_{\alpha,\beta}(S) \subseteq \Delta_\ell$ of size at most $\left({\left\lfloor {\frac{24\alpha{\ensuremath{k}}{\ensuremath{\ell}}}{\beta}} \right\rfloor}+6{\ensuremath{k}}{\ensuremath{\ell}}\right)^{{\ensuremath{\ell}}}$. Furthermore, if $\alpha \geq \operatorname{opt}^{(\infty)}_{{\ensuremath{k}},{\ensuremath{\ell}}}(P)$, then the generated set contains $k$ candidates $\tilde{C}=\tilde{c_1},\dots,\tilde{c_k}$ with $$\operatorname{cost}_{\infty}(P, \tilde{C}) \leq \alpha + \beta.$$ Let $C=c_1,\dots,c_{{\ensuremath{k}}}$ denote an optimal solution for $P$ and let $c_i=z_{i1},\dots,z_{i{{\ensuremath{\ell}}}}$ denote the vertices for each cluster center. Consider the union of intervals $$R= \bigcup_{i=1}^{{\ensuremath{k}}} \bigcup_{j=1}^{{\ensuremath{\ell}}} [z_{ij}-4\alpha,z_{ij}+4\alpha].$$ [Lemma \[lemma:nec:suff\]]{} implies that $R$ contains all elements of ${\ensuremath{\mathcal{V}}}$, the signature vertices computed by [Algorithm \[alg:candidate:generator:center\]]{}. Now consider the dual statement, namely, whether the vertices $z_{ij}$ are contained in the set $U$ computed by the algorithm. If there exists a $z_{ij}$ which is not contained in $U$, then [Theorem \[theo:remove:one\]]{} implies that we can omit $z_{ij}$ from the solution while not increasing the cost beyond $\alpha$. Therefore, let $\widehat{C}$ denote the solution where all vertices that lie outside $U$ have been omitted. Clearly, $U$ contains all remaining vertices of cluster centers in $\widehat{C}$. Therefore, $\Gamma^{{\ensuremath{k}},{\ensuremath{\ell}}}_{\alpha,\beta}$ must contain $k$ candidates $\tilde{C}=\tilde{c_1},\dots,\tilde{c_k}$ with $$\operatorname{cost}_{\infty}(P,\tilde{C}) \leq \alpha + \beta.$$ Note that $R$ consists of at most ${\ensuremath{k}}{\ensuremath{\ell}}$ intervals and has measure at most $\mu(R)=8\alpha{\ensuremath{k}}{\ensuremath{\ell}}$. Therefore, the measure of $U$ can be at most $\mu(U) \leq \mu(R) +(2{\ensuremath{k}}{\ensuremath{\ell}})8\alpha = 24\alpha{\ensuremath{k}}{\ensuremath{\ell}}.$ In the worst case a signature vertex lies at each boundary point of $R$. Furthermore, $U$ consists of at most ${\left\lceil {\frac{\mu(U)}{8\alpha}} \right\rceil}\leq 3{\ensuremath{k}}{\ensuremath{\ell}}$ intervals, since each interval has measure at least $8\alpha$. [[\[theo:k:l:center:main\]]{}]{} Let ${{\varepsilon}}>0$ and ${\ensuremath{k}},{\ensuremath{\ell}}\in {{\rm I\!\hspace{-0.025em} N}}$ be given constants. Given a set of curves $P=\{{{\ensuremath{{{\ensuremath{\tau}}}_{1}}}},\dots,{{\ensuremath{{{\ensuremath{\tau}}}_{n}}}}\}$ we can compute a $(1+{{\varepsilon}})$-approximation to $\operatorname{opt}^{(\infty)}_{{\ensuremath{k}},{\ensuremath{\ell}}}(P)$ in time $O{\!\left({nm\log m}\right)}$. We use [Algorithm \[alg:phase1\]]{} described in [Section \[sec:cf:approx:uni\]]{} to compute a constant-factor approximation. We obtain an interval $[\delta_{\min},\delta_{\max}]$ which contains $\operatorname{opt}^{(\infty)}_{{\ensuremath{k}},{\ensuremath{\ell}}}(P)$ and such that $\delta_{\max}/\delta_{\min}\leq 8$ by [Theorem \[theo:cf:approx:center:main\]]{}. We can now do a binary search in this interval. In each step of the binary search, we apply [Algorithm \[alg:candidate:generator:center\]]{} to $P$ a constant number of times and evaluate every candidate solution. More specifically, if we apply [Algorithm \[alg:candidate:generator:center\]]{} to $P$ with parameters $\alpha$ and $\beta={{\varepsilon}}\alpha$, by [Lemma \[lemma:candidate:generator:center\]]{} we gain the following knowledge: Either $\alpha < (1+{{\varepsilon}})\operatorname{opt}^{(\infty)}_{{\ensuremath{k}},{\ensuremath{\ell}}}(P)$, or $\alpha \geq \operatorname{opt}^{(\infty)}_{{\ensuremath{k}},{\ensuremath{\ell}}}(P)$ and we have computed a solution with cost at most $(1+{{\varepsilon}})\alpha$. In both cases, the outcome is correct. Since we want to take an exact decision during the binary search, we simply call the procedure twice with parameters $\alpha_1=\frac{\alpha}{1+{{\varepsilon}}/2},\beta_1=\frac{{{\varepsilon}}}{2}\alpha_1$ and $\alpha_2=\alpha,\beta_2={{\varepsilon}}\alpha_2$. Now there are three possible outcomes: Either $\operatorname{opt}^{(\infty)}_{{\ensuremath{k}},{\ensuremath{\ell}}}(P) < \alpha$, or $\operatorname{opt}^{(\infty)}_{{\ensuremath{k}},{\ensuremath{\ell}}}(P) > (1+{{\varepsilon}})\alpha$, or $\operatorname{opt}^{(\infty)}_{{\ensuremath{k}},{\ensuremath{\ell}}}(P) \in [\alpha,(1+{{\varepsilon}})\alpha]$. So either we can take an exact decision and proceed with the binary search, or we obtain a $(1+{{\varepsilon}})$-approximation to the solution and we stop the search. By [Lemma \[lemma:candidate:generator:center\]]{} we know that the size of the candidate set $\Gamma^{{\ensuremath{k}},{\ensuremath{\ell}}}_{\alpha,\beta}$ is $O(1)$ (where the constant depends on ${{\varepsilon}}, {\ensuremath{k}}$ and ${\ensuremath{\ell}}$. One execution of [Algorithm \[alg:candidate:generator:center\]]{} takes $O(nm)$ time for computing the $n$ signatures (using [Algorithm \[alg:low:pass\]]{}) and $O(1)$ time for generating the candidate set. Evaluating one candidate solution (consisting of $k$ centers from the candidate set) takes $kn$ [Fréchet]{}distance computations, where one [Fréchet]{}distance computation takes time $O(m \log m)$ using the algorithm by Alt and Godau [@ag-cfdbt-95]. The number of binary search steps depends only on the constant ${{\varepsilon}}$ and so is $O(1)$, which implies the running time. $({\ensuremath{k}},{\ensuremath{\ell}})$-median =============================================== In this section we will make use of a result by Ackermann [*et al.*]{}[@abs-cm-10] for computing an approximation to the $k$-median problem under an arbitrary dissimilarity measure $D:\Delta \times \Delta \rightarrow \mathbb{R}_0^+$ on a ground set of items $\Delta$, i.e. a function that satisfies $D(x,y) = 0$, iff $x \not= y$. The result roughly says that we can obtain an efficient $(1+\varepsilon)$-approximation algorithm for the $k$-median problem on input $P \subseteq \Delta$, if there is an algorithm that given a random sample of constant size returns a set of candidates for the $1$-median that contains with constant probability (over the choice of the sample) a $(1+\varepsilon)$-approximation to the $1$-median. We restate the sampling property defined by Ackermann [*et al.*]{} ([@abs-cm-10],Property 4.1). [\[def:sampling:property\]]{} We say a dissimilarity measure D satisfies the (weak) $[{{\varepsilon}},\lambda]$-sampling property iff there exist integer constants $m_{{{\varepsilon}},\lambda}$ and $t_{{{\varepsilon}},\lambda}$ such that for each $P\subseteq \Delta$ of size $n$ and for each uniform sample multiset $S\subseteq P$ of size $m_{{{\varepsilon}},\lambda}$ a set $\Gamma(S) \subseteq \Delta$ of size at most $t_{{{\varepsilon}},\lambda}$ can be computed satisfying $$\Pr\left[ \exists \tilde{c} \in \Gamma(S) : D(P,\tilde{c}) \leq (1+{{\varepsilon}}) \operatorname{opt}(P) \right] \geq 1-\lambda.$$ Furthermore, $\Gamma(S)$ can be computed in time depending on $\gamma,\delta$ and $m_{{{\varepsilon}},\lambda}$ only. It is likely that the sampling property ([Definition \[def:sampling:property\]]{}) does not hold for the [Fréchet]{}distance for arbitrary value of ${\ensuremath{\ell}}$. We will therefore prove a modified sampling property, which allows the size of the sample to depend on ${\ensuremath{\ell}}$. The following lemma intuitively says that curves that lie far away from a candidate median have little influence on the shape of the candidate median. [\[lemma:omit:far:away\]]{} Given a set of $n$ curves $P=\{\tau_1,\ldots, \tau_n\}$ and a polygonal curve $\pi$, it holds that $$\operatorname{cost}_{1}(P,\widehat{\pi}) \leq (1+{{\varepsilon}})\operatorname{cost}_{1}(P,\pi)$$ for a curve $\widehat{\pi}$ obtained from $\pi$ by omitting any subset of vertices lying outside the following union $R_S$:$$R_S = \bigcup_{ 1\leq i \leq n \atop x_i \leq \frac{2x_1}{{{\varepsilon}}} }\; \bigcup_{v \in {\ensuremath{\mathcal{V}}}(\sigma_i) } [v-4x_i,v+4x_i],$$ where the $\tau_i$ are sorted in increasing order of $x_i={\ensuremath{d_F{\!\left({{{\ensuremath{{{\ensuremath{\tau}}}_{i}}}},\pi}\right)}}}$ and where $\sigma_i$ is the $x_i$-signature of ${{\ensuremath{{{\ensuremath{\tau}}}_{i}}}} \in P$. By [Lemma \[lemma:nec:suff\]]{}, the curve $\pi$ has a vertex in each range centered at the vertices of $\sigma_i$. These will not be omitted, therefore it is ensured that $\widehat{\pi}$ has at least 2 vertices, i.e. defines a curve. By [Theorem \[theo:remove:one\]]{}, it holds that ${\ensuremath{d_F{\!\left({\widehat{\pi},{{\ensuremath{{{\ensuremath{\tau}}}_{i}}}}}\right)}}}\leq x_i$ for the curves ${{\ensuremath{{{\ensuremath{\tau}}}_{i}}}}$ with $x_i \leq \frac{2x_1}{{{\varepsilon}}}$, that is, for the curves that lie close to $\pi$. We now argue using the triangle inequality that the distances to the curves that lie further away are only altered by a factor of at most $(1+{{\varepsilon}})$. Consider any index $i$, such that $x_i > \frac{2x_1}{{{\varepsilon}}} = \widehat{x}$. By the triangle inequality, it holds that $$\begin{aligned} {\ensuremath{d_F{\!\left({\widehat{\pi},{{\ensuremath{{{\ensuremath{\tau}}}_{i}}}}}\right)}}} &\leq& {\ensuremath{d_F{\!\left({\widehat{\pi},{{\ensuremath{{{\ensuremath{\tau}}}_{1}}}}}\right)}}} + {\ensuremath{d_F{\!\left({{{\ensuremath{{{\ensuremath{\tau}}}_{1}}}},{{\ensuremath{{{\ensuremath{\tau}}}_{i}}}}}\right)}}} \\ &\leq& {\ensuremath{d_F{\!\left({\widehat{\pi},{{\ensuremath{{{\ensuremath{\tau}}}_{1}}}}}\right)}}} +{\ensuremath{d_F{\!\left({{{\ensuremath{{{\ensuremath{\tau}}}_{1}}}},\pi}\right)}}} + {\ensuremath{d_F{\!\left({\pi,{{\ensuremath{{{\ensuremath{\tau}}}_{i}}}}}\right)}}} \\ &\leq& x_1 + x_1 + x_i \\ &<& (1+{{\varepsilon}}) x_i\end{aligned}$$ Therefore, $$\begin{aligned} \operatorname{cost}_{1}(P,\widehat{\pi}) \leq \sum_{x_i \leq \widehat{x}} {\ensuremath{d_F{\!\left({\pi,{{\ensuremath{{{\ensuremath{\tau}}}_{i}}}}}\right)}}} + \sum_{x_i > \widehat{x}} (1+{{\varepsilon}}) {\ensuremath{d_F{\!\left({\pi,{{\ensuremath{{{\ensuremath{\tau}}}_{i}}}}}\right)}}} \leq (1+{{\varepsilon}}) \operatorname{cost}_{1}(P,\pi).\end{aligned}$$ The following lemma is in similar spirit as [Lemma \[lemma:omit:far:away\]]{}. We prove that the basic shape of a candidate median can be approximated based on a constant-size sample. [\[lemma:omit:low:prob\]]{} There exist integer constant $m_{{{\varepsilon}},\lambda,{\ensuremath{\ell}}}$ such that given a set of curves $P$ and a curve $\pi=u_1,\dots,u_{{\ensuremath{\ell}}}$ for each uniform sample multiset $S\subseteq P$ of size $m_{{{\varepsilon}},\lambda,{\ensuremath{\ell}}} \ge {\left\lceil {\frac{8{\ensuremath{\ell}}}{{{\varepsilon}}} \left(\log\left(\frac{1}{\lambda}\right)+\log({\ensuremath{\ell}})\right)} \right\rceil}$ it holds that $$\Pr\left[ \operatorname{cost}_{1}(P,\widehat{\pi}) \leq (1+{{\varepsilon}})\operatorname{cost}_{1}(P,\pi) \right] \geq 1-\lambda,$$ for a curve $\widehat{\pi}$ obtained from $\pi$ by omitting any subset of vertices lying outside the following union $R_S$: $$R_S = \bigcup_{ {{\ensuremath{{{\ensuremath{\tau}}}_{i}}}} \in S }\; \bigcup_{v \in {\ensuremath{\mathcal{V}}}(\sigma_i) } [v-4x_i,v+4x_i],$$ where the $\tau_i$ are sorted in increasing order of $x_i={\ensuremath{d_F{\!\left({{{\ensuremath{{{\ensuremath{\tau}}}_{i}}}},\pi}\right)}}}$ and where $\sigma_i$ is the $x_i$-signature of ${{\ensuremath{{{\ensuremath{\tau}}}_{i}}}} \in S$. If all vertices of $\pi$ are contained in $R_S$, then $\pi=\widehat{\pi}$ and the claim is implied. However, this is not necessarily the case. In the following, we consider a fixed vertex $u_j$ and we prove that it is either contained in $R_S$ with sufficiently high probability or ignoring it will not increase the cost of a solution significantly. For this purpose, let $T_j \subseteq P$ be the subset of curves ${{\ensuremath{{{\ensuremath{\tau}}}_{i}}}}$ with $$u_j \in \bigcup_{v\in {\ensuremath{\mathcal{V}}}(\sigma_i) } [v-4x_i,v+ 4x_i].$$ If any curve of $T_j$ is contained in our sample $S$, then $u_j$ is contained in $R_S$. We distinguish two cases. If $T_j$ is large enough then $u_j$ is contained in $R_S$ with high probability, or we argue that the total change in cost resulting from omitting $u_j$ from $\pi$ will be small. We will first argue that for all $j$ with $1\le j \le \ell$ and with $T_j > \frac{{{\varepsilon}}n}{4\ell}$ we obtain for our choice of $m_{{{\varepsilon}},\lambda,{\ensuremath{\ell}}}$ that at least one element of $T_j$ is contained in $S$. Indeed, we have $$\begin{aligned} \Pr[ T_j \cap S = \emptyset] &\leq (1-|T_j|/n)^{m_{{{\varepsilon}},\lambda,{\ensuremath{\ell}}}} \leq \left( 1 - \frac{{{\varepsilon}}}{4 {\ensuremath{\ell}}} \right) ^{m_{{{\varepsilon}},\lambda,{\ensuremath{\ell}}}}.\end{aligned}$$ We use the union bound, to estimate the probability that this event fails for at least one of the sets $T_j$ in question. We choose the parameter $m_{{{\varepsilon}},\lambda,{\ensuremath{\ell}}}$ large enough, such that it holds for the failure probability that $$\lambda \leq {\ensuremath{\ell}}\left( 1 - \frac{{{\varepsilon}}}{4 {\ensuremath{\ell}}} \right) ^{m_{{{\varepsilon}},\lambda,{\ensuremath{\ell}}}}.$$ For this, it suffices to choose $$\begin{aligned} m_{{{\varepsilon}},\lambda,{\ensuremath{\ell}}} &\ge {\left\lceil {\frac{8{\ensuremath{\ell}}}{{{\varepsilon}}} \left(\log\left(\frac{1}{\lambda}\right)+\log({\ensuremath{\ell}})\right)} \right\rceil}\end{aligned}$$ to obtain that with probability at least $1-\lambda$ for all $1\le j \le \ell$, simultaneously, we have that at least one element of $T_j$ is in $S$, if $|T_j| \ge \frac{{{\varepsilon}}n}{4 \ell}$. Now consider the set of curves $T = \{ T_j : 1\le j \le \ell, T_j \cap S = \emptyset\}$. By our previous considerations we have that with probability at least $1-\lambda$, $T \subseteq \left\{T_j \left| 1\leq j \leq{\ensuremath{\ell}}, |T_j| \leq \frac{{{\varepsilon}}n}{4 {\ensuremath{\ell}}} \right. \right\}$. We will assume in the following that this event happens. Let $\tilde{\pi}$ denote the curve obtained from $\pi$ by removing all vertices from $R_S$, which is equivalent to removing all vertices $u_j$ with $T_j \in T$. In the following, let $P_T = \bigcup_{T'\in T} T'$ be the set of input curves that are contained in one of the sets in $T$. By [Theorem \[theo:remove:one\]]{}, it holds for any curve ${{\ensuremath{{{\ensuremath{\tau}}}_{i}}}} \in P \setminus P_T$, that ${\ensuremath{d_F{\!\left({\tilde{\pi},{{\ensuremath{{{\ensuremath{\tau}}}_{i}}}}}\right)}}} \leq {\ensuremath{d_F{\!\left({\pi,{{\ensuremath{{{\ensuremath{\tau}}}_{i}}}}}\right)}}}=x_i$, i.e., their distances do not increase beyond $x_i$ by the removal. Let ${{\ensuremath{{{\ensuremath{\tau}}}_{q}}}}$ be the curve of this set with minimal distance to $\pi$ (i.e. with smallest index $q$). Since at least half of the input curves have to lie within a radius of $\frac{2}{n}\operatorname{cost}_{1}(P,\pi)$ from $\pi$ (two times the average distance of the input curves to $\pi$) and since the union of the sets from $T$ has size less than $n/2$ (with probability at least $1-\lambda$), this implies that $x_q \leq \frac{2}{n}\operatorname{cost}_{1}(P,\pi)$. Therefore, $$\begin{aligned} \operatorname{cost}_{1}(P,\tilde{\pi}) &= \operatorname{cost}_{1}(P\setminus P_T, \tilde{\pi}) + \operatorname{cost}_{1}(P_T, \tilde{\pi})\\ &\leq \operatorname{cost}_{1}(P\setminus P_T, \pi) + \sum_{{{\ensuremath{{{\ensuremath{\tau}}}_{}}}} \in P_T} \left( {\ensuremath{d_F{\!\left({{{\ensuremath{{{\ensuremath{\tau}}}_{}}}},\pi}\right)}}} + {\ensuremath{d_F{\!\left({\pi,{{\ensuremath{{{\ensuremath{\tau}}}_{q}}}}}\right)}}} + {\ensuremath{d_F{\!\left({{{\ensuremath{{{\ensuremath{\tau}}}_{q}}}},\tilde{\pi}}\right)}}} \right) \\ & \leq \operatorname{cost}_{1}(P\setminus P_T, \pi) + \operatorname{cost}_{1}(P_T, \pi) + 2|P_T|x_q\\ &\leq \operatorname{cost}_{1}(P,\pi) + \frac{{{\varepsilon}}n}{4} \cdot \frac{4\operatorname{cost}_{1}(P,\pi)}{n}\\ &= \left(1+{{{\varepsilon}}}\right)\operatorname{cost}_{1}(P,\pi) .\end{aligned}$$ Generating Candidate Solutions ------------------------------ Our next step is to define an algorithm that generates a set of candidate curves from the sample set. [\[alg:candidate:generator\]]{} For each ${{\ensuremath{{{\ensuremath{\tau}}}_{i}}}}$, let ${\ensuremath{\mathcal{V}}}_i$ be the vertex set of the signature of size at most $\ell+3$ ([Theorem \[theo:compute:signatures\]]{}) Compute the union $U$ of the intervals $r=\left[v-8\alpha,v+8\alpha\right]$ for $v \in {\ensuremath{\mathcal{V}}}=\bigcup_{i=1}^{s} {\ensuremath{\mathcal{V}}}_i$ Discretize $U$ with resolution $\beta$, thereby generating a set of vertices $\widehat{{\ensuremath{\mathcal{V}}}}$ Return all possible curves consisting of ${\ensuremath{\ell}}$ vertices from $\widehat{{\ensuremath{\mathcal{V}}}}$ We prove some properties of [Algorithm \[alg:candidate:generator\]]{} and of the candidate set generated by it. This proof serves as a basis for the proof of the sampling property in [Theorem \[theo:modified:sampling:prop\]]{}. [\[lemma:candidate:generator\]]{} Given a set of curves $S=\{{{\ensuremath{{{\ensuremath{\tau}}}_{1}}}},\dots,{{\ensuremath{{{\ensuremath{\tau}}}_{s}}}}\} $ and parameters $\alpha,\beta,{{\varepsilon}}> 0$, and ${\ensuremath{\ell}}\in {{\rm I\!\hspace{-0.025em} N}}$, with $\alpha \geq \min_{1\leq i\leq s} \frac{ {\ensuremath{d_F{\!\left({{{\ensuremath{{{\ensuremath{\tau}}}_{i}}}},c_s}\right)}}} }{{{\varepsilon}}}$, where $c_s$ denotes an optimal $(1,{\ensuremath{\ell}})$-median of $S$. There exist $\widehat{c} \in \Delta_{{\ensuremath{\ell}}}$ with $$\operatorname{cost}_{1}(S,\widehat{c}) \leq (1+{{\varepsilon}})\operatorname{opt}^{(1)}_{1,{\ensuremath{\ell}}}(S),$$ and [Algorithm \[alg:candidate:generator\]]{} computes a set of candidates $\Gamma^{{\ensuremath{\ell}}}_{\alpha,\beta}(S) \subseteq \Delta_\ell$ of size $\left({\frac{16\alpha s({\ensuremath{\ell}}+3)}{\beta}}\right)^{{\ensuremath{\ell}}}$ which contains an element $\tilde{c}$, such that $${\ensuremath{d_F{\!\left({\widehat{c},\tilde{c}}\right)}}}\leq \beta.$$ Let ${{\ensuremath{{{\ensuremath{\tau}}}_{1}}}},\dots,{{\ensuremath{{{\ensuremath{\tau}}}_{s}}}}$ denote the input curves in the increasing order of their distance denoted by $x_i={\ensuremath{d_F{\!\left({{{\ensuremath{{{\ensuremath{c}}}_{s}}}},{{\ensuremath{{{\ensuremath{\tau}}}_{i}}}}}\right)}}}$. For every ${{\ensuremath{{{\ensuremath{\tau}}}_{i}}}}$, consider its $x_i$-signature denoted by $\sigma_i$. By [Lemma \[lemma:nec:suff\]]{}, each vertex of ${{\ensuremath{{{\ensuremath{c}}}_{s}}}}$ lies within distance $4x_i$ to a vertex of some signature $\sigma_i$ otherwise we can omit it by [Theorem \[theo:remove:one\]]{}. Hence, there must be a solution where ${{\ensuremath{{{\ensuremath{c}}}_{s}}}}$ has its vertices in the union of the intervals. $$\bigcup_{i=1}^s \bigcup_{v \in {\ensuremath{\mathcal{V}}}(\sigma_i) } [v-4x_i,v+4x_i].$$ Since $x_i$ could be very large, we cannot cover this entire region with candidates. Instead, we consider the following union of intervals: $$R_S = \bigcup_{ x_i \leq \widehat{x}}\; \bigcup_{v \in {\ensuremath{\mathcal{V}}}(\sigma_i) } [v-4x_i,v+4x_i],$$ with $\widehat{x}=\frac{2x_1}{{{\varepsilon}}}$. Now, let $\widehat{{{\ensuremath{{{\ensuremath{c}}}_{}}}}}$ be the median obtained from ${{\ensuremath{{{\ensuremath{c}}}_{s}}}}$ by omitting all vertices which do not lie in $R_S$. [Lemma \[lemma:omit:far:away\]]{} implies $\operatorname{cost}_{1}(S,\widehat{{{\ensuremath{{{\ensuremath{c}}}_{}}}}}) \leq (1+{{\varepsilon}}) \operatorname{cost}_{1}(S,{{\ensuremath{{{\ensuremath{c}}}_{s}}}}).$ Clearly, the generated set contains a curve $\tilde{{{\ensuremath{{{\ensuremath{c}}}_{}}}}}$ which lies within [Fréchet]{}distance $\beta$ of $\widehat{{{\ensuremath{{{\ensuremath{c}}}_{}}}}}$. Indeed, by [Lemma \[lemma:canonical:signature\]]{}, the vertices of $\sigma_i$ are contained in the set of signature vertices computed by [Algorithm \[alg:candidate:generator\]]{}, since [Corollary \[cor:nec:suff\]]{} implies that $\ell \geq |{\ensuremath{\mathcal{V}}}{\!\left({{{\ensuremath{{{\ensuremath{c}}}_{s}}}}}\right)}| \geq |{\ensuremath{\mathcal{V}}}{\!\left({\sigma_i}\right)}|-2$. If a signature of size $\ell+3$ does not exists, then by the general position assumption, there must be a signature of size $\ell+2$. The algorithm sets $\widehat{x}= 2\alpha \geq 2 x_1/{{\varepsilon}}$. Therefore, the generated candidate set covers the region $R_S$ with resolution $\beta$. Before we prove the modified sampling property we prove the following two easy lemmas. [\[lemma:markov\]]{} Let $\lambda > 0$. Given a set of curves $P$, for each uniform sample multiset $S\subseteq P$ it holds that $$\begin{aligned} \Pr\left[\operatorname{cost}_1(S,c) \geq \frac{|S|}{\lambda n} \operatorname{opt}^{(1)}_{1,{\ensuremath{\ell}}}(P)\right] \leq \lambda,\end{aligned}$$ where $c$ is an optimal median of $P$. It holds that $$\operatorname{E}{\left[ {\operatorname{cost}_1(S,c)} \right]} = \frac{|S|}{n} \operatorname{opt}^{(1)}_{1,{\ensuremath{\ell}}}(P).$$ Since $\operatorname{cost}_1(S,c)$ is a nonnegative random variable we can apply Markov’s inequality and obtain $$\Pr\left[\operatorname{cost}_1(S,c) \geq \frac{\operatorname{E}{\left[ {\operatorname{cost}_1(S,c)} \right]}}{\lambda} \right] \leq \lambda,$$ which implies the claim. [\[lemma:diam:S\]]{} Let $\lambda > 0$. Given a set of curves $P$, for each uniform sample multiset $S\subseteq P$ of size at least ${\left\lceil {5\log(\frac{1}{\lambda})} \right\rceil}+1$ it holds that $$\Pr{\left[ { 12 \operatorname{opt}_{1,{\ensuremath{\ell}}}^{(1)}(S) \geq \min_{\tau \in P} {\ensuremath{d_F{\!\left({\tau,c}\right)}}} } \right]} \geq 1-\lambda,$$ where $c$ denotes an optimal $(1,{\ensuremath{\ell}})$-median of $P$. We analyze two cases. For the first case, assume that there exists a curve $q\in \Delta_{{\ensuremath{\ell}}}$, such that $${\left\lvert{\left\{ {\tau \in P: {\ensuremath{d_F{\!\left({q,\tau}\right)}}} \leq r} \right\}}\right\rvert} \geq \frac{5}{7} |P|,$$ where $r=\frac{{\ensuremath{d_F{\!\left({q,c}\right)}}}}{5}$. That is, a large fraction of $P$ lies within a small ball far away from the optimal center. We let $$Q= {\left\{ {\tau \in P: {\ensuremath{d_F{\!\left({q,\tau}\right)}}} < 2r} \right\}},$$ and we claim that $Q$ has size at most $\frac{6}{7}|P|$. Assume the opposite for the sake of contradiction. In this case, it follows by the triangle inequality that $$\begin{aligned} \operatorname{cost}_{1}(P,q) &\leq \operatorname{cost}_{1}(P,c) - |Q|r + |P\setminus Q| 5r \leq \operatorname{cost}_{1}(P,c) - \frac{1}{7} |P| r < \operatorname{cost}_1(P,c).\end{aligned}$$ This would imply that $c$ is not optimal. We analyze the event that at least one curve of $P$ lies within [Fréchet]{}distance $r$ of $q$ and at least one curve lies further than $2r$ from $q$. If this event happens, then again by the triangle inequality $$\operatorname{opt}^{(1)}_{1,{\ensuremath{\ell}}}(S)\geq \max_{\pi,\pi' \in S} {\ensuremath{d_F{\!\left({\pi,\pi'}\right)}}} \geq r \geq \frac{\min_{\tau \in P}{\ensuremath{d_F{\!\left({c,\tau}\right)}}}}{6}.$$ Clearly, it holds for the $i$th sample point $s_i$, that $$\Pr{\left[ { {\ensuremath{d_F{\!\left({s_i,q}\right)}}} \leq r } \right]} \geq \frac{5}{7}~~\text{and}~~ \Pr{\left[ { {\ensuremath{d_F{\!\left({s_i,q}\right)}}} \geq 2r } \right]} \geq \frac{1}{7}.$$ Using this, we can show that ${\left\lceil {5 \log(\frac{1}{\lambda})} \right\rceil}$ samples suffice to ensure that this event happens with probability of at least $(1-\lambda)$. Now, assume the second case that no such $q$ exists. Let $s_1$ be the first sample point and let $\widehat{s_1}$ be its minimum-error $\ell$-simplification ([Definition \[def:min:error:simp\]]{}). We need to prove the claim in the case that $$\label{assump1} 12 {\ensuremath{d_F{\!\left({s_1,\widehat{s_1}}\right)}}} < {\ensuremath{d_F{\!\left({s_1,c}\right)}}},$$ since $\operatorname{opt}^{(1)}_{1,{\ensuremath{\ell}}}(S)$ is lower-bounded by ${\ensuremath{d_F{\!\left({s_1,\widehat{s_1}}\right)}}}$. By the case analysis, it holds that $${\left\lvert{\left\{ {\tau \in P: {\ensuremath{d_F{\!\left({\widehat{s_1},\tau}\right)}}} \leq r} \right\}}\right\rvert} < \frac{5}{7} |P|,$$ for $r=\frac{{\ensuremath{d_F{\!\left({\widehat{s_1},c}\right)}}}}{5}$. Therefore, it holds for each of the remaining sample points $s_i$, for $1 < i \leq |S|$, that $$\Pr{\left[ { {\ensuremath{d_F{\!\left({\widehat{s_1},s_i}\right)}}} > r} \right]} \geq \frac{2}{7}.$$ In case this event happens, it holds by the triangle inequality $${\ensuremath{d_F{\!\left({s_1,s_i}\right)}}} \geq {\ensuremath{d_F{\!\left({s_i,\widehat{s_1}}\right)}}}-{\ensuremath{d_F{\!\left({\widehat{s_1},s_1}\right)}}} \geq r-{\ensuremath{d_F{\!\left({\widehat{s_1},s_1}\right)}}} \geq \frac{{\ensuremath{d_F{\!\left({c,s_1}\right)}}}-{\ensuremath{d_F{\!\left({s_1,\widehat{s_1}}\right)}}}}{5}-{\ensuremath{d_F{\!\left({\widehat{s_1},s_1}\right)}}} \geq \frac{{\ensuremath{d_F{\!\left({c,s_1}\right)}}}}{10}.$$ The analysis of this event is almost the same as in the first case. In this case, a total number of ${\left\lceil {5 \log(\frac{1}{\lambda})} \right\rceil} + 1$ samples suffices to ensure that this event happens with probability of at least $(1-\lambda)$. We are now ready to prove the modified sampling property. [[\[theo:modified:sampling:prop\]]{}]{} There exist integer constants $m_{{{\varepsilon}},\lambda,{\ensuremath{\ell}}}$ and $t_{{{\varepsilon}},\lambda,{\ensuremath{\ell}}}$ such that given a set of curves $P={\left\{ {{{\ensuremath{{{\ensuremath{\tau}}}_{1}}}},\dots,{{\ensuremath{{{\ensuremath{\tau}}}_{n}}}}} \right\}}$ for a uniform sample multiset $S\subseteq P$ of size $m_{{{\varepsilon}},\lambda,{\ensuremath{\ell}}}$ we can generate a candidate set $\Gamma(S) \subset \Delta_{{\ensuremath{\ell}}}$ of size $t_{{{\varepsilon}},\lambda,{\ensuremath{\ell}}}$ satisfying $$\Pr\left[ \exists q \in \Gamma(S): \operatorname{cost}_{1}(P,q) \leq (1+{{\varepsilon}})\operatorname{opt}^{(1)}_{1,{\ensuremath{\ell}}}(P) \right] \geq 1-\lambda.$$ Furthermore, we can compute $\Gamma(S)$ in time depending on ${\ensuremath{\ell}},\lambda$ and ${{\varepsilon}}$ only. Let $\lambda'=\frac{\lambda}{4}$ and ${{\varepsilon}}'=\frac{{{\varepsilon}}}{4}$. Let $c$ denote an optimal $(1,\ell)$-median of $P$ and let $c_s$ denote an optimal $(1,\ell)$-median of $S$. We use the algorithm described in [Section \[sec:cf:approx:uni\]]{} to compute a constant-factor approximation to $\operatorname{opt}^{(1)}_{1,{\ensuremath{\ell}}}(S) $ and obtain an interval $[\delta_S^{\min}, \delta_S^{\max}]$ which contains $\operatorname{opt}^{(1)}_{1,{\ensuremath{\ell}}}(S)$ and by [Theorem \[theo:cf:approx:median:main\]]{} it holds that $\delta_S^{\max}/\delta_S^{\min} \leq 65$. We apply [Algorithm \[alg:candidate:generator\]]{} to $S$ with parameters $$\alpha = \frac{6\delta_S^{\max}}{{{\varepsilon}}'} ~~ \text{ and } ~~ \beta = \frac{{{\varepsilon}}'\lambda' \delta_S^{min}}{|S|},$$ and obtain a set $\Gamma^{{\ensuremath{\ell}}}_{\alpha,\beta}$. With the help of [Lemma \[lemma:omit:low:prob\]]{}, we can now adapt the proof of [Lemma \[lemma:candidate:generator\]]{} to our probabilistic setting. Let ${{\ensuremath{{{\ensuremath{c}}}_{}}}}=z_1,\dots,z_{{\ensuremath{\ell}}}$ be an optimal $(1,\ell)$-median of $P$. Let ${{\ensuremath{{{\ensuremath{\tau}}}_{1}}}},\dots,{{\ensuremath{{{\ensuremath{\tau}}}_{n}}}}$ denote the input curves in the increasing order of their distance denoted by $x_i={\ensuremath{d_F{\!\left({{{\ensuremath{{{\ensuremath{c}}}_{}}}},{{\ensuremath{{{\ensuremath{\tau}}}_{i}}}}}\right)}}}$. For every ${{\ensuremath{{{\ensuremath{\tau}}}_{i}}}}$, consider its $x_i$-signature denoted by $\sigma_i$. By [Lemma \[lemma:nec:suff\]]{}, each vertex of ${{\ensuremath{{{\ensuremath{c}}}_{}}}}$ lies within distance $4x_i$ to a vertex of some signature $\sigma_i$, otherwise we can omit it by [Theorem \[theo:remove:one\]]{}. Hence, there must be a $(1,{\ensuremath{\ell}})$-median whose vertex set is contained in the union of the intervals $$\bigcup_{{{\ensuremath{{{\ensuremath{\tau}}}_{i}}}} \in P} \bigcup_{v \in {\ensuremath{\mathcal{V}}}(\sigma_i) } [v-4x_i,v+4x_i].$$ Let this solution be denoted ${{\ensuremath{{{\ensuremath{c}}}_{}}}}$. So, consider the following union of intervals: $$R_1 = \bigcup_{{{\ensuremath{{{\ensuremath{\tau}}}_{i}}}} \in S} \bigcup_{v \in {\ensuremath{\mathcal{V}}}(\sigma_i) } [v-4x_i,v+4x_i],$$ Let $\widehat{{{\ensuremath{{{\ensuremath{c}}}_{1}}}}}$ be the median obtained from ${{\ensuremath{{{\ensuremath{c}}}_{}}}}$ by omitting all vertices which do not lie in $R_1$. [Lemma \[lemma:omit:low:prob\]]{} implies $$\begin{aligned} \Pr\left[ \operatorname{cost}_{1}(P,\widehat{{{\ensuremath{{{\ensuremath{c}}}_{1}}}}}) \leq (1+{{\varepsilon}}') \operatorname{cost}_{1}(P,{{\ensuremath{{{\ensuremath{c}}}_{}}}})\right] \geq 1-\lambda',\end{aligned}$$ if we choose $|S| \geq {\left\lceil {\frac{8{\ensuremath{\ell}}}{{{\varepsilon}}} \left(\log\left(\frac{1}{\lambda'}\right)+\log({\ensuremath{\ell}})\right)} \right\rceil}$. So, now consider the following union of intervals: $$R_2 = \bigcup_{{{\ensuremath{{{\ensuremath{\tau}}}_{i}}}} \in P \atop x_i \leq \widehat{x}} \bigcup_{v \in {\ensuremath{\mathcal{V}}}(\sigma_i) } [v-4x_i,v+4x_i],$$ where $\widehat{x}=\frac{2 x_1}{{{\varepsilon}}'}$. Let $\widehat{{{\ensuremath{{{\ensuremath{c}}}_{2}}}}}$ be the median obtained from $\widehat{{{\ensuremath{{{\ensuremath{c}}}_{1}}}}}$ by omitting all vertices which do not lie in $R_2$. [Lemma \[lemma:diam:S\]]{} implies that if $|S| \geq {\left\lceil {5\log(\frac{1}{\lambda'})} \right\rceil}+1$, then it holds with a probability of at least $(1-\lambda')$ that $x_1 \leq 12 \operatorname{opt}^{{(1)}}_{1,{\ensuremath{\ell}}}(S).$ [Algorithm \[alg:candidate:generator\]]{} sets $\widehat{x} = 4\alpha$, therefore $$\widehat{x} = \frac{24 \delta_S^{\max}}{{{\varepsilon}}'} \geq \frac{24 \operatorname{opt}^{(1)}_{1,{\ensuremath{\ell}}}(S)}{{{\varepsilon}}'}\geq \frac{2x_1}{{{\varepsilon}}'}.$$ Thus, we can apply [Lemma \[lemma:omit:far:away\]]{} and obtain $$\begin{aligned} \operatorname{cost}_{1}(P,\widehat{{{\ensuremath{{{\ensuremath{c}}}_{2}}}}}) \leq (1+{{\varepsilon}}') \operatorname{cost}_{1}(P,\widehat{{{\ensuremath{{{\ensuremath{c}}}_{1}}}}}).\end{aligned}$$ Therefore, with probability $1-2\lambda'$, the generated set $\Gamma^{{\ensuremath{\ell}}}_{\alpha,\beta}$ contains a curve $\tilde{{{\ensuremath{{{\ensuremath{c}}}_{}}}}}$ which lies within [Fréchet]{}distance $\beta$ of $\widehat{{{\ensuremath{{{\ensuremath{c}}}_{2}}}}}$. [Lemma \[lemma:markov\]]{} implies that with probability at least $1-\lambda'$ it holds that $$\operatorname{opt}^{(1)}_{1,{\ensuremath{\ell}}}(S) \leq \operatorname{cost}_{1}(S,c) \leq \frac{|S|}{\lambda' n}\operatorname{opt}^{(1)}_{1,{\ensuremath{\ell}}}(P).$$ Thus, with the same probability it holds that $$\begin{aligned} \beta = \frac{{{\varepsilon}}'\lambda' \delta_S^{min}}{|S|} \leq \frac{{{\varepsilon}}'\lambda' \operatorname{opt}^{(1)}_{1,{\ensuremath{\ell}}}(S)}{|S|} \leq \frac{{{\varepsilon}}' \operatorname{opt}^{(1)}_{1,{\ensuremath{\ell}}}(P) }{n}.\end{aligned}$$ We conclude that with probability $1-3\lambda' > 1-\lambda$ (union bound) there exists a candidate in $\tilde{c} \in \Gamma^{{\ensuremath{\ell}}}_{\alpha,\beta}$ such that $$\begin{aligned} \operatorname{cost}_{1}(P,\tilde{c}) &\leq \operatorname{cost}_{1}(P,\widehat{c_2}) + \beta n \leq (1+{{\varepsilon}}')\operatorname{cost}_{1}(P,\widehat{c_1}) + \beta n\\ &\leq (1+{{\varepsilon}}')^2 \operatorname{cost}_{1}(P,c) + \beta n \leq ((1+{{\varepsilon}}')^2 + {{\varepsilon}}') \operatorname{opt}^{(1)}_{1,{\ensuremath{\ell}}}(P) \leq (1+{{\varepsilon}}) \operatorname{opt}^{(1)}_{1,{\ensuremath{\ell}}}(P). \end{aligned}$$ Furthermore, by [Lemma \[lemma:candidate:generator\]]{} the size of $\Gamma^{{\ensuremath{\ell}}}_{\alpha,\beta}$ is bounded as follows $$\begin{aligned} t_{{{\varepsilon}}',\lambda',{\ensuremath{\ell}}} \leq \left({\frac{16\alpha |S|({\ensuremath{\ell}}+3)}{\beta}}\right)^{{\ensuremath{\ell}}} = {\!\left({c_1 {{\varepsilon}}^{-4}\lambda^{-1} {\ensuremath{\ell}}^3 {\!\left({\log^2{\!\left({\frac{1}{\lambda}}\right)}+\log^2 {\ensuremath{\ell}}}\right)}}\right)}^{{\ensuremath{\ell}}},\end{aligned}$$ where $c_1$ is a sufficiently large constant. The following theorem follows from Ackermann [*et al.*]{} [@abs-cm-10] ([Theorem \[theo:modified:sampling:prop\]]{}). For this purpose, recall that the analysis of Ackermann [*et al.*]{}does not require the distance function to satisfy the triangle inequality. Therefore we can adopt the $(k,\ell)$-median formulation from [Section \[sec:problem\]]{} which uses the dissimilarity measure $D$ on the set $\Delta_{\ell} \cup P$. To achieve the running time we use Alt and Godau’s algorithm [@ag-cfdbt-95] for distance computations. [[\[theo:k:l:median:main\]]{}]{} Let ${{\varepsilon}}, k,\ell >0$ be constants. Given a set of curves $P={\left\{ {{{\ensuremath{{{\ensuremath{\tau}}}_{1}}}},\dots,{{\ensuremath{{{\ensuremath{\tau}}}_{n}}}}} \right\}} \subset \Delta_{m}$, there exists an algorithm that with constant probability returns a $(1+{{\varepsilon}})$-approximation to the $({\ensuremath{k}},{\ensuremath{\ell}})$-median problem for input instance $P$, and that has running time $O(n m \log m)$. Constant-factor approximation in various cases ============================================== [\[sec:cf:approx:uni\]]{} It is not difficult to compute a constant-factor approximation for our problem. We include the details for the sake of completeness. Our algorithm first simplifies the input curves before applying a known approximate clustering algorithm designed for general metric spaces. Note that an approximation scheme which first applies clustering and then simplification does not yield the same running time, since the distance computations are expensive. [\[alg:phase1\]]{} For each ${{\ensuremath{{{\ensuremath{\tau}}}_{i}}}}$ we compute an approximate minimum-error $\ell$-simplification ${{\ensuremath{\widehat{{{\ensuremath{\tau}}}_{i}}}}}$ ([Lemma \[lemma:apx:min:error:simp\]]{}) We apply a known approximation algorithm for clustering in general metric spaces on $\widehat{P}={{\ensuremath{\widehat{{{\ensuremath{\tau}}}_{1}}}}}, \dots ,{{\ensuremath{\widehat{{{\ensuremath{\tau}}}_{n}}}}}$ (i.e., Gonzales’ algorithm [@Gonzalez1985] for ${\ensuremath{k}}$-center and Chen’s algorithm [@c-kmc-09] for ${\ensuremath{k}}$-median) We return the resulting cluster centers $C={{\ensuremath{{{\ensuremath{c}}}_{1}}}},\dots,{{\ensuremath{{{\ensuremath{c}}}_{{\ensuremath{k}}}}}}$ with approximate cost $$\begin{aligned} D_{\infty}&=&\operatorname{cost}_{\infty}(C,\widehat{P})+ \max_{1\leq i\leq n} {\ensuremath{d_F{\!\left({{{\ensuremath{{{\ensuremath{\tau}}}_{i}}}},{{\ensuremath{\widehat{{{\ensuremath{\tau}}}_{i}}}}}}\right)}}} \\ D_1&=&\operatorname{cost}_{1}(C,\widehat{P})+ \sum_{1\leq i\leq n} {\ensuremath{d_F{\!\left({{{\ensuremath{{{\ensuremath{\tau}}}_{i}}}},{{\ensuremath{\widehat{{{\ensuremath{\tau}}}_{i}}}}}}\right)}}} \end{aligned}$$ for $({\ensuremath{k}},{\ensuremath{\ell}})$-center and $({\ensuremath{k}},{\ensuremath{\ell}})$-median, respectively [\[lemma:c:f:approx\]]{} The cost $D_{\infty}$ (resp., $D_{1}$) and solution $C$ computed by [Algorithm \[alg:phase1\]]{} constitute a $(\alpha+\beta+\alpha\beta)$-approximation to the $({\ensuremath{k}},{\ensuremath{\ell}})$-center problem (resp., the $({\ensuremath{k}},{\ensuremath{\ell}})$-median problem), where ${\ensuremath{\alpha}}$ is the approximation factor of the simplification step and ${\ensuremath{\beta}}$ is the approximation factor of the clustering step. We first discuss the case of $({\ensuremath{k}},{\ensuremath{\ell}})$-center. The $({\ensuremath{k}},{\ensuremath{\ell}})$-median will follow with a simple modification. First, we have that $$\begin{aligned} {\ensuremath{D}}_{\infty} &=& \operatorname{cost}_{\infty}(C,\widehat{P})+ \max_{1\leq i\leq n} {\ensuremath{d_F{\!\left({{{\ensuremath{{{\ensuremath{\tau}}}_{i}}}},{{\ensuremath{\widehat{{{\ensuremath{\tau}}}_{i}}}}}}\right)}}} \\ &=& \max_{1\leq i\leq n}\; \min_{{{\ensuremath{{{\ensuremath{c}}}_{}}}} \in C} {\ensuremath{d_F{\!\left({{{\ensuremath{\widehat{{{\ensuremath{\tau}}}_{i}}}}},{{\ensuremath{{{\ensuremath{c}}}_{}}}}}\right)}}} + \max_{1\leq i\leq n} {\ensuremath{d_F{\!\left({{{\ensuremath{{{\ensuremath{\tau}}}_{i}}}},{{\ensuremath{\widehat{{{\ensuremath{\tau}}}_{i}}}}}}\right)}}} \\ &\geq& \max_{1\leq i\leq n}\; \min_{{{\ensuremath{{{\ensuremath{c}}}_{}}}} \in C} {\!\left({{\ensuremath{d_F{\!\left({{{\ensuremath{\widehat{{{\ensuremath{\tau}}}_{i}}}}},{{\ensuremath{{{\ensuremath{c}}}_{}}}}}\right)}}} + {\ensuremath{d_F{\!\left({{{\ensuremath{{{\ensuremath{\tau}}}_{i}}}},{{\ensuremath{\widehat{{{\ensuremath{\tau}}}_{i}}}}}}\right)}}}}\right)} \\ &\geq& \max_{1\leq i\leq n}\; \min_{{{\ensuremath{{{\ensuremath{c}}}_{}}}} \in C} {\ensuremath{d_F{\!\left({{{\ensuremath{{{\ensuremath{\tau}}}_{i}}}},{{\ensuremath{{{\ensuremath{c}}}_{}}}}}\right)}}} \\ &\geq& \operatorname{cost}_{\infty}(C,P).\end{aligned}$$ Now, let ${\ensuremath{\delta^{*}}}$ be the optimal cost for a solution to the $({\ensuremath{k}},{\ensuremath{\ell}})$-center problem for $P=\{{{\ensuremath{{{\ensuremath{\tau}}}_{1}}}},\dots,{{\ensuremath{{{\ensuremath{\tau}}}_{n}}}}\}$. It holds that $$\begin{aligned} {\ensuremath{D}}_{\infty} &\leq& \operatorname{cost}_{\infty}(C,\widehat{P}) + {\ensuremath{\alpha}}{\ensuremath{\delta^{*}}}, \end{aligned}$$ since ${\ensuremath{\delta^{*}}}$ is lower bounded by the distance of any input time series to its optimal ${\ensuremath{\ell}}$-simplification and this is the minimal [Fréchet]{}distance the time series can have to any curve with at most ${\ensuremath{\ell}}$ vertices. Now, consider an optimal solution $C^{*}$ with cost ${\ensuremath{\delta^{*}}}$. We can relate it to ${\ensuremath{D}}_{\infty}$ as follows. In the following, let $p_i \in C^{*}$ be the center of this optimal solution which is closest to ${{\ensuremath{{{\ensuremath{\tau}}}_{i}}}}$. $$\begin{aligned} {\ensuremath{\delta^{*}}}&=& \max_{i=1,\ldots, n}\; {\ensuremath{d_F{\!\left({{{\ensuremath{{{\ensuremath{\tau}}}_{i}}}},p_i}\right)}}}\\ &\geq& \max_{i=1,\ldots, n}\; {\!\left({{\ensuremath{d_F{\!\left({{{\ensuremath{\widehat{{{\ensuremath{\tau}}}_{i}}}}},p_i}\right)}}} - {\ensuremath{d_F{\!\left({{{\ensuremath{{{\ensuremath{\tau}}}_{i}}}},{{\ensuremath{\widehat{{{\ensuremath{\tau}}}_{i}}}}}}\right)}}}}\right)}\\ &\geq& \max_{i=1,\ldots, n}\; {\ensuremath{d_F{\!\left({{{\ensuremath{\widehat{{{\ensuremath{\tau}}}_{i}}}}},p_i}\right)}}} - \max_{i=1,\ldots, n} {\ensuremath{d_F{\!\left({{{\ensuremath{{{\ensuremath{\tau}}}_{i}}}},{{\ensuremath{\widehat{{{\ensuremath{\tau}}}_{i}}}}}}\right)}}}\\ &\geq& \operatorname{cost}_{\infty}(C^{*},\widehat{P}) - \max_{i=1,\ldots, n} {\ensuremath{d_F{\!\left({{{\ensuremath{{{\ensuremath{\tau}}}_{i}}}},{{\ensuremath{\widehat{{{\ensuremath{\tau}}}_{i}}}}}}\right)}}}\\ &\geq& \frac{1}{{\ensuremath{\beta}}}{\!\left({ \operatorname{cost}_{\infty}(C,\widehat{P})}\right)} - {\ensuremath{\alpha}}{\ensuremath{\delta^{*}}}\\ &\geq& \frac{1}{{\ensuremath{\beta}}}{\!\left({ {\ensuremath{D}}_{\infty} - {\ensuremath{\alpha}}{\ensuremath{\delta^{*}}}}\right)} - {\ensuremath{\alpha}}{\ensuremath{\delta^{*}}}\end{aligned}$$ It follows that ${\ensuremath{D}}_{\infty} \leq ({\ensuremath{\alpha}}+{\ensuremath{\beta}}+{\ensuremath{\alpha}}{\ensuremath{\beta}}){\ensuremath{\delta^{*}}}$. [[\[theo:cf:approx:center:main\]]{}]{} Given a set of $n$ curves $P=\lbrace{{\ensuremath{{{\ensuremath{\tau}}}_{1}}}},\ldots , {{\ensuremath{{{\ensuremath{\tau}}}_{n}}}} \rbrace \subseteq \Delta_m$ and parameters ${\ensuremath{k}},{\ensuremath{\ell}}\in {{\rm I\!\hspace{-0.025em} N}}$, we can compute an $8$-approximation to $\operatorname{opt}^{(\infty)}_{{\ensuremath{k}},{\ensuremath{\ell}}}(P)$ and a witness solution in time $O({\ensuremath{k}}n m {\ensuremath{\ell}}\log(m {\ensuremath{\ell}}))$. The theorem follows by [Lemma \[lemma:c:f:approx\]]{} and by setting $\alpha=\beta=2$. We use [Lemma \[lemma:apx:min:error:simp\]]{} to obtain a $2$-approximate simplification for each curve. Then, we use Gonzales’ algorithm which yields a $2$-approximation for the simplifications. Each distance computation takes $O(m{\ensuremath{\ell}}\log(m{\ensuremath{\ell}}))$ time using Alt and Godau’s algorithm [@ag-cfdbt-95]. [[\[theo:cf:approx:median:main\]]{}]{} Given a set of $n$ curves $P=\lbrace {{\ensuremath{{{\ensuremath{\tau}}}_{1}}}},\ldots , {{\ensuremath{{{\ensuremath{\tau}}}_{n}}}} \rbrace \subseteq \Delta_m$ and parameters ${\ensuremath{k}},{\ensuremath{\ell}}\in {{\rm I\!\hspace{-0.025em} N}}$, we can compute a $65$-approximation to $\operatorname{opt}^{(1)}_{{\ensuremath{k}},{\ensuremath{\ell}}}(P)$ and a witness solution in time $O((nk+k^7\log^5 n)m\ell \log(m\ell))$. The theorem follows from [Lemma \[lemma:c:f:approx\]]{} and by setting $\alpha=2$ and $\beta=21$. We use [Lemma \[lemma:apx:min:error:simp\]]{} to obtain a $2$-approximate simplification for each curve. Then, we use the algorithm of Chen [@c-kmc-09] to solve the discrete version of the $k$-median problem on the simplifications. Each distance computation takes $O(m{\ensuremath{\ell}}\log(m{\ensuremath{\ell}}))$ time using Alt and Godau’s algorithm [@ag-cfdbt-95]. Chen’s algorithm yields an $10.5$-approximation for the discrete problem, where the centers are constrained to lie in $P$. Since the [Fréchet]{}distance satisfies the triangle inequality, this implies a $21$-approximation for our problem. Therefore, setting $\beta=21$ yields a correct bound. These results can be easily extended to a $({\ensuremath{k}},{\ensuremath{\ell}})$-means variant of the problem, as well as to multivariate time series, using known simplification algorithms, such as the algorithm by Abam [*et al.*]{} [@abh-sals-10]. On computing signatures ======================= [\[sec:computing:signatures\]]{} In this section we discuss how to compute signatures efficiently. Our signatures have a unique hierarchical structure as testified by [Lemma \[lemma:canonical:signature\]]{}. Together with the concept of vertex permutations ([Definition \[def:vtx:permutation\]]{}) this allows us to construct a data structure, which supports efficient queries for the signature of a given size ([Theorem \[theo:compute:signatures\]]{}). If the parameter $\delta$ is given, we can compute a signature in linear time using [Algorithm \[alg:low:pass\]]{}. Furthermore, we show that our signatures are approximate simplifications in [Lemma \[lemma:apx:min:error:simp\]]{}. [\[lemma:canonical:signature\]]{} Given a polygonal curve $\tau:[0,1]\rightarrow{{\rm I\!\hspace{-0.025em} R}}$ with vertices in general position, there exists a series of signatures $\sigma_1,\sigma_2,\dots,\sigma_k$ and corresponding parameters $0 = \delta_1 < \delta_2 < \dots < \delta_{k+1}$, such that $\sigma_i$ is a $\delta$-signature of $\tau$ for any $\delta \in [\delta_i,\delta_{i+1})$, the vertex set of $\sigma_{i+1}$ is a subset of the vertex set of $\sigma_{i}$, $\sigma_k$ is the linear interpolation of $\tau(0)$ and $\tau(1)$. We set $\sigma_1=\tau$ and obtain the desired series by a series of edge contractions. Clearly, $\sigma_1$ is a minimal $\delta$-signature for $\delta=0$. We now conceptually increase the signature parameter $\delta$ until a smaller signature is possible. In general, let $0=t_0 < t_1 < \dots < t_{\ell}=1$ be the series of parameters that defines $\sigma_i$. Let $$\label{min:edge} \delta_{i+1}=\min{\left\{ {|\tau(t_1)-\tau(t_2)|, |\tau(t_{\ell-1})-\tau(t_{\ell})|, \min_{2 \leq j \leq \ell-2} \frac{|\tau(t_j)-\tau(t_{j+1})|}{2}} \right\}}.$$ We contract the edge where the minimum is attained to obtain $\sigma_{i+1}$. By the general position assumption, this edge is unique. If the edge is connected to an endpoint, we only remove the interior vertex, otherwise we remove both endpoints of the edge. We now argue that the obtained curve $\sigma_{i+1}$ is a $\delta_{i+1}$-signature. Let $\overline{\tau(t_j)\tau(t_{j+1})}$ be the contracted edge and assume for now that $2\leq j \leq \ell-2 $. We prove the conditions in [Definition \[def:signature\]]{} in reverse order. Observe that $$\label{range} \tau(t_j),\tau(t_{j+1}) \in {\left\langle{\tau(t_{j-1}),\tau(t_{j+2})}\right\rangle},$$ since otherwise the contracted edge would not minimize the expression in (\[min:edge\]). By induction the *range condition* was satisfied for $\sigma_i$ and by the statement in (\[range\]) it cannot be violated by the edge contraction. The contracted edge was the shortest interior edge of $\sigma_i$ and by construction we have that $$\label{edge:length} |\tau(t_j)-\tau(t_{j+1})| = 2\delta_{i+1}$$ Therefore, the *minimum-edge-length* condition is also satisfied for $\sigma_{i+1}$. Since $\delta_i<\delta_{i+1}$, we have to prove the *direction-preserving* condition only for the newly established edge $\overline{\tau(t_{j-1})\tau(t_{j+2})}$ of $\tau_{i+1}$. For any $s,s'\in [t_j,t_{j+1}]$ it holds that $|\tau(s)-\tau(s')|\leq 2\delta_{i+1}$. Indeed, by induction, the range condition held true for the contracted edge and by Equation (\[edge:length\]) its length was $2\delta_{i+1}$. For any $s,s' \in [t_{j-1},t_j]$ the direction-preserving condition holds by induction, and the same holds for $s,s' \in [t_{j+1},t_{j+2}]$. The remaining case is $s,s' \in [t_{j-1},t_{j+2}]$ where the interval $[s,s']$ crosses the boundary of at least one of the edges. In this case, the direction-preserving condition holds by the range property of $\sigma_i$ and by Equation (\[edge:length\]). It remains to prove the *non-degeneracy* condition. Assume for the sake of contradiction that it would not hold, i.e., either that $\tau(t_{j-1})\in {\left\langle{\tau(t_{j-2}), \tau(t_{j+2})}\right\rangle}$, or that $\tau(t_{j+2})\in {\left\langle{\tau(t_{j-1}), \tau(t_{j+3})}\right\rangle}$. Since the two cases are symmetric, we only discuss the first one and the other case will follow by analogy. Then, (\[range\]) would imply that $\tau(t_{j-1}) \in {\left\langle{\tau(t_{j-2}),\tau(j)}\right\rangle}$, which contradicts the range property of $\sigma_i$. So far we proved the conditions of [Definition \[def:signature\]]{} in the case that an interior edge is being contracted. Now, assume that $j=1$ and again let the contracted edge be $\overline{\tau(t_j)\tau(t_{j+1})}$ (the case $j={\ensuremath{\ell}}-1$ is analogous). Again, we prove the conditions in reverse order. By induction, the range condition is satisfied for the first two edges of $\sigma_i$, as well as the non-degeneracy condition. Since it holds for the length of the second edge that $|\tau(t_2)-\tau(t_3)| > 2\delta_{i+1}$, it must be that ${\left\langle{\tau(t_1)-\delta_{i+1}, \tau(t_1)+\delta_{i+1}}\right\rangle} \cup {\left\langle{\tau(t_1),\tau(t_3)}\right\rangle}$ spans the range of values on $\tau[t_1,t_3]$. Thus, the range condition is implied for $\sigma_{i+1}$. Similarly, $|\tau(t_2)-\tau(t_3)| > 2\delta_{i+1}$ and $|\tau(t_1)-\tau(t_2)|=\delta_{i+1}$ implies the minimum-edge-length condition, i.e. that $|\tau(t_1)-\tau(t_3)|>\delta$. The arguments for the direction-preserving condition are the same as above for $j>1$. The non-degeneracy condition on the vertex at $t_2$ is not affected by the edge-contraction, since $\tau(t_2)$ stays a minimum (resp. maximum) in $\sigma_{i+1}$ if it was a minimum (resp. maximum) in $\sigma_{i}$. Otherwise, the contracted edge would not minimize the expression in (\[min:edge\]). By construction it is clear that the vertex set of the $\sigma_{i+1}$ is a subset of the vertex set of $\sigma_{i}$ for each $i$, as well as that $\sigma_k$ is the linear interpolation of $\tau(0)$ and $\tau(1)$. This completes the proof of the lemma. [\[def:vtx:permutation\]]{} Given a curve ${{\ensuremath{{{\ensuremath{\tau}}}_{}}}}:[0,1]\rightarrow{{\rm I\!\hspace{-0.025em} R}}$ with $m$ vertices in general position, consider its canonical signatures $\sigma_1,\dots,\sigma_k$ of [Lemma \[lemma:canonical:signature\]]{}. We call a permutation of the vertices of ${{\ensuremath{{{\ensuremath{\tau}}}_{}}}}$ canonical if for any two vertices $x,y$ of $\tau$ it holds that if $x \notin {\ensuremath{\mathcal{V}}}{\!\left({\sigma_{i}}\right)}$ (the vertex set of $\sigma_{i}$) and $y\in {\ensuremath{\mathcal{V}}}{\!\left({\sigma_{i}}\right)}$, for some $i$, then $x$ appears before $y$ in the permutation. Furthermore, we require that the permutation contains a token separator for every $\sigma_i$, for $1\leq i \leq k$, such that $\sigma_i$ consists of all vertices appearing after the separator. [\[lemma:compute:vtx:permutation\]]{} Given a curve ${{\ensuremath{{{\ensuremath{\tau}}}_{}}}}:[0,1]\rightarrow{{\rm I\!\hspace{-0.025em} R}}$ with $m$ vertices in general position, we can compute a canonical vertex permutation ([Definition \[def:vtx:permutation\]]{}) in $O(m \log m)$ time and $O(m)$ space. Let $w_1,\ldots,w_m$ be the vertices of the curve $\tau$. The idea is to simulate the series of edge contractions done in the proof of [Lemma \[lemma:canonical:signature\]]{}. We build a min-heap from the vertices $w_2,\dots,w_{m-1}$ using certain weights, which will be defined shortly.[^10] We then iteratively extract the (one or more) vertices with the current minimum weight from the heap and update the weights of their neighbors along the current signature curves. The extracted vertices are recorded in a list $L$ in the order of their extraction and will form the canonical vertex permutation in the end. Before every iteration we append a token separator to $L$. In this way, all vertices extracted during one iteration are placed between two token separators in $L$. After the last iteration we again append a token separator and at last the two vertices $w_1$ and $w_m$. More precisely, let $v_1,\dots,v_k$ denote the vertices contained in the heap in the beginning of one particular iteration, sorted in the order in which they appear along the curve $\sigma$. We call the curve $$\sigma = w_1,v_1,\dots,v_k,w_m,$$ the *current signature*. For every vertex we keep a pointer to the heap element which represents its current predecessor and successor along the current signature. We also keep these pointers to the virtual elements $w_1$ and $w_m$, which are not included in the heap. We define the weight $W(\cdot)$ for every vertex $v_i$ in the heap as follows: if $i=1$, then $W(v_i)=\min{\!\left({|w_1-v_1|, \frac{|v_1-v_{2}|}{2}}\right)}$, if $i=k$, then $W(v_i)=\min{\!\left({|w_{m}-v_{k}|, \frac{|v_{k}-v_{k-1}|}{2}}\right)}$, otherwise $W(v_i)=\min{\!\left({\frac{|v_i-v_{i-1}|}{2}, \frac{|v_i-v_{i+1}|}{2}}\right)}$. Initially, the current signature equals $\tau$ and initializing these weights takes $O(n)$ time in total. Following the argument in [Lemma \[lemma:canonical:signature\]]{}, we need to contract the edge(s) with minimum length (where exceptions hold for the first and last edge). This is captured by the choice of the weights above. Assume for simplicity that the minimum is attained for exactly one edge[^11] with endpoints $v_i$ and $v_{i+1}$ for some $i$. In this case, $v_{i}$ and $v_{i+1}$ are the next two elements to be extracted from the heap and their weight must be equal to $\frac{|v_i-v_{i+1}|}{2}$. Using the pointers to $v_{i-1}$ (unless $i=1$) and $v_{i+2}$ (unless $i=k$), we now update the weights of these neighbors and update the pointers such that $v_{i-1}$ (resp., $w_1$) becomes predecessor of $v_{i+2}$ (resp., $w_m$). Computing the new weight of one of these neighboring vertices can be done in $O(1)$ time, updating the weights in the heap takes $O(\log n)$ time per vertex. We can charge every update to the extraction of a neighboring vertex. Since every vertex is extracted at most once, we have $O(n)$ weight updates in total. [\[lemma:lb:vtx:permutation\]]{} Given a curve $\tau:[0,1]\rightarrow{{\rm I\!\hspace{-0.025em} R}}$ with $m$ vertices in general position, the problem of computing a canonical vertex permutation ([Definition \[def:vtx:permutation\]]{}) has time-complexity $\Theta(m\log m)$. By [Lemma \[lemma:compute:vtx:permutation\]]{}, we can compute this canonical vertex permutation in time $O(m \log m)$. We show the lower bound via a reduction from the problem of sorting a list of $M=\frac{m-2}{2}$ natural numbers. Let $a_1,\dots,a_{M}$ be the elements of the list in the order in which they appear in the list. We can determine the maximal element $a_{\max}$ in $O(M)$ time. We now construct a curve ${{\ensuremath{{{\ensuremath{\tau}}}_{}}}}$ as follows: $${{\ensuremath{{{\ensuremath{\tau}}}_{}}}} =~ 0,~ x_1,~ x_1-a_1,~ \dots,~ x_i,~ x_i-a_i,~ \dots, x_M, x_M-a_M, x_{M+1},$$ where $x_i=2ia_{\max}$. The constructed curve contains an edge of length $a_i$ for every $a_i$ of our sorting instance. We call these edges *variable edges*. The remaining edges of the ${{\ensuremath{{{\ensuremath{\tau}}}_{}}}}$ are called *connector edges*. All connector edges are longer than $a_{\max}$. A canonical vertex permutation of ${{\ensuremath{{{\ensuremath{\tau}}}_{}}}}$ would provide us the with variable edges in ascending order of their length. The following lemma testifies that we can query the canonical vertex permutation for a signature of a given size $\ell$. (Note that a canonical signature of size exactly $\ell$ may not exist.) [\[lemma:extract:signature\]]{} Given a canonical vertex permutation ([Definition \[def:vtx:permutation\]]{}) of a curve ${{\ensuremath{{{\ensuremath{\tau}}}_{}}}}$, we can in $O(\ell \log \ell)$ time extract the canonical signature of ${{\ensuremath{{{\ensuremath{\tau}}}_{}}}}$ of maximal size $\ell'$ with $\ell'\leq \ell$. Let $L'$ denote the suffix of the canonical vertex permutation which contains the last $\ell$ vertices. If there is no token separator at the starting position of $L'$, then we remove the maximal prefix of $L'$ which contains not token separator. In this way, we obtain the vertices of the canonical signature $\sigma$ of maximal size $\ell'$ with $\ell'\leq \ell$. We now sort the vertices in the order of their appearance along $\sigma$ and return the resulting curve. The following theorem follows from [Lemma \[lemma:compute:vtx:permutation\]]{} and [Lemma \[lemma:extract:signature\]]{}. Furthermore, [Lemma \[lemma:lb:vtx:permutation\]]{} testifies that the preprocessing time is asymptotically tight. [[\[theo:compute:signatures\]]{}]{} Given a curve ${{\ensuremath{{{\ensuremath{\tau}}}_{}}}}:[0,1]\rightarrow {{\rm I\!\hspace{-0.025em} R}}$ with $m$ vertices in general position, we can construct a data structure in time $O(m \log m)$ and space $O(m)$, such that given a parameter $\ell$ we can extract in time $O(\ell \log \ell)$ a canonical signature of maximal size $\ell'$ with $\ell'\leq \ell$. [\[alg:low:pass\]]{} $j=1$; $a=1$; $s_1=0$ ()[ $\tau(t_j) \notin {\ensuremath{[\tau(0)]_{\delta}}}$ or $j \geq m$ ]{}[$j=j+1$]{} $b=j$ ()[$i=j+1$ to $m$]{}[ ]{} $a=a+1$; $s_a=1$; [\[lemma:low:pass:correct\]]{} Given a curve ${{\ensuremath{{{\ensuremath{\tau}}}_{}}}}:[0,1]\rightarrow{{\rm I\!\hspace{-0.025em} R}}$ with $m$ vertices in general position, and given a parameter $\delta>0$, [Algorithm \[alg:low:pass\]]{} computes a $\delta$-signature $\sigma:[0,1]\rightarrow{{\rm I\!\hspace{-0.025em} R}}$ of ${{\ensuremath{{{\ensuremath{\tau}}}_{}}}}$ with ${\ensuremath{\ell}}$ vertices in $O(m)$ time and space. We prove that [Algorithm \[alg:low:pass\]]{} produces the values $s_1<\ldots < s_{\ensuremath{\ell}}$ that define a proper $\delta$-signature $\sigma=\tau(s_1),\tau(s_2),\dots,\tau(s_{{\ensuremath{\ell}}})$ of $\tau$ according to [Definition \[def:signature\]]{}. The algorithm operates in three phases: (1) lines 2-4, (2) lines 5-11, and (3) lines 12-14. In the first phase the algorithm finds the first vertex $\tau(t_j)$ of $\tau$ which lies outside the interval ${\ensuremath{[\tau(0)]_{\delta}}}$ and assigns its index to the variable $b$. In the trivial case, $\tau$ is entirely contained in the interval ${\ensuremath{[\tau(0)]_{\delta}}}$. In this case, the second phase is not executed and the condition in line 12 evaluates to false. The algorithm returns the correct signature, which has two vertices, $s_1=0$ and $s_2=1$. Otherwise, $\tau$ must leave the interval ${\ensuremath{[\tau(0)]_{\delta}}}$. We claim that the following invariants hold at the end of each iteration of the for-loop in phase 2 (lines 5-11): $\tau(s_1),\dots,\tau(s_a)$ is a correct prefix of the $\delta$-signature, for any $x\in [s_a,t_i]$ it holds: if $a>1$ then $\tau(x) \in {\left\langle{\tau(s_a),\tau(t_b)}\right\rangle}$ if $a=1$ then $\tau(x) \in [\tau(0)-\delta,\tau(t_b)]$ when $\tau(t_b)>\tau(0)$\ (resp., $\tau(x) \in [\tau(t_b),\tau(0)+\delta]$ when $\tau(t_b)<\tau(0)$). if $a>1$, then $|\tau(s_a)-\tau(t_b)|>2\delta$, if $a=1$ then $|\tau(0)-\tau(t_b)|>\delta$, if $t_i>t_b$, then for any $x \in [t_b, t_i]:$ $|\tau(t_b)-\tau(x)|\leq 2 \delta$, the direction-preserving property holds for the subcurve $\tau[s_a,t_b]$, We prove the invariants by induction on $i$. The base case happens after execution of line 4, before the first iteration of the for-loop. For ease of notation, we define $i=b$ for this case. Invariants (I1), (I3) and (I4) hold by construction. The other two invariants follow immediately from the observation that $\tau(t_b)$ is the first point outside the interval ${\ensuremath{[\tau(0)]_{\delta}}}$. Now we prove the induction step. One may have the following intuition. During the execution of the for-loop in lines 5-11, we implicitely maintain a general direction in which the curve $\tau$ is moving. This direction is *upwards* if $\tau(s_{a}) < \tau(s_b)$ and *downwards* otherwise. Furthermore, we maintain that $\tau(t_b)$ is the furthest point from $\tau(s_a)$ on the current signature edge (starting at $\tau(s_a)$) in the current general direction. Note that a new vertex is appended to the signature prefix only when $\tau$ has already moved in the opposite direction by a distance greater than $2\delta$. Only then, we say that the current general direction of the curve has changed. Consider an arbitrary iteration $i$ of the for-loop. There are three cases, line 7 is executed and $i$ becomes the new $b$\ (this happens if $\tau$ is moving in the current general direction beyond $\tau(s_b)$), lines 10 and 11 are executed and a new signature vertex is appended to the signature prefix\ (this happens if $\tau$ has changed its general direction), no assignments are being made\ (this happens if $\tau$ locally changes direction, but the current general direction does not change). For each invariant we consider each of the three cases above. (I1): If the signature prefix was not changed in the previous iteration (cases (i) and (iii)), then (I1) simply holds by induction. Otherwise, we argue that the new signature prefix is correct. By induction, $\tau(s_1),\dots,\tau(s_{a-1})$ is a correct signature prefix. The conditions of [Definition \[def:signature\]]{} for $\tau(s_1),\dots,\tau(s_{a})$ follow by the induction hypotheses (I2),(I3), and (I5) in the iteration step $i'<i$, in which the last value of $b$ was assigned. In particular, (I2) implies the range condition and the non-degeneracy, (I3) implies the minimum-edge-length condition, and (I5) implies the direction-preserving condition. (I2): Assume $a=1$ and $\tau(t_b)>\tau(0)$. Since $a=1$, we cannot be in case (ii). Furthermore, once we enter the for-loop, the current general direction is fixed until $a$ is incremented for the first time. Therefore, by (I2) we have for $x\in\left[ s_1, t_{i-1}\right]$ that $\tau(x)\in\left[ \tau(0)-\delta, \tau(t_{b'})\right]$, where $b'$ holds the value of $b$ before we entered the for-loop in the current iteration. Now, in case (i) the claim follows immediately. In case (iii) it follows from the (false) condition in line 9, that $\tau(t_i) > \tau(t_b)-2\delta \geq \tau(0)-\delta$, and by the (false) condition in line 6, that $\tau(t_i)<\tau(t_b)$. The case $a=1$ and $\tau(t_b)<\tau(0)$ is analogous. It remains to prove the invariant for $a>1$. Assume case (ii) and assume that the general direction changed from upwards to downwards (the opposite case is analogous). Let $a'$ and $b'$ be the values of $a$ and $b$ before the new assignment in lines 10 and 11. By (I2), we have $\tau(t_{b'}) \geq \tau(x)$ for any $x \in [t_{b'},t_{i-1}]$. By (I4), we have $\tau(t_{b'})-2\delta \leq \tau(x)$ for any $x \in [t_{b'},t_{i-1}]$. By the (true) condition in line 9, we have $\tau(t_i)<\tau(t_{b'})-2\delta$. Therefore, for any $x \in [t_{b'}, t_i]$, we have $\tau(x) \in [\tau(t_i),\tau(t_{b'})]$, which implies (I2) after the assignment in lines 10 and 11. Now, in case (i) and case (ii), we have by (I2) for $x\in[s_a,t_{i-1}]$ that $\tau(x)\in {\left\langle{\tau(s_a),\tau(t_b')}\right\rangle}$. The correctness in case (i) follows immediately. In case (iii), assume $\tau(t_b) > \tau(s_a)$ (the opposite case is analogous). It follows from the (false) condition in line 9 and by (I3), that $\tau(t_i) > \tau(t_b)-2\delta \geq \tau(s_a)$, and by the (false) condition in line 6, that $\tau(t_i)<\tau(t_b)$. (I3) holds in case $a=1$, since the for-loop was started after the curve $\tau$ left the interval ${\ensuremath{[\tau(0)]_{\delta}}}$ for the first time and by (I2) $\tau(t_b)$ is furthest point from $\tau(0)$. In case $a>1$, (I3) also holds, since we append the parameter $t_b$ to the signature prefix and re-initialize $b$ with $i$ only after the curve has moved by a distance of at least $2\delta$ (line 9) from $\tau(t_b)$. The distance is further maintained by (I2). (I4) is clearly satisfied in case (i) and (ii), since $b=i$ is assigned. In case (iii) it holds by the (false) condition in line 9 that $|\tau(t_b)-\tau(t_i)|\leq 2\delta$ and for the remaining curve $\tau\left[ t_b, t_{i-1}\right]$ it follows by induction. (I5) holds since we assign a new signature vertex with parameter $s_a=t_b$ as soon as the curve moves by more than $2\delta$ in the opposite direction (case (ii)). In phase 3, there are two cases. If the range condition is satisfied for the last signature edge from $\tau(s_a)$ to $\tau(1)$, the algorithm only appends the last vertex $\tau(1)$ of the curve $\tau$ to the signature $\sigma$. Otherwise, the algorithm appends $\tau(t_b)$ and $\tau(1)$ to the signature. In both cases, the conditions in [Definition \[def:signature\]]{} are satisfied also for the last part. If we use a linked-list for the signature, the running time and space requirements of the algorithm are linear in $m$, since the execution of one iteration of the for-loop takes constant time and there are at most $m$ such iterations. From [Lemma \[lemma:low:pass:correct\]]{} we obtain the following theorem. [[\[theo:computing:delta:signature\]]{}]{} Given a curve $\tau:[0,1]\rightarrow{{\rm I\!\hspace{-0.025em} R}}$ with $m$ vertices in general position, and given a parameter $\delta>0$, we can compute a $\delta$-signature of ${{\ensuremath{{{\ensuremath{\tau}}}_{}}}}$ in $O(m)$ time and space. [\[lemma:apx:min:error:simp\]]{} Given a curve $\tau:[0,1]\rightarrow{{\rm I\!\hspace{-0.025em} R}}$ with $m$ vertices in general position, and given a parameter $\ell \in {{\rm I\!\hspace{-0.025em} N}}$, we can compute in $O(m \log m)$ time a curve $\pi:[0,1]\rightarrow{{\rm I\!\hspace{-0.025em} R}}$ of at most $\ell$ vertices, such that ${\ensuremath{d_F{\!\left({{{\ensuremath{{{\ensuremath{\tau}}}_{}}}},\pi}\right)}}} \leq 2 {\ensuremath{d_F{\!\left({{{\ensuremath{{{\ensuremath{\tau}}}_{}}}},\pi^*}\right)}}}$, for $\pi^*$ being a minimum-error $\ell$-simplification of ${{\ensuremath{{{\ensuremath{\tau}}}_{}}}}$ ([Definition \[def:min:error:simp\]]{}). Let $\sigma_1,\dots,\sigma_k$ be the signatures of $\tau$ with corresponding parameters $\delta_1,\dots,\delta_k$ as defined in [Lemma \[lemma:canonical:signature\]]{}. [Lemma \[lemma:fd:signature\]]{} implies that ${\ensuremath{d_F{\!\left({\sigma_i,\tau}\right)}}} \leq \delta_i$. Consider the signature $\sigma_i$ with the maximal number of $\ell'\leq \ell$ vertices. We claim that $$\frac{\delta_i}{2}\leq {\ensuremath{d_F{\!\left({\pi^{*},\tau}\right)}}} \leq \delta_i .$$ The second inequality follows from the definition of $\pi^{*}$ and the fact that $\sigma_i$ consists of at most $\ell$ vertices. To see the first inequality, consider the signature $\sigma_{i-1}=w_1,\dots,w_{h}$ with $h>\ell$. By [Lemma \[lemma:canonical:signature\]]{}, for any ${{\varepsilon}}>0$, the signature $\sigma_{i-1}$ is a $\delta$-signature of $\tau$ for $\delta=\delta_i-{{\varepsilon}}$. By [Definition \[def:signature\]]{}, it holds for $$r_j = \left[w_j-\frac{\delta}{2}, w_j+\frac{\delta}{2}\right],$$ that for any $1\leq j \leq h-1: r_j \cap r_{j+1} = \emptyset.$ By the arguments in the proof of [Lemma \[lemma:nec:suff\]]{}, it follows that any curve $\pi$ with ${\ensuremath{d_F{\!\left({\pi,\tau}\right)}}}\leq \frac{\delta}{2}$ needs to consist of at least $h>\ell$ vertices. Therefore, the first inequality follows. We can compute the signature $\sigma_i$ in $O(m \log m)$ time using [Theorem \[theo:compute:signatures\]]{}. Hardness of clustering under the [Fréchet]{}distance ==================================================== [\[sec:nphard\]]{} A metric embedding is a function between two metric spaces which preserves the distances between the elements of the metric space. The embedding is called isometric if distances are preserved exactly. It is known that one can isometrically embed any bounded subset of a $d$-dimensional vector space equipped with the $\ell_{\infty}$-norm into $\Delta_{3d}$ [@IndMat04]. This immediately implies [**NP**]{}-hardness for $\ell \geq 6 $ knowing that the clustering problems we consider are [**NP**]{}-hard under the $\ell_{\infty}$ distance for $d\geq 2$. In this section we prove that the [**NP**]{}-hardness holds from $\ell=2$. This is achieved by preserving $\ell=d$ in the metric embedding. We begin by establishing the basic facts about clustering under the $\ell_{\infty}$-norm. The $k$-center problem under $\ell_{\infty}$ is [**NP**]{}-hard for $d \geq 2$ as shown by Feder and Greene [@feder1988optimal].[^12] Even approximating the optimal cost within a factor smaller than $2$ was shown to be [**NP**]{}-hard by the same authors. The $k$-median problem under $\ell_1$ was proven to be [**NP**]{}-hard for $d \geq 2$ by Megiddo and Supowit [@megiddo1984geo]. The following well-known observation implies that the $k$-median problem is also [**NP**]{}-hard under $\ell_{\infty}$ for $d=2$ (and therefore also for $d \geq 2$). For any two points $w$ and $v$ in $\mathbb{R}^2$ it holds that $\| w-v \|_{\infty} = {\| T(w)-T(v) \|_{1}},$ where $T$ is a rotation by $\frac{\pi}{4}$ followed by a uniform scaling with $\frac{1}{\sqrt{2}}$. We now describe the metric embedding used in the **NP**-hardness reduction. [\[lemma:embedding\]]{} Any metric space $(S,\ell_{\infty})$, where $S \subset {{\rm I\!\hspace{-0.025em} R}}^d$ is a bounded set, can be embedded isometrically into $\Delta_{d}$. Furthermore, if $S$ is discrete, the embedding and its inverse can be computed in time linear in $|S|$ and $d$. In the following, we view a list of reals $w={\left\langle{v_1,\dots,v_d}\right\rangle}$, $v_i \in {{\rm I\!\hspace{-0.025em} R}}$, from two different perspectives. We either (i) interpret $w$ as an element of ${{\rm I\!\hspace{-0.025em} R}}^d$, or (ii) interpret $w$ as a curve of $\Delta_d$. The interpretation we take should be clear from the context. Let $\delta= \max_{w \in S} \|w\|_{\infty}$, that is, the maximal norm of an element of $S$. Since $S$ is bounded, $\delta$ is well-defined. Note that $\delta$ also bounds the maximal coordinate value of an element of $S$ and likewise $-\delta$ bounds the minimal coordinate of an element of $S$. We define the translation vector $$T= {\!\left({6\delta,-6\delta,6\delta,-6\delta, \dots }\right)}.$$ Thus, $T$ translates every even coordinate by $6\delta$ and every odd coordinate by $-6\delta$. Let $w,x \in S$ be two elements of the metric space. We argue that $${\ensuremath{d_F{\!\left({T(w), T(x)}\right)}}} = \|w-x\|_{\infty}.$$ Note that by the triangle inequality $$\| w-x\|_{\infty} \leq \|w\|_{\infty} + \|x\|_{\infty} \leq 2\delta.$$ By [Observation \[obs:segments\]]{}, the [Fréchet]{}distance between $T(w)$ and $T(x)$ is at most $\| w-x\|_{\infty}$, since we can associate the $i$th coordinate of $w$ with the $i$th coordinate of $x$ to construct an eligible matching $f$. We claim that the matching $f$ is in fact optimal. Assume for the sake of contradiction that there exists a matching $g$ which is “better” than $f$, that is, $g$ matches each point of $T(w)$ to a point on $T(x)$ within a distance strictly smaller than $\| w-x\|_{\infty}$. It must be that $f$ and $g$ are structurally different from each other, in the sense that their corresponding paths in the free space diagram do not visit the same cells. This follows from our construction of $T$ which ensures that the path corresponding to $f$ is optimal among all paths which visit the same cells. So consider an arbitrary prefix curve $\widehat{w}$ where $f$ and $g$ structurally differ, that is, the image of $\widehat{w}$ under $g$ contains either fewer or more vertices than the image under $f$. Assume fewer (otherwise let $\widehat{w}$ be the corresponding suffix curve of $T(w)$). By our construction of $f$, the image of $\widehat{w}$ under $f$ has the same number of vertices as $\widehat{w}$. Let $\widehat{x}$ denote the image of $\widehat{w}$ under $g$. By our construction of $T$, it holds that the difference between any two consecutive coordinate values of $T(w)$ is at least $8\delta$. Therefore, the $(2\delta)$-signature of $\widehat{w}$ is equal to $\widehat{w}$. It follows from [Lemma \[lemma:nec:suff\]]{} that $\widehat{x}$ needs to have at least as many vertices as $\widehat{w}$. However, by our choice of $\widehat{w}$, the subcurve $\widehat{x}$ has fewer vertices than $\widehat{w}$. A contradiction. Thus $f$ is optimal and ${\ensuremath{d_F{\!\left({T(w), T(x)}\right)}}} = \| w-x\|_{\infty}$. The **NP**-hardness reduction takes an instance of the $k$-center (resp., $k$-median) problem under $\ell_{\infty}$ in ${{\rm I\!\hspace{-0.025em} R}}^d$ and embeds it into $\Delta_d$ using [Lemma \[lemma:embedding\]]{}. If we could solve the $(k,d)$-center (resp., $(k,d)$-median) problem described in [Section \[sec:problem\]]{}, then by [Lemma \[lemma:embedding\]]{}, we could apply the inverse embedding function to the solution to obtain a solution for the original problem instance. The same holds for approximate solutions. Note that the embedding given in [Lemma \[lemma:embedding\]]{} works for any point in the convex hull of $S$, therefore also for the centers (resp., medians) that form the solution. We obtain the following theorems. [[\[theo:center:nphard\]]{}]{} The $({\ensuremath{k}},{\ensuremath{\ell}})$-center problem (where ${\ensuremath{k}}$ is part of the input) is [**NP**]{}-hard for ${\ensuremath{\ell}}\geq 2$. Furthermore, the problem is [**NP**]{}-hard to approximate within a factor of $2$. [[\[theo:median:nphard\]]{}]{} The $({\ensuremath{k}},{\ensuremath{\ell}})$-median problem (where ${\ensuremath{k}}$ is part of the input) is [**NP**]{}-hard for ${\ensuremath{\ell}}\geq 2$. Doubling dimension of the metric space {#section:doublingdimension} ====================================== In this section we show that the unconstrained metric space of univariate time series under the [Fréchet]{}distance has unbounded doubling dimension ([Lemma \[lemma:doub:unlimit\]]{}). Even if we restrict the complexity of the curves to $\ell\geq 4$, the doubling dimension is unbounded ([Lemma \[lemma:doub:limit\]]{}). For lower complexities, one can easily show that the doubling dimension is bounded. Note that for $\ell=2$ the metric space $(\Delta_\ell, d_F)$ equals the metric space $({{\rm I\!\hspace{-0.025em} R}}^2,\ell_{\infty})$. Note that the infinity of the doubling dimension is simply caused by the fact that the metric space is incomplete. However, it remains that standard techniques for doubling spaces cannot be applied. In any metric space $(X,D)$ a ball of center $p\in X$ and radius $r\in {{\rm I\!\hspace{-0.025em} R}}$ is defined as the set $\{q\in X : D(p,q) \leq r\}$. The doubling dimension of a metric space is the smallest positive integer $d$ such that every ball of the metric space can be covered by $2^d$ balls of half the radius. \[lemma:doub:unlimit\] The doubling dimension of the metric space $(\Delta, d_F)$ is unbounded. Assume for the sake of contradiction that there exists a positive integer $d$ which equals the doubling dimension of the metric space $(\Delta, d_F)$. We show that such $d$ cannot exist by constructing $2^d+1$ elements of $\Delta$ which lie in a ball of radius $\frac{1}{8}$ while no two elements can be covered by a ball of radius $\frac{1}{16}$. However, by the pidgeon hole principle, at least one of the smaller balls would have to cover two different curves in the set. We construct a set of curves $P=\tau_{1},\dots,\tau_{2^d+1}$ with $\tau_i=0,i,i-\frac{1}{2},2^d+2$. The set $P$ is contained in the ball of radius $\frac{1}{8}$ centered at the curve $$c=0,\frac{7}{8}, \frac{5}{8}, \dots,i-\frac{1}{8},i-\frac{3}{8}, \dots, 2^d+\frac{7}{8}, 2^d+\frac{5}{8}, 2^d+2.$$ See [Figure \[fig:ddex\]]{} (left) for an example. Any two curves $\tau_i,\tau_j \in P$ have [Fréchet]{}distance $\frac{1}{4}$ to each other. Now, assume that a ball of radius $\frac{1}{16}$ exists which contains both $\tau_i$ and $\tau_j$. Let its center be denoted $c_{ij}$. We can derive a contradiction using the triangle inequality: $$\frac{1}{4}={\ensuremath{d_F{\!\left({\tau_i,\tau_j}\right)}}}\leq {\ensuremath{d_F{\!\left({\tau_i,c_{ij}}\right)}}} +{\ensuremath{d_F{\!\left({c_{ij},\tau_j}\right)}}}=\frac{1}{8}.$$ ![Examples of the constructed curves in [Lemma \[lemma:doub:unlimit\]]{} (left) and [Lemma \[lemma:doub:limit\]]{} (right). ](ddex.pdf) [\[fig:ddex\]]{} \[lemma:doub:limit\] For any integer $\ell>3$, the doubling dimension of the metric space $(\Delta_{\ell}, d_F)$ is unbounded. The proof is similar to the proof of [Lemma \[lemma:doub:unlimit\]]{}. However, this time we argue that no two curves in the set can be covered by a ball of half the radius because there exists no suitable center in $\Delta_{\ell}$, that is, the center would need to have complexity higher than $\ell$. As in the other proof, we define a set $P=\lbrace \tau_1,\dots,\tau_{2^d+1}\rbrace \subset \Delta_{\ell}$. For $s={\left\lfloor {\frac{\ell-2}{2}} \right\rfloor}$, let $$\tau_i = 0, s(i-1) + 1, s(i-1) + \frac{1}{2}, \dots, s(i-1) + j, s(i-1) + j - \frac{1}{2}, \dots, s i, s i - \frac{1}{2}, s(2^d+2).$$ See [Figure \[fig:ddex\]]{} (right) for an example with $\ell=8$. Clearly, each $\tau_i \in P$ is an element of $\Delta_{\ell}$, since its complexity is at most $\ell$. Furthermore, the set $P$ is contained in the ball with radius $\frac{1}{4}$ centered at $c=0,s(2^{d}+2)$. Note that the $(\frac{1}{8})$-signature of any $\tau_i\in P$ is equal to $\tau_i$ itself. Thus, by [Lemma \[lemma:nec:suff\]]{}, any center of a ball of radius at most $\frac{1}{8}$, which contains $\tau_i$ needs to have a vertex in each interval ${\left[ {v-\frac{1}{8},v+\frac{1}{8}} \right]}$ for each vertex $v$ of $\tau_i$. By construction, these intervals are pairwise disjoint for each curve and across all curves in $P$ (except for the intervals around the two endpoints). Therefore, such a ball with radius at most $\frac{1}{8}$ which would cover two different curves $\tau_i,\tau_j \in P$, would need to have more than $\ell$ vertices and is therefore not contained in $\Delta_{\ell}$. Indeed, the number of pairwise disjoint signature intervals induced by any $\tau_i,\tau_j \in P$ with $i\neq j$, is $2+4s = 2 \ell - 2 > \ell$. [^1]: Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands; `a.driemel``tue.nl`. Work on this paper was partially funded by NWO STW project “Context awareness in predictive analytics” and NWO Veni project “Clustering time series and trajectories (10019853)”. [^2]: Department of Computer Science, TU Dortmund, Germany; `amer.krivosija``tu-dortmund.de`. Work on this paper has been partly supported by DFG within the Collaborative Research Center SFB 876 “Providing Information by Resource-Constrained Analysis”, project A2. [^3]: Department of Computer Science, TU Dortmund, Germany; `christian.sohler``tu-dortmund.de`. Work on this paper has been partly supported by DFG within the Collaborative Research Center SFB 876 “Providing Information by Resource-Constrained Analysis”, project A2. [^4]: The conference version of this paper will be published at 26th ACM-SIAM Symposium on Discrete Algorithms (SODA) 2016. [^5]: Notice that this notation does not specify the points of time at which the measurements are taken. The reason is that the [Fréchet]{}distance only depends on the ordering of the measurements (and their values), but not on the exact points of time when the measurements are taken. [^6]: In the above definition, the [Fréchet]{} distance is only defined for functions with domain $[0,1]$. This is mostly for simplicity of analysis. We can easily extend it to other domains using an arbitrary homeomorphism that identifies this domain with $[0,1]$. The [Fréchet]{} distance is invariant under reparametrizations. All our algorithms use the parametrizations only implicitely. Any input time series is given as an ordered list of measurements, without explicit time-stamps. Therefore our definitions are without loss of generality. [^7]: Technically, the constructed matching is not strictly increasing. However, for any ${{\varepsilon}}>0$ it can be perturbed slightly to obtain a proper bijection. The result is then obtained in the limit. [^8]: The existence of such a matching $f$ is ensured since ${\ensuremath{d_F{\!\left({\tau,\pi}\right)}}} <\delta$. [^9]: Note that the constructed matching is not a bijection. However, for any ${{\varepsilon}}>0$, it can be perturbed to obtain a proper bijection. [^10]: A heap of the edges can be alternatively used. [^11]: The two other possible cases are as follows. It may be that multiple edges of the same length are contracted at once. In this case more than two vertices need to be extracted. Furthermore, it may be that only one vertex $v_1$ or $v_k$ is extracted at once. This corresponds to the case that an edge adjacent to $w_1$ or $w_m$ is being contracted. [^12]: Note that this result can also be obtained from an earlier result by Megiddo and Supowit [@megiddo1984geo]. They show that approximating the $k$-center problem under $\ell_1$ within a factor smaller than $2$ is [**NP**]{}-hard.

--- abstract: 'Navigation in unknown, chaotic environments continues to present a significant challenge for the robotics community. Lighting changes, self-similar textures, motion blur, and moving objects are all considerable stumbling blocks for state-of-the-art vision-based navigation algorithms. In this paper we present a novel technique for improving localization accuracy within a visual-inertial navigation system (VINS). We make use of training data to learn a model for the quality of visual features with respect to localization error in a given environment. This model maps each visual observation from a predefined prediction space of visual-inertial predictors onto a scalar weight, which is then used to scale the observation covariance matrix. In this way, our model can adjust the influence of each observation according to its quality. We discuss our choice of predictors and report substantial reductions in localization error on 4 km of data from the KITTI dataset, as well as on experimental datasets consisting of 700 m of indoor and outdoor driving on a small ground rover equipped with a Skybotix VI-Sensor.' author: - 'Valentin Peretroukhin, Lee Clement, Matthew Giamou, and Jonathan Kelly[^1]' bibliography: - 'probe\_abbrev.bib' title: '**PROBE: Predictive Robust Estimation for Visual-Inertial Navigation** ' --- INTRODUCTION ============ Robot navigation relies on an accurate quantification of sensor noise or uncertainty in order to produce reliable state estimates. In practice, this uncertainty is often fixed for a given sensor and experiment, whether by automatic calibration or by manual tuning. Although a fixed measure of uncertainty may be reasonable in certain static environments, dynamic scenes frequently exhibit many effects that corrupt a portion of the available observations. For visual sensors, these effects include, for example, self-similar textures, variations in lighting, moving objects, and motion blur. We assert that there may be useful information available in these observations that would normally be rejected by a fixed-threshold outlier rejection scheme. Ideally, we would like to retain some of these observations in our estimator, while still placing more trust in observations that do not suffer from such effects. In this paper we present PROBE, a Predictive ROBust Estimation technique that improves localization accuracy in the presence of such effects by building a model of the uncertainty in the affected visual observations. We learn the model in an offline training procedure and then use it online to predict the uncertainty of incoming observations as a function of their location in a predefined *prediction space*. Our model can be learned in completely unknown environments with frequent or infrequent ground truth data. The primary contributions of this research are a flexible framework for learning the quality of visual features with respect to navigation estimates, and a straightforward way to incorporate this information into a navigation pipeline. On its own, PROBE can produce more accurate estimates than a binary outlier rejection scheme like Random Sample Consensus (RANSAC) [@Fischler:1981cv] because it can simultaneously reduce the influence of outliers while intelligently weighting inliers. PROBE reduces the need to develop finely-tuned uncertainty models for complex sensors such as cameras, and better accounts for the effects observed in complex, dynamic scenes than typical fixed-uncertainty models. While we present PROBE in the context of visual feature-based navigation, we stress that it is not limited to visual measurements and could also be applied to other sensor modalities. ![PROBE maps image features into a prediction space to predict feature quality ($\alpha$). Feature quality is a function of the nearest neighbours from training data.[]{data-label="fig:feature_space"}](figs/inkscape_graphics/feature_space){width="50.00000%"} RELATED WORK {#sec:related_work} ============ Two important tasks must be performed to obtain accurate estimates from navigation systems: sensor noise characterization and outlier rejection. Standard calibration techniques can provide some insight into sensor noise, but in practice, noise parameters are often manually tuned to optimize estimator performance for a given environment. Outlier rejection typically relies on front-end procedures such as Random Sample Consensus (RANSAC) [@Fischler:1981cv; @VisualOdometry:n9UHQMsK], which attempts to remove outliers before the estimator can process them, or back-end procedures such as M-estimation, which uses robust cost functions to limit the influence of large errors [@Latif:2014dy]. For visual systems, a large body of literature has also focused on compensating for specific effects such as moving objects [@Wangsiripitak:2009bt], lighting variations [@McManus:2014ew], and motion blur [@Pretto:2009er] using carefully crafted models. More general approaches for attempting to select an optimal set of features has also been investigated in the literature. In [@Sim:1999ksa], visual features are parametrized in an attribute space that is used both to establish feature correspondences and to learn a model of feature reliability for pose estimation. Sufficiently reliable features are stored for use as landmarks during pose estimation, while unreliable features are discarded. Likewise, our method maps each feature in a prediction space to a corresponding scalar weight, but we use this weight in our estimator rather than as a criterion to discard data. In more recent work [@Carlone:2014dt], the authors forgo the concept of outliers in favour of finding the largest subset of coherent measurements. Once again, our method differs in that it reduces the influence of deleterious features without explicitly discarding them. Our work on predicting the general quality of visual features is inspired by recent research on predictive covariance estimation [@VegaBrown:ew], which estimates full measurement covariances by collecting empirical errors and storing them in a prediction space. For cameras, learning measurement covariances requires ground truth for the observed quantities, which is difficult to obtain for sparse visual landmarks. To relax this requirement, the authors of [@VegaBrown:2013fv] describe an expectation-maximization technique that can serve as a proxy for ground truth, but it is unclear whether this technique is applicable to sparse vision-based navigation. In contrast, our method requires ground truth for only a subset of position states and can be straightforwardly applied to standard sparse visual navigation systems. Finally, adaptive techniques exist that learn scalar weights to intelligently modify the influence of particular measurements within a Kalman filter estimator [@Ting:2007js]. We adopt a similar approach for a visual-inertial navigation pipeline, but do so predictively rather than reactively to respond to new measurements with minimal delay. VISUAL-INERTIAL NAVIGATION SYSTEM {#sec:vins} ================================= In this work, we evaluate PROBE as a way to improve navigation estimates from visual-inertial navigation systems (VINS). VINS have been used onboard many different robotic platforms including micro aerial vehicles (MAVs), ground vehicles, smartphones, and even wearable devices such as Google Glass [@Anonymous:m8ztJh1D]. Stereo cameras [@Leutenegger:2014hk] and monocular cameras [@Anonymous:5tilMnYo; @Indelman:2013bca] are common sensor choices for acquiring visual data in such systems. We choose to implement a stereo VINS adapted from a common stereo visual odometry (VO) framework used in numerous successful mobile robotics applications [@Cheng:2006kl; @Kelly:2008tr; @Geiger:2011jb]. Unlike other VINS [@Leutenegger:2014hk; @Anonymous:5tilMnYo], we choose not to use the linear accelerations reported by the IMU to drive a motion model. Instead, we opt to use only angular velocities to extract rotation estimates between successive poses. Although this choice reduces the potential accuracy of the method, it also confers several advantages that make our pipeline a solid foundation from which to evaluate PROBE. Specifically, we do not need to keep track of a linear velocity state, accelerometer biases, or orientation relative to gravity. Although seemingly minor, this simplification greatly reduces the complexity of the pipeline, obviates the need for a special starting state that is typically required to make gravity observable, and eliminates several sensitive tuning parameters. We use the relatively stable rotational rates reported by commercial IMUs to extract accurate relative rotation estimates over short time periods. Similar to [@Kneip:dw], we bootstrap a crude transformation estimate with the rotation estimate before carrying out a full non-linear point cloud alignment. Finally, because we assume a stereo camera, the metric scale of our solution is directly observable from the visual measurements, whereas monocular VINS must rely on noisy linear accelerations to determine metric scale [@Kelly:2011bw]. Observation Model ----------------- To begin, we outline our point feature observation model. We define two coordinate frames, $\cframe{a}$ and $\cframe{b}$, which represent the stereo camera pose at times $t_a$ and $t_b$ (where $t_b > t_a$), respectively. The coordinates $\mbf{p}_a$ of a landmark observed in $\cframe{a}$ can be transformed into its coordinates $\mbf{p}_b$ in $\cframe{b}$ as follows: $$\mbf{p}_b = \mbf{C}_{ba}(\mbf{p}_a - \mbf{r}_a^{ba}).$$ $\mbf{r}_a^{ba}$ is a vector from the origin of $\cframe{a}$ to the origin of $\cframe{b}$ expressed in $\cframe{a}$, and $\mbf{C}_{ba}$ is the rotation matrix from $\cframe{a}$ to $\cframe{b}$. Assuming rectified stereo images, point $\mbf p$ is projected from the camera frame (with its origin in the centre of the left camera) into image coordinates $\mbf y$ in the left and right cameras according to the camera model $$\begin{gathered} \label{eqn:imgCoord} \mbf{y} = \mbf f (\mbf p) = \bbm u_l \\ v_l \\ u_r \\ v_r \ebm = \frac{f}{z} \bbm x \\ y \\ x - b \\ y \\ \ebm + \bbm c_u \\ c_v \\ c_u \\ c_v \ebm.\end{gathered}$$ Here, $(x,y,z)$ are the components of $\mbf p$, $(u_l,u_r,v_l,v_r)$ are the horizontal and vertical pixel coordinates in each camera respectively, $b$ is the baseline of the stereo pair, $(c_u,c_v)$ is the principal point, and $f$ is the focal length. Direct Motion Solution ---------------------- Assuming all landmarks are tracked correctly, our goal is to calculate the transformation from $\cframe{a}$ to $\cframe{b}$, parametrized by $\mbf{C}_{ba}$ and $\mbf{r}_a^{ba}$. We obtain an estimate for $\mbf{C}_{ba}$, (denoted by $\overline{\mbf{C}}_{ba}$) by integrating angular velocities, $\mbs \omega_j$, from the IMU that are recorded between $t_a$ and $t_b$. Defining $\mbs \psi_j := \mbs \omega_j \Delta t$, $\psi_j := \abs{\mbs \psi_j}$, and $\mbshat \psi_j := \mbs \psi_j / \psi_j$, we have $$\begin{aligned} \mbs \Psi_j &= \cos \psi_j \mbf 1 + (1 - \cos \psi_j) \mbshat \psi_j \mbshat \psi_j^T - \sin \psi_j \mbshat \psi_j^\times, \\ \overline{\mbf{C}}_{ba} &= \mbf{C}_{cv} \mbs \Psi_{J} \cdots \mbs \Psi_2 \mbs \Psi_1 \mbf{C}_{cv}^T\end{aligned}$$ where $J$ is the number of IMU measurements between $t_a$ and $t_b$, $\mbf {C}_{cv}$ is the rotation from the IMU frame to the camera frame, $\One$ is the identity matrix, and $(\cdot)^\times$ is the skew-symmetric cross-product operator. Given our rotation estimate, we can compute an estimate for translation as $$\label{eqn:cab_umeyama} \overline{\mbf{r}}_a^{ba} = -\overline{\mbf{C}}_{ba}^T \mbf{u}_b + \mbf{u}_a,$$ where $\mbf{u}_a, \mbf{u}_b$ are the point cloud centroids given by $$\begin{aligned} \mbf{u}_a &= \frac{1}{N} \sum_{i=1}^{N} \mbf{p}_a^i & \text{and} && \mbf{u}_b &= \frac{1}{N} \sum_{i=1}^{N} \mbf{p}_b^i .\end{aligned}$$ Non-linear Motion Solution -------------------------- The direct motion solution forms an initial guess {$\overline{\mbf{C}}_{ba}$, $\overline{\mbf{r}}_a^{ba}$} for a non-linear optimization procedure. Given $N$ tracked points, we wish to minimize a 3D point cloud alignment error $$\begin{aligned} \label{eqn:matrixOptfun} \mathcal{L} = \frac{1}{2} \sum_{i=1}^{N} & \left( \mbf{p}_b^i - \mbf{C}_{ba}(\mbf{p}_a^i - \mbf{r}_a^{ba}) \right)^T \notag \\ & \mbs{\Gamma}^i \left( \mbf{p}_b^i - \mbf{C}_{ba}(\mbf{p}_a^i - \mbf{r}_a^{ba}) \right),\end{aligned}$$ with $$\label{eqn:gamma} \mbs{\Gamma}^i = \left( \mbf{G}_b^i \mbf{R}_b^{i} \mbf{G}_b^{i^T} + \mbf{C}_{ba} \mbf{G}_a^i \mbf{R}_a^{i^T} \mbf{G}_a^{i^T} \mbf{C}_{ba}^T \right)^{-1},$$ where $$\begin{aligned} \mbf{G}_a^i &= \frac{\partial \mbf{g} }{\partial \mbf y }\bigg|_{\mbf{f}(\mbf{p}^i_a)} & \text{and} && \mbf{G}_b^i &= \frac{\partial \mbf{g} }{\partial \mbf y }\bigg|_{\mbf{f}(\mbf{p}^i_b)}\end{aligned}$$ are the Jacobians of the inverse camera model $\mbf{g} := \mbf{f}^{-1}$, and $\mbf{R}_a^{i}$, $\mbf{R}_b^{i}$ are the covariance matrices of the $i^\text{th}$ point in image space. We proceed by perturbing the initial guess by a vector $\mbs \xi = \bbm\mbs{\epsilon}^T & \mbs{\phi}^T \ebm^T$, where $\mbs{\epsilon}$ is a translational perturbation and $\mbs{\phi}$ is a rotational perturbation: $$\begin{aligned} \mbf{r}_a^{ba} &= \mbf{\overline{r}}_a^{ba} + \mbs{\epsilon}, \label{eqn:trans_perturb} \\ \mbf{C}_{ba} &= e^{-\mbs{\phi}^{\times}} \mbfbar{C}_{ba} \approx (\mbf{1} - \mbs{\phi}^{\times})\mbfbar{C}_{ba}. \label{eqn:rot_perturb}\end{aligned}$$ Inserting and into , we arrive at a cost function that is quadratic in the perturbations: $$\begin{gathered} \label{eqn:matrixOptfunPerturbed} \mathcal{L} \approx \frac{1}{2} \sum_{j=1}^{N} \left( \mbfbar{e}^i + \mbfbar{E}^i \mbs{\xi} \right)^T \mbsbar{\Gamma}^i \left( \mbfbar{e}^i + \mbfbar{E}^i \mbs{\xi} \right),\end{gathered}$$ where $$\begin{aligned} \mbfbar{e}^i &= \mbf{p}_b^i - \mbfbar{C}_{ba}(\mbf{p}_a^i - \mbfbar{r}_a^{ba}), \\ \mbfbar{E}^i &= \left[\mbfbar{C}_{ba} ~ -\left(\mbfbar{C}_{ba}(\mbf{p}_a^i - \mbfbar{r}_a^{ba})\right)^\times \right],\end{aligned}$$ and $\mbsbar \Gamma^i$ indicates that $\mbfbar{C}_{ba}$ has replaced $\mbf{C}_{ba}$ in . Setting the derivative of with respect to $\mbs \xi$ to zero yields a system of linear equations in the optimal update step $\mbs \xi^*$: $$\begin{gathered} \label{eqn:matrixOptLinEqns} \sum_{j=1}^{N} \left( \mbfbar{E}^{i^T} \mbsbar{\Gamma}^i \mbfbar{E}^i \right) \mbs \xi^* = - \sum_{i=1}^{N} \mbfbar{E}^{i^T} \mbsbar{\Gamma}^i \mbfbar{e}^i.\end{gathered}$$ Once $\mbs \xi^*$ is determined, the state estimate can be repeatedly updated using: $$\begin{gathered} \mbfbar{C}_{ba} \leftarrow \exp((-\mbs{\phi^*})^{\times}) \mbfbar{C}_{ba}, \\ \mbfbar{r}_a^{ba} \leftarrow \mbfbar{r}_a^{ba} + \mbs{\epsilon^*}.\end{gathered}$$ In practice, is also modified with Levenberg-Marquardt damping to improve convergence. PROBE: PREDICTIVE ROBUST ESTIMATION {#sec:probe} =================================== The aim of PROBE is to learn a model for the quality of visual features, with the goal of reducing the impact of deleterious visual effects such as moving objects, motion blur, and shadows on navigation estimates. Feature quality is characterized by a scalar weight, $\beta_i$, for each visual feature in an environment. To compute $\beta_i$ we define a prediction space (similar to [@VegaBrown:ew]) that consists of a set of visual-inertial predictors computed from the local image region around the feature and the inertial state of the vehicle (Section \[sec:predictors\] details our choice of predictors). We then scale the image covariance of each feature ($\mbf{R}_a^{i}$, $\mbf{R}_b^{i}$ in ) by $\beta_i$ during the non-linear optimization. In a similar manner to M-estimation, PROBE achieves robustness by varying the influence of certain measurements. However, in contrast to robust cost functions that weight measurements based purely on estimation error, PROBE weights measurements based on their assessed quality. To learn the model, we require training data that consists of a traversal through a typical environment with some measure of ground truth for the path, but not for the visual features themselves. Like many machine learning techniques, we assume that the training data is representative of the test environments in which the learned model will be used. We learn the quality of visual features *indirectly* through their effect on navigation estimates. We define high quality features as those that result in estimates that are close to ground truth. Our framework is flexible enough that we do not require ground truth at every image and we can learn the model based on even a single loop closure error. Training -------- Training proceeds by traversing the training path, selecting a subset of visual features at each step, and using them to compute an incremental position estimate. By comparing the estimated position to the ground truth position, we compute the translational Root Mean Squared Error (RMSE), denoted by $ \alpha_{l,s} $ for iteration $l$ and step $s$, and store it at each feature’s position in the prediction space (we denote the set of predictors and associated RMSE value by $\Theta_{l,s}$). The full algorithm is summarized in Figure \[fig:ProbeTraining\]. Note that $\alpha_{l,s}$ can be computed at each step, at intermittent steps, or for an entire path, depending on the availability of ground truth data. $f_1, \dots, f_J \gets visualFeatureSubset(l)$ $\mbs\pi^1_l, \dots, \mbs\pi^J_l \gets predictors(f_1, \dots, f_J)$ $\bar{\mbf{C}}_{ba}, \bar{\mbf{r}}_a^{ba} \gets poseChange(f_1, \dots, f_J)$ $\mbf \alpha_{l,s} \gets computeRMSE(\bar{\mbf{r}}_a^{ba}, {\mbf{r}_{a}^{ba}}_{GT} )$ $ \Theta_{l,s} \gets \left\{\mbs\pi^1_{l,s}, \dots, \mbs\pi^J_{l,s}, \alpha_{l,s}\right\}$ **return** $\mbs \Theta = \{ \Theta_{l,s} \}$ Evaluation ---------- To use the PROBE model in a test environment, we compute the location of each observed visual feature in our prediction space, and then compute its relative weight $\beta_i$ as a function of its $K$ nearest neighbours in the training set. For efficiency, the $K$ nearest neighbours are found using a $k$-d tree. The final scaling factor $\beta_i$ is a function of the mean of the $\alpha$ values corresponding to the $K$ nearest neighbours, normalized by $\overline \alpha$, the mean $\alpha$ value of the entire training set. $\mbs \pi_i \gets predictors(f_i) $ $\alpha_1,...,\alpha_K \gets findKNN(\mbs \pi_i, K, \mbs\Theta)$ $\beta_i \gets \left(\frac{1}{\overline{\alpha} K} \sum_{k=1}^K \alpha_k \right)^{\gamma}$ **return** $\mbs \beta = \{\beta_i\}$ The value of $K$ can be determined through cross-validation, and in practice depends on the size of the training set and the environment. The computation of $\beta_i$ is designed to map small differences in learned $\alpha$ values to scalar weights that span several orders of magnitude. An appropriate value of $\gamma$ can be found by searching through a set range of candidate values and choosing the value that minimizes the average RMSE (ARMSE) on the training set. Prediction Space {#sec:predictors} ---------------- A crucial component of our technique is the choice of prediction space. In practice, feature tracking quality is often degraded by a variety of effects such as motion blur, moving objects, and textureless or self-similar image regions. The challenge is in determining predictors that account for such effects without requiring excessive computation. In our implementation, we use the following predictors, but stress that the choice of predictors can be tailored to suit particular applications and environments: - Angular velocity and linear acceleration magnitudes - Local image entropy - Blur (quantified by the blur metric of [@Anonymous:Ngi3VEEU]) - Optical flow variance score - Image frequency composition We discuss each of these predictors in turn. ### Angular velocity and linear acceleration While most of the predictors in our system are computed directly from image data, the magnitudes of the angular velocities and linear accelerations reported by the IMU are in themselves good predictors of image degradation (e.g., image blur) and hence poor feature tracking. ### Local image entropy Entropy is a statistical measure of randomness that can be used to characterize the texture in an image or patch. Since the quality of feature detection is strongly influenced by the strength of the texture in the vicinity of the feature point, we expect the entropy of a patch centered on the feature to be a good predictor of its quality. We evaluate the entropy $S$ in an image patch by sorting pixel intensities into $N$ bins and computing $$S = -\sum_{i=1}^N c_i \log_2(c_i),$$ where $c_i$ is the number of pixels counted in the $i^\text{th}$ bin. ### Blur Blur can arise from a number of sources including motion, dirty lenses, and sensor defects. All of these have deleterious effects on feature tracking quality. To assess the effect of blur in detail, we performed a separate experiment. We recorded images of 32 interior corners of a standard checkerboard calibration target using a low frame-rate (20 FPS) Skybotix VI-Sensor stereo camera and a high frame-rate (125 FPS) Point Grey Flea3 monocular camera rigidly connected by a bar (Figure \[fig:tricifix\]). Prior to the experiment, we determined the intrinsic and extrinsic calibration parameters of our rig using the Kalibr package [@Furgale:2013dm]. The apparatus underwent both slow and fast translational and rotational motion, which induced different levels of motion blur as quantified by the blur metric proposed by [@Anonymous:Ngi3VEEU]. ![The Skybotix VI-Sensor, Point Grey Flea3, and checkerboard target used in our motion blur experiments.[]{data-label="fig:tricifix"}](figs/tricifix2){width="40.00000%"}   We detected checkerboard corners in each camera at synchronized time steps, computed their 3D coordinates in the VI-Sensor frame, then reprojected these 3D coordinates into the Flea3 frame. We then computed the reprojection error as the distance between the reprojected image coordinates and the true image coordinates in the Flea3 frame. Since the Flea3 operated at a much higher frame rate than the VI-Sensor, it was less susceptible to motion blur and so we treated its observations as ground truth. We also computed a tracking error by comparing the image coordinates of checkerboard corners in the left camera of the VI-Sensor computed from both KLT tracking [@Lucas:1981uw] and re-detection. Figure \[fig:visensor\_histograms\] shows histograms and fitted normal distributions for both reprojection error and tracking error. From these distributions we can see that the errors remain approximately zero-mean, but that their variance increases with blur. This result is compelling evidence that the effect of blur on feature tracking quality can be accounted for by scaling the feature covariance matrix by a function of the blur metric. ### Optical flow variance score To detect moving objects, we compute a score for each feature based on the ratio of the variance in optical flow vectors in a small region around the feature to the variance in flow vectors of a larger region. Intuitively, if the flow variance in the small region differs significantly from that in the larger region, we might expect the feature in question to belong to a moving object, and we would therefore like to trust the feature less. Since we consider only the variance in optical flow vectors, we expect this predictor to be reasonably invariant to scene geometry. We compute this optical flow variance score according to $$\log \left( \frac{\bar{\sigma}^2_s}{\bar{\sigma}^2_l} \right),$$ where $\bar{\sigma}^2_s, \bar{\sigma}^2_l$ are the means of the variance of the vertical and horizontal optical flow vector components in the small and large regions respectively. Figure \[fig:flow\_variance\] shows sample results of this scoring procedure for two images in the KITTI dataset [@Geiger:2013kp]. Our optical flow variance score generally picks out moving objects such as vehicles and cyclists in diverse scenes. ![The optical flow variance predictor can help in detecting moving objects. Red circles correspond to higher values of the optical flow variance score (i.e., features more likely to belong to a moving object).[]{data-label="fig:flow_variance"}](figs/flowPredictorCombined.jpg){width="40.00000%"} ### Image frequency composition Reliable feature tracking is often difficult in textureless or self-similar environments due to low feature counts and false matches. We detect textureless and self-similar image regions by computing the Fast Fourier Transform (FFT) of each image and analyzing its frequency composition. For each feature, we compute a coefficient for the low- and high-frequency regimes of the FFT. Figure \[fig:high\_frequency\] shows the result of the high-frequency version of this predictor on a sample image from the KITTI dataset [@Geiger:2013kp]. Our high-frequency predictor effectively distinguishes between textureless regions (e.g., shadows and roads) and texture-rich regions (e.g., foliage). ![A high-frequency predictor can distinguish between regions of high and low texture such as foliage and shadows. Green indicates higher values.[]{data-label="fig:high_frequency"}](figs/highFreqPredictor.jpg){width="40.00000%"} EXPERIMENTAL RESULTS {#sec:results} ==================== [cccccccccccc]{} & & & & & & & & &\ & & & & & & & & ------------------------------------------------------------------------ \ Trial & Type & Path Length && ARMSE & Final Error & & ARMSE & Final Error & & ARMSE & Final Error ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ ------------------------------------------------------------------------ `26_drive_0051` & City & 251.1 m && 4.84 m & 12.6 m && 3.30 m & 8.62 m && 3.48 m & 8.07 m\ `26_drive_0104` & City & 245.1 m && 0.977 m & 4.43 m && 0.850 m & 3.46 m && 1.19 m & 3.61 m\ `29_drive_0071` & City & 234.0 m && 5.44 m & 30.3 m && 5.44 m & 30.4 m && 3.03 m & 12.8 m\ `26_drive_0117` & City & 322.5 m && 2.29 m & 9.07 m && 2.29 m & 9.07 m && 2.76 m & 9.08 m\ `30_drive_0027` & Residential & 667.8 m && 4.22 m & 12.2 m && 4.30 m & 10.6 m && 3.64 m & 4.57 m\ `26_drive_0022` & Residential & 515.3 m && 2.21 m & 3.99 m && 2.66 m & 6.09 m && 3.06 m & 4.99 m\ `26_drive_0023` & Residential & 410.8 m && 1.64 m & 8.20 m && 1.77 m & 8.27 m && 1.71 m & 8.13 m\ `26_drive_0027` & Road & 339.9 m && 1.63 m & 8.75 m && 1.63 m & 8.65 m && 1.40 m & 7.57 m\ `26_drive_0028` & Road & 777.5 m && 4.31 m & 16.9 m && 3.72 m & 13.1 m && 3.92 m & 13.2 m\ `30_drive_0016` & Road & 405.0 m && 4.56 m & 19.5 m && 3.33 m & 14.6 m && 2.76 m & 13.9 m\ UTIAS Outdoor & Snowy parking lot & 302.0 m && 7.24 m & 10.1 m && 7.02 m & 10.6 m && 6.85 m & 6.09 m\ UTIAS Indoor & Lab interior & 32.83 m && — & 0.854 m && — & 0.738 m && — & 0.617 m ------------------------------------------------------------------------ \ \[table:kitti\_data\] Trained using sequence `09_26_drive_0005`.  $^2$ Trained using sequence `09_26_drive_0046`.  $^3$ Trained using sequence `09_26_drive_0015`. This residential trial was evaluated with a model trained on a sequence from the city category because of several moving vehicles that were better represented in that training dataset. Datasets -------- {width="90.00000%"} ![Our four-wheeled skid-steered Clearpath Husky rover equipped with Skybotix VI-Sensor and Ashtech DGPS antenna used to collect the outdoor UTIAS dataset.[]{data-label="fig:huskypic"}](figs/husky2){width="40.00000%"} We trained and evaluated PROBE in two sets of experiments. The first set of experiments made use of 4.5 km of data from the City, Residential, and Road categories of the KITTI dataset [@Geiger:2013kp]. In the second set of experiments, we collected indoor and outdoor datasets at the University of Toronto Institute for Aerospace Studies (UTIAS) using a Skybotix VI-Sensor mounted on an Adept MobileRobots Pioneer 3-AT rover and a Clearpath Husky rover, respectively. In both cases, the camera recorded stereo images at 10 Hz while the IMU operated at 200 Hz. The outdoor dataset consisted of a 264 m training run followed by a 302 m evaluation run, with ground truth provided by RTK-corrected GPS. The indoor dataset consisted of a 32 m training run and a 33 m evaluation run through a room with varying lighting and shadows. For the indoor dataset, no ground truth was available, so we trained PROBE using only the knowledge that the training path should form a closed loop. We compare PROBE to what we call the nominal VINS, as well as a VINS with an aggressive RANSAC routine. In the nominal pipeline, we use RANSAC with enough iterations to be 99% confident that we select only inliers when as many as 50% of the features are outliers. In the aggressive case, we increase the confidence level to 99.99%. When PROBE is used, we apply a pre-processing step that makes use of the rotational estimate from the IMU to reject any egregious feature matches by thresholding the cosine distance between pairs of matched feature vectors. We assume small translations between frames and typically set the threshold to reject feature vectors that are separated by more than five degrees. For feature extraction, matching, and sparse optical flow calculations, we use the open source vision library `LIBVISO2` [@Geiger:2011jb]. `LIBVISO2` efficiently detects thousands of feature correspondences by finding stable features using blob and corner masks and matching them by comparing Sobel filter responses. For all prediction space calculations, we use features in the left image of the stereo pair. Training -------- Sample results of the training procedure described in Section \[sec:probe\] are illustrated in Figure \[fig:residential:training\] for data from the Residential category. As ground truth is available for each image, we compute the RMSE at every frame, and only iterate over the path 10 times. Note the large variance of training run error in the sharp turn in Figure \[fig:residential:training\], caused by a car that drives through the camera’s field of view. PROBE is able to distinguish features on the car and adequately reduce their influence with the final learned model. [figs/KITTI/Training/2011\_09\_26\_drive\_0046\_sync\_paths]{} (20,30)[![Training iterations for the Residential category, sequence `09_26_drive_0046`. The left turn is particularly problematic due to a moving car that comes into view. Although none of the training runs completely remove all features from the car, the path differences are enough for the learned PROBE model to adequately reduce the influence of the car on the final motion estimate.[]{data-label="fig:residential:training"}](figs/KITTI/Training/2011_09_26_drive_0046_car.png "fig:"){width="20.00000%"}]{} (34, 26.5)[Moving Vehicle]{} (70,46)[(3,1)[18]{}]{} [figs/KITTI/Test/2011\_09\_29\_drive\_0071\_sync\_comparison]{} (20,21)[![A 234 m KITTI test run in the City category, sequence `09_29_drive_0071` containing numerous pedestrians and dramatic lighting changes. PROBE is able to produce more accurate navigation estimates than even an aggressive RANSAC routine.[]{data-label="fig:city:test"}](figs/KITTI/Test/2011_09_29_drive_0071_flockOfLadies "fig:"){width="15.00000%"}]{} (31,17.5)[Pedestrians]{} (58,30)[(1,[0.2]{})[10]{}]{} [figs/KITTI/Test/2011\_09\_30\_drive\_0027\_sync\_comparison]{} (39,21)[![A 667 m test run in the Residential category, sequence `09_30_drive_0027`. PROBE is better able to deal with a static portion when a large shadow and moving vehicle cross the field of view of the camera.[]{data-label="fig:residential:test"}](figs/KITTI/Test/2011_09_30_drive_0027_truckshadow "fig:"){width="15.00000%"}]{} (43,17.5)[Vehicle with Shadow]{} (40,20)[(-2,-2)[3.5]{}]{} [figs/KITTI/Test/2011\_09\_26\_drive\_0027\_sync\_comparison]{} (36,65)[![A 440 m test run in the Road category, sequence `09_26_drive_0027`. PROBE is able to better predict a large moving vehicle, and extract higher quality features from dense foliage. Note the difference in scale between the two axes.[]{data-label="fig:road:test"}](figs/KITTI/Test/2011_09_26_drive_0027_truck "fig:"){width="14.00000%"}]{} (43,61.5)[Moving Vehicle]{} (48,60)[(0,-1)[35]{}]{} Evaluation ---------- To evaluate PROBE, we run the nominal VINS (cf. Section \[sec:vins\]) on a given test trial, tune the RANSAC threshold to achieve reasonable translation error ($< 5\%$ final drift), then repeat the trial with the aggressive RANSAC procedure. Finally, we run VINS again, this time disabling RANSAC completely and applying our trained PROBE model (with pre-processing) to each observed feature. Table \[table:kitti\_data\] compares the performance of each trained PROBE model to that of the nominal and aggressive-RANSAC VINS. In the best case, PROBE achieves a final translational error norm of less than half that of both reference VINS. Figures \[fig:city:test\], \[fig:residential:test\], and \[fig:road:test\] reveal problematic sections of the KITTI dataset where PROBE is able to significantly improve upon the performance of both reference VINS. Moving vehicles and pedestrians are the most obvious sources of error that PROBE is able to identify. The datasets also included more subtle effects such as motion blur (notable at the edge of images), slowly swaying vegetation, and shadows. [figs/Husky/2015-02-18-12-33-30\_comparison]{} (56,10)[![Test and training trials in the outdoor UTIAS environment. The rover traverses a snowy landscape and people walk though the field of view.[]{data-label="fig:husky"}](figs/Husky/2015-02-18-12-43-18_paths_small "fig:"){width="16.00000%"}]{} [figs/Pioneer/2015-02-25-16-58-23\_comparison\_top]{} (15,12)[![A 32.8 m test run in our indoor UTIAS dataset. The subplot in the bottom-right corner of the figure shows that PROBE reduces drift in the vertical (y) direction more than the nominal VINS and aggressive RANSAC.[]{data-label="fig:pioneer"}](figs/Pioneer/2015-02-25-16-56-01_paths_small "fig:"){width="10.00000%"}]{} (60,10)[![A 32.8 m test run in our indoor UTIAS dataset. The subplot in the bottom-right corner of the figure shows that PROBE reduces drift in the vertical (y) direction more than the nominal VINS and aggressive RANSAC.[]{data-label="fig:pioneer"}](figs/Pioneer/2015-02-25-16-58-23_comparison_small_vert.pdf "fig:"){width="14.00000%"}]{} DISCUSSION {#sec:discussion} ========== In most of the datasets we evaluated, PROBE performed as well as or better than a standard RANSAC routine in reducing the influence of deleterious features. In particular, PROBE has proven to be more robust than aggressive RANSAC in datasets that exhibit visual effects such as shadows, large moving objects, and self-similar textures. Often, PROBE can produce more accurate navigation estimates by intelligently weighting measurements to reflect their quality while still exploiting the information contained in low-quality measurements. In this sense, PROBE can be thought of as a soft outlier classification scheme, while RANSAC rejects outliers based on binary classification. Like with other machine learning approaches, a good training dataset is essential to producing an accurate and generalizable model. PROBE is flexible enough that it can be taught with varying frequency of ground truth data. For instance, in the outdoor dataset collected at UTIAS, GPS measurements were available at only 1 Hz, and PROBE was trained by evaluating the ARMSE over the entire path. For the indoor dataset, no ground truth was available and the model was learned by computing the loop closure error between the start and end points of the training path. CONCLUSIONS {#sec:conclusions} =========== In this work, we presented PROBE, a novel method for predicting the quality of visual features within complex, dynamic environments. By using training data to learn a mapping from a predefined space of visual-inertial predictors to a scalar weight, we can adjust the influence of individual visual features on the final navigation estimate. PROBE can be used in place of traditional outlier rejection techniques such as RANSAC, or combined with them to more intelligently weight inlier measurements. We explored a variety of potential predictors, and validated our technique using a visual-inertial navigation system on over 4 km of data from the KITTI dataset and 700 m of indoor and outdoor data collected at the University of Toronto Institute for Aerospace Studies. Our results show that PROBE outperforms RANSAC-based binary outlier rejection in many environments, even with only sparse ground truth available during the training step. In future work, we plan to examine a broader set of predictors, and extend the training procedure to incorporate online learning using intermittent ground truth measurements or loop closures detected by a place recognition thread. Further, we are interested in analyzing the amount of training data required for a given improvement in navigation accuracy, and in investigating PROBE’s effect on estimator consistency. ACKNOWLEDGEMENT {#acknowledgement .unnumbered} =============== This work was supported by the Natural Sciences and Engineering Research Council (NSERC) through the NSERC Canadian Field Robotics Network (NCFRN). \#1 [^1]: All authors are at the Institute for Aerospace Studies, University of Toronto, Canada. {v.peretroukhin, lee.clement, matthew.giamou}@mail.utoronto.ca, jkelly@utias.utoronto.ca.

--- abstract: | In this paper we examine the existence of bicomplexified inverse Laplace transform  as an extension of it’s complexified inverse version within the region of convergence of bicomplex Laplace transform. In this course we use the idempotent representation of bicomplex-valued functions as projections on the auxiliary complex spaces of the components of bicomplex numbers along two orthogonal,idempotent hyperbolic directions. **Keywords**: Bicomplex numbers, Laplace transform, Inverse Laplace transform. author: - | A. BANERJEE$^{1}$, S. K. DATTA$^{2}$ and MD. A. HOQUE$^{3}$\ $^{1}$Department of Mathematics, Krishnath College, Berhampore,\  Murshidabad 742101, India, E-mail: abhijit.banerjee.81@gmail.com\ $^{2}$Department of Mathematics, University of Kalyani, Kalyani, Nadia,\  PIN-741235,India, E-mail: sanjib\_kr\_datta@yahoo.co.in   \ $^{3}$Gobargara High Madrasah (H. S.), Hariharpara, Murshidabad\  742166, India,E-mail: mhoque3@gmail.com title: '**Inverse Laplace Transform for Bi-Complex Variables**' --- Introduction ============ The theory of bicomplex numbers is a matter of active research for quite a long time science the seminal work of Segre\[1\] in search of special algebra.The algebra of bicomplex numbers are widely used in the literature as it becomes a valiable commutative alternative \[2\] to the non-commutative skew field of quaternions introduced by Hamilton \[3\] (both are four- dimensional and generalization of complex numbers). A bicomplex number is defined as $\ $$$\ \xi =a_{0}+i_{1}a_{1}+i_{2}a_{2}+i_{1}i_{2}a_{3},$$where  $a_{0,}$ $a_{1},a_{2},a_{3}$ are real numbers, $i_{1}^{2}=i_{2}^{2}=-1$ and $$\ i_{1}i_{2}=i_{2}i_{1},(i_{1}i_{2})^{2}=1.$$ The set of bicomplex numbers,complex numbers and real numbers are denoted by $C_{2},C_{1},$and $C_{0}$ respectively. $C_{2}$ becomes a Real Commutative Algebra with identity $$\ \ 1=1+i_{1}\cdot 0+i_{2}\cdot 0+i_{1}i_{2}\cdot 0$$$\ \ \ \ \ $ with standard binary composition. There are  two non trivial elements $e_{1}=\frac{1+i_{1}i_{2}}{2}$ and $e_{2}=\frac{1-i_{1}i_{2}}{2}$  in $C_{2}$ with the properties $\ e_{1}^{2}=e_{1},e_{2}^{2}=e_{2},e_{1}\cdot e_{2}=e_{2}\cdot e_{1}=0$ and $\ e_{1}+e_{2}=1$ which means that $\ e_{1}$ and $e_{2}$ are idempotents (some times called also orthogonal idempotents). By the help of the idempotent elements $\ e_{1}$ and $e_{2}$ any bicomplex number$$\xi =a_{0}+i_{1}a_{1}+i_{2}a_{2}+i_{1}i_{2}a_{3}=(a_{0}+i_{1}a_{1})+i_{2}(a_{2}+i_{1}a_{3})=z_{1}+i_{2}z_{2}$$ where  $a_{0,}$ $a_{1},a_{2},a_{3}\epsilon R,$ $$z_{1}(=a_{0}+i_{1}a_{1}),z_{2}(=a_{2}+i_{1}a_{3})\epsilon C_{1}$$can be expressed as $\ $$$\xi =z_{1}+i_{2}z_{2}=\xi _{1}e_{1}+\xi _{2}e_{2}$$where $\xi _{1}(=z_{1}-i_{1}z_{2})$ and $\xi _{2}(=z_{1}+i_{1}z_{2})\epsilon C_{1}.$ This representation of a bicomplex number is known as the Idempotent Representation of $\xi $. $\xi _{1}$ and $\xi _{2}$ are called the Idempotent Components of the bicomplex number $\xi =z_{1}+i_{2}z_{2}$, resulting a pair of mutually complementary projections $$\ P_{1}:(z_{1}+i_{2}z_{2})\epsilon C_{2}\longmapsto (z_{1}-i_{1}z_{2})\epsilon C_{1}$$ and $$P_{2}:(z_{1}+i_{2}z_{2})\epsilon C_{2}\longmapsto (z_{1}+i_{1}z_{2})\epsilon C_{1}.$$The spaces $A_{1}=\{P_{1}(\xi ):\xi \epsilon C_{2}\}$ and $A_{2}=\{P_{2}(\xi ):\xi \epsilon C_{2}\}$ are called the auxiliary complex spaces of bicomplex numbers. An element $\xi =z_{1}+i_{2}z_{2}$ is singular if and only if $|z_{1}^{2}+z_{2}^{2}|=0$.The set of singular elements is denoted as $O_{2}$ and defined by $O_{2}=\{\xi \epsilon C_{2}:\xi $ is the collection of all -complex multiples of$\ e_{1}$ and $e_{2}$ $\}$ The norm the $||\cdot ||:C_{2}\longmapsto C_{0}^{+}$(set of all non negetive real numbers) of a bicomplex number is defined as $$||\xi ||=\sqrt{\{|z_{1}|^{2}+|z_{2}|^{2}\}}=\sqrt{a_{0}^{2}+a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}$$ Laplace transform ================= Let $f(t)$ be a real valued function of exponential order k. The  coplex version of Laplace Transform  \[5\] of $f(t)$ for $t\geq 0$ can be defined as$$L\{f(t)\}=F_{1}(\xi _{1})=\dint\limits_{0}^{\infty }f(t)e^{-\xi _{1}t}dt$$. Here $F_{1}(\xi _{1}:\xi _{1}\epsilon C_{1})$ exists and absolutely convergent for Re$(\xi _{1})$ $k.$Similarly $$F_{2}(\xi _{2})=\dint\limits_{0}^{\infty }f(t)e^{-\xi _{2}t}dt$$ converges absolutely for Re$(\xi _{2})$ $k$  .  Then the bicomplex Laplace Transform \[4\] of $f(t)$ for $t\geq 0$ can be defined as$$L\{f(t)\}=F(\xi )=\dint\limits_{0}^{\infty }f(t)e^{-\xi t}dt$$. Here $F(\xi )$ exists and convergent in the region $$D=\ \{\xi \epsilon C_{2}:\xi =\xi _{1}e_{1}+\xi _{2}e_{2}:Re(\xi _{1})>k,Re(\xi _{2})>k\}$$ in idempotent representation. Inverse Laplace Transform for Bicomplex variables ================================================= If $f(t)$ real valued function of exponential order k, defined on $t\geq 0$ ,its Laplace  transform $F_{1}(\xi _{1})$ in bicomplex variable $\xi _{1}=x_{1}+i_{1}y_{1}\epsilon C_{_{1}}$ is simply$$\begin{aligned} F_{1}(\xi _{1}) &=&\dint\limits_{0}^{\infty }f(t)e^{-\xi _{1}t}dt=\dint\limits_{0}^{\infty }f(t)e^{-(x_{1}+i_{1}y_{1})t}dt=\dint\limits_{0}^{\infty }e^{-x_{1}t}f(t)e^{-i_{1}y_{1}t}dt \\ &=&\dint\limits_{0}^{\infty }\{e^{-x_{1}t}f(t)\}e^{-i_{1}y_{1}t}dt=\dint\limits_{-\infty }^{\infty }g(t)e^{-i_{1}y_{1}t}dt=\psi (x_{1,}y_{1})\end{aligned}$$ which is Fourier transform of $g(t)$ where $$g(t)=f(t)e^{-x_{1}t},t\geq 0;and=0,t<0$$ in usual complex exponential form. $F_{1}(\xi _{1})$ converges for Re$(\xi _{1})$ $k$  and $$|\ F_{1}(\xi _{1})|\ <\infty \ \Rightarrow |\dint\limits_{0}^{\infty }f(t)e^{-\xi _{1}t}dt|=\dint\limits_{-\infty }^{\infty }|g(t)e^{-i_{1}y_{1}t}|dt=\dint\limits_{-\infty }^{\infty }|g(t)|dt<\infty$$ The later condition shows that $g(t)$ is absotulely integrable .Then by Laplace inverse transform in complex exponential form $$g(t)=\frac{1}{2\pi i_{1}}\dint\limits_{-\infty }^{\infty }e^{i_{1}y_{1}t}\psi (x_{1,}y_{1})dy_{1}\Rightarrow f(t)=\frac{1}{2\pi i_{1}}\dint\limits_{-\infty }^{\infty }e^{x_{1}t}e^{i_{1}y_{1}t}\psi (x_{1,}y_{1})dy_{1}.$$                                                                                                                Now if we integrate along a vertical line then x$_{1}$ is a constant and so for complex variable $\xi _{1}=x_{1}+i_{1}y_{1}\epsilon C_{_{1}}($that implies $d\xi _{1}=$ $dy_{1})$  the above inversion formula can be                                                                                                                          extended to complex Laplace inverse transform $$\begin{aligned} f(t) &=&\frac{1}{2\pi i_{1}}\dint\limits_{x_{1}-i_{1}\infty }^{x_{1}+i_{1}\infty }e^{(x_{1}+i_{1}y_{1})t}\psi (x_{1,}y_{1})dy_{1}=\frac{1}{2\pi i_{1}}\dint\limits_{x_{1}-i_{1}\infty }^{x_{1}+i_{1}\infty }e^{\xi _{1}t}\psi (x_{1,}y_{1})d\xi _{1} \\ &=&\frac{1}{2\pi i_{1}}\lim\limits_{y_{1}\rightarrow \infty }\dint\limits_{x_{1}-i_{1}y_{1}}^{x_{1}+i_{1}y_{1}}e^{\xi _{1}t}F(\xi _{1})d\xi _{1}...........(1)\end{aligned}$$ $\ \ \ \ $ Here the integration is to be performed along a vertical line in the complex $\ \xi _{1}$-plane employing contour integration method. We assume that $F_{1}(\xi _{1})$  is holomorphic in $x_{1}<k$ except for having a finite number of poles $\xi _{1}^{k}$ ,$k=1,2,3,................n$ therein. Taking  $R\rightarrow \infty $ we can guarantee that all these poles lie inside the contour $\Gamma _{R}$ .Since $e^{\xi _{1}t}$ never vanishes so the poles of $e^{\xi _{1}t}F(\xi _{1})$ and $F_{1}(\xi _{1})$  are same.Then by Cauchy residue theorem $$\lim\limits_{R\rightarrow \infty }\dint\limits_{\Gamma _{R}}e^{\xi _{1}t}F(\xi _{1})d\xi _{1}=2\pi i_{1}\sum \func{Re}s\{e^{\xi _{1}t}F(\xi _{1}):\xi _{1}=\xi _{1}^{k}\}.$$ Now since for $\xi $ on $C_{R}$ and $|F(\xi )$ $|<\frac{M}{|\xi |^{p}}$ \[6\] some $p>0$ and all $R>R_{0},$$$\ \lim\limits_{R\rightarrow \infty }\dint\limits_{C_{R}}e^{\xi _{1}t}F(\xi _{1})d\xi _{1}=0\text{ }for\text{ }t>0\ \ $$. so$$\dint\limits_{\Gamma _{R}}e^{\xi _{1}t}F(\xi _{1})d\xi _{1}=\dint\limits_{C_{R}}e^{\xi _{1}t}F(\xi _{1})d\xi _{1}+\dint\limits_{x_{1}-i_{1}R}^{x_{1}+i_{1}R}e^{\xi _{1}t}F(\xi _{1})d\xi _{1}=2\pi i_{1}\sum \func{Re}s\{e^{\xi _{1}t}F(\xi _{1}):\xi _{1}=\xi _{1}^{k}\}$$ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ .$ then for $R\rightarrow \infty $ we obtain $$\dint\limits_{x_{1}-i_{1}\infty }^{x_{1}+i_{1}\infty }e^{\xi _{1}t}F(\xi _{1})d\xi _{1}=2\pi i_{1}\sum \func{Re}s\{e^{\xi _{1}t}F(\xi _{1}):\xi _{1}=\xi _{1}^{k}\},t>0.$$ We first attend the right half plane D$_{1}=$Re$(\xi _{1})$ $k $ and $$\lim_{Re(\xi _{1})\longrightarrow \infty }F_{1}(\xi _{1})=0.$$ The inverse Laplace transform of $F_{1}(\xi _{1})$ will then a real valued function $$f(t)=\frac{1}{2\pi i_{1}}\dint\limits_{x_{1}-i_{1}\infty }^{x_{1}+i_{1}\infty }e^{\xi _{1}t}F_{1}(\xi _{1})d\xi _{1}\ \ ..............\ \ \ \ \ (2)$$where$\ \xi _{1}=x_{1}+i_{1}y_{1}\epsilon C_{_{1}}.$ In the right half plane  D$_{2}=$ Re$(\xi _{2})$ $k$ and $$\lim_{Re(\xi _{2})\longrightarrow \infty }F_{2}(\xi _{2})=0$$the inverse Laplace transform of $F_{2}(\xi _{2})$ will be $$f(t)=\frac{1}{2\pi i_{1}}\dint\limits_{x_{2}-i_{1}\infty }^{x_{2}+i_{1}\infty }e^{\xi _{2}t}F_{2}(\xi _{2})d\xi _{2},\xi _{2}=x_{2}+i_{1}y_{2}\epsilon C_{_{1}}..............(3)$$ Moreover in each case $f(t)$  is of exponential order k. Then $$\begin{aligned} f(t) &=&f(t)e_{1}+f(t)e_{2}=\frac{1}{2\pi i_{1}}\dint\limits_{D_{1}}e^{\xi _{1}t}F_{1}(\xi _{1})d\xi _{1}e_{1}+\frac{1}{2\pi i_{1}}\dint\limits_{D_{2}}e^{\xi _{2}t}F_{2}(\xi _{2})d\xi _{2}e_{2} \\ &=&\frac{1}{2\pi i_{1}}\dint\limits_{D=D_{1}\cup D_{2}}e^{\xi t}F(\xi )d\xi ..........(4)\end{aligned}$$where we use the fact that any real number $c$ can be written as $$c=c+i_{1}\cdot 0+i_{2}\cdot 0+i_{1}i_{2}\cdot 0=c_{1}e_{1}+c_{2}e_{2}.$$The bicomplex version of inverse Laplace transform thus can be defined as (4). Evidently, here also$$\lim_{Re(\xi _{1,2})\longrightarrow \infty }F(\xi )=0$$and $f(t)$ is of exponential order $k$ . Reversing this proces one can at once obtain f(t)  from the integration defined in (4). It guarantees the existance of inverse Laplace transform. Definition ---------- If $F(\xi )$ exists and is convergent in a region $D=D_{1}\cup D_{2}$ which are the right half planes $D_{1,2}=R(\xi _{1,2})>k$ together with$$\lim_{Re(\xi _{1,2})\longrightarrow \infty }F(\xi )=0$$ then the inverse Laplace transform of $F(\xi )$ can be defined as $$L^{-1}\{F(\xi )\}=\frac{1}{2\pi i_{1}}\dint\limits_{D=D_{1}\cup D_{2}}e^{\xi t}F(\xi )d\xi =f(t)$$The integral in each plane $D_{1}$and $D_{2}$ are taken along any straight line $R(\xi _{1,2})>k$ . As a result our object function $f(t)$ will be of exponential order $k$ ,in the principal value sense. Examples -------- - If we take $F(\xi )d\xi =\frac{1}{\xi }$, then it’s inverse Laplace transform is given by $$f(t)=\frac{1}{2\pi i_{1}}\dint\limits_{D=D_{1}\cup D_{2}}e^{\xi t}F(\xi )d\xi =\frac{1}{2\pi i_{1}}\dint\limits_{D_{1}}e^{\xi _{1}t}F_{1}(\xi _{1})d\xi _{1}e_{1}+\frac{1}{2\pi i_{1}}\dint\limits_{D_{2}}e^{\xi _{2}t}F_{2}(\xi _{2})d\xi _{2}e_{2}............(4)$$ $\ \ \ \ \ \ $ Now $$\frac{1}{2\pi i_{1}}\dint\limits_{D_{1}}e^{\xi _{1}t}F_{1}(\xi _{1})d\xi _{1}=\frac{1}{2\pi i_{1}}\dint\limits_{x_{1}-i_{1\infty }}^{x_{1}+i_{1\infty }}e^{\xi _{1}t}\frac{1}{\xi _{1}}d\xi _{1}=2\pi i_{1}\cdot 1=2\pi i_{1}$$ as $\xi _{1}=0$ is the only singular point therein,  so$$residue=\lim_{\xi _{1}\longrightarrow 0}(\xi -0)e^{\xi _{1}t}\frac{1}{\xi _{1}}=1.$$ In a similar way,$$\frac{1}{2\pi i_{1}}\dint\limits_{D_{2}}e^{\xi _{2}t}F_{2}(\xi _{2})d\xi _{2}=2\pi i_{1}$$  and those leads (4) to$$f(t)=e_{1}+e_{2}=1.$$ - In our procedure one may easily check a partial list....to name a few.... - $L^{-1}\{\frac{\omega }{\xi ^{2}+\omega ^{2}}\}=\sin \omega t,$ - $L^{-1}\{\frac{\xi }{\xi ^{2}+\omega ^{2}}\}=\cos \omega t,$ - $L^{-1}\{\frac{\xi +a}{(\xi +a)^{2}+\omega ^{2}}\}=e^{-at}\cos \omega t,$ - $L^{-1}\{\frac{\omega }{(\xi +a)^{2}+\omega ^{2}}\}=e^{-at}\sin \omega t.$ References: \[1\]C.Segre Math.Ann.:40,1892,pp:413 \[2\] N.Spampinato Ann.Math.Pura .Appl.:14,1936,pp:305 \[3\] W.R .Hamilton Lectures on quaternion Dublin:Hodges and Smith:1853 \[4\] A.Kumar,P.Kumar IJET : 3,2011,pp225. \[5\] Y.V.Sidorov,M.V.Fedoryuk,M.I.Shabunin Mir Publishers,Moscow: 1985. \[6\] Joel L.Schiff  Springer.

--- author: - 'Wei Gong$^\diamond$' - 'Hehu Xie$^\dag$' - 'Ningning Yan$^\ddag$' title: A multilevel correction method for optimal controls of elliptic equation --- [**Abstract:**]{} [[**Keywords:**]{} Optimal control problems, elliptic equation, control constraints, finite element method, multilevel correction method ]{} [**Subject Classification**]{}: 49J20, 49K20, 65N15, 65N30 Introduction ============ Optimal control problems [@Hinze09book; @Lions; @LiuYan08book] play a very important role in modern sciences and industries, and have many applications in such as chemical process, fluid dynamics, medicine, economics and so on. The finite element method is among the most important and popular numerical methods for solving control problems governed by partial differential equations. So far there have existed much work on the finite element method for the optimal control problems. The interested readers are referred to [@Falk; @Geveci; @LiuYan01SINUM; @LiuYan01ACM; @LiuYan08book] and books and papers cited therein. As we know, the control problems governed by partial differential equations [@Hinze09book; @Lions; @LiuYan08book] are generally nonlinear and result in large scale optimization problems which bring much more difficulties to design efficient solvers. It is also well known that the multigrid or multilevel method is the optimal solver for many partial differential equations discretized by the finite element method, finite difference method and so on (see, e.g., [@Shaidurovbook]). Naturally, it is an important issue how to construct the multilevel type numerical method for the optimal control problems governed by partial differential equations. So far, there is only few work in this direction, we refer to [@Borzi09SIRE] for an overview. Since the classical multilevel or multigrid method for the optimal control problem is designed to solve the linear algebraic systems formulated on each step of the optimization algorithm, it is not so easy to give the analysis on optimal error estimates with the optimal computational complexity [@Borzi09SIRE]. The aim of this paper is to propose a multilevel correction method for the optimal control problems governed by partial differential equations based on the multilevel correction idea introduced in [@LinXie; @Xie_IMA; @Xie_JCP]. In this method, solving the control problem will not be more difficult than solving the corresponding linear boundary value problems. The multilevel correction method for the control problem is based on a series of nested finite element spaces with different level of accuracy which can be built with the same way as the multilevel method for boundary value problems. The multilevel correction scheme can be described as follows: (1) solve the control problem in an initial coarse finite element space; (2) use the multigrid method to solve two additional linear boundary value problems which are constructed by using the previous obtained state, adjoint state and control approximations; (3) solve a control problem again on the finite element space which is constructed by combining the coarsest finite element space with the obtained state and adjoint state approximations in step (2). Then go to step (2) for the next loop until stop. In this method, we replace solving control problem on the finest finite element space by solving a series of linear boundary value problems with multigrid scheme in the corresponding series of finite element spaces and a series of control problems in the coarsest finite element space. The corresponding error and computational work estimates of the proposed multilevel correction scheme for the control problem will also be analyzed. Based on the analysis, the proposed method can obtain optimal errors with an almost optimal computational complexity. So our proposed multilevel correction method can improve the overall efficiency for solving the control problem as it does for linear boundary value problems. An outline of the paper goes as follows. In Section 2, we introduce the finite element method for the optimal control problem. The multilevel correction method for the control problem is given in Sections 3. In Section 4, we extend the multilevel correction method to the optimal control problems governed by semilinear elliptic equation. Section 5 is devoted to providing the numerical results to validate the efficiency of the proposed numerical scheme. Some concluding remarks are given in the last section. Finite element method for optimal control problem ================================================= Let $\Omega\subset \mathbb{R}^{d}$, $d=2,3$ be a bounded and convex polygonal or polyhedral domain. Let $\|\cdot\|_{m,s,\Omega}$ and $\|\cdot\|_{m,\Omega}$ be the usual norms of the Sobolev spaces $W^{m,s}(\Omega)$ and $H^m(\Omega)$ respectively. Let $|\cdot|_{m,s,\Omega}$ and $|\cdot|_{m,\Omega}$ be the usual seminorms of the above-mentioned two spaces respectively. In this section, we introduce the finite element method for the optimal control problem constrained by elliptic equations. The corresponding a priori error estimates will also be given. At first we consider the following linear-quadratic optimal control problem: $$\begin{aligned} \label{OCP} \min\limits_{u\in U_{ad}}\ \ J(y,u)={1\over 2}\|y-y_d\|_{0,\Omega}^2 + \frac{\alpha}{2}\|u\|_{0,\Omega}^2\end{aligned}$$ subject to $$\label{OCP_state} \left\{\begin{array}{llr} -\Delta y=f+u \ \ &\mbox{in}\ \Omega, \\ \ \ \ \ \ y=0 \ \ \ &\mbox{on}\ \partial\Omega. \end{array} \right.$$ The admissible control set is of box type: $$\begin{aligned} U_{ad}:=\Big\{u\in L^2(\Omega):\ a(x)\leq u(x)\leq b(x)\ \ &\mbox{a.e.\ in}\ \Omega\Big\}\label{control_set}\end{aligned}$$ with $a(x)< b(x)$ for a.e. $x\in\Omega$. We require $a,b\in L^\infty(\Omega)$. Since the state equation (\[OCP\_state\]) is affine linear with respect to the control $u$, we can introduce a linear operator $S:L^2(\Omega)\rightarrow H_0^1(\Omega)$ such that $y=Su+y_f$, where $y_f\in H_0^1(\Omega)$ is the solution of (\[OCP\_state\]) corresponding to the right hand side $f$. Then standard elliptic regularity theory gives $y\in H^2(\Omega)$. With this notation we can formulate a reduced optimization problem $$\begin{aligned} \label{OCP_Operator} \min\limits_{u\in U_{ad}} \hat J(u):=J(Su,u)={1\over 2}\|Su+y_f-y_d\|_{0,\Omega}^2 + \frac{\alpha}{2}\|u\|_{0,\Omega}^2.\end{aligned}$$ Since the above optimization problem is linear and strictly convex, there exists a unique solution $u\in U_{ad}$ (see [@Lions]). Moreover, the first order necessary and sufficient optimality condition can be stated as follows: $$\begin{aligned} \label{reduced_opt} J'(u)(v-u)=(\alpha u+S^*(Su+y_f-y_d),v-u)\geq 0, \ \ \ \ \ \forall v\in U_{ad},\end{aligned}$$ where $S^*$ is the adjoint of $S$ ([@Hinze09book]). Introducing the adjoint state $p=S^*(Su+y_f-y_d)\in H_0^1(\Omega)$, we are led to the following optimality condition $$\label{OCP_OPT} \left\{\begin{array}{llr}a(y,v)=(f+u,v),\ \ &\forall v\in H_0^1(\Omega),\\ a(w,p)=(y-y_d,w),\ \ &\forall w\in H_0^1(\Omega),\\ (\alpha u+p,v-u)\geq 0, \ \ &\forall v\in U_{ad}, \end{array} \right.$$ where we use the standard notations $$\begin{aligned} a(y,v):=\int_\Omega\nabla y\nabla vdx,\ \ &\forall y,v\in H_0^1(\Omega).\nonumber\\ (y,v):=\int_\Omega yvdx,\ \ &\forall y,v\in L^2(\Omega).\nonumber\end{aligned}$$ Hereafter, we call $u$, $y$ and $p$ the optimal control, state and adjoint state, respectively. With the admissible control set (\[control\_set\]) we can get the explicit representation of the optimal control $u$ through the adjoint state $p$ $$\begin{aligned} u(x)=P_{U_{ad}}\big\{-\frac{1}{\alpha}p(x)\big\},\label{p_to_u}\end{aligned}$$ where $P_{U_{ad}}$ is the orthogonal projection operator onto $U_{ad}$. Let $\mathcal{T}_h$ be a regular and quasi-uniform triangulation of $\Omega$ such that $\bar\Omega=\cup_{\tau\in\mathcal{T}_h}\bar\tau$. On $\mathcal{T}_h$ we construct the piecewise linear and continuous finite element space $V_{h}$ such that $V_h\subset C(\bar\Omega)\cap H_0^1(\Omega)$. Based on the finite element space $V_h$, we can define the finite dimensional approximation to the optimal control problem (\[OCP\])-(\[OCP\_state\]) as follows: Find $(\bar{u}_h,\bar{y}_h,\bar{p}_h)\in U_{ad}\times V_h\times V_h$ such that $$\begin{aligned} \label{OCP_h} \min\limits_{\bar u_h\in U_{ad}}J_h(\bar y_h,\bar u_h)={1\over 2}\|\bar y_h-y_d\|_{0,\Omega}^2 + \frac{\alpha}{2}\|\bar u_h\|_{0,\Omega}^2\end{aligned}$$ subject to $$\begin{aligned} \label{OCP_state_h} a(\bar y_h,v_h)=(f+\bar u_h,v_h),\ \ \ \ \forall v_h\in V_h.\end{aligned}$$ In this paper, we use the piecewise linear finite element to approximate the state $y$, and variational discretization for the optimal control $u$ (see [@Hinze05COAP]). Similar to the infinite dimensional problem (\[OCP\])-(\[OCP\_state\]), the above discretized optimization problem also admits a unique solution $\bar{u}_h\in U_{ad}$. The discretized first order necessary and sufficient optimality condition can be stated as follows: $$\label{OCP_OPT_h} \left\{\begin{array}{llr}a(\bar y_h,v_h)=(f+\bar u_h,v_h),\ \ &\forall v_h\in V_h,\\ a(w_h,\bar p_h)=(\bar y_h-y_d,w_h),\ \ &\forall w_h\in V_h,\\ (\alpha \bar u_h+\bar p_h,v_h-\bar u_h)\geq 0, \ \ \ &\forall v_h\in U_{ad}. \end{array} \right.$$ The above optimization problem can be solved by projected gradient method or semi-smooth Newton method, see [@Hintermueller03SIOPT], [@Hinze09book], [@Vierling] and [@LiuYan08book] for more details. Now we state the following error estimate results for the finite element approximation of the control problem and the proof can be found in [@Hinze05COAP]. \[Thm:2.1\] Let $(u,y,p)\in U_{ad}\times H_0^1(\Omega)\times H_0^1(\Omega)$ and $(\bar u_h,\bar y_h,\bar p_h)\in U_{ad}\times V_h\times V_h$ be the solutions of problems (\[OCP\])-(\[OCP\_state\]) and (\[OCP\_h\])-(\[OCP\_state\_h\]), respectively. Then the following error estimates hold $$\begin{aligned} \label{u_error} \|u-\bar u_h\|_{0,\Omega}+\|y-\bar y_h\|_{0,\Omega} +\|p-\bar p_h\|_{0,\Omega}\leq Ch^{2}.\end{aligned}$$ Multilevel correction method for optimal control problems ========================================================= In this section, we propose a type of multilevel correction method for the optimal control problem (\[OCP\_h\])-(\[OCP\_state\_h\]). In this scheme, solving the optimization problem on the finest finite element spaces is transformed to a series of solutions of linear boundary value problems by the multigrid method on multilevel meshes and a series of solutions of optimization problems on the coarsest finite element space. In order to introduce the multilevel correction scheme, we define a sequence of triangulations $\mathcal{T}_{h_k}$ of $\Omega$ determined as follows. Suppose a very coarse mesh $\mathcal{T}_{H}$ is given and let $\mathcal{T}_{h_k}$ be obtained from $\mathcal{T}_{h_{k-1}}$ via regular refinement (produce $\beta^d$ subelements) such that $$h_k\approx\frac{1}{\beta}h_{k-1}$$ for $k=1,\cdots, n$ and $\mathcal{T}_{h_0}:=\mathcal{T}_H$. Here $\beta\geq 2$ is a positive integer. Let $V_H$ denote the coarsest linear finite element space defined on the coarsest mesh $\mathcal{T}_H$. Besides, we construct a series of finite element spaces $V_{h_1}$, $V_{h_2}$, $\cdots$, $V_{h_n}$ defined on the corresponding series of multilevel meshes $\mathcal{T}_{h_k}$ ($k=1,2,\cdots,n$) such that $V_H\subseteq V_{h_1}\subset V_{h_2}\subset\cdots\subset V_{h_n}$. In order to design the multilevel correction method for the optimization problem, we first introduce an one correction step which can improve the accuracy of the given numerical approximations for the state, adjoint state and optimal control. This correction step contains solving two linear boundary value problems with multigrid method in the finer finite element space and an optimization problem on the coarsest finite element space. Assume that we have obtained an approximate solution $(u_{h_{k}},y_{h_{k}},p_{h_{k}})\in U_{ad}\times V_{h_{k}}\times V_{h_{k}}$ on the $k$-th level mesh $\mathcal{T}_{h_k}$. Now we introduce an one correction step to improve the accuracy of the current approximation $(u_{h_{k}},y_{h_{k}},p_{h_{k}})$. \[Alg:3.1\] One correction step: 1. Find $y_{h_{k+1}}^*\in V_{h_{k+1}}$ such that $$\label{step_2} a(y^*_{h_{k+1}},v_{h_{k+1}}) = (f+u_{h_{k}},v_{h_{k+1}}),\ \ \ \ \forall\ v_{h_{k+1}}\in V_{h_{k+1}}.$$ Solve the above equation with multigrid method to obtain an approximation $\hat y_{h_{k+1}}\in V_{h_{k+1}}$ with error $\|\hat y_{h_{k+1}}-y_{h_{k+1}}^*\|_{1,\Omega}\leq Ch_{h_k}^{2}$ and define $\hat{y}_{h_{k+1}}:=MG(V_{h_{k+1}},u_{h_k})$. 2. Find $p_{h_{k+1}}^*\in V_{h_{k+1}}$ such that $$\label{step_3} a(w_{h_{k+1}},p^*_{h_{k+1}}) = (\hat y_{h_{k+1}}-y_d,v_{h_{k+1}}),\ \ \ \ \forall\ v_{h_{k+1}}\in V_{h_{k+1}}.$$ Solve the above equation with multigrid method to obtain an approximation $\hat p_{h_{k+1}}\in V_{h_{k+1}}$ with error $\|\hat p_{h_{k+1}}-p_{h_{k+1}}^*\|_{1,\Omega}\leq Ch_{h_k}^{2}$ and define $\hat{p}_{h_{k+1}}:=MG(V_{h_{k+1}},\hat y_{h_{k+1}})$. 3. Define a new finite element space $V_{H,h_{k+1}}:=V_H+{\rm span}\{\hat y_{h_{k+1}}\}+{\rm span}\{\hat p_{h_{k+1}}\}$ and solve the following optimal control problem: $$\begin{aligned} \label{step_4} \min\limits_{u_{h_{k+1}}\in U_{ad},\ y_{h_{k+1}}\in V_{H,h_{k+1}}} J(y_{h_{k+1}},u_{h_{k+1}})={1\over 2}\|y_{h_{k+1}}-y_d\|_{0,\Omega}^2 + \frac{\alpha}{2}\|u_{h_{k+1}}\|_{0,\Omega}^2\end{aligned}$$ subject to $$\label{step_4_state} a(y_{h_{k+1}},v_{H,h_{k+1}}) = (f+u_{h_{k+1}},v_{H,h_{k+1}}), \ \ \ \ \forall\ v_{H,h_{k+1}}\in V_{H,h_{k+1}}.$$ The corresponding optimality condition reads: Find $(u_{h_{k+1}},y_{h_{k+1}},p_{h_{k+1}}) \in U_{ad}\times V_{H,h_{k+1}}\times V_{H,h_{k+1}}$ such that $$\label{step_4_OPT} \left\{\begin{array}{llr} a(y_{h_{k+1}},v_{H,h_{k+1}}) = (f+u_{h_{k+1}},v_{H,h_{k+1}}), \ \ &\forall\ v_{H,h_{k+1}}\in V_{H,h_{k+1}},\\ a(v_{H,h_{k+1}},p_{h_{k+1}}) = (y_{h_{k+1}}-y_d,v_{H,h_{k+1}}), \ \ &\forall\ v_{H,h_{k+1}}\in V_{H,h_{k+1}},\\ (\alpha u_{h_{k+1}}+p_{h_{k+1}},v-u_{h_{k+1}})\geq 0, \ \ &\forall\ v\in U_{ad}. \end{array} \right.$$ We define the output of above algorithm as $$\begin{aligned} \label{correction} (u_{h_{k+1}},y_{h_{k+1}},p_{h_{k+1}}) =\mbox{\rm Correction}(V_H,u_{h_{k}},y_{h_k},p_{h_k},V_{h_{k+1}}),\end{aligned}$$ where $V_H$ denotes the coarsest finite element space, $(u_{h_{k}}, y_{h_k}, p_{h_k})$ is the given approximation of the optimal control, the state and the adjoint state in the coarse finite element space $V_{h_k}$ and $V_{h_{k+1}}$ denotes the finer finite element space. In Algorithm \[Alg:3.1\], one needs to solve an optimization problem (\[step\_4\])-(\[step\_4\_state\]) on the finite element space $V_{H,h_{k+1}}$, there are several ways to provide a good initial guess for the optimization algorithm which may speed up the convergence. One option is to use $u_{h_{k}}$ as initial guess, while the other choice is $P_{U_{ad}}\{-\frac{1}{\alpha}\hat p_{h_{k+1}}\}$. In the following of this paper, we denote $(\bar u_{h_{k+1}},\bar y_{h_{k+1}},\bar p_{h_{k+1}})\in U_{ad} \times V_{h_{k+1}}\times V_{h_{k+1}}$ the finite element solution to the discrete optimal control problems (\[OCP\_h\])-(\[OCP\_state\_h\]) in the finite element space $V_{h_{k+1}}$. We are able to analyze the error estimates between solutions $(\bar u_{h_{k+1}},\bar y_{h_{k+1}},\bar p_{h_{k+1}})$ and the correction one $(u_{h_{k+1}},y_{h_{k+1}},p_{h_{k+1}})$ on mesh level $\mathcal{T}_{h_{k+1}}$. \[Thm:3.1\] Let $(\bar u_{h_{k+1}},\bar y_{h_{k+1}}, \bar p_{h_{k+1}})\in U_{ad}\times V_{h_{k+1}}\times V_{h_{k+1}}$ be the solution of problems (\[OCP\_h\])-(\[OCP\_state\_h\]) and $(u_{h_{k+1}},y_{h_{k+1}},p_{h_{k+1}})$ be the numerical approximation by Algorithm \[Alg:3.1\], respectively. Assume there exists a real number $\eta_{h_k}$ such that $(u_{h_k}, y_{h_k}, p_{h_k})$ have the following error estimates $$\begin{aligned} \label{assume_error} \|\bar u_{h_k}-u_{h_k}\|_{0,\Omega}+\|\bar y_{h_k}-y_{h_k}\|_{0,\Omega} +\|\bar p_{h_k}-p_{h_k}\|_{0,\Omega} = \eta_{h_k}.\end{aligned}$$ Then the following error estimates hold $$\begin{aligned} \label{correct_error} \|\bar u_{h_{k+1}}-u_{h_{k+1}}\|_{0,\Omega} +\|\bar y_{h_{k+1}}-y_{h_{k+1}}\|_{0,\Omega} +\|\bar p_{h_{k+1}}-p_{h_{k+1}}\|_{0,\Omega} \leq C\eta_{h_{k+1}},\end{aligned}$$ where $\eta_{h_{k+1}}=H(h_{k}^{2}+\eta_{h_{k}})$. Note that $$\begin{aligned} a(\bar y_{h_{k+1}},v_{h_{k+1}})=(f+\bar u_{h_{k+1}},v_{h_{k+1}}),\ \ \ \ \forall v_{h_{k+1}}\in V_{h_{k+1}},\end{aligned}$$ we conclude from (\[step\_2\]) that $$\begin{aligned} a(\bar y_{h_{k+1}}-y^*_{h_{k+1}},v_{h_{k+1}}) =(\bar u_{h_{k+1}}-u_{h_{k}},v_{h_{k+1}}),\ \ \ \ \forall v_{h_{k+1}}\in V_{h_{k+1}},\end{aligned}$$ which implies $$\begin{aligned} \|\bar y_{h_{k+1}}-y^*_{h_{k+1}}\|_{1,\Omega} &\leq& C\|\bar u_{h_{k+1}}-u_{h_{k}}\|_{0,\Omega}\nonumber\\ &\leq& C\|\bar u_{h_{k+1}}-\bar u_{h_{k}}\|_{0,\Omega} +C\|\bar u_{h_{k}}-u_{h_{k}}\|_{0,\Omega}\nonumber\\ &\leq& C(h_k^{2}+\eta_{h_k}).\end{aligned}$$ Note that $\hat y_{h_{k+1}}$ is obtained by the multigrid method with estimate $\|\hat y_{h_{k+1}}-y_{h_{k+1}}^*\|_{1,\Omega}\leq Ch_{h_k}^{2}$, triangle inequality yields $$\begin{aligned} \label{correct_error_2} \|\bar y_{h_{k+1}}-\hat y_{h_{k+1}}\|_{1,\Omega}&\leq& \|\bar y_{h_{k+1}}-y^*_{h_{k+1}}\|_{1,\Omega} +\|y^*_{h_{k+1}}-\hat y_{h_{k+1}}\|_{1,\Omega}\nonumber\\ &\leq& C(h_k^{2}+\eta_{h_k}).\end{aligned}$$ Similarly, we can prove $$\begin{aligned} \label{correct_error_3} \|\bar p_{h_{k+1}}-\hat p_{h_{k+1}}\|_{1,\Omega} &\leq&C(h_k^{2}+\eta_{h_k}).\end{aligned}$$ From (\[OCP\_OPT\_h\]) and (\[step\_4\_OPT\]), we have $$\begin{aligned} (\alpha \bar u_{h_{k+1}}+ \bar p_{h_{k+1}},v_{h_{k+1}}-\bar u_{h_{k+1}}) \geq 0,\ \ \ \ \forall v_{h_{k+1}}\in U_{ad}\end{aligned}$$ and $$\begin{aligned} (\alpha u_{h_{k+1}}+ p_{h_{k+1}},w_{h_{k+1}}-u_{h_{k+1}})\geq 0, \ \ \ \ \forall w_{h_{k+1}}\in U_{ad}.\end{aligned}$$ Setting $v_{h_{k+1}}=u_{h_{k+1}}$ and $w_{h_{k+1}}=\bar u_{h_{k+1}}$, adding the above two inequalities together we are led to $$\begin{aligned} \label{correct_error_4} &&\alpha\|\bar u_{h_{k+1}}-u_{h_{k+1}}\|_{0,\Omega}^2\nonumber\\ &\leq& (\bar u_{h_{k+1}}-u_{h_{k+1}},p_{h_{k+1}}-\bar p_{h_{k+1}})\nonumber\\ &=&\big(\bar u_{h_{k+1}}-u_{h_{k+1}},p_{h_{k+1}}-p_{h_{k+1}}(\bar y_{h_{k+1}})\big) +\big(\bar u_{h_{k+1}}-u_{h_{k+1}},p_{h_{k+1}}(\bar y_{h_{k+1}})-\bar p_{h_{k+1}}\big)\nonumber\\ &=&a\big(y_{h_{k+1}}(\bar u_{h_{k+1}})-y_{h_{k+1}},p_{h_{k+1}}-p_{h_{k+1}}(\bar y_{h_{k+1}})\big)\nonumber\\ &&\ \ \ \ +\big(\bar u_{h_{k+1}}-u_{h_{k+1}},p_{h_{k+1}}(\bar y_{h_{k+1}})-\bar p_{h_{k+1}}\big)\nonumber\\ &=&\big(y_{h_{k+1}}(\bar u_{h_{k+1}})-y_{h_{k+1}},y_{h_{k+1}}-\bar y_{h_{k+1}}\big)\nonumber\\ &&\ \ \ \ +\big(\bar u_{h_{k+1}}-u_{h_{k+1}},p_{h_{k+1}}(\bar y_{h_{k+1}})-\bar p_{h_{k+1}}\big),\end{aligned}$$ where $y_{h_{k+1}}(\bar u_{h_{k+1}})\in V_{H,h_{k+1}}$ and $p_{h_{k+1}}(\bar y_{h_{k+1}})\in V_{H,h_{k+1}}$ satisfy $$\begin{aligned} a(y_{h_{k+1}}(\bar u_{h_{k+1}}),v_{H,h_{k+1}})=(f+\bar u_{h_{k+1}},v_{H,h_{k+1}}), \ \ \ \ \forall v_{H,h_{k+1}}\in V_{H,h_{k+1}}\end{aligned}$$ and $$\begin{aligned} a(w_{H,h_{k+1}},p_{h_{k+1}}(\bar y_{h_{k+1}}))=(\bar y_{h_{k+1}}-y_d,w_{H,h_{k+1}}), \ \ \ \ \forall w_{H,h_{k+1}}\in V_{H,h_{k+1}}.\end{aligned}$$ Then triangle inequality and $\epsilon$-Young inequality yield $$\begin{aligned} \label{correct_error_5} &&\|\bar u_{h_{k+1}}-u_{h_{k+1}}\|_{0,\Omega}+ \|\bar y_{h_{k+1}}-y_{h_{k+1}}\|_{0,\Omega}\nonumber\\ &\leq& C\big(\|y_{h_{k+1}}(\bar u_{h_{k+1}})-\bar y_{h_{k+1}}\|_{0,\Omega} +\|p_{h_{k+1}}(\bar y_{h_{k+1}})-\bar p_{h_{k+1}}\|_{0,\Omega}\big).\end{aligned}$$ It is easy to see that $y_{h_{k+1}}(\bar u_{h_{k+1}})$ is the finite element approximation to $\bar y_{h_{k+1}}$ on $V_{H,h_{k+1}}$ because of $V_{H,h_{k+1}}\subset V_{h_{k+1}}$. Standard Ceá-lemma implies (cf. [@Brenner]) $$\begin{aligned} \|y_{h_{k+1}}(\bar u_{h_{k+1}})-\bar y_{h_{k+1}}\|_{1,\Omega} &\leq& C\inf\limits_{v_{H,h_{k+1}}\in V_{H,h_{k+1}}}\|\bar y_{h_{k+1}}-v_{H,h_{k+1}}\|_{1,\Omega}\nonumber\\ &\leq&C\|\bar y_{h_{k+1}}-\hat y_{h_{k+1}}\|_{1,\Omega}\nonumber\\ &\leq& C(h_k^{2}+\eta_{h_k}).\label{correct_error_6}\end{aligned}$$ From the following equation $$\begin{aligned} a(y_{h_{k+1}}(\bar u_{h_{k+1}})-\bar y_{h_{k+1}},v_{H,h_{k+1}})=0, \ \ \ \ \forall v_{H,h_{k+1}}\in V_{H,h_{k+1}},\end{aligned}$$ and Aubin-Nitsche technique (cf. [@Brenner]), we are able to prove the improved $L^2$-norm estimate $$\begin{aligned} \label{correct_error_7} \|y_{h_{k+1}}(\bar u_{h_{k+1}})-\bar y_{h_{k+1}}\|_{0,\Omega}&\leq& CH\|y_{h_{k+1}}(\bar u_{h_{k+1}})-\bar y_{h_{k+1}}\|_{1,\Omega}\nonumber\\ &\leq&CH(h_k^{2}+\eta_{h_k}).\end{aligned}$$ Similarly to (\[correct\_error\_6\])-(\[correct\_error\_7\]), we can derive $$\begin{aligned} \label{correct_error_8} \|p_{h_{k+1}}(\bar y_{h_{k+1}})-\bar p_{h_{k+1}}\|_{0,\Omega} &\leq&CH(h_k^{2}+\eta_{h_k}).\end{aligned}$$ Combining (\[correct\_error\_5\]), (\[correct\_error\_7\]) and (\[correct\_error\_8\]) leads to the following estimate $$\begin{aligned} \label{correct_error_9} \|\bar u_{h_{k+1}}-u_{h_{k+1}}\|_{0,\Omega}+\|\bar y_{h_{k+1}}-y_{h_{k+1}}\|_{0,\Omega} &\leq&CH(h_k^{2}+\eta_{h_k}).\end{aligned}$$ Using the triangle inequality, we obtain $$\begin{aligned} \|\bar u_{h_{k+1}}-u_{h_{k+1}}\|_{0,\Omega}+\|\bar y_{h_{k+1}}-y_{h_{k+1}}\|_{0,\Omega} +\|\bar p_{h_{k+1}}-p_{h_{k+1}}\|_{0,\Omega} \leq CH(h_k^{2}+\eta_{h_k}),\end{aligned}$$ which is the desired result (\[correct\_error\]) and the proof is complete. Based on the sequence of nested finite element spaces $V_{h_1}\subset V_{h_2}\subset\cdots\subset V_{h_n}$ and the one correction step defined in Algorithm \[Alg:3.1\], we can define the following multilevel correction method to solve the optimal control problem: \[Alg:3.2\] A multilevel correction method for optimal control problem: 1. Solve an optimal control problem in the initial finite element space $V_{h_1}$: $$\begin{aligned} \label{step_1} \min\limits_{u_{h_1}\in U_{ad},y_{h_1}\in V_{h_1}} J(y_{h_1},u_{h_1})={1\over 2}\|y_{h_1}-y_d\|_{0,\Omega}^2 + \frac{\alpha}{2}\|u_{h_1}\|_{0,\Omega}^2\end{aligned}$$ subject to $$\label{step_1_state} a(y_{h_1},v_{h_1}) = (f+u_{h_1},v_{h_1}),\ \ \ \ \forall\ v_{h_1}\in V_{h_1}.$$ The corresponding optimality condition reads: Find $(u_{h_1},y_{h_1},p_{h_1})\in U_{ad}\times V_{h_1}\times V_{h_1}$ such that $$\label{step_1_OPT} \left\{\begin{array}{llr} a(y_{h_1},v_{h_1}) = (f+u_{h_1},v_{h_1}),\ \ &\forall\ v_{h_1}\in V_{h_1},\\ a(v_{h_1},p_{h_1}) = (y_{h_1}-y_d,v_{h_1}),\ \ &\forall\ v_{h_1}\in V_{h_1},\\ (\alpha u_{h_1}+p_{h_1},v-u_{h_1})\geq 0,\ \ &\forall\ v\in U_{ad}. \end{array} \right.$$ 2. Do $k=1$, $\cdots$, $n-1$ Obtain a new optimal solution $(u_{h_{k+1}},y_{h_{k+1}},p_{h_{k+1}}) \in U_{ad}\times V_{h_{k+1}}\times V_{h_{k+1}}$ by Algorithm \[Alg:3.1\] $$\begin{aligned} (u_{h_{k+1}},y_{h_{k+1}},p_{h_{k+1}})={\rm Correction}(V_H,u_{h_{k}},y_{h_k},p_{h_k},V_{h_{k+1}}).\end{aligned}$$ end Do Finally, we obtain a numerical approximation $(u_{h_n},y_{h_n}, p_{h_n}) \in U_{ad}\times V_{h_{n}}\times V_{h_n}$ for problem (\[OCP\])-(\[OCP\_state\]). Now we are in the position to give the error estimates for the solution generated by the above multilevel correction scheme described in Algorithm \[Alg:3.2\]. \[Thm:3.2\] Let $(u,y,p)\in U_{ad}\times H_0^1(\Omega)\times H_0^1(\Omega)$ and $(u_{h_n},y_{h_n},p_{h_n})\in U_{ad}\times V_{h_n}\times V_{h_n}$ be the solution of problems (\[OCP\])-(\[OCP\_state\]) and the solution by Algorithm \[Alg:3.2\], respectively. Assume the mesh size $H$ satisfies the condition $CH\beta^2<1$. Then the following error estimates hold $$\begin{aligned} \label{multigrid_error} \|\bar{u}_{h_n}-u_{h_n}\|_{0,\Omega}+\|\bar{y}_{h_n}-y_{h_n}\|_{0,\Omega} +\|\bar{p}_{h_n}-p_{h_n}\|_{0,\Omega}\leq Ch_n^{2}.\end{aligned}$$ Finally, we have the following error estimates $$\begin{aligned} \label{final_error} \|u-u_{h_n}\|_{0,\Omega}+\|y-y_{h_n}\|_{0,\Omega}+\|p-p_{h_n}\|_{0,\Omega}\leq Ch_n^{2}.\end{aligned}$$ Since we solve the optimal control problem directly in the first step of Algorithm \[Alg:3.2\], we have the following estimates $$\begin{aligned} \label{Estimate_h_1} \|\bar{u}_{h_1}-u_{h_1}\|_{0,\Omega}+\|\bar{y}_{h_1}-y_{h_1}\|_{0,\Omega}+\|\bar{p}_{h_1}-p_{h_1}\|_{0,\Omega}=0.\end{aligned}$$ From Theorem \[Thm:3.1\] and its proof, the following estimates for $(u_{h_2}, y_{h_2}, p_{h_2})$ hold $$\begin{aligned} \label{Estimate_h_2} \|\bar{u}_{h_2}-u_{h_2}\|_{0,\Omega}+\|\bar{y}_{h_2}-y_{h_2}\|_{0,\Omega} +\|\bar{p}_{h_2}-p_{h_2}\|_{0,\Omega}\leq CH h_1^{2}.\end{aligned}$$ Then based on Theorem \[Thm:3.1\], the condition $CH\beta^2<1$ and recursive argument, we have $$\begin{aligned} \eta_{h_n}&\leq& CH\big(h_{n-1}^{2}+\eta_{h_{n-1}}\big)\leq CH\big(h_{n-1}^{2}+CH(h_{n-2}^{2}+\eta_{h_{n-2}})\big)\nonumber\\ &\leq& \sum_{k=1}^{n-1}(CH)^{(n-k)}h_{k}^{2} \leq \Big(\sum_{k=1}^{n-1}(CH)^{(n-k)}\beta^{2(n-k)}\Big)h_{n}^{2} \nonumber\\ &=& \Big(\sum_{k=1}^{n-1}(CH\beta^2)^{(n-k)}\Big)h_{n}^{2} \leq \frac{CH\beta^2}{1-(CH\beta^2)}h_n^{2}\leq Ch_n^{2}.\end{aligned}$$ This is the desired result (\[multigrid\_error\]) and the estimate (\[final\_error\]) can be derived by combining (\[multigrid\_error\]) and (\[u\_error\]). Then the proof is complete. Now, we come to analyze the computational work for the multilevel correction scheme defined in Algorithm \[Alg:3.2\]. Since the linear boundary value problems (\[step\_2\]) and (\[step\_3\]) in Algorithm \[Alg:3.1\] are solved by multigrid method, the corresponding computational work is of optimal order. We define the dimension of each level linear finite element space as $$\begin{aligned} N_k := {\rm dim}\ V_{h_k},\ \ \ k=1,\cdots,n.\end{aligned}$$ Then the following relation holds $$\begin{aligned} \label{relation_dimension} N_k \thickapprox\Big(\frac{1}{\beta}\Big)^{d(n-k)}N_n,\ \ \ k=1,\cdots,n.\end{aligned}$$ The estimate of computational work for the second step in Algorithm \[Alg:3.1\] is different from the linear eigenvalue problems [@LinXie; @Xie_IMA; @Xie_JCP]. In this step, we need to solve a constrained optimization problem (\[step\_4\_OPT\]). Always, some types of optimization methods are used to solve this problem. In each iteration step, we need to evaluate the orthogonal projection in the finite element space $V_{H,h_k}$ ($k=2,\cdots,n$) onto $U_{ad}$ which needs work $\mathcal{O}(N_k)$. Fortunately, this step always can be carried out in the parallel way. \[Thm:linear\] Assume that we solve Algorithm \[Alg:3.2\] with $m$ processors parallely, the optimization problem solving in the coarse spaces $V_{H,h_k}$ ($k=1,\cdots, n$) and $V_{h_1}$ need work $\mathcal{O}(M_H)$ and $\mathcal{O}(M_{h_1})$, respectively, and the work of multigrid method for solving the boundary value problems in $V_{h_k}$ is $\mathcal{O}(N_k)$ for $k=2,3,\cdots,n$. Let $\varpi$ denote the iteration number of the optimization algorithm when we solve the optimization problem (\[step\_4\_OPT\]) in the coarse space. Then in each computational processor, the work involved in Algorithm \[Alg:3.2\] has the following estimate $$\begin{aligned} \label{Computation_Work_Estimate} {\rm Total\ work}&=&\mathcal{O}\Big(\big(1+\frac{\varpi}{m}\big)N_n + M_H\log N_n+M_{h_1}\Big).\end{aligned}$$ In each computational processor, let $W_k$ denote the computational work for the correction step in the $k$-th finite element space $V_{h_k}$. Then from the description of Algorithm \[Alg:3.1\] we have $$\begin{aligned} \label{work_k} W_k&=&\mathcal{O}\left(N_k +M_H+\varpi\frac{N_k}{m}\right), \ \ \ \ {\rm for}\ k=2,\cdots,n.\end{aligned}$$ Iterating (\[work\_k\]) and using (\[relation\_dimension\]), we obtain $$\begin{aligned} \label{Work_Estimate} \text{Total work} &=& \sum_{k=1}^nW_k\nonumber = \mathcal{O}\left(M_{h_1}+\sum_{k=2}^n \Big(N_k + M_H+\varpi\frac{N_k}{m}\Big)\right)\nonumber\\ &=& \mathcal{O}\Big(\sum_{k=2}^n\Big(1+\frac{\varpi}{m}\Big)N_k + (n-1) M_H + M_{h_1}\Big)\nonumber\\ &=& \mathcal{O}\left(\sum_{k=2}^n \Big(\frac{1}{\beta}\Big)^{d(n-k)}\Big(1+\frac{\varpi}{m}\Big)N_n + M_H\log N_n+M_{h_1}\right)\nonumber\\ &=& \mathcal{O}\left(\big(1+\frac{\varpi}{m}\big)N_n + M_H\log N_n+M_{h_1}\right).\end{aligned}$$ This is the desired result and we complete the proof. Since we have a good enough initial solution $(\hat{y}_{h_{k+1}},\hat p_{h_{k+1}})$ in the second step of Algorithm \[Alg:3.1\], solving the optimization problem (\[step\_4\_OPT\]) always does not need too many iterations. Then the complexity in each computational node is always $\mathcal{O}(N_n)$ provided $M_H\ll N_n$ and $M_{h_1}\leq N_n$. Application to optimal controls of semilinear elliptic equation =============================================================== In this section, we will extend the multilevel correction method to optimal control problem governed by semilinear elliptic equation: $$\begin{aligned} \label{OCP_nonlinear} \min\limits_{u\in U_{ad}}\ \ J(y,u)={1\over 2}\|y-y_d\|_{0,\Omega}^2 + \frac{\alpha}{2}\|u\|_{0,\Omega}^2\end{aligned}$$ subject to $$\label{OCP_state_nonlinear} \left\{\begin{array}{llr} -\Delta y+\phi(\cdot,y)=f+u \ \ &\mbox{in}\ \Omega, \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ y=0 \ \ \ &\mbox{on}\ \partial\Omega, \end{array} \right.$$ where the function $\phi:\Omega\times\mathbb{R}\rightarrow \mathbb{R}$ is measurable with respect to $x\in \Omega$ for all $y\in \mathbb{R}$ and is of class $\mathcal{C}^2$ with respect to $y$, its first derivative with respect to $y$, denoted by $\phi'$ in this paper, is nonnegative for all $x\in \Omega$ and $y\in \mathbb{R}$. In the following, we will omit the first argument of $\phi(\cdot,y)$ and denote it by $\phi(y)$. For all $M>0$, we assume that there exists $C_M>0$ such that $$\begin{aligned} |\phi''(y_1)-\phi''(y_2)|\leq C_M|y_1-y_2|\nonumber\end{aligned}$$ for all $(y_1,y_2)\in [-M,M]^2$. It is well-known that the state equation (\[OCP\_state\_nonlinear\]) admits a unique solution $y\in H_0^1(\Omega)\cap L^\infty(\Omega)$ under the aforementioned conditions (see [@Arada]). Moreover, we have $y\in H_0^1(\Omega)\cap H^2(\Omega)$. Then we are able to introduce the control-to-state mapping $G: L^2(\Omega)\rightarrow H_0^1(\Omega)\cap L^\infty(\Omega)$, which leads to the reduced optimization problem $$\begin{aligned} \min\limits_{u\in U_{ad}}\ \ \hat J(u):= J(G(u),u).\label{reduced_semilinear}\end{aligned}$$ Similar to the linear case, it is easy to prove the existence of a solution to (\[OCP\_nonlinear\])-(\[OCP\_state\_nonlinear\]), see, e.g., [@Arada]. However, the uniqueness is generally not guaranteed. We can also derive the first order necessary optimality conditions as $$\begin{aligned} \label{OCP_OPT_nonlinear} \hat J'(u)(v-u)=(\alpha u+p,v-u)\geq 0,\ \ \ \forall v\in U_{ad},\end{aligned}$$ where the adjoint state $p\in H_0^1(\Omega)$ satisfies $$\label{adjoint_nonlinear} \left\{\begin{array}{llr} -\Delta p+\phi'(y)p=y-y_d \ \ &\mbox{in}\ \Omega,\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p=0 \ \ \ &\mbox{on}\ \partial\Omega. \end{array} \right.$$ Moreover, we assume the following second order sufficient optimality condition. \[Ass:ssc\] Let $u\in U_{ad}$ fulfil the first order necessary optimality conditions (\[OCP\_OPT\_nonlinear\]). We assume that there exists a constant $\gamma >0$ such that $$\begin{aligned} \hat J''(u)(v,v)\geq \gamma \|v\|_{0,\Omega}^2,\ \ \ \ \forall v\in L^2(\Omega).\nonumber\end{aligned}$$ We note that Assumption \[Ass:ssc\] is a rather strong second order sufficient optimality condition compared to the one presented in [@Arada], it is commonly used in the error estimates of nonlinear optimal control problems (see [@Kroner] and [@Neitzel]). For $u\in U_{ad}$ and $v_1,v_2\in L^2(\Omega)$, the second order derivative of $\hat J$ is given by (see [@Arada] and [@Hinze09book]) $$\begin{aligned} \hat J''(u)(v_1,v_2)=\int_\Omega(\alpha v_1v_2+ \tilde y_1\tilde y_2-p\phi''(y)\tilde y_1\tilde y_2)dx,\nonumber\end{aligned}$$ where $y=G(u)$, $\tilde y_i=G'(u)v_i$, $i=1,2$. Now we can show that the second order derivative of $\hat J$ is Lipschitz continuous in $L^2(\Omega)$ . \[La:4.2\] There exists a constant $C$ such that for all $u_1,u_2\in U_{ad}$ and $v\in L^2(\Omega)$ $$\begin{aligned} |\hat J''(u_1)(v,v)-\hat J'' (u_2)(v,v)|\leq C\|u_1-u_2\|_{0,\Omega}\|v\|_{0,\Omega}^2\nonumber\end{aligned}$$ holds. Let $y_i=G(u_i)$, $\tilde y_i=G'(u_i)v$, $i=1,2$, $p_1$ be the adjoint state associated with $u_1$ and $p_2$ be the adjoint state associated with $u_2$. Then from the definition of the second order derivative of $\hat J$ we have $$\begin{aligned} |\hat J''(u_1)(v,v)-\hat J'' (u_2)(v,v)|&=&\Big|\int_\Omega(\tilde y_1^2-\tilde y_2^2+p_2\phi''(y_2) \tilde y_2^2-p_1\phi''(y_1)\tilde y_1^2)dx\Big|\nonumber\\ &\leq&\int_\Omega|(\tilde y_1+\tilde y_2)(\tilde y_1-\tilde y_2)+(p_2-p_1)\phi''(y_1)\tilde y_1^2\nonumber\\ &&-p_2\phi''(y_2)(\tilde y_1^2-\tilde y_2^2)-p_2(\phi''(y_1)-\phi''(y_2))\tilde y_1^2|dx. \end{aligned}$$ This gives $$\begin{aligned} &&|\hat J''(u_1)(v,v)-\hat J'' (u_2)(v,v)|\nonumber\\ &\leq&(\|\tilde y_1\|_{0,\Omega}+\|\tilde y_2\|_{0,\Omega})\|\tilde y_1-\tilde y_2\|_{0,\Omega} +c\|\phi''(y_1)\|_{0,\infty,\Omega}\|p_2-p_1\|_{0,\Omega}\|\tilde y_1\|^2_{0,4,\Omega}\nonumber\\ &&+c\|p_2\|_{0,\infty,\Omega}\big(\|\phi''(y_2)\|_{0,\infty,\Omega}(\|\tilde y_1\|_{0,\Omega} +\|\tilde y_2\|_{0,\Omega})\|\tilde y_1-\tilde y_2\|_{0,\Omega}+\|y_1-y_2\|_{0,\Omega}\|\tilde y_1\|^2_{0,4,\Omega}\big).\nonumber\end{aligned}$$ It has been proved in [@Arada] that $$\begin{aligned} \|G(u)\|_{1,\Omega}\leq C\|u\|_{0,\Omega},\nonumber\\ \|G(u_1)-G(u_2)\|_{0,\Omega}\leq C\|u_1-u_2\|_{0,\Omega},\nonumber\\ \|G'(u_1)v-G'(u_2)v\|_{0,\Omega}\leq C\|u_1-u_2\|_{0,\Omega}\|v\|_{0,\Omega}.\nonumber\end{aligned}$$ Using the boundedness of $U_{ad}$ and $\phi''(\cdot)$, the embedding $H^1(\Omega)\hookrightarrow L^4(\Omega)$, we can obtain the desired result. \[La:4.3\] Let $u$ satisfy Assumption \[Ass:ssc\]. Then there exists an $\epsilon>0$ such that $$\begin{aligned} \hat J''(w)(v,v)\geq \frac{\gamma}{2}\|v\|_{0,\Omega}^2\nonumber\end{aligned}$$ holds for all $v\in L^2(\Omega)$ and $w\in U_{ad}$ with $\|w-u\|_{0,\Omega}\leq\epsilon$. From Assumption \[Ass:ssc\] and Lemma \[La:4.2\] we have $$\begin{aligned} \hat J''(w)(v,v)&=&\hat J''(u)(v,v)+\hat J''(w)(v,v)-\hat J''(u)(v,v)\nonumber\\ &\geq&\gamma\|v\|_{0,\Omega}^2-c\|w-u\|_{0,\Omega}\|v\|_{0,\Omega}^2\nonumber\\ &\geq&{\gamma\over 2}\|v\|_{0,\Omega}^2\nonumber\end{aligned}$$ with $\epsilon <{{\gamma}\over{2c} }$. This gives the result. With this estimate at hand we can prove the local convexity of the objective functional. \[La:4.4\] Let $u\in U_{ad}$ satisfy the first order necessary optimality condition (\[OCP\_OPT\_nonlinear\]) and Assumption \[Ass:ssc\]. Then there exist constants $\epsilon >0$ and $\gamma>0$ such that for all $v\in U_{ad}$ and $w\in U_{ad}$ satisfying $\|v-u\|_{0,\Omega}\leq \epsilon$ and $\|w-u\|_{0,\Omega}\leq \epsilon$, there holds $$\begin{aligned} {\gamma\over 2}\|v-w\|_{0,\Omega}^2\leq (\hat J'(v)-\hat J'(w),v-w).\nonumber\end{aligned}$$ We can conclude from Lemma \[La:4.3\] that for some $\theta\in [0,1]$ $$\begin{aligned} (\hat J'(v)-\hat J'(w),v-w)&=&\hat J''(\theta v+(1-\theta)w)(v-w,v-w)\nonumber\\ &\geq &{\gamma\over 2}\|v-w\|_{0,\Omega}^2,\nonumber\end{aligned}$$ this gives the desired result. Now we are ready to define the finite dimensional approximation to the optimal control problem (\[OCP\_nonlinear\])-(\[OCP\_state\_nonlinear\]): $$\begin{aligned} \label{OCP_nonlinear_h} \min\limits_{\bar u_h\in U_{ad}}J_h(\bar y_h,\bar u_h)={1\over 2}\|\bar y_h-y_d\|_{0,\Omega}^2 + \frac{\alpha}{2}\|\bar u_h\|_{0,\Omega}^2\end{aligned}$$ subject to $$\begin{aligned} a(\bar y_h,v_h)+(\phi(\bar y_h),v_h)=(f+\bar u_h,v_h),\ \ \ \ \forall v_h\in V_h. \label{OCP_state_nonlinear_h}\end{aligned}$$ Similar to the continuous case, we can define a discrete control-to-state mapping $G_h:L^2(\Omega)\rightarrow V_h$ and formulate a reduced discretised optimization problem $$\begin{aligned} \min\limits_{u_h\in U_{ad}}\hat J_h(u_h):=J_h(G_h(u_h),u_h).\nonumber\end{aligned}$$ The above discretised optimization problem admits at least one solution. The discretised first order necessary optimality condition can be stated as follows: $$\label{OCP_OPT_nonlinear_h} \left\{\begin{array}{llr}a(\bar y_h,v_h)+(\phi(\bar y_h),v_h) =(f+\bar u_h,v_h),\ \ &\forall v_h\in V_h,\\ a(w_h,\bar p_h)+(\phi'(\bar y_h)\bar p_h,v_h) =(\bar y_h-y_d,w_h),\ \ &\forall w_h\in V_h,\\ (\alpha \bar u_h+\bar p_h,v_h-\bar u_h)\geq 0, \ \ &\forall v_h\in U_{ad}. \end{array} \right.$$ Similar to the proof in [@Arada] we can prove the following a priori error estimates \[La:4.6\] For $u,v\in L^2(\Omega)$, assume that $G(u)\in H_0^1(\Omega)$ and $G_h(u)\in V_h$ be the solutions of the continuous and discretised state equation, $G'(u)v\in H_0^1(\Omega)$ and $G_h'(u)v\in V_h$ be the solutions of the continuous and discretised linearized state equation, respectively. Then the following error estimates hold $$\begin{aligned} \|G(u)-G_h(u)\|_{0,\Omega}+h\|G(u)-G_h(u)\|_{1,\Omega}\leq Ch^2\|u\|_{0,\Omega},\label{non_error_1}\\ \|G'(u)v-G_h'(u)v\|_{0,\Omega}+h\|G'(u)v-G_h'(u)v\|_{1,\Omega}\leq Ch^{2}\|v\|_{0,\Omega}.\label{non_error_2}\end{aligned}$$ Now we can formulate the following coercivity of the second order derivative of the discrete reduced objective functional \[La:4.5\] Let $u$ satisfy Assumption \[Ass:ssc\]. Then there exists an $\epsilon>0$ such that $$\begin{aligned} \hat J_h''(w)(v,v)\geq \gamma\|v\|_{0,\Omega}^2\nonumber\end{aligned}$$ holds for all $v\in L^2(\Omega)$ and $w\in U_{ad}$ with $\|w-u\|_{0,\Omega}\leq\epsilon$. Let $y=G(w)$, $y_h=G_h(w)$, $\tilde y = G'(w)v$ and $\tilde y_h=G_h'(w)v$, $p$ and $p_h$ be the continuous and discrete adjoint states associated with $w$, respectively. Similar to the proof of Lemma \[La:4.2\], using the explicit representations of $\hat J$ and $\hat J_h$ we have $$\begin{aligned} |\hat J''(w)(v,v)-\hat J_h'' (w)(v,v)|&=&\Big|\int_\Omega(\tilde y^2-\tilde y_h^2 +p_h\phi''(y_h)\tilde y_h^2-p\phi''(y)\tilde y^2)dx\Big|\nonumber\\ &\leq&\int_\Omega|(\tilde y+\tilde y_h)(\tilde y-\tilde y_h)+(p_h-p)\phi''(y)\tilde y^2\nonumber\\ &&-p_h\phi''(y_h)(\tilde y^2-\tilde y_h^2)-p_h(\phi''(y)-\phi''(y_h))\tilde y^2|dx,\nonumber\end{aligned}$$ this together with Lemma \[La:4.6\], the boundedness of $\phi''(\cdot)$ and the embedding $H^1(\Omega)\hookrightarrow L^4(\Omega)$ gives $$\begin{aligned} \label{Estimate_Semilinear_1} &&|\hat J''(w)(v,v)-\hat J_h'' (w)(v,v)|\nonumber\\ &\leq&(\|\tilde y\|_{0,\Omega}+\|\tilde y_h\|_{0,\Omega})\|\tilde y-\tilde y_h\|_{0,\Omega} +c\|\phi''(y)\|_{0,\infty,\Omega}\|p_h-p\|_{0,\Omega}\|\tilde y\|^2_{0,4,\Omega}\nonumber\\ &&+c\|p_h\|_{0,\infty,\Omega}\big(\|\phi''(y_h)\|_{0,\infty,\Omega}(\|\tilde y\|_{0,\Omega} +\|\tilde y_h\|_{0,\Omega})\|\tilde y-\tilde y_h\|_{0,\Omega}+\|y-y_h\|_{0,\Omega}\|\tilde y\|^2_{0,4,\Omega}\big)\nonumber\\ &\leq&Ch^2\|v\|_{0,\Omega}^2\nonumber\\ &\leq&{\gamma\over 2}\|v\|_{0,\Omega}^2\end{aligned}$$ for sufficiently small $h$. Combining (\[Estimate\_Semilinear\_1\]) with Lemma \[La:4.3\] we complete the proof. Now we are in the position to derive the a priori error estimates for the above finite element approximations \[Thm:priori\_semilinear\] Let $(u,y,p)\in U_{ad}\times H_0^1(\Omega)\times H_0^1(\Omega)$ and $(\bar u_h,\bar y_h,\bar p_h)\in U_{ad}\times V_h\times V_h$ be the solutions of problems (\[OCP\_nonlinear\])-(\[OCP\_state\_nonlinear\]) and (\[OCP\_OPT\_nonlinear\_h\]), respectively. Then the following error estimates hold $$\begin{aligned} \label{u_error_nonlinear} \|u-\bar u_h\|_{0,\Omega}+\|y-\bar y_h\|_{0,\Omega} +\|p-\bar p_h\|_{0,\Omega}\leq Ch^{2}.\end{aligned}$$ At first, from Proposition 4.3 and Theorem 4.4 in [@Arada] one can prove that $\bar u_h$ converges strongly to $u$. Then, from Lemma \[La:4.4\] we have $$\begin{aligned} {\gamma\over 2}\|u-\bar u_h\|_{0,\Omega}^2&\leq& (\hat J'(u)-\hat J'(\bar u_h), u-\bar u_h)\nonumber\\ &=&(\alpha u+p,u-\bar u_h)-(\alpha \bar u_h+ p(\bar u_h),u-\bar u_h)\nonumber\\ &\leq&(\alpha \bar u_h+ \bar p_h,\bar u_h-u)+(p(\bar u_h)-\bar p_h,\bar u_h-u)\nonumber\\ &\leq&C(\gamma)\|p(\bar u_h)-\bar p_h\|_{0,\Omega}^2+{\gamma\over 4}\|\bar u_h-u\|_{0,\Omega}^2,\label{Thm4.1-1}\end{aligned}$$ where $p(\bar u_h)\in H_0^1(\Omega)$ is the solution of the following systems $$\begin{aligned} a(y(\bar u_h),v)+(\phi(y(\bar u_h)),v)=(f+\bar u_h,v),\ \ \forall v\in H_0^1(\Omega),\nonumber\\ a(v, p(\bar u_h))+(\phi'(y(\bar u_h))p(\bar u_h),v)=(y(\bar u_h)-y_d,v),\ \ \forall v\in H_0^1(\Omega).\nonumber\end{aligned}$$ Now it remains to estimate $\|p(\bar u_h)-\bar p_h\|_{0,\Omega}$. We have the splitting $$\begin{aligned} \|p(\bar u_h)-\bar p_h\|_{0,\Omega}\leq \|p(\bar u_h)-p(\bar y_h)\|_{0,\Omega}+\|p(\bar y_h)-\bar p_h\|_{0,\Omega}\nonumber\end{aligned}$$ with $p(\bar y_h)\in H_0^1(\Omega)$ the solution of the following equation $$\begin{aligned} a(v, p(\bar y_h))+(\phi'(\bar y_h)p(\bar y_h),v)=(\bar y_h-y_d,v),\ \ \forall v\in H_0^1(\Omega).\nonumber\end{aligned}$$ Because $\phi'(\cdot)\geq 0$ and $\phi'$ is Lipschitz continuous, setting $v=p(\bar u_h)-p(\bar y_h)$ we have $$\begin{aligned} \|p(\bar u_h)-p(\bar y_h)\|_{1,\Omega}^2&\leq& a(p(\bar u_h)-p(\bar y_h),p(\bar u_h)-p(\bar y_h))\nonumber\\ &=&(y(\bar u_h)-\bar y_h,v)-(\phi'(y(\bar u_h))p(\bar u_h)-\phi'(\bar y_h)p(\bar y_h),v)\nonumber\\ &=&(y(\bar u_h)-\bar y_h,v)-((\phi'(y(\bar u_h))-\phi'(\bar y_h))p(\bar u_h),v)\nonumber\\ &&-(\phi'(\bar y_h)(p(\bar u_h)-p(\bar y_h)),v)\nonumber\\ &\leq&(y(\bar u_h)-\bar y_h,v)-((\phi'(y(\bar u_h))-\phi'(\bar y_h))p(\bar u_h),v)\nonumber\\ &\leq&\|y(\bar u_h)-\bar y_h\|_{0,\Omega}\|v\|_{0,\Omega}+C\|y(\bar u_h)-\bar y_h\|_{0,\Omega} \|p(\bar u_h)\|_{0,4,\Omega}\|v\|_{0,4,\Omega}\nonumber\\ &\leq&C\|y(\bar u_h)-\bar y_h\|_{0,\Omega}(1+\|p(\bar u_h)\|_{1,\Omega})\|v\|_{1,\Omega},\label{Thm4.1-2_new}\end{aligned}$$ where we used the Sobolev embedding theorem in the last inequality. Collecting the above estimates we arrive at $$\begin{aligned} \|u-\bar u_h\|_{0,\Omega}\leq C\|y(\bar u_h)-\bar y_h\|_{0,\Omega}+C\|p(\bar y_h)-\bar p_h\|_{0,\Omega}.\label{Thm4.1-3}\end{aligned}$$ Note that $\bar y_h$ and $\bar p_h$ are the standard finite element approximations of $y(\bar u_h)$ and $p(\bar y_h)$, respectively. Standard error estimates (cf. [@Arada]) yield $$\begin{aligned} \|u-\bar u_h\|_{0,\Omega}\leq Ch^{2}.\label{Thm4.1-5}\end{aligned}$$ Similar to (\[Thm4.1-2\_new\]) we can prove that $$\begin{aligned} \|y-y(\bar u_h)\|_{0,\Omega}+\|p-p(\bar u_h)\|_{0,\Omega}\leq C\|u-\bar u_h\|_{0,\Omega}.\nonumber\end{aligned}$$ Then triangle inequality implies that $$\begin{aligned} \|y-\bar y_h\|_{0,\Omega}+\|p-\bar p_h\|_{0,\Omega}\leq Ch^{2}.\label{Thm4.1-6}\end{aligned}$$ This completes the proof. Assume that we have obtained the approximate solution $(u_{h_{k}},y_{h_{k}},p_{h_{k}})\in U_{ad}\times V_{H,h_{k}}\times V_{H,h_{k}}$ on the $k$-th level mesh $\mathcal{T}_{h_k}$. Now we introduce an one correction step to improve the accuracy of the current approximation $(u_{h_{k}},y_{h_{k}},p_{h_{k}})$. \[Alg:4.1\]One correction step: 1. Find $y_{h_{k+1}}^*\in V_{h_{k+1}}$ such that $$\label{step_2_semilinear} a(y^*_{h_{k+1}},v_{h_{k+1}}) = (f+u_{h_{k}},v_{h_{k+1}})-(\phi(y_{h_k}),v_{h_{k+1}}),\ \ \ \ \forall v_{h_{k+1}}\in V_{h_{k+1}}.$$ Solve the above equation with multigrid method to obtain an approximation $\hat y_{h_{k+1}}\in V_{h_{k+1}}$ with error $\|\hat y_{h_{k+1}}-y_{h_{k+1}}^*\|_{1,\Omega}\leq Ch_{h_{k}}^{2}$ and define $\hat y_{h_{k+1}}:=MG(V_{h_{k+1}},u_{h_k})$. 2. Find $p_{h_{k+1}}^*\in V_{h_{k+1}}$ such that $$\label{step_3_semilinear} a(w_{h_{k+1}},p^*_{h_{k+1}})+(\phi'(\hat y_{h_{k+1}})p^*_{h_{k+1}},v_{h_{k+1}}) = (\hat y_{h_{k+1}}-y_d,v_{h_{k+1}}),\ \ \ \ \forall v_{h_{k+1}}\in V_{h_{k+1}}.$$ Solve the above equation with multigrid method to obtain an approximation $\hat p_{h_{k+1}}\in V_{h_{k+1}}$ with error $\|\hat p_{h_{k+1}}-p_{h_{k+1}}^*\|_{1,\Omega}\leq Ch_{h_{k}}^{2}$ and define $\hat p_{h_{k+1}}:=MG(V_{h_{k+1}},\hat y_{h_{k+1}})$. 3. Define a new finite element space $V_{H,h_{k+1}}:=V_H+{\rm span}\{\hat y_{h_{k+1}}\}+{\rm span}\{\hat p_{h_{k+1}}\}$ and solve the following optimal control problem: $$\begin{aligned} \label{step_4_semilinear} \min\limits_{u_{h_{k+1}}\in U_{ad},\ y_{h_{k+1}} \in V_{H,h_{k+1}}} J(y_{h_{k+1}},u_{h_{k+1}})&={1\over 2}\|y_{h_{k+1}}-y_d\|_{0,\Omega}^2 + \frac{\alpha}{2}\|u_{h_{k+1}}\|_{0,\Omega}^2\end{aligned}$$ subject to $$\label{step_4_state_semilinear} a(y_{h_{k+1}},v_{H,h_{k+1}})+(\phi(y_{h_{k+1}}),v_{H,h_{k+1}}) = (f+u_{h_{k+1}},v_{H,h_{k+1}}),\ \ \ \ \forall v_{H,h_{k+1}}\in V_{H,h_{k+1}}.$$ The corresponding optimality condition reads: Find $(u_{h_{k+1}},y_{h_{k+1}},p_{h_{k+1}})\in U_{ad} \times V_{H,h_{k+1}}\times V_{H,h_{k+1}}$ such that $$\label{step_4_OPT_semilinear} \left\{\begin{array}{lll} a(y_{h_{k+1}},v_{H,h_{k+1}}) +(\phi(y_{h_{k+1}}),v_{H,h_{k+1}}) = (f+u_{h_{k+1}},v_{H,h_{k+1}}), &\forall v_{H,h_{k+1}}\in V_{H,h_{k+1}},\\ a(v_{H,h_{k+1}},p_{h_{k+1}}) +(\phi'(y_{h_{k+1}})p_{h_{k+1}},v_{H,h_{k+1}}) = (y_{h_{k+1}}-y_d,v_{H,h_{k+1}}), &\forall v_{H,h_{k+1}}\in V_{H,h_{k+1}},\\ (\alpha u_{h_{k+1}}+p_{h_{k+1}},v-u_{h_{k+1}})\geq 0, &\forall v\in U_{ad}. \end{array} \right.$$ We define the output of above algorithm as $$\begin{aligned} \label{correction_semilinear} (u_{h_{k+1}},y_{h_{k+1}},p_{h_{k+1}})=\mbox{\rm Correction}(V_H,u_{h_{k}},y_{h_k}, p_{h_k}, V_{h_{k+1}}).\end{aligned}$$ As for the linear case, in Algorithm \[Alg:4.1\] one needs to solve a nonlinear optimization problem (\[step\_4\_semilinear\])-(\[step\_4\_state\_semilinear\]) on the finite element space $V_{H,h_{k+1}}$, there are several ways to provide a good initial guess for the optimization algorithm which may speed up the convergence. Also, the good initial guess would lead to the correct solution for the nonlinear optimization problem. One option is to use $u_{h_{k}}$ as initial guess, while the other choice is $P_{U_{ad}}\{-\frac{1}{\alpha}\hat p_{h_{k+1}}\}$. In the following of this paper, we denote $(\bar u_{h_{k+1}},\bar y_{h_{k+1}},\bar p_{h_{k+1}})\in U_{ad} \times V_{h_{k+1}}\times V_{h_{k+1}}$ the finite element solution to the discrete optimal control problems (\[OCP\_nonlinear\_h\])-(\[OCP\_state\_nonlinear\_h\]) in the finite element space $V_{h_{k+1}}$. We are able to analyze the error estimates between solutions $(\bar u_{h_{k+1}},\bar y_{h_{k+1}},\bar p_{h_{k+1}})$ and the correction one $(u_{h_{k+1}},y_{h_{k+1}},p_{h_{k+1}})$ on mesh level $\mathcal{T}_{h_{k+1}}$. \[Thm:4.1\] Let $(\bar u_{h_{k+1}},\bar y_{h_{k+1}}, \bar p_{h_{k+1}})\in U_{ad}\times V_{h_{k+1}}\times V_{h_{k+1}}$ be the solution of problems (\[OCP\_nonlinear\_h\])-(\[OCP\_state\_nonlinear\_h\]) and $(u_{h_{k+1}},y_{h_{k+1}},p_{h_{k+1}})$ be the numerical approximation by Algorithm \[Alg:4.1\], respectively. Assume there exists a real number $\eta_{h_k}$ such that $(u_{h_k}, y_{h_k}, p_{h_k})$ have the following error estimates $$\begin{aligned} \label{assume_error_nonlinear} \|\bar u_{h_k}-u_{h_k}\|_{0,\Omega}+\|\bar y_{h_k}-y_{h_k}\|_{0,\Omega} +\|\bar p_{h_k}-p_{h_k}\|_{0,\Omega} = \eta_{h_k}.\end{aligned}$$ Then the following error estimates hold $$\begin{aligned} \label{correct_error_nonlinear} \|\bar u_{h_{k+1}}-u_{h_{k+1}}\|_{0,\Omega} +\|\bar y_{h_{k+1}}-y_{h_{k+1}}\|_{0,\Omega} +\|\bar p_{h_{k+1}}-p_{h_{k+1}}\|_{0,\Omega} \leq C\eta_{h_{k+1}},\end{aligned}$$ where $\eta_{h_{k+1}}=H(h_{k}^{2}+\eta_{h_{k}})$. Setting $v=\bar y_{h_{k+1}}-y_{h_{k+1}}^*$, from the state equation approximation we have $$\begin{aligned} \|\bar y_{h_{k+1}}-y_{h_{k+1}}^*\|_{1,\Omega}^2&\leq& a(\bar y_{h_{k+1}}-y_{h_{k+1}}^*,\bar y_{h_{k+1}}-y_{h_{k+1}}^*)\nonumber\\ &=&(\bar u_{h_{k+1}}-u_{h_k},v)-(\phi(\bar y_{h_{k+1}})-\phi(y_{h_k}),v)\nonumber\\ &\leq&C(\|\bar u_{h_{k+1}}-u_{h_k}\|_{0,\Omega}+\|\bar y_{h_{k+1}}-y_{h_{k}}\|_{0,\Omega})\|v\|_{1,\Omega},\nonumber\end{aligned}$$ which together with Theorem \[Thm:priori\_semilinear\] implies that $$\begin{aligned} \|\bar y_{h_{k+1}}-y_{h_{k+1}}^*\|_{1,\Omega}&\leq& C\|\bar u_{h_{k+1}}-\bar u_{h_k}\|_{0,\Omega}+C\|\bar u_{h_{k}}-u_{h_k}\|_{0,\Omega}\nonumber\\ &&+C\|\bar y_{h_{k+1}}-\bar y_{h_k}\|_{0,\Omega}+C\|\bar y_{h_{k}}-y_{h_k}\|_{0,\Omega}\nonumber\\ &\leq&C(h_k^{2}+\eta_k).\label{Thm4.1-1}\end{aligned}$$ So we can derive $$\begin{aligned} \|\bar y_{h_{k+1}}-\hat y_{h_{k+1}}\|_{1,\Omega}&\leq&\|\bar y_{h_{k+1}}-y_{h_{k+1}}^*\|_{1,\Omega}+\|y_{h_{k+1}}^*-\hat y_{h_{k+1}}\|_{1,\Omega}\nonumber\\ &\leq&C(h_k^{2}+\eta_k).\label{Thm4.1-2}\end{aligned}$$ Setting $v=\bar p_{h_{k+1}}-p_{h_{k+1}}^*$, we conclude from the adjoint state equation and $\phi'(\cdot)\geq 0$ that $$\begin{aligned} \|\bar p_{h_{k+1}}-p_{h_{k+1}}^*\|_{1,\Omega}^2&\leq& a(\bar p_{h_{k+1}}-p_{h_{k+1}}^*,\bar p_{h_{k+1}}-p_{h_{k+1}}^*)\nonumber\\ &=&(\bar y_{h_{k+1}}-\hat y_{h_{k+1}},v)-(\phi'(\bar y_{h_{k+1}})\bar p_{h_{k+1}}-\phi'(\hat y_{h_{k+1}})p_{h_{k+1}}^*,v)\nonumber\\ &=&(\bar y_{h_{k+1}}-\hat y_{h_{k+1}},v)-((\phi'(\bar y_{h_{k+1}})-\phi'(\hat y_{h_{k+1}}))\bar p_{h_{k+1}},v)\nonumber\\ &&-(\phi'(\hat y_{h_{k+1}})(\bar p_{h_{k+1}}-p_{h_{k+1}}^*),v)\nonumber\\ &\leq&(\bar y_{h_{k+1}}-\hat y_{h_{k+1}},v)-((\phi'(\bar y_{h_{k+1}})-\phi'(\hat y_{h_{k+1}}))\bar p_{h_{k+1}},v)\nonumber\\ &\leq&C\|\bar y_{h_{k+1}}-\hat y_{h_{k+1}}\|_{0,\Omega}\|v\|_{0,\Omega} +\|\bar y_{h_{k+1}}-\hat y_{h_{k+1}}\|_{0,\Omega}\|\bar p_{h_{k+1}}\|_{0,4,\Omega}\|v\|_{0,4,\Omega}\nonumber\\ &\leq&C\|\bar y_{h_{k+1}}-\hat y_{h_{k+1}}\|_{0,\Omega}(1+\|\bar p_{h_{k+1}}\|_{1,\Omega})\|v\|_{1,\Omega},\nonumber\end{aligned}$$ which gives $$\begin{aligned} \|\bar p_{h_{k+1}}-p_{h_{k+1}}^*\|_{1,\Omega}\leq C\|\bar y_{h_{k+1}}-\hat y_{h_{k+1}}\|_{0,\Omega} \leq C(h_k^{2}+\eta_k).\label{Thm4.1-3}\end{aligned}$$ Similar to (\[Thm4.1-2\]) we have $$\begin{aligned} \|\bar p_{h_{k+1}}-\hat p_{h_{k+1}}\|_{1,\Omega}&\leq&\|\bar p_{h_{k+1}}-p_{h_{k+1}}^*\|_{1,\Omega} +\|p_{h_{k+1}}^*-\hat p_{h_{k+1}}\|_{1,\Omega}\nonumber\\ &\leq&C(h_k^{2}+\eta_k).\label{Thm4.1-4}\end{aligned}$$ From the coercivity of the second order derivative of the discrete reduced objective functional presented in Lemma \[La:4.4\], for some $\theta\in [0,1]$ we can derive $$\begin{aligned} &&\gamma\|\bar u_{h_{k+1}}-u_{h_{k+1}}\|_{0,\Omega}^2\nonumber\\ &\leq& \hat J_h''(\theta \bar u_{h_{k+1}}+(1-\theta)u_{h_{k+1}}) (\bar u_{h_{k+1}}-u_{h_{k+1}},\bar u_{h_{k+1}}-u_{h_{k+1}})\nonumber\\ &=&\hat J_h'(\bar u_{h_{k+1}})(\bar u_{h_{k+1}}-u_{h_{k+1}})-\hat J_h'(u_{h_{k+1}})(\bar u_{h_{k+1}}-u_{h_{k+1}})\nonumber\\ &=&(\alpha \bar u_{h_{k+1}}+p_{h_{k+1}}(\bar u_{h_{k+1}}),\bar u_{h_{k+1}}-u_{h_{k+1}}) -(\alpha u_{h_{k+1}}+p_{h_{k+1}},\bar u_{h_{k+1}}-u_{h_{k+1}})\nonumber\\ &\leq&(\alpha \bar u_{h_{k+1}}+\bar p_{h_{k+1}},\bar u_{h_{k+1}}-u_{h_{k+1}}) +(p_{h_{k+1}}(\bar u_{h_{k+1}})-\bar p_{h_{k+1}},\bar u_{h_{k+1}}-u_{h_{k+1}})\nonumber\\ &\leq&(p_{h_{k+1}}(\bar u_{h_{k+1}})-\bar p_{h_{k+1}},\bar u_{h_{k+1}}-u_{h_{k+1}}),\nonumber\end{aligned}$$ which implies that $$\begin{aligned} &&{\gamma\over 4}\|\bar u_{h_{k+1}}-u_{h_{k+1}}\|_{0,\Omega} \leq C\|p_{h_{k+1}}(\bar u_{h_{k+1}})-\bar p_{h_{k+1}}\|_{0,\Omega}\nonumber\\ &\leq&C\|p_{h_{k+1}}(\bar u_{h_{k+1}})-p_{h_{k+1}}(\bar y_{h_{k+1}})\|_{0,\Omega} +C\|p_{h_{k+1}}(\bar y_{h_{k+1}})-\bar p_{h_{k+1}}\|_{0,\Omega}\nonumber\\ &\leq&C\|y_{h_{k+1}}(\bar u_{h_{k+1}})-\bar y_{h_{k+1}}\|_{0,\Omega} +C\|p_{h_{k+1}}(\bar y_{h_{k+1}})-\bar p_{h_{k+1}}\|_{0,\Omega}.\label{Thm4.1-5}\end{aligned}$$ It is easy to see that $y_{h_{k+1}}(\bar u_{h_{k+1}})$ is the finite element approximation to $\bar y_{h_{k+1}}$ on $V_{H,h_{k+1}}$ for the semilinear elliptic equation because of $V_{H,h_{k+1}}\subset V_{h_{k+1}}$. A Ceá-lemma for semilinear elliptic equation implies $$\begin{aligned} \|y_{h_{k+1}}(\bar u_{h_{k+1}})-\bar y_{h_{k+1}}\|_{1,\Omega} &\leq& C\inf\limits_{v_{H,h_{k+1}}\in V_{H,h_{k+1}}}\|\bar y_{h_{k+1}}-v_{H,h_{k+1}}\|_{1,\Omega}\nonumber\\ &\leq&C\|\bar y_{h_{k+1}}-\hat y_{h_{k+1}}\|_{1,\Omega}\nonumber\\ &\leq& C(h_k^{2}+\eta_{h_k}).\label{Thm4.1-6}\end{aligned}$$ Now we prove the improved $L^2$-norm estimate by Aubin-Nitsche argument. Consider the following adjoint equation $$\label{duality} \left\{\begin{array}{llr} -\Delta \psi+\phi'(\bar y_{h_{k+1}})\psi = y_{h_{k+1}}(\bar u_{h_{k+1}})-\bar y_{h_{k+1}}\ \ &\mbox{in}\ \Omega, \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \psi=0 \ \ \ &\mbox{on}\ \partial\Omega. \end{array} \right.$$ Then we have $\|\psi\|_{2,\Omega}\leq C\|y_{h_{k+1}}(\bar u_{h_{k+1}})-\bar y_{h_{k+1}}\|_{0,\Omega}$. Setting $v=y_{h_{k+1}}(\bar u_{h_{k+1}})-\bar y_{h_{k+1}}$ we have $$\begin{aligned} &&\|y_{h_{k+1}}(\bar u_{h_{k+1}})-\bar y_{h_{k+1}}\|_{0,\Omega}^2 =(y_{h_{k+1}}(\bar u_{h_{k+1}})-\bar y_{h_{k+1}},v)\nonumber\\ &=&a(y_{h_{k+1}}(\bar u_{h_{k+1}})-\bar y_{h_{k+1}},\psi) +(\phi'(\bar y_{h_{k+1}})\psi,y_{h_{k+1}}(\bar u_{h_{k+1}})-\bar y_{h_{k+1}})\nonumber\\ &=&a(y_{h_{k+1}}(\bar u_{h_{k+1}})-\bar y_{h_{k+1}},\psi-\Pi_H\psi) +(\phi(y_{h_{k+1}}(\bar u_{h_{k+1}}))-\phi(\bar y_{h_{k+1}}),\psi-\Pi_H\psi)\nonumber\\ &&+(\phi'(\bar y_{h_{k+1}})\psi,y_{h_{k+1}}(\bar u_{h_{k+1}})-\bar y_{h_{k+1}}) +((\bar y_{h_{k+1}}-y_{h_{k+1}}(\bar u_{h_{k+1}}))\phi'(\theta)),\psi)\nonumber\\ &\leq&CH\|\psi\|_{2,\Omega}\|y_{h_{k+1}}(\bar u_{h_{k+1}})-\bar y_{h_{k+1}}\|_{1,\Omega} +C\|\psi\|_{0,\infty,\Omega}\|\phi''(\xi)\|_{0,\infty,\Omega}\|y_{h_{k+1}} (\bar u_{h_{k+1}})-\bar y_{h_{k+1}}\|_{0,\Omega}^2\nonumber\\ &\leq&C\|\psi\|_{2,\Omega}(H\|y_{h_{k+1}}(\bar u_{h_{k+1}})-\bar y_{h_{k+1}}\|_{1,\Omega} +\|y_{h_{k+1}}(\bar u_{h_{k+1}})-\bar y_{h_{k+1}}\|_{0,\Omega}^2),\nonumber\end{aligned}$$ where $\Pi_H$ denotes the interpolation of $\psi$ in the finite element space $V_H$, $\theta=a_1\bar y_{h_{k+1}}+(1-a_1)y_{h_{k+1}}(\bar u_{h_{k+1}})$ for some $a_1\in [0,1]$ and $\xi=a_2\bar y_{h_{k+1}}+(1-a_2)\theta$ for some $a_2\in [0,1]$. Since $\|y_{h_{k+1}}(\bar u_{h_{k+1}})-\bar y_{h_{k+1}}\|_{0,\Omega}\ll 1$, we can conclude that $$\begin{aligned} \label{Thm4.1-7} \|y_{h_{k+1}}(\bar u_{h_{k+1}})-\bar y_{h_{k+1}}\|_{0,\Omega}&\leq& CH\|y_{h_{k+1}}(\bar u_{h_{k+1}})-\bar y_{h_{k+1}}\|_{1,\Omega}\nonumber\\ &\leq&CH(h_k^{2}+\eta_{h_k}).\end{aligned}$$ Similar to (\[Thm4.1-6\])-(\[Thm4.1-7\]), we can derive $$\begin{aligned} \label{Thm4.1-8} \|p_{h_{k+1}}(\bar y_{h_{k+1}})-\bar p_{h_{k+1}}\|_{0,\Omega} &\leq&CH(h_k^{2}+\eta_{h_k}).\end{aligned}$$ Combining (\[Thm4.1-5\]), (\[Thm4.1-7\]) and (\[Thm4.1-8\]) leads to the following estimate $$\begin{aligned} \label{Thm4.1-9} \|\bar u_{h_{k+1}}-u_{h_{k+1}}\|_{0,\Omega} &\leq&CH(h_k^{2}+\eta_{h_k}).\end{aligned}$$ Using the triangle inequality, we obtain $$\begin{aligned} \|\bar u_{h_{k+1}}-u_{h_{k+1}}\|_{0,\Omega}+\|\bar y_{h_{k+1}}-y_{h_{k+1}}\|_{0,\Omega} +\|\bar p_{h_{k+1}}-p_{h_{k+1}}\|_{0,\Omega} \leq CH(h_k^{2}+\eta_{h_k}).\end{aligned}$$ This is the desired result (\[correct\_error\_nonlinear\]) and the proof is complete. Based on the sequence of nested finite element spaces $V_{h_1}\subset V_{h_2}\subset\cdots\subset V_{h_n}$ and the one correction step defined in Algorithm \[Alg:4.1\], we can define the multilevel correction method to solve the nonlinear optimal control problem: \[Alg:4.2\] A multilevel correction method for nonlinear optimal control problem: 1. Solve a nonlinear optimal control problem in the initial space $V_{h_1}$: $$\begin{aligned} \label{step_1_semilinear} \min\limits_{u_{h_1}\in U_{ad},\ y_{h_1}\in V_{h_1}}\ \ J(y_{h_1},u_{h_1})={1\over 2}\|y_{h_1}-y_d\|_{0,\Omega}^2 + \frac{\alpha}{2}\|u_{h_1}\|_{0,\Omega}^2\end{aligned}$$ subject to $$\label{step_1_state_semilinear} a(y_{h_1},v_{h_1}) +(\phi(y_{h_1}),v_{h_1})= (f+u_{h_1},v_{h_1}), \ \ \ \ \forall v_{h_1}\in V_{h_1}.$$ The corresponding optimality condition reads: Find $(u_{h_1},y_{h_1},p_{h_1})\in U_{ad}\times V_{h_1}\times V_{h_1}$ such that $$\label{step_1_OPT_semilinear} \left\{\begin{array}{llr} a(y_{h_1},v_{h_1})+(\phi(y_{h_1}),v_{h_1}) = (f+u_{h_1},v_{h_1}),\ \ &\forall v_{h_1}\in V_{h_1},\\ a(v_{h_1},p_{h_1}) +(\phi'(y_{h_1})p_{h_1},v_{h_1}) = (y_{h_1}-y_d,v_{h_1}),\ \ &\forall v_{h_1}\in V_{h_1},\\ (\alpha u_{h_1}+p_{h_1},v-u_{h_1})\geq 0,\ \ &\forall v\in U_{ad}. \end{array} \right.$$ 2. Do $k=1$, $\cdots$, $n-1$ Obtain a new optimal solution $(u_{h_{k+1}},y_{h_{k+1}},p_{h_{k+1}})\in U_{ad} \times V_{h_{k+1}}\times V_{h_{k+1}}$ by Algorithm \[Alg:4.1\] $$\begin{aligned} (u_{h_{k+1}},y_{h_{k+1}},p_{h_{k+1}})={\rm Correction}(V_H,u_{h_{k}},y_{h_k}, p_{h_k}, V_{h_{k+1}}).\end{aligned}$$ end Do Finally, we obtain a numerical approximation $(u_{h_n},y_{h_n}, p_{h_n}) \in U_{ad}\times V_{h_{n}}\times V_{h_n}$ for problem (\[OCP\_nonlinear\])-(\[OCP\_state\_nonlinear\]). Now we are in the position to give the error estimates for the solution generated by the above multilevel correction scheme described in Algorithm \[Alg:4.2\]. \[Thm:4.2\] Let $(u,y,p)\in U_{ad}\times H_0^1(\Omega)\times H_0^1(\Omega)$ and $(u_{h_n},y_{h_n},p_{h_n})\in U_{ad}\times V_{h_n}\times V_{h_n}$ be the solution of problems (\[OCP\_nonlinear\])-(\[OCP\_state\_nonlinear\]) and the solution by Algorithm \[Alg:4.2\], respectively. Assume the mesh size $H$ satisfies the condition $CH\beta^2<1$. Then the following error estimates hold $$\begin{aligned} \label{multigrid_error_nonlinear} \|\bar{u}_{h_n}-u_{h_n}\|_{0,\Omega}+\|\bar{y}_{h_n}-y_{h_n}\|_{0,\Omega} +\|\bar{p}_{h_n}-p_{h_n}\|_{0,\Omega}\leq Ch_n^{2}.\end{aligned}$$ Finally, we have the following error estimates $$\begin{aligned} \label{final_error_nonlinear} \|u-u_{h_n}\|_{0,\Omega}+\|y-y_{h_n}\|_{0,\Omega}+\|p-p_{h_n}\|_{0,\Omega}\leq Ch_n^{2}.\end{aligned}$$ The proof is similar to the proof of Theorem \[Thm:3.2\], we omit it here. Numerical Examples ================== To test the efficiency of our proposed algorithm, we in this section carry out some numerical experiments. All the computations are based on the C++ library AFEPack (see [@Li; @and; @Liu]). At first, we consider the following linear-quadratic optimal control problem: $$\begin{aligned} \min\limits_{u\in U_{ad}} J(y,u)={1\over 2}\|y-y_d\|_{0,\Omega}^2 + \frac{\alpha}{2}\|u\|_{0,\Omega}^2\nonumber\end{aligned}$$ subject to $$\label{example_state} \left\{\begin{array}{llr} -\Delta y=f+u \ \ &\mbox{in}\ \Omega, \\ \ y=0 \ \ \ &\mbox{on}\ \partial\Omega,\\ a(x)\leq u(x)\leq b(x)\ \ &\mbox{a.e.\ in}\ \Omega. \end{array} \right.$$ \[Exm:1.1\] We set $\Omega=[0,1]^2$. Let $a=-1$, $b=1$, $\alpha =0.1$, $g(x_1,x_2)=2\pi^2\sin(\pi x_1)\sin(\pi x_2)$. Then $f$ is chosen as $$\begin{aligned} \label{nonumber} f(x_1,x_2) =\left\{\begin{array}{llr} g(x_1,x_2)-a,\ \ \ \ \ &g(x_1,x_2)< a,\\ 0, \ \ \ \ \ &a\leq g(x_1,x_2)\leq b,\\ g(x_1,x_2)-b,\ \ \ \ \ &g(x_1,x_2)>b. \end{array}\right.\end{aligned}$$ Due to the state equation (\[example\_state\]), we obtain the exact optimal control $u$ $$\begin{aligned} \label{nonumber} u(x_1,x_2) =\left\{\begin{array}{llr} a,\ \ \ \ \ &g(x_1,x_2)< a;\\ g(x_1,x_2), \ \ \ \ \ &a\leq g(x_1,x_2)\leq b;\\ b,\ \ \ \ \ &g(x_1,x_2)>b. \end{array}\right.\end{aligned}$$ We also have $$\begin{aligned} y(x_1,x_2)=\sin(\pi x_1)\sin(\pi x_2),\nonumber\\ p(x_1,x_2)=-2\pi^2\alpha\sin(\pi x_1)\sin(\pi x_2).\end{aligned}$$ The desired state is given by $$\begin{aligned} y_d(x_1,x_2)=y(x_1,x_2)+4\pi^4\alpha\sin(\pi x_1)\sin(\pi x_2).\end{aligned}$$ At first, we consider the comparison of errors for the solutions by the direct solving of optimal control problem and the multilevel correction method defined by Algorithm \[Alg:3.2\], respectively, on the sequence of nested linear finite element spaces $V_{h_1}\subset V_{h_2}\subset V_{h_3}$ which are defined on the three level meshes $\mathcal{T}_{h_1}$, $\mathcal{T}_{h_2}$ and $\mathcal{T}_{h_3}$. Here we set $\mathcal{T}_H=\mathcal{T}_{h_1}$. The series of meshes $\mathcal{T}_{h_1}$, $\mathcal{T}_{h_2}$ and $\mathcal{T}_{h_3}$ are produced by regular refinements with $\beta=2$. The optimal control is discretized implicitly by variational discretization concept proposed by Hinze [@Hinze05COAP] and the discretized optimization problem is solved by projected gradient method (see [@LiuYan08book]). In Algorithm \[Alg:3.2\], we note that on the coarsest finite element space $V_{h_1}$ one needs to solve the optimization problem directly, while on finer finite element spaces $V_{h_2}$ and $V_{h_3}$ one only needs to solve two linear boundary value problems and one optimization problem in the coarsest finite element space $V_{h_1}$. From Tables \[table:1.1\] and \[table:1.2\], we can observe that with same degree of freedoms the comparable errors can be obtained on finer finite element spaces $V_{h_2}$ and $V_{h_3}$ but with greatly reduced computational complexity by the multilevel correction method. $\#V_{h_1}$ $\|u-u_{h_1}\|_{0,\Omega}$ order $\#V_{h_2}$ $\|u-u_{h_2}\|_{0,\Omega}$ order $\#V_{h_3}$ $\|u-u_{h_3}\|_{0,\Omega}$ order ------------- ---------------------------- -------- ------------- ---------------------------- -------- ------------- ---------------------------- -------- $139$ 1.781810e-2 $513$ 5.343963e-3 1.7374 513 5.343963e-3 $1969$ 1.316909e-3 2.0208 1969 1.316909e-3 2.0208 1969 1.316909e-3 $7713$ 3.289924e-4 2.0010 7713 3.289928e-4 2.0010 7713 3.289925e-4 2.0010 $30529$ 8.206814e-5 2.0032 30529 8.208219e-5 2.0029 30529 8.206746e-5 2.0032 $ 121473$ 2.051887e-5 1.9999 121473 2.051903e-5 2.0001 121473 2.051942e-5 1.9998 : Convergence history of $\|u-u_h\|_{0,\Omega}$ for Example \[Exm:1.1\].[]{data-label="table:1.1"} $\#V_{h_1}$ $\|y-y_{h_1}\|_{0,\Omega}$ order $\#V_{h_2}$ $\|y-y_{h_2}\|_{0,\Omega}$ order $\#V_{h_3}$ $\|y-y_{h_3}\|_{0,\Omega}$ order ------------- ---------------------------- -------- ------------- ---------------------------- -------- ------------- ---------------------------- -------- $139$ 6.809744e-3 $513$ 1.722517e-3 1.9831 513 1.722504e-3 $1969$ 4.322249e-4 1.9947 1969 4.322236e-4 1.9947 1969 4.322212e-4 $7713$ 1.081827e-4 1.9983 7713 1.081826e-4 1.9983 7713 1.081824e-4 1.9983 $30529$ 2.705474e-5 1.9995 30529 2.705475e-5 1.9995 30529 2.705475e-5 1.9995 $121473 $ 6.764356e-6 1.9999 121473 6.764348e-6 1.9999 121473 6.764373e-6 1.9999 : Convergence history of $\|y-y_h\|_{0,\Omega}$ for Example \[Exm:1.1\].[]{data-label="table:1.2"} Then we test our proposed multilevel correction algorithm on the sequence of multilevel meshes. Two initial meshes with $68$ and $139$ nodes as shown in Figure \[fig:1\] are used. We show the errors of the discretised optimal state $y_h$, the adjoint state $p_h$ and the optimal control $u_h$ in Figure \[fig:2\] on two sequences of meshes after $7$ and $6$ regular refinements with $\beta =2$, respectively. It is also observed that second order convergence rate holds for $y_h$, $p_h$ and $u_h$. We remark that the solutions by the multilevel correction method are almost the same as the results by the direct optimization problem solving on the same meshes. The algorithm is obviously more efficient if $\beta$ is as large as possible, i.e., the coarse finite element space is as coarse as possible. To support this we also consider two sequences of meshes after $4$ regular refinements with $\beta =4$ based on the above mentioned two initial meshes. We show the errors of the discretised optimal state $y_h$, the adjoint state $p_h$ and the optimal control $u_h$ in Figure \[fig:4\], second order convergence rates for the optimal control, the state and adjoint state can be observed. 0.8cm ![Errors of the multilevel correction algorithm for the discretised optimal state $y_h$, adjoint state $p_h$ and optimal control $u_h$ of Example \[Exm:1.1\] (the left figure corresponds to the left mesh in Figure \[fig:1\] with 68 nodes as the initial mesh and $7$ uniform refinements with $\beta=2$, the right figure corresponds to the right mesh in Figure \[fig:1\] with $139$ nodes as the initial mesh and 6 uniform refinements with $\beta=2$).[]{data-label="fig:2"}](error_1.eps "fig:"){width="7cm" height="7cm"} ![Errors of the multilevel correction algorithm for the discretised optimal state $y_h$, adjoint state $p_h$ and optimal control $u_h$ of Example \[Exm:1.1\] (the left figure corresponds to the left mesh in Figure \[fig:1\] with 68 nodes as the initial mesh and $7$ uniform refinements with $\beta=2$, the right figure corresponds to the right mesh in Figure \[fig:1\] with $139$ nodes as the initial mesh and 6 uniform refinements with $\beta=2$).[]{data-label="fig:2"}](error_2.eps "fig:"){width="7cm" height="7cm"} ![Errors of the multilevel correction algorithm for the discretised optimal state $y_h$, adjoint state $p_h$ and optimal control $u_h$ of Example \[Exm:1.1\] (the left figure corresponds to the left mesh in Figure \[fig:1\] with 68 nodes as the initial mesh and $4$ uniform refinements with $\beta=4$, the right figure corresponds to the right mesh in Figure \[fig:1\] with $139$ nodes as the initial mesh and 4 uniform refinements with $\beta=4$).[]{data-label="fig:4"}](error_5.eps "fig:"){width="7cm" height="7cm"} ![Errors of the multilevel correction algorithm for the discretised optimal state $y_h$, adjoint state $p_h$ and optimal control $u_h$ of Example \[Exm:1.1\] (the left figure corresponds to the left mesh in Figure \[fig:1\] with 68 nodes as the initial mesh and $4$ uniform refinements with $\beta=4$, the right figure corresponds to the right mesh in Figure \[fig:1\] with $139$ nodes as the initial mesh and 4 uniform refinements with $\beta=4$).[]{data-label="fig:4"}](error_6.eps "fig:"){width="7cm" height="7cm"} In the second example, we consider the following optimal controls of semilinear elliptic equation: $$\begin{aligned} \min\limits_{u\in U_{ad}}\ \ J(y,u)={1\over 2}\|y-y_d\|_{0,\Omega}^2 + \frac{\alpha}{2}\|u\|_{0,\Omega}^2\nonumber\end{aligned}$$ subject to $$\label{example_state_1} \left\{\begin{array}{llr} -\Delta y+y^3=f+u \ \ &\mbox{in}\ \Omega, \\ \ y=0 \ \ \ &\mbox{on}\ \partial\Omega,\\ a(x)\leq u(x)\leq b(x)\ \ &\mbox{a.e.\ in}\ \Omega. \end{array} \right.$$ \[Exm:1.2\] We set $\Omega=[0,1]^2$. Let $a=0$, $b=3$, $\alpha =0.01$, $g_1(x_1,x_2)=2\pi^2\sin(\pi x_1)\sin(\pi x_2)$, $g_2(x_1,x_2)=\sin^3(\pi x_1)\sin^3(\pi x_2)$. Then $f$ is chosen as $$\begin{aligned} \label{nonumber} f(x_1,x_2) =\left\{\begin{array}{llr} g_1(x_1,x_2)+g_2(x_1,x_2)-a,\ \ \ \ \ &g_1(x_1,x_2)< a;\\ g_2(x_1,x_2), \ \ \ \ \ &a\leq g_1(x_1,x_2)\leq b;\\ g_1(x_1,x_2)+g_2(x_1,x_2)-b,\ \ \ \ \ &g_1(x_1,x_2)>b. \end{array}\right.\end{aligned}$$ Due to the state equation (\[example\_state\_1\]), we obtain the exact optimal control $u$ $$\begin{aligned} \label{nonumber} u(x_1,x_2) =\left\{\begin{array}{llr} a,\ \ \ \ \ &g_1(x_1,x_2)< a;\\ g_1(x_1,x_2), \ \ \ \ \ &a\leq g_1(x_1,x_2)\leq b;\\ b,\ \ \ \ \ &g_1(x_1,x_2)>b. \end{array}\right.\end{aligned}$$ We also have $$\begin{aligned} y(x_1,x_2)=\sin(\pi x_1)\sin(\pi x_2),\nonumber\\ p(x_1,x_2)=-2\pi^2\alpha\sin(\pi x_1)\sin(\pi x_2).\end{aligned}$$ The desired state is given by $$\begin{aligned} y_d(x_1,x_2)=y(x_1,x_2)-3y^2p+4\pi^4\alpha\sin(\pi x_1)\sin(\pi x_2).\end{aligned}$$ We solve this nonlinear optimisation problem with standard SQP method (see [@Hinze09book]). We test our proposed multilevel correction algorithm for the above nonlinear optimal control problems on the sequence of multilevel meshes. We use the same sequence of meshes generated with $\beta=2$ as in the first example. It is also observed that second order convergence rate holds for $y_h$, $p_h$ and $u_h$ in the nonlinear case. We remark that the solutions by the multilevel correction method are almost the same as the results by the direct optimization problem solving on the same meshes. ![Errors of the multilevel correction algorithm for the discretised optimal state $y_h$, adjoint state $p_h$ and optimal control $u_h$ of Example \[Exm:1.2\] (the left figure corresponds to the left mesh in Figure \[fig:1\] with 68 nodes as the initial mesh and $7$ uniform refinements, the right figure corresponds to the right mesh in Figure \[fig:1\] with $139$ nodes as the initial mesh and 6 uniform refinements).[]{data-label="fig:3"}](error_3.eps "fig:"){width="7cm" height="7cm"} ![Errors of the multilevel correction algorithm for the discretised optimal state $y_h$, adjoint state $p_h$ and optimal control $u_h$ of Example \[Exm:1.2\] (the left figure corresponds to the left mesh in Figure \[fig:1\] with 68 nodes as the initial mesh and $7$ uniform refinements, the right figure corresponds to the right mesh in Figure \[fig:1\] with $139$ nodes as the initial mesh and 6 uniform refinements).[]{data-label="fig:3"}](error_4.eps "fig:"){width="7cm" height="7cm"} Concluding remarks ================== In this paper, we introduce a type of multilevel correction scheme to solve the optimal control problem. The idea here is to use the multilevel correction method to transform the solution of the optimal control problem on the finest finite element space to a series of solutions of the corresponding linear boundary value problems which can be solved by the multigrid method and a series of solutions of optimal control problems on the coarsest finite element space. The optimal control problem solving is more difficult than the linear boundary value problem solving which has already many efficient solvers. Thus, the proposed method can improve the overall efficiency for the optimization problem solving. With the complexity analysis, we can find that the multilevel correction scheme can obtain the optimal finite element approximation by the almost optimal computational work [@Xie_IMA; @Xie_JCP]. We can replace the multigrid method by other types of efficient solvers such as algebraic multigrid method and the domain decomposition method. Furthermore, the framework here can also be coupled with the parallel method and the adaptive refinement technique. The ideas can be extended to other types of linear and nonlinear optimal control problems. Acknowledgments {#acknowledgments .unnumbered} =============== The first author was supported by the National Basic Research Program of China under grant 2012CB821204, the National Natural Science Foundation of China under grant 11201464 and 91330115, and the scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. The second author gratefully acknowledges the support of the National Natural Science Foundation of China (91330202, 11371026, 11001259, 11031006, 2011CB309703), the National Center for Mathematics and Interdisciplinary Science, CAS and the President Foundation of AMSS-CAS. The third author was supported by the National Natural Science Foundation of China under grant 11171337. [99]{} N. Arada, E. Casas and F. Tröltzsch, Error estimates for the numerical approximation of a semilinear elliptic control problem, Comput. Optim. Appl., 23 (2002), pp. 201-229. S. Brenner, L. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, 1994. A. Borzi and V. Schulz, Multigrid methods for PDE optimization, SIAM Review, 51 (2009), pp. 361-395. F. S. Falk, Approximation of a class of optimal control problems with order of convergence estimates, J. Math. Anal. Appl. 44 (1973), pp. 28-47. T. Geveci, On the approximation of the solution of an optimal control problem governed by an elliptic equation, RAIRO Anal. Numer., 13 (1979), pp. 313-328. M. Hintermüller, K. Ito and K. Kunisch, The primal-dual active set strategy as a semismooth Newton method, SIAM J. Optim., 13 (2003), pp. 865-888. M. Hinze, A variational discretization concept in control constrained optimization: The linear-quadratic case, Comput. Optim. Appl., 30 (2005), pp. 45-63. M. Hinze, R. Pinnau, M. Ulbrich, S. Ulbrich, Optimization with PDE Constraints, Math. Model. Theo. Appl., 23, Springer, New York, 2009. M. Hinze, M. Vierling, The semi-smooth Newton method for variationally discretized control constrained elliptic optimal control problems: implementation, convergence and globalization, Optim. Methods Softw., 27 (2012), no. 6, pp. 933-950. A. Kröner, B. Vexler, A priori error estimates for elliptic optimal control problems with a bilinear state equation, J. Comput. Appl. Math., 230 (2009), no. 2, pp. 781-802. R. Li, On multi-mesh H-adaptive methods, J. Sci. Comput., 24 (2005), no. 3, pp. 321-341. Q. Lin, H. Xie, A multi-level correction scheme for eigenvalue problems, Math. Comput., http://dx.doi.org/10.1090/S0025-5718-2014-02825-1. J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York-Berlin, 1971. W. Liu, N. Yan, A posteriori error estimates for optimal boundary control, SIAM J. Numer. Anal., 39 (2001), pp. 73-99. W. Liu, N. Yan, A posteriori error estimates for distributed convex optimal control problems, Adv. Comput. Math., 15 (2001), pp. 285-309. W. Liu, N. Yan, Adaptive Finite Element Methods for Optimal Control Governed by PDEs, Science Press, Beijing, 2008. C. Meyer, R. Rösch, Superconvergence properties of optimal control problems, SIAM J. Control Optim., 43 (2004), pp. 970-985. I. Neitzel, B. Vexler, A priori error estimates for space-time finite element discretisation of semilinear parabolic optimal control problems, Numer. Math., 120 (2012), pp. 345-386. V. V. Shaidurov, Multigrid Methods for Finite Elements, Kluwer Academic Publics, Netherlands, 1995. H. Xie, A type of multilevel method for the Steklov eigenvalue problem, IMA J. Numer. Anal., 34 (2014), no. 2, pp. 592-608. H. Xie, A multigrid method for eigenvalue problem, J. Comput. Phys., 274 (2014), pp. 550-561.

--- abstract: 'In recent years, the equations defining secant varieties and their syzygies have attracted considerable attention. The purpose of the present paper is to conduct a thorough study on secant varieties of curves by settling several conjectures and revealing interaction between singularities and syzygies. The main results assert that if the degree of the embedding line bundle of a nonsingular curve of genus $g$ is greater than $2g+2k+p$ for nonnegative integers $k$ and $p$, then the $k$-th secant variety of the curve has normal Du Bois singularities, is arithmetically Cohen–Macaulay, and satisfies the property $N_{k+2, p}$. In addition, the singularities of the secant varieties are further classified according to the genus of the curve, and the Castelnuovo–Mumford regularities are also obtained as well. As one of the main technical ingredients, we establish a vanishing theorem on the Cartesian products of the curve, which may have independent interests and may find applications elsewhere.' address: - 'Department of Mathematics, University Illinois at Chicago, 851 South Morgan St., Chicago, IL 60607, USA' - 'Department of Mathematical Sciences, University of Arkansas, Fayetteville, AR 72701, USA' - 'Department of Mathematics, Sogang University, 35 Beakbeom-ro, Mapo-gu, Seoul 04107, Republic of Korea' author: - Lawrence Ein - Wenbo Niu - Jinhyung Park title: Singularities and syzygies of secant varieties of nonsingular projective curves --- [^1] [^2] Introduction ============ Throughout the paper, we work over an algebraically closed field $\Bbbk$ of characteristic zero. Let $$C \subseteq {\mathbb{P}}(H^0(C, L))={\mathbb{P}}^r$$ be a nonsingular projective curve of genus $g\geq 0$ embedded by the complete linear system of a very ample line bundle $L$ on $C$. For an integer $k\geq 0$, the *$k$-th secant variety* $$\Sigma_k=\Sigma_k(C, L) \subseteq {\mathbb{P}}^r$$ to the curve $C$ is defined to be the Zariski closure of the union of $(k+1)$-secant $k$-planes to $C$ in ${\mathbb{P}}^r$. One has the natural inclusions $$C=\Sigma_0 \subseteq \Sigma_1\subseteq \cdots \subseteq \Sigma_{k-1} \subseteq \Sigma_k\subset {\mathbb{P}}^r.$$ If $\deg L \geq 2g+2k+1$, then $$\dim \Sigma_k = 2k+1~~\text{ and }~~\operatorname{Sing}(\Sigma_k)=\Sigma_{k-1}.$$ Note that $\Sigma_{k-1}$ has codimension two in $\Sigma_k$. The geometric consequence of the condition $\deg L\geq 2g+2k+1$ is that any effective divisor on $C$ of degree $k+1$ spans a $k$-plane in ${\mathbb{P}}^r$. There has been a great deal of work on the secant varieties in the last three decades. The major part of the research focused on local properties, defining equations, and syzygies. Recently, classical questions on secant varieties find interesting applications to algebraic statistics and algebraic complexity theory. However, a lot of problems in this area are still widely open, and not much is known about general pictures. For the first secant variety of a curve, investigation has been conducted in a series of work by Vermeire [@Vermeire:SomeResultSecFlip; @Vermeire:RegPowers; @Vermeire:RegNormSecCurve; @Vermeire:EqSyzSec] and the work with his collaborator Sidman [@Vermeire:SyzSecantVarCurves; @Vermeire:EqSecGeoComp]. Among other things, the issue whether secant varieties are normal attracted special attention, as normality is critical in establishing many other important properties. Only for the first secant variety, the normality problem was settled by Ullery [@Ullery:SecantVar] fairly recently for a nonsingular projective variety of any dimension under suitable conditions on the embedding line bundle. Soon afterwards Chou and Song [@Chou:SingSecVar] further showed that the first secant variety has Du Bois singularities under the setting of Ullery’s study. On the other hand, the classical questions on the projective normality and the defining equations of secant varieties are the initial case of a more general picture involving higher syzygies, under the frame of Green’s pioneering work [@G:Kosz]. Keeping in mind that the curve can be viewed as its zeroth secant variety, the fundamental *Green’s $(2g+1+p)$–theorem* (see [@G:Kosz] and [@GL:Syz]) asserts that if the embedding line bundle $L$ has $\deg L \geq 2g+1+p$, then $C \subseteq {\mathbb{P}}^r$ is projectively normal and satisfies the property $N_{2,p}$, i.e., the curve is cut out by quadrics and the first $p$ steps of its minimal graded free resolution are linear (see Subsection \[subsec:prelim-syz\] for relevant definitions on syzygies). This result sheds the lights on understanding the full picture of syzygies of arbitrary order secant varieties. In this paper, we give a thorough study on singularities and syzygies of the $k$-th secant variety $\Sigma_k$ of the curve $C$ for arbitrary integer $k\geq 0$. The general philosophy guiding our research can be summarized as that singularities and syzygies interact each other in the way that the singularities of $\Sigma_{k}$ determine its syzygies while the syzygies of $\Sigma_{k-1}$ determine the singularities of $\Sigma_{k}$, and so on and so forth. It turns out that all the sufficient conditions that guarantee each basic property of secant varieties are satisfied if the embedding line bundle is positive enough beyond an effective bound. The first main result of the paper describes that the possible singularities of secant varieties are mild ones naturally appearing in birational geometry. We refer to Subsection \[subsec:prelim-sing\] for the definitions of singularities. \[main1:singularities\] Let $C$ be a nonsingular projective curve of genus $g$, and $L$ be a line bundle on $C$. For an integer $k \geq 0$, suppose that $$\deg L \geq 2g+2k+1.$$ Then $\Sigma_k=\Sigma_k(C, L)$ has normal Du Bois singularities. Furthermore, one has the following: 1. $g=0$ if and only if $\Sigma_k$ is a Fano variety with log terminal singularities. 2. $g=1$ if and only if $\Sigma_k$ is a Calabi–Yau variety with log canonical singularities but not log terminal singularities. 3. $g \geq 2$ if and only if there is no boundary divisor $\Gamma$ on $\Sigma_k$ such that $(\Sigma_k, \Gamma)$ is a log canonical pair. The theorem therefore completely solves the normality problems mentioned above (see Ullery’s conjecture [@Ullery:Thesis Conjecture E]), and answers Chou–Song’s question [@Chou:SingSecVar Question 1.6] for curves. The second main result gives a description on syzygies of the $k$-th secant variety. It reveals one full picture hiding in the Green’s $(2g+1+p)$–theorem aforementioned. \[main2:syzygies\] Let $C \subseteq {\mathbb{P}}(H^0(C, L))={\mathbb{P}}^r$ be a nonsingular projective curve of genus $g$ embedded by the complete linear system of a very ample line bundle $L$ on $C$. For integers $k,p \geq 0$, suppose that $$\deg L \geq 2g+2k+1+p.$$ Then one has the following: 1. $\Sigma_k=\Sigma_k(C, L) \subseteq {\mathbb{P}}^r$ is arithmetically Cohen–Macaulay. 2. $\Sigma_k \subseteq {\mathbb{P}}^r$ satisfies the property $N_{k+2, p}$. 3. $\operatorname{reg}({\mathscr{O}}_{\Sigma_k})=2k+2$ unless $g=0$, in which case $\operatorname{reg}({\mathscr{O}}_{\Sigma_k})=k+1$. 4. $h^0(\omega_{\Sigma_k})=\dim K_{r-2k-1, 2k+2}(\Sigma_k, {\mathscr{O}}_{\Sigma_k}(1))={g+k \choose k+1}$. The results in the theorem were conjectured by Sidman–Vermeire ([@Vermeire:SyzSecantVarCurves Conjecture 1.3], [@Vermeire:RegNormSecCurve Conjectures 5 and 6])). The conjectures were quite wide open. For $g\leq 1$, the conjectures were settled by Graf von Bothmer–Hulek [@GBH] and Fisher [@Fish]. By work of Vermeire [@Vermeire:RegPowers; @Vermeire:RegNormSecCurve; @Vermeire:EqSyzSec], Sidman–Vermeire [@Vermeire:SyzSecantVarCurves], and Yang [@Yang:Letter], the question about $N_{3,p}$ was finally settled for the first secant variety $\Sigma_1$. Theorem \[main2:syzygies\] gives a complete picture for syzygies of arbitrary order secant varieties of curves. If $\deg L\geq 2g+2k+1$, then $\Sigma_k \subseteq {\mathbb{P}}^r$ is indeed projectively normal. If $\deg L \geq 2g+2k+2$, then $\Sigma_k$ is ideal-theoretically cut out by the hypersurfaces of degree $k+2$, as it cannot be contained in a smaller degree hypersurface. Furthermore, if $\deg L \geq 2g+2k+1+p$, then the first $p$ steps of the minimal graded free resolution of $\Sigma_k$ are linear. We mention here several quick examples to show that the degree bounds on the line bundle $L$ in the theorems are optimal. (i) Assume $C$ has genus $g=4$ and take general points $p$, $q$, $r$, and $s$ on $C$. The line bundle $L=\omega_C(p+q+r+s)$ embeds $C$ in ${\mathbb{P}}^{g+2}$. Then the first secant variety $\Sigma_1$ is neither normal nor Cohen–Macaulay. See Example \[Ex:non-normal\] for non-normal higher secant varieties $\Sigma_k$ with $k \geq 2$. (ii) If $C$ is an elliptic curve and $\deg L=2k+3$, then the $k$-th secant variety $\Sigma_k$ in ${\mathbb{P}}^{2k+2}$ is a hypersurface of degree $2k+3$. (iii) If $C$ has genus $2$ and degree $12$ in ${\mathbb{P}}^{10}$, then $\Sigma_{1}$ satisfies $N_{3,5}$ but fails $N_{3,6}$, and $\Sigma_{2}$ satisfies $N_{4,3}$ but fails $N_{4,4}$. The last two examples are taken from [@GBH] and [@Vermeire:EqSecGeoComp], and one may find more examples there. To prove the main results of the paper, we utilize Bertram’s construction [@Bertram:ModuliRk2] to realize secant varieties as the images of projectivized vector bundles. To be more precise, we consider the $k$-th symmetric product $C_{k+1}$ of $C$. We have a canonical morphism $\sigma_{k+1} \colon C_k \times C \to C_{k+1}$ defined by sending $(x,\xi)$ to $x+\xi$ and the projection $p \colon C_k \times C \to C$. One defines the secant sheaf $$E_{k+1,L}:=\sigma_{k+1,*}(p^*L),$$ which is a vector bundle on $C_{k+1}$ of rank $k+1$, and the secant bundle $$B^{k}(L):={\mathbb{P}}(E_{k+1,L}).$$ Notice that $E_{k+1, L}$ parameterizes $(k+1)$-secant $k$-planes, i.e., the fiber of $E_{k+1,L}$ over $\xi \in C_{k+1}$ can be identified with $H^0(\xi, L|_{\xi})$. The complete linear system of the tautological line bundle of $B^k(L)$ determines a natural morphism to the projective space ${\mathbb{P}}^r$ such that the image is $\Sigma_k$. It gives rise to a resolution of singularities $$\beta \colon B^k(L) \longrightarrow \Sigma_k.$$ We then consider the $(k-1)$-th relative secant variety $Z_{k-1}$, which is actually a divisor in the smooth variety $B^k(L)$. Our strategy is to pass computation for codimension two situation $\Sigma_{k-1}\subseteq \Sigma_k$ to the codimension one situation $Z_{k-1}\subseteq B^k(L)$. The picture for the first secant variety is rather simple, and $Z_0$ is just $C\times C$. Thus one can easily transfer cohomological computation from $\Sigma_1$ to $C_2$ through $B^1(L)$. However, for higher secant varieties, such method does not work directly in that $Z_{k-1}$ is singular. Fortunately, after blowup consecutively along the stratification induced by the inclusions $C \subseteq \Sigma_1\subseteq \Sigma_2\subseteq \cdots \subseteq \Sigma_{k-1}$, as exhibited in [@Bertram:ModuliRk2], we then arrive at a birational morphism $$b_k \colon \operatorname{bl}_k(B^k(L)) \longrightarrow B^k(L),$$ which we prove to be a log resolution of the log pair $(B^k(L), Z_{k-1})$. Based on this setup, in Theorem \[main1:singularities\], for instance, to prove the normality of $\Sigma_k$ at a point $x$, we adapt the strategy of Ullery in [@Ullery:SecantVar] to consider the unique minimal $m$-secant plane containing $x$. It cuts the curve along a degree $m+1$ divisor $\xi$. By the formal function theorem, the normality of the $k$-th secant variety $\Sigma_k$ at $x$ follows from the normality and projective normality of the smaller order secant variety $\Sigma_{k-m-1}$ in the space ${\mathbb{P}}(H^0(C, L(-2\xi)))$. This leads us to study a general question on the property $N_{k+2,p}$ or higher syzygies of $\Sigma_k$. Turning to the proof of Theorem \[main2:syzygies\], we assume $\deg L\geq 2g+2k+1+p$, and consider the kernel bundle $M_{\Sigma_k}$ in the exact sequence $$0 \longrightarrow M_{\Sigma_k} \longrightarrow H^0(C, L) \otimes {\mathscr{O}}_{\Sigma_k} \longrightarrow {\mathscr{O}}_{\Sigma_k}(1) \longrightarrow 0,$$ induced by the evaluation map on the global sections of ${\mathscr{O}}_{\Sigma_k}(1)$. The critical observation we made here is that in order to establish the property $N_{k+2, p}$, one only needs cohomology vanishing involving the wedge product of $M_{\Sigma_k}$ tensored with $I_{\Sigma_{k-1}|\Sigma_k}(k+1)$. More precisely, it is sufficient to show the following cohomology vanishing $$\label{intro-eq1} H^i(\Sigma_k, \wedge^j M_{\Sigma_k} \otimes I_{\Sigma_{k-1}|\Sigma_k}(k+1))=0~\text{ for $i \geq j-p$, $i \geq 1$, $j \geq 0$}.$$ The next important technical step is to prove the following *Du Bois type conditions*: $$\label{DB_cond} R^i \beta_* {\mathscr{O}}_{\Sigma_k}(k+1)(-Z_{k-1}) = \begin{cases} I_{\Sigma_{k-1}|\Sigma_k} & \text{for $i=0$,}\vspace{0.2cm}\\ 0 & \text{for $i>0$}. \end{cases}$$ Then the cohomology groups in (\[intro-eq1\]) can be calculated on $B^k(L)$ by involving the sheaf $\beta^* {\mathscr{O}}_{\Sigma_k}(k+1)(-Z_{k-1})$. We observe that in fact this sheaf is the pullback of a line bundle $A_{k+1, L}$ on the symmetric product $C_{k+1}$ of the curve $C$. Therefore, once we use the exact sequence $$0 \longrightarrow M_{k+1, L} \longrightarrow H^0(C, L) \otimes {\mathscr{O}}_{C_{k+1}} \longrightarrow E_{k+1, L} \longrightarrow 0,$$ induced by the evaluation map on the global sections of $E_{k+1,L}$, we are able to further connect vanishing (\[intro-eq1\]) with the following cohomological vanishing $$\label{intro-eq2} H^i(C_{k+1}, \wedge^{j} M_{k+1, L} \otimes A_{k+1, L})=0~\text{ for $i \geq j-p$, $i \geq 1$, $j \geq 0$},$$ on the symmetric product $C_{k+1}$. As the final ingredient of the proof, inspired by Rathmann’s vanishing results in [@Rathmann], we show the following vanishing $$\label{intro-eq3} H^i\big(C^{k+1}, \wedge^j q^*M_{k+1, L} \otimes (\underbrace{L \boxtimes \cdots \boxtimes L}_{k+1~\text{times}}) (-\Delta) \big)=0~\text{ for $i \geq j-p$, $i \geq 1$, $j \geq 0$},$$ on the Cartesian product $C^{k+1}$ of the curve $C$, where $q \colon C^{k+1} \to C_{k+1}$ is the natural quotient map and $\Delta$ is the sum of all pairwise diagonals. Now, (\[intro-eq3\]) implies (\[intro-eq2\]), and hence, we finally obtain (\[intro-eq1\]). The vanishing result (\[intro-eq3\]) may have independent interests, and we hope that it will find other applications somewhere in the future. This paper is organized as follows. We begin in Section \[sec:prelim1\] with recalling basic definitions and properties of singularities and syzygies of algebraic varieties. In Section \[sec:prelim2\], we introduce several vector bundles on symmetric products of curves, review Bertram’s blowup constructions for secant bundles, and show some useful results for the main results of the paper. In Section \[sec:vanishing\], one of the main technical ingredients, a vanishing theorem on the Cartesian products of curves, is established. Section \[sec:prop\] is then devoted to the proofs of the main results of the paper. Finally, we discuss some open problems on secant varieties in Section \[sec:problem\]. [*Acknowledgment*]{}. The authors would like to thank Robert Lazarsfeld for helpful suggestions and useful comments. The authors also wish to express their gratitude to Adam Ginensky for bringing the problems considered in this paper to our attention and to Jürgen Rathmann for his work in the paper [@Rathmann]. The authors are very grateful to the referee for careful reading of the paper and valuable suggestions to help improve the exposition of the paper. Preliminaries {#sec:prelim1} ============= We recall relevant definitions and properties of singularities and syzygies of algebraic varieties. Singularities {#subsec:prelim-sing} ------------- The Deligne-Du Bois complex $\underline{\Omega}_X^{\bullet}$ for a singular variety $X$ is a generalization of the de Rham complex for a nonsingular variety (see [@K Chapter 6] for detail). There is a natural map $${\mathscr{O}}_X \longrightarrow \underline{\Omega}_X^{0}=Gr_{\text{filt}}^0 \underline{\Omega}_X^{\bullet}.$$ We say that $X$ has *Du Bois* singularities if the above map is a quasi-isomorphism. Let $X$ be a normal projective variety, and $\Delta$ be a boundary divisor on $X$ so that $K_X + \Delta$ is $\mathbb{Q}$-Cartier. Take a log resolution $f \colon Y \to X$ of the pair $(X, \Delta)$. We may write $$K_Y = f^*(K_X + \Delta) + \sum_{E:\text{ prime divisor on $Y$}} a(E; X, \Delta) E,$$ where $a(E; X, \Delta)$ is the discrepancy of the prime divisor $E$ over $X$. It is easy to check that the discrepancy is independent of the choice of log resolutions. We say that $(X, \Delta)$ is a *klt* (resp. *log canonical*) pair if $a(E; X,\Delta) > -1$ (resp. $a(E; X,\Delta) \geq -1$) for every prime divisor $E$ over $X$. We say that $X$ has *log terminal* (resp. *log canonical*) singularities if $(X, 0)$ is a klt (resp. log canonical) pair. Note that log terminal singularities are rational singularities and (semi-)log canonical singularities are Du Bois singularities. We refer to [@K] for more details of the various notions of singularities and log pairs. Syzygies {#subsec:prelim-syz} -------- Let $X \subseteq {\mathbb{P}}(H^0(X, L)) = {\mathbb{P}}^r $ be a projective variety embedded by the complete linear system of a very ample line bundle $L$ on $X$. Let $S$ be the homogeneous coordinate ring of ${\mathbb{P}}^r$, and $$R=R(X, L):=\bigoplus_{m \geq 0} H^0(X, mL)$$ be the graded section ring associated to $L$, viewed as an $S$-module. Then $R$ has a minimal graded free resolution $E_\bullet(X, L)$: $$\xymatrix{ 0 & R \ar[l]& \bigoplus S(-a_{0,j}) \ar[l] \ar@{=}[d]& \bigoplus S(-a_{1,j}) \ar[l] \ar@{=}[d]& \ar[l] \cdots & \ar[l] \bigoplus S(-a_{r,j}) \ar[l] \ar@{=}[d] & \ar[l]0 . \\ & & E_0 & E_1 & &E_r }$$ We define the *Koszul cohomology group* $$K_{p,q}(X, L) := \operatorname{Tor}_p^S(R, S/S_+)_{p+q},$$ where $S_+ \subseteq S$ denotes the irrelevant maximal ideal. Then we have $$E_p = \bigoplus_{q} K_{p,q}(X, L) \otimes_{\Bbbk} S(-p-q).$$ Notice that $X \subseteq {\mathbb{P}}^r$ is projectively normal if and only if $K_{0,j}(X, L)=0$ for all $j \geq 1$. The *Castelnuovo–Mumford regularity* of $R$, denoted by $\operatorname{reg}(R)$, is defined to be the minimal positive integer $q$ such that $K_{p,j}(X, L)=0$ for all $p\geq 0$ and $j\geq q+1$. We say that $R$ satisfies the *property $N_{d, p}$* for some integer $d \geq 2$ if $$K_{i,j}(X, L)=0~\text{ for $i \leq p$ and $j \geq d$}.$$ Assume that $X \subseteq {\mathbb{P}}^r$ is projectively normal. Then $R$ is the homogeneous coordinate ring of $X$ so that $R$ satisfies the property $N_{d,p}$ if and only if $X \subseteq {\mathbb{P}}^r$ satisfies the property $N_{d,p}$ in the sense of [@EGHP]. In this case, it satisfies the property $N_{d,1}$ if and only if the defining ideal of $X$ in ${\mathbb{P}}^r$ is generated in degrees $\leq d$. In general, the property $N_{d,p}$ means that up to $p$ stage, the $i$-th syzygy of the minimal graded free resolution $E_\bullet(X, L)$ is generated in degrees $\leq i-1+d$. Consider now the evaluation map $$\operatorname{ev}\colon H^0(X, L)\otimes {\mathscr{O}}_{X} \longrightarrow L,$$ which is surjective since $L$ is base point free. Denote by $M_{L}$ the kernel sheaf of the map $\operatorname{ev}$, then one obtains a short exact sequence of vector bundles $$0\longrightarrow M_{L} \longrightarrow H^0(X, L)\otimes {\mathscr{O}}_{X} \stackrel{\operatorname{ev}}{\longrightarrow} L\longrightarrow 0.$$ We use the following result to compute the Koszul cohomology group. \[koszh1\] Assume that $H^i(X, L^m)=0$ for $i >0$ and $m >0$. Then one has $$K_{p,q}(X, L)=H^1(X, \wedge^{p+1}M_L \otimes L^{q-1})~\text{ for $q \geq 2$}.$$ We conclude this section by reviewing Castelnuovo–Mumford regularity for a projective subscheme $X\subseteq {\mathbb{P}}^r$. We say that ${\mathscr{O}}_X$ (resp. $X\subseteq {\mathbb{P}}^r$) is *$m$-regular* if $H^i(X, {\mathscr{O}}_X(m-i))=0$ (resp. $H^i({\mathbb{P}}^r, I_{X|{\mathbb{P}}^r}(m-i))=0$) for $i > 0$. We say that $X\subseteq {\mathbb{P}}^r$ is *$m$-normal* if the natural restriction map $H^0({\mathbb{P}}^r, {\mathscr{O}}_{{\mathbb{P}}^r}(m)) \to H^0(X, {\mathscr{O}}_X(m))$ is surjective. Note that $X \subseteq {\mathbb{P}}^r$ is $(m+1)$-regular if and only if ${\mathscr{O}}_X$ is $m$-regular and $X \subseteq {\mathbb{P}}^r$ is $m$-normal. By Mumford’s regularity theorem, if ${\mathscr{O}}_X$ (resp. $X\subseteq {\mathbb{P}}^r$) is $m$-regular, then so is $(m+1)$-regular. We denote by $\operatorname{reg}({\mathscr{O}}_X)$ (resp. $\operatorname{reg}(X)$) the smallest integer $m$ such that ${\mathscr{O}}_X$ (resp. $X\subseteq {\mathbb{P}}^r$) is $m$-regular. Notice that $\operatorname{reg}({\mathscr{O}}_X)=\operatorname{reg}(R(X, {\mathscr{O}}_X(1)))$. We refer to [@Ein:SyzygyKoszul; @EL:AsySyz] and [@G:Kosz] for more details on syzygies and Koszul cohomology of algebraic varieties. Symmetric products, secant bundles, and secant varieties {#sec:prelim2} ======================================================== In this section, we review relevant facts on symmetric products and basic constructions of secant bundles and secant varieties. We also show some useful results on secant bundles, which play important roles in proving the main results of the paper. The reader may also look Bertram’s original paper [@Bertram:ModuliRk2 Sections 1 and 2] for more details. Throughout the section, we fix a nonsingular projective curve $C$ of genus $g\geq 0$ and a line bundle $L$ on $C$. For an integer $k\geq 1$, we write the $k$-th symmetric product of the curve $C$ as $C_k$ and the $k$-th Cartesian (or ordinary) product of the curve $C$ as $C^k$. We set $C^0=C_0=\emptyset$. Denote by $$q_k \colon C^k\longrightarrow C_k$$ the quotient morphism from $C^k$ to $C_k$. It is a finite flat surjective morphism of degree $k!$. We have the canonical morphism $$\sigma_{k+1} \colon C_{k} \times C\longrightarrow C_{k+1}$$ defined by sending $(x,\xi)$ to $x+\xi$. It is a finite flat surjective morphism of degree $k+1$. Lemmas on symmetric products {#subsec:symprod} ---------------------------- We begin with defining the secant sheaf on $C_{k+1}$ associated to a line bundle on $C$. For an integer $k\geq 1$, let $p \colon C_k \times C \rightarrow C$ be the projection to $C$. For a line bundle $L$ on $C$, we define the *secant sheaf* on $C_{k+1}$ associated to $L$ to be $$E_{k+1,L}:=\sigma_{k+1,*}(p^*L)=\sigma_{k+1,*} ({\mathscr{O}}_{C_k} \boxtimes L).$$ Notice that $E_{k+1, L}$ is a locally free sheaf on $C_{k+1}$ of rank $k+1$ and the fiber of $E_{k+1, L}$ over $\xi \in C_{k+1}$ can be identified with $H^0(\xi, L|_{\xi})$. Next, we introduce several line bundles on the symmetric product $C_{k+1}$, which play a central role in this paper (see also [@Ein:Gonality] and [@Rathmann] for the importance in the gonality conjecture). Let $k \geq 1$ be an integer. 1. Write $L^{\boxtimes k}:=\underbrace{L\boxtimes \cdots \boxtimes L}_{k~\text{times}}=p_1^*L \otimes \cdots\otimes p_k^*L$ on $C^{k}$, where $p_i \colon C^k \to C$ is the projection to the $i$-th component. The symmetric group ${\mathfrak{S}}_k$ acts on $L^{\boxtimes k}$ in a natural way: $\mu\in {\mathfrak{S}}_k$ sends a local section $s_1\otimes \cdots \otimes s_k$ to $s_{\mu(1)}\otimes \cdots \otimes s_{\mu(k)}$. Then $L^{\boxtimes k}$ is invariant under the action, so descends to a line bundle on $C_k$, denoted by $T_k(L)$. 2. Define $\delta_{k+1}$ to be a divisor on $C_{k+1}$ such that ${\mathscr{O}}_{C_{k+1}}(\delta_{k+1}):=\det \big(\sigma_{k+1,*}({\mathscr{O}}_{C\times C_k})\big)^*.$ 3. Define $N_{k+1,L}:=\det E_{k+1,L}$ on $C_{k+1}$. 4. Define $ A_{k+1,L} := T_{k+1}(L)(-2\delta_{k+1})$ on $C_{k+1}$. When $k=0$, we use the convention that $T_1(L)=E_{1,L}=L$ and $\delta_1=0$. Due to the lack of reference, we list several basic properties of the line bundles defined above. Those are well known to experts, and are not hard to prove. Let $k \geq 1$ be an integer. 1. $N_{k+1,L}=T_{k+1}(L)(-\delta_{k+1})$. 2. $H^0(C_{k+1}, T_{k+1}(L))=S^{k+1} H^0(C, L)$ and $H^0(C_{k+1}, N_{k+1}(L))=\wedge^{k+1}H^0(C, L).$ 3. $q_{k+1}^*{\mathscr{O}}_{C_{k+1}}(\delta_{k+1}) = {\mathscr{O}}_{C^{k+1}}(\Delta_{k+1})$, where $\Delta_{u,v}:=\{ (x_1, \ldots, x_k) \in C^{k+1} \mid x_u=x_v \}$ is the pairwise diagonal on $C^{k+1}$ and $\Delta_{k+1}:=\sum_{1 \leq u < v \leq k+1} \Delta_{u,v}$. When $k=1$, we let $\Delta_1=0$. 4. $\sigma_{k+1}^*{\mathscr{O}}_{C_{k+1}}(\delta_{k+1}) =({\mathscr{O}}_{C_k}(\delta_{k}) \boxtimes {\mathscr{O}}_C)(D_k)$, where $D_k$ is the divisor on $C_k \times C$ defined to be the image of the morphism $C_{k-1} \times C \to C_k \times C$ sending $(\xi, p)$ to $(\xi+p, p)$. 5. $q_k^*T_k(L) = p^*_1L\otimes \cdots \otimes p^*_kL = L^{\boxtimes k}$. Since $q_{k,*}{\mathscr{O}}_{C^k}$ contains ${\mathscr{O}}_{C_k}$ as a direct summand, $T_k(L)$ is a direct summand of $q_{k,*}L^{\boxtimes k}$. 6. For any two line bundles $L_1$ and $L_2$ on $C$, one has $T_k(L_1)\otimes T_k(L_2)= T_k(L_1\otimes L_2)$. 7. Given a point $p\in C$, the divisor $X_{p}$ on $C_{k+1}$ is defined to be the image of the morphism $C_{k}\rightarrow C_{k+1}$ sending $\xi$ to $\xi+p$. It is ample, and ${\mathscr{O}}_{C_{k+1}}(X_p) = T_{k+1}({\mathscr{O}}_C(p))$. For any line bundle $L$ on $C$, we have $T_{k+1}(L)|_{X_{p}}= T_k(L)$. (See the proof of Lemma \[A-restriction\].) 8. The canonical bundle of $C_{k+1}$ is given by $\omega_{C_{k+1}}= T_{k+1}(\omega_C)(-\delta_{k+1}) = N_{k+1,\omega_C}$. We now prove some useful lemmas. \[A-restriction\] Let $k \geq 1, m \geq 0$ be integers. Fix a degree $m+1$ divisor $\xi_{m+1}$ on $C$, and consider $C_{k-m}$ as a subscheme of $C_{k+1}$ embedded by sending a divisor $\xi$ to $\xi+\xi_{m+1}$. Then one has $$A_{k+1,L}|_{C_{k-m}}= A_{k-m,L(-2\xi_{m+1})}.$$ Fix a point $p\in \xi_{m+1}$ so that we can write $\xi_{m+1}=\xi_{m} +p$ for some degree $m$ divisor $\xi_{m}$ on $C$. Consider the embeddings $C_{k-m}\subseteq C_{k}\subseteq C_{k+1}$, where $C_{k}\subseteq C_{k+1}$ is embedded by sending a divisor $\xi$ to $\xi+p$ and $C_{k-m}\subseteq C_k$ is embedded by sending a divisor $\xi$ to $\xi+\xi_{m}$. Thus, inductively, we only need to show that $$\label{eq:05} A_{k+1,L}|_{C_k} = A_{k,L(-2p)}.$$ Regard $X_p=C_k$ as a divisor in $C_{k+1}$. Recall by definition that $A_{k+1,L}=T_{k+1}(L)(-2\delta_{k+1})$. Thus it suffices to prove the following: (1) $T_{k+1}(L)|_{X_p}=T_k(L)$ and (2) $\delta_{k+1}|_{X_p}=\delta_k+T_k(p)$. To see (1), we use the commutative diagram $$\xymatrix{ C^k\ar[d]_{q_k} \ar[r]^-{} & C^k\times C\ar[d]^-{\sigma_{k+1}}\\ X_p \ar@{^{(}->}[r] &C_{k+1},}$$ where the upper horizontal map is given by sending $(x_1,\ldots, x_k)$ to $(x_1,\ldots, x_k,p)$. We can check that $q_k^*(T_{k+1,L}|_{X_p})=L^{\boxtimes k}$, which proves (1) as $q_k^*$ is an injection on Picard groups. To see (2), we use the adjunction formula $K_{X_p}=(K_{C_{k+1}}+X_p)|_{X_p}$. Since $K_{C_{k+1}}=T_{k+1}(K_C)-\delta_{k+1}$ and $K_{X_p}=T_{k}(K_C)-\delta_k$, we deduce that $\delta_{k+1}|_{X_p}=\delta_{k}+X_p|_{X_p}$. Note that $X_p|_{X_p}=T_{k+1}(p)|_{X_p}=T_k(p)$. Thus (2) is proved. \[diagonal\] For any integer $k \geq 1$, the line bundle ${\mathscr{O}}_{C_{k+1}}(-\delta_{k+1})$ is a direct summand of the locally free sheaf $q_{k+1,*} {\mathscr{O}}_{C^{k+1}}$. We prove the lemma by the induction on $k$. For $k=1$, it is well known that $q_{2,*} {\mathscr{O}}_{C^2}$ splits as ${\mathscr{O}}_{C_2} \oplus {\mathscr{O}}_{C_2}(-\delta_2)$. Since the quotient map $q_{k+1} \colon C^{k+1} \to C_{k+1}$ factors through $C_k \times C$, one only needs to show that ${\mathscr{O}}_{C_{k+1}}(-\delta_{k+1})$ is a direct summand of $\sigma_{k+1,*} ({\mathscr{O}}_{C_k}(-\delta_k) \boxtimes {\mathscr{O}}_C)$. Observe that ${\mathscr{O}}_{C_{k+1}}(-\delta_{k+1})$ is a direct summand of $(\sigma_{k+1,*}{\mathscr{O}}_{C_k \times C})^*(-\delta_{k+1})$. By the relative duality with the relative canonical line bundle $\omega_{C_k \times C/C_{k+1}} = {\mathscr{O}}_{C_k \times C}(D_k)$, one obtains $(\sigma_{k+1, *}{\mathscr{O}}_{C_k \times C})^* = \sigma_{k+1,*}{\mathscr{O}}_{C_k \times C}(D_k)$, so $$(\sigma_{k+1,*}{\mathscr{O}}_{C_k \times C})^*(-\delta_{k+1}) = \sigma_{k+1,*}{\mathscr{O}}_{C_k \times C}(D_k) \otimes {\mathscr{O}}_{C_{k+1}}(-\delta_{k+1}).$$ Recall that $\sigma_{k+1}^* {\mathscr{O}}_{C_{k+1}}(-\delta_{k+1}) = ({\mathscr{O}}_{C_k}(-\delta_k) \boxtimes {\mathscr{O}}_C)(-D_k)$. By the projection formula, we have $$\sigma_{k+1,*}{\mathscr{O}}_{C_k \times C}(D_k) \otimes {\mathscr{O}}_{C_{k+1}}(-\delta_{k+1}) = \sigma_{k+1,*} ({\mathscr{O}}_{C_k}(-\delta_k) \boxtimes {\mathscr{O}}_C),$$ and thus, the lemma is proved. We give an alternative proof of Lemma \[diagonal\] by group actions, which may be of independent interest. Write the divisor $\delta=\delta_{k+1}$ and the structure sheaf ${\mathscr{O}}={\mathscr{O}}_{C_{k+1}}$. Let ${\mathfrak{A}}_{k+1}$ be the alternating subgroup of the symmetric group ${\mathfrak{S}}_{k+1}$, and $f \colon C^{k+1} \to Y$ be the quotient morphism under the natural induced action of ${\mathfrak{A}}_{k+1}$ on $C^{k+1}$. There is a natural degree two morphism $g \colon Y \rightarrow C_{k+1}$ through which the quotient map $q=q_{k+1} \colon C^{k+1} \to C_{k+1}$ factors, i.e., $q=g \circ f$. Note that $Y$ has quotient singularities, which are rational singularities. Thus $Y$ is Cohen–Macaulay, so the map $g$ is flat and $g_*{\mathscr{O}}_Y$ splits as ${\mathscr{O}}\oplus {\mathscr{O}}(-\delta')$ for some divisor $\delta'$ on $C_{k+1}$. We claim that $\delta'$ is actually linearly equivalent to $\delta$. To see this, notice that $f$ is unramified at codimension one points. Then $q^*{\mathscr{O}}(-2\delta)\cong q^*{\mathscr{O}}(-2\delta')$, which means that $\delta-\delta'$ is a $2$-torsion divisor. So if the genus of $C$ is zero, then $C_{k+1}$ has no nontrivial torsion line bundle and therefore ${\mathscr{O}}(\delta -\delta')={\mathscr{O}}$. If the genus of $C$ is positive, then since $H^0({\mathscr{O}}(\delta))=0$ and $g_*(g^*{\mathscr{O}}(\delta))={\mathscr{O}}(\delta)\oplus {\mathscr{O}}(\delta-\delta')$, we see that ${\mathscr{O}}(\delta -\delta')={\mathscr{O}}$ if and only if $H^0(g^*{\mathscr{O}}(\delta))\neq 0$. But this follows from the fact that the section defining $q^*\delta=\Delta$ is invariant under the group ${\mathfrak{A}}_{k+1}$, and therefore, it gives a nonzero global section of $g^*{\mathscr{O}}(\delta)$. Thus the claim is proved. Finally, note that ${\mathscr{O}}_Y$ is a direct summand of $f_*{\mathscr{O}}_{C^{k+1}}$. The lemma then follows. The following seems to be well known to experts, but we include the proof. \[K-vanishing\] For any integers $k \geq 1$ and $i\geq 0$, one has $$H^i(C_{k+1},T_{k+1}(L))\cong S^{k+1-i}H^0(C,L)\otimes \wedge^iH^1(C,L).$$ In particular, the following hold: $$\begin{array}{l} H^0(C_{k+1}, T_{k+1}(\omega_C)) \cong S^{k+1} H^0(C, \omega_C),\\ H^1(C_{k+1}, T_{k+1}(\omega_C)) \cong S^k H^0(C, \omega_C),\\ H^i(C_{k+1}, T_{k+1}(\omega_C))=0~\text{ for $i \geq 2$.} \end{array}$$ By [@Lazarsfeld:CohSymm Proposition 1.1], we have $$H^i(C_{k+1}, T_{k+1}(L))=H^i(C^{k+1}, L^{\boxtimes k+1})^{{\mathfrak{S}}_{k+1}}~\text{ for any $i\geq 0$},$$ where the right-hand-side is the invariant subspace under the action of ${\mathfrak{S}}_{k+1}$. By Künneth formula, the vector space $V:=H^i(C^{k+1}, L^{\boxtimes k+1})$ is a direct sum of the subspace $W:=T^{k+1-i}H^0(C,L)\otimes T^{i}H^1(C,L)$ with some other isomorphic summands, where the notation $T^a$ means the $a$-times tensor products. Write ${\mathfrak{G}}={\mathfrak{S}}_{k+1-i}\times {\mathfrak{S}}_i$ as the subgroup of ${\mathfrak{S}}_{k+1}$ fixing the subspace $W$. Then one has the following commutative diagram $$\xymatrix{ W \ar[r]^-{\beta} \ar@{^{(}->}[d] & W^{{\mathfrak{G}}}\ar@{^{(}->}[d]^-{\alpha}\\ V \ar[r]^-{\alpha} &V^{{\mathfrak{S}}_{k+1}},}$$ where $\displaystyle \alpha(x)=\frac{1}{(k+1)!}\sum_{g\in {\mathfrak{S}}_{k+1}}g(x)$ and $\displaystyle \beta(x) = \frac{1}{(k+1-i)!i!} \sum_{g \in {\mathfrak{G}}} g(x)$. Since every invariant cohomological class must be of the form $$s+g_1(s)+g_2(s)+\cdots$$ where $s\in W$ and $g_i$ are suitable elements in ${\mathfrak{S}}_{k+1}$, it follows that the right-hand-side vertical map $\alpha \colon W^{{\mathfrak{G}}} \to V^{{\mathfrak{S}}_{k+1}}$ in the above diagram is surjective. Hence $W^{{\mathfrak{G}}}=V^{{\mathfrak{S}}_{k+1}}$. But note that the action of the subgroup ${\mathfrak{G}}$ is symmetric on $T^{k+1-i}H^0(C,L)$ part but alternating on $T^{i}H^1(C,L)$ part of the space $W$. Therefore, the invariant subspace $H^i(C^{k+1}, L^{\boxtimes k+1})^{{\mathfrak{S}}_{k+1}}$ is isomorphic to $S^{k+1-i}H^0(C,L)\otimes \wedge^iH^1(C,L)$. The following theorem will be applied to checking the projective normality of higher secant varieties of curves. In [@Danila], Danila considers the Hilbert schemes of points on surfaces, but the proof smoothly works for the symmetric products of curves. \[danila\] For integers $k \geq 1$ and $1 \leq \ell \leq k+1$, one has $$H^0\big(C_{k+1}, E_{k+1, L}^{\otimes \ell}\big) \cong H^0(C, L)^{\otimes \ell},$$ where the isomorphism is ${\mathfrak{S}}_{k+1}$-equivariant. In particular, $$H^0\big(C_{k+1}, S^{\ell} E_{k+1, L}\big) \cong S^{\ell} H^0(C, L).$$ Secant varieties via secant bundles {#subsec:secvar} ----------------------------------- We first recall the following definition. We say that a line bundle $L$ on $C$ *separates $k$ points* (or equivalently, *$L$ is $(k-1)$-very ample*) for an integer $k\geq 1$ if the restriction map $$H^0(C, L)\longrightarrow H^0(\xi, L|_{\xi})$$ is surjective for all $\xi \in C_{k}$. For instance, $L$ separates 1 point if and only if $L$ is globally generated, and $L$ separates 2 points if and only if $L$ is very ample. By Riemann-Roch theorem, it is elementary to see that if $\deg L\geq 2g+k$, then $L$ separates $k+1$ points. It can be also shown that if $B$ is an effective line bundle and $x_1, \ldots, x_{g+2k+1}$ are general points on $C$, then $B\big( \sum_{i=1}^{g+2k+1} x_i \big)$ separates $k+1$ points. Directly from the definition of secant sheaves, one has $H^0(C_{k+1}, E_{k+1,L})=H^0(C, L)$. Recall that the fiber of $E_{k+1, L}$ over $\xi \in C_{k+1}$ is $H^0(\xi, L|_{\xi})$. We then see that if $L$ separates $k+1$ points, then $E_{k+1,L}$ is globally generated. Thus one obtains a short exact sequence of vector bundles $$0\longrightarrow M_{k+1,L} \longrightarrow H^0(C, L)\otimes {\mathscr{O}}_{C_{k+1}} \stackrel{\operatorname{ev}}{\longrightarrow} E_{k+1,L}\longrightarrow 0,$$ where $M_{k+1, L}$ is the kernel bundle of the evaluation map $\operatorname{ev}\colon H^0(C, L)\otimes {\mathscr{O}}_{C_{k+1}} \rightarrow E_{k+1,L}$ on the global sections of $E_{k+1,L}$. For an integer $k \geq 0$, define the *secant bundle of $k$-planes* over $C_{k+1}$ to be $$B^{k}(L):={\mathbb{P}}(E_{k+1,L})$$ equipped with the natural projection $\pi_k \colon B^{k}(L) \rightarrow C_{k+1}$. Suppose that $L$ separates $k+1$ points. Then the tautological bundle ${\mathscr{O}}_{{\mathbb{P}}(E_{k+1,L})}(1)$ of $B^k(L)$ is also globally generated, and therefore, it induces a morphism $$\beta_k \colon B^k(L)\longrightarrow {\mathbb{P}}(H^0(C, L)).$$ For $k\geq 0$, assume that a line bundle $L$ on the curve $C$ separates $k+1$ points. The *$k$-th secant variety* $\Sigma_k=\Sigma_k(C, L)$ of $C$ in ${\mathbb{P}}(H^0(C, L))$ is the image of the morphism $\beta_k \colon B^k(L)\rightarrow {\mathbb{P}}(H^0(C, L))$. We have a morphism $$\beta_k \colon B^k(L) \longrightarrow \Sigma_k.$$ We use the convention that $B^{-1}(L)=\Sigma_{-1}=\emptyset$. Geometrically, if the curve $C$ is embedded by the complete linear system $|L|$ in the projective space ${\mathbb{P}}(H^0(C, L))$, then the $k$-th secant variety $\Sigma_k$ is nothing but the variety swept out by the $(k+1)$-secant $k$-planes of $C$. If $L$ separates $k+1$ points, then a $(k+1)$-secant $k$-plane of $C$ is spanned by a divisor $\xi$ on $C$ of degree $k+1$. Assume that a line bundle $L$ on the curve $C$ separates $2k+2$ points for an integer $k \geq 0$. Let $m$ be an integer with $0 \leq m\leq k$, and $x\in \Sigma_m \setminus \Sigma_{m-1}$ be a point. Since $L$ also separates $2m+2$ points, the morphism $\beta_m \colon B^m(L)\rightarrow \Sigma_m$ is an isomorphism over $U^m(L)$. Hence $x$ can be viewed as a point in $B^m(L)$. Then projecting $x$ by $\pi_m \colon B^m(L)\rightarrow C_{m+1}$, one gets a divisor $\xi_{m+1, x}$ on $C$ of degree $m+1$. It is uniquely determined by $x$. We call $\xi_{m+1, x}$ the *degree $m+1$ divisor on $C$ determined by $x$*. The above definition can be interpreted geometrically. The $m$-plane in ${\mathbb{P}}(H^0(C, L))$ spanned by $\xi_{m+1, x}$ is the unique $(m+1)$-secant $m$-plane of $C$ containing $x$. Let $x \in \Sigma_k$ be a general point so that $\xi_{k+1,x}$ contains distinct $k+1$ general points of $C$. The classical Terracini’s lemma asserts that the projective tangent space of $\Sigma_k$ at $x$ in ${\mathbb{P}}^r$ is spanned by the projective tangent lines of $C$ at the points of $\xi_{k+1,x}$. Hence the conormal space of $\Sigma_k$ in ${\mathbb{P}}^r$ at $x$ is isomorphic to $H^0(C,L(-2\xi_{k+1, x}))$. We will prove a more general version of this statement in Proposition \[p:02\] below. For $0 \leq m\leq k$, there is a natural morphism $$\alpha_{k,m} \colon B^m(L)\times C_{k-m}\longrightarrow B^k(L)$$ defined in [@Bertram:ModuliRk2 p.432, line –5], which we recall here. For any $\xi_{m+1} \in C_{m+1}$ and $\xi_{k-m} \in C_{k-m}$, let $\xi:=\xi_{m+1} + \xi_{k-m} \in C_{k+1}$. Note that the $(m+1)$-secant $m$-plane ${\mathbb{P}}(H^0(L|_{\xi_{m+1}}))$ spanned by $\xi_{m+1}$ is naturally embedded in the $(k+1)$-secant $k$-plane ${\mathbb{P}}(H^0(L|_{\xi}))$ spanned by $\xi$. Fiberwisely, $\alpha_{k,m}$ maps ${\mathbb{P}}(H^0(L|_{\xi_{m+1}}))\times \xi_{k-m}$ into ${\mathbb{P}}(H^0(L|_{\xi}))$. Next, we define the *relative secant variety $Z_m^k$ of $m$-planes* in $B^k(L)$ to be the image of the morphism $\alpha_{k,m}\colon B^m(L)\times C_{k-m}\rightarrow B^k(L)$. If the number $k$ is clear from the context, then we simply write $Z_m$ instead of $Z^k_m$. Define $$U^k(L):=B^k(L) \setminus Z^k_{k-1},$$ which is the complement of the largest relative secant variety (see [@Bertram:ModuliRk2 p.434]) The morphism $\alpha_{k,m}$ is compatible with the morphisms $\beta_k$ and $\beta_m$, i.e., one has a commutative diagram $$\xymatrix{ B^m(L)\times C_{k-m} \ar[d]_{\pi_{B^m(L)}} \ar[r]^-{\alpha_{m,k}} & B^k(L)\ar[d]^{\beta_{k}}\\ B^m(L) \ar[r]_-{\beta_{m}} &{\mathbb{P}}(H^0(L)),}$$ where $\pi_{B^m(L)}$ is the projection. It has been showed in [@Bertram:ModuliRk2 Lemma 1.4(a) and Corollary followed] that if $L$ separates $2k+2$ points, the morphism $\beta_k \colon B^k(L) \to \Sigma_k$ is birational. In particular, the restricted morphism $$\beta_k|_{U^k(L)} \colon U^k(L)\longrightarrow {\mathbb{P}}(H^0(C, L))$$ is an immersion. Especially, $\Sigma_m \setminus \Sigma_{m-1}$ is isomorphic to $U^m(L)$ for $0\leq m\leq k$. It is clear that $\beta_k(Z_m)=\Sigma_m$, so one has a commutative diagram $$\xymatrix{ Z_0 \ar@{^{(}->}[r] \ar[d]& Z_1\ar@{^{(}->}[r]\ar[d] & \cdots \ar@{^{(}->}[r] &Z_{k-1}\ar@{^{(}->}[r]\ar[d] & B^k(L) \ar[d]^{\beta_k} & \\ C \ar@{^{(}->}[r] & \Sigma_1 \ar@{^{(}->}[r]& \cdots \ar@{^{(}->}[r]&\Sigma_{k-1} \ar@{^{(}->}[r]& \Sigma_k \ar@{^{(}->}[r] & {\mathbb{P}}(H^0(L)). }$$ It is easy to check that set-theoretically $\beta_k^{-1}(\Sigma_m)=Z_m$. The set of secant varieties $\{\Sigma_i\}_{i=0}^{k-1}$ gives a stratification of $\Sigma_k$, which in turn induces a stratification by relative secant varieties $\{Z_i\}_{i=0}^{k-1}$ for $B^k(L)$. Therefore, for a point $x\in \Sigma_k$, there exists a unique integer $m$ with $0 \leq m\leq k$ such that $x\in \Sigma_m \setminus \Sigma_{m-1}$. The following is the main result of this subsection. It plays an important role in proving the normality of higher secant varieties of curves. The crucial point is the computation of the conormal sheaf $N^*_{F_{x}/B^k(L)}$. The obstruction lies on the fact that $Z_m$ is quite singular. To overcome this difficulty, we work on suitable nonsingular open subset of $Z_m$. \[p:02\]Fix an integer $k\geq 1$, and suppose that a line bundle $L$ on the curve $C$ separates $2k+2$ points. Let $m$ be an integer with $0\leq m\leq k$. Then the following hold true: 1. The commutative diagram $$\xymatrix{ U^m(L)\times C_{k-m} \ar[d]_{\pi_{U^m(L)}} \ar[r]^-{\alpha_{m,k}} & B^k(L)\ar[d]^{\beta_{k}}\\ U^m(L) \ar[r]_-{\beta_{m}} &{\mathbb{P}}(H^0(C, L))={\mathbb{P}}^r}$$ is a fiber product diagram. 2. Let $x\in \Sigma_m \setminus \Sigma_{m-1}$ be a point, $\xi_{m+1,x}$ be the unique degree $m+1$ divisor determined by $x$, and $F_{x}:=\beta_{k}^{-1}(x)$ be the fiber over $x$. Then one has the following: 1. $F_{x}\cong C_{k-m}$. 2. $N^*_{\Sigma_m/{\mathbb{P}}^r}\otimes \Bbbk(x)\cong H^0(C,L(-2\xi_{m+1, x}))$. 3. $N^*_{Z_{m}/B^k(L)}\Big|_{F_{x}}=E_{k-m,L(-2\xi_{m+1,x})}.$ 4. $N^*_{F_{x}/B^k(L)}\cong {\mathscr{O}}^{\oplus 2m+1}_{F_{x}}\oplus E_{k-m,L(-2\xi_{m+1,x})}$. 5. The natural morphism $$T^*_{x}{\mathbb{P}}^r\longrightarrow H^0(F_x, N^*_{F_x/B^k(L)} )$$ is surjective, and is an isomorphism if $m\neq k$. \(1) Let $U:={\mathbb{P}}(H^0(C, L)) \setminus \Sigma_{m-1}$ which is an open subset of ${\mathbb{P}}(H^0(C, L))$, and $V:=\beta_{k}^{-1}(U)$. Then we obtain a commutative diagram $$\xymatrix{ U^m(L)\times C_{k-m} \ar[d]_{\pi_{U^m(L)}} \ar[r]^-{\alpha_{m,k}} & V\ar[d]^{\beta_{k}}\\ U^m(L) \ar[r]_-{\beta_{m}} &U}$$ in which $\alpha_{m,k}$ and $\beta_{m}$ are closed immersions by [@Bertram:ModuliRk2 Lemma 1.2]. Write $Z:=\beta_{k}^{-1}(U^m(L))$. Then we see that $U^m(L)\times C_{k-m} \subseteq Z$. First, we claim that set-theoretically, $U^m(L)\times C_{k-m} = Z$. To see this, let $x\in \Sigma_m\subseteq \Sigma_k$ be a point. Then every $(m+1)$-secant $m$-plane containing $x$ is spanned by a unique degree $m+1$ divisor $\xi_{m+1}$ on $C$. By letting $\xi_{k-m}$ run through all points in $C_{k-m}$, one creates all possible $(k+1)$-secant $k$-plane containing $x$ spanned by $\xi_{m+1}+\xi_{k-m}$. But such $(m+1)$-secant $m$-planes are parameterized by $\beta^{-1}_m(x)$. Hence $\beta^{-1}_k(x)$ is the image of $\beta^{-1}_m(x)\times C_{k-m}$ under $\alpha_{m,k}$ as sets. This proves the claim. Next, we shall show that scheme-theoretically, $U^m(L)\times C_{k-m} = Z$. To this end, it is enough to show the natural morphism $$\beta_{k}^*(N^*_{U^m(L)/U})\longrightarrow N^*_{U^m(L)\times C_{k-m}/V}$$ of conormal sheaves is surjective. Take $x\in U^m(L)$. By base change, it is enough to show that $$\label{eq:011} \pi_{B^m(L)}^*(N^*_{U^m(L)/U}\otimes \Bbbk(x))\longrightarrow N^*_{U^m(L)\times C_{k-m}/V}|_{\{ x \} \times C_{k-m}}$$ is surjective. Following notation in [@Bertram:ModuliRk2 Lemmas 1.3 and 1.4], we have $$N^*_{U^m(L)\times C_{k-m}/V}|_{\{x\} \times C_{k-m}}=N^*_{\alpha_{k,m}}(\{ x \} \times C_{k-m}) ~~\text{ and }~~ N^*_{U^m(L)/U}\otimes \Bbbk(x)=N^*_{\beta_{m}}(x).$$ The morphism in (\[eq:011\]) is the same as $$\mu_{m,k} \colon \pi^*_{B^m(L)}N^*_{\beta_m}(x)\longrightarrow N^*_{\alpha_{m,k}}(\{x\} \times C_{k-m})$$ Hence by [@Bertram:ModuliRk2 Lemma 1.4(c)], $\mu_{m,k}$ is surjective, which completes the proof. \(2) (a) This follows directly from (1). \(b) We identify $U^m(L)=\Sigma_m \setminus \Sigma_{m-1}$. Recall that if $x$ is a general point of $U^m(L)$ and $\xi_{m+1,x}$ contains distinct $m+1$ general points of $C$, then the classical Terracini’s lemma implies that $N^*_{\Sigma_m/{\mathbb{P}}^r}\otimes \Bbbk(x)\cong H^0(C,L(-2\xi_{m+1, x}))$. Next write $\pi_C$ and $\pi_{C_{m+1}}$ to be the projections from $C_{m+1}\times C$ to the indicated factors. Let $D_{m+1}\subseteq C_{m+1}\times C$ be the universal divisor over $C_{m+1}$. Consider the sheaf ${\mathscr{M}}=\pi_{C_{m+1},*}(\pi^*_C(L)(-2D_{m+1}))$ on $C_{m+1}$. We have $$\xymatrix{ & & \pi^*_m{\mathscr{M}}|_{U^m(L)} \ar[dr]^{\eta} \ar@{^{(}->}[d] \\ 0\ar[r] & N^*_{\Sigma_m/{\mathbb{P}}^r}(1)|_{U^m(L)} \ar[r] & H^0(C,L)\otimes {\mathscr{O}}_{U^m(L)}\ar[r] & P^1({\mathscr{O}}_{ \Sigma_{m}}(1))|_{U^m(L)}\ar[r] & 0 , }$$ where $P^1({\mathscr{O}}_{\Sigma_{m}}(1))$ is the first principal part bundle. As the map $\eta$ is generically zero, it is zero. This implies that $\pi^*_m{\mathscr{M}}\cong N^*_{\Sigma_m/{\mathbb{P}}^r}(1)|_{U^m(L)}$, and the result follows. \(c) This is included in the proof of [@Bertram:ModuliRk2 Lemma 1.3] implicitly. For reader’s convenience, we outline the proof here. For a positive integer $i$, write $$D_{i+1}=C\times C_i\subseteq C\times C_{i+1}$$ to be the universal family of divisors of degree $i+1$, embedded via $(x,\xi)\mapsto (x,x+\xi)$. In the space $C\times C_{m+1}\times C_{k-m}$, we define two divisors ${\mathscr{D}}_{m+1}$ and ${\mathscr{D}}_{k-m}$ as follows $${\mathscr{D}}_{m+1}:=D_{m+1}\times C_{k-m}, \text{ and }{\mathscr{D}}_{k-m}:=C_{m+1}\times D_{k-m}.$$ They are nonsingular and meet transversally. Let $\pi_C$, $\pi_{C_{m+1}}$, $\pi_{C_{k-m}}$ be the projections of $C\times C_{m+1}\times C_{k-m}$ to the indicated factors, and $\pi^C$, $\pi^{C_{m-1}}$, $\pi^{C_{k-m}}$ be the projections to the complement of the indicated factors. Then $B^m(L)\times C_{k-m}$ can be realized as a projectivized vector bundle over $C_{m+1}\times C_{k-m}$ with a projection $\pi$, i.e., $$\pi \colon B^m(L)\times C_{k-m}={\mathbb{P}}\Big(\pi^C_*(\pi^*_CL\otimes {\mathscr{O}}_{{\mathscr{D}}_{m+1}}) \Big)\longrightarrow C_{m+1}\times C_{k-m}.$$ Let ${\mathscr{O}}_{B^m(L) \times C_{k-m}}(1)$ be the tautological line bundle on ${\mathbb{P}}\Big(\pi^C_*(\pi^*_CL\otimes {\mathscr{O}}_{{\mathscr{D}}_{m+1}}) \Big)$. Consider the vector bundle $${\mathscr{H}}=\pi^C_*(\pi^*_CL\otimes {\mathscr{O}}_{{\mathscr{D}}_{k-m}}(-2{\mathscr{D}}_{m+1})).$$ The key point proved in [@Bertram:ModuliRk2 p.439] is that $$N^*_{Z_{m}/B^k(L)}|_{U^m(L)\times C_{k-m}}\cong \pi^*{\mathscr{H}}\otimes {\mathscr{O}}_{B^m(L) \times C_{k-m}}(-1)|_{U^m(L)\times C_{k-m}}.$$ Thus we obtain $$N^*_{Z_{m}/B^k(L)}\Big|_{F_{x}}=\pi^*{\mathscr{H}}\otimes {\mathscr{O}}_{B^m(L) \times C_{k-m}}(-1)|_{F_x}$$ as $F_x\subseteq U^m(L)\times C_{k-m}$. Since ${\mathscr{O}}_{B^m(L) \times C_{k-m}}(-1)|_{F_x}={\mathscr{O}}_{F_x}$ and $\pi^*{\mathscr{H}}|_{F_x}=E_{k-m,L(-2\xi_{m+1,x})}$ by base change, the result follows immediately. \(d) By (1), we see the morphism $$\beta_k \colon U^m(L)\times C_{k-m}=Z_m \setminus Z_{m-1}\longrightarrow U^m(L)=\Sigma_m \setminus \Sigma_{m-1}$$ is a smooth morphism with fibers $C_{k-m}$. Thus we have $$N^*_{F_x/Z_m}=T^*_x\Sigma_m\otimes {\mathscr{O}}_{F_x}={\mathscr{O}}_{F_x}^{\oplus 2m+1}$$ since $\Sigma_m$ is nonsingular at $x$ and has dimension $2m+1$. In particular, $H^0(N^*_{F_x/Z_m})=T^*_x\Sigma_m$. Consider the short exact sequence $$\label{eq:02} 0\longrightarrow N^*_{Z_m/B^k(L)}|_{F_x}\longrightarrow N^*_{F_x/B^k(L)}\longrightarrow N^*_{F_x/Z_m}\longrightarrow 0.$$ We claim that the above short exact sequence splits. To this end, consider the diagram $$\xymatrix{ T^*_x{\mathbb{P}}(H^0(C, L)) \ar[d] \ar@{->>}[r] & T^*_x\Sigma_m\ar[d]^{=}\\ H^0(F_x, N^*_{F_x/B^k(L)}) \ar[r] & H^0(F_x, N^*_{F_x/Z_m}). }$$ We see that the morphism $H^0(F_x, N^*_{F_x/B^k(L)}) \rightarrow H^0(F_x, N^*_{F_x/Z_m})$ is surjective. Thus the short exact sequence (\[eq:02\]) splits because $N^*_{F_x/Z_m}$ is a direct sum of ${\mathscr{O}}_{F_x}$. Hence, we obtain $$N^*_{F_x/B^k(L)} = N^*_{Z_m/B^k(L)}|_{F_x}\oplus N^*_{F_x/Z_m} = E_{k-m,L(-2\xi_{m+1,x})} \oplus {\mathscr{O}}^{\oplus 2m+1}_{F_{x}},$$ as desired. \(e) Now we use (b), (d) and the sequence (\[eq:02\]) to form the commutative diagram $$\xymatrix{ 0 \ar[r] & H^0(C, L(-2\xi_{m+1, x})) \ar[r] \ar[d]^-{=} & T^*_x{\mathbb{P}}^r \ar[r] \ar[d] & T^*_x\Sigma_m \ar[r] \ar[d]^-{=} & 0\\ 0 \ar[r]& H^0(C_{k-m}, E_{k-m,L(-2\xi_{m+1, x})}) \ar[r]& H^0(F_x, N^*_{F_x/B^k(L)}) \ar[r]& T^*_x\Sigma_m \ar[r] & 0. }$$ The result then follows immediately. In the proposition above, it is worth noting that $Z_m \setminus Z_{m-1}= U^m(L)\times C_{k-m}$ and $U^m(L)= \Sigma_m \setminus \Sigma_{m-1}$. Therefore, we actually obtain a fiber product diagram $$\xymatrix{ Z_{m} \setminus Z_{m-1} \ar[d] \ar@{^{(}->}[r] & B^k(L)\ar[d]^{\beta_{k}}\\ \Sigma_m \setminus \Sigma_{m-1} \ar@{^{(}->}[r] &{\mathbb{P}}(H^0(C, L)) }$$ which means that $Z_m \setminus Z_{m-1}$ is the scheme-theoretical preimage of $\Sigma_m\setminus \Sigma_{m-1}$. Blowup construction of secant bundles {#subsec:blowup} ------------------------------------- We keep assuming that $k\geq 1$ and $\deg L\geq 2g+2k+1$. We use the blowup construction of secant bundles established in [@Bertram:ModuliRk2 Propostitions 2.2, 2.3 and Corollary 2.4]. For each $0\leq m\leq k$, we will consecutively blowup $B^m(L)$ along smooth centers $m$-times to obtain smooth varieties $$\operatorname{bl}_1(B^m(L)), \ \operatorname{bl}_2(B^m(L)), \ \ldots, \ \operatorname{bl}_m(B^m(L)).$$ If $m=0$, then there is nothing to blowup. We simply set $\operatorname{bl}_0(B^0(L)):=B^0(L)=C$. Thus we now start with constructing $\operatorname{bl}_1(B^m(L))$ for $m\geq 1$. Notice that the natural morphism $\alpha_{m,0} \colon B^0(L)\times C_m\rightarrow B^m(L)$ is a closed embedding for $m\geq 1$. We then define $$\operatorname{bl}_1(B^m(L)):=\text{ blowup of }B^m(L) \text{ along }B^0(C)\times C_m.$$ If $m=1$, then we are done. Otherwise, if $m\geq 2$, then suppose that $\operatorname{bl}_i(B^m(L))$ has been defined for any $1\leq i\leq m-1$. By [@Bertram:ModuliRk2 Proposition 2.2] and its proof (for instance, the claim in the last two lines on page 444 of [@Bertram:ModuliRk2]), we see that the natural morphism $\operatorname{bl}_i(B^i(L))\times C_{m-i}\rightarrow \operatorname{bl}_i(B^m(L))$ is a closed embedding. We then define $$\operatorname{bl}_{i+1}(B^m(L)):=\text{ blowup of }\operatorname{bl}_i(B^m(L)) \text{ along }\operatorname{bl}_i(B^i(C))\times C_{m-i}.$$ This construction works for any integer $m$ with $0\leq m\leq k$. We write $$b_m \colon \operatorname{bl}_m(B^m(L))\longrightarrow B^m(L)$$ the composition map of blowups. Denote by $E_i$ for $0\leq i\leq m-1$ the exceptional divisor on $\operatorname{bl}_m(B^m(l))$ which is from the $(i+1)$-th blowup. Note that $\beta_m(b_m(E_i))=\Sigma_i$. It has been showed in [@Bertram:ModuliRk2] that in each stage of blowups, the exceptional divisors always meet transversally with the center of the next blowup. Therefore, the divisor $E_0 + \cdots + E_{m-1}$ on $\operatorname{bl}_m(B^m(L))$ has a simple normal crossing support. As proved in [@Bertram:ModuliRk2], we have $$E_i \cap E_{i+1} \cap \cdots \cap E_{m-1} = \operatorname{bl}_i(B^i(L)) \times C^{m-i}~\text{ for $0 \leq i \leq m-1$}.$$ For example, $E_{m-1} = \operatorname{bl}_{m-1}(B^{m-1}(L))\times C$ and $E_0\cap \cdots \cap E_{m-1}=\operatorname{bl}_0(B^0(L))\times C^m=C^{m+1}$. In particular, for $m=k$ we get the following diagram describing blowups of $B^k(L)$: $$\Small \xymatrix@R-7pt@C-11Pt{ & & & & & \operatorname{bl}_k(B^k(L))\ar[d]^{\cong} \\ & & & & \operatorname{bl}_{k-1}(B^{k-1}(L)) \times C \ar@{^{(}->}[r]\ar[d]& \operatorname{bl}_{k-1}(B^k(L))\ar[d] \\ & & & & \vdots\ar[d]& \vdots\ar[d] \\ & & \operatorname{bl}_2(B^2(L))\times C_{k-2} \ar[d] \ar@{^{(}->}[r] &\cdots\ar@{^{(}->}[r] & \operatorname{bl}_2(Z_{k-1})\ar[d]\ar@{^{(}->}[r]& \operatorname{bl}_2(B^k(L))\ar[d] \\ & \operatorname{bl}_1(B^1(L))\times C_{k-1} \ar[d] \ar@{^{(}->}[r] & \operatorname{bl}_1(Z_2) \ar[d]\ar@{^{(}->}[r] &\cdots\ar@{^{(}->}[r] & \operatorname{bl}_1(Z_{k-1})\ar[d]\ar@{^{(}->}[r]& \operatorname{bl}_1(B^k(L)) \ar[d]^{} \\ B^0(L)\times C_k \ar@{^{(}->}[r] \ar[d] & Z_1\ar[d]\ar@{^{(}->}[r] & Z_2 \ar@{^{(}->}[r]\ar[d] &\cdots\ar@{^{(}->}[r] &Z_{k-1}\ar[d]\ar@{^{(}->}[r]& B^k(L) \ar[d]^-{\beta_k} \\ C & \Sigma_1 & \Sigma_2 & \cdots &\Sigma_{k-1}&\Sigma_k . }$$ where $\operatorname{bl}_i(Z_{l})$ is the strict transform of the variety $Z_{l}$ in $\operatorname{bl}_i(B^k(L))$. The variety on the left end of each row in the diagram is the center of the blowup for the next step. If we focus on the final step of blowups of $B^k(L)$, we obtain the following digram $$\tiny \xymatrix@R-5pt@C-8Pt{ E_0\cap \cdots\cap E_{k-1}\ar@{=}[d] &E_1\cap \cdots\cap E_{k-1}\ar@{=}[d] & E_2\cap \cdots \cap E_{k-1}\ar@{=}[d] &\cdots & E_{k-1}\ar@{=}[d]& \\ \operatorname{bl}_0(B^0(L))\times C^k\ar@{^{(}->}[r]\ar[d] &\operatorname{bl}_1(B^1(L))\times C^{k-1}\ar@{^{(}->}[r]\ar[d] & \operatorname{bl}_2(B^2(L))\times C^{k-2} \ar@{^{(}->}[r]\ar[d] &\cdots\ar@{^{(}->}[r] & \operatorname{bl}_{k-1}(B^{k-1}(L))\times C \ar@{^{(}->}[r]\ar[d]&\operatorname{bl}_k(B^k(L))\ar[d]^{b_k}\ar[d]^{} \\ B^0(L)\times C_k \ar@{^{(}->}[r] \ar[d] & Z_1\ar[d]\ar@{^{(}->}[r] & Z_2 \ar@{^{(}->}[r]\ar[d] &\cdots\ar@{^{(}->}[r] &Z_{k-1}\ar@{^{(}->}[r]\ar[d]& B^k(L)\ar[d]^-{\beta_k}\\ C & \Sigma_1 & \Sigma_2 & \cdots & \Sigma_{k-1}&\Sigma_k . }$$ The following is the main result of this subsection. It plays a crucial role in the proofs of the main theorems of the paper. \[p:01\] Fix an integer $k \geq 1$, and let $L$ be a line bundle on the curve $C$ with $\deg L \geq 2g+2k+1$. Recall that $\pi_k \colon B^k(L)\rightarrow C_{k+1}$ is the canonical projection. Then the following hold true: 1. $Z_{k-1}$ is flat over $C_{k+1}$. 2. Let $H$ be the tautological divisor on $B^k(L)={\mathbb{P}}(E_{k+1,L})$ so that ${\mathscr{O}}_{B^k(L)}(H):=\beta_k^*{\mathscr{O}}_{\Sigma_k}(1)$. Then one has $$\def\arraystretch{1.5} \begin{array}{l} {\mathscr{O}}_{B^k(L)}((k+1)H-Z_{k-1}) = \pi_k^*A_{k+1,L}, \\[5pt] R^i\pi_{k,*}{\mathscr{O}}_{B^k(L)}(\ell H-Z_{k-1})=\left\{ \begin{array}{ll} 0 & \textrm{for }i\geq0,\ 0< \ell \leq k\vspace{0.2cm}\\ 0 & \textrm{for }i>0,\ \ell \geq k+1. \end{array} \right. \end{array}$$ 3. $b_k \colon \operatorname{bl}_k(B^k(L))\rightarrow B^k(L)$ is a log resolution of the pair $(B^k(L), Z_{k-1})$ such that $$\def\arraystretch{1.5} \begin{array}{l} K_{\operatorname{bl}_k(B^k(L))}=b^*_k(K_{B^k(L)}+Z_{k-1})-E_0-E_1-\cdots -E_{k-1},\\ b^*_{k} Z_{k-1}=kE_0+(k-1)E_1+\cdots +E_{k-1}. \end{array}$$ We keep using the blowup construction of secant varieties. \(1) Recall that $Z_{k-1}$ is the image of the map $\alpha_{k-1,k} \colon B^{k-1}(L)\times C\rightarrow B^k(L)$ and $\alpha_{k-1,k}$ is birational to $Z_{k-1}$ since $L$ separates $2k+2$ points (see [@Bertram:ModuliRk2 Lemma 1.2]). Hence $Z_{k-1}$ is an irreducible divisor in $B^k(L)$, and therefore, is Cohen–Macaulay. Now for any point $\xi\in C_{k+1}$, the fiber of the map $Z_{k-1}\rightarrow C_{k+1}$ over $\xi$, at least set-theoretically, is the union of the linear spaces spanned by the length $k$ subschemes of $\xi$. Hence the fiber over $\xi$ has dimension $k-1$. By [@Mat86 23.1], we see that $Z_{k-1}$ is flat over $C_{k+1}$. \(2) Take a general point $\xi\in C_{k+1}$. Without loss of generality, we may assume that $\xi=x_1+\cdots+x_{k+1}$ is a sum of distinct $k+1$ points on $C$. Write $F_\xi:=\pi^{-1}_k(\xi)$ the fiber over $\xi$. Note that $F_\xi={\mathbb{P}}^k$, which can be regarded as a linear subspace of ${\mathbb{P}}(H^0(C, L))$ spanned by $x_1, \ldots, x_{k+1}$. In other words, $F_{\xi}$ is the $k$-plane secant to $C$ along $x_1, \ldots, x_{k+1}$. Write $\widetilde{F}_\xi$ the strict transform of $F_\xi$ under the birational morphism $b_k$. Write $\Lambda_i=F_\xi\cap Z_i$ for $0\leq i\leq k-1$. We note that $$\def\arraystretch{2} \begin{array}{rcl} \displaystyle \Lambda_0 &=&\displaystyle F_{\xi}\cap Z_0 =F_\xi\cap B^0(L)\times C_k=\{x_1, x_2,\cdots,x_{k+1}\},\\ \displaystyle \Lambda_1 &=&\displaystyle F_\xi\cap Z_1=\bigcup_{i\neq j}\overline{x_ix_j},\\ &&\vdots\\ \displaystyle\Lambda_{k-1}&=&\displaystyle F_\xi\cap Z_{k-1}= \bigcup_{i_1\neq i_2\neq \cdots \neq i_k}\overline{x_{i_1}x_{i_2}\cdots x_{i_k}}. \end{array}$$ To obtain $\widetilde{F}_{\xi}$, we blowup $F_\xi$ along $\Lambda_0$ and then blowup along the strict transform of $\Lambda_1$, and so on. Now, the number of irreducible components of $\Lambda_{k-1}$ containing $\overline{x_{i_1} \cdots x_{i_m}}$ is ${k+1-m \choose k-m}$ for all $1 \leq m \leq k$. This allows us to calculate the total transform of $\Lambda_{k-1}$ in $\widetilde{F}_\xi$, which in turn implies that $$\label{eq:Z_{k-1}} b^*_k Z_{k-1}={k\choose k-1}E_0+{k-1\choose k-2}E_1+\cdots+{1\choose 0}E_{k-1}=kE_0+(k-1)E_1+\cdots +E_{k-1}$$ because $\widetilde{F}_{\xi}$ meets all the divisors $E_0, \ldots, E_{k-1}$ transversally and $\widetilde{F}_{\xi} \cap E_{m-1}$ is the union of strict transforms of the exceptional divisors over $\Lambda_{m-1}$ for all $1 \leq m \leq k$. For a coherent sheaf ${\mathscr{F}}$ (resp. a subscheme $Z$) on $B^k(L)$ and for a point $\xi'\in C_{k+1}$, we denote by ${\mathscr{F}}_{\xi'}$ (resp. $Z_{\xi'}$) the fiber over $\xi'$. In this notation, $Z_{k-1,\xi}=\Lambda_{k-1}$ is a union of $k+1$ distinct linear spaces ${\mathbb{P}}^{k-1}$ in $B^k(L)_\xi={\mathbb{P}}^k$. Therefore $Z_{k-1,\xi}$ is a degree $k+1$ divisor in $B^k(L)_\xi$. By the result (1), $Z_{k-1}$ is flat over $C_{k+1}$, so the degree of $Z_{k-1,\xi'}$ in $B^k(L)_{\xi'}$ is $k+1$ for all $\xi'\in C_{k+1}$. This implies that $${\mathscr{O}}_{B^k(L)}(\ell H-Z_{k-1})_{\xi'}\cong {\mathscr{O}}_{{\mathbb{P}}^k}(\ell-(k+1))~ \text{ for all } \ell \in {\mathbb{Z}}.$$ Hence the function $h^0({\mathscr{O}}_{B^k(L)}((k+1)H-Z_{k-1})_{\xi'})=1$ for all $\xi'\in C_{k+1}$. Thus $$A:=\pi_{k,*}{\mathscr{O}}_{B^k(L)}((k+1)H-Z_{k-1})$$ is a line bundle on $C_{k+1}$. Since $\pi_k \colon {\mathbb{P}}(E_{k+1,L}) \to C_{k+1}$ is the natural projection, we have $$\pi_k^*A\cong {\mathscr{O}}_{B^k(L)}((k+1)H-Z_{k-1}).$$ Similarly, if $0< \ell \leq k$, then $h^i({\mathscr{O}}_{B^k(L)}(\ell H-Z_{k-1})_{\xi'})=0$ for all $i\geq 0$, and if $\ell \geq k+1$, then $h^i({\mathscr{O}}_{B^k(L)}(\ell H-Z_{k-1})_{\xi'})=0$ for all $i> 0$. Thus we obtain the second result in (2). Next, we show that $A=A_{k+1,L}$. We focus on the following commutative diagram $$\xymatrix{ C^{k+1} \ar@{^{(}->}[r] \ar[rrrdd]_-{q:=q_{k+1}}& \operatorname{bl}_1(B^1(L))\times C^2 \ar@{^{(}->}[r]& \operatorname{bl}_2(B^2(L))\times C \ar@{^{(}->}[r] & \operatorname{bl}_k(B^k(L))\ar[d]^{b_k} \\ & & & B^k(L) \ar[d]^{\pi_k} \\ & & & C_{k+1}. }$$ We have $$\def\arraystretch{1.5} \begin{array}{l} b^*_k(\pi_k^*A)|_{C^{k+1}}=q^*A,\\ b^*_k((k+1)H-Z_{k-1})|_{C^{k+1}}=(k+1)H-(kE_0+(k-1)E_1+\cdots +E_{k-1})|_{C^{k+1}}, \end{array}$$ where by abuse of notation we write $H=b^*_kH|_{C^{k+1}}$. Hence, on $C^{k+1}$, we have $$(k+1)H-(kE_0+(k-1)E_1+\cdots +E_{k-1})|_{C^{k+1}}\sim_{\operatorname{lin}} q^*A.$$ Recall that $C^{k+1}$ is a complete intersection in $\operatorname{bl}_k(B^k(L))$ cut out by the divisors $E_0, E_1, \ldots, E_{k-1}$. Thus we have $$\det N^*_{C^{k+1}/\operatorname{bl}_k(B^k(L))}={\mathscr{O}}_{C^{k+1}}(-E_0-E_1-\cdots -E_{k-1}).$$ Using the formula $\det N^*_{C^{k+1}/\operatorname{bl}_k(B^k(L))}=\omega_{\operatorname{bl}_k(B^k(L))}|_{C^{k+1}} \otimes \omega_{C^{k+1}}^{-1}$, we get $$\label{eq:03} -(E_0+E_1+\cdots +E_{k-1})|_{C^{k+1}}=K_{\operatorname{bl}_k(B^k(L))}|_{C^{k+1}}-K_{C^{k+1}}.$$ Recall that $\operatorname{bl}_k(B^k(L))$ is obtained by consecutively blowing up the smooth centers $\operatorname{bl}_i(B^i(L))\times C_{k-i}$ which has codimension $k-i$. Thus we find $$\label{eq:04} -((k-1)\cdot E_0+\cdots +1\cdot E_{k-2}+0\cdot E_{k-1})=- K_{\operatorname{bl}_k(B^k(L))}+b^*_kK_{B^k(L)}.$$ Combining (\[eq:03\]) and (\[eq:04\]), we obtain $$-(kE_0+(k-1)E_1+\cdots +E_{k-1})|_{C^{k+1}}=-K_{C^{k+1}}+b_k^*K_{B^k(L)}|_{C^{k+1}}.$$ Recall that $B^k(L)={\mathbb{P}}(E_{k+1,L})$ is a projectivized vector bundle over $C_{k+1}$. Thus we have $$\begin{aligned} K_{B^k(L)} & = &-(k+1)H+\pi_k^*\det E_{k+1,L}+\pi_k^*K_{C_{k+1}}\nonumber \\ & = & -(k+1)H+\pi_k^*T_{k+1}(L)(-\delta_{k+1})+\pi_k^*T_{k+1}(K_C)(-\delta_{k+1})\nonumber. \end{aligned}$$ Finally, we compute $$\begin{aligned} & &(k+1)H-(kE_0+(k-1)E_1+\cdots +E_{k-1})|_{C^{k+1}} \nonumber \\ & =&(k+1)H-K_{C^{k+1}}+\pi_k^*K_{B^k(L)}|_{C^{k+1}}\nonumber\\ & =& (k+1)H-K_{C^{k+1}} +[-(k+1)H+q^*T_{k+1}(L)(-\delta_{k+1})+q^*T_{k+1}(K_C)(-\delta_{k+1})]\nonumber\\ & =& q^*(T_{k+1}(L)(-2\delta_{k+1})).\nonumber \end{aligned}$$ Thus $q^*A \cong q^*(T_{k+1}(L)(-2\delta_{k+1}))$. Since $q^* \colon \operatorname{Pic}C_{k+1}\rightarrow \operatorname{Pic}C^{k+1}$ is injective, one gets $A \cong T_{k+1}(L)(-2\delta_{k+1})=A_{k+1,L}$. This proves the first result of (2). \(3) Recall that $E_0 + \cdots + E_k$ has a simple normal crossing support. Thus the birational morphism $b_k \colon \operatorname{bl}_k(B^k(L))\rightarrow B^k(L)$ is a log resolution of the pair $(B^k(L), Z_{k-1})$. The remaining assertions follow from (\[eq:Z\_[k-1]{}\]) and (\[eq:04\]). A vanishing theorem on Cartesian products of curves {#sec:vanishing} =================================================== The aim of this section is to establish a vanishing theorem on the product of a curve. It is inspired by Rathmann’s vanishing results in [@Rathmann Section 3]. A similar result on $C^2$ has been proved by Yang [@Yang:Letter]. Let us keep the notations introduced in previous sections. Let $k\geq 0$ be an integer. Recall that given a line bundle $L$ on the curve $C$ separating $k+1$ points, there is a short exact sequence $$0\longrightarrow M_{k+1,L} \longrightarrow H^0(C, L)\otimes {\mathscr{O}}_{C_{k+1}} \longrightarrow E_{k+1,L}\longrightarrow 0$$ on $C_{k+1}$ (see Subsection \[subsec:secvar\]). Recall also the quotient morphism $q_{k+1} \colon C^{k+1} \to C_{k+1}$, the pairwise diagonal $\Delta_{u,v}:=\{ (x_1, \ldots, x_k) \in C^{k+1} \mid x_u=x_v \}$ on $C^{k+1}$, and $\Delta_{k+1}:=\sum_{1 \leq u < v \leq k+1} \Delta_{u,v}$. We define the locally free sheaf $$Q_{k+1, L}:=q_{k+1}^*M_{k+1,L}.$$ on the Cartesian product $C^{k+1}$ of the curve $C$. Note that $$Q_{k+1, L}=p_*\left( ({\mathscr{O}}_{C^{k+1}} \boxtimes L) \left(-\sum_{u=1}^{k+1}\Delta_{u, k+2} \right) \right),$$ where $p \colon C^{k+2} \to C^{k+1}$ is the projection to the first $k+1$ components. \[vanishing\] Let $C$ be a nonsingular projective curve of genus $g$, and $L$ be a line bundle on $C$. For an integer $k \geq 0$, let $B=B'\big(\sum_{i=1}^{g+2k+1}x_i \big)$ be a line bundle on $C$, where $B'$ is an effective line bundle and $x_1, \ldots, x_{g+2k+1}$ are general points on $C$. For integers $i>0$ and $j \geq 0$, suppose that $$\deg L \geq 2g+2k+1-i+j.$$ Then one has $$\label{eq-van} H^i\big(C^{k+1}, \wedge^j Q_{k+1, B} \otimes L^{\boxtimes k+1}\big(-\Delta_{k+1} \big)\big) = 0.$$ Suppose that $B' \neq {\mathscr{O}}_C$ so that $b:=\deg B' >0$. We can write $B'={\mathscr{O}}_C\big(\sum_{i=1}^{b}x_i'\big)$, where $x_1', \ldots, x_b'$ are (possibly non-distinct) points on $C$. We set $B_0:={\mathscr{O}}_C\big( \sum_{i=1}^{g+2k+1} x_i \big)$ and $B_{\ell}:=B_0\big(\sum_{i=1}^{\ell}x_i' \big)$ for $1 \leq \ell \leq b$. Then $B_{\ell}$ separates $k+1$ points for each $0 \leq \ell \leq b$, and $B_b=B$. For $0 \leq \ell \leq b-1$, we have an exact sequence $$0 \longrightarrow Q_{k+1, B_{\ell}} \longrightarrow Q_{k+1, B_{\ell+1}} \longrightarrow {\mathscr{O}}_{C}(-x_{\ell+1}')^{\boxtimes k+1} \longrightarrow 0,$$ which induces an exact sequence $$0 \longrightarrow \wedge^j Q_{k+1, B_{\ell}} \longrightarrow \wedge^j Q_{k+1, B_{\ell+1}} \longrightarrow \wedge^{j-1} Q_{k+1, B_{\ell}} \otimes {\mathscr{O}}_C(-x_{\ell+1}')^{\boxtimes k+1} \longrightarrow 0.$$ Then we see that the cohomology vanishing $$H^i\big(C^{k+1},\wedge^j Q_{k+1, B_{\ell+1}} \otimes L^{\boxtimes k+1}\big(-\Delta_{k+1}\big)\big)=0$$ follows from the cohomology vanishing $$\def\arraystretch{1.5} \begin{array}{l} H^i\big(C^{k+1},\wedge^j Q_{k+1, B_{\ell}} \otimes L^{\boxtimes k+1}\big(-\Delta_{k+1}\big)\big)=0,\\ H^i\big(C^{k+1},\wedge^{j-1} Q_{k+1, B_{\ell}} \otimes L(-x_{\ell+1}')^{\boxtimes k+1}\big(-\Delta_{k+1}\big)\big)=0. \end{array}$$ Note that $\deg L \geq 2g+2k+1-i+j$ and $\deg L(-x_{\ell+1}') \geq 2g+2k+1-i+(j-1)$. For each $k$, by the induction on $\ell$, we can conclude that the cohomology vanishing (\[eq-van\]) for $B=B_0$ (or equivalently, $B'={\mathscr{O}}_C$) implies the cohomology vanishing (\[eq-van\]) for arbitrary $B$. We now proceed by the induction on $k$. First, we consider the case that $k=0$ and $B'={\mathscr{O}}_C$. Since $B={\mathscr{O}}_C \big( \sum_{i=1}^{g+1} x_i \big)$ is base point free, we have an exact sequence $$0 \longrightarrow Q_{1, B} \longrightarrow H^0(C, B) \otimes {\mathscr{O}}_C \longrightarrow B \longrightarrow 0.$$ By Riemann-Roch theorem, we find $h^0(C, B)=2$, so $Q_{1,B}=B^{-1}$ is a line bundle. In this case, the required cohomology vanishing (\[eq-van\]) for $B=B_0$ is nothing but $$\def\arraystretch{1.5} \begin{array}{l} H^1(C, L)=0 ~\text{ when $i=1, ~j=0, ~\deg L \geq 2g$,}\\ H^1(C, L \otimes B^{-1})=0~\text{ when $i=1,~ j=1, ~\deg L \geq 2g+1$.} \end{array}$$ The first vanishing is trivial, and the second vanishing follows from that $\deg L \otimes B^{-1} \geq g$. Thus the cohomology vanishing (\[eq-van\]) holds for $B=B_0$, and so does for arbitrary $B$ when $k=0$. Suppose now that $k >0$. By the induction on $k$, for smaller $k$, we assume that the cohomology vanishing (\[eq-van\]) holds for arbitrary $B$. We consider the case that $B=B_0={\mathscr{O}}_C\big(\sum_{i=1}^{g+2k+1}x_i \big)$. Assume that $j=\operatorname{rank}(Q_{k+1, B})=k+1$. Note that $\det Q_{k+1, B} = (B^{-1})^{\boxtimes k+1} (\Delta_{k+1})$. Then the desired cohomology vanishing (\[eq-van\]) is nothing but $$H^i(C^{k+1}, (L\otimes B^{-1})^{\boxtimes k+1})=0~\text{ for $i>0$}.$$ Since $\deg L \geq 2g+2k+1-i+(k+1)$, we have $$\deg L\otimes B^{-1} \geq 2g+3k+2-i - (g+2k+1) = g+k+1-i \geq g.$$ Thus $H^1(C, L\otimes B^{-1})=0$. By Künneth formula, the above vanishing holds. Assume that $j < \operatorname{rank}(Q_{k+1, B})$. From the definition of $Q_{k,L}$ one can deduce a short exact sequence $$0 \longrightarrow Q_{k+1, B} \longrightarrow Q_{k, B} \boxtimes {\mathscr{O}}_C \longrightarrow ({\mathscr{O}}_{C^k}\boxtimes B) \left(-\sum_{u=1}^{k}\Delta_{u, k+1} \right) \longrightarrow 0.$$ The Koszul complex then gives rise to a resolution of $\wedge^j Q_{k+1, B}$: $$\cdots \to (\wedge^{j+2}Q_{k, B} \boxtimes B^{-2}) \left(2 \sum_{u=1}^k \Delta_{u, k+1} \right) \to (\wedge^{j+1}Q_{k, B} \boxtimes B^{-1}) \left( \sum_{u=1}^k \Delta_{u, k+1} \right) \to \wedge^j Q_{k+1, B} \to 0$$ (see also [@Rathmann Proposition 3.1]). Thus to show the required cohomology vanishing (\[eq-van\]), it suffices to check that $$\label{eq-van-inproof} H^{i+\ell} \left(C^{k+1}, \big( (\wedge^{j+\ell+1}Q_{k, B} \otimes L^{\boxtimes k}) \boxtimes (L\otimes B^{-\ell-1} )\big) \left( (\ell+1) \sum_{u=1}^{k} \Delta_{u,k+1} - \Delta_{k+1} \right) \right)=0$$ for $\ell \geq 0$. In the sequel, we establish (\[eq-van-inproof\]) under the assumption $\deg L \geq 2g+2k+1-i+j$ and $B=B_0={\mathscr{O}}_C\big(\sum_{i=1}^{g+2k+1}x_i \big)$. Consider the case that $i + \ell \leq 1$, i.e., $i=1, \ell=0$. In this case, we have $$\deg L \otimes B^{-1} \geq 2g+2k+1-1+j - (g+2k+1)=g-1+j \geq g-1$$ so that $H^1(C, L \otimes B^{-1})=0$. Note that $$\sum_{u=1}^k \Delta_{u, k+1} - \Delta_{k+1} = -\sum_{1 \leq u < v \leq k} \Delta_{u,v} = - \Delta_k.$$ Since we have $$\deg L \geq 2g+2k+j \geq 2g+2k-1+j = 2g+2(k-1)+1-1 + (j+1),$$ it follows from the induction on $k$ that $$H^1 \big(C^k, \wedge^{j+1}Q_{k,B}\otimes L^{\boxtimes k}(-\Delta_k) \big) = 0.$$ By Künneth formula, we obtain the desired vanishing (\[eq-van-inproof\]) $$H^{1} \big(C^{k+1}, \big( \wedge^{j+1}Q_{k,B}\otimes L^{\boxtimes k}(-\Delta_k) \big) \boxtimes (L \otimes B^{-1}) \big) =0.$$ Consider the case that $i+\ell \geq 2$. Let $\text{pr}_{k+1} \colon C^{k+1} \to C$ be the projection to the $(k+1)$-th component. The fiber of $$R^{i'}\text{pr}_{k+1, *} \left( \big( (\wedge^{j+\ell+1}Q_{k, B} \otimes L^{\boxtimes k} ) \boxtimes (L\otimes B^{-\ell -1} )\big) \left( (\ell+1) \sum_{u=1}^{k} \Delta_{u,k+1} - \Delta_{k+1}\right) \right)$$ over $x \in C$ is $$\label{eq-van-inproof2} H^{i'}\big(C^k, \wedge^{j+\ell+1}Q_{k,B} \otimes L(\ell x)^{\boxtimes k} (-\Delta_k) \big).$$ By considering the Leray spectral sequence for $\text{pr}_{k+1, *}$, to show the desired vanishing (\[eq-van-inproof\]) $$H^{i+\ell} \left(C^{k+1}, \big( (\wedge^{j+\ell+1}Q_{k, B} \otimes L^{\boxtimes k} ) \boxtimes (L \otimes B^{-\ell-1})\big) \left( (\ell+1) \sum_{u=1}^{k} \Delta_{u,k+1} - \Delta_{k+1} \right) \right)=0,$$ it is enough to prove that the cohomology (\[eq-van-inproof2\]) vanishes for $i' = i+\ell-1, i+\ell$. For this $i'$, we have $i' \geq i-1$, so we find $$\deg L(\ell x) \geq 2g+2k+1-i+j + \ell \geq 2g+2(k-1)+1-i' + (j+\ell+1).$$ By the induction on $k$, we see that the cohomology (\[eq-van-inproof2\]) vanishes for $i' = i+\ell-1, i+\ell$. Thus we obtain the desired vanishing (\[eq-van-inproof\]). Therefore, the cohomology vanishing (\[eq-van\]) for $B=B_0$ follows, and so does for arbitrary $B$. We complete the proof. Properties of secant varieties of curves {#sec:prop} ======================================== This section is devoted to the study of various properties of secant varieties of curves. In particular, we prove the main results of the paper; Theorem \[main1:singularities\] follows from Theorem \[normality\] and Proposition \[sing\], and Theorem \[main2:syzygies\] follows from Theorem \[normality\], Theorem \[p:12\], and Corollary \[acmreg\]. We keep using notations introduced before. Recall that $C$ is a nonsingular projective curve of genus $g$ embedded by a very ample line bundle $L$ in the space ${\mathbb{P}}(H^0(C, L))={\mathbb{P}}^r$. Consider the $k$-th secant variety $\Sigma_{k}=\Sigma_{k}(C,L)$ in ${\mathbb{P}}^r$. As ${\mathscr{O}}_{ \Sigma_{k}}(1)$ is globally generated by the linear forms of ${\mathbb{P}}^r$, the evaluation map on the global sections of ${\mathscr{O}}_{ \Sigma_{k}}(1)$ induces an short exact sequence $$\label{eq:08} 0\longrightarrow M_{\Sigma_{k}}\longrightarrow H^0(C,L)\otimes {\mathscr{O}}_{ \Sigma_{k}}\longrightarrow{\mathscr{O}}_{ \Sigma_{k}}(1)\longrightarrow 0,$$ where $M_{\Sigma_{k}}$ is the kernel bundle. Moreover, we also need to consider the $(k-1)$-th secant variety $\Sigma_{k-1}=\Sigma_{k-1}(C,L)$, and use the following exact sequence $$\label{eq:07} 0 \longrightarrow I_{\Sigma_{k-1}|\Sigma_k} \longrightarrow {\mathscr{O}}_{\Sigma_k} \longrightarrow {\mathscr{O}}_{\Sigma_{k-1}} \longrightarrow 0,$$ where $I_{\Sigma_{k-1}|\Sigma_k}$ is the defining ideal sheaf of $\Sigma_{k-1}$ in $\Sigma_k$. Recall the birational morphism $\beta_k \colon B^k(L)\rightarrow \Sigma_k$ and the relative secant variety $Z_{k-1}$ on $B^k(L)$. Suppose that $\Sigma_k$ is normal. By Zariski’s main theorem, $\beta_{k,*}{\mathscr{O}}_{B^k(L)} = {\mathscr{O}}_{\Sigma_k}$, and hence, $$\beta_{k,*}{\mathscr{O}}_{B^k(L)}(-Z_{k-1})=I_{\Sigma_{k-1}|\Sigma_k}.$$ The following lemma is a consequence of the vanishing theorem established in Section \[sec:vanishing\]. \[keylemma\] Let $k\geq 0$ and $p\geq 0$ be integers, and $L$ be a line bundle on $C$. Assume that $$\deg L\geq 2g+2k+1+p.$$ Consider the $k$-th secant variety $\Sigma_{k}=\Sigma_{k}(C,L)$ in the space ${\mathbb{P}}(H^0(C, L))={\mathbb{P}}^r$. If $\Sigma_k$ is normal and $R^i\beta_{k,*}{\mathscr{O}}_{B^k(L)}(-Z_{k-1})=0 \text{ for all }i>0$, then one has $$H^i(\Sigma_k, \wedge^j M_{\Sigma_k} \otimes I_{\Sigma_{k-1}|\Sigma_k}(k+1))=0~\text{ for $i \geq j-p,~ i \geq 1, ~j \geq 0$.}$$ Recall that $B^k(L)={\mathbb{P}}(E_{k+1, L})$ with the natural projection $\pi_k \colon B^k(L) \to C_{k+1}$. Let $H$ be the tautological divisor on $B^k(L)$ so that ${\mathscr{O}}_{B^k(L)}(H)={\mathscr{O}}_{B^k(L)}(1)=\beta_k^*{\mathscr{O}}_{\Sigma_k}(1)$. One can identify $H^0(B^k(L), {\mathscr{O}}_{B^k(L)}(H))=H^0(C_{k+1}, E_{k+1, L})=H^0(C, L)$. Write $M_H:= \beta_k^* M_{\Sigma_k}$. By the snake lemma, one can form the following commutative diagram $$\label{diag:B^k(L)} \xymatrix{ & & & 0 \ar[d] & \\ & 0 \ar[d] & & K \ar[d] & \\ 0 \ar[r] & \pi_k^*M_{k+1, L} \ar[d] \ar[r] & H^0(C, L) \otimes {\mathscr{O}}_{B^k(L)} \ar@{=}[d] \ar[r] & \pi_k^*E_{k+1, L} \ar[d] \ar[r] & 0 \\ 0 \ar[r] & M_H \ar[d] \ar[r] & H^0(C, L) \otimes {\mathscr{O}}_{B^k(L)} \ar[r] & {\mathscr{O}}_{B^k(L)}(H) \ar[r] \ar[d] & 0 \\ & K \ar[d] & & 0 & \\ & 0, & & & }$$ in which the right-hand-side vertical exact sequence is the relative Euler sequence. By Bott’s formula on projective spaces, we obtain $$\label{keyclaim} R^i \pi_{k, *}\wedge^j K=0~ \text{ for all }i \geq 0 \text{ and } j > 0.$$ Since $\Sigma_k$ is normal and $R^i\beta_{k,*}{\mathscr{O}}_{B^k(L)}(-Z_{k-1})=0 \text{ for all }i>0$, we have $$\label{eq-keylemma1} H^i(\Sigma_k, \wedge^j M_{\Sigma_k} \otimes I_{\Sigma_{k-1}|\Sigma_k}(k+1)) = H^i(B^k(L), \wedge^j M_H \otimes {\mathscr{O}}_{B^k(L)}((k+1)H-Z_{k-1}))$$ for $i \geq 0$ and $j \geq 0$. Now, the left-hand-side vertical exact sequence of (\[diag:B\^k(L)\]) induces a filtration $$\wedge^j M_H = F^0 \supseteq F_1 \supseteq \cdots \supseteq F^j \supseteq F^{j+1} = 0$$ such that $F^{\ell}/F^{\ell+1} = \pi_k^* \wedge^{\ell} M_{k+1, L} \otimes \wedge^{j-\ell}K$ for $0 \leq \ell \leq j$. By (\[keyclaim\]) and the projection formula, we find $$H^i(B^k(L), \pi_k^* \wedge^{\ell} M_{k+1, L} \otimes \wedge^{j-\ell}K) = H^i(C_{k+1}, \wedge^{\ell} M_{k+1, L} \otimes \pi_{k,*}\wedge^{j-\ell}K)=0$$ for $i \geq 0, ~j>0$ and $0 \leq \ell \leq j-1$. We have ${\mathscr{O}}_{B^k(L)}((k+1)H-Z_{k-1})=\pi^*_kA_{k+1,L}$ by Proposition \[p:01\] (2). Thus we see that $$\label{eq-keylemma5} H^i(C_{k+1}, \wedge^{j}M_{k+1, L} \otimes A_{k+1, L})=0 ~\text{ for $i \geq j-p,~ i \geq 1, ~j \geq 0$,}$$ implies the cohomology vanishing $$H^i(B^k(L), \wedge^j M_H \otimes {\mathscr{O}}_{B^k(L)}((k+1)H-Z_{k-1}))=0 ~\text{ for $i \geq j-p,~ i \geq 1, ~j \geq 0$.}$$ Hence by (\[eq-keylemma1\]), to prove the lemma, it suffices to show the cohomology vanishing (\[eq-keylemma5\]). To this end, we consider the natural quotient map $q_{k+1} \colon C^{k+1} \to C_{k+1}$. Note that $$q_{k+1}^* ( \wedge^j M_{k+1, L} \otimes N_{k+1, L} ) = \wedge^{j} Q_{k+1, L} \otimes L^{\boxtimes k+1}\big(-\Delta_{k+1}).$$ By projection formula, we have $$\wedge^j M_{k+1, L} \otimes N_{k+1, L} \otimes q_{k+1,*}{\mathscr{O}}_{C^{k+1}}=q_{k+1,*}\big( \wedge^{j} Q_{k+1, L} \otimes L^{\boxtimes k+1}\big(-\Delta_{k+1} \big) \big).$$ Recall that $A_{k+1, L} = N_{k+1, L}(-\delta_{k+1})$. Lemma \[diagonal\] implies that $\wedge^i M_{k+1, L} \otimes A_{k+1, L}$ is a direct summand of $\wedge^j M_{k+1, L} \otimes N_{k+1, L} \otimes q_{k+1,*}{\mathscr{O}}_{C^{k+1}}$. Thus the desired cohomology vanishing (\[eq-keylemma5\]) follows from $$H^i\big(C^{k+1}, \wedge^{j} Q_{k+1, L} \otimes L^{\boxtimes k+1}(-\Delta_{k+1}) \big)=0 ~\text{ for $i \geq j-p, ~i \geq 1, ~j \geq 0$}.$$ which is nothing but Theorem \[vanishing\] because $L\big(-\sum_{i=1}^{g+2k+1} x_i \big)$ is effective for general points $x_1, \ldots, x_{g+2k+1}$ on $C$. We finish the proof. Normality, projective normality, and property $N_{k+2,p}$ --------------------------------------------------------- The following is the main result of the paper. It is worth noting that all of the claimed properties in the theorem are proved at the same time to make the induction work. \[normality\] Let $k\geq 0$ and $p\geq 0$ be integers, and $L$ be a line bundle on $C$. Assume that $$\deg L\geq 2g+2k+1+p.$$ Consider the $k$-th secant variety $\Sigma_{k}=\Sigma_{k}(C,L)$ in the space ${\mathbb{P}}(H^0(C, L))={\mathbb{P}}^r$. Then one has the following: 1. $\Sigma_k$ is normal. 2. $R^i\beta_{k,*}{\mathscr{O}}_{B^k(L)}(-Z_{k-1})=0 \text{ for all }i>0.$ 3. $H^i(\Sigma_k, I_{\Sigma_{k-1}|\Sigma_k}(\ell))=H^i(\Sigma_{k}, {\mathscr{O}}_{\Sigma_k}(\ell))=0\text{ for all } i>0, \ell >0.$ 4. $\Sigma_k \subseteq {\mathbb{P}}^r$ is projectively normal, and satisfies the property $N_{k+2,p}$. We proceed by the induction on the number $k$. The statements (1), (2), (3) in the theorem are trivial for the case $k=0$ while the statement (4) is Green’s theorem. Thus, in the sequel, we assume that $k\geq 1$ and the theorem holds for smaller $k$. For a number $m$ with $0\leq m\leq k$, we let $\Sigma_m:=\Sigma_m(C,L)$. \(1) The proof here follows the proofs of Lemma 2.1 and Theorems D of [@Ullery:SecantVar]. The question is local. For a closed point $x\in \Sigma_k$, it is enough to show that $\Sigma_k$ is normal at $x$. As $\Sigma_k\setminus \Sigma_{k-1}$ is nonsingular, we assume that $x \in \Sigma_m \setminus \Sigma_{m-1}$ for some $0 \leq m \leq k-1$. Let $\xi:=\xi_{m+1,x} \in C_{m+1}$ be the degree $m+1$ divisor on $C$ determined by $x$. The morphism $\beta=\beta_k \colon B^k(L) \to \Sigma_k$ induces the morphisms for sheaves $$\xymatrix{ {\mathscr{O}}_{{\mathbb{P}}^r} \ar@/^1.5pc/[rr]|\ \ar@{->>}[r] & {\mathscr{O}}_{ \Sigma_{k}} \ar@{^{(}->}[r] & \beta_*{\mathscr{O}}_{B^k(L)}. }$$ Thus it suffices to prove that the natural morphism ${\mathscr{O}}_{{\mathbb{P}}^r}\rightarrow \beta_*{\mathscr{O}}_{B^k(L)}$ is surjective at $x \in \Sigma_m \setminus \Sigma_{m-1}$. Let $F:=\beta^{-1}(x)$ be the fiber over $x$. Then $F \cong C_{k-m}$ (Proposition \[p:02\] (2.a)). By the formal function theorem, it is sufficient to show that the induced morphism $$\Psi_x \colon \lim_{\longleftarrow}({\mathscr{O}}_{{\mathbb{P}}^r}/{\mathfrak{m}}^{\ell})\longrightarrow \lim_{\longleftarrow}H^0({\mathscr{O}}_{B^k(L)}/I^{\ell}_{F})$$ is surjective, where ${\mathfrak{m}}={\mathfrak{m}}_x$ is the ideal sheaf of $x\in {\mathbb{P}}^r$ and $I_{F}$ is the ideal sheaf of $F$ in $B^k(L)$. Using the commutative diagram $$\xymatrix{ 0\ar[r]& {\mathfrak{m}}^{\ell}/{\mathfrak{m}}^{\ell+1} \ar[d]^{\alpha_{\ell}} \ar[r] & {\mathscr{O}}_{{\mathbb{P}}^r}/{\mathfrak{m}}^{\ell+1} \ar[r]\ar[d] & {\mathscr{O}}_{{\mathbb{P}}^r}/{\mathfrak{m}}^{\ell}\ar[d] \ar[r] & 0\\ 0\ar[r]& H^0(I_F^{\ell}/I_F^{\ell+1}) \ar[r]& H^0({\mathscr{O}}_{B^k(L)}/I^{\ell+1}_F)\ar[r] & H^0({\mathscr{O}}_{B^k(L)}/I^{\ell}_F)\ar[r] &\cdots }$$ and the induction on $\ell$, we further reduce to show that the map $$\alpha_{\ell} \colon {\mathfrak{m}}^{\ell}/{\mathfrak{m}}^{\ell+1}\longrightarrow H^0(I_F^{\ell}/I_F^{\ell+1})$$ is surjective for all $\ell \geq 0$. Note that $${\mathfrak{m}}^{\ell}/{\mathfrak{m}}^{\ell+1}=S^{\ell}(T^*_x{\mathbb{P}}^r) \quad \text{ and } \quad I_F^{\ell}/I_F^{\ell+1}\cong S^{\ell}N^*_{F/B^k(L)}.$$ The map $\alpha_{\ell}$ factors as follows $$\xymatrix{ S^{\ell} (T^*_x{\mathbb{P}}^r) \ar[dr]_-{\alpha_{\ell}} \ar[r]^-{S^{\ell} \alpha_1} &S^{\ell} H^0\big(N^*_{F/B^k(L)} \big)\ar[d]^{\theta_l}\\ & H^0\big(S^{\ell} N^*_{F/B^k(L)} \big). }$$ But Proposition \[p:02\] (2.e) says that the map $\alpha_1 \colon T_x^*{\mathbb{P}}^r\rightarrow H^*(N^*_{F/B^k(L)})$ is an isomorphism. Thus in order to show that $\alpha_{\ell}$ is surjective, it suffices to show that the morphism $\theta_{\ell}$ is surjective. To this end, we use Proposition \[p:02\] (2.d), which says that $$N^*_{F/B^k(L)}\cong {\mathscr{O}}^{\oplus 2m+1}_{F}\oplus E_{n-m,L(-2\xi)}.$$ Thus the surjectivity of $\theta_{\ell}$ would follow from the surjectivity of the morphism $$S^iH^0(E_{k-m,L(-2\xi)})\longrightarrow H^0(S^iE_{k-m,L(-2\xi)})~ \text{ for }0\leq i\leq \ell.$$ But this follows from the inductive hypothesis because $\deg L(-2\xi)\geq 2g+2(k-m-1)+1+p$ and therefore the secant variety $\Sigma_{k-m-1}(C, L(-2\xi))$ in the space ${\mathbb{P}}(H^0(C, L(-2\xi)))$ is normal and projective normality. \(2) The question is local. For a closed point $x \in \Sigma_k$, we shall show that $R^i\beta_{*}{\mathscr{O}}_{B^k(L)}(-Z_{k-1})_x= 0$ for all $i>0$. Since $\beta \colon B^k(L) \to \Sigma_k$ is isomorphic over $x \in \Sigma_k \setminus \Sigma_{k-1}$, we may assume $x\in \Sigma_m \setminus \Sigma_{m-1}$ for some $0\leq m\leq k-1$. Let $\xi:=\xi_{m+1, x} \in C_{m+1}$ be the degree $m+1$ divisor on $C$ determined by $x$. Let $F:=\beta^{-1}(x)$ be the fiber of $\beta$ over $x$, and $I_{F}$ be the ideal sheaf of $F$ in $B^k(L)$. Recall that $F \cong C_{k-m}$ (Proposition \[p:02\] (2.a)). By the formal function theorem, it suffices to show that $$\lim_{\longleftarrow} H^i(F, {\mathscr{O}}_{B^k(L)}(-Z_{k-1})\otimes {\mathscr{O}}_{B^k(L)}/I_{F}^{\ell})=0~\text{ for $i>0$}.$$ To this end, we need to prove that $$H^i(F, {\mathscr{O}}_{B^k(L)}(-Z_{k-1})\otimes {\mathscr{O}}_{B^k(L)}/I_{F}^{\ell})=0~ \text{ for }i>0 \text{ and } \ell \geq 1.$$ which can be deduced from the vanishing $$\label{eq:06} H^i(F, {\mathscr{O}}_{B^k(L)}(-Z_{k-1})\otimes I^{\ell}_{F}/I_{F}^{\ell+1})=0~ \text{ for }i>0 \text{ and } \ell \geq 0.$$ One can calculate that ${\mathscr{O}}_{B^k(L)}(-Z_{k-1})|_{F}=A_{k+1,L}|_{F}=A_{k-m,L(-2\xi)}$ by Lemma \[A-restriction\] and that $I^{\ell}_{F}/I_{F}^{\ell+1}=S^{\ell} N^*_{F/B^k(L)} $ for $\ell\geq 0$, where $N^*_{F/B^k(L)}\cong {\mathscr{O}}^{\oplus 2m+1}_{F}\oplus E_{k-m,L(-2\xi)}$ by Proposition \[p:02\] (2.d). Thus vanishing (\[eq:06\]) can be reduced further to show $$\label{eq-R2} H^i(C_{k-m}, A_{k-m,L(-2\xi)}\otimes S^\ell E_{k-m,L(-2\xi)})=0~ \text{ for }i>0 \text{ and } \ell\geq 0.$$ Now, as $\deg L(-2\xi) \geq 2g+2(k-m-1)+1+p$, the line bundle $L(-2\xi)$ is very ample. Accordingly, we consider the secant varieties $\Sigma'_{k-m-1}:=\Sigma_{k-m-1}(C, L(-2\xi))$ and $\Sigma'_{k-m-2}:=\Sigma'_{k-m-2}(C, L(-2\xi))$ in the space $H^0(C,L(-2\xi))$. By inductive hypothesis, the proposition holds for $\Sigma'_{k-m-1}$. Recall that $B^{k-m-1}(L(-2\xi))={\mathbb{P}}(E_{k-m, L(-2\xi)})$ with the projection $\pi_{k-m-1}$ to $C_{k-m}$ and there is a birational morphism $\beta_{k-m-1} \colon B^{k-m-1}(L(-2\xi)) \to \Sigma'_{k-m-1}$. Write $H$ to be the tautological divisor on $B^{k-m-1}(L(-2\xi))$. Notice that $$\def\arraystretch{1.5} \begin{array}{l} \pi_{k-m-1,*}{\mathscr{O}}_{B^{k-m-1}(L(-2\xi))}((k-m)H-Z_{k-m-2}) = A_{k-m, L(-2\xi)},\\ \beta_{k-m-1,*}{\mathscr{O}}_{B^{k-m-1}(L(-2\xi))}(-Z_{k-m-2}) = I_{\Sigma'_{k-m-2}|\Sigma'_{k-m-1}}. \end{array}$$ By applying the inductive hypothesis for $\Sigma'_{k-m-1}$, we have $$\def\arraystretch{1.5} \begin{array}{l} H^i(C_{k-m}, S^{\ell-k+m}E_{k-m, L(-2\xi)} \otimes A_{k-m, L(-2\xi)}) \\ =H^i(B^{k-m-1}(L(-2\xi)), {\mathscr{O}}_{B^{k-m-1}(L(-2\xi))}(\ell H - Z_{k-m-2})) \\ =H^i(\Sigma'_{k-m-1}, I_{\Sigma'_{k-m-2}|\Sigma'_{k-m-1}}(\ell)) \end{array}$$ for all $i\geq 0$ and $\ell \in \mathbb{Z}$. Hence, vanishing (\[eq-R2\]) follows from the vanishing for $I_{\Sigma'_{k-m-2}|\Sigma'_{k-m-1}}$, which holds by the inductive hypothesis. This completes the proof of (2). \(3) By the inductive hypothesis, we have $H^i(\Sigma_{k-1}, {\mathscr{O}}_{\Sigma_{k-1}}(\ell))=0$ for $i>0$ and $\ell >0$. Grant for the time being the following claim: $$\label{eq-strshf1} H^i(\Sigma_k, I_{\Sigma_{k-1}|\Sigma_k}(\ell))=0~\text{ for all $i>0$ and $1 \leq \ell \leq 2k+2-i$}.$$ Chasing through the associated long exact sequence to the short exact sequence (\[eq:07\]), we obtain $$H^i(\Sigma_k, {\mathscr{O}}_{\Sigma_k}(\ell))=0~\text{ for all $i>0$ and $1 \leq \ell \leq 2k+2-i$}.$$ In particular, ${\mathscr{O}}_{\Sigma_k}$ is $(2k+2)$-regular, so the assertion (3) follows. We next turn to the proof of the claim (\[eq-strshf1\]). Let $H$ be the tautological divisor on $B^k(L)={\mathbb{P}}(E_{k+1,L})$. By (1), $\Sigma_k$ is normal. Thus we have $$\beta_{k,*}{\mathscr{O}}_{B^k(L)}(-Z_{k-1})=I_{\Sigma_{k-1}|\Sigma_k}~~\text{ and }~~\pi_{k,*}{\mathscr{O}}_{B^k(L)}((k+1)H-Z_{k-1})=A_{k+1, L}.$$ By (2), $R^i\beta_{k,*}{\mathscr{O}}_{B^k(L)}(-Z_{k-1})=0$ for $i>0$, so we obtain $$H^i(\Sigma_k, I_{\Sigma_{k-1}|\Sigma_k}(\ell)) = H^i(B^k(L), {\mathscr{O}}_{B^k(L)}(\ell H - Z_{k-1})) = H^i(C_{k+1}, S^{\ell-k-1}E_{k+1, L} \otimes A_{k+1, L}).$$ Thus (\[eq-strshf1\]) holds automatically when $i \geq k+2$ or $1 \leq \ell \leq k$. It only remains to consider the case that $1 \leq i \leq k+1$ and $k+1 \leq \ell \leq 2k+2-i$. Now, the short exact sequence (\[eq:08\]) induces a short exact sequence $$0 \longrightarrow \wedge^{j+1} M_{\Sigma_k} \longrightarrow \wedge^{j+1} H^0(C, L) \otimes {\mathscr{O}}_{\Sigma_k} \longrightarrow \wedge^{j} M_{\Sigma_k} \otimes {\mathscr{O}}_{\Sigma_k}(1) \longrightarrow 0.$$ Tensoring with $I_{\Sigma_{k-1}|\Sigma_k}$, we obtain a short exact sequence $$0 \longrightarrow \wedge^{j+1} M_{\Sigma_k} \otimes I_{\Sigma_{k-1}|\Sigma_k} \longrightarrow \wedge^{j+1} H^0(C, L) \otimes I_{\Sigma_{k-1}|\Sigma_k} \longrightarrow \wedge^{j} M_{\Sigma_k} \otimes I_{\Sigma_{k-1}|\Sigma_k}(1) \longrightarrow 0.$$ This gives a long exact sequence of cohomology groups $$\cdots \longrightarrow \wedge^{j+1} H^0(C, L) \otimes H^i(\Sigma_k, I_{\Sigma_{k-1}|\Sigma_k}(\ell)) \longrightarrow H^i(\Sigma_k, \wedge^{j} M_{\Sigma_k} \otimes I_{\Sigma_{k-1}|\Sigma_k}(\ell+1)) \quad \quad\quad \quad \quad$$ $$\hspace{7.8cm} \longrightarrow H^{i+1}(\Sigma_k, \wedge^{j+1} M_{\Sigma_k} \otimes I_{\Sigma_{k-1}|\Sigma_k}(\ell))\longrightarrow \cdots.$$ It follows that the statement $$\begin{aligned} H^i(\Sigma_k, \wedge^{j} M_{\Sigma_k} \otimes I_{\Sigma_{k-1}|\Sigma_k}(\ell))=0 ~\text{ for }i\geq 1, \ j\geq 0 \text{ and } i\geq j-p \tag*{$(*)_{\ell}$} \end{aligned}$$ implies the corresponding statement $(*)_{\ell+1}$. Since Lemma \[keylemma\] says that $(*)_{k+1}$ is true, we conclude that $(*)_{\ell}$ holds for $\ell\geq k+1$, i.e., $$\label{eq:01} H^i(\Sigma_k, \wedge^{j} M_{\Sigma_k} \otimes I_{\Sigma_{k-1}|\Sigma_k}(\ell))=0~ \text{ for }i\geq 1, \ j\geq 0,\ i\geq j-p \text{ and } \ell\geq k+1.$$ When $j=0$, this implies (\[eq-strshf1\]) for $i \geq 1$ and $\ell \geq k+1$. This finishes the proof of (3). \(4) We first show that $\Sigma_k \subseteq {\mathbb{P}}^r$ is projectively normal. By Danila’s theorem (Theorem \[danila\]), $$H^0({\mathbb{P}}^r, {\mathscr{O}}_{{\mathbb{P}}^r}(\ell))=S^{\ell}H^0(C, L)=H^0(B^k(L), {\mathscr{O}}_{B^k(L)}(\ell))=H^0(\Sigma_k, {\mathscr{O}}_{ \Sigma_{k}}(\ell)) ~\text{ for } 0\leq \ell\leq k+1.$$ For $0\leq \ell\leq k+1$, this implies that $H^0({\mathbb{P}}^r, I_{\Sigma_{k}}(\ell))=H^1({\mathbb{P}}^r, I_{\Sigma_{k}}(\ell))=0$, where $I_{\Sigma_m}=I_{\Sigma_m|{\mathbb{P}}^r}$ is the defining ideal sheaf of $\Sigma_m$ in ${\mathbb{P}}^r$ for $0 \leq m \leq k$. We have a short exact sequence $$\label{eq:sesidealsheaves} 0 \longrightarrow I_{\Sigma_k} \longrightarrow I_{\Sigma_{k-1}} \longrightarrow I_{\Sigma_{k-1}|\Sigma_k} \longrightarrow 0.$$ We then obtain $H^0({\mathbb{P}}^r, I_{\Sigma_{k-1}}(\ell))=H^0(\Sigma_k, I_{\Sigma_{k-1}|\Sigma_k}(\ell))$ for $0\leq \ell\leq k+1$. For $\ell \geq k+1$, consider the following commutative diagram $$\label{eq:multimap} \xymatrix{ S^{\ell -k-1}H^0(C, L)\otimes H^0(\Sigma_k, I_{\Sigma_{k-1}}(k+1)) \ar@{=}[d] \ar[r] & H^0(\Sigma_k, I_{\Sigma_{k-1}}(\ell)) \ar[d]\\ S^{\ell -k-1}H^0(C, L)\otimes H^0(\Sigma_k, I_{\Sigma_{k-1}|\Sigma_k}(k+1)) \ar[r] & H^0(\Sigma_k, I_{\Sigma_{k-1}|\Sigma_k}(\ell)). }$$ By (\[eq:01\]), $H^1(\Sigma_k, M_{\Sigma_k}\otimes I_{\Sigma_{k-1}|\Sigma_k}(\ell))=0$ for $\ell\geq k+1$. Then the multiplication map in the bottom of (\[eq:multimap\]) is surjective, and hence, the right vertical map of (\[eq:multimap\]) is surjective. We then conclude that the map $H^0({\mathbb{P}}^r, I_{\Sigma_{k-1}}(\ell))\rightarrow H^0(\Sigma_k, I_{\Sigma_{k-1}|\Sigma_k}(\ell))$ is surjective for $\ell\geq 0$. By induction, $\Sigma_{k-1} \subseteq {\mathbb{P}}^r$ is projectively normal, so $H^1({\mathbb{P}}^r, I_{\Sigma_{k-1}}(\ell))=0$ for $\ell \geq 0$. Therefore, by considering (\[eq:sesidealsheaves\]), we obtain $H^1({\mathbb{P}}^r, I_{\Sigma_k}(\ell))=0$ for $\ell\geq 0$, which means that $\Sigma_k \subseteq {\mathbb{P}}^r$ is projectively normal. Next we show that $\Sigma_k \subseteq {\mathbb{P}}^r$ satisfies $N_{k+2,p}$. Recall from (3) that $H^i(\Sigma_k, {\mathscr{O}}_{ \Sigma_{k}}(\ell))=0$ for $i\geq 1$ and $\ell \geq 1$. By Proposition \[koszh1\], we only need to show that $H^1(\Sigma_k, \wedge^jM_{\Sigma_{k}}\otimes{\mathscr{O}}_{ \Sigma_{k}}(\ell))=0$ for $\ell \geq k+1$ and $1\leq j\leq p+1$. Consider the short exact sequence $$0\longrightarrow \wedge^j M_{\Sigma_k} \otimes {I}_{\Sigma_{k-1}|\Sigma_{k}}\longrightarrow \wedge^j M_{\Sigma_k}\longrightarrow \wedge^j M_{\Sigma_{k-1}}\longrightarrow 0.$$ Since $\deg L\geq2g+1+2(k-1)+1+p+2$, we may assume by induction that $\Sigma_{k-1} \subseteq {\mathbb{P}}^r$ satisfies $N_{k+1,p+2}$. So by Proposition \[koszh1\], we have $H^1(\Sigma_{k-1}, \wedge^jM_{\Sigma_{k-1}}(\ell))=0$ for $\ell \geq k$ and $1\leq j\leq p+3$. Combine this with (\[eq:01\]), we get $H^1(\Sigma_k, \wedge^jM_{\Sigma_{k}}(\ell))=0$ for $1\leq j\leq p+1$ and $\ell \geq k+1$ as desired. We have seen in the above proof that Danila’s theorem (Theorem \[danila\]) shows $H^0({\mathbb{P}}^r, {\mathscr{O}}_{{\mathbb{P}}^r}(\ell))=H^0(\Sigma_k, {\mathscr{O}}_{\Sigma_k}(\ell))$ for all $1 \leq \ell \leq k+1$. This in particular implies that the defining ideal of the $k$-th secant variety $\Sigma_k$ in ${\mathbb{P}}^r$ has no forms of degree $\leq k+1$. Singularities {#subsec:sing} ------------- \[sing\] Let $k\geq 0$ be an integer, and $L$ be a line bundle on $C$. Assume that $$\deg L\geq 2g+2k+1.$$ Consider the $k$-th secant variety $\Sigma_{k}=\Sigma_{k}(C,L)$ in the space ${\mathbb{P}}(H^0(C, L))$. Then one has the following: 1. $\Sigma_k$ has normal Du Bois singularities. 2. $g=0$ if and only if there exists a boundary divisor $\Gamma$ on $\Sigma_k$ such that $(\Sigma_k, \Gamma)$ is a klt pair. In this case, $\Sigma_k$ is a Fano variety with log terminal singularities and of Picard rank one. 3. $g=1$ if and only if there exists a boundary divisor $\Gamma$ on $\Sigma_k$ such that $(\Sigma_k, \Gamma)$ is a log canonical pair but it cannot be a klt pair. In this case, $\Sigma_k$ is a Calabi–Yau variety with log canonical singularities. In particular, $g \geq 2$ if and only if there is no boundary divisor $\Gamma$ on $\Sigma_k$ such that $(\Sigma_k, \Gamma)$ is a log canonical pair. \(1) By Theorem \[normality\] (1), we know that $\Sigma_k$ is normal. By proceeding by the induction on $k$, we show that $\Sigma_k$ has Du Bois singularities. If $k=0$, then $\Sigma_0=C$ so that the assertion is trivial. In the sequel, we assume that $k \geq 1$ and the assertion (1) holds for $k-1$. By [@K Corollary 6.28], it suffices to check the following: 1. $\Sigma_{k-1}$ has Du Bois singularities. 2. $Z_{k-1}$ has Du Bois singularities. 3. $\beta_{k,*}{\mathscr{O}}_{B^k(L)}(-Z_{k-1}) = {I}_{\Sigma_{k-1}|\Sigma_k}$ and $R^i \beta_{k,*}{\mathscr{O}}_{B^k(L)}(-Z_{k-1})=0$ for $i > 0$. By inductive hypothesis, (a) holds. For (b), consider the composition map $b_k \colon \text{bl}_k(B^k(L)) \to B^k(L)$ of blowups (see Subsection \[subsec:blowup\]). Recall from Proposition \[p:01\] (3) that $$K_{ \text{bl}_k(B^k(L))} = b_k^* ( K_{B^k(L)} + Z_{k-1}) - (E_0 + \cdots + E_{k-1}).$$ Thus the log pair $(B^k(L), Z_{k-1})$ is log canonical, and hence, $Z_{k-1}$ has semi-log canonical singularities. Then, by [@K Corollary 6.32], $Z_{k-1}$ has Du Bois singularities, i.e., (b) holds. Finally, (c) holds by Theorem \[normality\]. (2), (3) Recall that $\beta_k \colon B^k(L) \to \Sigma_k$ is a resolution of singularities and $\Sigma_k$ is normal. For a general point $x \in \Sigma_{k-1} \setminus \Sigma_{k-2}$, we denote by $F_x:=\beta_k^{-1}(x)$ the fiber of $\beta_k$ over $x$. Note that $F_x \cong C$. Let $H$ be the tautological divisor on $B^k(L)={\mathbb{P}}(E_{k+1, L})$, i.e., ${\mathscr{O}}_{B^k(L)}(H)={\mathscr{O}}_{B^k(L)}(1)$. Recall from Proposition \[p:01\] (2) that $Z_{k-1} \sim_{\operatorname{lin}} (k+1)H-\pi_k^*(T_{k+1}(L)-2\delta_{k+1})$. We can easily check that $$\label{eq-K} K_{B^k(L)} + Z_{k-1} \sim_{\operatorname{lin}} \pi_k^*(K_{C_{k+1}}+\delta_{k+1}) =\pi_k^*T_{k+1}(K_C).$$ We first prove (2). Suppose that $C={\mathbb{P}}^1$. It is well known that $C_{n+1} \cong {\mathbb{P}}^{n+1}$. For a sufficiently small rational number $\epsilon > 0$, by (\[eq-K\]), we have $$-(K_{B^k(L)}+(1-\epsilon) Z_{k-1}) \sim_{\mathbb{Q}\text{-lin}} \epsilon (k+1)H + \pi_k^*(T_{k+1}(-K_C - \epsilon L)+2\epsilon \delta_{k+1}).$$ We may assume that $T_{k+1}(-K_C - \epsilon L)+2\epsilon \delta_{k+1}$ is ample on $C_{k+1}$. Now, $B^k(L)$ has Picard rank two, and the nef cone of $B^k(L)$ is generated by $H$ and $\pi_k^*(T_{k+1}(-K_C - \epsilon L)+2\epsilon \delta_{k+1})$. Thus $-(K_{B^k(L)}+(1-\epsilon) Z_{k-1})$ is ample. By considering the log resolution of $(B^k(L), (1-\epsilon) Z_{k-1})$ in Proposition \[p:01\] (3), we see that $(B^k(L), (1-\epsilon) Z_{k-1})$ is a klt pair. Hence $B^k(L)$ is of Fano type. By [@FG Theorem 5.1], $\Sigma_k$ is also of Fano type. Now, $\Sigma_k$ has Picard rank one. Therefore, it is a Fano variety with log terminal singularities. For the converse, suppose that there exists a boundary divisor $\Gamma$ such that $(\Sigma_k, \Gamma)$ is a klt pair. By [@HM Corollary 1.5], $F_x \cong C$ is rationally chain connected, so $C$ is a rational curve. We finally prove (3). Suppose that $C$ is an elliptic curve. By (\[eq-K\]), we have $$K_{B^k(L)}+Z_{k-1} \sim_{\operatorname{lin}} \pi_k^* T_{k+1}(K_C) = 0.$$ Then the ‘only if’ direction immediately follows from [@FG Lemma 1.1]. In this case, we actually have $K_{\Sigma_k}=\beta_{k,*}(K_{B^k(L)}+Z_{k-1})=0$. Thus $\Sigma_k$ is a Calabi–Yau variety with log canonical singularities. For the converse, suppose that there exists a boundary divisor $\Gamma$ such that $(\Sigma_k, \Gamma)$ is a log canonical pair. We have $$K_{B^k(L)}+Z_{k-1} + \beta_k^{-1}\Gamma = \beta_k^*(K_{\Sigma_k}+\Gamma) + (1+a)Z_{k-1},$$ where $a=a(Z_{k-1}; \Sigma_k, \Gamma) \geq -1$ is the discrepancy of the $\beta_k$-exceptional divisor $Z_{k-1}$. By restricting the above divisor to $F_x \cong C$, we obtain $$K_C + (\beta_k^{-1}\Gamma)|_C = -(1+a)(L-2\xi),$$ where $\xi:=\xi_{k,x}$ is the degree $k$ divisor on $C$ determined by $x$. Then $$-K_C=(1+a)(L-2\xi) + (\beta_k^{-1}\Gamma)|_C$$ is effective so that $C$ is either a rational curve or an elliptic curve. This proves the converse direction, and hence, we complete the proof. It is easy to check that $g=0$ if and only if $\Sigma_k$ has rational singularities (cf. [@Vermeire:RegNormSecCurve Proposition 9]). When $g=1$, we see that $\Sigma_k$ is Gorenstein with $\omega_{\Sigma_k}\cong {\mathscr{O}}_{ \Sigma_{k}}$ (this is also proved in [@GBH 8.14]). In the next subsection, we show that $\Sigma_k \subseteq {\mathbb{P}}(H^0(C, L))$ is arithmetically Cohen–Macaulay, and therefore, its cone is Gorenstein. For instance, one can deduce that the $k$-th secant variety $\Sigma_k$ of an elliptic curve embedded by a degree $2k+4$ line bundle is a complete intersection in ${\mathbb{P}}^{2k+3}$. \[rmk:ellipic\] In contrast to the smaller genus case, if $g \geq 2$, then $\Sigma_k$ is not $\mathbb{Q}$-Gorenstein, i.e., $K_{\Sigma_k}$ is not $\mathbb{Q}$-Cartier. To show this, suppose that $K_{\Sigma_k}$ is $\mathbb{Q}$-Cartier. For a sufficiently divisible integer $m > 0$, we have $mK_{B^k(L)}-maZ_{k-1} \sim_{\operatorname{lin}} \beta_k^*(mK_{\Sigma_k})$, where $a=a(Z_{k-1};\Sigma_k,0) < -1$ is the discrepancy of $Z_{k-1}$. By restricting to $\beta_k^{-1}(x) \cong C_k$ for any point $x \in C \subseteq \Sigma_k$, we see that $$m\big(T_k(K_C+(1-a)L - 2(1-a)x \big) -2(1-a)\delta_k \sim_{\operatorname{lin}} 0.$$ Thus we obtain $2m(1-a)x \sim_{\operatorname{lin}} 2m(1-a)y$ for any points $x,y \in C$, but it is impossible. Arithmetic Cohen–Macaulayness and Castelnuovo–Mumford regularity {#subsec:acm} ---------------------------------------------------------------- \[p:12\] Let $k\geq 0$ be an integer, and $L$ be a line bundle on $C$. Assume that $$\deg L\geq 2g+2k+1.$$ Consider the $k$-th secant variety $\Sigma_{k}=\Sigma_{k}(C,L)$ in the space ${\mathbb{P}}(H^0(C, L))={\mathbb{P}}^r$. Then one has the following: 1. $H^i(\Sigma_k, {\mathscr{O}}_{\Sigma_k}(-\ell))=0$ for $1 \leq i \leq 2k$ and $\ell \geq 0$. 2. $H^{2k+1}(\Sigma_k, {\mathscr{O}}_{\Sigma_k})=S^{k+1}H^0(C, \omega_C)^*$. In particular, $\Sigma_k \subseteq {\mathbb{P}}^r$ is arithmetically Cohen–Macaulay. We first recall from Proposition \[sing\] (1) that $\Sigma_k$ has Du Bois singularities. By [@K Theorem 10.42], we have $$h^i(\Sigma_k, {\mathscr{O}}_{\Sigma_k}(-\ell)) = h^i(\Sigma_k, {\mathscr{O}}_{\Sigma_k}(-1)) ~\text{ for $1 \leq i \leq 2k$ and $\ell \geq 1$.}$$ Therefore, the result (1) is equivalent to the cohomology vanishing $$H^i(\Sigma_k, {\mathscr{O}}_{\Sigma_k}(-\ell))=0 ~\text{ for $1 \leq i \leq 2k$ and $\ell=0,1$}.$$ We now proceed by the induction on $k$. Note that the case with $k=0$ is trivial. For $k \geq 1$, we assume that $\Sigma_{k-1} \subseteq {\mathbb{P}}^r$ has results (1) and (2). Concerning the cohomological long exact sequence associated to the short exact sequence (\[eq:07\]), we make the following: \[claim:acm1\]$ $ 1. $H^i(\Sigma_k, I_{\Sigma_{k-1}|\Sigma_k}(-\ell))=0~\text{ for $1 \leq i \leq 2k-1$ and $\ell=0,1$}.$ 2. The connection map $\tau_{\ell}$ of the cohomological groups $$\cdots \longrightarrow H^{2k-1}({\mathscr{O}}_{\Sigma_{k-1}}(-\ell)) \stackrel{\tau_{\ell}}{\longrightarrow} H^{2k}(I_{\Sigma_{k-1}|\Sigma_k}(-\ell)) \longrightarrow \cdots$$ is an isomorphism for $\ell=0,1$. Granted the claim for the moment, using inductive hypothesis on $\Sigma_{k-1}$ and chasing through the long exact sequence associated to (\[eq:07\]), we immediately obtain from (a) that $$H^i(\Sigma_k, {\mathscr{O}}_{\Sigma_k}(-\ell))=0 ~\text{ for $1 \leq i \leq 2k-2$ and $\ell=0,1$}.$$ Furthermore, we arrive at an exact sequence involving the connection map $\tau_\ell$ as follows @size[10]{}@mathfonts $$0 \longrightarrow H^{2k-1}({\mathscr{O}}_{\Sigma_k}(-\ell)) \longrightarrow H^{2k-1}({\mathscr{O}}_{\Sigma_{k-1}}(-\ell)) \stackrel{\tau_{\ell}}{\longrightarrow} H^{2k}(I_{\Sigma_{k-1}|\Sigma_k}(-\ell)) \longrightarrow H^{2k}({\mathscr{O}}_{\Sigma_k}(-\ell)) \longrightarrow 0.$$ The statement (b) then implies that $$H^i(\Sigma_k, {\mathscr{O}}_{\Sigma_k}(-\ell))=0 ~\text{ for $2k-1 \leq i \leq 2k$ and $\ell=0,1$},$$ which proves (1). For the result (2), chasing through the long exact sequence would yield $$H^{2k+1}(\Sigma_k, {\mathscr{O}}_{\Sigma_k})=H^{2k+1}(\Sigma_k, I_{\Sigma_{k-1}|\Sigma_k}).$$ By Theorem \[normality\] (2) and Serre duality, for any $i$ and $\ell$, we have $$H^{i}(I_{\Sigma_{k-1}|\Sigma_k}(-\ell))=H^{i}({\mathscr{O}}_{B^k(L)}(-\ell H-Z_{k-1})) = H^{2k+1-i}( {\mathscr{O}}_{B^k(L)}(K_{B^k(L)} + Z_{k-1}+\ell H))^*,$$ where $H$ is the tautological divisor on $B^k(L)={\mathbb{P}}(E_{k+1,L})$. Recall from (\[eq-K\]) that $$K_{B^k(L)} + Z_{k-1} \sim_{\operatorname{lin}} \pi_k^*(K_{C_{k+1}}+\delta_{k+1}) =\pi_k^*T_{k+1}(K_C).$$ Thus we obtain $$\label{eq-cohacm} H^{i}(\Sigma_k, I_{\Sigma_{k-1}|\Sigma_k}(-\ell)) = H^{2k+1-i}(C_{k+1}, S^{\ell} E_{k+1, L} \otimes T_{k+1}(\omega_C))^*.$$ In particular, when $i=2k+1$, we find $$H^{2k+1}(\Sigma_k, {\mathscr{O}}_{\Sigma_k})=H^{2k+1}(\Sigma_k, I_{\Sigma_{k-1}|\Sigma_k}) = H^0(C_{k+1}, T_{k+1}(\omega_C))^*.$$ By Lemma \[K-vanishing\], we get the result (2). We now prove Claim \[claim:acm1\] (a). Assume that $\ell=0$. As calculated in (\[eq-cohacm\]), we have $$H^i(\Sigma_k, I_{\Sigma_{k-1}|\Sigma_k})=H^{2k+1-i}(C_{k+1}, T_{k+1}(\omega_C) )^*.$$ Then Lemma \[K-vanishing\] implies Claim \[claim:acm1\] (a) for $\ell=0$. Assume that $\ell=1$. By (\[eq-cohacm\]), we have $$H^i(\Sigma_k, I_{\Sigma_{k-1}|\Sigma_k}(-1))=H^{2k+1-i}(C_{k+1}, E_{k+1, L} \otimes T_{k+1}(\omega_C) )^*.$$ Recall that we have a canonical morphism $\sigma_{k+1} \colon C_k \times C \to C_{k+1}$. We observe that $$\sigma_{k+1,*} (T_k(\omega_C) \boxtimes (\omega \otimes L)) = E_{k+1,L} \otimes T_{k+1}(\omega_C).$$ Then we find $$\label{eq-cohacm2} H^{2k+1-i}(C_{k+1}, E_{k+1,L} \otimes T_{k+1}(\omega_C))=H^{2k+1-i}(C_k \times C, T_k(\omega_C) \boxtimes (\omega_C \otimes L)).$$ For $1 \leq i \leq 2k-1$, we have $2k+1-i \geq 2$. By Lemma \[K-vanishing\] and Künneth formula, we get $$H^{2k+1-i}(C_k \times C, T_k(\omega_C) \boxtimes (\omega_C\otimes L))=0.$$ This implies Claim \[claim:acm1\] (a) for $\ell=1$. We next turn to the proof of Claim \[claim:acm1\] (b). By Theorem \[normality\] (2) for both $\Sigma_k$ and $\Sigma_{k-1}$ and calculation in (\[eq-cohacm\]), we recall that $$\def\arraystretch{1.5} \begin{array}{l} H^{2k}(I_{\Sigma_{k-1}|\Sigma_k}(-\ell))^*=H^1(\omega_{B^k(L)}(Z_{k-1}+\ell H)) = H^{1}(S^{\ell}E_{k+1, L} \otimes T_{k+1}(\omega_C)),\\ H^{2k-1}({\mathscr{O}}_{\Sigma_{k-1}}(-\ell))^* =H^0(\omega_{B^{k-1}(L)}(Z_{k-2}+\ell H))=H^0(S^{\ell}E_{k,L} \otimes T_k(\omega_C)). \end{array}$$ For $\ell=0$, by Lemma \[K-vanishing\], we have $h^{2k}(\Sigma_k, I_{\Sigma_{k-1}|\Sigma_k})=h^{2k-1}(\Sigma_{k-1}, {\mathscr{O}}_{\Sigma_{k-1}}).$ For $\ell=1$, by (\[eq-cohacm2\]) and Künneth formula, we see that $$\label{eq-cohacm3} \def\arraystretch{1.5} \begin{array}{l} H^1(E_{k+1, L} \otimes T_{k+1}(\omega_C)) = H^1(T_k(\omega_C) \boxtimes (\omega_C \otimes L)) = H^1(T_k(\omega_C)) \otimes H^0(\omega_C \otimes L),\\ H^0(E_{k, L} \otimes T_k(\omega_C))=H^0(T_{k-1}(\omega_C) \boxtimes (\omega_C \otimes L))=H^0(T_{k-1}(\omega_C)) \otimes H^0(\omega_C \otimes L). \end{array}$$ Lemma \[K-vanishing\] then implies that $h^{2k}(\Sigma_k, I_{\Sigma_{k-1}|\Sigma_k}(-\ell))=h^{2k-1}(\Sigma_{k-1}, {\mathscr{O}}_{\Sigma_{k-1}}(-\ell)).$ Thus, to show Claim \[claim:acm1\] (b), it is sufficient to show that $\tau_{\ell}$ is injective for $\ell=0,1$. To this end, recall that we have the following commutative diagram $$\xymatrix{ C_k \times C \ar[rr]^-{\sigma_{k+1}} & & C_{k+1}\\ B^{k-1}(L) \times C \ar[u]^-{\pi_k \times \operatorname{id}_C} \ar[r]^-{\alpha_{k, k-1}} & Z_{k-1} \ar[d]_-{\beta_k|_{Z_{k-1}}} \ar@{^{(}->}[r] & B^k(L) \ar[u]_-{\pi_k} \ar[d]^-{\beta_k} \\ & \Sigma_{k-1} \ar@{^{(}->}[r] & \Sigma_k. }$$ Note that $\alpha_{k,k-1}^*\omega_{Z_{k-1}} = \omega_{B^{k-1}(L)}(Z_{k-2}) \boxtimes \omega_C$ and there is a natural injection $$H^0(B^{k-1}(L), \omega_{B^{k-1}(L)}(Z_{k-2}+\ell H)) \hookrightarrow H^1(B^{k-1}(L) \times C, \omega_{B^{k-1}(L)}( Z_{k-2}+\ell H)) \boxtimes \omega_C).$$ Then we obtain the following commutative diagram $$\xymatrix{ H^1(S^{\ell}E_{k_1} \otimes T_{k+1}(\omega_C))\ar@{=}[d] \ar[rr] & & H^1(S^{\ell}E_{k, L} \otimes T_k(\omega_C) \boxtimes \omega_C) \ar@{=}[d] \\ H^1(\omega_{B^k(L)}(Z_{k-1}+\ell H )) \ar@{=}[d] \ar[r] & H^1(\omega_{Z_{k-1}}(\ell H)) \ar[r] \ar[d] & H^1(\omega_{B^{k-1}(L)}( Z_{k-2}+\ell H)) \boxtimes \omega_C) \\ H^{2k}(I_{\Sigma_{k-1}|\Sigma_k}(-\ell))^* \ar[r]^-{\tau^{*}_{\ell}} & H^{2k-1}({\mathscr{O}}_{\Sigma_{k-1}}(-\ell))^* \ar@{=}[r] & H^0(\omega_{B^{k-1}(L)}(Z_{k-2}+\ell H)). \ar@{^{(}->}[u] }$$ It is enough to check that the map on the top is injective. This is clear for $\ell=0$. For $\ell=1$, by (\[eq-cohacm3\]) and Lemma \[K-vanishing\], we have the following injection $$H^1(E_{k+1} \otimes T_{k+1}(\omega_C)) \cong H^0(E_{k,L} \otimes T_k(\omega_C)) \hookrightarrow H^1(E_{k, L} \otimes T_k(\omega_C) \boxtimes \omega_C).$$ Thus the map on the top for $\ell=1$ is injective as required. Finally, recall the well known fact that a projective variety $X \subseteq {\mathbb{P}}^r$ is arithmetically Cohen–Macaulay if and only if the following hold: 1. $X \subseteq {\mathbb{P}}^r$ is projectively normal. 2. $H^i(X, {\mathscr{O}}_X(\ell))=0$ for $0 < i < \dim X$ and $\ell \in \mathbb{Z}$. By Theorem \[normality\] (3), (4) and the vanishing property (1) imply that $\Sigma_k \subseteq {\mathbb{P}}^r$ is arithmetically Cohen–Macaulay. We complete the proof. \[acmreg\] Let $k\geq 0$ be an integer, and $L$ be a line bundle on $C$. Assume that $$\deg L\geq 2g+2k+1.$$ Consider the secant variety $\Sigma_{k}=\Sigma_{k}(C,L)$ in the space ${\mathbb{P}}(H^0(C, L))={\mathbb{P}}^r$. Then one has the following: 1. $\displaystyle h^0(\omega_{\Sigma_k})=\dim K_{r-2k-1, 2k+2}(\Sigma_k, {\mathscr{O}}_{\Sigma_k}(1))={g+k \choose k+1}. $ 2. If $g=0$, then $\operatorname{reg}({\mathscr{O}}_{\Sigma_k})=k+1$ and $\operatorname{reg}(\Sigma_k)=k+2$. 3. If $g \geq 1$, then $\operatorname{reg}({\mathscr{O}}_{\Sigma_k})=2k+2$ and $\operatorname{reg}(\Sigma_k)=2k+3$. \(1) As $\Sigma_k \subseteq {\mathbb{P}}(H^0(C, L))={\mathbb{P}}^r$ is arithmetically Cohen–Macaulay by Theorem \[p:12\], dualizing the minimal graded free resolution of $R(\Sigma_k, {\mathscr{O}}_{\Sigma_k}(1))$ and shifting by $-r-1$ gives the minimal graded free resolution of the canonical module. This implies that $$\dim K_{r-2k-1, 2k+2}(\Sigma_k, {\mathscr{O}}_{\Sigma_k}(1)) = h^0(\Sigma_k, \omega_{\Sigma_k}).$$ By the Serre duality and Theorem \[p:12\], we obtain $$h^0(\Sigma_k, \omega_{\Sigma_k})=h^{2k+1}(\Sigma_k, {\mathscr{O}}_{\Sigma_k}) = \dim S^{k+1} H^0(C, \omega_C) = {g+k \choose k+1}.$$ (2), (3) By Theorem \[normality\] (3), (4), we see that $$\operatorname{reg}(\Sigma_k)=\operatorname{reg}({\mathscr{O}}_{\Sigma_k})+1 \leq 2k+3.$$ By Theorem \[normality\] (3) and Theorem \[p:12\] (1), we know that $H^i(\Sigma_k, {\mathscr{O}}_{\Sigma_k}(\ell))=0$ for $1 \leq i \leq 2k$ and $\ell \in \mathbb{Z}$. Thus we only have to consider the (non)vanishing of $H^{2k+1}(\Sigma_k, {\mathscr{O}}_{\Sigma_k}(\ell))$. For (2), suppose that $g=0$. It is enough to show that $H^{2k+1}(\Sigma_k, {\mathscr{O}}_{\Sigma_k}(-k))=0$ and $H^{2k+1}(\Sigma_k, {\mathscr{O}}_{\Sigma_k}(-k-1))\neq 0$. By Proposition \[sing\] (2), $\Sigma_k$ has log terminal singularities, and hence, it has rational singularities, i.e., $R^i \beta_{k,*}{\mathscr{O}}_{B^k(L)}=0$ for $i > 0$. Then we obtain $$H^{2k+1}(\Sigma_k, {\mathscr{O}}_{\Sigma_k}(\ell))=H^{2k+1}(B^k(L), {\mathscr{O}}_{B^k(L)}(\ell))=H^0(B^k(L), \omega_{B^k(L)}(-\ell))^*.$$ It is elementary to see that $H^0(B^k(L), \omega_{B^k(L)}(k))=0$ but $H^0(B^k(L), \omega_{B^k(L)}(k+1)) \neq 0$. For (3), suppose that $g \geq 1$. It is enough to prove that $H^{2k+1}(\Sigma_k, {\mathscr{O}}_{\Sigma_k})\neq 0$. By Theorem \[p:12\] (2), we find $H^{2k+1}(\Sigma_k, {\mathscr{O}}_{\Sigma_k}) = S^{k+1}H^0(C, \omega_C) \neq 0$. We finish the proof. Further properties of secant varieties -------------------------------------- We have shown the main theorems of the paper. In this subsection, we discuss further properties of secant varieties of curves. Let $k \geq 0$ be an integer, and $L$ be a line bundle on $C$. Assume that $$\deg L \geq 2g+2k+1.$$ Consider the $k$-th secant variety $\Sigma_k=\Sigma_k(C, L)$ in the space ${\mathbb{P}}(H^0(C, L))={\mathbb{P}}^r$. Then one has the following: 1. The degree of $\Sigma_k \subseteq {\mathbb{P}}^r$ is given by $$\deg \Sigma_k =\sum_{i=0}^{\min(k+1, g)} {\deg L -g-k-i \choose k+1-i} {g \choose i}.$$ 2. The multiplicity of $\Sigma_k$ at a point $x \in \Sigma_m \setminus \Sigma_{m-1}$ with $0 \leq m \leq k$ is given by $$\operatorname{mult}_x \Sigma_k = \deg \Sigma_{k-m-1}(C, L(-2\xi_{m+1,x}))= \sum_{i=0}^{\min(k-m,g)}{\deg L -g-m-1-k-i \choose k-m-i }{g \choose i} .$$ \(1) follows from [@Soule Proposition 1]. In fact, $\deg \Sigma_k$ is the Segre class $s_{k+1}(E_{k+1,L}^*)$. For (2), notice that $\operatorname{mult}_x \Sigma_k$ is the Segre class $s_0(\{ x \}, \Sigma_k)$, which is invariant under a birational morphism. Recall that $F:=\beta_k^{-1}(x) \cong C_{k-m}$ and $N_{F/B^k(L)} \cong {\mathscr{O}}_{F}^{\oplus 2m+1} \oplus E_{k-m,L(-2\xi_{m+1,x})}^*$ (Proposition \[p:02\] (2.a, 2.d)). Thus we have $$\operatorname{mult}_x \Sigma_k = s_{k-m}(F, B^k(L))=s_{k-m}(N_{F/B^k(L)})=s_{k-m}(E_{k-m,L(-2\xi_{m+1,x})}^*).$$ Consider the secant variety $\Sigma_{k-m-1}(C, L(-2\xi_{m+1,x}))$ in the space ${\mathbb{P}}(H^0(C, L(-2\xi_{m+1,x})))$. Then we obtain $$s_{k-m}(E_{k-m,L(-2\xi_{m+1,x})}^*) = \deg \Sigma_{k-m-1}(C, L(-2\xi_{m+1,x})),$$ which completes the proof by (1) since $\deg L(-2 \xi_{m+1,x}) \geq 2g+2(k-m-1)+1$. Next, we show that $B^k(L)$ is the normalization of the blowup of $\Sigma_{k}$ along $\Sigma_{k-1}$. For this purpose, we prove the following lemma. \[lem:Agg\] For any integer $k\geq 0$, one has the following: 1. $A_{k+1,L}$ is globally generated if $\deg L\geq 2g+2k$. 2. $A_{k+1,L}$ is globally generated and ample if $\deg L\geq 2g+2k+1$. For a point $p\in C$, consider the short exact sequence $$0\longrightarrow A_{k+1,L}(-X_p)\longrightarrow A_{k+1,L}\longrightarrow A_{k+1,L}|_{X_p}\longrightarrow 0.$$ Note that $A_{k+1,L}|_{X_p}=A_{k,L(-2p)}$ and $A_{k+1,L}(-X_p)=A_{k+1,L(-p)}$. By induction on $k$, we only need to show $H^1(C_{k+1},A_{k+1,L(-p)})=0$. Pulling back the involved line bundle to $C^{k+1}$ and applying Lemma \[diagonal\], we can reduce the problem to prove the following cohomology vanishing $$\label{eq:cohvanAgg} H^1(C^{k+1}, L^{\boxtimes k+1}(-\Delta_{k+1})) = 0~~ \text{ if }\deg L\geq 2g + 2k-1.$$ If $k=0$, then (\[eq:cohvanAgg\]) is clear. Assume $k \geq 1$. Then $L$ separates $k$ points. Let $p \colon C^{k+1}\rightarrow C^k$ be the projection to the first $k$ components. Then $$p_*L^{\boxtimes k+1}(-\Delta_{k+1}) = Q_{k,L}\otimes L^{\boxtimes k}(-\Delta_k)$$ so that $H^1(C^{k+1}, L^{\boxtimes k+1}(-\Delta_{k+1})) = H^1(C^k,Q_{k,L}\otimes L^{\boxtimes k}(-\Delta_k))$. As $\deg L\geq 2g+2k-1 = 2g+2(k-1)+1$, the desired cohomology vanishing (\[eq:cohvanAgg\]) follows from Theorem \[vanishing\], proving (1). For (2), notice that $A_{k+1,L}=A_{k+1,L(-p)} \otimes T_{k+1}({\mathscr{O}}_C(p))$. By (1), $A_{k+1,L(-p)} $ is globally generated, and we know that $T_{k+1}({\mathscr{O}}_C(p))$ is ample. Hence (2) follows. Let $k\geq 0$ be an integer, and $L$ be a line bundle on $C$. Assume that $$\deg L\geq 2g+2k+1.$$ Consider the $k$-th secant variety $\Sigma_{k}=\Sigma_{k}(C,L)$ in the space ${\mathbb{P}}(H^0(C, L))={\mathbb{P}}^r$. Then one has the following: 1. $\beta_k \colon B^k(L)\rightarrow \Sigma_k$ factors through the blowup $\operatorname{Bl}_{\Sigma_{k-1}}\Sigma_k$ of $\Sigma_k$ along $\Sigma_{k-1}$. 2. $B^k(L)$ is the normalization of $\operatorname{Bl}_{\Sigma_{k-1}}\Sigma_k$. 3. $\beta_{k,*}{\mathscr{O}}_{B^k(L)}(-mZ_{k-1}) = \overline{I^m_{\Sigma_{k-1}| \Sigma_k}}$ for $m\geq 0$, where $\overline{{\mathfrak{a}}}$ denotes the integral closure of an ideal sheaf ${\mathfrak{a}}$. Recall the projection $\pi_k \colon B^k(L)\rightarrow C_{k+1}$. We write ${\mathscr{O}}_{B^k(L)}(H)$ to be the tautological bundle of $B^k(L)$, which also equals to $\beta^*_k{\mathscr{O}}_{{\mathbb{P}}^r}(1)$. For simplicity, we set $I:=I_{\Sigma_k|\Sigma_{k-1}}$ and $Y:=\operatorname{Bl}_{\Sigma_{k-1}}\Sigma_k$. \(1) It is enough to show that the natural morphism $\beta^*_k I \rightarrow {\mathscr{O}}_{B^k(L)}(-Z_{k-1})$ is surjective. Thus we only have to show $I\cdot{\mathscr{O}}_{B^k(L)}={\mathscr{O}}_{B^k(L)}(-Z_{k-1})$. As we have seen in Proposition \[p:01\] (2) that ${\mathscr{O}}_{B^k(L)}((k+1)H-Z_{k-1})=\pi^*_kA_{k+1,L}$, we can form the following commutative diagram $$\xymatrix{ H^0(I(k+1)) \ar@{=}[r] \ar[d] & H^0({\mathscr{O}}_{B^k(L)}((k+1)H-Z_{k-1})) \ar[d]\\ I\cdot {\mathscr{O}}_{B^k(L)}((k+1)H) \ar[r] & \pi^*_kA_{k+1,L}~. }$$ But $A_{k+1,L}$ is globally generated by Lemma \[lem:Agg\]. Therefore $I\cdot {\mathscr{O}}_{B^k(L)}((k+1)H)=\pi^*A_{k+1,L}$, which implies $I\cdot {\mathscr{O}}_{B^k(L)}={\mathscr{O}}_{B^k(L)}(-Z_{k-1})$ as desired. \(2) We have the following factorization $$\xymatrix{ &Y=\operatorname{Bl}_{\Sigma_{k-1}}\Sigma_k \ar[d]^-\varphi\\ B^k(L) \ar[r]_-{\beta_k} \ar[ru]^-{\alpha_k} &\Sigma_{k}. }$$ Let $E$ be the exceptional divisor on $Y$. As $I(k+1)$ is globally generated, $\varphi^*{\mathscr{O}}_{ \Sigma_{k}}(k+1)(-E)$ is globally generated, and $\varphi^*{\mathscr{O}}_{ \Sigma_{k}}(k+2)(-E)$ is very ample. For any point $x\in \Sigma_m \setminus \Sigma_{m-1}$, the fiber $\beta^{-1}_k(x) \cong C_{k-m}$ (Proposition \[p:02\] (2.a)). Let $\alpha_{k,x} \colon \beta^{-1}_k(x)\rightarrow \varphi^{-1}(x)$ be the induced morphism on fibers. We see that $$\alpha^*_{k,x}(\varphi^*{\mathscr{O}}_{ \Sigma_{k}}(k+2)(-E))\cong A_{k+1,L}|_{C_{k-m}}\cong A_{k-m-1},L(-2\xi_{m+1,x}),$$ where $\xi_{m+1,x}$ is the unique degree $m+1$ divisor on $C$ determined by $x$. But the last line bundle is ample by Lemma \[lem:Agg\]. So $\alpha_{k,x}$ is finite, and therefore, $\alpha_{k}$ is finite. Hence $B^k(L)$ is the normalization of $Y$. \(3) This is a direct consequence of (2). Finally, we construct secant varieties of curves which are neither normal nor Cohen–Macaulay when $\deg L=2g+2k < 2g+2k+1$. This shows that the degree bounds on embedding line bundle in Theorem \[main1:singularities\] and Theorem \[main2:syzygies\] are optimal. \[Ex:non-normal\] Let $k\geq 1$ be an integer, and $C$ be a nonsingular projective curve of genus $g\geq 2k+2$. Take an effective divisor $D$ consisting of $2k+2$ general points of $C$ such that $h^0(C,{\mathscr{O}}_C(D))=1$. Consider a very ample line bundle $$L=\omega_C(D) \text{ with }\deg L=2g+2k.$$ Observe that $L$ separates $2k+1$ points, and $L$ separates $2k+2$ points except of $D$. We show that the $k$-th secant variety $$\Sigma_k=\Sigma_k(C,L)\subseteq {\mathbb{P}}(H^0(C, L))={\mathbb{P}}^{g+2k}$$ is neither normal nor Cohen–Macaulay. For any effective divisor $\xi$ on $C$, we denote by $\Lambda_{\xi}$ the linear space spanned by $\xi$ in the space ${\mathbb{P}}^{g+2k}$. Let $D_1$ and $D_2$ be two effective divisors of degree $k+1$ such that $D_1+D_2=D$. By Riemann-Roch, $h^0(C, L(-D_1-D_2))=g$. Thus $D_1+D_2$ span a linear space $\Lambda_{D_1+D_2}$ of dimension $2k$. This means that $\Lambda_{D_1}$ and $\Lambda_{D_2}$ span $\Lambda_{D_1+D_2}$ and intersect at a single point $q\in \Sigma_k \setminus C$. Let $Z$ be an effective divisor of degree $k+1$, and suppose $D_1+Z\neq D$. Then $L$ separates $D_1+Z$, and therefore, the space $\Lambda_{D_1+Z}$ has dimension $2k+1$. Hence $\Lambda_{D_1}\cap \Lambda_Z=\emptyset$. This implies that $q\in \Sigma_k \setminus \Sigma_{k-1}$ and except of $\Lambda_{D_1}$ and $\Lambda_{D_2}$, there is no any other $(k+1)$-secant $k$-plane of $C$ passing through $q$. For any two degree $k+1$ effective divisors $D_1'$ and $D_2'$ such that $D_1'+D_2'=D$, the $k$-secant planes $\Lambda_{D_1'}$ and $\Lambda_{D_2'}$ intersect at a single point in $\Sigma_k \setminus \Sigma_{k-1}$. Let $Q$ be the set of all such intersection points. Then $Q$ contains only finitely many points. Consider the morphism $\beta_k \colon B^k(L)\rightarrow \Sigma_k$. Let $x\in \Sigma_k \setminus \Sigma_{k-1}$. If $x\in Q$, then the fiber $\beta^{-1}_k(x)$ contains two points. If $x\notin Q$, then the fiber $\beta^{-1}_k(x)$ contains only one point $y$. In this case, we can show that the induced morphism $\beta^\#_k \colon T_x^*{\mathbb{P}}^r\longrightarrow {\mathfrak{m}}_{B^k(L),y}/{\mathfrak{m}}^2_{B^k(L),y}$ on cotangent spaces is surjective. Therefore $\beta_k$ is unramified at $y$, so it is isomorphic over $x$. In conclusion, $\beta_k$ is an isomorphism over $\Sigma_k \setminus (\Sigma_{k-1} \cup Q)$. Then we have the short exact sequence $$0\longrightarrow {\mathscr{O}}_{ \Sigma_{k}}\longrightarrow \beta_{k,*}{\mathscr{O}}_{B^k(L)}\longrightarrow {\mathscr{Q}}\longrightarrow 0,$$ where the support of the quotient sheaf ${\mathscr{Q}}$ has zero-dimensional components supported on $Q$. This means that $\Sigma_k$ is not normal at any point in $Q$. Moreover, $H^1(\Sigma_k, {\mathscr{O}}_{\Sigma_k}(-\ell)) \neq 0$ for all $\ell \geq 0$, so $\Sigma_k$ is not Cohen–Macaulay. Open problems {#sec:problem} ============= To conclude this paper, we present a number of open problems. We keep using notations introduced before; thus $C$ is a nonsingular projective curve of genus $g$ embedded by a very ample line bundle $L$ in the space ${\mathbb{P}}(H^0(C,L))={\mathbb{P}}^r$. One of critical steps in the proof of the main results is to establish the Du Bois type condition (\[DB\_cond\]). We have shown that $B^k(L)$ is the normalization of the blowup of $\Sigma_k$ along $\Sigma_{k-1}$. For better understanding of the geometry of $B^k(L)$, one observes that if $k=1$, then the variety $B^1(L)$ is indeed the blowup of $\Sigma_1$ along the curve $C$. This leads us to ask the following: Can the secant bundle $B^k(L)$ be realized as the blowup of $\Sigma_{k}$ along $\Sigma_{k-1}$? The Danila’s theorem (Theorem \[danila\]) handles the initial steps of projectively normality of secant varieties. It gives precise values of global sections of the symmetric products of the secant bundle $E_{k+1,L}$. On the other hand, the techniques used in Section \[sec:vanishing\] may offer an alternative approach to compute cohomology groups of the symmetric products of $E_{k+1,L}$. As an independent question, we wonder if one can deal with the following: Compute cohomology groups of the symmetric products of the secant bundle $E_{k+1,L}$ on $C_{k+1}$. If we view the classic theorem of Ein–Lazarsfeld [@Ein:SyzygyKoszul] as a higher dimensional generalization of Green’s result in [@G:Kosz], then we may ask a similar generalization of the results of the present paper to higher dimensional varieties. For a nonsingular projective variety $X$, consider the adjoint line bundle $L=K_X+dA$ where $A$ is an ample line bundle and $d$ is a natural number. For $d$ sufficiently large, $L$ embeds $X$ into a projective space. We expect that in this case the secant varieties of $X$ would have nice geometric and algebraic properties. Extend the results of present paper to secant varieties of a nonsingular projective variety $X$ embedded in a projective space by a sufficiently positive line bundle. This problem has two major essential difficulties. First of all, there is no a good construction involving secant bundles as the one in Betram’s work [@Bertram:ModuliRk2]. Secondly, the projectively normality of $X$ embedded by the adjoint line bundle is still unsolved. One may further impose the condition that $A$ is very ample so [@Ein:SyzygyKoszul] can be applied or may follow the idea in [@EL:AsySyz] to study the asymptotic behavior of secant varieties. However, the surface case seems a reasonable starting point toward the arbitrary dimensional case. Study secant varieties of a surface $X$ embedded by the ajoint line bundle $K_X+dA$ where $A$ is ample and $d$ is a large integer. [1]{} Aaron Bertram. Moduli of rank-[$2$]{} vector bundles, theta divisors, and the geometry of curves in projective space. , 35(2): 429–469, 1992. Chih-Chi Chou and Lei Song. Singularities of secant varieties. , (9): 2844–2865, 2018. Gentiana Danila. Sections de la puissance tensorielle du fibrè tautologique sur le schéma de Hilbert des points d’une surface. , 39(2): 311–316, 2007. Lawrence Ein and Robert Lazarsfeld. Syzygies and [K]{}oszul cohomology of smooth projective varieties of arbitrary dimension. , 111(1):51–67, 1993. Lawrence Ein and Robert Lazarsfeld. Asymptotic syzygies of algebraic varieties. ,190: 603–646, 2012. Lawrence Ein and Robert Lazarsfeld. The gonality conjecture on syzygies of algebraic curves of large degree. , 122: 301–313, 2015. David Eisenbud, Mark Green, Klaus Hulek, and Sorin Popescu. Restricting linear syzygies: algebra and geometry. , 141: 1460–1478, 2005. Tom Fisher. The higher secant varieties of an elliptic normal curve. preprint, 2006. Osamu Fujino and Yoshinori Gongyo. On canonical bundle formulas and subadjunctions. , 60: 255–264, 2012. Hans-Christian Graf von Bothmer and Klaus Hulek. Geometric syzygies of elliptic normal curves and their secant varieties. , 113 (1): 35–68, 2004. Mark Green. Koszul cohomology and the geometry of projective varieties. , 19: 125–171, 1984. Mark Green and Robert Lazarsfeld. Some results on the syzygies of finite sets and algebraic curves. , 67: 301–314, 1988. Christoper Hacon and James M^c^Kernan. On Shokurov’s rational connectedness conjecture. 138: 119–136, 2007. János Kollár. Singularities of the minimal model program. , 200, 2013. Robert Lazarsfeld. Cohomology on symmetric products, syzygies of canonical curves, and a theorem of [K]{}empf. In [*Einstein metrics and [Y]{}ang-[M]{}ills connections ([S]{}anda, 1990)*]{}, volume 145 of [*Lecture Notes in Pure and Appl. Math.*]{}, pages 89–97. Dekker, New York, 1993. Hideyuki Matsumura. , volume 8 of [*Cambridge Studies in Advanced Mathematics*]{}. Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. Jürgen Rathmann. An effective bound for the gonality conjecture. preprint, arXiv:1604.06072 Christophe Soulé. Secant varieties and successive minima. , 13: 323–341, 2004. Jessica Sidman and Peter Vermeire. Syzygies of the secant variety of a curve. , 3(4): 445–465, 2009. Jessica Sidman and Peter Vermeire. Equations defining secant varieties: geometry and computation. In [*Combinatorial aspects of commutative algebra and algebraic geometry*]{}, volume 6 of [*Abel Symp.*]{}, pages 155–174. Springer, Berlin, 2011. Brooke Ullery. Tautological bundles on the Hilbert scheme of points and the normality of secant varieties. Ph.D Thesis, University of Michigan, 2014. Brooke Ullery. On the normality of secant varieties. , 288: 631–647, 2016. Peter Vermeire. Some results on secant varieties leading to a geometric flip construction. , 125(3): 263–282, 2001. Peter Vermeire. On the regularity of powers of ideal sheaves. , 131(2): 161–172, 2002. Peter Vermeire. Regularity and normality of the secant variety to a projective curve. , 319(3): 1264–1270, 2008. Peter Vermeire. Equations and syzygies of the first secant variety to a smooth curve. , 140(8): 2639–2646, 2012. Ruijie Yang. A letter about syzygies of secant varieties. 2016. [^1]: L. Ein was partaily support NSF grant DMS-1801870. [^2]: J. Park was partially supported by NRF-2016R1C1B2011446 and the Sogang University Research Grant of 201910002.01.

--- abstract: 'UV frequency metrology has been performed on the [$a^3\Pi$]{} - [$X^1\Sigma^+$]{} (0,0) band of various isotopologues of CO using a frequency-quadrupled injection-seeded narrow-band pulsed Titanium:Sapphire laser referenced to a frequency comb laser. The band origin is determined with an accuracy of 5 MHz ($\delta \nu/\nu = 3 \cdot 10^{-9}$), while the energy differences between rotational levels in the [$a^3\Pi$]{} state are determined with an accuracy of 500 kHz. From these measurements, in combination with previously published radiofrequency and microwave data, a new set of molecular constants is obtained that describes the level structure of the [$a^3\Pi$]{} state of $^{12}$C$^{16}$O and $^{13}$C$^{16}$O with improved accuracy. Transitions in the different isotopologues are well reproduced by scaling the molecular constants of $^{12}$C$^{16}$O via the common mass-scaling rules. Only the value of the band origin could not be scaled, indicative of a breakdown of the Born-Oppenheimer approximation. Our analysis confirms the extreme sensitivity of two-photon microwave transitions between nearly-degenerate rotational levels of different $\Omega$-manifolds for probing a possible variation of the proton-to-electron mass ratio, $\mu=m_p/m_e$, on a laboratory time scale.' author: - 'Adrian J. de Nijs' - 'Edcel J. Salumbides' - 'Kjeld S.E. Eikema' - Wim Ubachs - 'Hendrick L. Bethlem' title: 'UV frequency metrology on CO ([$a^3\Pi$]{}); isotope effects and sensitivity to a variation of the proton-to-electron mass ratio' --- Introduction ============ The [$a^3\Pi$]{} state of CO is one of the most extensively studied triplet states of any molecule. The transitions connecting the [$a^3\Pi$]{} state to the [$X^1\Sigma^+$]{} ground state were first observed by Cameron in 1926 [@Cameron]. Later, the [$a^3\Pi$]{} state of the $^{12}$C$^{16}$O isotopologue was studied using radio frequency (rf) [@Freund; @Wicke:1972], microwave (mw) [@Saykally:1987; @Carballo; @Wada], infrared [@Havenith; @Davies], optical [@Effantin:1982] and UV spectroscopy [@Field]. The $^{13}$C$^{16}$O isotopologue was studied using rf [@Gammon:1971] and mw [@Saykally:1987] spectroscopy. Recently, Bethlem and Ubachs [@Bethlem:2009] identified metastable CO as a probe for detecting a temporal variation of the proton-to-electron mass ratio, $\mu=m_p/m_e$, on a laboratory time scale. Two-photon microwave transitions between nearly-degenerate rotational levels in different $\Pi_{\Omega}$ spin-orbit manifolds were shown to be very sensitive to a possible variation of $\mu$. As a measure of the inherent sensitivity of a transition to a drifting $\mu$, the sensitivity coefficient, $K_{\mu}$, is defined via: $$\frac{\Delta\nu}{\nu} = K_{\mu} \frac{\Delta\mu}{\mu}. \label{eq:detect_variation}$$ Transitions between the $J=8,\, \Omega=0$, the $J=6,\, \Omega=1$ and the $J=4, \, \Omega=2$ levels display sensitivities ranging from $K_{\mu}=-300$ to $+200$ [@Bethlem:2009]. For an overview on the topic of varying physical constants, we refer the reader to [@Uzan] and [@UbachsBuning]. In this paper, we present high-precision UV measurements of the [$a^3\Pi$]{} - [$X^1\Sigma^+$]{} (0,0) band in CO. In total 38 transitions in all six naturally occurring isotopes have been measured with MHz accuracy. All three $\Omega$-manifolds have been probed, with $J$ up to eight. A comprehensive fit of the optical data combined with previously published rf [@Wicke:1972] and mw [@Carballo; @Wada] measurements was performed. The molecular constants found for $^{12}$C$^{16}$O are mass scaled and compared with the measured transitions in other isotopologues. Level structure of CO ===================== The [$a^3\Pi$]{} state is the first electronically excited state of CO, lying 6 eV above the [$X^1\Sigma^+$]{} ground state. CO in the [$a^3\Pi$]{} state has two unpaired electrons, leading to a nonzero electronic spin, $\vec{S}$, and orbital angular momentum $\vec{L}$. For low rotational levels, the [$a^3\Pi$]{} state is best described in a Hund’s case (a) coupling scheme, with the good quantum numbers $\Lambda$ and $\Sigma$, the projection of $\vec{L}$ and $\vec{S}$ on the molecular axis, respectively. The projections of the total angular momentum $\vec{J}$ on the molecular axis and on the space-fixed axis lead to the good quantum numbers $\Omega$ and $M$. The basis functions are $|n\Lambda \rangle|v\rangle|S\Sigma\rangle|J\Omega M\rangle$, representing the electronic orbital, vibrational, electronic spin and rotational components of the wave function, respectively. For higher rotational levels, the spin decouples from the electronic angular momentum and a Hund’s case (b) coupling scheme becomes more appropriate. In Hund’s case (b), the different $\Omega$-manifolds are mixed. The energies of the lower rotational levels of the [$a^3\Pi$]{} and [$X^1\Sigma^+$]{} states are shown in Fig. \[fig:lvlscheme\], together with selected transitions. The transitions are denoted by $\Delta J_{\Omega+1}(J")$, where transitions with $\Delta J=-1,0$ and $1$ are denoted by $P$, $Q$ and $R$, respectively. As the parity changes in a one-photon transition, the upper lambda-doublet component of rotational levels in the [$a^3\Pi$]{} can only be reached via $Q$-transitions, whereas the lower lambda-doublet components can only be reached via $P$ or $R$ transitions. The separation of the electronic motion and nuclear motion is not exact, leading to a splitting into lambda-doublet states of opposite parity, as indicated, not to scale, in Fig. \[fig:lvlscheme\]. The lambda doubling in the $\Omega=0$ state is large and relatively independent of $J$. The lambda doubling in the $\Omega=1$ and $\Omega=2$ manifolds is much smaller. In Fig. \[fig:lvlscheme\] the total parity, *i.e.*, the product of the symmetries of the rotational and electronic parts of the wavefunction, is indicated by the - and + signs. The electronic part of the wave function of the upper (lower) lambda-doublet levels has $f$ ($e$) symmetry. ![Energy level diagram of the [$X^1\Sigma^+$]{} ($v=0$) ground state and the [$a^3\Pi$]{} ($v=0$) state of $^{12}$C$^{16}$O. Rotational quantum numbers and total parity are listed for each level. For the [$a^3\Pi$]{} state, the value of the band origin, $E_\Pi$, is subtracted from the energy scale. The [$a^3\Pi$]{} state has three $\Omega$-manifolds, arising from spin-orbit coupling, and shows lambda-type doubling, as illustrated, not to scale. A number of transitions are indicated by the vertical arrows.[]{data-label="fig:lvlscheme"}](fig1-lvlscheme.pdf){width="\linewidth"} The spin-forbidden [$a^3\Pi$]{} - [$X^1\Sigma^+$]{} system becomes weakly allowed due to spin-orbit mixing of singlet electronic character into [$a^3\Pi$]{}, most significantly of the $A^1\Pi$ state lying 2 eV above the [$a^3\Pi$]{} state. As the $A^1\Pi$ state consists of a single, $\Omega=1$, manifold, it only couples to the $\Omega=1$ levels in [$a^3\Pi$]{}. Transitions to the $\Omega=0, J>0$ and $\Omega=2$ manifolds become weakly allowed by mixing of the different $\Omega$-manifolds. The $\Omega=0,J=0$ level is not mixed with the other $\Omega$-manifolds, hence, the transition to this level, *i.e.* the $P_1(1)$, does not obtain transition strength via coupling to the $A^1\Pi$ state. The [$a^3\Pi$]{} ($v=0$) state can only decay to the ground state, hence, the radiative lifetimes of the different rotational are inversely proportional to the transition strengths. The lifetimes are thus strongly dependent on $J$ and $\Omega$. For example the $J=2,\, \Omega=2$ level has a lifetime of 140 ms, whereas the $J=1, \, \Omega=1$ level has a lifetime of 2.6 ms [@Gilijamse]. Isotopes with an odd number of nucleons have a non-zero nuclear spin that leads to hyperfine structure. The two relevant odd-nucleon-number nuclei are $^{13}$C and $^{17}$O, with nuclear spin $I=1/2$ and $I=5/2$, respectively. Due to its zero electronic angular and spin momentum the hyperfine splitting in the ground-state of CO is small ($\approx$50 kHz) [@Klapper:2000]. The hyperfine splittings in the [$a^3\Pi$]{} state vary between 30 and 500 MHz for the measured transitions. Experimental setup ================== ![A schematic drawing of the experimental setup. A pulsed beam of CO is produced by expanding CO gas into vacuum, using a solenoid valve. After passing through a 1 mm skimmer, the molecular beam is crossed at right angles with laser radiation at 206 nm. After being excited to the [$a^3\Pi$]{} state, the molecules fly 60 cm downstream before hitting an electron multiplier tube, where they are detected. The laser interaction region is built in a Sagnac interferometer to correct for Doppler shifts.[]{data-label="fig:sagnac"}](fig2-COBeam2.pdf){width="\linewidth"} The molecular beam setup used for frequency metrology on CO is schematically depicted in Fig. \[fig:sagnac\]. A pulsed beam of CO is produced by expanding CO gas into vacuum, using a solenoid valve (General Valve series 9). A backing pressure of 2 bar was used for recording transitions at low $J$, while for recording transitions at higher $J$, the backing pressure was reduced to 0.5 bar. For recording transitions in $^{13}$C$^{16}$O, isotopically enriched CO (Linde Gas) was used. The enriched sample also contained slightly enhanced fractions of $^{13}$C$^{18}$O and $^{13}$C$^{17}$O, sufficient for obtaining signals. Spectra of $^{12}$C$^{17}$O and $^{12}$C$^{18}$O were measured using a natural CO sample. After passing through a 1 mm skimmer, the molecular beam is crossed at right angles with laser radiation tunable near 206 nm. In the interaction region, a magnetic field of up to 200 Gauss can be applied by two coils in Helmholtz configuration. After being excited to the [$a^3\Pi$]{} state, the molecules fly 60 cm downstream before being detected by an Electron Multiplier Tube (EMT). The resulting signal is recorded using a digital oscilloscope and the integrated signal is stored. The absolute detection efficiency of this method for metastable CO (6 eV internal energy) is estimated to be on the order of 10$^{-3}$ [@Jongma:JCP]. Note that the time of flight (750 $\mu$s) is short compared to the lifetime of the metastable state ($>$2.6 ms). An adjustable slit is mounted in front of the EMT to limit the divergence of the beam that reaches the detector, thereby limiting the Doppler width of the recorded transitions. ![(Color online) A schematic drawing of the laser system. A ring laser is used to injection seed an oscillator cavity, which is pumped by a Nd:YAG laser at 10 Hz. The produced IR pulses are amplified in a multi-pass amplifier and subsequently quadrupled in two consecutive BBO crystals. The absolute frequency of the CW seed laser is determined using a fiber comb laser. OI: optical isolator; GP: glass plate; OC: output coupler; HR: high reflective mirror; PA: Piezo Actuator; HC-lock: Hänsch-Couillaud lock; SMF: single mode fiber; AOM: acousto-optical modulator.[]{data-label="fig:lasersetup"}](fig3-lasersetup.pdf){width="\linewidth"} The spectroscopic measurements are performed with a narrow-band frequency-quadrupled titanium:sapphire (Ti:Sa) pulsed laser described in detail by Hannemann *et al.* [@Hannemann:PRA1; @Hannemann:PRA2]. A schematic drawing of the laser setup is shown in Fig. \[fig:lasersetup\]. A continuous wave (CW) Ti:Sa ring laser (Coherent 899) produces around 700 mW of laser power tunable near 824 nm. Its output is split into three parts, with all parts having approximately the same power. One part of the light is sent as a seed frequency to a pulsed Ti:Sa ring oscillator that is pumped at 10 Hz with 50 mJ of pulsed 532 nm light from an injection seeded Nd:YAG laser (Spectra Physics LAB-170). The oscillator is locked to the CW seed light using a Hänsch-Couillaud scheme. The pulsed IR light emanating from the oscillator is amplified in a bow-tie multi-pass Ti:Sa amplifier pumped with 300 mJ of pulsed 532 nm light from the same Nd:YAG laser that pumps the oscillator. After nine passes the laser power of the IR-beam is around 70 mJ in a 100 ns pulse. These pulses are then frequency doubled twice in two consecutive BBO crystals, resulting in pulses of 20 $\mu$J at 206 nm. In order to determine the absolute frequency of the CW Ti:Sa ring laser, the CW light is mixed with the light from an erbium-doped fiber frequency-comb laser (Menlo systems MComb at 250 MHz repetition frequency) that is locked to a global positioning system (GPS) disciplined Rb-clock standard. The optical interference beat signal is measured with a photodiode and an Agilent 53132A counter. The obtained beat frequency is then transferred via ethernet to the central computer at which the data is analyzed. Further details on the absolute frequency calibration can be found in Sec. \[subsubsec:comb\]. By making a small portion of the pulsed light interfere with part of the CW light, possible small differences between the frequencies of the CW seed laser and the central frequency of the pulsed output of the bow-tie amplifier are measured and corrected for [@Hannemann:PRA1]. In order to have a good fringe visibility, and to ascertain that the full wavefront of the pulsed output is mapped onto the CW reference beam, both beams are sent through a short single-mode fiber. The beat pattern is detected using a fast photodiode in combination with an oscilloscope and analyzed online. The UV laser beam is split into two parts and sent through the molecular beam machine from opposite sides to limit the Doppler shift due to a possible imperfect perpendicular alignment of the laser beam. In order to ensure that the two beams are perfectly counterpropagating, the two laser beams are recombined after passing through the machine, forming a Sagnac interferometer. The paths through the molecular beam machine are aligned such that the two beams interfere destructively at the exit port (a dark fringe). The transition frequency is measured twice, using either the laser beam from the left or from the right hand side [@Hannemann:OptLett]. Experimental results ==================== ![(Color online) Recordings of the $R_2(0)$ transition in $^{12}$C$^{16}$O measured using the laser beam propagating from the right hand side, upper panel, and the left hand side, lower panel. The frequency of the seed laser is scanned while the signal from the EMT is recorded. Each point represents a single laser pulse. For each pulse the beat frequency of both the fiber comb and the CW-pulse offset measurement setup are recorded. A single scan takes around 20 minutes.[]{data-label="fig:typical"}](fig4-typical2.pdf){width="1.1\linewidth"} ------------------ --------------- ---------------- ----------- ----------- Isotopologue Transition Observed (MHz) Residuals Residuals Fitted Scaled $^{12}$C$^{16}$O $ R_1(0) $ 1452 065 305.5 0.4 — $ P_1(1) $ 1451 857 145.0 0.9 — $ Q_1(1) $ 1452 002 039.3 1.3 — $ R_1(7) $ 1452 112 469.1 2.1 — $ Q_1(8) $ 1451 232 745.8 2.4 — $ R_2(0) $ 1453 348 525.5 1.1 — $ Q_2(1) $ 1453 233 648.6 1.2 — $ R_2(5) $ 1453 607 338.8 -4.8 — $ Q_2(6) $ 1452 922 384.0 -6.6 — $ P_2(7) $ 1452 109 212.5 -6.2 — $ R_3(1) $ 1454 488 881.8 1.8 — $ Q_3(2) $ 1454 258 350.7 1.4 — $ R_3(3) $ 1454 683 187.7 2.2 — $ Q_3(4) $ 1454 222 240.4 2.7 — $^{12}$C$^{17}$O $ R_2(0) $ 1453 426 677.0 — — $ $ 1453 426 924.0 — — $ $ 1453 427 108.2 — — $^{12}$C$^{18}$O $ R_2(0) $ 1453 498 161.1 — — $^{13}$C$^{16}$O $ R_1(2) $ 1452 325 025.5 -0.2 0.2 $ $ 1452 325 217.9 -0.3 0.7 $ R_1(7) $ 1452 263 496.1 -1.6 -13.1 $ $ 1452 263 906.8 2.3 -8.0 $ Q_1(8) $ 1451 425 012.3 -0.3 -9.9 $ $ 1451 425 321.6 0.5 -10.1 $ R_2(0) $ 1453 491 411.3 0.8 3.4 $ $ 1453 491 449.2 1.8 4.4 $ Q_2(1) $ 1453 381 555.9 0.4 2.5 $ $ 1453 381 613.6 -0.2 2.4 $ R_2(1) $ 1453 570 841.9 -0.5 3.8 $ $ 1453 570 876.4 -0.6 3.4 $ R_2(5) $ 1453 740 818.2 0.7 5.2 $ $ 1453 740 854.3 -0.9 2.8 $ Q_2(6) $ 1453 085 785.3 -0.6 0.1 $ $ 1453 085 898.7 -0.6 0.9 $ P_2(7) $ 1452 308 565.9 -0.3 4.2 $ $ 1452 308 602.9 -0.9 2.8 $ R_3(1) $ 1454 635 451.9 -1.2 -3.1 $ $ 1454 636 019.2 -1.0 -2.7 $ R_3(3) $ 1454 818 557.3 0.9 2.8 $ $ 1454 818 676.7 0.8 2.6 $ Q_3(4) $ 1454 377 868.2 0.8 2.6 $ $ 1454 377 989.2 0.1 2.0 $^{13}$C$^{17}$O $ R_2(0) $ 1453 571 485.7 — — $ $ 1453 571 732.5 — — $ $ 1453 571 918.5 — — $^{13}$C$^{18}$O $ Q_1(3) $ 1452 210 864.9 — -12.3 $ $ 1452 211 002.6 — -12.4 $ R_2(0) $ 1453 643 585.2 — -1.7 $ $ 1453 643 626.6 — -1.4 $ Q_2(1) $ 1453 539 184.1 — -4.4 $ $ 1453 539 244.9 — -4.6 $ R_2(5) $ 1453 882 578.9 — 15.9 $ $ 1453 882 612.1 — 13.8 $ Q_2(6) $ 1453 259 995.8 — 7.6 $ $ 1453 260 099.7 — 4.4 $ R_3(3) $ 1454 963 029.8 — -0.5 $ $ 1454 963 165.2 — 1.7 $ Q_3(4) $ 1454 544 280.2 — -2.6 $ $ 1454 544 414.5 — -3.4 ------------------ --------------- ---------------- ----------- ----------- : Measured transition frequencies for various isotopologues of CO. The rightmost two columns are the residuals from the fit and the mass-scaling procedure discussed in Sec. \[sec:ana\]. \[tab:fitresults\] In Table \[tab:fitresults\] the frequencies are listed for the measured transitions in the [$a^3\Pi$]{} - [$X^1\Sigma^+$]{} (0,0) band. In view of the time-consuming measurement procedure, only a selection of transitions has been investigated. In total, 38 transitions have been recorded in all six stable isotopologues of CO. In $^{12}$C$^{16}$O, $^{13}$C$^{16}$O and $^{13}$C$^{18}$O, we have recorded low $J$ transitions to each $\Omega$-manifold and each parity level, to be able to give a full analysis of the level structure. Also, transitions to the $J=8,\,\Omega=0$, $J=6,\,\Omega=1$ and $J=4,\,\Omega=2$, the near degenerate levels of interest to the search for $\mu$ variation, were measured. Furthermore, the $R_2(0)$ transition, the most intense line under our conditions, has been measured in the $^{12}$C$^{17}$O, $^{12}$C$^{18}$O and $^{13}$C$^{17}$O isotopologues. Note that the observed signal strengths for the different transitions varies over more than three orders in magnitude. In Fig. \[fig:typical\] a typical recording of the $R_2(0)$ transition in $^{12}$C$^{16}$O is shown. The upper and lower graph show the spectra obtained with the laser beam propagating through either path of the Sagnac interferometer. For each pulse of the pulsed laser, the frequency of the CW laser is determined using the fiber comb laser and a possible shift between the pulsed laser and the CW laser is determined using the online CW-pulse offset detection. Subsequently, these data are combined with the metastable CO signal from the EMT. Typically, the recorded scans are not perfectly linear, resulting in an uneven distribution of the data points along the frequency-axis. This has no influence on the peak determination. At the peak of the transition, the observed signal corresponds to typically a few thousand detected metastable CO molecules per laser pulse. Note that the fluctuations in the signal shown in Fig. \[fig:typical\] are due to pulse-to-pulse variations of the molecular beam and the UV power and not due to counting (Poisson) statistics. The solid lines in the figure show a Gaussian fit to the spectra. The transition frequency of the $R_2(0)$ transition is determined by taking the average of the measurements taken with the laser beam propagating from either side. ![(Color online) Recording of the $P_2(7)$ transition and the $R_1(7)$ transition in $^{12}$C$^{16}$O. Both transitions originate from the same ground state level and connect to one of two nearly degenerate levels. Therefore, the combination difference of these two transition frequencies corresponds to the frequency of the transition between the two nearly-degenerate levels.[]{data-label="fig:longscan"}](fig5-longscan.pdf){width="1.1\linewidth"} Transitions to the six near-degenerate levels in both $^{12}$C$^{16}$O and $^{13}$C$^{16}$O were measured. For $^{13}$C$^{18}$O transitions to four of the six near degenerate levels were obtained. In Fig. \[fig:longscan\] a recording of the $P_2(7)$ and $R_1(7)$ transitions in $^{12}$C$^{16}$O is shown. Both transitions originate from the $J=7$ ground state level, thus the combination difference is equal to the frequency of the $J=6,\,\Omega =1,\,+ \rightarrow J=8,\,\Omega=0,\,+$ transition, which is measured to be 3256.6 MHz. As the transition strengths of transitions in the spin forbidden [$a^3\Pi$]{} - [$X^1\Sigma^+$]{} system originate from mixing of [$a^3\Pi$]{}  $\Omega=1$ with $A^{1}\Pi$ [@Gilijamse], the transition strengths of the different transitions are proportional to (the square of) the $\Omega=1$ character of the final rotational level and the Hönl-London factor. From this, we expect the $P_2(7)$ to be 3.2 times more intense than the $R_1(7)$. Experimentally, we find a ratio of 2.4 to 1. The deviation is explained by the fact that the $J=8,\, \Omega=0$ has a longer lifetime than the $J=6,\, \Omega=1$ (15.7 ms vs. 3.5 ms), and consequently, a smaller fraction of the metastable molecules decays back to the ground state before reaching the EMT. Taking the lifetime into account, we expect a ratio of 2.7 to 1, in reasonable agreement with the experiment. ![(Color online) A recording of the $P_1(1)$ transition in $^{12}$C$^{16}$O. If only the coupling to the $A^1\Pi$ state is considered, this transition has zero transition strength.[]{data-label="fig:P11"}](fig6-p11.pdf){width="1.1\linewidth"} The $J=0,\, \Omega=0$ is the only $J=0$ level, and is therefore not mixed with the $\Omega=1$ and $\Omega=2$ manifolds. Consequently, the $P_1(1)$ transition, connecting the [$X^1\Sigma^+$]{} $J=1$ with the [$a^3\Pi$]{} $\Omega=0,J=0$, + parity level does not obtain any transition strength from coupling to the $A^{1}\Pi$ state. Nevertheless, we were able to observe this transition, shown in Fig. \[fig:P11\], albeit with low signal to noise. From our measurements, we estimate that the $P_1(1)$ transition is about 65 times weaker than $Q_1(1)$ and about $10^{4}$ times weaker than $Q_{2}(1)$, resulting in a lifetime of 8(1) s [@groenenboomprivatecommunications]. The transition strength is ascribed to mixing of the [$a^3\Pi$]{} state with a $^{1}\Sigma^{+}$ state, most likely the [$X^1\Sigma^+$]{} ground state [@Minaev]. ![(Color online) A recording of the $Q_2(1)$ transition in $^{13}$C$^{16}$O. The labels indicate the value of the total angular momentum $\vec{F}=\vec{J}+ \vec{I}$ in the excited state.[]{data-label="fig:hyperfine"}](fig7-hyperfine.pdf){width="1.1\linewidth"} The rotational levels of isotopologues with one or two odd-numbered nuclei show hyperfine splitting. In Fig. \[fig:hyperfine\] a recorded spectrum of the $Q_2(1)$ transition in $^{13}$C$^{16}$O is shown. The transitions to the hyperfine sublevels $F=J-I=1/2$ and $F=J+I=3/2$ of the excited state are clearly resolved. Measurement uncertainties ========================= In this paragraph the main sources of uncertainty in the transition frequencies are discussed. The uncertainty budget is summarized in Table \[tab:error\]. Zeeman effect ------------- The ground state of CO is a $^1\Sigma$ state, hence, its Zeeman shift is small. The [$a^3\Pi$]{} state on the other hand has both electronic angular momentum and spin, and experiences a considerable Zeeman shift. In the $\Omega=2$ state the effect due to the electronic angular momentum and due to spin will add up, whereas in the $\Omega=0$ and $\Omega=1$ state these effects will partly cancel. Hence, we expect the largest Zeeman shift to occur in the $\Omega=2$ levels, in particular in the $J=2,\Omega=2$ level. In Fig. \[fig:Zeeman\] the recorded spectra are shown for the $R_{1}(1)$, $R_{2}(1)$ and $R_{3}(1)$ transitions in $^{12}$C$^{16}$O in a magnetic field of 170 Gauss. In these measurements, the polarization of the 206 nm light is parallel to the applied magnetic field, hence, only $\Delta M_{J}=0$ transitions are allowed. As all three measured transitions originate from the [$X^1\Sigma^+$]{} $J=1$ level only the $M_J= \pm1,0$ components of the probed [$a^3\Pi$]{} levels are observed. The Zeeman shift of the $J=2,\,\Omega=2,\,M_{J}=1$ state is 1.2 MHz/Gauss, while the shifts of the $J=2,\,\Omega=1,\,M_{J}=1$ and $J=2,\,\Omega=0,\,M_{J}=1$ are about 5 times smaller. It is observed that the transitions to the $M_J=-1$ and $M_J=1$ states are approximately equally strong, implying that the polarization of the 206 nm is nearly perfectly linear, estimated to be better than 99%, as expected from light that has been quadrupled in non-linear crystals. In the earth magnetic field, approximately $0.5$ Gauss, the $J=2,\,\Omega=1,\,M_{J}=-2$ and $M_{J}=2$, will be shifted in opposite directions by less than 1 MHz. As the polarization is nearly perfect, the shift of the line center is estimated to be less than 1 kHz. ![The $R_1(1)$, $R_2(1)$ and $R_3(1)$ transitions in $^{12}$C$^{16}$O measured in an applied magnetic field of 170 Gauss, showing Zeeman splitting.[]{data-label="fig:Zeeman"}](fig8-zeeman.pdf){width="1.1\linewidth"} Stark effect ------------ The Stark shift in the ground state of CO is negligibly small as the dipole moment is only 0.1 Debye (corresponding to $0.17\cdot10^{-2}$cm$^{-1}$/(kV/cm)) and mixing occurs between rotational levels. The Stark shift in the [$a^3\Pi$]{} state on the other hand is larger as the dipole moment is 1.37 Debye and mixing occurs between the lambda-doublet components [@Jongma:CPL]. We estimate that the electric field in the excitation region is below 1 V/cm. This corresponds to a Stark shift of 30 kHz for the $J=2,\,\Omega=2,\,M\Omega=4$ and less for all other levels. AC-Stark effect --------------- A priori, the AC-Stark shift is difficult to estimate. We have measured the $R_{2}(0)$ transition in $^{12}$C$^{16}$O for different laser powers, reducing the laser power by over an order of magnitude, but found no significant dependence of the transition frequency on laser power. Thus, we estimate the AC-Stark shift to be less than 100 kHz. Uncertainty in absolute frequency determination {#subsubsec:comb} ----------------------------------------------- The absolute frequency of the CW light is calibrated by mixing this light with the output of a frequency-comb laser and counting the resulting beat frequency $f_{\mathrm{beat}}$. The frequency $f_{\mathrm{CW}}$ is then obtained by the relation $$f_{\mathrm{CW}} = n \cdot f_{\mathrm{rep}} + f_{0} + f_{\mathrm{beat}},$$ with $n$ the mode number of the frequency comb that is nearest to the frequency of the CW light, and $f_{\mathrm{rep}}$ and $f_{0}$ the repetition and the carrier-envelope offset frequencies of the frequency comb, respectively. $f_{\mathrm{rep}}$ is tunable over a small range around 250 MHz and $f_{0}$ is locked at 40 MHz. We infer that the sign of $f_{\mathrm{beat}}$ is positive from the observation that $f_{\mathrm{beat}}$ increases when the CW laser is scanned towards higher frequency. Likewise, the sign of $f_{0}$ is positive from the observation that $f_{\mathrm{beat}}$ increases when $f_{0}$ is decreased. The beat note is averaged over a period of 100 ms. On this time scale, the accuracy of the Rb-clock standard is $10^{-10}$, equivalent to 150 kHz at the used frequencies. This uncertainty enters separately in each data point taken in a frequency scan. As each scan is approximately 1000 data points, this uncertainty averages out. The integer mode number $n$ is determined by measuring the transition frequency of the $R_{2}(0)$ transition in $^{12}$C$^{16}$O using the frequency comb at different repetition frequencies and then determining at which transition frequency these three measurements coincide. This method gives an unambiguous transition frequency provided that the change in repetition frequency is much larger than the measurement uncertainty [@Witte:Science]. In our case this condition is well met as the uncertainty between two consecutive measurements is on the order of 500 kHz, while the repetition frequency of the comb is varied by 2 MHz. The absolute value obtained for the $R_{2}(0)$ transition of $^{12}$C$^{16}$O is used to calibrate a wavelength meter (Burleigh WA-1500, 30 MHz precision) on a daily basis. This wavelength meter is then used to determine the mode number for the measurements of the other transitions. Doppler effect {#subsubsec:sagnac} -------------- ![(Color online) The $Q_2(1)$ transition measured using slit widths of 10 mm and 3 mm. When the width of the slit is decreased the transition becomes narrower, down to 23 MHz for a slit width of 3 mm. The 23 MHz spectral width is attributed to the line width of the UV-laser.[]{data-label="fig:dopplerwidth"}](fig9-dopplerwidth2.pdf){width="1.1\linewidth"} In a molecular beam experiment, the first-order Doppler effect is reduced by aligning the laser-beam perpendicular to the molecular beam. Two residual effects remain: (i) The finite transverse temperature of the molecular beam leads to a broadening of the transition, (ii) A possible imperfect perpendicular alignment of the laser beam leads to a shift of the center frequency. To limit the Doppler width, we have placed a variable slit in front of the EMT, which limits the divergence of the beam hitting the detector. In Fig. \[fig:dopplerwidth\] recordings of the $Q_2(1)$ transition of $^{12}$C$^{16}$O using a slit width of 10 mm and 3 mm are shown. When the slit width is reduced below 3 mm the width of the transition remains the same while the signal decreases further. The minimum full width at half maximum (FWHM) observed is 23 MHz, which is attributed to the line width of the UV-radiation. In our measurements, we have used a slit width of 6 mm, as a compromise between signal intensity and line width. In order to eliminate the Doppler shift in the measurement, the UV laser beam is aligned in the geometry of a Sagnac interferometer. The angle between the two counterpropagating laser beams may be estimated to be smaller than $\lambda/d$, with $\lambda$ being the wavelength of the light in nm and $d$ being the diameter of the laser beam in mm [@Hannemann:OptLett]. In our case this results in a maximum Doppler shift of 200 kHz. We have verified this by comparing measurements of the $R_2(0)$ transition of $^{12}$C$^{16}$O in a pure beam of CO (velocity of 800 m/s) and a beam of 5% CO seeded in He (longitudinal velocity of 1500 m/s). The second-order Doppler shift is sub-kHz. The recoil shift is $\approx$ 25 kHz. Uncertainty in peak determination --------------------------------- The number of metastable molecules detected (at the peak of) the $R_2(0)$ transition of $^{12}$C$^{16}$O, the strongest transition observed, is on the order of $10^{4}$ per laser pulse, while at the $P_1(1)$ transition of $^{12}$C$^{16}$O, the weakest transition observed, it is only a few molecules per laser pulse. The ability to determine the line center from the measurement is limited by pulse-to-pulse variations of the molecular beam and the UV power. Typically, the uncertainty in the peak determination is 300 kHz (corresponding to 1% of the line-width). Frequency chirp in the pulsed laser system {#subsubsec:chirp} ------------------------------------------ The frequency of the pulsed IR light differs slightly from the frequency of the CW seed laser mainly due to two effects: (i) Mode pulling in the oscillator and (ii) Frequency chirp and shift in the amplifier. The pump laser induces a change of the refractive index of the Ti:Sa crystal and therefore a variation of the optical path length of the cavity on a time scale much shorter than the response time of the electronic system used for locking the cavity. Therefore, at the instant the pulse is produced the cavity resonances are shifted from the frequency they are locked to. The size of the frequency shift due to mode pulling depends on the pump power. The optical properties of the Ti:Sa crystal in the bow-tie amplifier change depending on the population inversion, which decreases during the multi-step amplification of the IR pulse. The combined effect of both phenomena results in a shift on the order of a few MHz. We measure and compensate for these effects by making a part of the pulsed light interfere with part of the CW light. The CW-pulse offset can be determined within 200 kHz. However, as a result of wavefront distortions of the pulsed beam, it depends very critically on the alignment. We have measured the chirp at different positions in the wavefront by moving a small pinhole through the pulsed beam, and observed deviations of a few MHz. These deviations are amplified in the harmonic generation stages. As it is unclear which part of the pulsed laser beam gives rise to the observed metastable CO signal, we cannot compensate for this effect. Consequently, we find rather large day-to-day deviations in the measurements; measured transition frequencies taken on the same day, with a specific alignment and setting of the Ti:Sa oscillator and bow-tie amplifier, agree within a 500 kHz. Measured transition frequencies taken on different days, on the other hand, may deviate by a few MHz. In order to compensate for this systematic effect, we chose the $R_{2}(0)$ transition in $^{12}$C$^{16}$O as an anchor for each measurement session. With this anchoring, measurements of any given transition (other than the $R_{2}(0)$ transition) taken on different days agree within 500 kHz, showing that the consistency of the positions of the rotational levels of [$a^3\Pi$]{} is sub-MHz. The relatively large systematic uncertainty due to mode pulling and chirp will only enter in the value of the band origin. The root-mean-square deviation of 14 measurements of the $^{12}$C$^{16}$O $R_{2}(0)$ transition is about 2 MHz. We have set the uncertainty of the $R_{2}(0)$ transition, and hence the systematic uncertainty on all transitions, conservatively at 5 MHz. Note that Salumbides *et al.* [@Salumbides] have circumvented this systematic offset of the absolute calibration by putting a small pinhole in the pulsed beam, therewith selecting a portion of the laser beam with smaller wave-front distortions. We could not apply this method here, as it resulted in a too large decrease in signal. Source Uncertainty(MHz) ------------------------------------------- ------------------ Relative:     Zeeman effect $<$0.1     Stark effect $<$0.03     AC-Stark effect $<$0.1     Comb absolute frequency determination $\ll$0.15     Doppler effect $<$0.2     Peak determination 0.3     CW-pulse offset 0.2 Absolute:     CW-pulse offset 5 : The uncertainty budget of the measured transitions.[]{data-label="tab:error"} Analysis {#sec:ana} ======== Effective Hamiltonian and least square fitting for $^{12}$C$^{16}$O and $^{13}$C$^{16}$O {#subsec:C12O16} ---------------------------------------------------------------------------------------- ----------------------------------------- ------------------------------------------------------------------------------------------- $\langle ^3\Pi_0 | H | ^3\Pi_0 \rangle$ $E_\Pi+B(x+1)-D(x^2+4x+1)-A-2A_J(x+1)-C\mp C_{\delta}-\gamma+xB_0+(1\mp1)(4B_1^+-2B_0^+)$ $\langle ^3\Pi_1 | H | ^3\Pi_1 \rangle$ $E_\Pi+B(x+1)-D(x^2+6x-3)+2C-2\gamma-2B_0^++4B_1^+ $ $\langle ^3\Pi_2 | H | ^3\Pi_2 \rangle$ $E_\Pi+B(x-3)-D(x^2-4x+5)+A+2A_J(x-3)-C-\gamma+B_0^+(x-2)$ $\langle ^3\Pi_0 | H | ^3\Pi_1 \rangle$ $-\sqrt{2x}\left[ B-2D(x+1)-A_J-0.5\gamma+(1\mp2)B_1^+ \right] $ $\langle ^3\Pi_0 | H | ^3\Pi_2 \rangle$ $ -\sqrt{x(x-2)}\left[ 2D \pm B_0^+\right] $ $\langle ^3\Pi_1 | H | ^3\Pi_2 \rangle$ $-\sqrt{2(x-2)}\left[ B-2D(x-1)+A_J-0.5\gamma+B_1^+\right] $ ----------------------------------------- ------------------------------------------------------------------------------------------- Carballo *et al.* Brown and Merer ------------------- -------------------- $E_\Pi$ $T+B-D-q/2$ $B$ $B-2D-q/2$ $D$ $D$ $A$ $A+A_D+\gamma+p/2$ $A_j$ $A_D/2$ $C$ $-2/3\lambda$ $C_{\delta}$ $o$ $\gamma$ $\gamma+p/2$ $B_0^+$ $q/2$ $B_1^+$ $p/4+q/2$ : The conversion factors between the molecular constants used in the effective Hamiltonian of Carballo *et al.* [@Carballo] and Brown and Merer [@BrownMerer].[]{data-label="tab:rosetta"} The effective Hamiltonian for a $^3 \Pi$ state has been derived by several authors [@Field; @BrownMerer]. We have used the effective Hamiltonian from Field *et al.* [@Field] with the additions and corrections discussed by Carballo *et al.* [@Carballo]. The matrix elements for this effective Hamiltonian are listed in Table \[tab:matrix\]. As discussed by Carballo *et al.* [@Carballo], this Hamiltonian is equivalent to the effective Hamiltonian derived by Brown and Merer [@BrownMerer], but the molecular constants used in these Hamiltonians have a slightly different physical meaning, which will have consequences for the mass scaling discussed in Sec. \[subsec:Isotopes\]. In Table \[tab:rosetta\], relations between the constants in the Hamiltonian of Brown and Merer and those in the Hamiltonian of Field *et al.* are listed for clarity. A least-squares fitting routine was written in Mathematica to obtain the molecular constants of the effective Hamiltonian. We have verified that our fitting routine exactly reproduces the results of Carballo *et al.* [@Carballo] and that it is consistent with PGopher [@pgopher]. For $^{12}$C$^{16}$O, we have fitted our optical data simultaneously with lambda-doubling transition frequencies in the rf domain measured by Wicke *et al.* [@Wicke:1972] and the rotational transition frequencies in the mw domain measured by Carballo *et al.* [@Carballo] and Wada and Kanamori [@Wada]. The fitted set consists of 9 rf transitions, 28 mw transitions and 14 optical transitions. The different data sets were given a weight of one over the square of the measurement uncertainties, taken as 50 kHz for the mw and rf data and 1 MHz for the optical data. The molecular constants for the [$X^1\Sigma^+$]{} state of $^{12}$C$^{16}$O are taken from Winnewisser *et al.* [@Winnewisser:1997]. As discussed by Carballo *et al.* [@Carballo], $\gamma$ and $A_J$ can not be determined simultaneously from data of a single isotopologue, therefore $\gamma$ was fixed to zero. The first column of Table \[tab:fitpars\] lists the different constants obtained from our fit for $^{12}$C$^{16}$O. The deviations between the observed and fitted transition frequencies are listed in Table \[tab:fitresults\]. -------------- ------------- ------------------ ------------------ ------------------ ------------------ Molecular $K_{\mu}^X$ $^{12}$C$^{16}$O $^{13}$C$^{16}$O $^{13}$C$^{16}$O $^{13}$C$^{18}$O Constant Fitted Fitted Scaled Scaled $E$ 0 1453190243.1(8) 1453340486.4(7) 1453340489.2(9) 1453500584(3) $B$ -1 50414.24(3) 48198.28(7) 48197.73 45797.49 $D$ -2 0.1919(3) 0.1861(13) 0.1753 0.1582 $A$ 0 1242751.3(10) 1242807.6(10) 1242806.1 1242866.7 $A_j$ -1 -5.732(8) -5.51(2) -5.479 -5.206 $C$ 0 -538.4(6) -536.8(3) -538.8 -539.5 $C_{\delta}$ 0 26040.4(14) 26042.6(11) 26044.6 26049.4 $\gamma$ -1 0 0 0 0 $B_0^+$ -2 0.841(9) 0.775(7) 0.768 0.693 $B_1^+$ -1 39.67(5) 37.81(4) 37.94 36.08 $a$ — — 162.2(3) 161.9(10) 161.9 $b$ — — 638.0(6) 638(2) 638 $c$ — — 8.3(3) 8.5(10) 8.5 $d$ — — 107.3(10) 105(3) 105 -------------- ------------- ------------------ ------------------ ------------------ ------------------ All measured transitions could be fitted to approximately their respective uncertainties. However, the total root mean square (rms) of the residuals of the rf and mw data is significantly increased when the optical data is included; Carballo reported rms residuals of 27 kHz, while we find rms residuals of 52 kHz. The rms of the residuals of the fitted $^{12}$C$^{16}$O optical transitions is equal to 3.3 MHz. The molecular constants found from the fit to the combined data agree well with the constants found from a fit to the rf and mw data alone, but with largely decreased uncertainties. The uncertainty of $A$, $C$ and $C_{\delta}$, which are poorly constrained by the rotational and lambda-doubling transitions alone, are reduced by more than a factor of 10. Somewhat unexpectedly, the uncertainty of several other constants, including $B$, are also substantially decreased by the fit to the combined data. This can be understood from the fact that the mw data does not constrain $B$, but rather a combination of $A$ and $B$. This also explains why the uncertainty of $B$ as obtained from a fit to the rotational and lambda-doubling transitions is $\sim$300 kHz, whereas the rotational transitions have a quoted uncertainty of 5-8 kHz and are fitted with an rms uncertainty of 27 kHz [@Carballo]. Our optical data directly probe $A$, and the more precise value of $A$ results in turn in a more precise value of $B$. Adding the optical data results in an uncertainty in $B$ of 30 kHz, much closer to the value one would expect from the precision of the recorded mw transitions. Altogether, the new set of constants is more balanced and adequately describes the [$a^3\Pi$]{} state. As seen from Table \[tab:fitresults\], the residuals of the optical transitions probing the $\Omega=1$ manifold are larger than those to the $\Omega=0$ and $\Omega=2$ manifolds, which is surprising as these transitions are the strongest transitions in the spectra and are measured with a higher signal-to-noise ratio than the other transitions. We have investigated whether this might be explained by perturbations arising from the $a'^3\Sigma^+$ or $D^1\Delta$ state. When perturbations with the $a'^3\Sigma^+$ state were included, using the perturbation parameters from Carballo *et al.* [@Carballo], the residuals decreased marginally. A slight improvement was obtained by including a perturbation with the $D^1\Delta$ state. However, as the perturbation parameters of the coupling between the [$a^3\Pi$]{} and $D^1\Delta$ states are unknown, it is unclear if this improvement is genuine. A similar analysis has been performed for $^{13}$C$^{16}$O. We have included the hyperfine interaction in the effective Hamiltonian following Brown *et al.* [@Brown:1977]. Only terms that are diagonal in $J$ were included, since contributions from off-diagonal terms are estimated to be smaller than 100 kHz [@Gammon:1971]. We have fitted our optical data simultaneously with lambda-doublet transition frequencies measured by Gammon *et al.* [@Gammon:1971] and the rotational transition frequencies measured by Saykally *et al.* [@Saykally:1987]. The fitted set consisted of 19 rf transitions, 4 mw transitions and 24 optical transitions. The molecular constants for the [$X^1\Sigma^+$]{} are taken from Klapper *et al.* [@Klapper:2000]. The molecular constants resulting from the fit are listed in Table \[tab:fitpars\]. The difference between the observed transition frequencies and the frequencies from the fit are listed in Table \[tab:fitresults\]. The rms of the residuals of the fitted $^{13}$C$^{16}$O optical transitions is equal to 0.9 MHz. For $^{13}$C$^{18}$O, as for the other isotopologues, no previous measurements on the [$a^3\Pi$]{} were found in the literature, except for four mw-transitions in $^{12}$C$^{18}$O [@Saykally:1987]. Hence, no fit has been attempted. Mass scaling {#subsec:Isotopes} ------------ An important motivation for this work was to validate the mass scaling of the energy levels of the [$a^3\Pi$]{} state, and to confirm the sensitivity to a possible variation of the proton-to-electron mass ratio for a selected number of level spittings. In the literature, the reduced mass of the molecule is frequently denoted by the symbol $\mu$. In this paper, we will use $\mu$ to denote the proton-to-electron mass ratio, and will denote the reduced mass of the molecule by $\mu_{red}$. As we will see in Sec. \[subsec:deltamu\], $\mu_{red}$ is linearly proportional to $\mu$ which is defined for the various isotopologues $^x$C$^y$O as $$\mu_{red}^{x,y}=\frac{m_C^x \cdot m_O^y}{m_C^x + m_O^y}. \label{Eq:mured}$$ The molecular constants as determined from the fits are effective molecular constants for the $v=0$ level of the [$a^3\Pi$]{} state. In general, an effective molecular constant $X_{e,v}$ can be expressed as $$\begin{gathered} X_{e,v} = X_{e} + \alpha_{X,1}(v+1/2) \\ + \alpha_{X,2}(v+1/2)^2 + \ldots . \label{Eq:vserie}\end{gathered}$$ Note that for the constants $A$ and $B$ the second term of Eq. (\[Eq:vserie\]) has a minus sign by convention [@Havenith]. Thus, the mass dependence of every constant consists of the mass dependence of $X_{e}$ and a correction due to the vibrational dependence of $X_{e,v}$. The second column of Table \[tab:fitpars\] lists the dependence of the molecular constants, $X_{e}$, on the reduced mass of the molecule, $\mu_{red}$. The effective molecular constant, $X'_{e,v}$, of an isotopologue with a reduced mass $\mu'_{red}$ then becomes $$\begin{split} X'_{e,v} = &\left(\frac{\mu'_{red}}{\mu_{red}}\right)^{K^{X}_{\mu}}X_{e}\\ + &\left(\frac{\mu'_{red}}{\mu_{red}}\right)^{K^{X}_{\mu}+\frac{1}{2}} \alpha_{X,1}(v+1/2) \\ +&\left(\frac{\mu'_{red}}{\mu_{red}}\right)^{K^{X}_{\mu}+1} \alpha_{X,2}(v+1/2)^2 + \ldots , \end{split} \label{Eq:scaling}$$ where $X_{e}$, $\alpha_{X,1}$ and $\alpha_{X,2}$ are the constants for the isotopologue with reduced mass $\mu_{red}$ and $$K^{X}_{\mu} = \frac{\mu}{X_{e}}\frac{\partial X_{e}}{\partial \mu}. \label{Eq:KXmu}$$ Note that we use $\mu$ in Eq. (\[Eq:KXmu\]) rather than $\mu_{red}$, see below. As we have only measured transitions in the $v=0$ band, the vibrational dependencies of the molecular constants cannot be extracted from our data. Hence, we have used the ratios between $X_{e}$, $\alpha_{X,1}$ and $\alpha_{X,2}$ determined by Havenith *et al.* [@Havenith] to scale our constants. The molecular constants for $^{13}$C$^{16}$O and $^{13}$C$^{18}$O, found by scaling the constants of $^{12}$C$^{16}$O via the outlined procedure, are listed in the third and fourth column of Table \[tab:fitpars\], respectively. For $^{13}$C$^{16}$O, the value of the band origin and the hyperfine constants were determined by fitting the data while the other constants were fixed at the scaled values. For $^{13}$C$^{18}$O, the hyperfine constants were taken to be identical to those of $^{13}$C$^{16}$O. The data for the [$X^1\Sigma^+$]{} state was calculated from Puzzarini *et al.* [@Puzzarini:2003] and only the value of the band origin was fitted. The differences between the observed transition frequencies and the frequencies calculated using the scaled molecular constants are listed in Table \[tab:fitresults\]. As is seen the correspondence is satisfactory. The rms of the residuals of the $^{13}$C$^{16}$O and $^{13}$C$^{18}$O data with the frequencies found by scaling the molecular constants is equal to 5.1 MHz and 8.3 MHz, respectively. ![(Color online) In the upper panel, the measured values of the band origins of the [$a^3\Pi$]{} state of the six stable isotopologues of CO are plotted as a function of the reduced mass. The solid line shows the value of the band origin scaled with respect to $^{12}$C$^{16}$O. In the lower panel the difference between the measured and calculated values of the band origins are plotted.[]{data-label="fig:isotopeshift"}](fig10-isotopeshift.pdf){width="1.1\linewidth"} Until now the value of the band origin, $E_\Pi$, was treated as a free parameter without considering its proper mass scaling. The value of the band origin consists of: (i) A pure electronic part, that scales as $(\mu_{red})^0$ except for a small correction due to the finite mass of the nuclei, known as the normal mass shift, or Bohr-shift, which is proportional to the reduced mass of the nuclei-electron system. (ii) A vibronic part that can be expanded in a power series of $(v+1/2)$ and contains the difference in zero-point energies in the [$a^3\Pi$]{} and [$X^1\Sigma^+$]{} states. (iii) A rotational part, equal to $B - D$, that was absorbed in the value of the band origin in our definition of the effective Hamiltonian. (iv) The specific mass shift, dependent on the electron correlation function. (v) Nuclear-size effects, dependent on the probability density function of the electrons at the nucleus. In the upper panel of Fig. \[fig:isotopeshift\], the derived values of the band origin of the [$a^3\Pi$]{} state of the six stable isotopologues of CO are plotted as a function of the reduced mass. The solid line shows how the value of the band origin scales when effects (i-iii), which are expected to be dominant, are included. The used formulas for the normal mass shift, the vibrational and the rotational parts are: $$\begin{split} \Delta E_\Pi=\Delta E_{nms}+\Delta E_{vib}+\Delta E_{rot} \end{split} \label{Eq:Escaling}$$ with $$\begin{split} \Delta E_{nms}=E_{\Pi \mathrm{0}}\left[\left(\frac{\mu'_{red} \cdot m_{el}}{\mu'_{red} + m_{el}}\right)/\left(\frac{\mu_{red} \cdot m_{el}}{\mu_{red} + m_{el}}\right)-1\right], \end{split} \label{Eq:Escaling1}$$ $$\begin{split} \Delta E_{vib}&= \frac{1}{2} \left(\omega _{e\Pi}- \omega _{e\Sigma}\right)\left(\left(\frac{\mu_{red}}{\mu'_{red}}\right)^{1/2}-1\right)\\ -&\frac{1}{4} \left(\omega _{e\Pi}x_{e\Pi}- \omega _{e\Sigma}x_{e\Sigma}\right) \left(\frac{\mu_{red}}{\mu'_{red}}-1\right)\\ +&\frac{1}{8} \left(\omega _{e\Pi}y_{e\Pi}- \omega _{e\Sigma}y_{e\Sigma}\right) \left(\left(\frac{\mu_{red}}{\mu'_{red}}\right)^{3/2}-1\right), \end{split} \label{Eq:Escaling2}$$ $$\begin{split} \Delta E_{rot}=B_{\mathrm{0}}\left(\frac{\mu_{red}}{\mu'_{red}}-1\right)-D_{\mathrm{0}}\left(\left(\frac{\mu_{red}} {\mu'_{red}}\right)^2-1\right) \end{split} \label{Eq:Escaling3}$$ where $\Delta E_\Pi$ is the shift in the value of the band origin as a function of the reduced mass, $\mu'_{red}$, with respect to a given isotopologue with reduced mass $\mu_{red}$, band origin $E_{\Pi \mathrm{0}}$ and rotational constants $B_{\mathrm{0}}$ and $D_{\mathrm{0}}$. $m_{el}$ is the mass of the electron. The shift was calculated with respect to $^{12}$C$^{16}$O. For the [$X^1\Sigma^+$]{} state the vibrational constants were obtained from fitting to the data from Coxon *et al.* [@Coxon], while for the [$a^3\Pi$]{} state the constants from Havenith *et al.* [@Havenith] were used. In the lower panel, the difference between the experimental and calculated values of the band origins for the different isotopologues are plotted. The experimental and scaled values of the band origins deviate by a few GHz, which is more than expected given the precision of the data used in this analysis. Most surprising is the difference between the values of the band origins of $^{13}$C$^{16}$O and $^{12}$C$^{18}$O, which have nearly equal reduced masses (7.18 vs. 7.21 amu, respectively). This suggests a breakdown of the Born-Oppenheimer approximation. We have re-analyzed experimental data pertaining to all six stable isotopologues of CO for the E$^1\Pi$ ($v=1$) state [@Ubachs:2000] and the C$^1\Sigma^+$ ($v=1$) state [@Ubachs:2001] and found a similar deviation as found for the [$a^3\Pi$]{}, both in size and in direction. This suggests that the ground state is probably the source of the discrepancy. Calculations using the LEVEL program [@level] were performed to estimate the effects of the breakdown of the Born-Oppenheimer approximation in the ground-state of CO following the approach by Coxon and Hajigeorgiou [@Coxon]. The calculated energy difference between the $^{12}$C$^{18}$O and $^{13}$C$^{16}$O isotopologues was approximately 50 times smaller than observed in our measurements and is thus insufficient to explain the observed effect [@LeRoyprivatecommunications]. The specific mass shift was not included in our analysis. It is however proportional to $\mu_{red}$, and can thus not explain the observed large difference between $^{12}$C$^{18}$O and $^{13}$C$^{16}$O. Nuclear size effects could in principle cause a similar isotope shift as observed, but the difference in nuclear charge radius between $^{17}$O and $^{18}$O is approximately 10 times larger than the difference between $^{16}$O and $^{17}$O, whereas the observed difference between the values of the band origins in the isotopologues with these oxygen isotopes is similar in size [@Angeli]. It is therefore unlikely that the observed isotopic effect is due to a nuclear-size effect. CO([$a^3\Pi$]{}) as a target system for probing $\partial \mu / \partial t$. {#subsec:deltamu} ---------------------------------------------------------------------------- [cl D..[1]{} D..[1]{} D..[2]{}]{} Isotopologue & Transition & & \_. & K\_\ $^{12}$C$^{16}$O & $J=6,~\Omega =1,+ \rightarrow J=4,~\Omega=2,+$ &19270.1 & 3.5 & 27.8\   & $J=6,~\Omega =1,- \rightarrow J=4,~\Omega=2,-$ & 16057.7 & 4.7 & 33.7\   & $J=6,~\Omega =1,+ \rightarrow J=8,~\Omega=0,+$ & -1628.3 & -3.3 & -334\   & $J=6,~\Omega =1,- \rightarrow J=8,~\Omega=0,-$ & -19406.7 & 4.5 & -27.3\ $^{13}$C$^{16}$O & $J=6,~\Omega =1,+,F=3.5 \rightarrow J=4,~\Omega=2,+,F=6.5$ & 43005.8 & 0.1 &12.9\   & $J=6,~\Omega =1,-,F=3.5 \rightarrow J=4,~\Omega=2,-,F=6.5$ & 39988.0 & 0.7 &12.2\   & $J=6,~\Omega =1,+,F=5.5 \rightarrow J=8,~\Omega=0,+,F=8.5$ & 22329.6 & 6.0 &23.5\   & $J=6,~\Omega =1,-,F=5.5 \rightarrow J=8,~\Omega=0,-,F=8.5$ & 4003.4 & 5.0 &128\ $^{13}$C$^{18}$O & $J=6,~\Omega =1,+,F=3.5 \rightarrow J=4,~\Omega=2,+,F=6.5$ & 69062.0 & -7.1 &7.22\   & $J=6,~\Omega =1,-,F=3.5 \rightarrow J=4,~\Omega=2,-,F=6.5$ & 66277.5 & -3.6 &7.60\ ![(Color online) The frequencies of two-photon mw transitions between four near degeneracies as a function the reduced mass. The solid lines show the values obtained from mass scaling the molecular constants of $^{12}$C$^{16}$O. The crosses indicate the calculated frequencies of the transitions in the six stable isotopologues, whereas the boxes show the values obtained from differences between measured frequencies. The four different transitions are listed in the legend as they appear in the graph, from top to bottom.[]{data-label="fig:ndg"}](fig11-ndg.pdf){width="1.1\linewidth"} In Fig. \[fig:ndg\], the energies of the two-photon mw transitions between the near degenerate levels of the different $\Omega$-manifolds are shown as a function of $\mu_{red}$. The solid lines show the values obtained from scaling the molecular constants of $^{12}$C$^{16}$O using the mass-scaling relations discussed in Sec.\[subsec:Isotopes\]. The crosses indicate the reduced masses of the six stable isotopologues, whereas the boxes show the values directly obtained from the measurements. The strong energy dependence on the reduced mass is indicative of a high sensitivity to the proton-to-electron mass ratio. The measured and calculated values for the two-photon microwave transitions between the nearly-degenerate levels are listed in Table \[tab:ndg\]. Our measurements reduce the uncertainties of the two-photon transitions to $\approx$1 MHz, a factor 20 better than before. The mass of the proton is much larger than the masses of the constituent quarks, consequently, the mass of the proton is related to the strength of the forces between the quarks; $\Lambda_{QCD}$, the scale of quantum chromodynamics [@Flambaum:2004]. As the same argument holds for the neutron-to-electron mass ratio, a possible variation of the proton-to-electron mass ratio is expected to be accompanied by a similar variation of the neutron-to-electron mass ratio [@Dent:2007]. With this assumption, it follows that $$K_{\mu} = \frac{\mu}{\nu}\frac{\partial \nu}{\partial \mu}= \frac{\mu_{red}}{\nu}\frac{\partial \nu}{\partial \mu_{red}}. \label{Eq:trivial}$$ The sensitivity of a transition to a possible variation of the proton-to-electron mass ratio $\mu$ can now be calculated using the mass-scaling relations discussed before. The sensitivities for the two-photon microwave transitions are listed in the last column of Table \[tab:ndg\]. These coefficients have been calculated with an accuracy of 0.3-0.05%. It is instructive to compare the sensitivity to what one would expect in a pure Hund’s case (a). For the transition at 1628.3 MHz we expect a sensitivity of $K_\mu=A/2\nu=1242751.3/2\times1628.3\approx382$ which is 12% larger than calculated using the model that includes the coupling between the different $\Omega$-manifolds [^1]. Conclusion ========== UV frequency metrology has been performed on the [$a^3\Pi$]{} - [$X^1\Sigma^+$]{} (0,0) band of various isotopologues of CO using a frequency-quadrupled injection-seeded narrow-band pulsed Ti:Sa laser referenced to a frequency comb laser. We have fitted our optical data for $^{12}$C$^{16}$O together with the lambda-doubling transitions of Wicke *et al.* [@Wicke:1972] and the rotational transitions of Carballo *et al.* [@Carballo] and Wada and Kanamori [@Wada]. Adding the optical data resulted in a large decrease of the uncertainties of $A$, $C$ and $C_{\delta}$, and a smaller decrease in uncertainty in the other constants. From our measurements we obtain the value of the band origin with an uncertainty of 5 MHz, a 30-fold improvement compared to the value obtained from absorption measurements by Field *et al.* [@Field]. We have also measured the value of the band origin in different isotopes and found an unexpected behavior of the isotope shifts, probably due to a breakdown of the Born-Oppenheimer approximation. Our main motivation for this study was to obtain more accurate values for the 2-photon transitions between near degenerate rotational levels in different $\Omega$-manifolds and validate the large sensitivity coefficients predicted for these transitions. The calculated values of the transitions agree to within a few MHz with the measured values, giving confidence in the calculated values of $K_\mu$. Acknowledgements ================ We thank Robert J. Le Roy for his calculations on breakdown of the Born-Oppenheimer approximation in the ground state of CO and Gerrit Groenenboom for his calculations on the lifetime of the $J=0,\,\Omega=0$ level. This work is financially supported by the Netherlands Foundation for Fundamental Research of Matter (FOM) (project 10PR2793 and program “Broken mirrors and drifting constants”). H.L.B. acknowledges financial support from NWO via a VIDI-grant, and from the ERC via a Starting Grant. [39]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [**]{},  (),  @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [**]{},  (),  @noop @noop [****, ()]{} @noop [****,  ()]{} @noop [****,  ()]{} [^1]: Note that Bethlem and Ubachs [@Bethlem:2009] erroneously used $K_\mu=A/\nu$ instead of $K_\mu=A/2\nu$.

--- abstract: 'We generalise the Lieb-Robinson theorem to systems whose Hamiltonian is the sum of local operators whose commutators are bounded.' author: - 'Isabeau Prémont-Schwarz' - Alioscia Hamma - Israel Klich - 'Fotini Markopoulou-Kalamara' title: 'Lieb-Robinson bounds for commutator-bounded operators ' --- =1 The principle of locality is at the heart of the foundations of all modern physics. In quantum field theory, the principle of locality is enforced by an exact light cone. Whenever two (bosonic) observables are spacelike separated, they have to commute, so that neither can have any causal influence on the other. In ordinary quantum mechanics, no explicit request for locality is imposed, and it is, in principle, possible to signal between arbitrarily far apart points in an arbitrarily short time. Nevertheless, a simple perturbation analysis shows that such an influence must decay exponentially with the distance between the observables. The seminal work by Lieb and Robinson [@lieb] has made this statement rigourous for nonrelativistic spin systems. In essence, it states that any quantum system whose Hilbert space is composed of a tensor product of local, finite-dimensional Hilbert spaces and whose Hamiltonian is the sum of local operators will have an approximately maximum speed of signals. Here, local just means that every operator has as a support the tensor product of few degrees of freedom. The approximation consists of the fact that outside the effective light cone there is an exponentially decaying tail. Recently, Lieb-Robinson bounds (LRBs) have received renewed interest in both the fields of theoretical condensed matter and quantum information theory [@Bravyi:2006zz; @eisertetal; @cramer; @kitaev2003fault; @clustering; @hastings; @Eisert:2006zz; @schuch; @boso; @plenio; @sims; @lsm; @locality; @anharmonic]. In particular, the LRB has been used to prove that a nonvanishing spectral gap implies an exponential clustering in the ground state [@hastings; @clustering; @schuch]. Further developments can be found in [@sims], where the LRB is used also to argue about the existence of dynamics. The LRB has also been instrumental in the recent extension of the Lieb-Schultz-Mattis theorem to higher dimensions [@hastingsa; @lsm]. In [@Eisert:2006zz; @hastingsb], it has been shown how the Lieb-Robinson bounds can be exploited to find general scaling laws for entanglement. In [@Bravyi:2006zz], these techniques have been exploited to characterise the creation of topological order. The locality of dynamics has important consequences on the simulability of quantum spin systems. In [@osborne1; @osborne2] it has been shown that one-dimensional gapped spin systems can be efficiently simulated. A review of some of the most relevant aspects of the locality of dynamics for quantum spins systems can be found in [@locality]. Other developments of significant interest include [@boso; @plenio] which show that it is possible to entangle macroscopically separated nanoelectromechanical oscillators of the oscillator chain and that the resulting entanglement is robust to decoherence. Such a system is of great interest for its possible application as a quantum channel and as a tool to investigate the boundary between the classical and quantum worlds. The LRBs have found a more exotic use in the field of emergent gravity, where one wants to study locality, geometry and Lorentz symmetry as emergent phenomena [@Konopka:2008hp; @Konopka:2008ds]. An example of the usefulness of the Lieb-Robinson bounds can be found in [@LRB1], where it was shown that in spin systems with emergent electromagnetism [@wenlight], the speed of light is also the maximum speed of signals, without imposing from the beginning any Lorentz invariance. This raises the issue whether even Lorentz invariance could be emergent. One problem with the Lieb-Robinson bounds is that it is difficult to obtain bounds for unbounded Hamiltonians. In the usual Lieb-Robinson settings, the Hamiltonian must be a sum of local *bounded* operators. If the unbounded terms in the Hamiltonian are completely local, that is, if they are on-site terms, it is possible to prove a Lieb-Robinson theorem using the usual technique [@anharmonic]. In the specific case of coupled harmonic oscillators on a graph with local interactions, it was proven in [@eisertetal] that the Lieb-Robinson bound is valid for canonical and Weyl operators and a proof for a generalisation to general operators is outlined. Algebraic suppression (instead of the usual exponential suppression of the Lieb-Robinson Bound) is shown to result from nonlocal algebraic interactions. As an interesting corollary, [@eisertetal] shows how the approximate locality implied by the Lieb-Robinson bound becomes exact in the continuous limit for the Klein-Gordon field. In this paper, we will show how one can find a bound to the maximum speed of interactions in the case of a class of unbounded spin Hamiltonians. It is not true that for any unbounded Hamiltonian, a Lieb-Robinson bound exists [@supersonic]. Here, we want to show that one can derive a Lieb-Robinson bound if the Hamiltonian is the sum of local operators, *whose commutators are bounded*. Therefore, there is no necessity for even the nonlocal terms to be bounded, as long as their commutators are. More specifically, we show that for quantum systems whose Hilbert space is the tensor product of local Hilbert spaces associated with vertices and edges of a graph, if the Hamiltonian is the sum of local operators $\Phi_i$, each with a support on a region of the graph with a diameter less than a fixed number $R$,if each of these local operators $\Phi_i$ is noncommuting with less than $\nu$ other local operator terms of the Hamiltonian $\Phi_j$, and if for any two of these operators we have $\norm{\com{\Phi_i}{\Phi_j}}<K$ and for any three operators $\norm{\com{\Phi_i}{\com{\Phi_j}{\Phi_k}}}<Q$ for two positive numbers $K$ and $Q$, then we have that for any two local operators $\Phi_i$ and $\Phi_i$ which are terms in the Hamiltonian and whose support is separated by a graph distance $d$, that ,\[almostf\] where $\tilde{\tilde{M}}$ is a constant and $v_{LR}$, the limit on the speed of propagation of information, depends only on local operators of the Hamiltonian as it is they who affect the propagation. This bound can be generalised to any local observables $O_P$ and $O_Q$ with supports $P$ and $Q$ respectively, that satisfy the following local observable operator conditions:$(i)$ The graph distance $d$ separating $P$ and $Q$ is greater than $R$. $(ii)$ The number of terms $\Phi_i$ of the Hamiltonian whose support has nonempty intersection with $P$ is $n_P<\infty$. $(iii)$ There exists $F_P$ and $F_Q$ such that for all terms $\Phi_i$ and $\Phi_j$ of the Hamiltonian the inequalities $\norm{\com{O_P}{\Phi_i}}<F_P K$, $\norm{\com{O_Q}{\Phi_i}}< F_Q K$ and $\norm{\com{O_Q}{\com{\Phi_i}{\Phi_j}}}<F_Q Q$ are satisfied. The generalised bound is then && F\_P F\_Q && n\_P(n\_P+1) . \[finalb\] To motivate our discussion, let us start by the most trivial example: Consider the case of a Hamiltonian $H=\sum h_i$ which is composed of a sum of local terms $h_i$ which are commuting, such as the quantum Ising model without the transverse field. In such a case, there is simply no propagation of signals: Indeed, for any local operator $O_A$ we have $O_A(t)=e^{itH}O_Ae^{-iHt}= e^{itH_A}O_Ae^{-iH_At}$, where $H_A=\sum_{i:[h_i,O_A]\neq 0} h_i$, since there is a finite number of $h_i$ in $H_A$, and they are of finite range; $O_A(t)$ is also strictly local for arbitrary long times $t$, irrespective of the norm of the $h_i$ operators. This suggests that it is desirable to find Lieb-Robinson bounds in terms of the norm of the commutators rather than the norm of the local terms $h_i$. Let us outline a simple example. Consider a system of parallel quantum wires. We place fermions on the wires, and these are usually described by one-dimensional Luttinger liquids, and have approximately a linear dispersion relation. We place a density-density interaction between the wires. Labeling the wires by the index $j$, the system can be described by the following Hamiltonian: $$H_{wires} = \sum_j \left(-i\frac{\partial}{\partial x_j} + V(x_j - x_{j+1})\right),$$ which is commutator bounded in the sense of this paper as long as both $| \partial_{x_j} V|$ and $| \partial_{x_i}\partial_{x_j} V|$ are bounded. Another example involves a generalised Dicke model, describing an array of spins interacting with a boson field via $$\begin{aligned} H=\sum h_n\,\,\,\,;\,\,\,\, h_n=\sigma^z_n(b_n^{\dag}+b_n+ib_{n+1}^{\dag}-ib_{n+1})\end{aligned}$$ where $b_n$ are boson creation operators and $\sigma_n^z$ is the $n$th spin. It is easy to check that in this case the commutator $[h_n,h_{n+1}]=-2 i \sigma^z_n\sigma^z_{n+1}$ is bounded. \[In fact, this particular Hamiltonian can also be written as a sum of commuting terms $\tilde{h}_n=b_n(\sigma^z_n-i\sigma^z_{n-1})+h.c.$\]. We consider Hamiltonians that are the sum of two different types of operators $\Phi_0$ and $\Phi_1$ : $$\begin{aligned} H \equiv \sum_{i\in S_0} h_0 \Phi_0^i + \sum_{j\in S_1} h_1 \Phi_1^j \label{deuxint}.\end{aligned}$$ Here, $S_0,S_1$ are two sets of labels, $h_0$ and $h_1$ are two coupling constants, and $[\Phi_0^i,\Phi_0^j]=[\Phi_1^i,\Phi_1^j]=0$ for every $i,j$. As an example, consider the Ising model. Then $\Phi_0^i = \sigma^x_i \sigma^x_{i+1}$ and $\Phi_1^i = \sigma^z_i$. We call the subgraph which is the support of the operator $\Phi_q^m,\ \Gamma(q,m)$ and for $(a,b)\in\{0,1\}^2$, we define $$\begin{aligned} K_{a \ b}^{i \ j}(t) \equiv \com{\Phi_a^{i}(t)}{\Phi_b^j} . \label{f}\end{aligned}$$ We consider what we will refer to as commutator-bounded $R$-local quantum systems. For such systems, the commutators and the commutators of commutators of operators of the Hamiltonian are uniformly bounded while the operators themselves may be unbounded and the operators of the Hamiltonian have support on subgraphs of size less than $R$, for $R$ an arbitrary natural number. Explicitly, the diameter of all $\Gamma(q,m)$ is less than $R$ and for any three operators $\Phi_a^i$, $\Phi_b^j$, and $\Phi_c^k$ appearing in the Hamiltonian with the coupling constants $h_a$, $h_b$, and $h_c$, we have that $ h_a h_b\norm{\com{\Phi_a^i}{\Phi_b^j}}< K\ $ and $h_a h_b h_c\norm{\com{\com{\Phi_a^i}{\Phi_b^j}}{\Phi_c^k}}< Q$, where $K$ and $Q$ are positive real numbers. Note that a bounded system, which is uniformly bounded by $\tilde{K}$, must satisfy $K\leq 2\tilde{K}^2$ as well as $Q\leq 4\tilde{K}^3$, and thus boundedness implies commutatorboundedness. By taking the derivative of with respect to $t$, we obtain $({K_{a \ b}^{i_1 \ j}}(t))^\prime = \com{\com{-i H(t)}{\Phi_a^{i}(t)}}{\Phi_b^j}$, after keeping only the terms in $H(t)$ which do not commute with $\Phi_a^{i}(t)$, and after some algebra, we get \[here and in the following by $a+1$ we mean $a+1\ mod(2)$\] $$\begin{aligned} \nonumber ({K_{a \ b}^{i_1 \ j}}(t))^\prime = \com{{K_{a \ b}^{i_1 \ j}}(t)}{\Big(-i h_{a+1} \sum_{i_2\in Z_{i_1}} \Phi_{a+1}^{i_2}(t)\Big)} \\ + (-i h_{a+1}) \sum_{i_2\in Z_{i_1}}\com{\Phi_a^{i_1}(t)}{\com{\Phi_{a+1}^{i_2}(t)}{\Phi_b^j}}, \label{fpp}\end{aligned}$$ where, if $i\in S_0$, then $Z_i$ is the finite subset of $S_1$ such that $j\in Z_i \Leftrightarrow \Gamma(0,i)\bigcap \Gamma(1,j)\neq \emptyset$ and vice versa for $i\in S_1$. Taking the second derivative, and using the fact that $\com{\Phi_{a}^{i}(t)}{\Phi_{a}^{j}(t)} = 0$, we obtain, after some algebraic manipulation, $$\begin{aligned} \nonumber {K_{a \ b}^{i_1 \ j}}''(t) = && -i \com{{K_{a \ b}^{i_1 \ j}}'(t)}{\sum_{j_2\in Z_{i_1}} h_{a+1}\Phi_{a+1}^{j_2}(t) + \sum_{i_3\in Z_{i_2}} h_{a}\Phi_{a}^{i_3}(t)} - h_{a+1}^2 \sum_{j_2\in Z_{i_1}}\sum_{i_2\in Z_{i_1}} \com{\com{\Phi_a^{i_1}(t)}{\Phi_{a+1}^{i_2}(t)}}{\com{\Phi_{a+1}^{j_2}(t)}{\Phi_b^j}} \\ &&- h_{a} h_{a+1}\sum_{i_2\in Z_{i_1}}\sum_{i_3\in Z_{i_2}} \com{\com{\Phi_a^{i_1}(t)}{\Phi_{a+1}^{i_2}(t)}}{\com{\Phi_{a}^{i_3}(t)}{\Phi_b^j}}. \label{fqtt}\end{aligned}$$ Defining the following unitary operator $U_{2+3}(t)\equiv e^{-it \left(\sum_{j_2\in Z_{i_1}} h_{a+1}\Phi_{a+1}^{j_2}(t) + \sum_{i_3\in Z_{i_2}} h_{a}\Phi_{a}^{i_3}(t)\right)}$ and its associated unitary evolution $T_{2+3}(t) O \equiv U_{2+3}(t)^\dag O U_{2+3}(t)$, integrating , and taking the norm, we obtain, after some manipulations, $$\begin{aligned} \nonumber \norm{{K_{a \ b}^{i_1 \ j}}'(t)} \leq && h_{a+1} \sum_{i_2\in Z_{i_1}} \norm{\com{\com{\Phi_a^{i_1}}{\Phi_{a+1}^{i_2}}}{\Phi_b^j}} \\ && +\int_0^t ds \left( 2 h_{a+1}^2K \sum_{j_2\in Z_{i_1}}\sum_{i_2\in Z_{i_1}} \norm{\com{\Phi_{a+1}^{j_2}(s)}{\Phi_b^j}} \right. + \left. 2 h_{a} h_{a+1} K \sum_{i_2\in Z_{i_1}}\sum_{i_3\in Z_{i_2}}\norm{\com{\Phi_{a}^{i_3}(s)}{\Phi_b^j}}\right), \label{leqy} $$ where, we used the fact that $\norm{\com{\Phi_a^{i}(t)}{\Phi_{a+1}^{j}(t)}}\leq K$. By integrating , we get $$\begin{aligned} \nonumber \norm{K_{a b}^{i_1 j}(t)} & \leq \norm{\com{\Phi_a^{i_1}}{\Phi_b^j}} + h_{a+1} \sum_{i_2\in Z_{i_1}}\norm{\com{\com{\Phi_a^{i_1}}{\Phi_{a+1}^{i_2}}}{\Phi_b^j}} t \\ & + \int_0^t ds\int_0^s dl \left( 2 h_{a+1}^2 K \sum_{i_2\in Z_{i_1}}\sum_{j_2\in Z_{i_1}} \norm{K_{a+1 b}^{j_2 j}(l)} + 2 h_{a} h_{a+1} K \sum_{i_2\in Z_{i_1}}\sum_{i_3\in Z_{i_2}} \norm{K_{a b}^{i_3 j}(l)}\right). \label{KUN}\end{aligned}$$ Since the commutators are bounded, we have $\norm{\com{\Phi_a^{i_1}}{\Phi_b^j}} \leq K$ and $\norm{\com{\com{\Phi_a^{i_1}}{\Phi_{a+1}^{i_2}}}{\Phi_b^j}} \leq Q$ for some $K,Q>0$. Noting that $\norm{\com{\Phi_a^{i_1}}{\Phi_b^j}}= 0$ if $\Gamma(a,i_1)$ and $\Gamma(b,j)$ do not overlap and $\norm{\com{\com{\Phi_a^{i_1}}{\Phi_{a+1}^{i_2}}}{\Phi_b^j}} =0$ if $\Gamma(b,j)$ does not overlap with either $\Gamma(a,i_1)$ or $\Gamma(a+1,i_2)$, we see that implies $$\begin{aligned} \norm{K_{a b}^{i_1 j}(t)} & \leq K \delta_{i_1}^{j} + \sum_{i_2\in Z_{i_1}}h_{a+1} Q \delta_{i_1\cup i_2}^{j} t + \int_0^t ds\int_0^s dl \left( 2 h_{a+1}^2 \sum_{j_2\in Z_{i_1}}\sum_{i_2\in Z_{i_1}}\norm{K_{a+1 b}^{j_2 j}(l) } + 2 h_{a} h_{a+1} \sum_{i_2\in Z_{i_1}}\sum_{i_3\in Z_{i_2}}\norm{ K_{a b}^{i_3 j}(l)}\right), \label{nino}\end{aligned}$$ where we have used the following symbol: $$\begin{aligned} \delta_{i}^k := \left\{ \begin{array}{cc} 1 & \mbox{if } \Gamma(a_i,i)\cap \Gamma(a_k,k)\neq \emptyset , \\ 0 & \mbox{otherwise}. \end{array}\right. \label{eqn_delta}\end{aligned}$$ Solving for $ \norm{K_{a b}^{i_1 j}(t)}$, we find $$\begin{aligned} \norm{K_{a b}^{i_1 j}(t)} & \leq K \delta_{i_1}^{ j} + \sum_{i_2\in Z_{i_1}} Q h_{a+1} \delta_{i_1\cup i_2}^{ j} t + \sum_{j_2\in Z_{i_1}}\sum_{i_2\in Z_{i_1}} 2 h_{a+1}^2 K \delta_{j_2}^{ j} \frac{t^2}{2!} + \sum_{i_2\in Z_{i_1}}\sum_{i_3\in Z_{i_2}} 2 h_{a} h_{a+1} K \delta_{i_3}^{ j} \frac{t^2}{2!}\nn & + \sum_{j_2\in Z_{i_1}}\sum_{i_2\in Z_{i_1}}\sum_{i_3\in Z_{j_2}} 2 h_{a+1}^2 h_a K Q \delta_{j_2\cup i_3}^{ j} \frac{t^3}{3!} + \sum_{i_2\in Z_{i_1}}\sum_{i_3\in Z_{i_2}}\sum_{i_4\in Z_{i_3}} 2 h_{a+1} h_a^2 K Q \delta_{i_3\cup i_4}^{ j} \frac{t^3}{3!} \nn & + \int_0^t dv\int_0^v du\int_0^u ds\int_0^s dl \Bigg( (2 h_{a+1}h_a)^2 K^2 \sum_{i_2\in Z_{i_1}}\sum_{j_2\in Z_{i_1}}\sum_{i_3\in Z_{j_2}}\sum_{j_3\in Z_{j_2}} \norm{K_{a b}^{j_3 j}(l)}\nn & + \ (2 h_{a+1}^2)(2 h_{a+1}h_a) K^2\sum_{i_2\in Z_{i_1}}\sum_{j_2\in Z_{i_1}}\sum_{i_3\in Z_{j_2}}\sum_{i_4\in Z_{i_4}} \norm{K_{a+1 b}^{j_4 j}(l)}\nn & + (2 h_{a+1}h_a)^2 K^2 \sum_{i_2\in Z_{i_1}}\sum_{i_2\in Z_{i_1}}\sum_{i_3\in Z_{i_2}}\sum_{i_4\in Z_{i_3}} \norm{K_{a+1 b}^{i_4 j}(l)} + \left. (2 h_{a}^2)(2 h_{a+1}h_a) K^2 \sum_{i_2\in Z_{i_1}}\sum_{i_3\in Z_{i_2}}\sum_{i_3\in Z_{1_2}}\sum_{j_3\in Z_{i_2}} \norm{K_{a b}^{j_3 j}(l)}\right).\label{K3}\end{aligned}$$ Iterating this procedure we obtain by induction $$\begin{aligned} \norm{K_{a b}^{i_1 j}(t)} \leq M \sum_{n=0}^\infty \sqrt{2 h_0 h_1 K}^n \frac{\abs{t}^n}{n!} c_n \label{K40},\end{aligned}$$ where $M = \sqrt{2} \max\{\frac{h_0}{h_1},\frac{h_1}{h_0}\}\times\max\{\frac{1}{K},1\}\times\max\{\frac{\sqrt{K}}{Q},1\}$ and where $c_n$ is a combinatorial factor counting the number of linking operator chains of $n$ operators between $\Gamma(a,i_1)\ $and $\Gamma(b,j)$. What we call an operator chain is heuristically a sequence of intersecting operators linking the initial and final operators. The process of constructing the sequence of operators forming the chain is as follows: The $2j^{th}$ operator in the chain has to be noncommuting with the $(2j-1)^{th}$ one. This imposes that the two consecutive operators of a chain have to (1) be of a different interaction type and (2) have overlapping support. For the odd-numbered operators, there is an extra choice: The $(2k+1)^{th}$ operator can be an operator that does not commute with the $2k^{th}$ operator (as for the even case) *or* an operator that does not commute with the $(2k-1)^{th}$ operator. That is, even operators in the sequence must be noncommuting with the previous operator in the sequence and odd operators in the sequence must be noncommuting with either of the two previous operators in the sequence. From the recursive , we see that if we start with an operator $i_1$ of type $a$, the next operator in the chain must be an operator $i_2\in Z_{i_1}$ of the other type ($a+1$). The fact that it is in $Z_{i_1}$ means that its support overlaps with $i_1$’s, which is similar to what was found in the bounded case. However if we look at the operator that comes after $i_2$, we see that we have two distinct possibilities. The first \[second double sum under the integrals of \] is that it can be $i_3\in Z_{i_2}$ an operator of type $a$ (different from $a+1$) whose support overlaps with $i_2$’s; if this were the only possibility, we would have exactly the same situation as we had for bounded systems, but the first double sum under the integrals of adds another possibility. That second possibility (first double sum) is choosing an operator $j_2\in Z_{i_1}$ after the operator $i_2$, $j_2$ is, like $i_2$, an operator of type $a+1$ which (by virtue of being in $Z_{i_1}$) has a support overlapping $i_1$’s. To find the next operator after that, we reiterate and thus, like the first operator after $i_1$, we need to choose an operator which is of a different type than the last one (be it $j_2$ or $i_3$) and has overlapping support with the last one; thus, at this point, we cannot “change our mind". We can thus see the process of building the chain as, for every two choices, we must choose an operator that links with the previous one, but every other choice, we can also choose an operator that links to the penultimate one instead. Because every two choices in building up the chain we must choose an operator of a different type than the previous one, in the end, the chain contains the same number of operators of type $0$ as of type $1$ (plus or minus one). This means that there will be the same number of factors of $h_1$ as of $h_0$ in every term; hence, we can pull them out of the sums over chains and simply write an overall factor of $\sqrt{h_0 h_1}$ in front while passing from to . Furthermore, we can always find a bound of the following type for $c_n$: c\_n \^n e\^[( - d)]{} \[cng1\], where $\lambda$ is an arbitrary positive real number. This is because the $\Gamma(a,i)$’s have a diameter of $R$ or less. Hence, if the distance $d$ between the initial and final points is greater than $R^n$, then there are no possible linking operator chains of $n$ local operators between the initial and final points. Furthermore, since at every odd step along the chain there is a choice of at most $\nu$ local operators to choose from for the next operator in the chain and at every even step there is at maximum $2\nu$ operators to choose from, there is, at most, $(\sqrt{2}\nu)^n$ possible local operator chains of $n$ operators starting from any given position. Thus, we certainly have that c\_n \^n \^n e\^[(R [n]{} - d)]{} \[cngex\], where $\lambda$ is arbitrary. Using with , we obtain the LRB of , ,\[LRBUB\] where $\tilde{\tilde{M}} = \tilde{M}M$. To obtain the generalisation to local operators $O_P$ and $O_Q$ satisfying the local observable operator conditions enounced in the introduction, we introduce $\tilde{K}_{a}^{i_1}(t)\equiv \com{\Phi_a^{i}(t)}{O_Q(0)}$. Using exactly the same procedure used to obtain , we get && && \_0\^t ds ( 2 h\_[b]{}h\_[a]{} K F\_P \_[j\_2Z\_[P]{}]{}\_[i\_2Z\_[P]{}]{} . && + . 2 h\_[a]{} h\_[a+1]{} K F\_P \_[i\_2Z\_[P]{}]{}\_[i\_3Z\_[i\_2]{}]{})&& 2 {h\_0\^2, h\_1\^2} n\_P(n\_P+1)\_0\^t ds , \[leqyfier\] where $Z_P$ is the set of of labels of the terms of the Hamiltonian which do not commute with $O_P$ ($\norm{Z_P}=n_P$), where $k$ is such that $\int_0^t ds \norm{\tilde{K}_{a}^{k}(s)} = \max_{i\in Z_P} \int_0^t ds \norm{\tilde{K}_{a}^{i}(s)}$ and where, unlike in , the terms containing no integrals do not appear here because of the condition that $d>R$. $\norm{\tilde{K}_{a}^{k}(s)}$ can then be treated in exactly the same way as $ \norm{K_{a b}^{i_1 j}(s)}$ was, with the only exception that while bounding the final commutators \[i.e., when we place the $\delta$ of \], we will need an extra factor of $F_Q$. Thus, we obtain : && F\_P F\_Q n\_P(n\_P && +1) .\[LRBUBG\] Optimising for $\lambda$, we have that the Lieb-Robinson speed is thus v\_[L R]{} = 2 e .\[vlrunb\] We can compare with the bound obtained for Hamiltonians composed of bounded local operators and for bounded local observables $O_P$ and $O_Q$, which is [@LRB1] && n\_P && .\[LRBUBounded\] To summarize, in this paper, we have shown that a Lieb-Robinson bound exists for those Hamiltonians that are the sum of local operators whose commutator is bounded. This allows for treating a class of systems with unbounded operators. *Acknowledgments.—* Research at Perimeter Institute for Theoretical Physics is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MRI. [99]{} E. H. Lieb, and D. W. Robinson, Comm. Math. Phys. **28**, 251 (1972). S. Bravyi, M.B.Hastings, and F. Verstraete,, 050401 (2006). M. Cramer, A. Serafini, J. Eisert, *Quantum Information and Many Body Quantum Systems*, M. Ericsson and S. Montangero eds, Edizioni della Normale, Pisa,pp. 51-72 2008. \[arXiv:0803.0890v2\]. M., Cramer and J., Eisert, New J. Phys. 8, 71 (2006) A. Kitaev, Ann. Phys. (NY), 303 (2003) \[arXiv:quant-ph/9707021\] B. Nachtergaele et al., Comm. Math. Phys. [**265**]{}, 119 (2006). J. Eisert et al., Phys. Rev. Lett.  [**97**]{}, 150404 (2006) . N. Schuch et al., Comm. Math. Phys. [**267**]{}, 65 (2006). B. Nachtergaele, Y. Ogata, and R. Sims, J. Stat. Phys. **124**, 1 (2006) \[arXiv:math-ph/0603064v1\]. B. Nachtergaele, H. Raz, B. Schlein, and R. Sims, Comm. Math. Phys. [**286**]{}, 1073-1098 (2009) M. B. Hastings, , 104431 (2004) M. B. Hastings, , 140402 (2004) \[arXiv:cond-mat/0406348\]. M.B. Hastings et al., Comm. Math. Phys. **265**, 781 (2006). B. Nachtergaele, and R. Sims, Comm. Math. Phys. [**276**]{}, 437-472 (2007). B. Nachtergaele, and R. Sims, \[arXiv:0712.3318\]. J. Eisert et al., Phys. Rev. Lett. **93**, 190402 (2004). M.B. Plenio, et al., New J. Phys. **6**, 36 (2004). J. Eisert, D. Gross, arXiv:0808.3581v3 X.-G. Wen, , 065003 (2003). T.J. Osborne, Phys. Rev. Lett. [**97**]{}, 157202 (2006) T.J. Osborne, Phys. Rev. A [**75**]{}, 032321 (2007) T. Konopka, F. Markopoulou and S. Severini, Phys. Rev.  D [**77**]{}, 104029 (2008). T. Konopka, Phys. Rev.  D [**78**]{}, 044032 (2008). A. Hamma, F. Markopoulou, I. Prémont-Schwarz and S. Severini, Phys. Rev. Lett.  [**102**]{}, 017204 (2009).

--- abstract: 'Starting from the averaged null energy condition (ANEC) in Minkowski we show that conformal symmetry implies the ANEC for a conformal field theory (CFT) in a de Sitter and anti-de Sitter background. A similar and novel bound is also obtained for a CFT in the Lorentzian cylinder. Using monotonicity of relative entropy, we rederive these results for dS and the cylinder. As a byproduct we obtain the vacuum modular Hamiltonian and entanglement entropy associated to null deformed regions of CFTs in (A)dS and the cylinder. A third derivation of the ANEC in dS is shown to follow from bulk causality in AdS/CFT. Finally, we use the Tomita-Takesaki theory to show that Rindler positivity of Minkowski correlators generalizes to conformal theories defined in dS and the cylinder.' bibliography: - 'sample.bib' --- =10000 [**Global aspects of conformal symmetry\ and the ANEC in dS and AdS**]{} [**Felipe Rosso\ **]{} Department of Physics and Astronomy\ University of Southern California\ Los Angeles, CA 90089, USA\ \ Introduction and summary {#sec:intro} ======================== The main focus of this work is the averaged null energy condition (ANEC), defined for an arbitrary quantum field theory (QFT) on a fixed space-time $g_{\mu \nu}$ as $$\label{eq:210} \int_{-\infty}^{+\infty}d\lambda\, k^\mu k^\nu T_{\mu \nu}\ge 0\ ,$$ where $T_{\mu \nu}$ is the stress tensor operator and $k^\mu$ is the tangent vector over a complete null geodesic with affine parameter $\lambda$. The original motivation for considering this condition comes from general relativity, where it is a reasonable substitute for the null energy condition $k^\mu k^\nu T_{\mu \nu}\ge 0$, known to fail in quantum theories. The ANEC can be used to rule out space-times with certain unwanted features [@Morris; @Ori:1993eh; @Alcubierre:1994tu], as well as for proving classic theorems in general relativity [@Tipler:1978zz; @Borde:1987qr; @Roman:1986tp]. Even in the simplest case of a QFT in Minkowski, the ANEC has been applied to obtain very interesting results such as the conformal collider bounds of Ref. [@Hofman:2008ar]. Although the ANEC in Minkowski has been proven for general QFTs in Refs. [@Faulkner:2016mzt; @Hartman:2016lgu; @Longo:2018obd], the question still remains whether it is a true statement of quantum theories defined in more general backgrounds. In this work we take a few steps in this direction and prove the ANEC for arbitrary conformal field theories (CFTs) defined on fixed de Sitter and anti-de Sitter space-times. Moreover, for a CFT in the Lorentzian cylinder $\mathbb{R}\times S^{d-1}$ we obtain a similar condition given by $$\label{eq:209} \int_{-\pi/2}^{\pi/2}d\bar{\lambda}\, \cos^d(\bar{\lambda}) k^\mu k^\nu T_{\mu \nu}\ge 0\ ,$$ where $\bar{\lambda}$ is affine and the null geodesic is not complete but goes between antipodal points in the spatial sphere $S^{d-1}$. The stress tensor in (\[eq:209\]) is vacuum subtracted $T_{\mu \nu}\equiv T_{\mu \nu}-\bra{0}T_{\mu \nu}\ket{0}$ in order to avoid a trivial violation due to some constant Casimir energy.[^1] We start in Sec. \[sec:ANEC\_mapping\], where we derive the three constraints in (A)dS and the cylinder in a simple way. Given that the ANEC in Minkowski has been well established for general QFTs [@Faulkner:2016mzt; @Hartman:2016lgu; @Longo:2018obd], we start from this condition and apply certain conformal transformations from Minkowski to these space-times.[^2] After the mapping, the resulting constraint gives the ANEC in (A)dS and the bound (\[eq:209\]) for the cylinder. To implement these transformations appropriately we must carefully deal with the fact that the conformal group is only globally well defined in the Lorentzian cylinder.[^3] Since this plays an important role in this work, let us briefly explain its significance. The Lorentzian cylinder $\mathbb{R}\times S^{d-1}$ can be represented by an infinite strip in the $(\sigma/R,\theta)$ plane, where $\sigma\in \mathbb{R}$ is the time coordinate and $\theta\in[0,\pi]$, with the end points corresponding to the poles of the spatial sphere $S^{d-1}$ of radius $R$, see Fig. \[fig:14\]. The conformal transformations relating the cylinder, Minkowski and (A)dS are essentially given by different ways of cutting out regions of this infinite strip. When mapping a curve (or surface) from one space-time into another it is crucial that we keep track of this, since a given curve may not fit inside some of the sections of the strip shown in Fig. \[fig:14\]. The key technical feature of (A)dS that enables the derivation of the ANEC is that a complete and affinely parametrized null geodesic in Minkowski is also complete and affine in (A)dS. Since this is not true for the Lorentzian cylinder, we do not obtain the ANEC in this case but the constraint in (\[eq:209\]). In Sec. \[sec:null\_energy\_bounds\] we investigate whether an independent proof of these results can be obtained from monotonicity of relative entropy, as done in Ref. [@Faulkner:2016mzt] for the Minkowski ANEC. We do so by first computing the vacuum modular Hamiltonians of null deformed regions in these space-times, which we obtain by conformally mapping the Minkowski modular operator associated to null deformations of Rindler [@Casini:2017roe]. The appropriate conformal transformations are a slight modification from the ones used in Sec. \[sec:ANEC\_mapping\]. The case of dS is particularly simple, where we show that the modular Hamiltonian associated to null deformations of the static patch is given by $$\label{eq:213} K_{\rm dS}= 2\pi R^{d-2} \int_{S^{d-2}}d\Omega(\vec{x}_\perp) \int_{\bar{A}(\vec{x}_\perp)}^{+\infty} d\eta\, \left( \eta-\bar{A}(\vec{x}_\perp) \right) T_{\eta \eta}(\eta,\vec{x}_\perp)\ ,$$ where for fixed $\vec{x}_\perp$, $\eta$ is an affine parameter in dS and the stress tensor is projected along this direction. For $\bar{A}(\vec{x}_\perp)=0$ the integral is over the future horizon of the de Sitter static patch, while arbitrary $\bar{A}(\vec{x}_\perp)$ corresponds to null deformations. Using this together with monotonicity of relative entropy gives the ANEC in dS. Although a similar procedure results in the bound in the cylinder (\[eq:209\]), it does not generalize to the AdS case due to some technical issues related to our previous comment on the global definition of the conformal group. We finish Sec. \[sec:null\_energy\_bounds\] by computing the universal terms of the entanglement entropy associated to the null deformed modular Hamiltonians in (A)dS and the cylinder. The details of the computations are summarized in App. \[zapp:entanglement\], where we build on some results of Ref. [@Casini:2018kzx] using AdS/CFT. We continue in Sec. \[sec:ref\_positivity\], where we explore some aspects that would be necesary to generalize the causality proof of the Minkowski ANEC [@Hartman:2016lgu] to these curved space-times. In particular, we study one of its crucial ingredients, the “wedge reflection positivity” or “Rindler positivity”, which for two scalar operators can be written as $$\label{eq:212} \bra{0} \mathcal{O}^\dagger(\widetilde{X}^\mu) \mathcal{O}(X^\mu)\ket{0}>0\ , \qquad \qquad \widetilde{X}^\mu(X^\mu)=(-T,-X,\vec{Y})\ ,$$ where $X^\mu=(T,X,\vec{Y})$ are Cartesian coordinates in Minkowski and $X^\mu$ must satisfy $X>|T|$. This property was derived in Ref. [@Casini:2010bf] from the Tomita-Takesaki theory [@Haag:1992hx; @Witten:2018lha]. Using the conformal transformations of Sec. \[sec:null\_energy\_bounds\] we map the Bisognano-Wichmann Tomita operator [@Bisognano:1976za] to the CFTs in the Lorentzian cylinder and de Sitter, and show that a generalized version of (\[eq:212\]) holds in these backgrounds. The resulting property for the cylinder is particularly interesting since unlike (\[eq:212\]), the transformation $\widetilde{X}^\mu$ is non-linear.[^4] ![Diagrams illustrating the effect of the conformal transformations given in table \[table:2\] when applied to the Lorentzian cylinder. The whole infinite strip on the left diagram corresponds to the cylinder, with the North and South pole at $\theta=0$ and $\theta=\pi$ respectively.[]{data-label="fig:14"}](Conformal_relations.pdf){height="2.45"} The third (and last) independent proof of the ANEC in de Sitter is based on AdS/CFT and given in appendix \[zapp:ANEC\_holography\]. We show that the approach of Ref. [@Kelly:2014mra] used to derive the Minkwoski ANEC for holographic theories described by Einstein gravity can be naturally extended to de Sitter. We should mention that while this work was in preparation Ref. [@Iizuka:2019ezn] used a similar method to derive the bound in the Lorentzian cylinder (\[eq:209\]) for space-time dimensions $d=3,4,5$ and holographic CFTs dual to Einstein gravity. We finish in Sec. \[sec:Discussion\] with a discussion of our results and several future research directions. In particular we comment on the connection between these bounds and the quantum null energy condition (QNEC). Using the modular Hamiltonian in (\[eq:213\]), we point out that the QNEC in de Sitter can be written in terms of the second order variation of relative entropy. ANEC in (A)dS from conformal symmetry {#sec:ANEC_mapping} ===================================== In this section we map the null plane in Minkowski to the Lorentzian cylinder, de Sitter and anti-de Sitter. After describing the geometric aspects of the transformation we apply it to the ANEC operator in Minkowski space-time. This allows us to obtain the ANEC for CFTs in (A)dS and a similar novel bound for theories defined in the cylinder. Taking the null plane on a conformal journey - Take I ----------------------------------------------------- The conformal transformations relating Minkowski, the cylinder and (A)dS have been known for a long time [@Candelas:1978gf]. The simplest way to introduce them is to start from the metric in the Lorentzian cylinder $\mathbb{R}\times S^{d-1}$ written as $$\label{eq:170} ds^2_{LC}= -d\sigma^2+R^2\left( d\theta^2+\sin^2(\theta)d\Omega^2(\vec{v}\,) \right)\ ,$$ where $\sigma\in \mathbb{R}$ is the time coordinate and $\theta\in [0,\pi]$, with the end points corresponding to the North and South pole of the spatial sphere $S^{d-1}$ of radius $R$. The line element $d\Omega^2(\vec{v}\,)$ is given by $$\label{eq:152} d\Omega^2(\vec{v}\,)= \left( \frac{2L}{L^2+|\vec{v}\,|^2} \right)^2d\vec{v}.d\vec{v}\ ,$$ which corresponds to a unit sphere $S^{d-2}$ in stereographic coordinates $\vec{v}\in \mathbb{R}^{d-2}$. The length scale $L$ can be any, not necessarily related to $R$.[^5] This cylinder manifold can be represented by an infinite strip in the $(\sigma/R,\theta)$ plane, as shown in the first diagram of Fig. \[fig:14\], where the North and South pole are given by the vertical lines at $\theta=0$ and $\theta=\pi$ respectively. Other values of $\theta\in(0,\pi)$ in this diagram corresponds to a unit sphere $S^{d-2}$. Conformal transformations in the cylinder are essentially given by different ways of cutting this infinite strip. The cutting is implemented by a change of coordinates which puts the metric of the cylinder in the form $ds_{LC}^2=w^2d\bar{s}^2$, followed by a Weyl rescaling which removes the conformal factor $w^2$. Effectively, this maps a section of the Lorentzian cylinder to the space-time $d\bar{s}^2$. Through this procedure we can obtain Minkowski and (A)dS.[^6] The appropriate change of coordinates and conformal factors in each case are indicated in table \[table:2\]. From this it is straightforward to see that each of the transformations cuts the infinite strip as given in Fig. \[fig:14\]. For instance, in the Minkowski case we see that $r_\pm \in \mathbb{R}$ translates into $\theta_\pm \in [-\pi,\pi]$ together with the implicit constraint $\theta\in [0,\pi]$. [-.8in]{}[-.8in]{} The way in which we have written the metrics in (A)dS in table \[table:2\] is (probably) the most familiar form but not the most convenient to describe null surfaces, which is ultimately what we are interested in. A more suitable description of these space-times is given directly in terms of the coordinates in the cylinder $$\label{eq:154} \begin{aligned} ds^2_{\rm dS}&= \frac{-d\sigma^2+R^2\left( d\theta^2+\sin^2(\theta)d\Omega^2(\vec{v}\,) \right)}{\cos^2(\sigma/R)}\ ,\\ ds^2_{\rm AdS}&= \frac{-d\sigma^2+R^2\left( d\theta^2+\sin^2(\theta)d\Omega^2(\vec{v}\,) \right)}{\cos^2(\theta)}\ . \end{aligned}$$ Changing to $t_s$ and $\rho$ given in table \[table:2\], we obtain the more familiar forms of (A)dS. Notice that due to the denominators in (\[eq:154\]) the range of $\sigma$ is restricted to $|\sigma/R|\le \pi/2$ for dS while $\theta\in[0,\pi/2]$ in AdS. This implements the cutting of the infinite strip as sketched in Fig. \[fig:14\]. Let us now consider the null plane in $d$-dimensional Minkowski and analyze its transformation properties under these mappings. Taking Cartesian coordinates $X^\mu=(T,X,\vec{Y})$ in Minkowski, the null plane ${X_-=X-T=0}$ can be parametrized in terms of $(\lambda,\vec{x}_\perp)$ as $$\label{eq:50} \mathcal{N}_{\rm plane}= \left\lbrace X^\mu \in \mathbb{R}\times \mathbb{R}\times \mathbb{R}^{d-2}: \quad X^\mu(\lambda,\vec{x}_\perp)= (\lambda,\lambda,\vec{x}_\perp)\ , \quad (\lambda,\vec{x}_\perp)\in \mathbb{R}\times \mathbb{R}^{d-2} \right\rbrace .$$ For fixed $\vec{x}_\perp$ the curve $X^\mu(\lambda)$ trivially satisfies the geodesic equation $$\label{eq:63} \frac{d^2X^\mu }{d\lambda^2}+ \Gamma^\mu_{\alpha \beta} \frac{dX^\alpha}{d\lambda} \frac{dX^\beta}{d\lambda}=0\ ,$$ since the connection $\Gamma^\mu_{\alpha \beta}$ vanishes in these coordinates. This means that $\lambda$ is an affine parameter while we can think of $\vec{x}_\perp$ as a label going through the different geodesics. Since the transformation from Minkowski to the cylinder in table \[table:2\] is given in terms of radial null coordinates $r_\pm=r\pm t$, it is convenient to first change from the Cartesian spatial coordinates $(X,\vec{Y})$ to spherical. We can do this by defining $(r,\vec{v}\,)$ according to[^7] $$\label{eq:187} r=\big(X^2+|\vec{Y}|^2\big)^{1/2}\ , \qquad \qquad \vec{v}= \frac{2R\vec{Y}}{X +\big(X^2+|\vec{Y}|^2\big)^{1/2}}\ .$$ Using this together with (\[eq:50\]) we can write the null plane in spherical coordinates, where the Minkowski metric is ${ds^2=-dt^2+dr^2+r^2d\Omega^2(\vec{v}\,)}$.[^8] The conformal mapping from Minkowski to the cylinder is then applied by writing $r_\pm=R\tan(\theta_\pm/2)$ with ${\theta_\pm=\theta\pm \sigma/R}$, so that the null surface in the cylinder coordinates $v^\mu=(\theta_+,\theta_-,\vec{v}\,)$ becomes $$\label{eq:188} v^\mu(\lambda,\vec{x}_\perp)= \left( \theta_+(\lambda,\vec{x}_\perp), \theta_-(\lambda,\vec{x}_\perp), \frac{2R\vec{x}_\perp} {\lambda+\sqrt{\lambda^2+|\vec{x}_\perp|^2}} \right)\ , \qquad (\lambda,\vec{x}_\perp)\in \mathbb{R}\times \mathbb{R}^{d-2}\ ,$$ where $$\label{eq:189} \theta_\pm(\lambda,\vec{x}_\perp)= 2\,{\rm arctan}\left( \frac{\sqrt{\lambda^2+|\vec{x}_\perp|^2}\pm \lambda}{R} \right)\ .$$ If we evaluate the conformal factor associated to this transformation and given in table \[table:2\] along the surface we find $$\label{eq:191} w^2(\lambda,\vec{x}_\perp)= \frac{4R^2\lambda^2+(R^2+|\vec{x}_\perp|^2)^2} {4R^4}\ .$$ To understand the surface let us analyze its behavior for fixed values of $\vec{x}_\perp$. The geodesic equation (\[eq:63\]) is not invariant under the conformal transformations since the connection transforms with an additional term under the Weyl rescaling, and becomes $$\label{eq:64} \frac{d^2v^\mu}{d\lambda^2}+ \bar{\Gamma}^\mu_{\alpha \beta} \frac{dv^\alpha}{d\lambda} \frac{dv^\beta}{d\lambda}= \left[ -2\frac{d}{d\lambda} \ln\left(w(\lambda)\right) \right]\frac{dv^\mu }{d\lambda}\ ,$$ where $\bar{\Gamma}^\mu_{\alpha \beta}$ is the connection in the cylinder. One can explicitly check that the curve (\[eq:188\]) has a null tangent vector which satisfies this equation for any value of $\vec{x}_\perp$. Altogether, this means that $v^\mu(\lambda,\vec{x}_\perp)$ is (as expected) a null geodesic, even though $\lambda$ is not affine anymore due to the non-vanishing term on the right-hand side of (\[eq:64\]). This additional term can be canceled by defining an appropriate affine parameter $\bar{\lambda}(\lambda)$ according to $$\label{eq:78} \bar{\lambda}''(\lambda)= \left[ -2\frac{d}{d\lambda} \ln\left(w(\lambda)\right) \right] \bar{\lambda}'(\lambda) \qquad \Longrightarrow \qquad \bar{\lambda}(\lambda)= c_0 \int \frac{d\lambda}{w^2(\lambda)}+ c_1\ ,$$ where $c_0$ and $c_1$ are integration constants which can depend on the transverse coordinates $\vec{x}_\perp$. Using (\[eq:191\]) we can evaluate this explicitly and obtain an affine parameter in the cylinder $$\label{eq:190} \lambda(\bar{\lambda},\vec{x}_\perp)= \frac{R^2+|\vec{x}_\perp|^2}{2R} \tan(\bar{\lambda})\ , \qquad \qquad |\bar{\lambda}|\le \pi/2\ ,$$ where we have conveniently fixed the integration constants $c_0$ and $c_1$. Let us analyze the behavior of each of these geodesics. For any value of $\vec{x}_\perp$ all the curves begin and end at the same space-time points, given by $$\label{eq:194} (\sigma/R,\theta,|\vec{v}|\,) \big|_{\rm initial}= \left( -\frac{\pi}{2},\frac{\pi}{2},+\infty \right)\ , \qquad \qquad \left(\sigma/R,\theta,|\vec{v}|\,\right) \big|_{\rm final}= \left( \frac{\pi}{2},\frac{\pi}{2},0 \right)\ .$$ Remember that the $S^{d-2}$ in the cylinder metric (\[eq:170\]) is parametrized in stereographic coordinates $\vec{v}$, so that $|\vec{v}|$ equal to zero and infinity correspond to antipodal points in the $S^{d-2}$. This means that both the initial and final points lie on the equator $\theta=\pi/2$ of the spatial sphere $S^{d-1}$, but on opposite sides. As the affine parameter takes values in $\bar{\lambda}\in(-\pi/2,\pi/2)$, the curves travel between these points without intersecting and covering the whole sphere. Some special values of $\vec{x}_\perp$ have particularly simple trajectories. For instance, the geodesics with $|\vec{x}_\perp|=R$ always stay on the equator $\theta=\pi/2$, and are parametrized according to $$\label{eq:192} {\rm For\,\,}|\vec{x}_\perp|=R \qquad \Longrightarrow \qquad v^\mu(\bar{\lambda},\vec{x}_\perp)= \left( \frac{\pi}{2}+\bar{\lambda}, \frac{\pi}{2}-\bar{\lambda}, \frac{2\vec{x}_\perp} {\tan(\bar{\lambda})+\sec(\bar{\lambda})} \right)\ .$$ Other simple curves are given by $|\vec{x}_\perp|$ equal to zero or infinity, which corresponds to trajectories that go through the North and South pole of $S^{d-1}$ respectively. Their motion in the $\vec{v}$ coordinate is always constant expect at the pole where it discontinuously changes from zero to infinity. ![Geodesics in the Lorentzian cylinder and de Sitter for several values of $\vec{x}_\perp$. The diagram in the left does not contain all the information since it is missing the motion in the coordinates $\vec{v}$ on the $S^{d-2}$. In the center diagram, we plot some trajectories for the case $d=3$ where the spatial section of the cylinder is given by $S^2$. Equal colors in each diagram correspond to the same geodesics. To the right we have the geodesics in the $(\sigma/R,\theta)$ plane together with the region covered by de Sitter. Since the topology of dS is the same as the cylinder, the trajectories in dS are also given by the center diagram.[]{data-label="fig:15"}](Cylinder_dS_geodesics.pdf){height="2.7"} For all other values of $|\vec{x}_\perp|$ the curves travel along other possible paths in the sphere without intersecting. In the center diagram of Fig. \[fig:15\] we show some trajectories for the case $d=3$, where the spatial section of the cylinder is an $S^2$.[^9] For higher dimensions we can represent the geodesics in the $(\sigma/R,\theta)$ plane as shown in the left diagram of that figure. Although all these curves are null, they are not necessarily at an angle of $\pi/4$ since they have a non-trivial motion in the coordinate $\vec{v}$. Only for $|\vec{x}_\perp|$ equal zero and infinity the coordinate $\vec{v}$ remains constant and the curves have an angle of $\pi/4$ in the $(\sigma/R,\theta)$ plane. The mapping of this surface to (A)dS is straightforward since it only involves the Weyl rescaling in (\[eq:154\]). Using this, the conformal factors connecting Minkowski to (A)dS evaluated along the null surface can be computed from (\[eq:189\]) and (\[eq:191\]) $$\label{eq:197} w^2_{\rm dS}(\vec{x}_\perp)= \left( \frac{R^2+|\vec{x}_\perp|^2}{2R^2} \right)^2\ , \qquad \qquad w^2_{\rm AdS}(\vec{x}_\perp)=\left( \frac{R^2-|\vec{x}_\perp|^2}{2R^2} \right)^2\ .$$ Note that in both cases the results are independent of $\lambda$. This apparently innocent observation will have very deep consequences. In particular, it means that the affine parameter $\lambda$ in the null plane is also affine in (A)dS, since the right-hand side of the geodesic equation (\[eq:64\]) automatically vanishes. For de Sitter we plot the geodesics in the $(\sigma/R,\theta)$ plane in the right diagram of Fig. \[fig:15\]. All curves fit exactly inside in the space-time, traveling from the boundary at past infinity to future infinity. Since the topology of de Sitter is the same as the cylinder $\mathbb{R}\times S^{d-1}$, with a time dependent radius $S^{d-1}$, the trajectories are the same as for the cylinder shown in the center diagram of Fig. \[fig:15\]. The difference is that the curves in de Sitter cannot be extended beyond their initial and final points, since they encounter the dS boundaries at $|\sigma/R|=\pi/2$. ![In the left diagram we plot the AdS geodesics in the $(\sigma/R,\theta)$ plane. Comparing with Fig. \[fig:15\] we see that only half of the curves with $|\vec{x}_\perp|<R$ fit inside the space-time. In the right diagram we plot the trajectories in a cross section of the solid cylinder for the case of $d=3$. Equal colors in each diagram correspond to the same geodesics.[]{data-label="fig:16"}](AdS_geodesics){height="3.0"} The AdS case is quite different, since there are geodesics that lie outside the space-time, as we see in the left diagram of Fig. \[fig:16\]. Only curves with $|\vec{x}_\perp|<R$ lie inside AdS. The critical geodesic that has a vertical path in the $(\sigma/R,\theta)$ plane is given by $|\vec{x}_\perp|=R$ in (\[eq:192\]), and travels exactly along the AdS boundary. This is in accordance with the vanishing of the conformal factor in (\[eq:197\]), which is signaling something important since the conformal transformation is not invertible around that point. For $d=3$ we plot the trajectories of the AdS geodesics in a cross section of the solid cylinder, so that we get the right diagram in Fig. \[fig:16\].[^10] Different values of $\vec{x}_\perp$ follow distinct paths in AdS. This is in contrast to the cylinder and dS where all the geodesics are equivalent up to a rotation of the sphere $S^{d-1}$. The maximum depth in AdS reached by each geodesic is given at $\lambda=0$, and can be written in terms of the AdS radial coordinates $\rho=R\tan(\theta)$ in table \[table:2\] as $$\rho_{\rm min}= \left( \frac{2R^2}{R^2-|\vec{x}_\perp|^2} \right) |\vec{x}_\perp|\ , \qquad \qquad |\vec{x}_\perp|\in[0,R]\ .$$ The maximum depth corresponds to $|\vec{x}_\perp|=0$ where the geodesic reaches the center of AdS, while for $|\vec{x}_\perp|=R$ the geodesics travel along the AdS boundary and $\rho_{\rm min}$ diverges. Mapping the Minkowski ANEC -------------------------- Let us now apply the mapping of the Minkowski null plane to obtain some interesting results regarding the energy measured along null geodesics. Consider the ANEC in Minkowski, proven for general QFTs in Refs. [@Faulkner:2016mzt; @Hartman:2016lgu; @Longo:2018obd] and given by $$\label{eq:193} \mathcal{E}(\vec{x}_\perp)\equiv \int_{-\infty}^{+\infty}d\lambda\, T_{\lambda \lambda}(\lambda,\vec{x}_\perp)\ge 0\ .$$ The integral is over a null geodesic in the null plane (\[eq:50\]), parametrized by $\lambda$ and labeled by $\vec{x}_\perp$. The stress tensor $T_{\mu \nu}$ is projected along this null path according to $$\label{eq:200} T_{\lambda \lambda}= \frac{d X^\mu }{d\lambda} \frac{d X^\nu }{d\lambda} T_{\mu \nu}\ ,$$ where $X^\mu=X^\mu(\lambda,\vec{x}_\perp)$ in (\[eq:50\]). To map the integral operator in (\[eq:193\]) we require the transformation of the stress tensor. Given the Hilbert space $\mathcal{H}$ associated to the field theory in Minkowski, the unitary operator $U:\mathcal{H}\rightarrow \bar{\mathcal{H}}$ implements the mapping to $\bar{\mathcal{H}}$, the Hilbert space of the transformed CFT. Since $T_{\mu \nu}$ is a quasi-primary operator with spin $\ell=2$ and scaling dimension $\Delta=d$ it transforms under the adjoint action of $U$ as $$\label{eq:184} U T_{\mu \nu}U^\dagger= |w(v^\mu)|^{2-d} \frac{\partial v ^\alpha}{\partial X^\mu} \frac{\partial v ^\beta}{\partial X^\nu} \left( \bar{T}_{\alpha \beta}-\bar{S}_{\alpha \beta} \right)\ .$$ The anomalous term $\bar{S}_{\alpha \beta}$ is proportional to the identity operator and non-vanishing for even $d$. For $d=2$ it can be written in terms of the Schwartzian derivative. Assuming that $T_{\mu \nu}$ has vanishing expectation value in the Minkowski vacuum $\ket{0}$,[^11] we can determine the anomalous contribution $\bar{S}_{\alpha \beta}$ as $$\label{eq:27} 0=\bra{0}T_{\mu \nu}\ket{0} \qquad \Longrightarrow \qquad \bar{S}_{\alpha \beta}= \bra{\bar{0}}\bar{T}_{\alpha \beta} \ket{\bar{0}}\ ,$$ where we have used that $\bar{S}_{\alpha \beta}$ is proportional to the identity operator. The effect of the anomalous term is to ensure that the mapped stress tensor $\bar{T}_{\alpha\beta}$ vanishes when evaluated in the new vacuum state $\ket{\bar{0}}$. For the most part we leave this vacuum substraction implicit and simply write ${\bar{T}_{\alpha \beta}\equiv \bar{T}_{\alpha \beta}-\bra{\bar{0}}\bar{T}_{\alpha \beta}\ket{\bar{0}}}$. Using this we can write the transformation of the operator $T_{\lambda \lambda}$ appearing in (\[eq:193\]) as $$\label{eq:5} UT_{\lambda \lambda}(\lambda,\vec{x}_\perp) U^\dagger= |w(\lambda,\vec{x}_\perp)|^{2-d} \left[ \frac{d v^\alpha }{d\lambda} \frac{dv^\beta }{d\lambda} \bar{T}_{\alpha \beta}(\lambda,\vec{x}_\perp) \right]= |w(\lambda,\vec{x}_\perp)|^{2-d} \bar{T}_{\lambda \lambda}(\lambda,\vec{x}_\perp)\ ,$$ where the components of $\bar{T}_{\lambda \lambda}$ are now computed from the null surface $v^\mu(\lambda,\vec{x}_\perp)$ in (\[eq:188\]). In this way, the mapping of the Minkowski ANEC in (\[eq:193\]) is in general given by $$\label{eq:198} U\mathcal{E}(\vec{x}_\perp)U^\dagger= \int_{-\infty}^{+\infty}d\lambda\, |w(\lambda,\vec{x}_\perp)|^{2-d} \bar{T}_{\lambda\lambda}(\lambda,\vec{x}_\perp) \ge 0 \ .$$ This gives a non-trivial constraint for the CFTs defined on the cylinder and (A)dS implied by conformal symmetry and the ANEC in Minkowski. Since we are using the same coordinates $v^\mu=(\theta_+,\theta_-,\vec{v}\,)$ to describe all of these space-times, the geodesics are always given by (\[eq:188\]). ### Weighted average in Lorentzian cylinder For the case of the Lorentzian cylinder the conformal factor is given by (\[eq:191\]). Since it has a non-trivial dependence in $\lambda$, we change the integration variable to $\bar{\lambda}(\lambda)$, the affine parameter in (\[eq:190\]), which gives $$U\mathcal{E}(\vec{x}_\perp)U^\dagger= \frac{1}{R} \left( \frac{2R^2}{R^2+|\vec{x}_\perp|^2} \right)^{d} \int_{-\pi/2}^{\pi/2}d\bar{\lambda}\, \cos^{d}(\bar{\lambda}) \bar{T}_{\bar{\lambda}\bar{\lambda}} (\bar{\lambda},\vec{x}_\perp) \ ,$$ where we remember to consider the hidden factors of $d\lambda$ in the definition of $T_{\lambda \lambda}$ when changing the integration variable. The positivity of the Minkowski ANEC implies a novel bound for the null energy of a CFT in the cylinder[^12] $$\label{eq:195} \int_{-\pi/2}^{\pi/2}d\bar{\lambda}\, \cos^{d}(\bar{\lambda}) \bar{T}_{\bar{\lambda}\bar{\lambda}} (\bar{\lambda},\vec{x}_\perp) \ge 0\ .$$ Before analyzing its features, let us rewrite it in a more convenient way. Even though this inequality seems simple enough, the coordinate description of the geodesics in (\[eq:188\]) is complicated. However their trajectories in Fig. \[fig:15\] are very simple. A more convenient description of the same geodesics can be obtained by taking advantage of the rotation symmetry of the sphere. In particular we can rotate the coordinates in $S^{d-1}$ such that the initial and final points (\[eq:194\]) are instead given by the North and South pole. This has the advantage that every geodesic has a constant value of $\vec{v}$ along its trajectory, instead of the complicated dependence in (\[eq:188\]). The geodesics in the rotated frame are described in terms of the space-time coordinates $v^\mu=(\theta_+,\theta_-,\vec{v}\,)$ as $$\label{eq:196} v^\mu(\bar{\lambda},\vec{x}_\perp)= (2\bar{\lambda}+\pi/2,\pi/2,\vec{x}_\perp)\ , \qquad \qquad (\bar{\lambda},\vec{x}_\perp)\in [-\pi/2,\pi/2]\times \mathbb{R}^{d-2}\ .$$ These curves start and end at the same time as (\[eq:194\]) but at different spatial points of the sphere, given by the North and South pole. The tangent vector is clearly null and one can check that it satisfies the geodesic equation with affine parameter $\bar{\lambda}$. In Sec. \[sec:null\_energy\_bounds\] we rederive the bound (\[eq:195\]) from relative entropy directly in terms of a geodesic equivalent to (\[eq:196\]). Let us now comment on the most interesting features of (\[eq:195\]). The bound (\[eq:195\]) is not equivalent to the ANEC in the Lorentzian cylinder. To start, the condition is along a finite length geodesic which is not complete. Although we can obtain a bound for a complete geodesic going around the sphere $S^{d-1}$ an infinite number of times by applying (\[eq:195\]) to each section, it is not equivalent to the ANEC due to the non trivial weight function $\cos^d(\bar{\lambda})$.[^13] This weight function is required so that the operator (\[eq:195\]) is well defined. In the integration range $|\bar{\lambda}|\le \pi/2$, the function $\cos^d(\bar{\lambda})$ is non-negative, smooth and vanishes at the boundaries. The rapid decay of the function at $|\bar{\lambda}|=\pi/2$ is crucial, given that it is precisely at the boundary of a sharply integrated operator, where large amounts of negative energy can acumulate.[^14] Let us also recall that the stress tensor appearing in (\[eq:195\]) is normalized so that it vanishes in the vacuum state of the cylinder. This arises due to the anomalous transformation of the stress tensor under the conformal map (see the discussion around (\[eq:27\])). The operator in the inequality is then given by $\bar{T}_{\mu \nu}\equiv \bar{T}_{\mu \nu}-\bra{\bar{0}}\bar{T}_{\mu \nu}\ket{\bar{0}}$, where $\ket{\bar{0}}$ is the vacuum of the CFT in the cylinder. This vacuum contribution has been explicitly computed in Ref. [@Herzog:2013ed] for arbitrary CFTs, where it is shown to vanish when $d$ is odd while for even $d$ it is given by $$\label{eq:22} \bra{\bar{0}}\bar{T}_{\bar{\lambda} \bar{\lambda}}\ket{\bar{0}}= \frac{4(-1)^{d/2}A_d} {(d-1)R^{d-2}{\rm Vol}(S^d)}\ ,$$ with $A_d$ the trace anomaly coefficient, see Ref. [@Myers:2010tj] for conventions. The vacuum substraction ensures that the inequality (\[eq:195\]) is not trivially violated by come constant negative Casimir energy. Finally let us comment in the large $d$ limit, which is particularly interesting since the function $\cos^d(\bar{\lambda})$ localizes at $\bar{\lambda}=0$. Although $\bar{\lambda}=0$ in (\[eq:196\]) corresponds to the equator of $S^{d-1}$ we can always rotate the coordinates system so that the integral localized around an arbitrary point. This means we can write the bound directly in terms of the space-time coordinates $v^\mu$ in the large $d$ limit as the following local constraint $$\label{eq:169} \bar{T}_{--} (\theta_+,\theta_-,\vec{v}\,) \ge \lim_{d\rightarrow +\infty} \frac{(-1)^{d/2}A_d} {(d-1)R^{d-2}{\rm Vol}(S^d)}\ ,$$ where we have projected the stress tensor in the null coordinate $\theta_-$. Evaluating the limit on the right-hand side is not as simple as it might seem since the coefficient $A_d$ vanishes for $d$ odd and has a non-trivial dependence when $d$ is even. Although the explicit dependence of $A_d$ for even $d$ can be computed for free or holographic theories [@Cappelli:2000fe; @Myers:2010tj], the question still remains regarding how to deal with the factor $R^{d-2}$. Whatever the case may be, there are only two possible outcomes for the limit in (\[eq:169\]): it is either undetermined or it converges to zero. While an undetermined result means that there is something funny going on with large $d$ limit in (\[eq:195\]), if it goes to zero it implies that the stress tensor is locally a positive operator in the cylinder. This is an interesting result which we hope to further investigate in future work. ### ANEC in (A)dS Let us now consider the mapping to (A)dS, where the conformal factors evaluated on the null surface are given in (\[eq:197\]). Since these are independent of $\lambda$ the mapping of the Minkowski ANEC (\[eq:198\]) is given by $$U\mathcal{E}(\vec{x}_\perp)U^\dagger= |w_{\rm (A)dS}(\vec{x}_\perp)|^{2-d} \int_{-\infty}^{+\infty}d\lambda\, \bar{T}_{\lambda\lambda}(\lambda,\vec{x}_\perp)\ ,$$ which implies $$\label{eq:199} \mathcal{E}_{\rm (A)dS}(\vec{x}_\perp)\equiv \int_{-\infty}^{+\infty}d\lambda\, \bar{T}_{\lambda \lambda}(\lambda,\vec{x}_\perp)\ge 0\ .$$ Let us explain what are the features that allows us to identify this as the ANEC in both de Sitter and anti-de Sitter. The first crucial fact is that $w_{\rm (A)dS}(\vec{x}_\perp)$ is independent of $\lambda$, so that the right hand side of the geodesic equation (\[eq:64\]) vanishes and implies that $\lambda$ is an affine parameter in (A)dS.[^15] Moreover, this allows to remove it from the $\lambda$ integral in (\[eq:198\]) so that there is no weight function along the trajectory, as we had for the case of the Lorentzian cylinder (\[eq:195\]). Another important feature is that the geodesics in both dS and AdS are complete, *i.e.* they cannot be extended beyond $\lambda \in \mathbb{R}$. This is certainly the case as the curves start and end at the (A)dS boundaries. Altogether, this allows us to identify (\[eq:199\]) as the ANEC in (A)dS, valid for any conformal theory. Similarly to the case of the cylinder, for dS we can use the spatial symmetry to describe the null geodesics in (\[eq:188\]) in a more convenient way. Since de Sitter space-time is topologically given by $\mathbb{R}\times S^{d-1}$, we can use the same reasoning around (\[eq:196\]) to describe the geodesics in de Sitter as $$v^\mu(\lambda,\vec{x}_\perp)= \left( 2\arctan(\lambda)+\pi/2,\pi/2,\vec{x}_\perp \right)\ , \qquad (\lambda,\vec{x}_\perp)\in \mathbb{R}\times \mathbb{R}^{d-2}\ .$$ In Sec. \[sec:null\_energy\_bounds\] we rederive the ANEC in de Sitter from relative entropy directly in terms of a null geodesic equivalent to this one. For AdS we do not have a symmetry argument to simplify the description of the geodesics in (\[eq:188\]). As we see in the right diagram of Fig. \[fig:16\] the geodesics for different values of $\vec{x}_\perp$ are distinct and travel through the space-time in different ways. Before moving on let us recall that the stress tensor appearing in (\[eq:199\]) contains a substraction with respect to the (A)dS vacuum, *i.e.* $\bar{T}_{\mu \nu}\equiv \bar{T}_{\mu \nu}-\bra{0_{\rm (A)dS}}\bar{T}_{\mu \nu}\ket{0_{\rm (A)dS}}$. However, there is an important distinction in this case given by the fact that (anti-)de Sitter is a maximally symmetric space-time. This implies that the vacuum expectation value of the stress tensor is proportional to the (A)dS metric,[^16] which results in $$\label{eq:181} \bra{0_{\rm (A)dS}} \bar{T}_{\mu \nu}\ket{0_{\rm (A)dS}}\propto g_{\mu \nu}^{\rm (A)dS} \qquad \Longrightarrow \qquad \bra{0_{\rm (A)dS}} \bar{T}_{\lambda \lambda} \ket{0_{\rm (A)dS}}\propto g_{\mu \nu}^{\rm (A)dS} \frac{dv^\mu }{d\lambda} \frac{dv^\nu }{d\lambda}=0\ .$$ Therefore, the Casimir energy of (A)dS makes no contribution to the ANEC in (\[eq:199\]). Null energy bounds from relative entropy {#sec:null_energy_bounds} ======================================== In the previous section we showed that the ANEC in (A)dS and a similar bound for the Lorentzian cylinder follow from the Minkowski ANEC and conformal symmetry. The aim of this section is to investigate whether these results can also be obtained from relative entropy, as done in Ref. [@Faulkner:2016mzt] for the Minkowski ANEC. Let us start by briefly review the approach used in that paper. Consider a smooth curve in the null plane (\[eq:50\]) defined by $\lambda=A(\vec{x}_\perp)$ which splits the surface in two regions ${\mathcal{N}_{\rm plane}=\mathcal{A}^+\cup \mathcal{A}^-}$, where $\mathcal{A}^\pm$ are given by $\lambda\ge \pm A(\vec{x}_\perp)$. Given a QFT in $d$-dimensional space-time $X^\mu$ we take the space-time region $\mathcal{DA}^+$ for which $\mathcal{A}^+$ is its future horizon, and analogously for $\mathcal{DA}^-$. A diagram of the setup is given in Fig. \[fig:3\]. For these space-time regions let us consider the reduced density operator $\rho_{\mathcal{A}^{\pm}}$ associated to the vacuum state $\ket{0}$. We can define $\rho_{\mathcal{A}^\pm}$ as the operator which satisfies the following property $$\label{eq:67} \bra{0}\mathcal{O}_{\mathcal{A}^\pm}\ket{0}= {\rm Tr}\big( \rho_{\mathcal{A}^\pm}\, \mathcal{O}_{\mathcal{A}^\pm} \big)\ ,$$ for $\mathcal{O}_{\mathcal{A}^\pm}$ any operator (not necessarily local) supported exclusively in $\mathcal{DA}^\pm$. Given a reduced density operator its logarithm defines the modular Hamiltonian $K_{\mathcal{A}^\pm}=-\ln(\rho_{\mathcal{A}^\pm})+{\rm const}$, where the constant is fixed by normalization. ![Plot of $\mathcal{DA}^+$ (green) and $\mathcal{DA}^-$ (red) in the $(T,X)$ plane in Minkowski space-time. The function $A(\vec{x}_\perp)$ (blue dot) changes along the null plane $X-T=0$ as we move through the transverse directions $\vec{x}_\perp$ outside of the page. A very nice 3D picture of this region can be found in Fig. 1 of Ref. [@Casini:2018kzx].[]{data-label="fig:3"}](Deformed_wedge.pdf){height="2.0"} For this setup the modular Hamiltonian of the vacuum state was computed in Ref. [@Casini:2017roe] (see also Refs. [@Faulkner:2016mzt; @Wall:2011hj; @Koeller:2017njr]) and shown to have the following simple local expression $$\label{eq:66} K_{\mathcal{A}^\pm}=\pm 2\pi \int_{\mathcal{A}^\pm}dS \left(\lambda-A(\vec{x}_\perp)\right) T_{\lambda \lambda}(\lambda,\vec{x}_\perp)\ ,$$ where $dS=d\vec{x}_\perp d\lambda$ is the induced surface element on the null plane and $T_{\lambda \lambda}$ is defined in (\[eq:200\]). When $A(\vec{x}_\perp)=0$ the regions in Fig. \[fig:3\] corresponds to the Rindler wedge and its complement, so that (\[eq:66\]) follows from the Bisognano-Wichmann theorem [@Bisognano:1976za]. In this case the modular Hamiltonian can be written as a local integral over any Cauchy surface in $\mathcal{DA}^\pm$, not necesarily along the null horizons. This is not true when $A(\vec{x}_\perp)$ is a non-trivial function, since the operator has a local expression only along the null surface $\mathcal{A}^\pm$ [@Casini:2017roe]. It is useful to also consider the full modular Hamiltonian $\hat{K}_{\mathcal{A}^+}$, defined for a generic space-time region $V$ as $$\label{eq:161} \hat{K}_V=K_V-K_{V'}\ ,$$ where $V'$ is the causal complement of $V$. Using the expressions in (\[eq:66\]) we find $$\label{eq:177} \hat{K}_{\mathcal{A}^+}= 2\pi \int_{\mathcal{N}_{\rm plane}} dS\,\left( \lambda-A(\vec{x}_\perp) \right) T_{\lambda \lambda}(\lambda,\vec{x}_\perp)\ ,$$ where the integral is now over the full null plane. This operator has the advantage that it is globally defined in the Hilbert space, without any ambiguities that can arise in (\[eq:66\]) from the boundary of integration. In the context of the Tomita-Takesaki theory that we review in Sec. \[sec:ref\_positivity\], $\hat{K}_V$ determines the modular operator. To prove the Minkowski ANEC, Ref. [@Faulkner:2016mzt] combined the full modular Hamiltonian in (\[eq:177\]) together with relative entropy, that is defined as $$\label{eq:180} S(\rho||\sigma)= {\rm Tr}\left( \rho\ln(\rho) \right)- {\rm Tr}\left( \rho\ln(\sigma) \right)\ge 0\ ,$$ where $\rho$ and $\sigma$ are any two density operators. The monotonicity property of relative entropy implies that given any two space-time regions such that $A \supseteq B$, the reduced operators satisfy the inequality $S\left(\rho_A||\sigma_A\right)\ge S\left(\rho_B||\sigma_B\right)$. Taking $\sigma$ as a pure state and starting from this inequality and an analogous one for the complementary regions, it is straightforward to prove following constraint [@Blanco:2013lea] $$\label{eq:1} \hat{K}_{A}-\hat{K}_B\ge 0\ , \qquad {\rm for} \qquad A \supseteq B\ ,$$ where $\hat{K}_{A/B}$ is the full modular Hamiltonian of $\sigma$.[^17] Using (\[eq:177\]) we can explicitly write the inequality for null deformations of Rindler, which gives $$\hat{K}_A-\hat{K}_B= 2\pi\int_{-\infty}^{+\infty}d\lambda \int_{\mathbb{R}^{d-2}} d\vec{x}_\perp\left( B(\vec{x}_\perp)-A(\vec{x}_\perp) \right) T_{\lambda \lambda}(\lambda,\vec{x}_\perp)\ge 0\ ,$$ where we must have $B(\vec{x}_\perp)\ge A(\vec{x}_\perp)$. Taking $A(\vec{x}_\perp)=0$ and $B(\vec{x}_\perp)=\delta(\vec{x}_\perp-\vec{x}_\perp^{\,0})$ for any $\vec{x}_\perp^{\,0}$, gives the ANEC in Minkowski (\[eq:193\]) as derived in Ref. [@Faulkner:2016mzt]. Our strategy for extending this proof is simple. Using conformal transformations we map the modular Hamiltonian in (\[eq:177\]) to (A)dS and the Lorentzian cylinder. From this we can explicitly write the inequality (\[eq:1\]) coming from relative entropy and obtain a bound for the energy along null geodesics. We shall see that this procedure is non-trivial and while it works for de Sitter and the Lorentzian cylinder, it fails to give the ANEC in the anti-de Sitter case. Along the way we obtain several new modular Hamiltonians and compute their associated entanglement entropy. Taking the null plane on a conformal journey - Take II {#sec:geometry} ------------------------------------------------------ Since our aim is to map the modular Hamiltonian (\[eq:66\]), given by an integral over a region of the null plane, we start by discussing the geometric transformation of the null plane. Although we have already analyzed this in the previous section, the resulting surface (\[eq:188\]) has a complicated coordinate description which is not the most convenient. We now consider a slightly different conformal transformation that is more useful for writing the modular Hamiltonians. Instead of mapping the null plane directly to the cylinder, we first consider a conformal transformation mapping the Minkowski space-time $X^\mu=(T,X,\vec{Y}\,)$ into itself $x^\mu=(t,x,\vec{y}\,)$. This transformation is given by $$\label{eq:3} x^\mu(X^\mu)= \frac{X^\mu+(X\cdot X)C^\mu} {1+2(X\cdot C)+(X\cdot X)(C\cdot C)} -D^\mu\ ,$$ where $(X\cdot X)=\eta_{\mu \nu}X^\mu X^\nu$. It gives a space-time translation in the $D^\mu=(R,R,\vec{0}\,)$ direction together with a special conformal transformation with parameter $C^\mu=(0,1/(2R),\vec{0}\,)$. The Minkowski metric in the new coordinates becomes $ds^2=w^2(x^\mu)\eta_{\mu \nu}dx^\mu dx^\nu$, where the conformal factor is given by $$w^2(X^\mu)= \big(1+2(X\cdot C)+(X\cdot X)(C\cdot C)\big)^2\ .$$ Evaluating this along the null plane (\[eq:50\]) we find $$\label{eq:79} w^2(\lambda,\vec{x}_\perp)= \left( \frac{\lambda+ p(\vec{x}_\perp)}{R} \right)^2 \qquad \,\, {\rm where} \qquad \,\, p(\vec{x}_\perp)= \frac{|\vec{x}_\perp|^2+4R^2}{4R}\ .$$ The mapped suface can be found by evaluating (\[eq:3\]) in the parametrization of the null plane in (\[eq:50\]) $$\label{eq:23} x^\mu(\lambda,\vec{x}_\perp)= R\left( \frac{p(\vec{x}_\perp)} {\lambda+p(\vec{x}_\perp)} \right) \big(-1,\vec{n}(\vec{x}_\perp)\big)\ , \qquad \quad \vec{n}(\vec{x}_\perp)=\left( \frac{|\vec{x}_\perp|^2-4R^2} {|\vec{x}_\perp|^2+4R^2}, \frac{4R\vec{x}_\perp} {|\vec{x}_\perp|^2+4R^2} \right)\ ,$$ where $\vec{n}\in \mathbb{R}^{d-1}$ is a unit vector $|\vec{n}(\vec{x}_\perp)|=1$. This surface corresponds to a future and past null cone starting from the origin $x^\mu=0$. Although $\lambda$ is not affine anymore, we can define an affine parameter $\alpha$ according to $\lambda(\alpha)=p(\vec{x}_\perp)(R/\alpha-1)$,[^18] so that the surface is given by $$\label{eq:14} x^\mu(\alpha,\vec{x}_\perp)=\alpha(-1,\vec{n}(\vec{x}_\perp))\ , \qquad \qquad (\alpha,\vec{x}_\perp)\in \mathbb{R}\times \mathbb{R}^{d-2}\ .$$ Positive $\alpha$ corresponds to the past null cone of the origin $x^\mu=0$, while negative $\alpha$ gives the future cone. The transverse coordinates $\vec{x}_\perp$ parametrize a unit sphere $S^{d-2}$ in stereographic coordinates, as can be seen by computing the induced metric on the surface and finding $\alpha^2d\Omega^2(\vec{x}_\perp)$ with $d\Omega(\vec{x}_\perp)$ in (\[eq:152\]) (where $L=2R$). There is a subtlety in this transformation that we must be careful with. As we can see from the description in terms of $\lambda$ in (\[eq:23\]), there is a discontinuity in the mapping when $\lambda=-p(\vec{x}_\perp)$, that is precisely where the conformal factor (\[eq:79\]) vanishes. Similarly to the previous mapping to AdS in (\[eq:197\]), this is signaling a failure of the transformation, which is somewhat expected given that special conformal transformations are not globally defined in Minkowski but on its conformal compactification, the Lorentzian cylinder. To properly interpret the surface (\[eq:14\]) we must go to the cylinder. Since a single copy of Minkowski is not enough to cover the whole cylinder, we consider an infinite number of Minkowski space-times $\mathcal{M}_{n}$ and $\mathcal{M}_m$ labeled by the integers $(n,m)$, so that the whole cylinder manifold $\mathcal{M}_{LC}$ is obtained from $$\mathcal{M}_{LC}= \Big(\bigcup_{n\in \mathbb{Z}} \mathcal{M}_{n}\Big) \cup \Big(\bigcup_{m\in \mathbb{Z}} \mathcal{M}_{m}\Big)\ .$$ To each of the Minkowski copies we apply a slightly different conformal transformation $$\label{eq:10} r_\pm^{(n)}(\theta_\pm)= R\tan(\theta_\pm/2)\ , \qquad \qquad r_\pm^{(m)}(\theta_\pm)= -R\tan(\theta_\mp/2)\ ,$$ where the domain of the coordinates $\theta_\pm$ in each case is given by $$\label{eq:114} \begin{aligned} D_n&= \big\lbrace \theta_\pm \in \mathbb{R}:\quad (\theta_\pm\mp 2n\pi)\in[-\pi,\pi]\ , \quad \theta\in[0,\pi] \big\rbrace\ , \\[8pt] D_m&= \big\lbrace \theta_\pm \in \mathbb{R}:\quad (\theta_\pm\mp (2m-1)\pi)\in[0,2\pi]\ , \quad \theta\in[0,\pi] \big\rbrace\ . \end{aligned}$$ In every case, the transformations acts in the same way as in table \[table:2\] but mapping $\mathcal{M}_{(n,m)}$ to different sections of the Lorentzian cylinder. These are given in the $(\sigma/R,\theta)$ plane by the shaded blue and orange regions in the second diagram of Fig. \[fig:9\]. The main difference between the $n$ (blue) and $m$ (orange) patches is that the $n$ series maps the Minkowski origin to the North pole, while for $m$ the origin is mapped to the South. Let us now use these relations to map the null plane across the special conformal transformation and into the cylinder. From (\[eq:23\]) we can write the null radial coordinates $r_\pm$ on the surface as $$\label{eq:56} r_\pm(\lambda,\vec{x}_\perp)= \frac{2Rp(\vec{x}_\perp)} {|\lambda+p(\vec{x}_\perp)|} \Theta\big[ \mp \left(\lambda+p(\vec{x}_\perp)\right) \big]\ .$$ Applying the transformation associated to the patch $n=0$ in (\[eq:10\]) to the region of the null plane ${\lambda>-p(\vec{x}_\perp)}$ and $m=0$ to $\lambda<-p(\vec{x}_\perp)$ we find $$\label{eq:57} \begin{aligned} \lambda & >-p(\vec{x}_\perp) \qquad \Longrightarrow \qquad r_\pm^{(n=0)}: \left( \theta_+=0, \tan(\theta_-/2)=\frac{2p(\vec{x}_\perp)} {\lambda+p(\vec{x}_\perp)} \right)\ , \qquad \theta_-\in [0,\pi]\ ,\\[4pt] \lambda & <-p(\vec{x}_\perp) \qquad \Longrightarrow \qquad r_\pm^{(m=0)}: \left( \theta_+=0, \tan(\theta_-/2)= \frac{2p(\vec{x}_\perp)} {\lambda+p(\vec{x}_\perp)} \right)\ , \qquad \theta_-\in[\pi,2\pi]\ , \end{aligned}$$ where the range of $\theta_-$ in each case is obtained from (\[eq:114\]). Notice that the surface across the two patches is continuous as $\theta_-\rightarrow \pi$. Moreover, the singularity that is present in the Minkowski space $x^\mu$ at $\lambda\rightarrow -p(\vec{x}_\perp)$ is smoothed out in the cylinder by the tangent function. This completely determines the mapping of the null plane in the Minkowski coordinates $X^\mu$ to the Lorentzian cylinder, which we sketch in Fig. \[fig:9\]. ![Mapping of the null plane in Minkowski to the null surfaces in the $(\sigma/R,\theta)$ plane. The different diagrams indicate the sections of the cylinder covered by each of the space-times. We see that discontinuity at $\lambda=-p(\vec{x}_\perp)$ in the Minkowski case is nothing more than the surface going from the Minkowski patch $\mathcal{M}_{n=0}$ to $\mathcal{M}_{m=0}$ in the second diagram. For de Sitter we consider a slight variation of the Weyl rescaling so that the surface fits in the space-time, see discussion above (\[eq:202\]).[]{data-label="fig:9"}](Surface_rel){height="2.3"} We can now reinterpret the discontinuity in the Minkowski null cone $x^\mu$ in (\[eq:23\]) from the perspective of the Lorentzian cylinder. As we see in Fig. \[fig:9\], this discontinuity is nothing more than the null surface going from the Minkowski copy $\mathcal{M}_{n=0}$ to $\mathcal{M}_{m=0}$. The future null cone in appears to come from infinity, that is precisely what happens from the perspective of $\mathcal{M}_{m=0}$ in Fig. \[fig:9\]. This means that the future and past null cones in (\[eq:23\]) are not in the same Minkowski patch, since the mapping of the full null plane does not fit in the Minkowski space-time $x^\mu$. Shortly, this will play an important role when computing the modular Hamiltonian associated to the null cone. The conformal factor relating the Minkowski space-time $X^\mu$ with the cylinder is obtained by taking the product of (\[eq:79\]) and the expression in table \[table:2\] evaluated at (\[eq:56\]), which gives $$\label{eq:201} w(\lambda,\vec{x}_\perp)^2= \left[\frac{(\lambda+p(\vec{x}_\perp))^2}{R^2} \right] \times \left[\frac{4p^2(\vec{x}_\perp)+(\lambda+p(\vec{x}_\perp))^2} {4(\lambda+p(\vec{x}_\perp))^2} \right]= \frac{4p^2(\vec{x}_\perp)+(\lambda+p(\vec{x}_\perp))^2} {4R^2}\ .$$ Using this to solve the integral in (\[eq:78\]), we find an affine parameter $\beta=\beta(\lambda)$ for the surface in the cylinder $$\label{eq:116} \tan(\beta)=\frac{2p(\vec{x}_\perp)}{\lambda+p(\vec{x}_\perp)}\ ,$$ where we have conveniently fixed the integration constants to $c_0$ and $c_1$. Comparing with (\[eq:57\]) we identify $\beta=\theta_-/2$, so that the null surface in the cylinder coordinates $v^\mu=(\theta_+,\theta_-,\vec{v}\,)$ has the following simple description $$\label{eq:62} v^\mu(\beta,\vec{x}_\perp)= \left( 0,2\beta,\vec{x}_\perp \right)\ , \qquad \qquad (\beta,\vec{x}_\perp) \in [0,\pi]\times \mathbb{R}^{d-2}\ .$$ The surface goes from the South pole of the $S^{d-1}$ all the way to the North pole. Up to a time translation and rotation of the $S^{d-1}$, it is equivalent to the surface obtained through the mapping of the previous section in (\[eq:188\]) (see Fig. \[fig:15\]) but with a much simpler description. Let us now apply the transformation to (A)dS given by the Weyl rescaling in (\[eq:154\]). Since the surface in the cylinder (\[eq:62\]) has a range in $\sigma/R$ given by $\sigma/R\in[-\pi,0]$ we consider a slightly different Weyl rescaling for the de Sitter case, given by changing the conformal factor in (\[eq:154\]) to ${\cos^2(\sigma/R)\rightarrow \sin^2(\sigma/R)}$. This allows us to take the range of the time coordinate in dS $\sigma/R\in[-\pi,0]$ so that the surface (\[eq:62\]) fits in the space-time, as we see in Fig. \[fig:9\]. In the same figure we see that the null surface does not fit in a single copy of AdS. Evaluating the conformal factor relating the Minkowski space-time $X^\mu$ to (A)dS using (\[eq:201\]) and (\[eq:62\]) we find $$\label{eq:202} w_{\rm dS}(\vec{x}_\perp)= p(\vec{x}_\perp)/R\ , \qquad \qquad w_{\rm AdS}(\lambda,\vec{x}_\perp)^2= \left( \frac{\lambda+p(\vec{x}_\perp)}{2R} \right)^2\ .$$ For de Sitter the conformal factor is independent of $\lambda$ and similar to the one obtained from the conformal transformation in Sec. (\[eq:197\]). This means that $\lambda$ is an affine parameter in dS. We still find it convenient to apply an affine transformation by defining $\eta$ according to $\lambda(\eta)=p(\vec{x}_\perp)(2\eta-1)$ so that using (\[eq:116\]) the surface in dS has a simple description. Writing $\beta$ in (\[eq:62\]) in terms of $\eta$ we find $$\label{eq:203} v^\mu(\eta,\vec{x}_\perp)= \left( 0,2\,{\rm arccot}(\eta),\vec{x}_\perp \right)\ , \qquad \qquad (\eta,\vec{x}_\perp) \in \mathbb{R}\times \mathbb{R}^{d-2}\ ,$$ where since $\beta={\rm arccot}(\eta)$, the image of ${\rm arccot}(\eta)$ is taken in $[0,\pi]$. For anti-de Sitter the conformal factor depends on $\lambda$, which means $\lambda$ is not affine after the transformation. This is quite different to the mapping considered in the previous section, where it was independent of $\lambda$ (\[eq:197\]). An affine parameter in AdS $\zeta$ can be easily found by solving the integral in (\[eq:78\]), which gives $\lambda(\zeta)=p(\vec{x}_\perp)(2/\zeta-1)$. Writing $\beta$ in (\[eq:62\]) in terms of $\zeta$, the surface in AdS is given by $$v^\mu(\zeta,\vec{x}_\perp)= \left( 0,2\,{\rm arctan}(\zeta),\vec{x}_\perp \right)\ , \qquad \qquad (\eta,\vec{x}_\perp) \in \mathbb{R}_{\ge 0}\times \mathbb{R}^{d-2}\ .$$ As $\zeta\rightarrow +\infty$ the surface reaches $\theta=\pi/2$ corresponding to the AdS boundary and the conformal factor (\[eq:202\]) vanishes. The full surface does not fit in a single copy of AdS. [ Sl | Sl | Sl | Sl ]{} **Mapping of** & **Affine parameter** & **Induced** & **Fits inside**\ **null plane to** & **along geodesic** & **metric** & **space-time?**\ Minkowski null cone & $\lambda(\alpha)=p(\vec{x}_\perp)(R/\alpha-1)$ & $\alpha^2 d\Omega^2(\vec{x}_\perp)$ &\ Lorentzian cylinder & $\lambda(\beta)=p(\vec{x}_\perp)(2\cot(\beta)-1)$ & $R^2\sin^2(\beta) d\Omega^2(\vec{x}_\perp)$ &\ De Sitter & $\lambda(\eta)=p(\vec{x}_\perp)(2\eta-1)$ & $ R^2d\Omega^2(\vec{x}_\perp) $ &\ Anti-de Sitter & $\lambda(\zeta)= p(\vec{x}_\perp)(2/\zeta-1)$ & $R^2\zeta^2 d\Omega^2(\vec{x}_\perp)$ &\ Modular Hamiltonians of null deformed regions in curve backgrounds ------------------------------------------------------------------ Now that we have a simple description of the mapping of the null plane we can apply these conformal transformations on the modular Hamiltonian $K_{\mathcal{A}^\pm}$ in (\[eq:66\]) and explicitly write the constraint (\[eq:1\]) coming from relative entropy. We summarize the most important aspects of the mapping of the null plane in table \[table:1\]. A general conformal transformation given by a change of coordinates $z^\mu(X^\mu)$ induces a geometric transformation of the null surface $\mathcal{A}^\pm\rightarrow \bar{\mathcal{A}}^\pm$, while the Hilbert space is mapped by a unitary operator $U:\mathcal{H}\rightarrow \bar{\mathcal{H}}$. Consider an arbitrary primary operator $\mathcal{O}_a(X^\mu)$ of spin $\ell\in \mathbb{N}_0$, where the label $a$ contains all the Lorentz indices, *i.e.* $a=(\mu_1,\dots,\mu_\ell)$. An arbitrary matrix $R_a^{\,\,\,b}$ is obtained from $R_\mu^{\,\,\,\nu}$ as $$\label{eq:82} R_a^{\,\,\,b}\equiv R_{\mu_1}^{\,\,\,\nu_1}\dots R_{\mu_\ell}^{\,\,\,\nu_\ell}\ .$$ Since $\mathcal{O}_a(X^\mu)$ is primary, it transforms according to $$\label{eq:88} U\mathcal{O}_a(X^\mu)U^\dagger= |w(z^\mu)|^{\ell-\Delta} \frac{\partial z^b}{\partial X^a} \bar{\mathcal{O}}_b(z^\mu)\ ,$$ where $\bar{\mathcal{O}}_a(z^\mu)$ acts on the Hilbert space $\bar{\mathcal{H}}$. To obtain the transformation property of the reduced density operator $\rho_{\mathcal{A}^\pm}$ we consider its defining property (\[eq:67\]). Writing this relation for a primary operator $\mathcal{O}_a(X^\mu)$ and using its simple transformation law (\[eq:88\]) we find $$\label{eq:68} \bra{\bar{0}} \bar{\mathcal{O}}_b(z^\mu) \ket{\bar{0}}= {\rm Tr}\big( U \rho_{\mathcal{A}^\pm} U^\dagger \bar{\mathcal{O}}_b(z^\mu) \big)\ , \qquad \quad z^\mu \in \mathcal{D\bar{A}}^\pm\ ,$$ where $\ket{\bar{0}}=U\ket{0}$ is the vacuum state in the mapped CFT. We have canceled the conformal factors appearing on both sides as well as the Jacobian matrices, which are invertible since conformal transformations can be inverted. The location of the mapped operator is given by $z^\mu \in \mathcal{D\bar{A}}^\pm$. This relation allows us to identify the reduced density operator associated to the causal domain of the mapped null surface $\bar{\mathcal{A}}^\pm$ as $\bar{\rho}_{\mathcal{\bar{A}}^\pm}=U\rho_{\mathcal{A}^\pm}U^\dagger$. Although (\[eq:68\]) only involves primary operators of integer spin, we can differentiate it to obtain its descendants, while an analogous transformation property to (\[eq:88\]) gives the equivalent relation for primary operators of half-integer spin. Altogether, this means that the modular Hamiltonian transforms in the expected way given by the adjoint action of $U$ as $\bar{K}_{\bar{\mathcal{A}}^\pm}=UK_{\mathcal{A}^\pm}U^\dagger$. Since the modular Hamiltonian of the null plane (\[eq:66\]) is written as an integral of the stress tensor, we can directly use the transformation of $T_{\lambda \lambda}$ in (\[eq:5\]). The modular Hamiltonian associated to $\bar{\mathcal{A}}^\pm$ is then given by $$\label{eq:19} \bar{K}_{\bar{\mathcal{A}}^\pm}= \pm 2\pi \int_{\bar{\mathcal{A}}^\pm} d\bar{S}\, (\lambda-A(\vec{x}_\perp))\bar{T}_{\lambda\lambda}(\lambda,\vec{x}_\perp)\ .$$ We have absorbed the factor $|w(z)|^{2-d}$ into the surface element $d\bar{S}=d\vec{x}_\perp d\lambda\sqrt{\bar{h}}$, where $\bar{h}$ is the determinant of the induced metric of the mapped surface in the new space-time. Although $\bar{\mathcal{A}}^\pm$ is a $(d-1)$ dimensional surface, its surface element scales as $(d-2)$ because it is null. Applying a simple change of integration variables we can write the integral in terms of a generic affine parameter $\bar{\lambda}$ as $$\label{eq:71} \bar{K}_{\bar{\mathcal{A}}^\pm}= \pm 2\pi \int_{\bar{\mathcal{A}}^\pm} d\bar{S}\, \left[ \frac{\lambda(\bar{\lambda})-A(\vec{x}_\perp)} {\lambda'(\bar{\lambda})} \right] \bar{T}_{\bar{\lambda}\bar{\lambda}} (\bar{\lambda},\vec{x}_\perp)\ ,$$ where we took into account the $\lambda$ derivatives in the definition of $T_{\lambda \lambda}$. In an analogous way, the full modular Hamiltonian in (\[eq:177\]) transforms according to $$\label{eq:160} \hat{K}_{\bar{\mathcal{A}}^+}= 2\pi \int_{\bar{\mathcal{N}}} \,d\bar{S}\,\left[ \frac{\lambda(\bar{\lambda})-A(\vec{x}_\perp)} {\lambda'(\bar{\lambda})} \right] \bar{T}_{\bar{\lambda}\bar{\lambda}} (\bar{\lambda},\vec{x}_\perp)\ ,$$ where $\bar{\mathcal{N}}=\bar{\mathcal{A}}^+\cup \bar{\mathcal{A}}^-$. Using these relations and the results of the previous section summarized in table \[table:1\] we can easily write these operators explicitly. In Fig. \[fig:6\] we plot the null horizons $\bar{\mathcal{A}}^\pm$ and their causal regions for the different space-times in the $(\sigma/R,\theta)$ plane. ![Plot of the mapped surfaces $\bar{\mathcal{A}}^\pm$ and their causal domains $\mathcal{D\bar{A}}^\pm$ in the $(\sigma/R,\theta)$, where we indicate the region of the cylinder covered by each of the space-times.[]{data-label="fig:6"}](Conf_regions.pdf){height="2.5"} ### Minkowski null cone Let us start by taking $\bar{\mathcal{A}}^+$ as a region of the past null cone in Minkowski (\[eq:14\]), given by $$\label{eq:120} \bar{\mathcal{A}}^+=\left\lbrace x^\mu \in \mathbb{R}\times\mathbb{R}^{d-1}: \quad x^\mu(\alpha,\vec{x}_\perp)= \alpha(-1,\vec{n}(\vec{x}_\perp))\ , \quad (\alpha,\vec{x}_\perp) \in [0,\bar{A}(\vec{x}_\perp)]\times \mathbb{R}^{d-2} \right\rbrace\ ,$$ where $\vec{n}(\vec{x}_\perp)$ is given in (\[eq:23\]). The affine parameter $\alpha$ is obtained from the relation $\lambda(\alpha)$ in table \[table:1\], while the entangling surface $\lambda=\bar{A}(\vec{x}_\perp)\ge 0$ is determined from $A(\vec{x}_\perp)=p(\vec{x}_\perp)(R/\bar{A}(\vec{x}_\perp)-1)$. Using (\[eq:71\]) and the results in table \[table:1\] the modular Hamiltonian associated to the region $\bar{\mathcal{A}}^+$ is given by $$\label{eq:72} \bar{K}_{\bar{\mathcal{A}}^+}= 2\pi \int_{S^{d-2}}d\Omega(\vec{x}_\perp) \int_{0}^{\bar{A}(\vec{x}_\perp)} d\alpha\,\alpha^{d-2} \left[ \frac{\alpha(\bar{A}(\vec{x}_\perp)-\alpha)} {\bar{A}(\vec{x}_\perp)} \right] \bar{T}_{\alpha \alpha} (\alpha,\vec{x}_\perp)\ .$$ Fixing $\bar{A}(\vec{x}_\perp)=R$, the space-time region $\mathcal{D\bar{A}}^+$ corresponds to the causal domain of a ball of radius $R$ centered at $t=-R$, whose modular Hamiltonian has been long known [@Hislop:1981uh; @Casini:2011kv]. For an arbitrary function $\bar{A}(\vec{x}_\perp)$ it gives the modular Hamiltonian associated to null deformations of the ball.[^19] This operator was previously considered in Ref. [@Casini:2017roe] but the result in that paper is incorrect, as can be seen by noting that it does not reproduce the correct result when $\bar{A}(\vec{x}_\perp)=R$.[^20] The integral in (\[eq:72\]) can also be written directly in terms of the space-time coordinates using that $r_-=2\alpha$ and $\vec{x}_\perp=\vec{v}$. For the complementary space-time region $\mathcal{D\bar{A}}^-$ we cannot write the modular Hamiltonian since the null surface $\bar{\mathcal{A}}^-$ does not fit inside Minkowski, see Fig. \[fig:6\]. This means we cannot write the full modular Hamiltonian $\hat{K}_{\bar{\mathcal{A}}^+}$ and derive a null energy bound from the monotonicity of relative entropy. An exception to this is given by the case of the ball where $\bar{A}(\vec{x}_\perp)=R$ implies $A(\vec{x}_\perp)=0$. As previously discussed, for this particular case the modular Hamiltonian becomes the Bisognano-Wichmann result, meaning that it can be written as a local integral over any Cauchy surface in the region $\mathcal{D\bar{A}}^-$. We can use this freedom to chose a surface which fits in Minkowski, starting from $\bar{A}(\vec{x}_\perp)=R$ (blue dot in second diagram of Fig. \[fig:6\]) and finishing at space-like infinity ${(\sigma/R,\theta)=(0,\pi)}$. Using this we can write the modular Hamiltonian corresponding to the complementary region of a ball in Minkowski, as done for example in Ref. [@Blanco:2013lea]. This analysis clarifies the validity of such expression. ### Lorentzian cylinder We now consider the transformation to the Lorentzian cylinder, where the null surface $\bar{\mathcal{A}}^+$ is written in the coordinates $u^\mu=(\sigma/R,\theta,\vec{v}\,)$ as $$\label{eq:73} \bar{\mathcal{A}}^+= \left\lbrace u^\mu\in \mathbb{R}\times [0,\pi]\times \mathbb{R}^{d-2}: \quad u^\mu= \left( -\beta,\beta,\vec{x}_\perp \right)\ , \quad (\beta,\vec{x}_\perp) \in [0,\bar{A}(\vec{x}_\perp)] \times \mathbb{R}^{d-2} \right\rbrace\ .$$ The entangling surface $\beta=\bar{A}(\vec{x}_\perp)\in[0,\pi]$ is given by the function $\bar{A}(\vec{x}_\perp)$, which can be written from the relation $\lambda(\beta)$ in table \[table:1\] as $A(\vec{x}_\perp)=p(\vec{x}_\perp)(2\cot(\bar{A}(\vec{x}_\perp))-1)$. The modular Hamiltonian is obtained from (\[eq:71\]) and table \[table:1\], so that we find $$\label{eq:75} \bar{K}_{\bar{\mathcal{A}}^+}= 2\pi \int_{S^{d-2}}d\Omega(\vec{x}_\perp) \int_0^{\bar{A}(\vec{x}_\perp)}d\beta\, \left(R\sin(\beta)\right)^{d-2} \left[ \frac{\sin(\beta)\sin(\bar{A}(\vec{x}_\perp)-\beta)}{\sin(\bar{A}(\vec{x}_\perp))}\right] \bar{T}_{\beta\beta}(\beta,\vec{x}_\perp)\ .$$ For $\bar{A}(\vec{x}_\perp)=\theta_0$ the region $\mathcal{D\bar{A}}^+$ corresponds to the causal domain of a cap region centered at the North Pole on the spatial sphere $S^{d-1}$ and agrees with the result obtained in [@Casini:2011kv]. The operator can be written in terms of the space-time coordinates using that $\theta_-=2\beta$ and $\vec{x}_\perp=\vec{v}$. Since the whole null surface fits in the cylinder, we can write the operator associated to the complementary region or equivalently, we can directly express the full modular Hamiltonian using (\[eq:160\]) as $$\label{eq:162} \hat{K}_{\mathcal{\bar{A}}^+}= 2\pi\int_{S^{d-2}}d\Omega(\vec{x}_\perp) \int_0^\pi d\beta\, \left(R\sin(\beta)\right)^{d-2} \left[ \frac{ \sin(\beta) \sin(\bar{A}(\vec{x}_\perp)-\beta)} {\sin(\bar{A}(\vec{x}_\perp))} \right] \bar{T}_{\beta\beta}(\beta,\vec{x}_\perp)\ .$$ From this we can explicitly write the constraint (\[eq:1\]) coming from relative entropy and obtain a bound on the null energy. Since the two regions are determined by the functions $\bar{A}(\vec{x}_\perp)$ and $\bar{B}(\vec{x}_\perp)$ the constraint becomes $$\label{eq:168} \int_0^{\pi}d\beta\,\sin^{d}(\beta) \int_{\mathbb{R}^{d-2}} d\vec{x}_\perp \left[ \frac{\cot(\bar{B}(\vec{x}_\perp))- \cot(\bar{A}(\vec{x}_\perp))} {p(\vec{x}_\perp)^{d-2}} \right] \bar{T}_{\beta \beta}(\beta,\vec{x}_\perp)\ge 0\ ,$$ where $\bar{A}(\vec{x}_\perp)\ge \bar{B}(\vec{x}_\perp)$ so that the condition for the regions in (\[eq:1\]) is satisfied. We have also written the surface element $d\Omega(\vec{x}_\perp)$ explicitly in terms of $p(\vec{x}_\perp)$. It is now convenient to fix the functions $\bar{A}(\vec{x}_\perp)$ and $\bar{B}(\vec{x}_\perp)$ to $$\label{eq:175} \bar{A}(\vec{x}_\perp)=\pi/2\ , \qquad {\rm and} \qquad \cot(\bar{B}(\vec{x}_\perp))= p(\vec{x}_\perp)^{d-2} \delta(\vec{x}_\perp-\vec{x}_\perp^{\,0})\ ,$$ where $\vec{x}_\perp^{\,0}$ is any fixed vector in $\mathbb{R}^{d-2}$. Although the condition for $\bar{B}(\vec{x}_\perp)$ involving the Dirac delta might seem unusual due to the cotangent function, $\bar{B}(\vec{x}_\perp)$ is determined by the original function $B(\vec{x}_\perp)$ from $B(\vec{x}_\perp)=p(\vec{x}_\perp)(2\cot(\bar{B}(\vec{x}_\perp))-1)$. The behavior of $\bar{B}(\vec{x}_\perp)$ implied by (\[eq:175\]) is qualitatively given by $$\bar{B}(\vec{x}_\perp)\sim \begin{cases} \quad \pi/2 \quad\ , \quad{\rm for\,\,} \vec{x}_\perp \neq \vec{x}_\perp^{\,0}\\ \quad \,\,\,0 \quad \,\,\,\, \ , \quad{\rm for\,\,} \vec{x}_\perp = \vec{x}_\perp^{\,0}\ . \end{cases}$$ Using this we can solve the integral in $\vec{x}_\perp$ in (\[eq:168\]) and find $$\label{eq:31} \int_0^\pi d\beta\,\sin^d(\beta) \, \bar{T}_{\beta\beta}(\beta,\vec{x}_\perp) \ge 0\ ,$$ where the affine parameter $\beta$ describes the geodesic in (\[eq:62\]). Up to a translation of the geodesic, this is equivalent to the constraint derived in the previous section (\[eq:195\]). ### De Sitter For de Sitter, the null surface $\bar{\mathcal{A}}^+$ is given in the $u^\mu=(\sigma/R,\theta,\vec{v}\,)$ coordinates by $$\label{eq:15} \mathcal{\bar{A}}^+= \left\lbrace u^\mu\in [-\pi,0]\times [0,\pi]\times \mathbb{R}^{d-2}: \quad u^\mu= \left( -\beta(\eta),\beta(\eta),\vec{x}_\perp \right)\ , \quad (\eta,\vec{x}_\perp)\in [\bar{A}(\vec{x}_\perp),+\infty)\times \mathbb{R}^{d-2} \right\rbrace\ ,$$ where $\eta(\beta)=\cot(\beta)$. The entangling surface $\eta=\bar{A}(\vec{x}_\perp)\in \mathbb{R}$ is obtained from the relation $\lambda(\eta)$ in table \[table:1\] as $A(\vec{x}_\perp)=p(\vec{x}_\perp)(2\bar{A}(\vec{x}_\perp)-1)$. Using (\[eq:71\]) and the results in table \[table:1\] we can write the associated modular Hamiltonian as $$\label{eq:24} \bar{K}_{\bar{\mathcal{A}}^+}= 2\pi R^{d-2} \int_{S^{d-2}}d\Omega(\vec{x}_\perp) \int_{\bar{A}(\vec{x}_\perp)}^{+\infty} d\eta\, \left( \eta-\bar{A}(\vec{x}_\perp) \right) \bar{T}_{\eta \eta} (\eta,\vec{x}_\perp)\ ,$$ which has a similar structure to that of the Minkowski null plane (\[eq:66\]). When $\bar{A}(\vec{x}_\perp)=0$ we have $\beta=\pi/2$ so that the space-time regions $\mathcal{D\bar{A}}^\pm$ correspond to the left and right static patches of de Sitter, see Fig. \[fig:6\]. For general $\bar{A}(\vec{x}_\perp)$ it is given by null deformations of these regions. Since the whole null surface fits inside de Sitter, we can write the modular Hamiltonian of the complementary region and therefore the full modular Hamiltonian, which from (\[eq:160\]) is given by $$\label{eq:164} \hat{K}_{\bar{\mathcal{A}}^+}= 2\pi R^{d-2}\int_{S^{d-2}} d\Omega(\vec{x}_\perp) \int_{-\infty}^{+\infty}d\eta\, \left( \eta-\bar{A}(\vec{x}_\perp) \right) \bar{T}_{\eta \eta}(\eta,\vec{x}_\perp)\ .$$ From this we can explicitly write the constraint (\[eq:1\]) coming from monotonicity of relative entropy. Taking the regions as determined by the two functions $\bar{A}(\vec{x}_\perp)$ and $\bar{B}(\vec{x}_\perp)$, the general inequality in (\[eq:1\]) implies $$\label{eq:26} \int_{-\infty}^{+\infty}d\eta \int_{{\mathbb{R}}^{d-2}} d\vec{x}_\perp\, \left[ \frac{\bar{B}(\vec{x}_\perp)- \bar{A}(\vec{x}_\perp)} {p(\vec{x}_\perp)^{d-2}} \right] \bar{T}_{\eta \eta}(\eta,\vec{x}_\perp)\ge 0\ ,$$ where $\bar{B}(\vec{x}_\perp)\ge \bar{A}(\vec{x}_\perp)$ so that the condition for the regions in (\[eq:1\]) is satisfied. We have also written the integral over $S^{d-2}$ explicitly in terms of $\vec{x}_\perp$. Fixing the regions such that $$\bar{A}(\vec{x}_\perp)=0\ , \qquad {\rm and} \qquad \bar{B}(\vec{x}_\perp)= p(\vec{x}_\perp)^{d-2} \delta(\vec{x}_\perp-\vec{x}_\perp^{\,0})\ ,$$ we can trivially solve the integral and obtain the ANEC for a CFT in de Sitter $$\mathcal{E}_{\rm dS}(\vec{x}_\perp)= \int_{-\infty}^{+\infty}d\eta\, \bar{T}_{\eta \eta}(\eta,\vec{x}_\perp)\ge 0\ ,$$ where the geodesic is given by (\[eq:203\]). ### Anti-de Sitter Finally let us consider the conformal transformation to AdS, where the null surface $\bar{\mathcal{A}}^+$ is written in the coordinates $u^\mu=(\sigma/R,\theta,\vec{v}\,)$ as $$\label{eq:158} \mathcal{\bar{A}}^+= \left\lbrace u^\mu\in \mathbb{R}\times [0,\pi/2]\times \mathbb{R}^{d-2}: \quad u^\mu= \left( -\beta(\zeta),\beta(\zeta),\vec{x}_\perp \right)\ , \quad (\zeta,\vec{x}_\perp)\in[0,\bar{A}(\vec{x}_\perp)]\times \mathbb{R}^{d-2} \right\rbrace\ ,$$ where $\zeta(\beta)=\tan(\beta)$ and $\bar{A}(\vec{x}_\perp)\in \mathbb{R}_{\ge 0}$ is obtained from the relation $\lambda(\zeta)$ in table \[table:1\] as $A(\vec{x}_\perp)=p(\vec{x}_\perp)\left(2/\bar{A}(\vec{x}_\perp)-1\right)$. The modular Hamiltonian associated to $\bar{\mathcal{A}}^+$ is obtained from (\[eq:71\]) and the results in table \[table:1\] $$\label{eq:18} \bar{K}_{\bar{\mathcal{A}}^+}= 2\pi R^{d-2} \int_{S^{d-2}} d\Omega^2(\vec{x}_\perp) \int_0^{\bar{A}(\vec{x}_\perp)} d\zeta\,\zeta^{d-2} \left[ \frac{\zeta(\bar{A}(\vec{x}_\perp)-\zeta)} {\bar{A}(\vec{x}_\perp)} \right] \bar{T}_{\zeta \zeta}(\zeta,\vec{x}_\perp)\ .$$ Notice that it has the same structure as the modular Hamiltonian on the deformed null cone (\[eq:72\]). If the function $\bar{A}(\vec{x}_\perp)$ is constant, the space-time region $\mathcal{D\bar{A}}^+$ corresponds to the causal domain of a ball in AdS. We can see this noting that the usual AdS radial coordinate $\rho$ in table \[table:2\] is given by $\rho=R\tan(\theta)$. Since the full null surface $\mathcal{\bar{N}}=\bar{\mathcal{A}}^+\cup \bar{\mathcal{A}}^-$ does not fit inside the whole AdS space-time we cannot write the full modular Hamiltonian and the constraint (\[eq:1\]) coming from relative entropy. This means that while the ANEC in dS can be derived from relative entropy, this is not true for AdS, as a consequence of the fact that the Minkowski null plane does not fit inside AdS. Entanglement entropy -------------------- Since we have derived some new modular Hamiltonians for CFTs in the Lorentzian cylinder and (A)dS, we would like to compute their associated entanglement entropy. In Ref. [@Casini:2018kzx] the entropy of the regions in the null plane and cone in Minkowski were computed using two independent approaches; the first one based on some symmetry considerations and the second on the HRRT holographic prescription [@Ryu:2006bv; @Hubeny:2007xt]. We follow the holographic approach since it is the simplest, although in future work it would be interesting to study the generalization of the other procedure. The details of the calculations are summarized in App. \[zapp:entanglement\]. The final result for the entanglement entropy can be written in every case as $$\label{eq:171} S=\frac{\mu_{d-2}}{\epsilon^{d-2}}+ \dots+ a_d^*\times \begin{cases} \displaystyle \frac{4(-1)^{\frac{d-2}{2}}} {{\rm Vol}(S^{d-2})} \int_{S^{d-2}} d\Omega(\vec{v}\,) \ln\left[ \frac{2R}{\epsilon}b_0(\vec{v}\,) \right]\ , \quad d{\rm \,\, even} \vspace{6pt}\\ \qquad \qquad \qquad \displaystyle 2\pi(-1)^{\frac{d-1}{2}} \qquad \qquad \quad \,\,\,\ , \quad \, d{\rm \,\, odd}\ , \end{cases}$$ where ${\rm Vol}(S^{d-1})=2\pi^{d/2}/\Gamma(d/2)$, $\epsilon$ is a short distance cut-off and $a_d^*$ is given by [@Nishioka:2018khk] $$\label{eq:155} a_d^*= \begin{cases} \qquad \qquad \,\,\,\,\, A_d \qquad \quad \,\,\,\, \ , & {\rm for\,\,d\,\,even} \vspace{6pt}\\ \,\,(-1)^{\frac{d-1}{2}}\ln[ Z(S^d)]/2\pi\ , & {\rm for\,\,d\,\,odd}\ . \\ \end{cases}$$ The coefficient of the Euler density in the stress tensor trace anomaly is given by $A_d$ (see Ref. [@Myers:2010tj] for conventions) while $Z(S^d)$ is the regularized vacuum partition function of the CFT placed on a unit $d$-dimensional sphere (see Ref. [@Pufu:2016zxm] for some examples in free theories). The entanglement entropy (\[eq:171\]) has a divergent expansion in $\epsilon$ with a leading area term, whose coefficient $\mu_{d-2}$ is non-universal (depends on the regularization procedure). The only universal term is indicated in (\[eq:171\]) and depends on the value of $d$. For odd space-times it is the same in every setup, while for even $d$ the function $b_0(\vec{v}\,)$ is given in each case by $$b_0(\vec{v}\,)= \begin{cases} \,\,\,\, \displaystyle \sin(\bar{A}(\vec{v}\,))\in(0,1] \,\,\,\,\ , \qquad {\rm Lorentzian\,\,cylinder}\\ \displaystyle \,\,\, 1/\bar{A}(\vec{v}\,)\in(0,+\infty) \,\,\,\ , \qquad\, {\rm de\,\, Sitter}\\ \displaystyle \quad \bar{A}(\vec{v}\,)\in(0,+\infty) \quad\ ,\, \qquad {\rm anti-de\,\, Sitter}\ . \end{cases}$$ We have indicated the range of $b_0(\vec{v}\,)$ given by the fact that the functions $\bar{A}(\vec{v}\,)$ are different in each setup, see the definition of the null surfaces above. Based on the arguments given in Ref. [@Casini:2018kzx], we expect this calculation for the entanglement entropy to hold to every order in the holographic CFT. Notice that for the case of de Sitter we have restricted $\bar{A}(\vec{v}\,)>0$ despite of the fact that the mapping of the null plane fits in the space-time for $\bar{A}(\vec{v}\,)\in \mathbb{R}$ (see (\[eq:15\]) and Fig. \[fig:6\]). The issue with $\bar{A}(\vec{v})\le 0$ is that the associated space-time region $\mathcal{D\bar{A}}^+$ lies outside of de Sitter. The entanglement entropy is a non-local quantity that captures this so that the holographic calculation breaks down in this regime, see App. \[zapp:entanglement\] for details. Wedge reflection positivity in curved backgrounds {#sec:ref_positivity} ================================================= In the previous sections we derived interesting bounds for the null energy along a complete geodesic for CFTs in (A)dS and the Lorentzian cylinder. We now want to investigate whether these results can be obtained from the causality arguments used in Ref. [@Hartman:2016lgu] to derive the Minkowski ANEC. One of the crucial ingredients in this proof from causality is the so called “Rindler positivity” or “wedge reflection positivity” (we use these terms interchangeably). This is a general property proved in Ref. [@Casini:2010bf] that implies the positivity of certain correlation functions in Minkowski. The aim of this section is to show that wedge reflection positivity generalizes to CFTs in dS and the Lorentzian cylinder, but not to AdS. Let us start by reviewing some general aspects of the Tomita-Takesaki theory [@Haag:1992hx; @Witten:2018lha] that is the central formalism used in this section. Given a QFT and a space-time region $W$ in Minkowski we can identify a Von Neumann algebra $\mathcal{W}$, given by all the bounded operators supported in $W$ that close under hermitian conjugation and the weak operator topology.[^21] From this algebra we can construct its commutant $\mathcal{W}'$, that is also a Von Neumann algebra formed by all the operators that commute with every element in $\mathcal{W}$. The Tomita-Takesaki theory starts by assuming that we can find a cyclic and separating vector $\ket{\psi}$ with respect to the Von Neumann algebra $\mathcal{W}$.[^22] For a particular choice of $\ket{\psi}$ and $\mathcal{W}$ we define the Tomita operator $S$ according to $$S\,\mathcal{O}\ket{\psi}= \mathcal{O}^\dagger \ket{\psi}\ , \qquad \forall \,\,\mathcal{O}\in \mathcal{W}\ .$$ Since $\ket{\psi}$ is cyclic this defines the action of $S$ on every vector of the Hilbert space. The Tomita operator can be written in terms of its polar decomposition as $S=J\Delta^{1/2}$ with $J$ anti-unitary and $\Delta^{1/2}$ hermitian and positive semi-definite. Moreover, since $S$ has an inverse $S^{-1}=S$, the choice of $J$ is unique and $\Delta^{1/2}$ is positive definite. The operator $J$ is called the modular conjugation and $\Delta$ the modular operator. Without too much effort, they can be shown to satisfy the following properties (*e.g.* see Ref. [@Witten:2018lha]) $$\label{eq:76} J=J^\dagger=J^{-1}\ , \qquad \quad J\Delta^{1/2} J=\Delta^{-1/2}\ , \qquad \quad J\ket{\psi}=\Delta\ket{\psi}=\ket{\psi}\ ,$$ where the definition of the hermitian conjugate for an anti-unitary operator is $\bra{\alpha}J\ket{\beta}=\bra{\beta}J^\dagger\ket{\alpha}$. The key properties satisfied by $J$ and $\Delta$ which amounts to the Tomita-Takesaki theorem are given by $$\label{eq:35} J\,\mathcal{W}J=\mathcal{W}'\ ,$$ $\Delta^{is}\mathcal{W}\Delta^{-is}=\mathcal{W}$ and $\Delta^{is}\mathcal{W}'\Delta^{-is}=\mathcal{W}'$ where $s\in \mathbb{R}$. The modular conjugation $J$ maps the algebra into its commutant, while $\Delta^{is}$ transforms each algebra into itself. Given $\mathcal{O}\in \mathcal{W}$ we define the “reflected” operator $\widetilde{\mathcal{O}}$ as $\widetilde{\mathcal{O}}\equiv J\mathcal{O}J \,\in \mathcal{W}'$. From this formalism follows a very general inequality which bounds the expectation value of $\widetilde{\mathcal{O}}\mathcal{O}$ in the state $\ket{\psi}$ $$\bra{\psi} \widetilde{\mathcal{O}} \mathcal{O} \ket{\psi}= \bra{\psi} \mathcal{O} \Delta^{1/2}S\mathcal{O} \ket{\psi}^*= \bra{\psi} \mathcal{O} \Delta^{1/2}\mathcal{O}^\dagger \ket{\psi}= \bra{\alpha} \Delta^{1/2}\ket{\alpha}\ ,$$ where we have define $\ket{\alpha}=\mathcal{O}^\dagger\ket{\psi}$. Using that $\Delta^{1/2}$ is positive definite we arrive at the central inequality $$\label{eq:34} \bra{\psi}\widetilde{\mathcal{O}}\mathcal{O}\ket{\psi}>0\ , \qquad \mathcal{O}\in \mathcal{W}\ .$$ For a generic setup the reflected operator $\widetilde{\mathcal{O}}$ is related to $\mathcal{O}$ in a very complicated way. The only certainty we have regarding $\widetilde{\mathcal{O}}$ is that it is in the commutant algebra of $\mathcal{W}$, which follows from the Tomita-Takesaki theorem (\[eq:35\]). This means that extracting useful information from (\[eq:34\]) might be very challenging. There is however a particular setup in which the action of $J$ becomes simple enough. Taking the Minkowski space-time coordinates $X^\mu=(T,X,\vec{Y})$, consider the right Rindler wedge $$\label{eq:80} W=\left\lbrace X^\mu \in \mathbb{R}\times \mathbb{R}\times \mathbb{R}^{d-2}: \quad X_\pm=X\pm T>0 \right\rbrace\ .$$ For the Von Neumann algebra associated to this wedge and the Minkowski vacuum state $\ket{0}$,[^23] Bisognano and Wichmann [@Bisognano:1976za] proved that the modular operator $\Delta$ is given by $\Delta=e^{-\hat{K}_W}$, where $\hat{K}_W$ is the full modular Hamiltonian defined in (\[eq:161\]), which can be written as $$\label{eq:172} \hat{K}_W= 2\pi \int_{\mathcal{N}_{\rm plane}} dS\,\lambda\,T_{\lambda \lambda}(\lambda,\vec{x}_\perp)\ ,$$ where the integral is over the full null plane in (\[eq:50\]) with $dS=d\vec{x}_\perp d\lambda$. Moreover, they showed that the modular conjugation $J$ is obtained from the consecutive discrete transformations $J={\rm CRT}$, where the operators ${\rm T}$ and ${\rm R}$ reflect the coordinates $T$ and $X$ respectively while ${\rm C}$ implements charge conjugation. Starting from a QFT that is invariant under the Poincare group without assuming invariance under any discrete symmetry, it can be shown that the vacuum is invariant under the combination ${\rm CRT}$, *i.e.* ${\rm CRT}\ket{0}=\ket{0}$. The proof is analogous to the ${\rm CPT}$ theorem for $d=4$, see the discussion in Refs. [@Witten:2018lha; @Manoharan1969]. This gives a very simple description of the modular conjugation $J$, whose action on an arbitrary operator $\mathcal{O}_a(X^\mu)$ of integer spin $\ell$ is given by [@Casini:2010bf] $$\label{eq:89} J\mathcal{O}_a(X^\mu)J= \frac{\partial \widetilde{X}^b} {\partial X^a} \mathcal{O}_b^\dagger(\widetilde{X}^\mu)\ , \qquad {\rm with} \qquad \widetilde{X}^\mu(X^\mu)=(-T,-X,\vec{Y})\ ,$$ where we are using the notation $a=(\nu_1,\dots,\nu_\ell)$ and the jacobian matrix is written in the convention (\[eq:82\]).[^24] If the operator $\mathcal{O}_a(X^\mu)$ is inserted in the right wedge $W$, the action of $J$ translates it to the complementary region $W'$, the left wedge $$W'=\left\lbrace X^\mu \in \mathbb{R}\times \mathbb{R}\times \mathbb{R}^{d-2}: \quad X_\pm=X\pm T<0 \right\rbrace\ .$$ For this reason, we call the geometric action $\widetilde{X}^\mu(X^\mu)$ a reflection. Using this we can explicitly write the general inequality (\[eq:34\]) and obtain Rindler positivity as derived in Ref. [@Casini:2010bf] $$\label{eq:90} (-1)^P \bra{0} \mathcal{O}^\dagger_{\mu_1\dots\mu_\ell} (\widetilde{X}^\mu) \mathcal{O}_{\nu_1\dots\nu_\ell}(X^\mu) \ket{0}>0\ , \qquad \qquad X^\mu \in W\ ,$$ where $P$ is the number of $T$ indices plus $X$ indices. Although we have only written the expression for a single operator this property holds for an arbitrary number of operators, where notice that the order of the reflected operator is not inverted, *i.e.* $\widetilde{\mathcal{O}_1\mathcal{O}_2}=\widetilde{\mathcal{O}}_1\widetilde{\mathcal{O}}_2$. Moreover, since the expectation values of operators in Lorentzian signature are not functions but distributions, this is a constraint on a distribution. Conformal transformation of Tomita operator ------------------------------------------- The strategy for generalizing (\[eq:90\]) is simple. Using the conformal transformations discussed in Sec. \[sec:null\_energy\_bounds\] we can map the Tomita operator, explicitly write the general inequality (\[eq:34\]) and obtain wedge reflection positivity in these curved backgrounds. Consider a generic conformal transformation implemented in the space-time by a change of coordinates $z^\mu(X^\mu)$ that maps the right Rindler wedge in Minkowski $W$ to some other region $\bar{W}$ in the new space-time. The transformation of the Hilbert space is implemented by a unitary operator $U:\mathcal{H}\rightarrow \bar{\mathcal{H}}$, so that the algebra $\mathcal{W}$ is mapped by the adjoint action of $U$ according to $U\mathcal{W}U^\dagger=\bar{\mathcal{W}}$. Using that $\mathcal{W}$ is a Von Neumann algebra it is straightforward to show that this is also true for $\bar{\mathcal{W}}$. Although every local operator in $W$ is mapped to a local operator in $\bar{W}$ under the action of $U$, only primary operators have a simple transformation law. The vacuum state $\ket{0}\in \mathcal{H}$ is mapped to $U\ket{0}=\ket{\bar{0}}\in \bar{\mathcal{H}}$ which can be shown to be cyclic and separating with respect to $\bar{\mathcal{W}}$, using that this is true for $\ket{0}$ and $\mathcal{W}$. This means we can construct the Tomita operator $\bar{S}$ associated to $\ket{\bar{0}}$ and the algebra $\bar{\mathcal{W}}$ in the usual way $$\bar{S}\bar{\mathcal{O}}\ket{\bar{0}}= \bar{\mathcal{O}}^\dagger \ket{\bar{0}}\ , \qquad \qquad \bar{\mathcal{O}}\in \bar{\mathcal{W}}\ .$$ The mapped Tomita operator $\bar{S}$ is related to $S$ in the Rindler wedge through the adjoint action of $U$, so that the mapped modular operator and conjugation are given by $$\bar{\Delta}= \exp\big( -U\hat{K}_WU^\dagger \big) \ , \qquad \qquad \bar{J}=U({\rm CRT})U^\dagger\ ,$$ where $\hat{K}_W$ is the boost generator in (\[eq:172\]). The mapping of the modular operator $\Delta$ is completely determined by the transformation of the full modular Hamiltonian $\hat{K}_W$. Since we already analyzed the mapping of this operator in Sec. \[sec:null\_energy\_bounds\] we focus on the modular conjugation.[^25] The action of the modular conjugation $\bar{J}$ can be found by applying $U$ to the ${\rm CRT}$ action in (\[eq:89\]). If we restrict to bosonic primary operators $\mathcal{O}_a(X^\mu)$ and use that they transform according to (\[eq:88\]), we find $$\label{eq:46} \bar{J} \mathcal{\bar{O}}_a(z^\mu) \bar{J}= \left| \frac{w(z^\mu)}{w(\tilde{z}^\mu)} \right|^{\Delta-\ell} \frac{\partial\tilde{z}^b} {\partial z^a} \bar{\mathcal{O}}_b^\dagger(\tilde{z}^\mu)\ , \qquad {\rm with} \qquad \tilde{z}^\mu=z^\mu(\widetilde{X}^\mu)\ ,$$ where we used that the jacobian matrix is invertible since this is true for the conformal mapping. The action of $\bar{J}$ is similar to that of ${\rm CRT}$, since the local operator inserted at $z^\mu$ is geometrically reflected to $\tilde{z}^\mu$. However, notice that (\[eq:46\]) only holds for primary operators while the action of ${\rm CRT}$ in (\[eq:89\]) is for arbitrary operators. From this we can write the general positivity inequality (\[eq:34\]) coming from the Tomita-Takesaki theory and find $$\label{eq:107} \frac{\partial \tilde{z}^c}{\partial z^b} \bra{\bar{0}} \bar{\mathcal{O}}_c^\dagger(\tilde{z}^\mu) \bar{\mathcal{O}}_a(z^\mu) \ket{\bar{0}}>0\ , \qquad \bar{\mathcal{O}}_a:{\rm primary}\ , \quad z^\mu \in \bar{W}\ .$$ This gives a positivity constraint on the correlators of the mapped CFT that is analogous to Rindler positivity in (\[eq:90\]). In the following, we explicitly write this for CFTs in the Lorentzian cylinder and de Sitter and show that it can be expressed as in (\[eq:90\]). Before moving on, let us note that $\bar{J}$ gives an interesting discrete symmetry of the vacuum ${\bar{J}\ket{\bar{0}}=\ket{\bar{0}}}$ which might not be evident from first principles. In particular, it relates two point functions of primary operators according to $$\label{eq:108} \bra{\bar{0}} \mathcal{\bar{O}}_a(z_1) \mathcal{\bar{O}}_b(z_2) \ket{\bar{0}}= \left| \frac{w(z_1)w(z_2)} {w(\tilde{z}_1)w(\tilde{z}_2)} \right|^{\Delta-\ell} \left( \frac{\partial \tilde{z}_1^c} {\partial z_1^a} \frac{\partial \tilde{z}_2^d} {\partial z_2^b} \right) \bra{\bar{0}} \bar{\mathcal{O}}_c(\tilde{z}_1) \bar{\mathcal{O}}_d(\tilde{z}_2) \ket{\bar{0}}\ .$$ This gives a simple non-trivial way of checking our calculations. Lorentzian cylinder ------------------- Let us start by considering the conformal transformation relating Minkowski to the Lorentzian cylinder. Using a more rigorous approach, the mapping of the Tomita operator under this transformation was analyzed in Ref. [@Hislop:1981uh] for a massless scalar and more generally in Ref. [@Brunetti:1992zf] for an arbitrary CFT. As a first step, consider the special conformal transformation in (\[eq:3\]) with the slight modification $D^\mu=(R,R,\vec{0}\,)\rightarrow (0,R,\vec{0}\,)$. The right Rindler wedge $W$ in (\[eq:80\]) is mapped to the causal domain of a ball of radius $R$ centered at the origin of the $x^\mu=(t,x,\vec{y}\,)$ coordinates [@Haag:1992hx] $$\mathcal{DB} =\left\lbrace x^\mu \in \mathbb{R}\times \mathbb{R}\times \mathbb{R}^{d-2}: \quad R-\sqrt{x^2+|\vec{y}\,|^2}>|t| \right\rbrace\ .$$ The mapping of the ${\rm CRT}$ operator is characterized by the geometric reflection $\tilde{x}^\mu=x^\mu(-T,-X,\vec{Y})$, that from the change of coordinates in (\[eq:3\]), can be easily found to be given by $$\label{eq:36} \tilde{x}^\mu(x^\mu)= \frac{R^2}{(x\cdot x)} x_\rho \delta^{\rho \mu}\ ,$$ where $(x\cdot x)=\eta_{\mu \nu}x^\mu x^\nu$ and $x_\mu=(-t,x,\vec{y})$. As first noted in Ref. [@Hislop:1981uh] this corresponds to the composition of an inversion $x^\mu \rightarrow R^2x^\mu/(x\cdot x)$ with a time reflection $t\rightarrow -t$, meaning that the ${\rm CRT}$ operator is mapped to $$U({\rm CRT})U^\dagger= ({\rm CIT})\ ,$$ where ${\rm I}$ is the inversion operator. In appendix \[zapp:Mod\_conj\] we show that the discrete transformation ${\rm CIT}$ is part of the Euclidean conformal group in the same way as ${\rm CRT}$ belongs to the Euclidean Poincare group. The action of ${\rm CIT}$ on a primary operator of integer spin $\ell$ can be obtained from (\[eq:46\]) using that[^26] $$\label{eq:91} \frac{w(x^\mu)} {w(\tilde{x}^\mu)}= \frac{R^2}{|(x\cdot x)|}\ , \qquad \qquad \frac{\partial \tilde{x}^\mu } {\partial x^\nu}= \frac{R^2}{(x\cdot x)} \left[ \eta_{\rho \nu}-\frac{2x_\rho x_\nu}{(x\cdot x)} \right]\delta^{\rho\mu}\ .$$ Let us analyze the geometric action of ${\rm CIT}$ in the causal domain of the ball, which is supposed to give the modular conjugation $\bar{J}$. To do so it is convenient to write the reflection transformation in (\[eq:36\]) in terms of the null radial coordinates $r_\pm=r\pm t$, which gives $$\label{eq:32} \tilde{r}_\pm(r_\pm)= \frac{R^2}{r_\pm}\Theta(r_+r_-)- \frac{R^2}{r_\mp}\Theta(-r_+r_-)\ .$$ Since this transformation is discontinuous and not well defined in the future and past null cone ${(x\cdot x)=r_+r_-=0}$, there are three regions in $\mathcal{DB}=A\cup B\cup C$ where $\tilde{r}_\pm$ acts in a distinct way (depending on the sign of $r_\pm$). In the left diagram of Fig. \[fig:10\] we plot the three regions and their behavior under the ${\rm CIT}$ transformation in the $(t,r)$ plane. ![On the left we have a diagram of the $(t,r)$ Minkowski plane with the causal domain of the ball $\mathcal{DB}=A\cup B \cup C$, where each region is marked with a different color and vertical lines. The reflected regions under the action of $\tilde{r}_\pm$ in (\[eq:32\]) are marked with horizontal lines and the corresponding colors. On the right we have the mapping of the same regions but from the perspective of the Lorentzian cylinder with coordinates $(\sigma/R,\theta)$.[]{data-label="fig:10"}](Modular_conjugation.pdf){height="2.8"} The immediate observation is that $\widetilde{\mathcal{DB}}=\tilde{A}\cup \tilde{B}\cup\tilde{C}$ is a disconnected space-time region. This is problematic for the action of the modular conjugation since according to the Tomita-Takesaki theorem (\[eq:35\]), $\bar{J}$ should map the algebra to its commutant. The regions $\tilde{B}$ and $\tilde{C}$ are causally connected to $\mathcal{DB}$, meaning that operators with support in $\mathcal{DB}$ and $\widetilde{\mathcal{DB}}$ do not commute with each other. Altogether this means that the mapping of the modular conjugation $J$ under this conformal transformation fails. The origin of the problem is the same as the one discussed in Sec. \[sec:null\_energy\_bounds\]: special conformal transformations are not well defined in Minkowski but on its conformal compactification, the Lorentzian cylinder. To obtain a well defined action for the modular conjugation $\bar{J}$, we must apply another mapping that takes the ${\rm CIT}$ operator to the cylinder. We can do this by using the conformal transformation in table \[table:2\], which we slightly modify by introducing the constant $\theta_0\in[0,\pi]$ according to $$\label{eq:112} r_\pm(\theta_\pm)= R\frac{\tan(\theta_\pm/2)} {\tan(\theta_0/2)} \qquad \Longrightarrow \qquad w=\frac{\cot(\theta_0/2)} {2\cos(\theta_+/2)\cos(\theta_-/2)}\ ,$$ where $w$ is the conformal factor and $\theta_\pm=\theta\pm \sigma/R$ are the null coordinates in the cylinder (\[eq:170\]). The advantage of introducing $\theta_0$ is that $\theta_\pm=\theta_0$ corresponds to the boundary of $\mathcal{DB}$, so that the causal domain of the ball is mapped to the region in the cylinder $$\label{eq:109} \bar{W}_{\theta_0}=\left\lbrace (\sigma/R,\theta,\vec{v}\,) \in \mathbb{R}\times[0,\pi] \times \mathbb{R}^{d-2}: \quad |\theta_\pm|<\theta_0 \right\rbrace\ .$$ Although the space-time region is given by the causal domain of a cap of size $\theta_0$ around the North pole, the region in parameter space $(\sigma/R,\theta)$ is given by a wedge, see right diagram in Fig. \[fig:10\]. We now need to obtain the mapping of $\bar{W}_{\theta_0}$ under the reflection transformation induced by ${\rm CIT}$ in (\[eq:36\]). One way of doing this is using the change of coordinates in (\[eq:10\]), which take into account that a single Minkowski copy does not cover the entire cylinder. Although this is certainly possible, it is technically and conceptually more clear to take a different route based on the embedding formalism of the conformal group. In appendix \[zapp:Mod\_conj\] we use this to show that the geometric action of the modular conjugation $\bar{J}$ in the cylinder is given by the following relation $$\label{eq:110} \tan(\tilde{\theta}_\pm/2)= \tan^2(\theta_0/2)\cot(\theta_\pm/2)\ .$$ This transformation leaves the wedge $\theta_\pm=\theta_0$ fixed and if we apply it to $\bar{W}_{\theta_0}$ in (\[eq:109\]) we find $$\label{eq:111} \bar{W}_{\theta_0}'= \left\lbrace (\sigma/R,\theta,\vec{v}\,) \in \mathbb{R}\times[0,\pi] \times \mathbb{R}^{d-2}: \quad |\theta_\pm|>\theta_0 \right\rbrace\ .$$ We plot the transformation $\bar{W}_{\theta_0}\rightarrow \bar{W}_{\theta_0}'$ in the right diagram of Fig. \[fig:10\]. The reflection in the cylinder is exactly what we could have guessed: it reflects across a wedge in parameter space obtained by splitting the cylinder at $\theta=\theta_0$. From Fig. \[fig:10\] we see that the issues that arise from the action of ${\rm CIT}$ in Minkowski are resolved from the perspective of the cylinder. The space-time regions $\bar{W}_{\theta_0}=A\cup B\cup C$ and $\bar{W}_{\theta_0}'=\tilde{A}\cup \tilde{B}\cup \tilde{C}$ are the causal complements of each other, as required for the action of the modular conjugation $\bar{J}$ by the Tomita-Takesaki theory (\[eq:35\]). The transformation in (\[eq:110\]) can only be explicitly solved when we split the cylinder in two wedges of equal size, *i.e.* $\theta_0=\pi/2$ $$\label{eq:113} \tilde{\theta}_\pm(\theta_\pm)\big|_{\theta_0=\pi/2}= \pi-\theta_\pm\ .$$ For $\theta_0\neq \pi/2$ the transformation is non-linear, as expected by the fact that it relates wedges of different sizes. We can still solve (\[eq:110\]) numerically and plot it in Fig. \[fig:11\], where we explicitly see its non-linear behavior. ![Plot of the reflected coordinate $\tilde{\theta}_\pm$ as a function of $\theta_\pm$ for several values of $\theta_0$. Only for $\theta_0=\pi/2$ (purple line in the center) the transformation is linear.[]{data-label="fig:11"}](Reflection_cylinder.pdf){height="2.0"} Now that we understand the mapping of the Tomita operator to the cylinder we can write the general inequality (\[eq:107\]) and obtain wedge reflection positivity. To do so, let us first analyze the action of the modular conjugation $\bar{J}$ on primary operators, which can be obtained from the general relation (\[eq:46\]). The conformal factor appearing in this expression is the one relating the Minkowski coordinates $X^\mu$ to the cylinder, which is given by the product of (\[eq:91\]) with (\[eq:112\]), so that we find $$\label{eq:173} \left| \frac{w(\theta_\pm)} {w(\tilde{\theta}_\pm)} \right|= \left| \frac{R^2}{r_+r_-}\times \frac{\cos(\tilde{\theta}_+/2) \cos(\tilde{\theta}_-/2)} {\cos(\theta_+/2)\cos(\theta_-/2)} \right|= f(\theta_+)f(\theta_-)\ ,$$ where in the second equality we have used (\[eq:112\]) and (\[eq:110\]) and defined $f(\theta_\pm)$ as $$\label{eq:174} f(\theta_\pm)= \frac{\tan(\theta_0/2)} {\cos(\theta_\pm/2) \sqrt{\tan^2(\theta_\pm/2)+\tan^4(\theta_0/2)}} \ge 0\ .$$ This is non-negative since $\theta_0\in[0,\pi]$ and $\theta_\pm\in[-\pi,\pi]$ for $\bar{W}$. When the wedges are of equal size $\theta_0=\pi/2$, this function equals to one. The Jacobian matrix associated to the reflection transformation (\[eq:110\]) can be written in terms of the space-time coordinates $v^\mu=(\theta_+,\theta_-,\vec{v}\,)$ using that the only non-trivial components are given by $$\label{eq:204} \frac{\partial \tilde{\theta}_\pm} {\partial \theta_\pm}=-f(\theta_\pm)^2\ .$$ Using all this in (\[eq:46\]) we can explicitly write the action of the modular conjugation on a primary field of integer spin $\ell$. Moreover, the general positivity relation (\[eq:107\]) becomes $$\label{eq:211} (-1)^P\bra{\bar{0}} \bar{\mathcal{O}}^\dagger_{\mu_1\dots \mu_\ell}(\tilde{v}^\mu) \bar{\mathcal{O}}_{\nu_1\dots \nu_\ell} (v^\mu) \ket{\bar{0}}>0\ , \qquad v^\mu \in \bar{W}_{\theta_0}\ ,$$ where $P$ is the sum of $\theta_+$ indices plus $\theta_-$ indices. This proves the wedge reflection positivity of correlators in the Lorentzian cylinder. It is somewhat more interesting that Rindler positivity given that for $\theta_0\neq \pi/2$ the reflection transformation $\tilde{\theta}_\pm(\theta_\pm)$ is non-linear. As a simple check of our calculations we can verify the validity of the identity (\[eq:108\]) implied by $\bar{J}\ket{\bar{0}}=\ket{\bar{0}}$. Using that the two point function of scalar primary operators of scaling dimension $\Delta$ in the cylinder is given by[^27] $$\bra{\bar{0}} \bar{\mathcal{O}}(v_1^\mu) \bar{\mathcal{O}}(v_2^\mu) \ket{\bar{0}}= \left| 4R^2 \sin(\Delta \theta_+/2) \sin(\Delta \theta_-/2) \right|^{-\Delta},$$ it is straightforward to check that (\[eq:108\]) holds for arbitrary values of $\theta_0\in[0,\pi]$. De Sitter --------- The generalization to a CFT in de Sitter space-time is straightforward, since the conformal mapping is just given by the Weyl rescaling in (\[eq:154\]). Since we keep the same space-time coordinates, the geometric action of the modular conjugation is still given by (\[eq:110\]). However, the value of $\theta_0$ is restricted to $\theta_0=\pi/2$, since for other values one of the wedges in the dS diagram of Fig. \[fig:11\] necessarily lies outside of de Sitter. The modular conjugation in dS is then characterized by $\tilde{\theta}_\pm(\theta_\pm)=\pi-\theta_\pm$, which corresponds to a reflection between the left and right de Sitter static patches. Using that $f(\theta_\pm)\big|_{\theta_0=\pi/2}=1$ and the expressions (\[eq:173\]) and (\[eq:204\]) we can explicitly write the action of the modular conjugation $\bar{J}$ on any bosonic primary operator from (\[eq:46\]).[^28] Moreover, the wedge reflection positivity in de Sitter (\[eq:107\]) is given by $$\label{eq:205} (-1)^P\bra{\bar{0}} \bar{\mathcal{O}}^\dagger_{\mu_1\dots \mu_\ell}(\tilde{v}^\mu) \bar{\mathcal{O}}_{\nu_1\dots \nu_\ell} (v^\mu) \ket{\bar{0}}>0\ , \qquad v^\mu \in \bar{W}_{\theta_0=\pi/2}\ ,$$ where $P$ is the sum of $\theta_+$ indices plus $\theta_-$ indices. Discussion and future directions {#sec:Discussion} ================================ In this work we derived the ANEC for general CFTs in (A)dS and a similar novel bound for the Lorentzian cylinder. By thoroughly studing the connection of these conditions with the previous derivations of the Minkowski ANEC in Refs. [@Faulkner:2016mzt; @Hartman:2016lgu; @Kelly:2014mra] we have obtained other useful technical results. This includes null deformed modular Hamiltonians and their associated entanglement entropies in Sec. \[sec:null\_energy\_bounds\], as well as an extension of Rindler positivity to curved backgrounds in Sec. \[sec:ref\_positivity\]. Let us comment on some future research directions that would be interesting to pursue. #### ANEC in (A)dS beyond conformal theories: Since our derivation of these conditions relies heavily on conformal symmetry, a natural question is whether they can be extended to general quantum field theories. For de Sitter, following Refs. [@Faulkner:2016mzt; @Hartman:2016lgu] would require to show that the full modular Hamiltonian (\[eq:164\]) or the wedge reflection positivity (\[eq:205\]) are still true beyond CFTs. Since our methods used to derive both of these results rely on conformal symmetry, one would require more powerful tools to do so. For the AdS case, we have seen that both aproaches used in Refs. [@Faulkner:2016mzt; @Hartman:2016lgu] fail even for CFTs, which suggests that a general proof of the ANEC in AdS calls for a completely new procedure. #### ANEC in AdS from holography: In App. \[zapp:ANEC\_holography\] we have shown how the ANEC in de Sitter can be derived for holographic CFTs described by Einstein gravity. The bound for the CFT in the cylinder (\[eq:195\]) has also been recently obtained through this method in Ref. [@Iizuka:2019ezn] for $d=3,4,5$. This suggests it might be possible to extend the holographic proof to the AdS case, although we have seen some examples where the generalization of certain results to AdS is delicate and does not work. #### Vacuum susbstracted ANEC in the cylinder: We have shown that a CFT in the Lorentzin cylinder satisfies the novel bound given in (\[eq:195\]). Although we have stressed that this condition is not equivalent to the ANEC, it is still possible that the vacuum substracted ANEC is a true statement of QFTs defined in the cylinder. For the particular case of a free scalar in $\mathbb{R}\times S^{1}$ this was explicitly shown in Ref. [@Ford:1994bj]. In future work it would be interesting to explore other methods that could allow to extend this to more general setups.[^29] #### Constraint on higher spin operators: In the causality proof of the Minkowski ANEC in Ref. [@Hartman:2016lgu] the following positivity constraint for higher spin null integrated operators was derived $$\label{eq:178} \mathcal{E}^{(\ell)}(\vec{x}_\perp)\equiv \int_{-\infty}^{+\infty}d\lambda\, T^{(\ell)}_{\lambda\dots \lambda}(\lambda,\vec{x}_\perp) \ge 0\ ,$$ where $T^{(\ell)}_{\mu_1\dots \mu_\ell}$ is the lowest dimension operator of even spin $\ell\ge 2$ and $(\lambda,\vec{x}_\perp)$ are the coordinates in the null plane (\[eq:50\]). Applying the conformal transformations in Sec. \[sec:ANEC\_mapping\] we can obtain the analogous constraint in (A)dS. Since $T^{(\ell)}_{\mu_1\dots \mu_\ell}$ is a primary operator it transforms in similar way to $T_{\lambda \lambda}$ in (\[eq:5\]) $$U T^{(\ell)}_{\lambda\dots \lambda}U^\dagger= |w_{\rm (A)dS}(\vec{x}_\perp)|^{\ell-\Delta} \bar{T}^{(\ell)}_{\lambda\dots \lambda}\ .$$ Integrating over $\lambda\in \mathbb{R}$, the left hand-side becomes (\[eq:178\]) and we get $$U \mathcal{E}^{(\ell)}(\vec{x}_\perp)U^\dagger= |w_{\rm (A)dS}(\vec{x}_\perp)|^{\ell-\Delta} \int_{-\infty}^{+\infty}d\lambda\, \bar{T}^{(\ell)}_{\lambda\dots \lambda} (\lambda,\vec{x}_\perp)\ ,$$ where we have used that the conformal factors $w_{\rm (A)dS}(\vec{x}_\perp)$ in (\[eq:197\]) are independent of $\lambda$. Since $\lambda$ is an affine parameter in (A)dS, the higher spin Minkowski ANEC (\[eq:178\]) implies the analogous constraint for (A)dS. The geodesics are given in (\[eq:188\]) where in the AdS case $\vec{x}_\perp$ is constrained to $|\vec{x}_\perp|<R$, so that the curves lie in the space-time. A completely analogous calculation using (\[eq:191\]) and (\[eq:190\]) also generalizes the bound obtained for the Lorentzian cylinder $$U \mathcal{E}^{(\ell)}(\vec{x}_\perp)U^\dagger \propto \int_{-\pi/2}^{\pi/2} d\bar{\lambda} \left(\cos(\bar{\lambda})\right)^{\Delta+\ell-2}\, \bar{T}^{(\ell)}_{\bar{\lambda}\dots \bar{\lambda}} (\bar{\lambda},\vec{x}_\perp)\ge 0\ ,$$ where the proportionality constant is positive for $\ell$ even. For the cylinder and de Sitter it should be possible to derive these higher spin constraints using the wedge reflection positivity proved in Sec. \[sec:ref\_positivity\]. Moreover, it would be interesting to analyze the generalization of these conditions to continuous spin, as obtained for Minkowski in Ref. [@Kravchuk:2018htv]. #### Witt algebra in de Sitter: In Ref. [@Casini:2017roe] it was shown that it is possible to define some null integrated operators in the Minkowski null plane which satisfy the Witt algebra. More precisely, the operators[^30] $$L^{(n)}(\vec{x}_\perp)= i\int_{-\infty}^{+\infty}d\lambda\, \lambda^{n+1}\, T_{\lambda \lambda}(\lambda,\vec{x}_\perp)\ ,$$ where shown to satisfy the following algebra $$\label{eq:179} \left[ L^{(n)}(\vec{x}_\perp), L^{(m)}(\vec{y}_\perp) \right]=(n-m) \delta(\vec{x}_\perp-\vec{y}_\perp) L^{(n+m)}(\vec{x}_\perp)\ .$$ We can apply the conformal transformation of Sec. \[sec:null\_energy\_bounds\] from Minkowski to dS, so that using (\[eq:5\]) the operators $L^{(n)}(\vec{x}_\perp)$ transform as $$UL^{(n)}(\vec{x}_\perp)U^\dagger= \frac{R^{d-2}}{p(\vec{x}_\perp)^{d-2}} i\int_{-\infty}^{+\infty}d\lambda\, \lambda^{n+1}\bar{T}_{\lambda \lambda}(\lambda,\vec{x}_\perp)= \frac{R^{d-2}}{p(\vec{x}_\perp)^{d-2}} L_{\rm dS}^{(n)}(\vec{x}_\perp)\ ,$$ where $p(\vec{x}_\perp)=(|\vec{x}_\perp|+4R^2)/4R$ and we have defined $L_{\rm dS}^{(n)}(\vec{x}_\perp)$ in terms of $\lambda$, which is affine in de Sitter. Using this in (\[eq:179\]), the operators $L_{\rm dS}^{(n)}(\vec{x}_\perp)$ satisfy the following algebra $$\left[ L_{\rm dS}^{(n)}(\vec{x}_\perp), L_{\rm dS}^{(m)}(\vec{y}_\perp) \right]= (n-m) \left[ \frac{p(\vec{x}_\perp)^{d-2}} {R^{d-2}}\delta(\vec{x}_\perp-\vec{y}_\perp) \right] L_{\rm dS}^{(n+m)}(\vec{x}_\perp)\ .$$ The term between square brackets in the right hand side is nothing more than the Dirac delta associated to the induced metric in the null surface in de Sitter, see table \[table:1\]. Hence, the operators $L_{\rm dS}^{(n)}(\vec{x}_\perp)$ in this surface also satisfy the Witt algebra. It would be interesting to further explore this in the context of the calculations in Refs. [@Cordova:2018ygx; @Huang:2019fog]. #### Entanglement entropy beyond holography: In appendix \[zapp:entanglement\] we computed the entanglement entropy associated to the null deformed regions in the Lorentzian cylinder and (A)dS using AdS/CFT. Although these results are valid to all orders in the boundary CFT, it would be instructive to recover the same expressions directly in field theory. One way of doing so is by applying a similar approach as the one used in Ref. [@Casini:2017roe] to compute the entanglement entropy associated to the null plane and cone in Minkowski. #### Other conformally related space-times: In this work we have focused on the conformal transformations relating Minkowski, the Lorentzian cylinder and (A)dS. However, Ref. [@Candelas:1978gf] describes some additional space-times that are connected through conformal mappings which might be interesting to further explore. For instance, for a CFT in $\mathbb{R}\times \mathbb{H}^{d-1}$, with $\mathbb{H}$ a hyperbolic plane, one could use similar methods to compute both the modular Hamiltonian and associated entanglement entropy of null deformed regions. #### Negative energy in large $d$ limit: The energy condition obtained for the CFT in the Lorentzian cylinder (\[eq:195\]) has a very interesting behavior in the large space-time dimension limit, where it gives a local constraint on the null projection of the stress tensor (\[eq:169\]). This suggest that the study of negative energy in this regime might give some interesting insights. To our knowledge, the large $d$ limit of negative energy in QFT has not been systematically investigated in the literature. Since we have not been able to completely determine the limit in (\[eq:195\]) this is an interesting result that deserves further study. #### Wedge reflection positivity and entropy inequalities: In Sec. \[sec:ref\_positivity\] we derived the Rindler positivity for CFTs in the Lorentzian cylinder and de Sitter, the case of the cylinder being particularly interesting since the transformation is non-linear. Following a similar approach as in Refs. [@Casini:2010nn; @Blanco:2019gmt] it would be interesting to explore the consequences of these properties regarding entanglement entropy inequalities. Comment on the QNEC ------------------- The quantum null energy condition (QNEC) is a local constraint on the null projection of the stress tensor that has recently attracted much interest [@Bousso:2015mna]. For a general QFT in Minkowski the QNEC has been proven in Ref. [@Balakrishnan:2017bjg] and more interestingly in Ref. [@Ceyhan:2018zfg], where it was shown to follow from the Minkowski ANEC. The results of this paper raise the question of whether there is a similar connection to be made between the conditions in (A)dS. To do so let us first review the statement of the QNEC in Minkowski from the perspective of relative entropy. Consider the relative entropy between the vacuum $\sigma=\ket{0}\bra{0}$ and an arbitrary state $\rho$ reduced to null deformations of the Rindler region. Using that the modular Hamiltonian is given by (\[eq:66\]), the relative entropy (\[eq:180\]) can be written as $$\label{eq:206} S(\rho||\sigma)= 2\pi \int_{\mathbb{R}^{d-2}} d\vec{x}_\perp \int_{A(\vec{x}_\perp)}^{+\infty}d\lambda\, \left( \lambda-A(\vec{x}_\perp) \right) \big[ \langle T_{\lambda \lambda} \rangle_\rho - \langle T_{\lambda \lambda} \rangle_{\ket{0_M}} \big] - \big[ S(\rho) -S(\ket{0_M}) \big]\ ,$$ where $S(\rho)$ and $S(\ket{0_M})$ are the entanglement entropy of each state reduced to the null deformed region. Now let us consider a one parameter family of deformations labeled by $\kappa$ and given by $A(\vec{x}_\perp;\kappa)=A(\vec{x}_\perp)+\kappa \dot{A}(\vec{x}_\perp)$ with $\dot{A}(\vec{x}_\perp)\ge 0$. The QNEC in Minkowski can be formulated as the statement that the second derivative of the relative entropy with respect to $\kappa$ is positive $\partial^2_{\kappa}S(\rho||\sigma)\ge 0$. The derivative of (\[eq:206\]) can be further simplified using that $\langle T_{\lambda \lambda} \rangle_{\ket{0_M}}$ vanishes since Minkowski is a maximally symmetric space-time (see discussion around (\[eq:181\])). Furthermore, some symmetry considerations regarding Minkowski and the null plane given in Ref. [@Casini:2018kzx] show that the vacuum entanglement entropy $S(\ket{0_M})$ is independent of $A(\vec{x}_\perp)$. Altogether, the QNEC in Minkowski is given by $$\label{eq:182} \frac{d^2}{d\kappa^2} S(\rho||\sigma)\ge 0 \qquad \Longleftrightarrow \qquad 2\pi \int_{\mathbb{R}^{d-2}} d\vec{x}_\perp \dot{A}(\vec{x}_\perp)^2 \langle T_{\lambda \lambda} \rangle_\rho \ge \frac{d^2}{d\kappa^2} S(\rho)\ .$$ This was proven for general QFTs in Refs. [@Balakrishnan:2017bjg; @Ceyhan:2018zfg]. The local version of the bound is obtained by taking $\dot{A}(\vec{x}_\perp)^2=\delta(\vec{x}_\perp-\vec{x}_\perp^{\,0})$. Let us now discuss the case of de Sitter. The first thing we might try is to directly map the inequality on the right of (\[eq:182\]) by applying the conformal transformation from Minkowski to dS discussed in Sec. \[sec:null\_energy\_bounds\]. Using the transformation property of the stress tensor $T_{\lambda \lambda}$ in (\[eq:5\]) and the conformal factor (\[eq:197\]) we can map the left-hand side of the inequality and find $$\label{eq:208} 2\pi R^{d-2} \int_{S^{d-2}} d\Omega(\vec{x}_\perp) \dot{A}(\vec{x}_\perp)^2 \langle \bar{T}_{\lambda \lambda} \rangle_{\bar{\rho}} \ge \frac{d^2}{d\kappa^2} S(\rho)\ ,$$ where $d\Omega(\vec{x}_\perp)=d\vec{x}_\perp/p(\vec{x}_\perp)^{d-2}$ and $\bar{\rho}=U\rho \,U^\dagger$. The mapping of the right-hand side is more complicated since it involves the entanglement entropy. Although the entanglement entropy in quantum mechanics is invariant under a unitary transformation, this is not true in QFTs given that the entropy requires a cut-off $\epsilon$ which transforms in a non-trivial way. To our knowledge there is no standard general prescription for the transformation of the entanglement entropy. For the particular case of holographic theories dual to Einstein gravity, Ref. [@Koeller:2015qmn] obtained some interesting results by using some earlier observations from Ref. [@Graham:1999pm]. Applying this to the mapping of Minkowski to de Sitter in Sec. \[sec:null\_energy\_bounds\], their results suggest that the transformation of the right-hand side of (\[eq:208\]) is given by $$\label{eq:207} \frac{d^2}{d\kappa^2} S(\rho) \qquad \longrightarrow \qquad \frac{d^2}{d\kappa^2} \big[ S(\bar{\rho})-S(\ket{0_{\rm dS}}) \big]\ ,$$ where $S(\bar{\rho})$ is the entropy of the the mapped state $\bar{\rho}$ in the null deformed region of dS. A first argument supporting (\[eq:207\]) is that it implies the saturation of the QNEC in de Sitter when evaluated in the vacuum $\ket{0_{\rm dS}}$, which we expect to be true given that it is in Minkowski. If we did not have the vacuum substraction in (\[eq:207\]) the QNEC would not saturate given that the vacuum entanglement entropy of de Sitter (\[eq:171\]) has a non-trivial dependence on the entangling surface. Another argument in favor of (\[eq:207\]) comes from relative entropy. Using the modular Hamiltonian in dS (\[eq:24\]), we can explicitly write the the relative entropy between the states $\bar{\rho}$ and $\bar{\sigma}=\ket{0_{\rm dS}}\bra{0_{\rm dS}}$ and take its second derivative with respect to $\kappa$, so that we find $$\frac{d^2}{d\kappa^2} S(\bar{\rho}||\bar{\sigma})\ge 0 \qquad \Longleftrightarrow \qquad 2\pi R^{d-2} \int_{S^{d-2}}d\Omega(\vec{x}_\perp) \dot{A}(\vec{x}_\perp)^2 \langle \bar{T}_{\lambda \lambda}\rangle_{\bar{\rho}} \ge \frac{d^2}{d\kappa^2} \big[ S(\bar{\rho})-S(\ket{0_{\rm dS}}) \big]\ .$$ To obtain this, we have written the modular Hamiltonian (\[eq:24\]) in terms of the affine parameter $\lambda$ using $\lambda(\eta)=p(\vec{x}_\perp)(2\eta-1)$ from table \[table:1\]. The negativity of the second derivative of the relative entropy in dS implies precisely the same transformation property of the entropy given in (\[eq:207\]). For the other space-times and surfaces studied in this paper, the treatment becomes more obscure. For AdS we have the issue that the mapping of the whole null plane does not fit inside the space-time, so that the conformal transformation of (\[eq:182\]) becomes even more ambiguous. Moreover, the QNEC is obtained from the quantum focusing conjecture [@Bousso:2015mna] applied to a point $p$ and a hypersurface orthogonal surface that is locally stationary through $p$. A straightforward computation of the expansion of the null congruence of each surface considered in Sec. \[sec:null\_energy\_bounds\], show that this is only true for the case of de Sitter. This is also evident when computing the relative entropy from the modular Hamiltonians in Sec. \[sec:null\_energy\_bounds\]. Since the operators in AdS (\[eq:18\]) and the Lorentzian cylinder (\[eq:75\]) have a much more complicated structure, their second derivative with respect to $\kappa$ is not as simple as in (\[eq:182\]). I thank Clifford V. Johnson for comments on the manuscript and the organizers of TASI 2019 where I learned many of the tools used in this paper. This work is partially supported by DOE grant DE-SC0011687. ANEC in de Sitter from holography {#zapp:ANEC_holography} ================================= In this section we give a proof of the ANEC for a holographic conformal field theory in de Sitter, dual to Einstein gravity. We follow the approach of Ref. [@Kelly:2014mra], where the Minkowski ANEC was derived under the assumption that the gravity dual has good causal properties. More precisely, the assumption is that for two boundary points connected by a boundary null geodesic, there is no causal curve (*i.e.* time-like or null) through the bulk which travels faster than the boundary null geodesic. General features of bulk AdS with de Sitter boundary ---------------------------------------------------- Let us start by discussing some general notions regarding AdS/CFT and asymptotically AdS$_{d+1}$ space-time. An asymptotically AdS space-time can be written in Fefferman-Graham coordinates as $$\label{eq:138} ds^2=(L/z)^2\left[ dz^2+g_{\mu \nu}(z,v)dv^\mu dv^\nu \right]\ ,$$ where the AdS radius is $L$, the boundary is at $z=0$ and $z>0$ corresponds to the bulk interior. The $d$-dimensional metric $g_{\mu \nu}(z,v)$ admits an expansion in powers of $z$ given by [@deHaro:2000vlm] $$g_{\mu \nu}(z,v)= g_{\mu \nu}^{(0)}(v) +z^2g_{\mu \nu}^{(2)}(v)+ \dots+z^d \ln(z^2/L^2) h_{\mu \nu}(v) + z^dg_{\mu \nu}^{(d)}(v) +o(z^{d})\ ,$$ where $h_{\mu\nu}$ is non-zero only for even $d$ and $o(z^d)$ means terms that vanish strictly faster than $z^d$. The first term in this expansion $g_{\mu \nu}^{(0)}$ gives the space-time in which the boundary CFT is defined. Since in this case we are interested in a de Sitter background, we have from (\[eq:154\]) $$\label{eq:146} g_{\mu \nu}^{(0)}dv^\mu dv^\nu= \frac{R^2} {\sin^2\left[ (\theta_+-\theta_-)/2\right]} \left[ d\theta_+d\theta_- +\sin^2\left(\theta\right) d\Omega^2(\vec{v}\,) \right] \ ,$$ where $v^\mu=(\theta_+,\theta_-,\vec{v}\,)$ with the null coordinates $\theta_\pm=\theta\pm \sigma/R$. We have written dS with the conformal factor $\sin^2(\sigma/R)$ so that $\sigma/R\in[-\pi,0]$. The higher order terms $h_{\mu \nu}$ and $g_{\mu \nu}^{(n)}(v)$ with $n<d$ can be obtained by perturbately solving Einstein’s equations. They are all written in terms of geometric quantities built from the boundary metric $g_{\mu \nu}^{({\rm dS})}$ [@deHaro:2000vlm], *i.e.* they are a complicated functions of the Riemann, Ricci and curvature tensor of $g_{\mu \nu}^{({\rm dS})}$ and their covariant derivatives. For instance, the first order term is given by $$\label{eq:150} g_{\mu \nu}^{(2)}= \frac{1}{d-2}\left[ \frac{\mathcal{R}}{2(d-1)}g_{\mu \nu}^{({\rm dS})}- \mathcal{R}_{\mu \nu} \right]\ ,$$ where the Ricci tensor and scalar on the right-hand side are computed from the metric $g_{\mu \nu}^{({\rm dS})}$. Given that in this particular case we are considering a de Sitter boundary, we can use the fact that it is maximally symmetric, so that the Riemann tensor is completely fixed by the metric $$\mathcal{R}_{\mu \nu\rho \sigma}= \frac{1}{R^2}\left( g_{\mu \rho}g_{\nu \sigma}- g_{\mu \sigma}g_{\nu \rho} \right) \ .$$ From this we see that (\[eq:150\]) is proportional to the boundary metric ${g_{\mu \nu}^{(2)}=-g_{\mu \nu}^{({\rm dS})}/(2R^2)}$.[^31] The powerful observation is that this is true for all the higher order terms $h_{\mu \nu}$ and $g^{(n)}_{\mu \nu}$ with $n<d$.[^32] Although the actual proportionality constants $m_n$ cannot be computed for arbitrary $d$, it will be enough to use that they are proportional $$h_{\mu \nu}=m_dg_{\mu \nu}^{({\rm dS})}\ , \qquad \qquad g_{\mu \nu}^{(n)}=m_ng_{\mu \nu}^{({\rm dS})}\ , \qquad n<d \ .$$ Using this, we can write any asymptotically AdS metric with a de Sitter boundary as $$\label{eq:144} ds^2=(L/z)^2\left[ dz^2+ \left(m(z) g_{\mu \nu}^{({\rm dS})}(v) +z^dg_{\mu \nu}^{(d)}(v) \right) dv^\mu dv^\nu+ o(z^{d}) \right]\ ,$$ where the function $m(z)$ satisfies $m(z=0)=1$ and is determined from the coefficients $m_n$ $$m(z)=1+m_2z^2+\dots+m_dz^d\ln(z^2/L^2)\ .$$ This expansion to order $o(z^{d})$ will be enough for our purposes. The higher order terms are determined by the particular state in the boundary CFT. The first undetermined contribution $g_{\mu\nu}^{(d)}$ is related to the expectation value of the stress tensor of the dual CFT according to the standard AdS/CFT dictionary $$\label{eq:143} \langle T_{\mu \nu} \rangle= \frac{dL^{d-1}}{16\pi G} g^{(d)}_{\mu \nu}(v)+X_{\mu \nu}[g^{(n<d)}]\ ,$$ where $X_{\mu \nu}$ gives the anomalous term of the stress tensor in the CFT and $G$ is Newton’s constant. Although in a general setup $X_{\mu \nu}$ is a functional of $g_{\mu \nu}^{(n)}$ with $n<d$, we can use the same observation as before to conclude that the anomalous terms is also proportional to the boundary metric ${X_{\mu \nu}=x_dg_{\mu \nu}^{({\rm dS})}}$. If we project the stress tensor along the null direction $\theta_-$, the anomalous terms drops out and we find $$\label{eq:148} \langle T_{--} \rangle= \frac{dL^{d-1}}{16\pi G} g_{--}^{(d)}(v)\ , \qquad \qquad g_{--}^{(d)}= \frac{dv^\mu }{d\theta_-} \frac{dv^\nu }{d\theta_-} g_{\mu \nu}^{(d)}\ .$$ Curve ansatz and no bulk shortcut --------------------------------- Let us now describe the setup that will allow us to obtain the ANEC. Consider a null geodesic in the boundary moving along the $\theta_-$ direction $$\label{eq:136} v^\mu(\theta_-)= (0,\theta_-,\vec{v}\,)\ , \qquad \theta_-\in[\pi-\theta_0,\pi +\theta_0]\ ,$$ where $\vec{v}$ is fixed and the null tangent vector is given by $(0,1,\vec{0}\,)$. The parameter $\theta_0\in[0,\pi]$ determines the initial and final points of the geodesic. For $\theta_0=\pi$ the geodesic is complete, going from the South pole of de Sitter at past infinity to the North at future infinity, while for $\theta_0=0$ it is a single point. In the left diagram of Fig. \[fig:13\] we sketch this curve in blue in the $(\sigma/R,\theta)$ plane. Although $\theta_-$ is not an affine parameter in dS, it is convenient to describe the geodesic in this way. We now wish to construct a bulk curve which starts at the same point as (\[eq:136\]) at the boundary, goes into the bulk and ends in some other point at the boundary (not necessarily the same one as (\[eq:136\])). Consider the curve given by $$x^A(\theta_-)= (z,v^\mu)= ( f_z(\theta_-),f_+(\theta_-), \theta_-,\vec{v}\, )\ , \qquad \theta_-\in[\pi-\theta_0,\pi +\theta_0]\ ,$$ which has a tangent vector equal to $$\label{eq:151} \frac{dx^A}{d\theta_-}= (f'_{z}(\theta_-),f'_+(\theta_-),1,\vec{0}\,)= (f'_{z}(\theta_-),k^\mu(\theta_-))\ .$$ The functions $f_z(\theta_-)$ and $f_+(\theta_-)$ must satisfy the following boundary conditions $$\label{eq:139} f_z(\pi\pm \theta_0)=0 \ , \qquad \qquad f_+(\pi+\theta_0)=0\ ,$$ which ensures that the bulk curve behaves in the way we just described. A sketch of two bulk curves in red and green are shown in the left diagram of Fig. \[fig:13\]. ![On the left we have a diagram of the setup in the $(\sigma/R,\theta)$ plane, with the shaded blue region corresponding to de Sitter. The boundary curve is shown in blue, while in green and red we show to different bulk curves going out of the page which satisfy the boundary conditions (\[eq:139\]). The no bulk shortcut property implies that if these bulk curves are causal, the red trajectory is forbidden. On the right we plot our ansatz for the function $f_z(\theta_-)$ which shows how the bulk curve goes into the bulk for different values of $\epsilon L$. As $\epsilon\rightarrow 0$ the depth of the curve decreases and since $\theta_0=\pi-\epsilon^{d-1}$, its range in the boundary goes to $\theta_-\in[0,2\pi]$.[]{data-label="fig:13"}](Bulk_Causality.pdf){height="2.4"} The final position of this curve in the boundary is determined from $\theta_+^{\rm final}=f_+(\pi-\theta_0)$. The no bulk shortcut property is the statement that there is no bulk causal curve $x^A(\theta_-)$ whose end point at the boundary is at the past of the end point of (\[eq:136\]). More concretely, it implies the following $$\label{eq:140} {\rm If\,\,\,} g_{AB}\, \frac{dx^A} {d\theta_-} \frac{dx^B} {d\theta_-}\le 0 \qquad \Longrightarrow \qquad \theta_+^{\rm final}= f_+(\pi-\theta_0)\ge 0\ ,$$ where $g_{AB}$ is the full bulk metric given in (\[eq:144\]). This forbids a causal curve as the red one shown in the left diagram of Fig. \[fig:13\]. Violation of the no bulk shortcut property would result in causality and locality problems of the boundary theory (see Refs. [@Kelly:2014mra; @Iizuka:2019ezn; @Gao:2000ga; @Witten:2019qhl] for related discussions). The strategy is to construct a particular causal bulk curve given in (\[eq:151\]), such that the no bulk shortcut property (\[eq:140\]) gives the ANEC for the boundary theory. From the expansion of the bulk metric in (\[eq:144\]), the curve is causal as long as it satisfies the following constraint $$\label{eq:145} (f'_z(\theta_-))^2+ \left[m(f_z(\theta_-)) g_{\mu \nu}^{({\rm dS})}(\theta_-) +f_z(\theta_-)^dg_{\mu \nu}^{(d)}(\theta_-) \right] k^\mu(\theta_-)k^\nu(\theta_-)+ o(z^{d})\le 0 \ ,$$ where $k^\mu(\theta_-)$ is given in (\[eq:151\]). We will consider a particular bulk curve whose maximum depth in the bulk is given by $\epsilon L$ with $\epsilon$ a dimensionless quantity, and expand to leading order in $\epsilon\ll 1$. This curve must satisfy the boundary conditons in (\[eq:139\]) to every order in $\epsilon$ as well as the causality constraint (\[eq:145\]) to leading order. Our ansatz is inspired by the calculations in Refs. [@Kelly:2014mra; @Iizuka:2019ezn]. For the function giving the $z$ coordinates $f_z(\theta_-)$ we choose $$f_z(\theta_-)= \epsilon L\left( \frac{\tan(\theta_0/2)-|\cot(\theta_-/2)|} {\tan(\theta_0/2)} \right)\ .$$ We plot this in the right diagram of Fig. \[fig:13\] for several values of $\epsilon$. The function is positive in the range of $\theta_-\in (\pi-\theta_0,\pi+\theta_0)$ and vanishes at the end points, so that it satisfies the boundary conditions (\[eq:139\]). The maximum depth of the curve into the bulk is given by $\epsilon L$. To obtain the ANEC we relate the parameter $\theta_0$ to $\epsilon$ according to $$\label{eq:141} \theta_0=\pi-\epsilon^{d-1}\ .$$ The limit of $\epsilon\ll 1$ then corresponds to a bulk curve near the boundary which covers a complete null geodesic in de Sitter (\[eq:136\]), see Fig \[fig:13\]. If we expand for small $\epsilon$ we find $$f_z(\theta_-)= \epsilon- \frac{|\cot(\theta_-/2)|}{2}\epsilon^d+ \mathcal{O}(\epsilon^{2d})\ .$$ From this we see that the function $f_z(\theta_-)$ is of order $\epsilon$ while its derivative goes like $\epsilon^d$. This is one of the crucial properties of the ansatz, since it ensures that the first positive term in the causality constraint (\[eq:145\]) is subleading in the $\epsilon$ expansion. For the remaining function $f_+(\theta_-)$ we consider the following $$f_+(\theta_-)= \frac{\epsilon^d}{R^2} \int_{\theta_-}^{\pi+\theta_0} d\theta'_- \sin^2(\theta_-'/2) g_{--}^{(d)}(0,\theta_-',\vec{v}\,)+ Q(\theta_-) \left( \frac{\tan(\theta_0/2)+\cot(\theta_-/2)} {\tan(\theta_0/2)} \right)\epsilon^{d+\delta} \ ,$$ where $Q(\theta_-)$ is any regular function and $\delta$ a small positive number. This function satisfies the boundary condition in (\[eq:139\]) since it vanishes at the initial point $\theta_-=\pi+\theta_0$. From this, we can write the tangent vector for the boundary components $k^\mu(\theta_-)$ in (\[eq:151\]) in an expansion in $\epsilon$ as $$\label{eq:147} k^\mu(\theta_-)= \frac{dv^\mu }{d\theta_-}+\left[ - \frac{\sin^2(\theta_-/2)}{R^2} g_{--}^{(d)}(0,\theta_-,\vec{v}\,) \epsilon^d+ Q'(\theta_-)\epsilon^{d+\delta}\right] (1,1,\vec{0}\,) + \mathcal{O}(\epsilon^{d+1})\ .$$ Now that we have a bulk curve which satisfies the boundary conditions in (\[eq:139\]), we check that it is causal to leading order in $\epsilon$, *i.e.* that it satisfies (\[eq:145\]). Expanding this constraint we find $$\frac{R^2 m(f_z(\theta_-))} {\sin^2\left[ (f_+(\theta_-)-\theta_-)/2 \right]} \left[ -\frac{\sin^2(\theta_-/2)}{R^2} g_{--}^{(d)}(0,\theta_-,\vec{v}\,) \epsilon^d+ Q'(\theta_-)\epsilon^{d+\delta}\right] +\epsilon^d g_{--}^{(d)}(0,\theta_-,\vec{v}\,) + o(\epsilon^{d})\le 0 \ ,$$ where we have used that the de Sitter metric is given by (\[eq:146\]) so that the contraction of the term $du^\mu/d\theta_-$ vanishes. Using that the function $m(f_z(\theta_-))=1+\mathcal{O}(\epsilon^2)$ and expanding the sine in the denominator we find $$\left[ -g_{--}^{(d)}(0,\theta_-,\vec{v}\,) \epsilon^d+ \frac{R^2Q'(\theta_-)} {\sin^2\left(\theta_-/2\right)} \epsilon^{d+\delta}\right] +\epsilon^d g_{--}^{(d)}(0,\theta_-,\vec{v}\,) + o(\epsilon^{d})\le 0 \ .$$ The leading order in $\epsilon^d$ involving the metric $g_{--}^{(d)}$ cancels and the first non-vanishing contribution is given by $\epsilon^{d+\delta}$. Recall that $o(\epsilon^d)$ means terms that vanish strictly faster than $\epsilon^d$. This means that for any fixed bulk space-time (corresponding to a state in the boundary CFT) we can fix $\delta>0$ to be small enough so that it is the leading contribution in $\epsilon$ when compared to the unknown terms $o(\epsilon^d)$. In this way, the causality constraint reduces to the following condition on the function $Q(\theta_-)$ $$\label{eq:149} Q'(\theta_-)\le 0\ .$$ By fixing this function such that it satisfies this property we are guaranteed to have a causal curve. Now that we have constructed the bulk causal curve, we can investigate the consequences of imposing the no bulk shortcut property in (\[eq:140\]). Writing this explicitly we find $$\theta_+^{\rm final}= \frac{\epsilon^d}{R^2} \int_{\pi-\theta_0}^{\pi+\theta_0} d\theta'_- \sin^2(\theta_-'/2) g_{--}^{(d)}(0,\theta_-',\vec{v}\,)+ 2Q(\pi-\theta_0)\epsilon^{d+\delta} \ge 0\ .$$ Since the bulk curve is causal only in the limit of $\epsilon\ll 1$ we must expand in $\epsilon$. Doing so, and using that the boundary stress tensor is related to $g_{--}^{(d)}$ according to (\[eq:148\]) we find $$\int_{0}^{2\pi} d\theta'_- \sin^2(\theta_-'/2) \langle T_{--}(0,\theta_-',\vec{v}\,) \rangle \ge - \left( \frac{2R^2dL^{d-1}}{16\pi G} \right) \lim_{\epsilon\rightarrow 0}\left[ Q(\epsilon^{d-1})\epsilon^{\delta}+ \dots \right]\ .$$ There are three possibilities for the value of the function $Q(\theta_-)$ as $\theta_-\rightarrow 0$. The least interesting case is when it diverges to $+\infty$ faster that $\epsilon^\delta$ goes to zero so that the bound becomes trivial. On the contrary, if it diverges to $-\infty$ then the causality condition $Q'(\theta_-)\le 0$ in (\[eq:149\]) is not verified and the curve is not causal. The most interesting case is when $Q(\theta_-)$ goes to a constant value, so that the right hand side vanishes and we obtain a non-trivial condition given by $$\int_{0}^{2\pi} d\theta'_- \sin^2(\theta_-'/2) \langle T_{--}(0,\theta_-',\vec{v}\,) \rangle \ge 0\ .$$ This is actually the ANEC in de Sitter, as can be seen by remembering that the parameter $\theta_-$ is not affine. If we change the integration variable to an affine parameter $\eta(\theta_-)=\cot(\theta_-/2)$ (see table \[table:1\] noting that $\beta=\theta_-/2$), we obtain $$\mathcal{E}_{\rm dS}(\vec{v}\,)= \int_{-\infty}^{+\infty} d\eta\, T_{\eta \eta}(\eta,\vec{v}\,)\ge 0 \ .$$ Entanglement entropy of null deformed regions {#zapp:entanglement} ============================================= In this appendix we compute the entanglement entropy associated to the modular Hamiltonians obtained in Sec. \[sec:null\_energy\_bounds\]. The case of the null plane and null cone in Minkowski space-time have already been considered in Ref. [@Casini:2018kzx]. Using a similar approach we obtain explicit expressions for the entanglement entropy of null deformed regions associated to the Lorentzian cylinder and (A)dS. Review: Minkowski null cone --------------------------- Let us start by considering the entanglement entropy of the vacuum associated to an arbitrary surface in the null cone of Minkowski, given in (\[eq:120\]) and following the holographic calculation in Ref. [@Casini:2018kzx]. Since the global state in the CFT is the Minkowski vacuum, we must consider pure AdS in Poincare coordinates $$\label{eq:132} ds^2= \left(L/z\right)^2 \left(dz^2 -dt^2+dr^2+r^2d\Omega^2(\vec{v}\,) \right)\ ,$$ where $L$ is the AdS radius and $d\Omega^2(\vec{v})$ is the metric on a unit sphere $S^{d-2}$ parametrized by stereographic coordinates $\vec{v}\in \mathbb{R}^{d-2}$ (\[eq:152\]). According to the HRRT prescription [@Ryu:2006bv; @Hubeny:2007xt], the entanglement entropy is obtained from the area of the extremal bulk surface that intersects with the boundary $z= 0$ on the entangling surface of (\[eq:120\]). Since the surface lies on the null cone, it is convenient to define null coordinates in the bulk $\hat{r}_\pm=\hat{r}\pm t$ obtained from $$\label{eq:39} z=\hat{r}\sin(\psi)\ , \qquad \qquad r=\hat{r}\cos(\psi)\ ,$$ where $\hat{r}\ge 0$ and $\psi \in[0,\pi/2]$. The AdS metric in these coordinates becomes $$\label{eq:2} ds^2= \left( \frac{L}{\hat{r}\sin(\psi)} \right)^2 \left( -dt^2+d\hat{r}^2+\hat{r}^2 \left(d\psi^2+ \cos^2(\psi)d\Omega^2(\vec{v}\,)\right) \right)\ ,$$ so that the $d$-dimensional Minkowski boundary is located at $\psi\rightarrow 0$. Since in this limit $\hat{r}\rightarrow r$, the coordinates $\hat{r}_\pm$ become the null coordinates in Minkowski $r_\pm=r\pm t$. Moreover, given that the entangling surface in the CFT (\[eq:120\]) is located at $(r_+,r_-)=(0,2\bar{A}(\vec{v}\,))$, the boundary condition for the bulk extremal surface can be easily written as $$\label{eq:125} \lim_{\psi\rightarrow 0} \left(\hat{r}_+,\hat{r}_-,\vec{v}\,\right)= \left(0,2\bar{A}(\vec{v}\,),\vec{v}\,\right)\ ,$$ where $\bar{A}(\vec{v}\,)>0$. To obtain the entanglement entropy we must find the extremal codimension two surface subject to this constraint. This was computed exactly in Refs. [@Neuenfeld:2018dim; @Casini:2018kzx] where it was shown to satisfy $\hat{r}_+=0$ not only at the boundary but at every point in the bulk. This means that the area of the extremal surface is obtained from the induced metric (\[eq:2\]) at $\hat{r}_+=0$ $$\label{eq:128} \left.ds^2\right|_{\hat{r}_+=0}= L^2\left( \frac{d\psi^2+ \cos^2(\psi)d\Omega^2(\vec{v}\,)} {\sin^2(\psi)} \right)\ .$$ Using that the entropy is related to the area according to $S=2\pi {\rm Area}/\ell_p^{d-1}$, the entanglement entropy is given by $$\label{eq:121} S= \frac{4\pi a_d^*}{{\rm Vol}(S^{d-1})} \int_{S^{d-2}} d\Omega^2(\vec{v}\,) \int_{0}^{\pi/2} d\psi \frac{\cos^{d-2}(\psi)}{\sin^{d-1}(\psi)}\ ,$$ where we have conveniently defined the factor $a^*_d$ in Einstein gravity according to $$2a_d^*={\rm Vol}(S^{d-1})(L/\ell_p)^{d-1} \qquad {\rm with} \qquad {\rm Vol}(S^{d-1})=2\pi^{d/2}/\Gamma(d/2)\ .$$ For the boundary CFT this factor is mapped to (\[eq:155\]). Note that the integral in (\[eq:121\]) seems to be insensitive to the details of the extremal surface (given by $\hat{r}_-(\psi,\vec{v}\,)$) and therefore independent of the entangling surface $\bar{A}(\vec{v})$. However, this is not true since the extremal surface plays a role in regulating the integral in (\[eq:121\]), which diverges in the limit of $\psi\rightarrow 0$. To reproduce a divergent field theory quantity through a bulk computation we must be careful about the choice of the cut-off, since different regularizations yield distinct results. For instance, if we regulate (\[eq:121\]) with $\psi_{\rm min}=\epsilon$ we incorrectly conclude that the entanglement entropy in the null cone is independent of the entangling surface. A field theory computation shows that this is incorrect [@Casini:2018kzx]. The appropriate cut-off is dictated by holographic renormalization [@Skenderis:2002wp], in which we must first write the bulk metric in Fefferman-Graham coordinates $$\label{eq:133} ds^2=(L/z)^2\left[ dz^2+g_{\mu \nu}(z,x^\mu)dx^\mu dx^\nu \right]\ ,$$ where the boundary is described by $x^\mu$ and located at $z\rightarrow 0$. The metric $g_{\mu \nu}(z,x^\mu)$ admits an expansion in $z$ given by $g_{\mu \nu}(z,x^\mu)=g_{\mu \nu}^{(0)}(x^\mu)+z^2g_{\mu \nu}^{(2)}(x^\mu)+\dots$, where $g_{\mu\nu}^{(0)}(x^\mu)$ corresponds to the space-time in which the boundary CFT is defined. The appropriate cut-off $\epsilon$ is obtained from the $z$ coordinate according to $z_{\rm min}=\epsilon$. In this case, the AdS metric as written in (\[eq:132\]) is already in Fefferman-Graham coordiantes. We can relate $\psi$ to $z$ using (\[eq:39\]), so that the cut-off $\epsilon$ is given by $$\label{eq:123} z_{\rm min}=\epsilon= \sin(\psi) \hat{r}_-(\psi,\vec{v}\,)/2\ ,$$ where we have evaluated at the extremal surface $\hat{r}_+=0$ and $\hat{r}_-(\psi,\vec{v}\,)$. To compute the entanglement entropy in terms of $\epsilon$, we must invert this relation to get an expansion for $\psi(\epsilon,\vec{v}\,)$ and solve the integral in (\[eq:121\]). The function $\hat{r}_-(\psi,\vec{v}\,)$ is determined from the details of the extremal surface and has some expansion near the boundary as $\psi\rightarrow 0$ $$\label{eq:126} \hat{r}_-(\psi,\vec{v}\,)= 2\bar{A}(\vec{v})+b_1(\vec{v}\,)\psi+ b_2(\vec{v}\,)\psi^2+\dots\ ,$$ where the first order term is fixed by the boundary condition (\[eq:125\]) and the coefficients $b_i(\vec{v}\,)$ determine the higher order contributions. They can be obtained from the exact expressions of the extremal surface given in Refs. [@Neuenfeld:2018dim; @Casini:2018kzx]. Using this in (\[eq:123\]) we can invert the relation and find the expansion for $\psi(\epsilon,\vec{v}\,)$ $$\label{eq:124} \psi(\epsilon,\vec{v}\,)= \frac{\epsilon}{\bar{A}(\vec{v}\,)} -\frac{b_1(\vec{v}\,)}{2\bar{A}(\vec{v}\,)^3} \epsilon^2+\dots \ .$$ With this expression we regulate the integral (\[eq:121\]) and obtain the entanglement entropy. As usual, the entanglement entropy is dominated by a divergent area term and subleading contributions. We only compute the universal terms, *i.e.* contributions that are independent of the regularization procedure. For even $d$ this is given by a logarithmic term, while for odd $d$ it is a constant term. Using (\[eq:124\]) in (\[eq:121\]) we find the following expansion as derived in Ref. [@Casini:2018kzx] $$\label{eq:43} S=\frac{\mu_{d-2}}{\epsilon^{d-2}}+ \dots+ a_d^*\times \begin{cases} \displaystyle \frac{4(-1)^{\frac{d-2}{2}}} {{\rm Vol}(S^{d-2})} \int_{S^{d-2}} d\Omega(\vec{v}\,) \ln\left( 2\bar{A}(\vec{v}\,)/\epsilon \right)\ , \quad d{\rm \,\, even} \vspace{6pt}\\ \qquad \qquad \qquad \displaystyle 2\pi(-1)^{\frac{d-1}{2}} \qquad \qquad \quad \,\ , \quad \, d{\rm \,\, odd}\ , \end{cases}$$ where $\mu_i$ are non-universal coefficients. If we take $\bar{A}(\vec{v})=R$ we recover the well known result for the entanglement of a ball [@Casini:2011kv]. For odd $d$, the universal term is independent of the entangling surface $\bar{A}(\vec{v}\,)$. In Ref. [@Casini:2018kzx] it was argued that this feature is not modified by quantum and higher curvature corrections in the bulk, meaning that the entanglement entropy in (\[eq:43\]) is valid to all orders in the dual field theory. Notice that the higher order terms in the expansion of the null surface (\[eq:126\]) play no role in determining the universal term of the entanglement entropy. This means that the only non-trivial information we used to obtain (\[eq:43\]) is that the whole surface satisfies $\hat{r}_+=0$. This will simplify the calculation for the curved backgrounds we consider in the following. Lorentzian cylinder ------------------- We can apply a similar procedure to obtain the entanglement entropy associated to the null surface (\[eq:73\]) in the Lorentzian cylinder. Since the state is still given by the CFT vacuum, the bulk geometry is also pure AdS. However, we must consider a different set of coordinates which give a different conformal frame at the boundary. To do so, we define the following coordinates $$\label{eq:4} \hat{r}_\pm=\hat{r}\pm t=R \tan(\hat{\theta}_\pm/2) \ , \qquad \qquad \hat{\theta}_\pm=\hat{\theta}\pm \sigma/R\ ,$$ so that the AdS metric (\[eq:2\]) becomes $$\label{eq:127} ds^2= \frac{L^2}{ \sin^2(\psi) \sin^2(\hat{\theta})} \left[ -(d\sigma/R)^2+d\hat{\theta}^2 +\sin^2(\hat{\theta}) \left( d\psi^2 +\cos^2(\psi)d\Omega^2(\vec{v}\,) \right) \right]\ .$$ As we take the boundary limit $\psi\rightarrow 0$ we recover the metric $\mathbb{R}\times S^{d-1}$, where the bulk coordinates $\hat{\theta}_\pm$ become the null coordinates in the boundary ${\hat{\theta}_\pm\rightarrow \theta_\pm}$. To find the entanglement entropy we look for the extremal surface with boundary conditions fixed by the entangling surface in (\[eq:73\]), so that we have $$\label{eq:129} \lim_{\psi\rightarrow 0} (\hat{\theta}_+,\hat{\theta}_-,\vec{v}\,)= (0,2\bar{A}(\vec{v}\,),\vec{v}\,)\ ,$$ where $\bar{A}(\vec{v}\,)\in (0,\pi)$.[^33] Instead of computing the extremal surface from scratch we use the results obtained for the Minkowski null cone. The extremal surface of the Minkowski null cone is mapped under the change of coordinates (\[eq:4\]) so that the condition $\hat{r}_+=0$ translates into $\hat{\theta}_+=0$. The induced metric in (\[eq:127\]) under the constraint $\hat{\theta}_+=0$ is the same as in (\[eq:128\]), meaning that the entanglement entropy is again determined by the integral in (\[eq:121\]). The difference comes from the regularization procedure. To find the appropriate cut-off $z_{\rm min}=\epsilon$ we write the space-time metric (\[eq:127\]) in Fefferman-Graham coordinates (\[eq:133\]). The appropriate change of coordinates is given by $$\cot(\psi)= \left(\frac{4R^2-z^2} {4Rz}\right)\sin(\theta) \ , \qquad \qquad \cos(\hat{\theta})= \left( \frac{4R^2-z^2}{4R^2+z^2} \right) \cos(\theta) \ ,$$ where $z\in [0,2R]$ and $\theta\in[0,\pi]$. Inverting these relations $$\label{eq:12} \frac{z}{2R}=\frac{1- \sqrt{1-\sin^2(\psi)\sin^2(\hat{\theta})}} {\sin(\psi)\sin(\hat{\theta})} \ , \qquad \qquad \tan(\theta)=\cos(\psi)\tan(\hat{\theta})\ ,$$ and applying to the AdS metric in (\[eq:127\]) we find $$ds^2=(L/z)^2\left[ dz^2-\left(\frac{4R^2+z^2}{4R^2}\right)^2d\sigma^2+ \left(\frac{4R^2-z^2}{4R^2}\right)^2 R^2\left( d\theta^2+\sin^2(\theta)d\Omega^2(\vec{v}\,) \right) \right]\ ,$$ that is precisely in the Fefferman-Graham form with the cylinder metric at the boundary. Setting $z_{\rm min}=\epsilon$ in (\[eq:12\]) we can find the relation between $\psi$ and the cut-off by evaluating the right-hand side on the extremal surface $\hat{\theta}=\hat{\theta}_-(\psi,\vec{v}\,)/2$. This has a near boundary expansion given by $$\hat{\theta}_-(\psi,\vec{v}\,)= 2\bar{A}(\vec{v}\,)+b_1(\vec{v}\,)\psi+\dots\ ,$$ where the first order term is fixed by the boundary condition (\[eq:129\]). Using this in (\[eq:12\]) with $z_{\rm min}=\epsilon$ and inverting we find $$\label{eq:131} \psi(\epsilon,\vec{v}\,)= \frac{(\epsilon/R)}{\sin(\bar{A}(\vec{v}\,))}- \frac{b_1(\vec{v}\,)} {\sin^2(\bar{A}(\vec{v}\,)) \tan(\bar{A}(\vec{v}\,))}(\epsilon/R)^2 +\dots\ .$$ From this we can regulate and solve the integral in (\[eq:121\]) to obtain the universal terms of the entanglement entropy. Comparing with the expansion of $\psi(\epsilon,\vec{v}\,)$ that we got for the case of the null cone (\[eq:124\]) we immediately see that the entropy only gets a universal contribution from the linear term in (\[eq:131\]). For odd $d$ it is also given by (\[eq:43\]) while for even $d$ we obtain $$S=\frac{\mu_{d-2}}{\epsilon^{d-2}}+ \dots+ (-1)^{\frac{d-2}{2}} \frac{4a_d^*} {{\rm Vol}(S^{d-2})} \int_{S^{d-2}} d\Omega(\vec{v}\,) \ln\left[ \frac{2R}{\epsilon} \sin(\bar{A}(\vec{v}\,)) \right]\ .$$ Same as with the Minkowski null cone, we expect this result to be valid to all orders in the dual CFT. For the particular case in which $\bar{A}(\vec{v})=\theta_0$, we recover the result for a cap region of angular size $\theta_0$ [@Casini:2011kv]. De Sitter --------- A similar story holds for the entanglement in de Sitter associated to the null surface in (\[eq:15\]). Since the coordinates in the boundary are also given by $(\sigma/R,\theta,\vec{v}\,)$ we can still work with pure AdS as written in (\[eq:127\]). To get de Sitter at the boundary we simply have to take the limit $\psi\rightarrow 0$ with the additional factor of $\sin^2(\sigma/R)$ in the conformal factor. The boundary condition of the extremal surface is obtained from (\[eq:15\]) using this $\eta(\beta)=\cot(\beta)$ so that we find $$\label{eq:135} \lim_{\psi\rightarrow 0} ( \hat{\theta}_+,\hat{\theta}_-,\vec{v}\, )= \left(0, 2\,{\rm arcot} \left( \bar{A}(\vec{v}\,) \right), \vec{v}\,\right)\ ,$$ where in principle $\bar{A}(\vec{v})\in \mathbb{R}$.[^34] The entanglement entropy is still given by the integral in (\[eq:121\]). To obtain the relation between the cut-off $\epsilon$ and $\psi$ we must write the metric (\[eq:127\]) in Fefferman-Graham coordinates with $g_{\mu \nu}^{(0)}$ given by the de Sitter metric. Since the relation between the coordinates is fairly complicated it is convenient to break it up in two steps.[^35] First let us consider the coordinates $\varrho\ge 0$ and $\theta\in[0,\pi]$ defined according to $$\label{eq:159} \cot(\psi)= \frac{\varrho}{L}\sin(\theta)\ , \qquad \qquad \cos(\hat{\theta})= \frac{\varrho} {\sqrt{\varrho^2+L^2}}\cos(\theta)\ ,$$ which has an inverse given by $$\varrho=L \frac{\sqrt{1-\sin^2(\psi)\sin^2(\hat{\theta})}} {\sin(\psi)\sin(\hat{\theta})}\ , \qquad \qquad \tan(\theta)=\cos(\psi)\tan(\hat{\theta})\ .$$ The metric (\[eq:127\]) takes the standard form of global AdS $$\label{eq:156} ds^2=- \left( \frac{\varrho^2+L^2}{R^2} \right)d\sigma^2+ \left( \frac{L^2}{\varrho^2+L^2} \right)d\varrho^2+ \varrho^2\left( d\theta^2+\sin^2(\theta)d\Omega^2(\vec{v}\,) \right)\ .$$ From this we can define the new coordinates $z\in [0,2R]$ and $\hat{\sigma}/R\in[-\pi,0]$ according to $$\varrho= - \left(\frac{4R^2-z^2}{4Rz}\right) \frac{L}{\sin(\hat{\sigma}/R)}\ , \qquad \qquad \tan(\sigma/R)=- \left(\frac{4R^2-z^2}{4R^2+z^2}\right) \cot(\hat{\sigma}/R) \ .$$ Since $\hat{\sigma}$ has a finite domain we are only taking a section of the range of $\sigma$, given by $|\sigma|\le \pi R/2$. The inverse can be computed and written as $$\frac{z}{2R}= \sqrt{ (\varrho/L)^2+1} \cos(\sigma/R)- \sqrt{(\varrho/L)^2\cos^2(\sigma/R) -\sin^2(\sigma/R)}\ ,$$ $$\cos(\hat{\sigma}/R)= \sqrt{\frac{\varrho^2+L^2}{\varrho^2}} \sin(\sigma/R) \ .$$ The metric (\[eq:156\]) then becomes $$\label{eq:176} ds^2= (L/z)^2 \left[ dz^2+ \left( \frac{4R^2-z^2}{4R^2\sin(\hat{\sigma}/R)} \right)^2 \left[ -d\hat{\sigma}^2+R^2\left( d\theta^2+\sin^2(\theta)d\Omega^2(\vec{v}\,) \right) \right] \right]\ ,$$ which is in Fefferman-Graham coordinates with a dS boundary. Using the relations between the different coordinates we can obtain an expression for $z$ in terms of the coordinates $(\psi,\hat{\theta}_\pm)$. Imposing also the constraint $\hat{\theta}_+=0$ which is satisfied by the extremal surface we find $$\label{eq:157} \frac{z}{2R}= \frac{\cot(\hat{\theta}_-/2)- \sqrt{\cot^2(\hat{\theta}_-/2)-\sin^2(\psi)}} {\sin(\psi)}\ ,$$ where $\hat{\theta}_-(\psi,\vec{v}\,)$ determines the extremal surface. This has a near boundary expansion given by $$\hat{\theta}_-(\psi,\vec{v}\,)= 2\,{\rm arccot}\left( \bar{A}(\vec{v}\,) \right)+b_1(\vec{v}\,)\psi+\dots\ ,$$ with the first order term determined from the boundary condition (\[eq:135\]). Evaluating this in (\[eq:157\]) with $z_{\rm min}=\epsilon$ we get an expansion that we can invert to get $\psi(\epsilon,\vec{v}\,)$ and find $$\psi(\epsilon,\vec{v}\,)= \begin{cases} \quad \displaystyle \bar{A}(\vec{v}\,) (\epsilon/R)+\mathcal{O}(\epsilon/R)^2\ , \qquad \bar{A}(\vec{v}\,)>0\ , \vspace{5pt}\\ \quad \displaystyle 4\bar{A}(\vec{v}\,)(R/\epsilon) +\mathcal{O}(1) \quad\,\ , \qquad \bar{A}(\vec{v}\,)<0\ . \end{cases}$$ From this we see that the case in which $\bar{A}(\vec{v}\,)<0$ is anomalous since the limit $\epsilon\rightarrow 0$ gives a divergence in $\psi$. We can understand this by noting that for $\bar{A}(\vec{v}\,)<0$ the corresponding space-time region $\mathcal{D\bar{A}}^+$ in de Sitter lies outside the space-time (see Fig. \[fig:6\]). Since the entanglement entropy is a non-local quantity which captures global information about the region it is no surprise that the calculation breaks down in this regime. The case $\bar{A}(\vec{v}\,)=0$ is also anomalous from this perspective and corresponds to taking the space-time region $\mathcal{D\bar{A}}^+$ as the de Sitter static patch. If we restrict to the $\bar{A}(\vec{v}\,)>0$ case, we find that the entanglement entropy for even $d$ can be written as $$S=\frac{\mu_{d-2}}{\epsilon^{d-2}}+ \dots+ (-1)^{\frac{d-2}{2}} \frac{4a_d^*} {{\rm Vol}(S^{d-2})} \int_{S^{d-2}} d\Omega(\vec{v}\,) \ln\left[ \frac{2R}{\epsilon \bar{A}(\vec{v}\,)} \right]\ .$$ Anti-de Sitter -------------- Finally we consider the entanglement entropy for a CFT in a fixed AdS$_d$ background given by the space-time region associated to the null surface (\[eq:158\]). The boundary condition for the extremal surface is now given by $$\lim_{\psi\rightarrow 0} (\hat{\theta}_+,\hat{\theta}_-,\vec{v}\,)= (0,2\,{\rm arctan}\left(\bar{A}(\vec{v}\,)\right),\vec{v}\,)\ ,$$ where $\bar{A}(\vec{v}\,)>0$. Considering the AdS$_{d+1}$ bulk metric as in (\[eq:127\]) we get an AdS$_d$ boundary by taking the limit $\psi\rightarrow 0$ with the additional conformal factor $\cos^2(\theta)$, so that we get (\[eq:154\]). The entropy is still given by (\[eq:121\]) where we must regulate with an appropriate cut-off obtained from the Fefferman-Graham bulk coordinates. To find these coordinates we first apply the transformation in (\[eq:159\]) to (\[eq:127\]) so that we get pure AdS$_{d+1}$ in the standard coordinates in (\[eq:156\]). Once we have the metric in this form we can define the new coordinates $(\rho,z)$ given by $$\varrho^2= \left( \frac{4R^2+z^2}{4 R z} \right)^2 (\rho^2+L^2) -L^2\ , \qquad \qquad \tan(\theta)= \frac{\rho}{L} \left(\frac{4R^2+z^2}{4R^2-z^2}\right)\ ,$$ where $z\in[0,2R]$ and we restrict $\theta\in[0,\pi/2]$. The inverse transformation can be written as $$\frac{z}{2R} =-(\varrho/L)\cos(\theta)+ \sqrt{(\varrho/L)^2\cos^2(\theta)+1}\ , \qquad \qquad \rho= \frac{L\varrho\sin(\theta)} {\sqrt{\varrho^2\cos^2(\theta)+L^2}}\ .$$ This takes the AdS$_{d+1}$ bulk metric in (\[eq:156\]) to the appropriate Fefferman-Graham coordinates $$ds^2= (L/z)^2\left[ dz^2+ \left(\frac{4R^2+z^2}{4 L R}\right)^2 \left[ -\left( \frac{\rho^2+L^2}{R^2}\right)d\sigma^2+ \left(\frac{L^2}{\rho^2+L^2}\right)d\rho^2+ \rho^2d\Omega^2(\vec{v}\,) \right] \right]\ .$$ From this we can write the relation between $z$ and the coordinates $(\psi,\hat{\theta})$ as $$\frac{z}{2R}= \frac{\sqrt{1+\sin^2(\hat{\theta})\tan^2(\hat{\theta})\sin^2(\psi)\cos^2(\psi)} -\sqrt{1-\sin^2(\hat{\theta})\sin^2(\psi)}} {\sin(\hat{\theta})\sin(\psi) \sqrt{1+\tan^2(\hat{\theta})\cos^2(\psi)}}\ .$$ Evaluating at $z_{\rm min}=\epsilon$ and expanding for the extremal surface near the boundary $$\hat{\theta}(\psi,\vec{v}\,)={\rm arctan}\left(\bar{A}(\vec{v}\,)\right)+b_1(\vec{v}\,)\psi+\mathcal{O}(\psi^2)\ ,$$ we can invert the relation and find the appropriate cut-off to regulate the integral (\[eq:121\]) $$\psi(\epsilon,\vec{v}\,)= \frac{(\epsilon/R)}{\bar{A}(\vec{v}\,)} +\mathcal{O}(\epsilon/R)^2\ .$$ The entanglement entropy for even values of $d$ is then given by $$S=\frac{\mu_{d-2}}{\epsilon^{d-2}}+ \dots+ (-1)^{\frac{d-2}{2}} \frac{4a_d^*} {{\rm Vol}(S^{d-2})} \int_{S^{d-2}} d\Omega(\vec{v}\,) \ln\left[ \frac{2R}{\epsilon}\bar{A}(\vec{v}\,) \right]\ .$$ Modular conjugation in cylinder from embedding formalism {#zapp:Mod_conj} ======================================================== In this appendix we use the embedding formalism of the conformal group to map the geometric action of ${\rm CIT}$ operator in (\[eq:36\]) to the Lorentzian cylinder. The main idea of the embedding space formalism is to embed the space-time of the CFT into a larger space where conformal transformations act linearly. Since the conformal group is isomorphic to ${\rm SO}(d,2)$ we define the embedding coordinates $\xi\in\mathbb{R}^{d,2}$ $$\xi=(\xi^0,\xi^i,\xi^d,\xi^{d+1})\ ,$$ in the space $$\label{eq:94} ds^2= -(d\xi^0)^2 +\sum_{i=1}^{d-1}(d\xi^i)^2 +\left[ (d\xi^{d})^2-(d\xi^{d+1})^2 \right] \ .$$ Every group element $g\in {\rm SO}(d,2)$ has a representation in terms of a matrix $M_g$ which has a linear action in the embedding coordinates given by ordinary matrix multiplication $\xi'=M_g\, \xi$. The relation with the $d$-dimensional space-time of the CFT is obtained as follows. We first define the projective null cone as $$\label{eq:97} \mathcal{PC}= \frac{\left\lbrace \xi\in \mathbb{R}^{2,d}\,\,: \quad (\xi \cdot \xi)=0 \right\rbrace} {\xi\sim c \,\xi\ ,\,\, c\in \mathbb{R}_{+}}\ ,$$ where $(\xi\cdot \xi)$ is computed using the embedding metric (\[eq:94\]). The denominator means that there is a gauge redundancy in the scaling of $\xi$. To obtain the $d$-dimensional Minkowski space-time we use this gauge freedom to fix $\xi^+=\xi^d+\xi^{d+1}=R$ (called the Poincare section) with $R$ an arbitrary length scale. With this gauge choice we can parametrize $\xi\in {\mathcal{PC}}$ as $$\label{eq:95} \xi(x)= \left(x^\mu,\frac{R^2-(x\cdot x)}{2R}, \frac{R^2+(x\cdot x)}{2R}\right)\ ,$$ where $x^\mu=(t,\vec{x})$ and $(x\cdot x)=\eta_{\mu \nu}x^\mu x^\nu$. Using this we compute the induced metric in $\mathcal{PC}$ and obtain $d$-dimensional Minkowski $ds^2=d\xi(x)\cdot d\xi(x)=\eta_{\mu \nu}dx^\mu dx^\nu$. By considering a different section of the projective null cone in which $\xi^+=R/w(x)$ we can obtain a different $d$-dimensional space-time that is conformally related to Minkowski. Let us now describe how a conformal transformation is induced by the linear action of ${M_g\in {\rm SO}(d,2)}$. Since $M_g\,\xi$ might take us off the section of the projective cone that we started from (*i.e.* $\xi(x)^+\neq (M_g\,\xi(x))^+$) we must also apply a rescaling, so that the overall transformation is given by $$\label{eq:98} \xi(x) \qquad \longrightarrow \qquad M_g\,\xi(x) \qquad \longrightarrow \qquad \frac{\xi(x)^+}{(M_g\,\xi(x))^+} M_g\,\xi(x)= \xi(x')\ .$$ This induces a transformation from $x\rightarrow x'$ that corresponds to a conformal transformation in the $d$-dimensional space-time of the CFT. We now want to apply this formalism to obtain the linear transformation in the embedding space $\xi$ which implements the action of ${\rm CIT}$ (\[eq:36\]) in the Poincare section. Consider the matrix $M_{\tilde{g}}$ which implements a rotation of angle $\pi$ between the embedding coordinates $(\xi^0,\xi^d)$, so that we have $$\label{eq:96} M_{\tilde{g}}\,\xi= (-\xi^0,\xi^i,-\xi^d,\xi^{d+1})\ .$$ Following the prescription described in (\[eq:98\]) the transformation in the embedding coordinates is given by $$\tilde{\xi}(x)= \frac{R^2}{(x\cdot x)} \left( -t,\vec{x},\frac{-R^2+(x\cdot x)}{2R}, \frac{R^2+(x\cdot x)}{2R} \right)\ ,$$ where the rescaling by $R^2/(x\cdot x)$ is such that $\tilde{\xi}(x)^+=R$. Comparing this expression with $\xi(\tilde{x})$ in (\[eq:95\]) we find that the induced transformation in $x^\mu$ is given by $$\tilde{x}^\mu(x)= \frac{R^2}{(x\cdot x)}(-t,\vec{x})\ ,$$ that is precisely the ${\rm CIT}$ reflection in (\[eq:36\]). This shows that $M_{\tilde{g}}$ given in (\[eq:96\]) implements the reflection transformation in the embedding space. Notice that although $M_{\tilde{g}}$ does not correspond to a conformal transformation since $M_{\tilde{g}}\not \in {\rm SO}(d,2)$, it belongs to the Euclidean conformal group ${{\rm SO}(d+1,1)}$. This is analogous to what happens with the ${\rm CRT}$ operator that is not in the Lorentz group but is part of the Euclidean group. Using this we can easily obtain the action of ${\rm CIT}$ applied to the Lorentzian cylinder $\mathbb{R}\times S^{d-1}$. To do so, we consider a different section of the projective null cone $\mathcal{PC}$ obtained from the following parametrization $$\label{eq:100} \xi(\sigma,\theta,\vec{n})= R \big( \sin(\sigma/R),\sin(\theta)\vec{n}, \cos(\theta),\cos(\sigma/R) \big)\ ,$$ where $\vec{n}\in \mathbb{R}^{d-1}$ such that $|\vec{n}|^2=1$. This is a vector in the projective null cone in the section given by $$\label{eq:99} \xi^+= 2R \cos(\theta_+/2)\cos(\theta_-/2)\ ,$$ where $\theta_\pm=\theta\pm \sigma/R$. The $d$-dimensional induced metric in (\[eq:94\]) is given by $$\label{eq:102} ds^2=d\xi(\sigma,\theta,\vec{n}) .d\xi(\sigma,\theta,\vec{n})= -d\sigma^2+R^2\left( d\theta^2+\sin^2(\theta)ds^2_{S^{d-2}} \right)\ ,$$ that is the Lorentzian cylinder $\mathbb{R}\times S^{d-1}$. Considering the action of $M_{\tilde{g}}$ in (\[eq:96\]) and following the procedure in (\[eq:98\]), we find $$\tilde{\xi}(\sigma,\theta,\vec{n})= \frac{R}{\tan(\theta_+/2)\tan(\theta_-/2)} \big( -\sin(\sigma/R),\sin(\theta)\vec{n}, -\cos(\theta),\cos(\sigma/R) \big)\ ,$$ where the rescaling ensures that we remain in the section (\[eq:99\]). Comparing with $\xi(\tilde{\sigma},\tilde{\theta},\vec{n})$[^36] in (\[eq:100\]) the reflection transformation in the cylinder is given by $$\label{eq:101} \tan(\tilde{\theta}_\pm)=-\tan(\theta_\pm) \qquad \Longrightarrow \qquad \tilde{\theta}_\pm(\theta_\pm)=\pi-\theta_\pm\ ,$$ where to get rid of the trigonometric functions we imposed $\theta_++\theta_-=2\theta\in[0,2\pi]$. This is a simple linear relation that reflects points across the wedge $\theta_\pm=\pi/2$, that is the fixed point of the transformation. Since there should not be anything special about $\theta_0=\pi/2$, we would like to generalize (\[eq:101\]) to arbitrary values of $\theta_0\in[0,\pi]$. Inspired by some calculations in [@Casini:2011kv], we can do so by considering a slight variation of the parametrization (\[eq:100\]), given by boosting the embedding coordinates in the $(\xi^d,\xi^{d+1})$ direction $$\begin{aligned} \xi^0&=R\sin(\sigma/R)\ , \qquad \qquad \,\, \xi^i=R\sin(\theta)n^i\ , \\ \xi^d&=R\cosh(\gamma)\cos(\theta)+R\sinh(\gamma)\cos(\sigma/R)\ ,\\ \xi^{d+1}&=R\cosh(\gamma)\cos(\sigma/R)+R\sinh(\gamma)\cos(\theta)\ , \end{aligned}$$ where $\gamma\in \mathbb{R}$ is the boost parameter. Since the boost is an isometry of the embedding space, the vector $\xi$ is still null and gives the same induced metric as in (\[eq:102\]). However the gauge condition $\xi^+$ is slightly different $$\label{eq:105} \xi^+= 2Re^{\gamma}\cos(\theta_+/2)\cos(\theta_-/2)\ .$$ Repeating the calculation leading to (\[eq:101\]) but for $\gamma\neq 0$, we find that the induced reflection transformation is now given by $$\label{eq:103} \tan(\tilde{\theta}_\pm)= \frac{-\sin(\theta_\pm)} {\cos(\theta_\pm)\cosh(2\gamma)+\sinh(2\gamma)}\ .$$ The value of $\gamma$ determines the size of the wedge $\theta_0$ in the cylinder where the reflection is applied. The relation between $\gamma$ and $\theta_0$ can be found by looking at the fix point of the transformation (\[eq:103\]), so that we get $e^{\gamma}=\tan(\theta_0/2)$. Since we cannot analytically solve $\tilde{\theta}_\pm(\theta_\pm)$ for arbitrary $\theta_0\in [0,\pi]$ we compute it numerically and obtain the diagram in Fig. \[fig:11\]. Although the relation in (\[eq:103\]) written in terms of $\theta_0$ is quite complicated, it is straightforward to check that the same reflection transformation is obtained from the following simpler relation[^37] $$\label{eq:104} \tan(\tilde{\theta}_\pm/2)= \tan^2(\theta_0/2)\cot(\theta_\pm/2)\ .$$ [^1]: Given that (A)dS are maximally symmetric space-times, this is not necessary for the ANEC in (\[eq:210\]). For more details see discussion around (\[eq:181\]). [^2]: See Refs. [@Visser:1994jb; @Urban:2009yt] for previous studies on the behavior of the ANEC under conformal transformations. [^3]: For a pedagogical introduction see David Simmons-Duffin’s lecture notes on TASI 2019 (although to this date the notes are not complete, they are still very useful). Another useful explanation is given in the first secion of Ref. [@Brunetti:1992zf]. [^4]: The wedge reflection positivity for the CFT in the Lorentzian cylinder and de Sitter for operators of arbitrary even spin are given in (\[eq:211\]) and (\[eq:205\]) respectively. [^5]: To obtain the $S^{d-2}$ in terms of the usual angles we describe the vector $\vec{v}\in \mathbb{R}^{d-2}$ in spherical coordinates and then parametrize its radius according to $|\vec{v}\,|=L\tan(\phi/2)$ with $\phi\in[0,\pi]$. [^6]: Starting from the Lorentzian cylinder, Ref. [@Candelas:1978gf] discusses some additional conformal relations. Although in this work we restrict to Minkowski and (A)dS, a similar treatment is possible in these other cases. [^7]: The inverse transformation is given by $(X,\vec{Y})=r \left(|\vec{v}\,|^2-4R^2,4R\vec{v} \right)/\left(|\vec{v}\,|^2+4R^2\right)$. [^8]: The metric in the unit sphere $d\Omega^2(\vec{v}\,)$ is given in (\[eq:152\]) with $L=2R$. [^9]: To plot the curves on the $S^2$ it is useful to write the stereographic coordinate $v\in \mathbb{R}$ as $v=2R\tan(\phi/2)$ with $|\phi|\le \pi$ and then consider Cartesian coordinates $(x,y,z)$ in terms of the spherical angles $(\theta,\phi)$. Using (\[eq:188\]) this gives $(x,y,z)$ in terms of $(\lambda,\vec{x}_\perp)$ so that the curves always lie on the surface of the $S^2$. [^10]: To obtain this plot we write the Cartesian coordinates $(x,y)$ as $(x,y)=\theta(\cos(\phi),\sin(\phi))$ where $\phi$ is obtained from $v=2R\tan(\phi/2)$. Using the description of the geodesics in (\[eq:188\]) we get $(x,y)$ as a function of $(\lambda,\vec{x}_\perp)$. [^11]: For our purpose this assumption is not strictly necessary. Although Poincare symmetry of the vacuum only implies $\bra{0}T_{\mu \nu}\ket{0}\propto \eta_{\mu \nu}$, when projecting the stress tensor along the null direction $T_{\lambda \lambda}$ this constant factor drops out. [^12]: While this work was in preparation Ref. [@Iizuka:2019ezn] appeared where this inequality was derived for $d=3,4,5$ and strongly coupled holographic CFTs described by Einstein gravity. This derivation show that the bound is valid in a more general setup. [^13]: It is important that the inequality (\[eq:195\]) is written in terms of the *affine* parameter of the geodesic, since we could always define a new parameter which absorbs the weight function $\cos^d(\bar{\lambda})$ in the integral. [^14]: See section 4.2.4 of Ref. [@Fewster:2004nj] for an explicit example of this feature in two dimensional CFTs. [^15]: It important that the integral in the ANEC is written in terms of an affine parameter. While the condition in (\[eq:193\]) is clearly invariant under affine transformations $\lambda \rightarrow a\lambda+b$, it changes its form under a more general transformation, *e.g.* $\lambda \rightarrow L\sinh(\lambda/L)$. [^16]: We can explicitly check this from equation (21) in Ref. [@Herzog:2013ed] using that the Riemann tensor of (A)dS is determined from its metric. [^17]: The inequality implied by relative entropy is more general than (\[eq:1\]) and given by $${\rm Tr}\big[ \rho\big(\hat{K}_A-\hat{K}_B\big)\big] \ge 2S_f(A,B)\ ,$$ where $S_f(A,B)$ is the free entropy of the state $\rho$. This is a non-negative and UV finite quantity constructed from the entanglement entropy $2S_f(A,B)=\left(S_A-S_{A'}\right)-\left(S_B-S_{B'}\right)$, see Ref. [@Blanco:2013lea]. If $\rho$ is a pure state, the free entropy vanishes and we recover (\[eq:1\]). [^18]: This expression for $\alpha$ can be obtained from the integral in (\[eq:78\]), using $w^2(\lambda,\vec{x}_\perp)$ in (\[eq:79\]) and conveniently fixing the integration constants $c_1$ and $c_0$. [^19]: For a nice 3D picture of the setup see Fig. 2 of Ref. [@Casini:2018kzx]. [^20]: A correct expression for the modular Hamiltonian that is equivalent to (\[eq:72\]) was previously given in Ref. [@Neuenfeld:2018dim] without proof. [^21]: Given a sequence of operators $\mathcal{O}_n$ the weak operator topology defines the limit $\mathcal{O}_n\rightarrow \mathcal{O}$ according to $|\bra{\alpha}\left(\mathcal{O}_n-\mathcal{O}\right)\ket{\beta}|<\epsilon_n$, where $\ket{\alpha}$ and $\ket{\beta}$ are any two vectors in the Hilbert space. [^22]: The vector $\ket{\psi}$ is cyclic if the set $\left\lbrace \mathcal{O}\ket{\psi},\,\forall\,\mathcal{O}\in \mathcal{W} \right\rbrace$ is dense in $\mathcal{H}$, while it is separating if ${\mathcal{O}\ket{\psi}=0}$ for $\mathcal{O}\in \mathcal{W}$ implies $\mathcal{O}\equiv 0$. [^23]: The Minkowski vacuum state is cyclic and separating as a consequence of the Reeh-Schlieder theorem. [^24]: For half-integer spin the action of ${\rm CRT}$ is more complicated and there are some subtelties regarding the inequality (\[eq:34\]), see Ref. [@Casini:2010bf]. [^25]: In Sec. \[sec:null\_energy\_bounds\] we analyze the transformation of the full modular Hamiltonian for more general regions given by arbitrary null deformations of the Rindler wedge. Here we restrict to the case in which we have no deformations. [^26]: The conformal factor $w(x^\mu)$ obtained from applying the conformal transformation in (\[eq:3\]) with ${D^\mu=(0,R,\vec{0})}$ is given by $$w^2(x^\mu)=\left[ \frac{4R^2}{-t^2+(R-x)^2+|\vec{y}|^2}\right]^2\ .$$ [^27]: We have chosen the coordinate system so that the position of the two points in the unit sphere $S^{d-2}$ is the same, *i.e.* $\vec{v}_1=\vec{v}_2$. Moreover, this is the correlator for space-like separated points since for the time-like case we have an additional phase $e^{\pm i \Delta}$ depending on the ordering. [^28]: Notice that the conformal factor $w^2(\sigma)=\cos^2(\sigma/R)$ satisfies $w^2(\sigma)/w^2(\tilde{\sigma})=1$. [^29]: For a field theory on a generic space-time Ref. [@Graham:2007va] proposed that the ANEC (without any vacuum energy subtraction) must hold along achronal null geodesics, *i.e.* curves which do not contain points connected by a time-like path. Several references in the literature find evidence supporting this proposal [@Wall:2009wi; @Kontou:2012ve; @Kontou:2015yha], while other claim to obtain counter examples [@Urban:2009yt; @Ishibashi:2019nby]. [^30]: See section 4.6 of Ref. [@Kologlu:2019bco] for a discussion regarding some aspects of the definition of these operators. [^31]: When the bulk is pure AdS the metric is Fefferman-Graham metric is given by (\[eq:176\]) to all orders and we can explicitly check the proportionality factor $-1/(2R^2)$. [^32]: Since the Riemann is proportional to the metric, the terms in $g_{\mu \nu}^{(n)}$ involving covariant derivatives vanish. [^33]: The function $\bar{A}(\vec{v})$ is not the same as the one for the null cone. They are related through the coordinate change (\[eq:4\]). [^34]: The inverse of the cotangent function is defined so that its image is in the range $[0,\pi]$. [^35]: The easiest way to obtain these coordinate transformations is to use the embedding description of AdS and analyze the relation between the different parametrizations. [^36]: We find that it is consistent to assume that the unit vector $\vec{n}$ is not changed by the transformation. [^37]: This relation was obtained from the action of ${\rm CIT}$ in Minkwoski given in (\[eq:32\]) and applying the conformal transformation to the cylinder $r_\pm(\theta_\pm)=R\tan(\theta_\pm/2)/\tan(\theta_0/2)$. Notice that this works for the Minkowski region in which $r_+r_->0$. For the regions $r_+r_-<0$ one must carefully analyze other coordinates to the cylinder, given by (\[eq:10\]) with $R\rightarrow R/\tan(\theta_0/2)$. Doing so, one obtains the same result as in (\[eq:104\]).

--- abstract: 'We study exactly both the ground-state fidelity susceptibility and bond-bond correlation function in the Kitaev honeycomb model. Our results show that the fidelity susceptibility can be used to identify the topological phase transition from a gapped A phase with Abelian anyon excitations to a gapless B phase with non-Abelian anyon excitations. We also find that the bond-bond correlation function decays exponentially in the gapped phase, but algebraically in the gapless phase. For the former case, the correlation length is found to be $1/\xi=2\sinh^{-1}[\sqrt{2J_z -1}/(1-J_z)]$, which diverges around the critical point $J_z=(1/2)^+$.' author: - Shuo Yang - 'Shi-Jian Gu' - 'Chang-Pu Sun' - 'Hai-Qing Lin' title: 'Fidelity susceptibility and long-range correlation in the Kitaev honeycomb model' --- Introduction ============ Quite recently, a great deal of effort [@HTQuan2006; @Zanardi06; @Pzanardi0606130; @Buonsante1; @PZanardi032109; @WXG; @PZanardi0701061; @WLYou07; @HQZhou07; @LCVenuti07; @SChen07; @SJGu072; @MFYang07; @NPaunkovic07; @zhq; @AHamma07] has been devoted to the role of fidelity, a concept borrowed from quantum information theory [@Nielsen1], in quantum phase transitions(QPTs)[@Sachdev]. The motivation is quite obvious. Since the fidelity is a measure of similarity between two states, the change of the ground state structure around the quantum critical point should result in a dramatic change in the fidelity across the critical point. Such a fascinating prospect has been demonstrated in many correlated systems. For example, in the one-dimensional XY model, the fidelity shows a narrow trough at the phase transition point [@Zanardi06]. Similar properties were also found in fermionic [@Pzanardi0606130] and bosonic systems [@Buonsante1]. The advantage of the fidelity is that, since the fidelity is a space geometrical quantity, no a priori knowledge of the order parameter and symmetry-breaking is required in studies of QPTs. Nevertheless, the properties of the fidelity are mainly determined by its leading term [@PZanardi0701061; @WLYou07], i.e., its second derivative with respect to the driving parameter (or the so-called fidelity susceptibility [@WLYou07]). According to the standard perturbation method, it has been shown that the fidelity susceptibility actually is equivalent to the structure factor (fluctuation) of the driving term in the Hamiltonian [@WLYou07]. For example, if we focus on the thermal phase transitions and choose the temperature as the driving parameter, the fidelity susceptibility, extracted from the mixed state fidelity between two thermal states[@WXG], is simply the specific heat[@PZanardi0701061; @WLYou07]. From this point of view, the fidelity approach to QPTs seems still to be within the framework of the correlation functions approach, which is intrinsically related to the local order parameter. However, some systems cannot be described in a framework built on the local order parameter. This might be due to the absence of preexisting symmetry in the Hamiltonian, such as topological phase transitions [@wen-book] and Kosterlitz-Thouless phase transitions [@JMKosterlitz73]. For the latter, since the transition is of infinite-order, it has already been pointed out that the fidelity might fail to identify the phase transition point [@WLYou07; @SChen07]. Therefore, it is an interesting issue to address the role of fidelity in studying the topological phase transition. The Kitaev honeycomb model was first introduced by Kitaev in search of topological order and anyonic statistics. The model is associated with a system of 1/2 spins which are located at the vertices of a honeycomb lattice. Each spin interacts with three nearest neighbor spins through three types of bonds, called “$x$($y,z$)-bonds" depending on their direction. The model Hamiltonian [@Kitaev] is as follows: $$\begin{aligned} H &=& -J_{x}\sum_{x\text{-bonds}}\sigma _{j}^{x}\sigma _{k}^{x}-J_{y}\sum_{y \text{-bonds}}\sigma _{j}^{y}\sigma _{k}^{y}-J_{z}\sum_{z\text{-bonds} }\sigma _{j}^{z}\sigma _{k}^{z},\nonumber \\ &=& -J_x H_x - J_y H_y - J_z H_z. \label{eq:Hamiltonian}\end{aligned}$$ where $j, k$ denote the two ends of the corresponding bond, and $J_a, \sigma^a(a=x,y,z)$ are dimensionless coupling constants and Pauli matrices respectively. Such a model is rather artificial. However, its potential application in topological quantum computation has made it a focus of research in recent years [@Kitaev; @Wen; @preskill9; @pachos; @Sarma; @XYFeng07; @CHD; @YuYue; @YS; @spKou; @model3D; @Dusuel]. The ground state of the Kitaev honeycomb model consists of two phases, i.e., a gapped A phase with Abelian anyon excitations and a gapless B phase with non-Abelian anyon excitations. The transition has been studied by various approaches. For example, it has been shown that a kind of long range order exists in the dual space [@XYFeng07], such that basic concepts of Landau’s theory of continuous phase transitions might still be applied. In real space, however, the spin-spin correlation functions vanishes rapidly with increasing distance between two spins. Therefore, the transition between the two phases is believed to be of topological type due to the absence of a local order parameter in real space [@Kitaev]. In this work, we *firstly* try to investigate the topological QPT occurring in the ground state of the Kitaev honeycomb model in terms of the fidelity susceptibility. We find that the fidelity susceptibility can be used to identify the topological phase transition from a gapped phase with Abelian anyon excitations to gapless phase with non-Abelian anyon excitations. Various scaling and critical exponents of the fidelity susceptibility around the critical points are obtained through a standard finite-size scaling analysis. *These observations from the fidelity approach are a little surprising*. Our earlier thought was that the fidelity susceptibility, which is a kind of structure factor obtained by a combination of correlation functions, can hardly be related to the topological phase transition, since the latter cannot be described by the correlation functions of local operators. So our *second* motivation following from the first one is to study the dominant correlation function appearing in the definition of the fidelity susceptibility, i.e., the bond-bond correlation function. We find that the correlation function decays algebraically in the gapless phase, but exponentially in the gapped phase. For the latter, the correlation length takes the form $1/\xi=2\sinh^{-1}[\sqrt{2J_z -1}/(1-J_z)]$ along a given evolution line. Therefore, the divergence of the correlation length around the critical point $J_z=(1/2)^+$ is also a signature of the QPT. We organize our work as follows. In Sec. \[sec:gsfs\], we introduce briefly the definition of the fidelity susceptibility in the Hamiltonian parameter space, then we diagonalize the Hamiltonian based on Kitaev’s approaches and obtain the explicit forms of the Riemann metric tensor, from which the fidelity susceptibility along any direction can be obtained. The critical and scaling behaviors of the fidelity susceptibility are also studied numerically. In Sec. \[sec:lrc\], we explicitly calculate the bond-bond correlation functions in both phases. Its long range behavior and the correlation length in the gapped phase are studied both analytically and numerically. Sec. \[sec:sum\] includes a brief summary. Fidelity susceptibility in the ground state {#sec:gsfs} =========================================== To study the fidelity susceptibility, we notice that the structure of the parameter space of the Hamiltonian (\[eq:Hamiltonian\]) is three dimensional. In this space, we can always let the ground state of the Hamiltonian evolves along a certain path in the parameter space, i.e., $$\begin{aligned} J_a=J_a(\lambda),\end{aligned}$$ where $\lambda$ is a kind of driving parameter along the evolution line. We then extend the definition of fidelity to this arbitrary line in high-dimensional space. Following Ref. [@Zanardi06], the fidelity is defined as the overlap between two ground states $$\begin{aligned} F=|\langle \Psi_0(\lambda) |\Psi_0(\lambda+\delta\lambda)\rangle|,\end{aligned}$$ where $\delta\lambda$ is the magnitude of a small displacement along the tangent direction at $\lambda$. Then the fidelity susceptibility along this line can be calculated as $$\begin{aligned} \chi_F = \lim_{\delta\lambda\rightarrow 0}\frac{-2\ln F_i}{\delta\lambda^2} = \sum_{a b} g_{a b} n^a n^b, \label{eq:deffsmetric}\end{aligned}$$ where $n^a=\partial J_a /\partial \lambda$ denotes the tangent unit vector at the given point, and $g_{ab}$ is the Riemann metric tensor introduced by Zanardi, Giorda, and Cozzini[@PZanardi0701061]. For the present model, we have $$\begin{aligned} g_{ab}=\sum_n \frac{\langle \Psi_n(\lambda) |H_a|\Psi_0(\lambda) \rangle \langle \Psi_0(\lambda) |H_b| \Psi_n(\lambda) \rangle}{(E_n -E_0)^2}, \label{eq:metric}\end{aligned}$$ where $|\Psi_n(\lambda) \rangle$ is the eigenstate of the Hamiltonian with energy $E_n$. Clearly, $g_{ab}$ does not depend on the specific path along which the system evolves. However, once $g_{ab}$ are obtained, the fidelity susceptibility is just a simple combination of $g_{ab}$ together with a unit vector which defines the direction of system evolution in the parameter space. According to Kitaev [@Kitaev], the Hamiltonian (\[eq:Hamiltonian\]) can be diagonalized exactly by introducing Majorana fermion operators to represent the Pauli operators as $$\begin{aligned} \sigma ^{x}=\text{i}b^{x}c, \; \sigma ^{y}=\text{i}b^{y}c,\; \sigma ^{z}=\text{i}b^{z}c, \label{eq:spinmajorana}\end{aligned}$$ where the Majorana operators satisfy $A ^{2}=1$, $AB =-BA $ for $A ,B \in \left\{ b^{x},b^{y},b^{z},c\right\} $ and $A \neq B $, and also $b^{x}b^{y}b^{z}c\left\vert \psi \right\rangle =\left\vert \psi \right\rangle $ to ensure the commutation relations of spin operators. Then the Hamiltonian can be written as $$\begin{aligned} H= \frac{\text{i}}{2} \sum_{j,k}\widehat{u}_{jk}J_{a_{jk}}c_{j}c_{k}.\end{aligned}$$ Since the operators $ \widehat{u}_{jk}=\text{i}b_{j}^{a _{jk}}b_{k}^{a _{jk}}$ satisfy $\left[ \widehat{u}_{jk},H\right] =0$, $\left[ \widehat{u}_{jk},\widehat{u}_{ml}% \right] =0$, and $\widehat{u}_{jk}^{2}=1$, they can be regarded as generators of the $Z_{2}$ symmetry group. Therefore, the whole Hilbert space can be decomposed into common eigenspaces of $\widehat{u}_{jk}$, each subspace is characterized by a group of $u_{jk}=\pm 1$. The spin model is transformed to a quadratic Majorana fermionic Hamiltonian $$\begin{aligned} H=\frac{\text{i}}{2} \sum_{j,k}u_{jk}J_{a _{jk}}c_{j}c_{k}.\end{aligned}$$ Here we restrict ourselves to only the vortex free subspace with translational invariants, i.e., all $u_{jk}=1$. After Fourier transformation, we get the Hamiltonian of a unit cell in the momentum representation [@Kitaev], $$H=\sum_{\textbf{q}}\left(\begin{array}{c}a_{-\textbf{q},1} \\a_{-\textbf{q},2}\end{array} \right) ^{\mathrm{T}}\left(\begin{array}{cc}0 & \text{i}f\left( \textbf{q}\right) \\ -\text{i}f\left( \textbf{q}\right) ^{\ast } & 0\end{array} \right) \left( \begin{array}{c}a_{\textbf{q},1} \\a_{\textbf{q},2}\end{array}\right) ,$$ where $\textbf{q}=(q_x, q_y)$, $$a_{\textbf{q},\gamma }=\frac{1}{\sqrt{2L^{2}}}\sum_{\mathbf{r}}e^{-\text{i}\mathbf{q}\cdot \mathbf{r}}c_{\mathbf{r},\gamma },\label{eq:aqcq}$$ $\mathbf{r}$ refers to the coordinate of a unit cell, $\gamma $ to a position type inside the cell, and $$\begin{aligned} f\left( \textbf{q}\right) &=&\epsilon _{\textbf{q}}+\text{i}\Delta _{\textbf{q}}, \nonumber \\ \epsilon _{\textbf{q}} &=&J_{x}\cos q_{x}+J_{y}\cos q_{y}+J_{z}, \nonumber \\ \Delta _{\textbf{q}} &=&J_{x}\sin q_{x}+J_{y}\sin q_{y}.\end{aligned}$$ Here, we set $L$ to be an odd integer, then the system size is $N=2L^{2}$. The momenta take the values $$q_{x\left( y\right) }=\frac{2n\pi }{L},n=-\frac{L-1}{2},\cdots ,\frac{L-1}{2} .$$ The above Hamiltonian can be rewritten using fermionic operators as $$H=\sum_{\textbf{q}}\sqrt{\epsilon _{\textbf{q}}^{2}+\Delta _{\textbf{q}}^{2}}\left( C_{\textbf{q},1}^{\dag }C_{\textbf{q},1}-C_{\textbf{q},2}^{\dag }C_{\textbf{q},2}\right) .$$ Therefore, we have the ground state $$\begin{aligned} \left\vert \Psi _{0}\right\rangle &=&\prod_{\textbf{q}}C_{\textbf{q},2}^{\dag }\left\vert 0\right\rangle \nonumber \\ &=&\prod_{\textbf{q}}\frac{1}{\sqrt{2}}\left( \frac{\sqrt{\epsilon _{\textbf{q}}^{2}+\Delta _{\textbf{q}}^{2}}}{\Delta _{\textbf{q}}+\text{i}\epsilon _{\textbf{q}}}a_{-\textbf{q},1}+a_{-\textbf{q},2}\right) \left\vert 0\right\rangle ,\label{eq:ground}\end{aligned}$$ with the ground state energy $$E_{0}=-\sum_{\textbf{q}}\sqrt{\epsilon _{\textbf{q}}^{2}+\Delta _{\textbf{q}}^{2}}.$$ ![(Color online) Fidelity susceptibility as a function of $J_z$ along the dashed line shown in the triangle for various system sizes $L=101, 303, 909$. Both upper insets correspond to enlarged pictures of two small portions.[]{data-label="figure_fs"}](fig1.eps){width="8.5"} The fidelity of the two ground states at $\lambda$ and $\lambda'$ can be obtained as $$\begin{aligned} F^2 &=&\prod_{\textbf{q}}\frac{1}{2}\left( 1+\frac{\Delta _{\textbf{q}}\Delta _{\textbf{q}}^{\prime }+\epsilon _{\textbf{q}}\epsilon _{\textbf{q}}^{\prime }}{E_{\textbf{q}}E_{\textbf{q}}^{\prime }} \right), \nonumber \\ &=&\prod_{\textbf{q}}\cos ^{2}\left( \theta _{\textbf{q}}-\theta _{\textbf{q}}^{\prime }\right).\end{aligned}$$ with $$\begin{aligned} \cos \left( 2\theta _{\textbf{q}}\right) &=&\frac{\epsilon _{\textbf{q}}}{E_{\textbf{q}}},\sin \left( 2\theta _{\textbf{q}}\right) =\frac{\Delta _{\textbf{q}}}{E_{\textbf{q}}}, \nonumber \\ \cos \left( 2\theta _{\textbf{q}}^{\prime }\right) &=&\frac{\epsilon _{\textbf{q}}^{\prime }}{% E_{\textbf{q}}^{\prime }},\sin \left( 2\theta _{\textbf{q}}^{\prime }\right) =\frac{\Delta _{\textbf{q}}^{\prime }}{E_{\textbf{q}}^{\prime }}.\end{aligned}$$ The Riemann metric tensor can be expressed as $$\begin{aligned} g^{ab} =\sum_{\textbf{q}}\left( \frac{\partial \theta _{\textbf{q}}}{\partial J_{a}}% \right) \left( \frac{\partial \theta _{\textbf{q}}}{\partial J_{b}}\right),\end{aligned}$$ where $$\begin{aligned} \frac{\partial \left( 2\theta _{\textbf{q}}\right) }{\partial J_{x}} &=&\frac{% J_{z}\sin q_{x}+J_{y}\sin \left( q_{x}-q_{y}\right) }{\epsilon _{\textbf{q}}^{2}+\Delta _{\textbf{q}}^{2}}\cdot \frac{\Delta _{\textbf{q}}}{\left\vert \Delta _{\textbf{q}}\right\vert }, \nonumber \\ \frac{\partial \left( 2\theta _{\textbf{q}}\right) }{\partial J_{y}} &=&-\frac{% J_{x}\sin \left( q_{x}-q_{y}\right) -J_{z}\sin q_{y}}{\epsilon _{\textbf{q}}^{2}+\Delta _{\textbf{q}}^{2}}\cdot \frac{\Delta _{\textbf{q}}}{\left\vert \Delta _{\textbf{q}}\right\vert }, \nonumber \\ \frac{\partial \left( 2\theta _{\textbf{q}}\right) }{\partial J_{z}} &=&-\frac{% J_{x}\sin q_{x}+J_{y}\sin q_{y}}{\epsilon _{\textbf{q}}^{2}+\Delta _{\textbf{q}}^{2}}\cdot \frac{\Delta _{\textbf{q}}}{\left\vert \Delta _{\textbf{q}}\right\vert }.\end{aligned}$$ Clearly, with these equations, we can in principle calculate the fidelity susceptibility along any direction in the parameter space according to Eq. (\[eq:deffsmetric\]). Here, we would like to point out that the same results can be obtained from the generalized Jordan-Wigner transformation used firstly by Feng, Zhang, and Xiang[@XYFeng07]. ![ (Color online) Finite size scaling analysis for the case of power-law divergence for system sizes $L=201, 301, \dots, 901$. The fidelity susceptibility, considered as a function of system size and driving parameter is a function of $L^\nu (J_z -J_z^{\rm max})$ only, and has the critical exponent $\nu=0.96$.[]{data-label="figure_scale"}](fig2.eps){width="8.5"} Following Kitaev [@Kitaev], we restrict our studies to the plane $J_x+J_y+J_z=1$ (see the large triangle in Fig. \[figure\_fs\]). According to his results, the plane consists of two phases, i.e., a gapped A phase with Abelian anyon excitations and a gapless B phase with non-Abelian excitations. The two phases are separated by three transition lines, i.e. $J_x=1/2$, $J_y=1/2$, and $J_z=1/2$ which form a small triangle in the B phase. Generally, we can define an arbitrary evolution line on the plane. Without loss of generality, we first choose the line as $J_{x}=J_{y}$ (see the dashed line in the triangle of Fig. \[figure\_fs\]). Then the fidelity susceptibility along this line can be simplified as $$\begin{aligned} \chi _{F} = \frac{1}{16}\sum_{\textbf{q}}\left[ \frac{\sin q_{x}+\sin q_{y}}{\epsilon _{\textbf{q}}^{2}+\Delta _{\textbf{q}}^{2}}\right] ^{2}.\end{aligned}$$ The numerical results of different system sizes are shown in Fig. \[figure\_fs\]. First of all, the fidelity susceptibility per site, i.e. $\chi_F/N$ diverges quickly with increasing system size around the critical point $J_z=1/2$. This property is similar to the fidelity susceptibility in other systems, such as the one-dimensional Ising chain [@Zanardi06] and the asymmetric Hubbard model[@SJGu072]. Secondly, $\chi_F/N$ is an intensive quantity in the A phase ($J_z>1/2$), while in the B phase, the fidelity susceptibility also diverges with increasing system size. Thirdly, the fidelity susceptibility shows many peaks in the B phase, the number of peaks increases linearly with the system size $L$ (see the left upper inset of Fig. \[figure\_fs\]). The phenomena of fidelity susceptibility per site in the B phase have not been found in other systems previously, to our knowledge, so that they are rather impressive. ![(Color online) Fidelity susceptibility and a few low-lying excitations as a function of $J_z$ in a small portion of the evolution line for system size $L=51$.[]{data-label="figure_fsen"}](fig3.eps){width="8.5"} To study the scaling behavior of the fidelity susceptibility around the critical point, we perform a finite-size scaling analysis. Since the fidelity susceptibility in the A phase is an intensive quantity, the fidelity susceptibility in the thermodynamic limit, scales as [@SJGu072] $$\begin{aligned} \frac{\chi_F}{N} \propto \frac{1}{|J_z -J_z^c|^\alpha}.\end{aligned}$$ around $J_z^c=1/2$. Meanwhile, the maximum point of $\chi_F$ at $J_z=J_z^{\rm max}$ for a finite sample behaves as $$\begin{aligned} \frac{\chi_F}{N} \propto L^\mu,\end{aligned}$$ with $\mu=0.507\pm 0.0001$ (see the inset of Fig. \[figure\_scale\]). According to the scaling ansatz, the rescaled fidelity susceptibility around its maximum point at $J_z^{\rm max}$ is just a simple function of the rescaled driving parameter, i.e., $$\begin{aligned} \frac{\chi_F^{\rm max}-\chi_F}{\chi_F} =f[L^\nu (J_z - J_z^{\rm max})].\end{aligned}$$ where $f(x)$ is a universal scaling function and does not depend on the system size, and $\nu$ is the critical exponent. The function $f(x)$ is shown in Fig. \[figure\_scale\]. Clearly, the rescaled fidelity susceptibilities of various system sizes fall onto a single line for a specific $\nu=0.96\pm 0.005$. Then the critical exponent $\alpha$ can be obtained as $$\begin{aligned} \alpha=\frac{\mu}{\nu}=0.528\pm 0.001.\end{aligned}$$ ![(Color online) Fidelity susceptibility as a function of $J_x=2/3-J_y$ along the dashed line shown in the triangle for various system sizes $L=101, 303, 909$. []{data-label="figure_fs2"}](fig4.eps){width="8"} One of the most interesting observations is that a huge number of peaks appear in the B phase. The scaling analysis shows that the number of peaks is proportional to the system size. Physically, a peak means that the ground state can not adiabatically evolve from one side of the peak to the other side easily because the two ground states have distinct features. From this point of view, the ground state in the B phase might be stable to a adiabatic perturbation. Moreover, the existence of many peaks can also be reflected by reconstruction of the energy spectra. For this purpose, we choose a small portion of the evolution line and plot both the fidelity susceptibility and a few low-lying excitations in Fig. \[figure\_fsen\]. Since the fidelity is inversely proportional to the energy gap \[Eq. (\[eq:metric\])\], the location of each peak corresponds to a gap minimum. Similarly, we can also choose the system evolution line as $J_z=1/3$, the fidelity susceptibility then takes the form $$\begin{aligned} \chi _{F} = \frac{1}{36}\sum_{\textbf{q}}\left[ \frac{\left( \sin q_{x}-\sin q_{y}\right) +2\sin \left( q_{x}-q_{y}\right) }{\epsilon _{\textbf{q}}^{2}+\Delta _{\textbf{q}}^{2}}\right] ^{2}.\end{aligned}$$ The numerical results for this case are shown in Fig. \[figure\_fs2\]. The results are qualitatively similar to those of previous cases. In the B phase, there still exist many peaks. Both the number and the magnitude of the peaks increase with the system size, while in the A phase, the fidelity susceptibility becomes an intensive quantity. Long-range correlation and fidelity susceptibility {#sec:lrc} ================================================== Follow You, [*et al.*]{} [@WLYou07], the fidelity susceptibility is a combination of correlation functions. Precisely, for a general Hamiltonian $$\begin{aligned} H=H_0+\lambda H_I,\end{aligned}$$ the fidelity susceptibility can be calculated as $$\begin{aligned} \chi_F=\int\tau\left[ \langle \Psi_0 |H_I(\tau) H_I(0)|\Psi_0\rangle -\langle\Psi_0|H_I|\Psi_0\rangle^2\right] d\tau\label{eq:fidelityfnal},\end{aligned}$$ with $\tau$ being the imaginary time and $$H_I(\tau)=e^{H(\lambda)\tau} H_I e^{-H(\lambda)\tau}.$$ Therefore, the divergence of the fidelity susceptibility at the critical point implies the existence of a long-range correlation function. Without loss of generality, if we still restrict ourselves to the plane $J_x+J_y+J_z=1$ and choose $J_z$ ($J_{x}=J_{y}$) as the driving parameter, the bond-bond correlation function is defined as $$\begin{aligned} C\left( \mathbf{r}_{1},\mathbf{r}_{2}\right) &=&\left\langle \sigma _{ \mathbf{r}_{1},1}^{z}\sigma _{\mathbf{r}_{1},2}^{z}\sigma _{\mathbf{r} _{2},1}^{z}\sigma _{\mathbf{r}_{2},2}^{z}\right\rangle \nonumber \\ &&-\left\langle \sigma _{\mathbf{r}_{1},1}^{z}\sigma _{\mathbf{r} _{1},2}^{z}\right\rangle \left\langle \sigma _{\mathbf{r}_{2},1}^{z}\sigma _{ \mathbf{r}_{2},2}^{z}\right\rangle \label{eq:correlationdef}.\end{aligned}$$ Here the subscripts $\mathbf{r}_{1},1$ and $\mathbf{r}_{1},2$ denote the two ends of the single $z$-bond at $\mathbf{r}_{1}$=($x, y$). In the vortex-free case, through Eqs. (\[eq:spinmajorana\]), (\[eq:aqcq\]), and (\[eq:ground\]), the spin operators $\sigma _{ \mathbf{r}_{1},1}^{z}\sigma _{\mathbf{r}_{1},2}^{z}$ can be expressed in the form of fermion operators. So we finally get $$\begin{aligned} \left\langle \sigma _{\mathbf{r}_{1},1}^{z}\sigma _{\mathbf{r} _{1},2}^{z}\right\rangle =\left\langle \sigma _{\mathbf{r}_{2},1}^{z}\sigma _{\mathbf{r}_{2},2}^{z}\right\rangle =\frac{1}{N}\sum_{\mathbf{q}}\frac{ \epsilon _{\mathbf{q}}}{E_{\mathbf{q}}}\end{aligned}$$ and $$\begin{aligned} &&\left\langle \Psi _{0}\right\vert \sigma _{\mathbf{r}_{1},1}^{z}\sigma _{% \mathbf{r}_{1},2}^{z}\sigma _{\mathbf{r}_{2},1}^{z}\sigma _{\mathbf{r}% _{2},2}^{z}\left\vert \Psi _{0}\right\rangle \nonumber \\ &=&\frac{1}{N^{2}}\sum_{\mathbf{q},\mathbf{q}^{\prime }}\left\{ \cos \left[ \left( \mathbf{q}-\mathbf{q}^{\prime }\right) \left( \mathbf{r}_{1}-\mathbf{r% }_{2}\right) \right] -1\right\} \nonumber \\ &&\times \frac{\left( \Delta _{\mathbf{q}}\Delta _{\mathbf{q}^{\prime }}-\epsilon _{% \mathbf{q}}\epsilon _{\mathbf{q}^{\prime }}\right)}{ E_{\mathbf{q}}E_{\mathbf{% q}^{\prime }}} \label{eq:Correlation4343}\end{aligned}$$ with $\mathbf{q}\neq\mathbf{q}^{\prime }$ and $\mathbf{r}_{1}\neq\mathbf{r}_{2}$. The same results can also be obtained by using the Jordan-Wigner transformation method [@XYFeng07; @CHD]. ![(Color online) Bond-bond correlation function as a function of distance $r$ for various $J_z$ and a finite sample of $L=100$, where $\textbf{r}_1 -\textbf{r}_2=(r,r)$. Downward peaks in top lines are due to zero-point crossing.[]{data-label="figure_correlation"}](fig5.eps){width="8.5"} ![(Color online) Fidelity susceptibility and the correlation function at $\textbf{r}_1 -\textbf{r}_2=(L/2, L/2) $ as a function of $J_z$ for a finite sample of $L=100$. []{data-label="figure_fscorr"}](fig6.eps){width="8.5"} We show the dependence of the correlation function Eq. (\[eq:correlationdef\]) on the distance for a finite sample of $L=100$ in Fig. \[figure\_correlation\]. Obviously, the lines can be divided into two groups. In the gapless phase ($J_z<1/2$), the correlation function decays algebraically, while in the gapped phase ($J_z>1/2$), it decays exponentially. If $J_z<1/2$, the denominator in Eq. (\[eq:Correlation4343\]) has two zero points, which are of order $1/N$ in the large $N$ limit. Their contribution causes the summation to be finite in the thermodynamic limit. Then using the stationary phase method, we can evaluate the exponents of the correlation function at long distance to be 4, i.e., $$\begin{aligned} C\left(\mathbf{r}_1,\mathbf{r}_2\right) \propto \frac{1}{|\mathbf{r}_1-\mathbf{r}_2|^4}.\end{aligned}$$ From Fig. \[figure\_correlation\], the average slope of the top three lines around $r=10$ is estimated to be $4.05$, which is slightly different from $4$. Nevertheless, we would rather interpret the difference as due to both finite size effects and numerical error. On the other hand, if $J_z>1/2$, the phase is gapped and the denominator in Eq. (\[eq:Correlation4343\]) does not have zero point on the real axis. Therefore, the whole summation is strongly suppressed except for the case of small $|\mathbf{r}_{1}-\mathbf{r}_{2}|$, whose range actually defines the correlation length. In order to evaluate the correlation, we need to extend the integrand (in the thermodynamic limit) in Eq. (\[eq:Correlation4343\]) to the whole complex plane, where we can find two singular points. Using the steepest descent method, we can evaluate the correlation length to be $$\begin{aligned} \frac{1}{\xi}=2\sinh^{-1}\frac{\sqrt{2J_z -1}}{1-J_z}.\end{aligned}$$ Obviously, the correlation length becomes divergent as $J_z\rightarrow 0.5^+$. This property can also be used to signal the QPT occurring in the Kitaev honeycomb model in addition to the fidelity and Chern number [@Kitaev]. The correlation length we obtained is the same as that of the string operators [@CHD], which, however, is a non-local operator. Although it is not easy to calculate the fidelity susceptibility from the correlation function directly due to the dynamic term in Eq. (\[eq:fidelityfnal\]), our conjecture is confirmed for the present model. That is the divergence of the fidelity susceptibility is related to the long-range correlations. Fig. \[figure\_fscorr\] is illustrative. The correlation function at $\textbf{r}_1 -\textbf{r}_2=(L/2, L/2)$, in spite of its smallness, remains nonzero in the region $J_z < 1/2$, but it vanishes in $J_z > 1/2$. For the former, the oscillating structures of the two lines meet each other. summary and discussion {#sec:sum} ====================== In summary, we have studied the critical behavior of the fidelity susceptibility where a topological phase transition occurrs in the honeycomb Kitaev model. Though no symmetry breaking exists and no local order parameter in real space can be used to describe the transition, the fidelity susceptibility definitely can indicate the transition point. We found that the fidelity susceptibility per site is an intensive quantity in the gapped phase, while in the gapless phase, the huge number of peaks reflects frequent spectral reconstruction along the evolution line. We also studied various scaling and critical exponents of the fidelity susceptibility around the critical points. Based on the conclusions from the fidelity, we further studied the bond-bond correlation function in both phases. We found that the bond-bond correlation function, which plays a dominant role in the expression for the fidelity susceptibility, decays exponentially in the gapped phase, but algebraically in the gapless phase. The critical exponents of the correlation function in both the gapless and gapped phases are calculated numerical and analytically. Therefore, in addition to the topological properties of the Kitaev honeycomb model, say, the Chern number, we found that both the fidelity susceptibility and the bond-bond correlation functions can be used to witness the QPT in the model. *Note added*. After finishing this work, we noticed that a work on the fidelity per site instead of the fidelity susceptibility in a similar model appeared[@JHZhao0803]. We thank Xiao-Gang Wen, Yu-Peng Wang, Guang-Ming Zhang, and Jun-Peng Cao for helpful discussions. This work is supported by CUHK (Grant No. A/C 2060344) and NSFC. [99]{} H. T. Quan, Z. Song, X. F. Liu, P. Zanardi, and C. P. Sun, Phys. Rev. Lett. [**96**]{}, 140604 (2006). P. Zanardi and N. Paunkovi¡¬c, Phys. Rev. E **74**, 031123 (2006). P. Zanardi, M. Cozzini, and P. Giorda, J. Stat. Mech. L02002 (2007); M. Cozzini, P. Giorda, and P. Zanardi, Phys. Rev. B **75**, 014439 (2007); M. Cozzini, R. Ionicioiu, and P. Zanardi, Phys. Rev. B **76**, 104420 (2007). P. Buonsante and A. Vezzani, Phys. Rev. Lett. **98**, 110601 (2007). P. Zanardi, H. T. Quan, X. Wang, and C. P. Sun, Phys. Rev. A [**75**]{}, 032109 (2007). P. Zanardi, H. T. Quan, X. G. Wang, and C. P. Sun, Phys. Rev. A [**75**]{}, 032109 (2007). P. Zanardi, P. Giorda, and M. Cozzini, Phys. Rev. Lett. **99**, 100603 (2007). W. L. You, Y. W. Li, and S. J. Gu, Phys. Rev. E [**76**]{}, 022101 (2007). H. Q. Zhou and J. P. Barjaktarevic, arXiv: cond-mat/0701608; H. Q. Zhou, J. H. Zhao, and B. Li, arXiv:0704.2940; L. C. Venuti and P. Zanardi, Phys. Rev. Lett. **99**, 095701 (2007). S. Chen, L. Wang, S. J. Gu, and Y. Wang, Phys. Rev. E [**76**]{} 061108 (2007); S. Chen, L. Wang, Y. Hao, and Y. Wang, arXiv:0801.0020. S. J. Gu, H. M. Kwok, W. Q. Ning, and H. Q. Lin, arXiv:0706.2495. M. F. Yang, Phys. Rev. B **76**, 180403 (R) (2007); Y. C. Tzeng and M. F. Yang, Phys. Rev. A [**77**]{}, 012311 (2008). N. Paunkovic, P. D. Sacramento, P. Nogueira, V. R. Vieira, and V. K. Dugaev, arXiv:0708.3494. H. Q. Zhou, R. Orus, and G. Vidal, Phy. Rev. Lett. [**100**]{} 080601 (2008). A. Hamma, W. Zhang, S. Haas, and D. A. Lidar, arXiv:0705.0026. M. A. Nilesen and I. L. Chuang, [*Quantum Computation and Quantum Information*]{} (Cambridge University Press, Cambridge, England, 2000). S. Sachdev, [*Quantum Phase Transitions*]{} (Cambridge University Press, Cambridge, England, 1999). X. G. Wen, *Quantum Field Theory of Many-Body Systems* (Oxford University, New York, 2004). J. M. Kosterlitz and D. J. Thouless, J. Phys. C [**6**]{}, 1181(1973). A. Kitaev, Ann. Phys. **303**, 2 (2003); Ann. Phys. **321**, 2 (2006). X. G. Wen, Phys. Rev. Lett. **90**, 016803 (2003); M. A. Levin and X. G. Wen, Phys. Rev. B **71**, 045110 (2005). J. Preskill, *Topological quantum computation*, http:// www.theory.caltech.edu/people/preskill/ph229/ (2004). J. K. Pachos, IJQI **4**, 947 (2006); Ann. Phys. **322**, 1254 (2007). S. D. Sarma, M. Freedman, C. Nayak, S. H. Simon, A. Stern, Rev. Mod. Phys., accepted. X. Y. Feng, G. M. Zhang, and T. Xiang, Phys. Rev. Lett. **98**, 087204 (2007); D. H. Lee, G. M. Zhang, and T. Xiang, Phys. Rev. Lett. **99**, 196805 (2007). H. D. Chen and J. P. Hu, arXiv: cond-mat/0702366; H. D. Chen and Z. Nussinov, J. Phys. A: Math. Theor. **41**, 075001 (2008); Z. Nussinov, G. Ortiz, arXiv: cond-mat/0702377. Y. Yu, arXiv: 0704.3829; Y. Yu, Z. Q. Wang, arXiv: 0708.0631; T. Y. Si and Y. Yu, arXiv:0709.1302; T. Y. Si and Y. Yu, arXiv: 0712.4231 . S. Yang, D. L. Zhou, and C. P. Sun, Phys. Rev. B [**76**]{}, 180404(R) (2007). S. P. Kou and X. G. Wen, arXiv: 0711.0571. S. Mandal and N. Surendran, arXiv: 0801.0229. K. P. Schmidt, S. Dusuel, and J. Vidal, Phy. Rev. Lett. [**100**]{}, 057208 (2008); S. Dusuel, K. P. Schmidt, and J. Vidal, arXiv:0802.0379. J. H. Zhao and H. Q. Zhou, arXiv:0803.0814.

--- abstract: 'Multiple image gravitational lens systems, and especially quads are invaluable in determining the amount and distribution of mass in galaxies. This is usually done by mass modeling using parametric or free-form methods. An alternative way of extracting information about lens mass distribution is to use lensing degeneracies and invariants. Where applicable, they allow one to make conclusions about whole classes of lenses without model fitting. Here, we use approximate, but observationally useful invariants formed by the three relative polar angles of quad images around the lens center to show that many smooth elliptical+shear lenses can reproduce the same set of quad image angles within observational error. This result allows us to show in a model-free way what the general class of smooth elliptical+shear lenses looks like in the three dimensional (3D) space of image relative angles, and that this distribution does not match that of the observed quads. We conclude that, even though smooth elliptical+shear lenses can reproduce individual quads, they cannot reproduce the quad population. What is likely needed is substructure, with clump masses larger than those responsible for flux ratio anomalies in quads, or luminous or dark nearby perturber galaxies.' author: - | Addishiwot G. Woldesenbet$^{1}$[^1] and Liliya L. R. Williams$^{1}$\ $^{1}$ School of Physics and Astronomy, University of Minnesota, 116 Church Street SE, Minneapolis, MN 55455 title: 'Model-free analysis of quadruply imaged gravitationally lensed systems and substructured galaxies' --- \[firstpage\] gravitational lensing: strong – galaxies: fundamental parameters – dark matter Introduction ============ The field of multiple image gravitational lensing was born about 35 years ago with the discovery of Q0957+561 [@w79] and PG1115+08 [@w80], which were followed by discoveries of many more doubly and quadruply imaged quasars. The theoretical understanding of these systems grew up alongside the growing body of observations. The earliest works found that simple parametric forms for the mass distribution in the lens can account for the systems’ observables [@young80], and that image properties, namely positions, shapes, time delays and flux ratios can be derived from the suitably defined lensing potential [@bn86; @sef92]. It was also soon realized that strong lensing has degeneracies and invariants, and that these theoretical insights prove useful when lensing is used as a tool. For example, mass sheet degeneracy, and some other degeneracies where different lenses reproduce exactly the same image observables have been well studied [@fgs85; @gsf88; @s00; @ld12], especially in relation to mass modeling, where degeneracies affect the conclusions about the mass profiles of lenses. The first lensing invariant discovered was the magnification invariant. It states that the sum of signed magnifications for all lensed images of any source within the caustic of a given lens is a constant. This constant can be the same for more than one lensing potential, and is often independent of the parameters characterizing the potential [@wm95; @d98]. [@wm00] and [@he01] showed that image positions can also be used in forming invariants. Aside from their intrinsic theoretical value, the invariants allow us to rule out some lens models without even fitting the lens. To quote [@he01], “The major application of lensing invariants is to shortcut the modeling process.” Because observational uncertainties on the image observables are never zero, not only exact but also approximate degeneracies and invariants are important for the practical work. For example, [@ss14] start with an exact invariance transformation that applies to axisymmetric lenses, and show that it survives in an approximate, but still useful form when applied to a wider range of lens models. Our work in the present paper is in the same vein: we study near invariants that provide useful insights without mass modeling, and have important consequences for mass substructure in lenses. Our analysis is of quads; it involves image positions only, and does not rely on magnifications. The relative image locations of a quad are specified by six numbers, say, $(x,y)$ coordinates of any of the three images with respect to the 4th. Lens mass modeling—parametric or free-form—attempts to reproduce all six numbers at once. In [@w08] and [@us] (hereafter WW12) we started developing a new way of looking at quads, that considers only the three polar angular coordinates of quads, and disregards the three $r$ coordinates. We showed that information about the mass distribution of certain lens models contained in the azimuthal, or angular coordinates of images around the center of the lens is almost independent of that contained in the radial coordinates, and that the angular coordinates of images show approximate degeneracies. Specifically, if a lens with double mirror symmetry can reproduce the relative images angles of a quad, then all lenses with double mirror symmetry can reproduce the relative image angles of that quad, though for different locations of the (unobservable) source. Here we continue developing the theory of relative image angles of quads by extending it to lenses represented by purely or approximately elliptical mass profiles with external shear. The lenses in this general class of mass models are important to study because they fit observed quads well, and are common in parametric lens modeling. Since we study a wide class of such lenses, and identify similarities, we are able to make conclusions that are independent of a specific lens model. Our analysis suggests that even though these common models are able to reproduce observed quads one at a time, they are unable to reproduce them [*as a population*]{}. Instead, substructured lenses or luminous or dark nearby perturbers are likely needed. Because of their importance in the $\Lambda$CDM cosmological model [@Moore; @k99], substructure has been the focus of many recent papers. The main method employed for finding substructure uses flux ratios of close pairs or triples of images of quad lenses. In the absence of substructure clumps, i.e. when the potential is smooth, the magnifications of these images obey certain relations [@bn86; @sw92]. Substructure induces deviations from these relations, or anomalies [@ms98; @mz02]. The substructure finding methods based on flux anomalies and image positions are complementary because the former are sensitive to small clumps located close to the images, while the latter are less easily perturbed and respond to larger clumps, $\gtrsim 10^9-10^{10} M_\odot$, located anywhere around the Einstein radius. The two methods also differ in other respects. Image fluxes are subject to different effects, depending on wavelength, like microlensing by stars in the optical and near infra-red, and propagation effects in the radio, and it is still unclear how major a role these factors play in the observed flux anomalies [@xu14]. Image positions are affected only by relatively massive clumps. This paper is organized as follows. In Section \[typeII\] we define our classification of lens types based on the caustics they produce. In Sections \[powerlaw\] and \[nonpowerlaw\] we discuss the detailed properties of lenses that belong to what we call Type II lenses, which are the focus of this paper. Readers interested primarily in substructure can skip directly to Section \[realquads\]. Section \[conc\] summarizes our findings. Type II lens models {#typeII} =================== Parametric modeling, where parameters such as ellipticity, shear, position angle, etc. identify a lens model, is a commonly used method to theoretically represent gravitational lenses. Simple models obeying certain symmetries are used to fit observed properties of lens systems, such as image positions, time delays, and magnifications. The symmetries could be either in the isopotential or isodensity contours of lenses with different radial density profiles. Examples of such axisymmetric models that are widely used are Elliptic Mass Distribution (EMD) and Elliptic Potential (EP)[@KassKov] with different radial profile such as Isothermal and NFW. The amount of constraining information that can be obtained from a given set of observed images is very limited, and inadequate to fully describe the details of the lensing mass. This means the lens equation is severely under constrained allowing it to admit multiple solutions. But any symmetry in how images are distributed on the lens plane is a direct consequence of the underlying symmetry of the lens and the source position in the source plane. For example, any axially symmetric lens, independent of the radial density profile, would give rise to an Einstein ring for a central source. Therefore categorizing and studying lenses that share sets of symmetries using parameters is a practical approach that does not contradict observational data. In previous work by the authors (WW12) three classes of lenses were introduced. The classification is based on azimuthal symmetries of isopotential or isodensity contours of lenses, and is independent of the radial density profile. Lens models obeying twofold symmetry (symmetric about two orthogonal axes) in the lens plane were termed Type I lenses, and breaking this symmetry once resulted in Type II lenses. Type III encompassed all other models. A more precise categorization, which we will adopt in this paper, can be achieved by using the symmetries of the diamond caustics. Type I lenses are those that give rise to caustics with twofold rotational symmetry[^2] and double mirror symmetries, while Type II lenses are the ones that give rise to caustics obeying only twofold rotational symmetry. In this new definition of Type I and II we also limit ourselves to lenses with diamond caustics and exclude those that produce caustics with higher order catastrophes. Such constraint is justified for our analysis which aims to understand the population of observed lens systems. Higher order catastrophes can produce lenses with more than five images, which are not observed in galaxy lenses. Type III lenses include everything except Type I and Type II, for example, substructured lenses. This paper examines Type II models in detail. Elliptical potentials or mass distributions of arbitrary ellipticity and density profile have twofold and double mirror symmetries. By introducing external shear at a non-zero angle with respect to the ellipticity axes, the double mirror symmetry is broken giving rise to Type II lenses. In this paper we look at Type IIs with power law and NFW density profiles. We identify a lens by three parameters: ellipticity, $\epsilon$, external shear, $\gamma$, and the angle, $\beta$, between shear and the ellipticity principal axis. By definition, $\beta$ lies between $0^{\circ}$ and $90^{\circ}$. Our basic analysis method uses the relative angular distribution of the four images of quads[^3] about the center of lens. It is applicable to any lens system, such as quasar-galaxy or galaxy-galaxy, where the center of the lens is known and the four images are point like. This approach was introduced in [@w08] and refined in WW12. The method is statistical in nature in the sense that it works with quads as a class to draw conclusions about the mass distribution in galaxy hosts of quad lenses. We will show that properties related to quad image angles are approximately independent of the radial mass density profile of the lens, and that Type I and II lenses cannot account for the observed population of quads. Power law potentials as examples of Type II {#powerlaw} =========================================== Type II lenses are a wide class. Our strategy is to study a few representative examples of Type IIs in detail, note their similarities and differences, and draw conclusions based on these. Our examples are chosen to be representative of the models used in the lensing literature and to resemble the real galaxy lenses. Furthermore, for computational ease, we use models whose lensing potentials are expressed by simple analytical functions. Power law lensing potentials are one example that satisfy all the above criteria. They have a general form, $$\phi(r,\theta) = b r^\alpha \sqrt{1-\epsilon \cos ( 2 \theta)}+\frac{\gamma}{2} r^2 \cos ( 2[\theta-\beta]). \label{eq1}$$ In the above expression $b$ is the normalization factor, $\epsilon$ is an ellipticity parameter (henceforth ellipticity)[^4], and $r$ is the sky-projected distance from the lens center. The second term is external shear of strength $\gamma$ oriented at an angle of $\beta$ relative to the ellipticity’s principal axis. The exponent $\alpha$ in the first term is physically constrained to be between zero ($\alpha<0$ implies decreasing total enclosed mass with increasing $r$), and two ($\alpha>2$ implies that the density is increasing with $r$), but observations constrain it even further to be around $1$ [@SLACS3]. Theoretically, we can use any values of $\gamma$ as long as the lens gives rise to quads, whereas $\epsilon$ is limited to be between 0 and 1 by definition. In addition, $\epsilon$ and $\gamma$ are also restricted from above by the requirement that the corresponding isodensity contours are not peanut shaped, which happens, for example, for $\alpha\approx 1.2$ and $\epsilon\approx 0.25$, or $\alpha\approx 1$ and $\epsilon\approx 0.15$. Singular Isothermal Elliptical Potential with External Shear (SIEP+shear) {#siep} ------------------------------------------------------------------------- In this section we use singular isothermal elliptical potential with external shear to demonstrate detailed properties of Type II lenses. SIEP+shear, given by eq. \[eq1\] with $\alpha=1$, is chosen because it is the simplest form of power law potential that is semi-analytically treatable. Different combinations of $\epsilon$, $\gamma$ and $\beta$ give rise to different diamond caustics (based on orientation, elongation of cusps and curving of folds) but all obey the twofold rotational symmetry that identifies Type II lenses. One interesting property of Type IIs concerns the isopotential contours. The main galaxy, i.e. first term in equation \[eq1\] with $\alpha=1$ gives rise to contours all of which have the same ellipticity axis, but the addition of external shear results in slightly twisting isopotential contours, i.e. angular orientation of the principal axis of each contour of the total lens potential changes with radius. This is true of the contours in the first panel of Figure \[fig1\], but is hard to see because the degree of twisting is small. We parametrize a given diamond caustic by angles formed by its diagonals and the ratio of the length of the diagonals. We do not consider the curvatures of the folds, i.e. there could be two diamond caustics of the same diagonal ratios and angles but with different fold curvatures. In the case of Type I lenses, the caustic diagonals are perpendicular to each other while for Type II caustics, the diagonals form angles other than $90^{\circ}$; see the middle panel of Figure \[fig1\]. Another interesting property of SIEP+shear is that the singularity at the center is not a true critical point; mapping it to the source position using the lens equation gives rise to a set of points forming an oval which acts like an oval caustic, and is called a ‘cut’ [@Kov]. Even though this set of points does not satisfy the common definition of a caustic, where the determinant of the Jacobian of the lens mapping is zero, it still acts like one; crossing the cut results in a change of image multiplicity. Quads are formed when a source is within the diamond caustic and the cut. The angular distribution of the four images about the center of the lens can be uniquely represented by three relative angles $\theta_{ij}$ (the acute angle between image $i$ and image $j$), where image $i$ is the $i^{th}$ arriving image. Image ordering for synthetic lenses is always known. Images in observed quads can be correctly time ordered in most cases using image morphology [@sw03]. In this paper we use the same set of relative angles as in WW12: $\theta_{12}$, $\theta_{34},$ and $\theta_{23}$. A 3D space can now be formed using these angles, where a point in the space represents a single quad. Given a lens characterized by ($\epsilon,\gamma,\beta$), one can generate a large number of quads arising from different source positions. Numerical calculation shows that the normalization factor, $b$, has no impact on the angular distribution of the images. This also holds true for the NFW lens discussed later in this paper. The distribution of the corresponding points in the 3D angle space can be used as an alternative way of characterizing the lens. Quads produced by Type I lenses lie on a slightly curved surface, which we called the Fundamental Surface of Quads (FSQ) with a peak at $(\theta_{12},~\theta_{34},~\theta_{23})=(180^\circ,~180^\circ,~90^\circ)$. Quads from all Type I lenses lie very close to the FSQ, making it a nearly invariant surface, and a useful reference. On the other hand, quads arising from Type II lenses form two separate surfaces, each usually confined to two different portions in the 3D angle space separated by the FSQ. Based on the positions relative to the FSQ, the two surfaces are identified as upper or lower surface. Like the FSQ, each of these surfaces has two well defined edges which corresponds to $\theta_{12}=180^\circ$ and $\theta_{34}=180^\circ$. These two edges of each of the two surfaces meet to form two peaks, one above and one below the FSQ peak as shown in the left panel of Figure \[fig2\]. Each surface sits at the farthest point relative to the FSQ at its peak, i.e. near $\theta_{23}\sim 90^\circ$. The edges of the surfaces come closer to each other at lower values of angle $\theta_{23}$ while their middle parts stay farther apart (right panel of Figure \[fig2\]). The two $\theta_{23}$ angles of the peaks of the two surfaces of an SIEP+shear lens (or any Type II lens) are supplementary angles (i.e. they add to $180^\circ$). We use the $\theta_{23}$ value of the peak quad of the upper of the two surfaces, $\theta_{23,p}$, as an index to parametrize the surfaces of a given lens in the 3D space. ### Potential, caustic and 3D angles space {#genprops} As described above, Type II lenses can be characterized in three different spaces (Figure \[fig1\]). The potential space, where $\epsilon$, $\gamma$ and $\beta$ parametrize the lens, the caustic space where the angles between caustic diagonals and the ratio between their lengths characterize the caustic, and the 3D angles space where $\theta_{23,p}$ value of the peak of the top surface is the characteristic parameter. Now, we describe how the three spaces and their corresponding parameters are related. [**The central source.**]{}  The source at the center of the lens corresponds to the center of the diamond caustic and is mapped to two pairs of images with each image of a given pair having the same arrival time. We arbitrarily choose one of the first arrivals as image 1 and the other image 2. The same is done with images 3 and 4. The two images with the same arrival time sit at $180^\circ$ of each other on the lens plane, therefore $\theta_{12}=\theta_{34}=180^\circ$. For Type I’s, the line connecting opposite images (those with the same arrival time) are orthogonal, so $\theta_{23}=90^\circ$, but this is not the case for Type IIs, where $\theta_{23}$ is either acute or obtuse, depending on the choice of ordering, but the two possibilities are always supplementary angles. The maximum angular separation of images 2 and 3 is attained when the source is at the center of the caustic, and this separation decreases as the source moves radially out from the center. In the 3D angles space, the central source corresponds to two degenerate quads, each located at the peak of each of the two surfaces, at $(180^\circ,180^\circ,\theta_{23,p})$, and $(180^\circ,180^\circ,180^\circ-\theta_{23,p})$. SIEP+shear has an interesting property connecting the caustic and the images arising from the central source. The two supplementary angles formed by the diagonals of a diamond caustic are same as the $\theta_{23,p}$ angles of the 3D space. The symbolic expressions for the angles formed by the diagonals of the diamond caustics and $\theta_{23,p}$ are rather large when expressed in terms of the parameters $\epsilon$, $\gamma$, and $\beta$. Therefore, we compared them by calculating the difference between them using Wolfram Mathematica for 10,000 different combinations of ($\epsilon$, $\gamma$, and $\beta$) which resulted in exactly zero up to 13 decimal place precision. (In addition, the ratio of the caustic diagonal lengths is the same as the ratio of the radial positions of images 2 and 3 from the center of the lens, but we do not consider the image distance ratios in this paper, so we will not explore this relation any further.) [**Sources on the caustic diagonals.**]{}  The images of sources on the diagonals of the caustic form the two outer edges of the sheets, ($180^\circ,\theta_{34},\theta_{23}$) and ($\theta_{12},180^\circ,\theta_{23}$). Just like the central source, sources on the diagonals results in degenerate images in terms of arrival time. It is important to note that this degeneracy is only for source positions along the diagonals. All other sources within the caustic are mapped to quads that form the body of the 3D surfaces, with each image arriving at distinctly different time from the rest. Therefore, the two sheets are completely different and not a result of ordering choice of the equally arriving images. [**The rest of the sources.**]{}  The rest of the two sheets in the 3D angles space comprises quads arising from sources within the body of the diamond caustic. As shown in the top panels of Figure \[fig3\] there is a bifurcation about the diagonal of the caustic. Continuously crossing a diagonal, i.e. moving from one quadrant of the diamond caustics to another (for example, from the black region to the yellow, lighter shade, one) results in a jump between the two sheets of the 3D angles space. Sources on straight lines of constant source position angle in the source plane (various colored lines in the caustic shown in the lower left panel of Figure \[fig3\]) result in quads that form non-crossing monotonic, but not necessarily straight curves on the 3D surfaces (lower right panel). [**The inversion symmetry of Type II lenses.**]{}  The twofold (inversion) symmetry of the Type II lens diamond caustic also applies to the potential, and is further reflected in the 3D angles space. Source positions from the two opposite quadrants of the caustic generate quads that form one of the sheets. The remaining two quadrants produce the second sheet. In Figure \[fig3\] the two yellow, lighter shade, (black) quadrants of the caustic shown in the upper left panel generate the yellow, lighter shade, (black) surface in the 3D angles space shown in the upper right panel. So the inversion symmetry reduces the number of distinct caustic quadrants from four to two. [*The existence of the two surfaces in the 3D space of relative image angles is the most important characteristic of Type II lenses.*]{} Note that the properties described in this subsection are not unique to SIEP+shear lenses, but are common to all Type IIs. The inversion symmetry of Type II, which is their defining property, naturally predicts the two surfaces in the 3D angles space. We have tested the existence of the two sheets on many more lenses than are presented in this paper (for example, several Sersic type models), and have confirmed the qualitative behavior of the two sheets. At large $\theta_{23}$ the two surfaces are always found on the opposite sides of the FSQ in the 3D angles space (left panel of Figure \[fig2\]), and the separation of the two peaks depends on how much the lens potential deviates from that of Type I. At small $\theta_{23}$ the edges of the two sheets approach each other and the FSQ (right panel of Figure \[fig2\]). ### Near degeneracies of lens models with different sets of ($\epsilon$, $\gamma$, $\beta$) {#degen} As mentioned earlier, we chose to parametrize the distribution of quads in the 3D angles space by $\theta_{23,p}$, the angle between the 2nd and 3rd arriving images of the source located at the center of the lens. The full distribution of quads from a single lens are the two surfaces or sheets (right panel of Figure \[fig1\], or Figure \[fig2\]). One may ask if it is adequate to represent the surfaces with just a single point, $\theta_{23,p}$? Figure \[fig4\] shows contour surfaces of constant $\theta_{23,p}$ in $\gamma$ vs. $\epsilon$ vs. $\beta$ space. Any two points on a given surface represent two different lenses in the potential space, each characterized by a different set of ($\epsilon,~\gamma,~\beta$), but sharing the same $\theta_{23,p}$. The two contour surfaces shown have $\theta_{23,p}=91^\circ$ and $\theta_{23,p}=92^\circ$. In the caustic space these lenses have some commonalities: the diagonals of the caustics intersect at the same angle, since, as previously discussed, these angles are the same as $\theta_{23,p}$. However the folds and cusps of these caustics look different. Having matched different lenses based on their $\theta_{23,p}$ value, we would like to see how the corresponding surfaces in the 3D angle space compare. We carry out the comparison in two steps: first we compare the edges and then the bodies of the surfaces. To carry out these comparisons we picked four different lens models, each belonging to the same $\theta_{23,p} = 92^\circ$ contour. We identify each lens as L($\epsilon,~\gamma,~\beta,~\theta_{23,p}$). Table \[table1\] summarizes these lenses. For the first comparison, the ($180^\circ,\theta_{34},\theta_{23}$) edge of the lower of the two 3D surfaces for lenses A-D are plotted in Figure \[fig5\]. Visually they seem to form the same line. But in order to quantify any difference between the edges, we compare fit equations of the lines. The fit equations for Lenses A-D are $$\begin{array}{l} \theta_{23}=0.48617\theta_{34}+0.25271\\ \theta_{23}=0.48617\theta_{34}+0.252017\\ \theta_{23}=0.48621\theta_{34}+0.247357\\ \theta_{23}=0.48627\theta_{34}+0.240303 \end{array}$$ respectively, and the angles are expressed in degrees. The average of the median observational error of the three relative angles is $\sim 0.8^\circ$ (WW12) which is at least two orders of magnitude greater than the difference between the intercepts of the above equations. Similarly, the median observational error of the slope $\theta_{23}/\theta_{34}$ is $\sim 0.006$ which is again two orders of magnitude greater than the difference between the slopes of the four equations above. A corresponding comparison for the other edge of the 3D surface, the ($\theta_{12},180^\circ,\theta_{23}$), yields similar results. Therefore, the difference between the edges of these four, and presumably other different SIEP+shear lenses in the 3D space is negligible for all practical and observational purposes. For the second comparison we generate two sets of random source positions on one of the two surfaces of each lens. The quads from the first set are used to obtain an interpolation fit equation, which is then compared to the quads of the second set. This is our calibration, or cross-validation procedure. The comparison is done with the root mean square of errors, RMSE. The cross-validation for Lens A yields RMSE of $0.005^\circ$. The RMSE’s of Lenses B-D, when compared to the fit equation of Lens A are $0.011^\circ$, $0.029^\circ$, and $0.017^\circ$, respectively (see Table 1). A visual indication of how close these surfaces look is provided by Figure \[fig10\], which plots a certain projection of the 3D angles space (explained in detail in Section \[realquads\]) for Lenses A-D. The largest deviation is for Lens C, represented by (green) diamond symbols, which extend lower along the vertical axis by about $0.1$ degree compared to the other three lenses. In general, lenses with largest deviations from the rest of their family members are those with larger ratios of $\epsilon/\gamma$. The RMSE values and Figure \[fig10\] show that the surfaces are not mathematically identical, but because the deviations between lenses sharing the same $\theta_{23,p}$ are smaller than the typical observational error (even for Lens C), observationally speaking these four SIEP+shear lenses of different lens parameters ($\epsilon,~\gamma,~\beta$) are degenerate in the 3D angles space, i.e. in terms of the relative image angles. In other words, if a lens model described by ($\epsilon_1,~\gamma_1,~\beta_1$) is found to fit the relative angles of an observed quad, then any other lens characterized by the same $\theta_{23,p}$, but a different set ($\epsilon_2,~\gamma_2,~\beta_2$) will also be able to reproduce the image angles, though for a different location of the source. So the answer to the question raised at the top of this section is [*yes*]{}, a single point in the 3D angles space, $\theta_{23,p}$, adequately represents the rest of the surfaces. We note an interesting consequence of the degeneracy described above. Taking a $\beta$=const two dimensional slice through the surfaces in Figure \[fig4\] we get contours plotted in Figure \[fig6\]. These contours are symmetric about $\epsilon=\gamma$ line, which implies that $\epsilon$ and $\gamma$ are interchangeable. This is unexpected because this symmetry is not at all obvious from the equation of the lens potential, eq. \[eq1\] with $\alpha=1$. Later we will see that this symmetry applies to other potentials if the isopotential contours are purely elliptical (as defined in Section \[Non-iso\]). Note that in this paper we do not consider image distances from the lens center; two lenses degenerate in image angles need not be degenerate in image distances. Non-isothermal Elliptical Potentials with External Shear {#Non-iso} -------------------------------------------------------- As discussed when introducing the general power law potential, eq. \[eq1\], $\alpha$ is observationally constrained to be $\sim 1$ [@SLACS3]. Therefore, we extend our discussion of Type II lenses only somewhat beyond this value; we use $\alpha=0.9$ and $\alpha=1.2$. We expand $\alpha$ up to 1.2 so that we can explore lenses with higher values of ellipticity. Going beyond these limits results in the mass density contours becoming peanut shaped even for moderate values of ellipticity $\epsilon$. For $\alpha$ = 1.2 the two sheets in the 3D space starts crossing for small values of $\theta_{23}$ when the ratio $\epsilon/\gamma \gtrsim 1$. We note that the way $\alpha$ is introduced in eq. \[eq1\] creates lenses whose isopotential contours due to the main galaxy are not exactly elliptical; we call these hybrid potentials. The pure elliptical potentials are obtained by raising what we call the elliptical radius, $r\sqrt{1-\epsilon \cos{2 \theta}}$, to the power of $\alpha$ instead of raising just $r$. But the angular distribution of quads generated by pure elliptical potentials are not distinctly different from that of SIEP+shear, therefore our discussion in this section focuses on hybrid potentials. The $\gamma$ vs. $\epsilon$ contour is not symmetric about $\gamma = \epsilon$ line, with the contours rotated clockwise for $\alpha= 0.9$ and counter clockwise for $\alpha= 1.2$; see Figure \[fig5\]. But for pure elliptical lenses defined above, the contours remain symmetric independent of $\alpha$ and this symmetry extends to NFW profiles discussed in the next section, as already alluded to at the end of Section \[degen\]. We would like to know if the degeneracy property of SIEP+shear (Section \[degen\]) applies to non-isothermal lenses, i.e. if these lenses can be adequately characterized by just $\theta_{23,p}$. For that purpose we generate quad distributions for Lenses F - I; see Table \[table1\]. Lens F and G have $\alpha=1.2$, while H and I have $\alpha=0.9$ and all share the same peak of $\theta_{23,p}=93^\circ$. We use the second of the two tests introduced in Section \[degen\]. We find fit equations for one of the two surfaces of Lenses F and H, and compared these to the actual surfaces of Lenses G and I. The cross-validation RMSE for Lenses F and H are $0.013^\circ$ and $0.003^\circ$. The RMSE for Lens I as predicted by H is $0.264^\circ$. The RMSE for Lens G as predicted by F is $0.551^\circ$. However, the high $\epsilon$ to $\gamma$ ratio of Lens F ($\epsilon/\gamma=4.5$) results in its two surfaces in 3D angles space crossing each other, which makes it hard to separate out a single surface and find a fit equation for it. Therefore the latter value can be compared only approximately to other RMSE values. This comparison of different lenses within the same family (F vs. G and H vs. I) shows that non-isothermal hybrid potential lenses sharing the same peak $\theta_{23,p}$ are approximately degenerate in the 3D angles space (all RMSEs quoted above are smaller than observational uncertainty), but are not as similar to each other as those within the SIEP+shear family. We note, however, that this comparison might be affected by the fact that these calculations were done numerically; SIEP+shear potential is simple enough to be amenable to semi-analytical calculations, but non-isothermal hybrid potentials require fully numerical computations, and there are two places where numerical errors can creep in: (i) when selecting ($\epsilon,~\gamma,~\beta$) sets from the same $\theta_{23,p}$ contour, and (ii) when doing forward lensing of sources to produce images and measure their polar angles. These numerical errors could contribute to the differences in the 3D surfaces. In the next section, where these errors are also an issue, we carry out a test to assess the error arising from (i). As with isothermal lenses, the models that differ the most from the rest of the family are those with larger $\epsilon/\gamma$ ratios. For $\alpha=1.2$ larger $\epsilon/\gamma$ ratio also make the two surfaces in the 3D angle space cross the FSQ and each other. Nevertheless, the differences quoted above are at most of the order of observational errors. Therefore observations based on the relative image angles of quads can not discriminate between any of the lenses within the family given by eq. \[eq1\], as long as $\epsilon/\gamma\lesssim 2-2.5.$ Non Power law potentials as examples of Type II {#nonpowerlaw} =============================================== In this section we broaden our exploration of Type II lenses to include models with non power law density profiles. We chose NFW radial density profile [@NFW], which has an analytical form for its lensing potential [@NFWPotential]. Though the central density profiles of lensing galaxies are significantly affected by baryons and so typically have steeper than NFW slopes, we use this profile to explore how non power law profiles behave as Type II lenses. In order to make a Type II lens with NFW radial density profile we introduce external shear and ellipticity to the potential, $$\phi(r,\theta) = b\Biggl (\ln^2{\frac{r}{2}}-\frac{1}{4} \ln^2{\frac{1+\sqrt{1-r^2}}{1-\sqrt{1-r^2}}}\Biggr) +\frac{\gamma}{2} r^2 \cos ( 2[\theta-\beta]) \label{eqNFW}$$ where $r={x \sqrt{1-\epsilon \cos(2 \theta)}}/{r_{s}}$, $x$ is the polar radius in the lens plane, $r_{s}$ is the scale factor of NFW density profile, and $b$ is a normalization that is a function of $r_{s}$ and the characteristic density. The rest of the variables are as defined in previous sections. This potential is defined for $r<1$, i.e for $x<r_{s}$. The diamond caustic of this potential has the symmetry of Type II lenses, and gives rise to two surfaces in the 3D space of relative image angles of quads. As shown in upper right panel of Figure \[fig6\], $\gamma$ vs. $\epsilon$ contour map of constant $\theta_{23,p}$ is symmetric with respect to $\gamma=\epsilon$ line. Lenses J and K in Table \[table1\] are two different lenses in potential space whose two surfaces each have the same peak, $\theta_{23,p}$, in the 3D angles space. To check if these surfaces of two different lenses are degenerate in 3D angles space, the second test of Section \[degen\] is implemented. A cross-validation RMSE for Lens J is $0.006^\circ$ while the RMSE of Lens K as predicted by Lens J is $0.181^\circ$. Again, numerical errors could contribute to the difference between the 3D angles space surfaces of these two lenses. To show that this is likely to be the case we made use of the symmetry with respect to the $\gamma=\epsilon$ line mentioned in the previous paragraph. We generated 3D angle space surfaces for sets of lenses related by this symmetry: Lenses K and KK, and J and JJ in Table 1 are two examples. All four lenses share the same $\theta_{23,p}=93^\circ$, but the $\epsilon$ and $\gamma$ parameters of J and K were picked from numerically generated output, whereas those of K and KK (J and JJ) were obtained by swapping $\epsilon$ and $\gamma$ values, which yields exact $\epsilon$ and $\gamma$ parameters. The RMSE from the comparison of Lenses J and JJ (K and KK) yields $0.007^\circ$ ($0.036^\circ$), values which are considerably smaller that $0.181^\circ$ quoted in the previous paragraph. Whatever the source of the discrepancy between lenses such as J and K, the difference is still less than the observational errors, so elliptical NFW+shear lenses with the same $\theta_{23,p}$ are nearly degenerate in terms of image relative angles. Implications for the Observed Quads and Substructure in Lens Galaxies {#realquads} ===================================================================== In this section we rely on the basic property of Type II lenses, namely that the quads of each lens generate two distinct surfaces in the 3D space of image relative angles, such as shown in Figure \[fig2\] (Section \[genprops\]), and that the two sheets are qualitatively similar for all Type II lenses that are suitable as models for observed quads. -0.1in Why degeneracies are useful --------------------------- To draw conclusions about the broad class of lenses used as models one needs to explore a wide range of Type II lens models that are used in the literature to model quads. This task is made easier by the results of Sections \[powerlaw\]-\[nonpowerlaw\], where we explored detailed properties of three different representative classes of parametric models as examples of Type II lenses, and showed that lenses within the same class, like SIEP+shear, or elliptical NFW+shear exhibit approximate degeneracies, in the sense that lenses with the same angle $\theta_{23,p}$ have very similar distribution of quad relative angles in the 3D angle space. As mentioned earlier, we also verified that other commonly used profiles, most notably Sersic, also share these properties. Furthermore, we now show that this degeneracy extends to lenses with different radial density profiles sharing the same peak, $\theta_{23,p}$, as long as the ratio $\epsilon/\gamma$ is smaller than $2-2.5$. To show this we introduce another lens, Lens E (SIEP+shear), which has the same $\theta_{23,p}$ as Lenses F - K. Interpolation fit for Lens E gives a cross-validation RMSE=0.003 degrees. Then, fitting Lenses G (hybrid with $\alpha =1.2$), H (hybrid with $\alpha =0.9$), and J (NFW) give RMSE of 0.146, 0.011, and 0.085 degrees, respectively (Table 1), which are all smaller or of the order of observational uncertainty. This is an important result: most Type II lenses which have the same angle between 2nd and 3rd arriving images of the central source are nearly degenerate in terms of their image angles, for all sources. The existence of this degeneracy, even if approximate, reduces the number of lens models one has to consider. It follows that we have to consider only a set of lenses with different $\theta_{23,p}$, and any density profile (power law or curved in log-log space, like NFW, with the projected density slope not too different from 1 and any set of ($\epsilon,~\gamma,~\beta$), as long as $\epsilon/\gamma\lesssim 2-2.5$. The restrictions on $\alpha$ and $\epsilon$ are because the degeneracies break down outside of the specified ranges. However, almost all of the lens models used to fit observed quad systems belong to the set of degenerate lenses. It is also important to stress that the general shape of the surfaces formed by quads in the 3D angles space is the same for all Type II lenses, even if they are not degenerate. -0.1in Type II lenses cannot fit the population of observed quads ---------------------------------------------------------- Now we are ready to compare Type II lenses with the observed quad population. We will work in two dimensions instead of the three dimensions of the 3D angles space. We use the FSQ as a reference surface, and plot the vertical “distance” between a quad in the 3D relative angles space and the FSQ, called $\Delta\theta_{23}$, versus $\theta_{23}$. This plane is shown in Figure \[fig7\], together with the quads data (black filled symbols with error-bars) originally presented in WW12. FSQ in this plot is the $\Delta\theta_{23}=0$ line. Using this same diagnostic tool, Type I lenses were shown to be inconsistent with the observed quad population (WW12). Specifically, the deviation of even extreme Type I lens models from the FSQ is not enough to account for the wide spread of observed quads in $\Delta\theta_{23}$ vs. $\theta_{23}$ plane, especially at the lower values of $\theta_{23}$. Gray regions in Figure \[fig7\] represent four examples (with two sheets per lens) of Type II lenses. Larger values of $\gamma$ or $\epsilon$ move the gray surfaces further away from the FQS. Looking at the gray surfaces shown in that Figure, and remembering that many more similar surfaces can be drawn, both closer and further away from the FSQ, one may conclude that the whole wide class of Type II lenses—the most commonly used parametric lens models—should be able to reproduce the quad data. This impression comes about because one can find a Type II model, i.e. a set of ($\epsilon,~\gamma,~\beta$) that would reproduce any individual quad (with the possible exception of the extreme quads at $|\Delta\theta_{23}|\gtrsim 15^\circ$), or in the parlance of Figure \[fig7\], one can find a Type II model whose surfaces go through any given quad. However, this impression that Type IIs can reproduce the quad population is false, due to two reasons, with the second one being more important. \(1) In order to reproduce quads at small $\theta_{23}$ Type II lenses would need to have high values of ellipticity and shear. For example, the full SLACS lens samples were fit with SIE mass profiles plus external shear model with median (maximum) values of $\gamma$ and $\epsilon$ of 0.05(0.27) and 0.21 (0.49), respectively [@SLACS5], and the corresponding values from [@sluse12] for 14 quads are 0.12 (0.33) and 0.20 (0.49). The higher end of these parameter ranges appear to be larger than what should be expected in a realistic lensing case. Using monte carlo simulation and considering the effects of environment [@Wong] have found the typical value of total shear to be 0.08 with highest value of 0.17, and [@Dalal1] calculated a typical value for shear of 0.03.[^5] Similar results were obtained by [@k97] and [@hs03]. These works imply that Type II lenses used as models for some observed quads have higher $\gamma$ values than realistically expected. \(2) In Figure \[fig7\] the relative angles of all quads at $|\Delta\theta_{23}|< 12^\circ$ and $\theta_{23}<45^\circ$ can be modeled with Type II lenses shown in that plot as gray surfaces. However, these same Type II models also predict a large population of quads at $\theta_{23}\gtrsim 50^\circ$ and $|\Delta\theta_{23}|\gtrsim5^\circ$. In fact, any Type II model predicts that about $1/3$ of its quads should have $\theta_{23}\!>\!60^\circ$. This is in contradiction to the observations which show virtually no quads in that region. We conclude that even though Type II lenses are able to model most quads individually, they introduce a serious problem because they predict the existence of quads at large $\theta_{23}$ and $\Delta\theta_{23}$ of at least a few degrees. -0.1in Is our conclusion affected by a lens selection bias? ---------------------------------------------------- Since the population of quads we use does not represent a homogeneous sample, one may wonder if selection biases are responsible for the distribution of observed quads in the 3D angle space and hence in the $\Delta\theta_{23}$ vs. $\theta_{23}$ plane. In particular, can selection effects negate the argument in (2) above? To see that this is not the case let us recall how angle $\theta_{23}$ of a quad is related to the source’s location within the caustic. Quads with large $\theta_{23}$ originate from the central regions of caustics, while those with small $\theta_{23}$, from the outer regions, adjacent to the folds and cusps. To account for the distribution of observed quads described in (2) one would need to have the central quads arising from Type II lenses suppressed by some selection bias, while not suppressing the central quads arising from Type I lenses. It would be hard for a selection bias to accomplish that. For example, magnification bias would bias would suppress central quads because they have smaller magnifications. However, it could not distinguish between Type I and Type II lenses, which is what would be needed to explain the distribution seen in Figure \[fig7\]. -0.1in Modeling individual quads vs. modeling quad population ------------------------------------------------------ We stress that our analysis presented above differs significantly from lens modeling of individual quads. Unlike lens mass modeling, parametric or free-form, which is done with one lens system at a time, we study the population properties of quads, and compare them to the generic properties of Type II lenses. This is why we refer to our method as model-free analysis of quads. As pointed out above, and can be seen from lens modeling literature, most individual quads can be successfully modeled with Type II lenses. However, these same lens models also predict the existence of quads which are not observed, and whose absence would be hard to explain by a selection effect. This predicted, but unobserved population of quads is most clearly seen in the space of relative images angles, as is done in Figure \[fig7\]. -0.1in Type III lenses --------------- This leaves us with Type III lenses, for example substructured lenses. A preliminary analysis gives promising results. As depicted in Figure \[fig8\], a better match to the distribution of quads in the $\Delta\theta_{23}$ vs. $\theta_{23}$ plane is achieved with the introduction of substructure in the main lens. By increasing the clumpiness of the lens, the gaps in the peak region, near large $\theta_{23}$, of the two surfaces of quads decreased significantly and looks more like the observational data. However, too much clumpiness could lead to higher order catastrophes in the caustic which would result in observationally unsupported systems with higher image multiplicities. In the examples used in Figure \[fig8\] this is not a concern since even the clumpiest of the two substructured lenses (bottom row) produce more than five images for less than $1\%$ of the total sources. Figure \[fig9\] shows that one does not need to use a highly substructured lens to reproduce the population distribution of observed quads. Here we start with a substructure-less Type I lens (top row) that gives rise to quads which are closely packed on the horizontal axis, i.e. the FSQ, failing to match the spread of observed quads. The introduction of substructure (bottom left) does not perturb the shape of the outer isodensity contours, but the resulting distribution of quads is dispersed in the $\Delta\theta_{23}$ vs. $\theta_{23}$ plane in the same fashion as the observed quads (bottom right). This is just a limited foray into Type III lenses. The conclusion of this Section is that Type II lenses cannot reproduce the distribution of image angles of observed quads, and that possibly substuctured lenses, or additional nearby perturber galaxies are needed. We leave quantitative work on these lenses, and detained comparison with the observed quad distribution for a later paper. Conclusions {#conc} =========== In this paper we presented a technique to explore the properties of quadruple image gravitational lens systems. In general, the relative distribution of quad’s four images about the center of a lens can be fully described in 6D space of three relative angles and three relative radial distances in the lens plane. Our technique uses only the 3D subspace of relative angles which is orthogonal to the remaining 3D subspace of relative distances. We classify lenses into three main categories based on whether their diamond caustic obey twofold and double mirror symmetries. The caustics of Type I lenses fulfills both symmetries, those of Type II obey only the twofold symmetries while the caustics of Type IIIs obey neither. In this paper we focused on Type II lenses. Our main working space is the 3D space spanned by the three relative image angles of a quad, $\theta_{12}$, $\theta_{23}$, and $\theta_{34}$. One of our main conclusions is that this space is a useful tool in studying quad lenses. We showed how the distribution of the three angles of a given lens relates to its lensing potential and the caustic. The two fold symmetry of the Type II lens caustic is reflected in the fact that in the 3D angle space the distribution of the relative angles is always confined to two surfaces (Figure \[fig2\]). This is the defining feature of Type II lenses. Of the three potentials we studied the closest quantitative connections between the lens potential, caustic and 3D angle space are exhibited by the SIEP+shear potential (eq. \[eq1\] with $\alpha=1$). The angle between the second and third arriving images of the source located at the center of the lens, $\theta_{23,p}$ is equal to the angle between the caustic diagonals. (Though we do not study the images distances from lens center in this paper, we note that the distance ratio of these two images is equal to the ratio of the two caustic diagonals.) SIEP+shear lenses that have the same angle $\theta_{23,p}$ have nearly the same distributions of quads in the 3D angle space, implying that $\theta_{23,p}$ is sufficient to specify the shape of the surfaces of all these lenses, and that these surfaces are degenerate. Other lenses also show close similarity between lenses of the same $\theta_{23,p}$, but not as close as for SIEP+shear. The similarity extended to lenses of different radial density profiles: quads of Type II lenses with ellipticities $\epsilon/\gamma\lesssim 2-2.5$ that have the same $\theta_{23,p}$ share approximately the same two surfaces in the 3D angles space. In addition to the lens potentials discussed in detail in this paper we explored more general forms such as the Sersic profiles. The commonalities in the distribution of quads in the 3D angle space (the general shape and location of the two surfaces) persist in all the models considered. This should not come as a surprise since it is a direct reflection of the two-fold symmetry of the caustics of Type IIs which identify the class. Furthermore, the near degeneracies described above persist across all the models studied and we conjuncture that they extend to all Type IIs since there is no evidence to the contrary. Because these observations apply to a wide range of models and allow us to draw conclusions about Type II lenses as a class, we call this approach model-free analysis. We conclude that, even though Type II lenses can successfully model individual quad lenses, they cannot reproduce the relative image angle distribution of the population of the observed quads. We present two examples of Type III substructured lenses that have distributions of relative image angles that are generally consistent with the observed distribution. These examples are not meant to be a close match to observations. Future work will need to look at different types of substructure, as well as luminous and dark secondary perturber galaxies more carefully to ascertain what type and amount of substructure is necessary to fit the observed quad population. Finally, we note that the substructure mass clumps one would need to explain image angles are likely to be much larger in mass and extent compared to those thought to be responsible for the flux ratio anomalies of quads. [99]{} Blandford, R. & Narayan, R. 1986, , 310, 568 Bolton, A. S., Burles, S., Koopmans, L. V. E., et al. 2008, , 682, 964 Bourassa, R. R. & Kantowski, R. 1975, , 195, 13 Dalal, N. & Watson, C. R. 2005, 25 Years After the Discovery: Some Current Topics on Lensed QSOs, e-Edited by Luis J. Goicoechea Dalal, N. 1998, , 509, L13 Falco, E. E., Gorenstein, M. V. & Shapiro, I. I. 1985, , 289, L1 Golse, G., & Kneib, J.-P. 2002, , 390, 821 Gorenstein, M. V., Shapiro, I. I. & Falco, E. E. 1988, , 327, 693 Holder, G. P. & Schechter, P. L. 2003, ApJ, 589, 688 Hunter, C. & Evans, N. W. 2001, , 554, 1227 Kassiola, A., & Kovner, I. 1993, ApJ, 417, 450 Keeton, C. R., Kochanek, C. S. & Seljak, U. 1997, , 482, 604 Klypin, A., Kravtsov, A. V., Valenzuela, O. & Prada, F. 1999, , 522, 82 Koopmans, L. V. E., Treu, T., Bolton, A. S., Burles, S., & Moustakas, L. A. 2006, , 649, 599 Kovner, I. 1987, , 312, 22 Liesenborgs, J. & De Rijcke, S. 2012, MNRAS, 425, 1772 Mao, S. & Schneider, P. 1998, MNRAS, 295, 587 Metcalf, R. B. & Zhao, H.S. 2002, , 567, L5 Moore, B., Ghigna, S., Governato, F., et al. 1999, , 524, L19 Navarro, J. F., Frenk, C. S., & White, S. D. M. 1997, , 490, 493 Saha, P. 2000, , 120, 1654 Saha, P. & Williams, L.L.R. 2003, AJ, 125, 2769 Schneider, P. & Sluse, D. 2014, A&A, 564, 103 Schneider, P. & Weiss, A. 1992, A&A, 260, 1 Schneider, P., Ehlers, J. & Falco, E. E. 1992, “Gravitational Lenses”, Springer-Verlag Berlin Heidelberg New York. Sluse, D., Chantry, V., Magain, P., Courbin, F. & Meylan, G. 2012, A&Ap, 538, 99 Walsh et al. 1980, Nature, 285, 641 Walsh, D., Carswell, R. F. & Weymann, R. J. 1979, Nature, 279, 381 Williams, L. L. R., Foley, P., Farnsworth, D. & Belter, J. 2008, , 685, 725 Witt H. J. & Mao, S. 1995, , 447, L105 Witt H. J. & Mao, S. 2000, MNRAS, 311, 689 Woldesenbet,A.G., & Williams,L.R. 2012, , 420, 2944 (WW12) Wong, K. C., Keeton, C. R., Williams, K. A., Momcheva, I. G., & Zabludoff, A. I. 2011, , 726, 84 Young, P., Gunn, J. E., Kristian, J., Oke, J. B. & Westphal, J. A. 1980, , 241, 507 Xu, D., Sluse, D., Gao, L., Wang, J., Frenk, C., Mao, S., Schneider, P. & Springel, V. 2015, MNRAS, 447, 3189 ![Lens Parametrization in three types of spaces: *Left:* Potential space. A Type II elliptical lens potential contours with external shear oriented along the line (inclined, and going through the center). The four images are numbered based on their arrival time *Middle:* Caustic space, parametrized by the ratio of its diagonals (red lines) and the angle they make at their crossing. **S** is the source that gives rise to the images shown in the left panel. *Right:* 3D space of quad relative image angles: three dimensional space formed by the three relative angles $\theta_{12}$, $\theta_{34}$ and $\theta_{23}$. Each point in this space represents a quad. The surfaces are generated by images of many randomly distributed sources within the diamond caustic and the cut. []{data-label="fig1"}](LenPar.png) ![3D angles space: Type II lenses produce quads that form two sheets on either side of the FSQ, shown as the smooth surface in both panels. (Note that it looks dark red from the left and bright red from the right). Quads belonging to the Type II lens are represented by light blue dots. *Left*: The top portion, $\theta_{23}>75^\circ$, of the 3D space and the surfaces. The peak of the FSQ is at $\theta_{23,p}=90^\circ$. The peaks of the two surfaces of Type II lens are at $\theta_{23,p}\sim 94^\circ$, and $\theta_{23,p}\sim 86^\circ$. *Right*: The lower portion of the same 3D space, for $\theta_{23}<20^\circ$.[]{data-label="fig2"}](Type2VsFSQ.png) ![Mapping of the caustic space to the 3D space of relative image angles. *Top*: Each pair of opposite quadrants of the diamond caustic (for example, the two black quadrants) map into one of the two surfaces in the 3D space (black). This is a reflection of the twofold symmetry of Type II lenses. *Bottom*: Sources along constant position angle in the caustic plane (various colored lines) are mapped to non-crossing curves in the 3D angles space (similarly colored lines).[]{data-label="fig3"}](3DVsCaustic.png) ![Contour surfaces of constant $\theta_{23,p}$ for SIEP+shear lens in $\gamma$ vs. $\epsilon$ vs. $\beta$ space. $\beta$ is in radians. The outer (inner) surface is for $\theta_{23,p}= 91^\circ$ ($\theta_{23,p}=92^\circ$). The jagged line corresponding to $\beta=0$ and $\gamma=\epsilon$ is the result of no quads being formed for those parameters. Two orientations of the same space are shown.[]{data-label="fig4"}](twoLayersContour.png) ![The $\theta_{12}=180 ^\circ$ edge of the lower of the two surfaces, in the 3D space of relative angles, for four SIEP+shear lenses sharing the same peak $\theta_{23,p}$. *Left:* Bottom end of the edge. *Right:* Top end of the edge. The parameters of the four lenses A-D are given in Table 1.[]{data-label="fig5"}](3Dedges.png) ![Pojection of Lenses A-D with parameters as given in Table \[table1\] . Lens C, diamond (green), slightly deviate from the other three lenses by saging at the folds.[]{data-label="fig10"}](plotfun.png) ![Contours of constant $\theta_{23,p}$ in $\gamma$ vs. $\epsilon$ plane for lenses of four different radial density profiles. The contours are labeled by $\theta_{23,p}$. For SIEP+shear (*Top left*) and NFW (*Top right*) lenses the contour lines are symmetric about $\gamma=\epsilon$ line. For hybrid power law potentials the contours are rotated, as compared to that of SIEP+shear, clockwise for $\alpha<1$ (*Bottom left*) and counterclockwise for $\alpha>1$ (*Bottom right*). []{data-label="fig6"}](contours.png) ![Distribution of observed quads (black filled circles) and quads from four Type II lenses (gray distributions) in $\theta_{23}$ vs. $ \Delta\theta_{23}$ plane. The quad data and the errorbars were taken from WW12. $\Delta\theta_{23}=0$ is FSQ. Each lens has two gray surfaces which are about equally close to, but on opposite sides of the FSQ. From the outermost to the innermost surfaces, the lens are: NFW ($\beta=0.1$rad, $\epsilon =0.27, \gamma =0.25$), Power Law ($\alpha=1.2, \beta=0.2$rad, $\epsilon =0.25, \gamma =0.25$), Power Law ($\alpha=1, \beta=0.2$rad, $\epsilon=0.15, \gamma=0.17$), and Power Law ($\alpha=0.9, \beta=0.15$rad, $\epsilon=0.14, \gamma=0.13$), respectively. []{data-label="fig7"}](TypeIIVsData0.png) ![*Left panels:* All three have isothermal radial density profile with ellipticity $\epsilon=0.25$ and shear $\gamma=0.2$ oriented at an angle $\beta=\pi/6$ to each other. The $\kappa=1$ contour is shown by the dashed, red, contour. *Right panels:* The corresponding distribution of quads from these lenses in the $\theta_{23}$ vs. $\Delta\theta_{23}$ plane are shown as the the light shade, yellow, distributions. The observed quads (same as in Figure \[fig7\]) are red (black) dots. *Top:* A Type II lens with no substructure. *Middle:* A Type III lens with randomly distributed clumps that account for 4.38% of the total mass within the window. *Bottom:* A Type III lens with randomly distributed clumps that account for 17.96% of the total mass within the window.[]{data-label="fig8"}](type3witherror.png) ![Similar to Figure \[fig8\], but different lenses. *Top panels:* A Type I lens with elliptical isothermal radial density profile with $\epsilon=0.25$, and $\gamma=0$. *Bottom panels:* A Type III substructured lens.[]{data-label="fig9"}](TypeISub.png) [|l|l|l|l|l|]{} Lens & Radial Profile & L ($\beta,\epsilon,\gamma,\theta_{23,P}$) & Cross-validation & Lens comparison\ name & & & RMSE (deg.) & RMSE (deg.)\ A & SIEP & L (1.2, 0.09, 0.10366, 92) & 0.005 &\ B & & L (1.2, 0.06603, 0.19, 92) & & A: 0.011\ C & & L (0.5, 0.14, 0.036752, 92) & & A: 0.029\ D & & L (0.5, 0.034944, 0.06, 92) & & A: 0.017\ E & & L ($\frac{\pi}{4}$, 0.069197, 0.08, 93) & 0.003 &\ F & Power Law $\alpha$=1.2 & L ($\frac{\pi}{6}$, 0.2, 0.0446, 93) & 0.013 &\ G & & L ($\frac{\pi}{6}$, 0.06536, 0.18, 93)& & E: 0.146, F: 0.551\ H & Power Law $\alpha$=0.9 & L ($\frac{\pi}{6}$, 0.04706, 0.11, 93)& 0.003 & E: 0.011\ I & & L ($\frac{\pi}{6}$, 0.095, 0.058, 93) & & H: 0.264\ J & NFW & L ($\frac{\pi}{6}$, 0.054, 0.075, 93) & 0.006 & E: 0.085\ JJ & & L ($\frac{\pi}{6}$, 0.075, 0.054, 93) & & J: 0.007\ K & & L ($\frac{\pi}{6}$, 0.13, 0.05265, 93) & 0.002 & J: 0.181\ KK & & L ($\frac{\pi}{6}$, 0.05265, 0.13, 93) & & K: 0.036\ \[lastpage\] [^1]: E-mail: woldesenbet@physics.umn.edu (AGW); llrw@astro.umn.edu (LLRW) [^2]: Objects with n-fold rotational symmetry look the same when rotated by $\frac{360}{n}$ degrees. Twofold symmetry is identical to inversion symmetry through the origin. [^3]: Technically, quads are five image configuration with one of the images invisibly sitting near the very bright center of lenses. For all purposes of this paper, the fifth central image is ignored. [^4]: Note that $\epsilon$ is related to the standard definition of ellipticity\ (1 - axes ratio) as $1-\frac{b}{a} = 1- \sqrt{\frac{1-\epsilon}{1+\epsilon}}$ [^5]: Note that these ellipticity values refer to mass, so cannot be compared directly to our $\epsilon$.

--- author: - 'Mohamed M. Anber,' - Erich Poppitz bibliography: - 'ReferencesADW.bib' title: Deconfinement on axion domain walls --- Introduction ============= Understanding of the phase structure of gauge theories is an interesting and unsolved problem. Due to strong coupling, determining the renormalization group (RG) flow to the infrared (IR) is difficult and few general constraints on possible IR behaviors exist. The ’t Hooft anomaly matching [@tHooft:1979rat] stands out as an exact constraint on the IR behavior: the anomaly is RG invariant and any proposed IR phase has to produce the same anomalies as those of the ultraviolet (UV) theory. The idea of anomaly matching is not new and it has been an important tool constraining the behavior of strongly coupled theories. Anomaly matching has recently attracted renewed interest due to the injection of substantial new insight [@Gaiotto:2014kfa; @Gaiotto:2017yup; @Gaiotto:2017tne]: that introducing background fields for all global symmetries—internal or spacetime, continuous or discrete, $0$- or higher-form—consistent with their faithful action, can yield new nontrivial constraints on the possible IR phases. We refer to these constraints as “generalized ’t Hooft anomalies." Their study is evolving too rapidly to allow us to do justice to all interesting aspects currently investigated; we only note that complementary aspects of theories closely related to the ones we consider here are the subject of [@Komargodski:2017smk; @Tanizaki:2018wtg; @Wan:2018bns; @Cordova:2019jnf; @Cordova:2019uob; @Bolognesi:2019fej; @Wan:2019soo; @Wan:2019oax; @Cordova:2019bsd; @Hason:2019akw; @Cordova:2019jqi; @Anber:2019nfu]. In this paper, we continue our study of generalized ’t Hooft anomalies in $SU(N_c)$ gauge theories with vectorlike fermions in representations of nonzero $N$-ality. These theories have exact discrete chiral symmetries. Earlier [@Anber:2019nze], we showed that introducing general baryon, flavor, and color (BCF) backgrounds, consistent with the faithful action of the global symmetries, leads to a mixed anomaly between the discrete chiral symmetry and the BCF background. We used this “BCF anomaly" to rule out certain kinds of IR behavior—notably, we showed that some gauge theories cannot flow to an IR theory of massless composite fermions only. Here, we consider the same class of theories in cases when the discrete chiral symmetry is broken. Concretely, we focus on the breaking of the chiral symmetry and the matching of the BCF anomaly after coupling the theory to an axion. We take the axion scale $v$ larger than the strong coupling scale $\Lambda$ of the theory, so that the IR theory is that of only a light axion, of mass $\Lambda^2 \over v$. We argue that the BCF anomaly is matched by axion domain walls that arise from the breakdown of the chiral symmetry. Anomaly matching also requires nontrivial physics on their worldvolume: Wilson loops are expected to obey perimeter law on the domain walls, corresponding to the breaking of an (emergent) $1$-form center symmetry and the deconfinement of quarks on the domain wall worldvolume. Elucidating the microscopic nature of the worldvolume physics is the main goal of this paper. This is not straightforward on $\R^4$, as the physics determining the domain wall behavior is not semiclassically accessible, contrary to naive expectations—despite the axion’s lightness, axion domain walls probe also the much shorter distance scales of order the confinement scale, representing a subtle failure of effective field theory (EFT) that has been anticipated earlier. In particular, it was argued, using large-$N_c$ arguments (see [@Gabadadze:2002ff] for a review and [@Komargodski:2018odf] for recent related work) that as the axion wall is traversed, a rearrangement of the hadronic degrees of freedom should take place. We show that the deconfinement and rearrangement of heavy degrees of freedom on the axion domain wall are intertwined. We use a calculable compactification [@Unsal:2007jx; @Unsal:2008ch] to $\R^3 \times \S^1$ to study the mechanism of deconfinement, following the observations of our work with Sulejmanpasic [@Anber:2015kea] and the recent remarks on anomaly matching [@Tanizaki:2019rbk]. Here, the physics of confinement is semiclassical and the failure of the axion EFT can be traced to the multi-branch nature of its potential (which, in this setup, is seen at any finite $N_c$). We show that elementary axion domain walls necessarily involve “branch hopping," and as a result they acquire a multi-scale structure, probing both the axion scale and the much shorter confinement scale. This “layered" structure of the axion domain wall is ultimately responsible for the nontrivial worldvolume physics leading to quark deconfinement, which is otherwise similar to [@Anber:2015kea; @Cox:2019aji]. Our results give an explicit realization of domain wall anomaly inflow, yield further confidence both in the generalized anomaly arguments and the “adiabatic continuity" of $\R^3\times \S^1$ compactifications [@Dunne:2016nmc], and suggest that multi-scale structures on axion domain walls also appear on $\R^4$ at finite $N_c$. In Section \[bcfsection\], we describe the class of theories we study, their coupling to the axion, the symmetries, and the expected axion dynamics on $\R^4$. We also review the BCF anomaly, show that it is matched in the IR by axion domain walls, and discuss the nontrivial physics and deconfinement on their worldvolume. In Section \[calculabledeformation\], we study the same theories on $\R^3 \times \S^1$, with $\S^1$ size $L$ obeying $L N_c \Lambda \ll 1$, a condition making semiclassical analysis possible. We review the salient features of such compactifications and use them to study the BCF anomaly matching, the vacuum structure, and domain walls of the axion theory. We then explain the multi-scale structure on axion domain walls and the physics responsible for the deconfinement of quarks on the domain walls. Many interesting questions remain unanswered. For example, a possible phase of these theories, with the fermions taken light or massless, is also one of broken chiral symmetry. Thus, it would be interesting to study the behaviour of the theory, in the bulk and on the domain walls, as the axion scale is lowered, see [@Choi:2018tuh] for related work. The finite temperature phases are also of interest. Our results here also lead us to expect that physics on walls between non-neighboring vacua may have interesting properties. Further, formulating these theories on more general spacetime manifolds than the ones considered here may also lead to more stringent generalized ’t Hooft anomalies. Finally, various aspects of the physics discussed may be relevant for “hidden sector" models of dark matter and models of natural inflation. We hope to return to these questions in the future. The BCF anomaly with axions {#bcfsection} =========================== In this paper, we consider $SU(N_c)$ gauge theories with $N_f$ flavors of vectorlike fermions $(\psi, \tilde\psi)$, both taken as two-component left-handed Weyl fermions, transforming in the representation $(\cal{R}, \cal{\overline{R}})$ of $SU(N_c)$ and the (anti) fundamental $(\square, \overline\square)$ representation of the vectorlike $U(N_f) = {SU(N_f) \times U(1)_B \over \Z_{N_f}}$ flavor symmetry. We take the $U(1)_B$ charges of $(\psi, \tilde\psi)$ to be $(+1, -1)$. We denote by $n_c$ the $N$-ality of the representation $\cal{R}$, the number of boxes in the Young tableau of $\cal{R}$ modulo $N_c$. We focus on representations of nonzero $N$-ality. The fermions are further coupled to a complex Higgs field $\Phi = \rho e^{i a}$, neutral under the $SU(N_c)$ gauge group. The Higgs field has potential $V(\Phi) = \lambda (|\Phi|^2 - v^2)^2$ and a Yukawa coupling to the fermions,[^1] $L_{Y} = y \Phi \tilde\psi \cdot \psi + {\rm h.c.}$ The Yukawa coupling respects both the $U(N_f)$ global symmetry and the classical $U(1)$ global chiral symmetry: $\Phi \rightarrow e^{2 i \alpha} \Phi$, $\psi \rightarrow e^{- i \alpha} \psi$, $\tilde\psi \rightarrow e^{- i \alpha} \tilde\psi$. The $U(1)$ chiral symmetry is broken by the anomaly to $\Z_{2 N_f T_{\cal{R}}}$, leaving only invariance under $$\label{globalchiral} \Z_{2 N_f T_{\cal{R}}}: ~ \Phi \rightarrow e^{i {4 \pi k \over 2 N_f T_{\cal{R}}} }~ \Phi,~~ \psi \rightarrow e^{- i {2 \pi k \over 2 N_f T_{\cal{R}}} } ~\psi,~~ \tilde\psi \rightarrow e^{- i {2 \pi k \over 2 N_f T_{\cal{R}}} } ~\tilde\psi~.$$ The Dynkin index $T_{\cal{R}}$ of the representation $\cal{R}$ is normalized so that for the fundamental representation $T_{\square} = 1$. In addition, when gcd$(N_c,n_c) = p>1$ these theories have an exact $\Z_p^{(1)}$ 1-form center symmetry. It is useful to keep in mind a few simple benchmark cases: [**i.)**]{} when $\cal{R}$ is the fundamental (F) representation, the theory we study is $N_f$-flavor $SU(N_c)$ QCD(F) coupled to the scalar $\Phi$, with a $\Z_{2 N_f}$ anomaly free global chiral symmetry (\[globalchiral\]).[^2] and [**ii.)**]{} when $\cal{R}$ is the two-index symmetric (S) or antisymmetric (AS) representation, the theory is $N_f$-flavor $SU(N_c)$ QCD(S/AS), coupled to $\Phi$, with a $\Z_{2 N_f (N + 2)}$ (S) or $\Z_{2 N_f (N - 2)}$ (AS) anomaly free symmetry (\[globalchiral\]). For $N_c$-even, the theories of type [**ii.**]{} have a $1$-form $\Z_2^{(1)}$ center symmetry, reflecting the fact that two-index representation fields can not screen fundamental quarks. As stated in the Introduction, we shall be concerned with the regime $v \gg \Lambda, y \sim {\cal{O}}(1)$, with $\Lambda$ the strong coupling scale of the gauge theory. The Higgs field, of charge 2 under $\Z_{2 N_f T_{\cal{R}}}$ (as per (\[globalchiral\])), acquires an expectation value $\langle \Phi \rangle = v$, breaking the global symmetry $\Z_{2 N_f T_{\cal{R}}} \rightarrow \Z_2$, where $\Z_2$ is the fermion number. At long distance, only the axion $a$ survives (as $\Phi \approx v e^{i a}$). In the regime of interest, the fermions can be integrated out, giving rise to an effective theory at scales $\Lambda \ll \mu \ll v$: $$\label{effective1} L_{\Lambda \ll \mu \ll v} = {v^2 \over 2} (\partial_\mu a)^2 + N_f T_{\cal{R}} \; a \; q^c + L_{gauge}~+ \ldots~.$$ Here we denoted by $q^c$ and $L_{gauge}$ the topological charge density[^3] and kinetic term of the $SU(N_c)$ gauge field, respectively, and the dots denote higher dimensional operators suppressed by the Higgs and fermion mass. In the absence of an anomaly, the axion $a$ would be a $2\pi$-periodic Goldstone field of the spontaneously broken $U(1)$ symmetry. The coupling to the topological charge density $q^c$ breaks the axion shift symmetry to the discrete subgroup (\[globalchiral\]) $$\label{axionshift} a \rightarrow a + {2 \pi \over N_f T_{\cal{R}}}~.$$ At scales $\mu \le \Lambda$, we can also integrate out the gauge field fluctuations, and one expects an effective Lagrangian for the axion field only: $$\label{effective2} L_{ \mu \ll \Lambda} = {v^2 \over 2} (\partial_\mu a)^2 + \Lambda^4\;(1- \cos( a \; N_f T_{\cal{R}})) + \ldots~,$$ where dots denote other terms in the (non-calculable) periodic axion potential. The theory (\[effective2\]) has $N_f T_{\cal{R}}$ vacua corresponding to the $\Z_{2 N_f T_{\cal{R}}} \rightarrow \Z_2$ symmetry breaking. The vacua are gapped, with the axion mass of order $m_a \sim {\Lambda^2 N_f T_{\cal{R}}\over v}$. The potential in (\[effective2\]) is—at best, see below—only a model for the long-distance dynamics. Nonetheless, it is natural to expect that any potential with the proper periodicity will exhibit this symmetry breaking pattern, giving rise to $N_f T_{\cal{R}}$ vacua and the associated domain walls (DW). Taking (\[effective2\]) at face value, one infers that the DW width scales as $\delta_{DW} \sim {1\over \Lambda} {v \over \Lambda N_f T_{\cal{R}}}\gg {1\over \Lambda}$. Thus, in the regime we study, the naive axion effective action (\[effective2\]) indicates that the DW dynamics involves distance scales larger than the inverse strong-coupling scale $\Lambda$. However, as we shall see, the naive action (\[effective2\]) does not account for important physical phenomena detected by long-distance probes. In particular, anomaly considerations will lead us to expect that the DW structure probes also distance scales of order $\Lambda^{-1}$, contrary to the naive expectation from (\[effective2\]).[^4] As we shall see in Section \[calculabledeformation\], this expectation is explicitly confirmed in a calculable deformation of the axion theory. Postponing momentarily a discussion of these topics, we continue by describing the dynamics at longer length scales. At distances longer than the axion Compton wavelength $m_a^{-1}$ (formally, in the $v \rightarrow \infty$ limit with fixed ${v\over \Lambda} \gg 1$) the $N_f T_{\cal{R}}$ vacua of (\[effective2\]) can be described by means of a $\Z_{N_f T_{\cal{R}}}$ TQFT. The fields in the TQFT are a compact scalar $\phi^{(0)}$ and a compact 3-form field $a^{(3)}$ (such that $\oint d \phi^{(0)}\in 2 \pi \Z$, $\oint da^{(3)}\in 2 \pi \Z$, where the integrals are over closed manifolds of the appropriate dimensionality) with Minkowskian Lagrangian $$\label{bulkTQFT} L_{TQFT} = {N_f T_{\cal{R}} \over 2 \pi}\; \phi^{(0)} \wedge d a^{(3)}~.$$ The TQFT (\[bulkTQFT\]) has a $0$-form $\Z_{N_f T_{\cal{R}}}$ global symmetry $\phi^{(0)} \rightarrow \phi^{(0)} + {2 \pi \over N_f T_{\cal{R}}}$ and a $3$-form $\Z_{N_f T_{\cal{R}}}^{(3)}$ global symmetry: $a^{(3)} \rightarrow a^{(3)} + {1 \over N_f T_{\cal{R}}}\epsilon^{(3)}$, where $\oint \epsilon^{(3)} \in 2 \pi \Z$ is a constant $3$-form with integral periods. There are $N_f T_{\cal{R}}$ vacua[^5] corresponding to the breaking of the $0$-form $\Z_{N_f T_{\cal{R}}}$ symmetry. The objects charged under the $3$-form symmetry are the DW between these vacua. We note that the TQFT carries no information on the energy and length scales associated with the DW, which are determined by the UV theory, e.g. (\[effective2\]) or its generalization. Likewise, the $3$-form symmetry, absent in (\[effective2\]), is emergent at long distances (according to the general criteria of [@Gaiotto:2014kfa], it is also broken—the operators $e^{i \oint a^{(3)}}$, with the integral taken over closed $3$-manifolds, obey a “$3$-volume" law, the analogue of perimeter law for Wilson loops). The theory (\[bulkTQFT\]) will be useful to study the IR matching of ’t Hooft anomalies of the global symmetries. Recall that in [@Anber:2019nze] we coupled the gauge theory considered here to background fields of the global flavor symmetries. In fact, the backgrounds considered corresponded to gauging the $U(N_f) \over \Z_{N_c}$ global symmetry, where $\Z_{N_c}$ denotes the center of the gauge group. The modding by the center of the gauge group arises because some discrete transformations of $U(1)_B$, acting on $\psi$ and $\tilde\psi$ with opposite phases, are really part of the gauge group. Explicitly, when the theory is considered on the four-torus, these backgrounds are ’t Hooft fluxes for the $SU(N_c), SU(N_f)$, and $U(1)_B$ gauge fields. These ’t Hooft fluxes are chosen to be consistent with the torus transition functions for the fermions ($\psi$, $\tilde\psi$) in the representation ($\cal{R}, \cal{\overline{R}}$). We shall not give the explicit form of the ’t Hooft flux backgrounds here and refer the reader to [@Anber:2019nze]. The important point to make is that these $U(N_f) \over \Z_{N_c}$ global symmetry backgrounds carry fractional topological charges $Q^c$ of $SU(N_c)$, $Q^f$ of $SU(N_f)$, and $Q^B$ of $U(1)_B$:$$\begin{aligned} \label{backgrounds} Q^c = m\; m' \left( 1 - {1 \over N_c}\right), ~ Q^f = k\; k' \left(1 - {1 \over N_f}\right),~ Q^B = \left( n_c {m \over N_c} + {k \over N_f}\right) \left( n_c {m' \over N_c} + {k' \over N_f}\right), \end{aligned}$$ where $m, m'$ are integers defined modulo $N_c$ (likewise, $k, k' \in \Z$ are defined modulo $N_f$). The importance of the global symmetry backgrounds (\[backgrounds\]) is that under the anomaly-free discrete chiral $\Z_{ 2 N_f T_{\cal{R}} }$ transformation (\[globalchiral\]), the partition function $Z$ of the UV theory transforms as $$\begin{aligned} \label{BCF1} \Z_{2 N_f T_{\cal{R}}}: \; Z &\rightarrow& Z \; e^{i {2 \pi \over N_f T_{\cal{R}} }\left[ N_f T_{\cal{R}} Q^c + d_{\cal{R}} Q^f + N_f d_{\cal{R}} Q^B\right]}, \end{aligned}$$ where $d_{\cal{R}}$ denotes the dimension of the representation $\cal{R}$. It is easy to see that the phase on the r.h.s. of (\[BCF1\]) is nontrivial in the background (\[backgrounds\]).[^6] Thus, the discrete chiral $\Z_{2N_f T_{\cal{R}}}$ symmetry has a mixed ’t Hooft anomaly with the $U(N_f) \over \Z_{N_c}$ symmetry. This mixed anomaly was termed the “BCF anomaly" in [@Anber:2019nze]. The anomaly was shown to restrict the possible IR phases of these gauge theories. Coming back to the theory with axions, in the background with topological charges (\[backgrounds\]), the axion acquires also couplings to the topological charges of the various background fields:$$\label{axiontop} L_{top.} = a \; (N_f T_{\cal{R}} \; q^c + d_{\cal{R}} \; q^f + N_f d_{\cal{R}} \;q^B),~$$ where $q^f$ and $q^B$ are the topological charge densities of the flavor and baryon number fields and we included the coupling to $q^c$ from (\[effective1\]). The couplings of the axion to the topological charge densities in (\[axiontop\]) match the BCF anomaly at the energy scales where (\[effective1\]) is valid: under a $\Z_{N_f T_{\cal{R}}}$ shift of the axion (\[axionshift\]), in the backgrounds (\[backgrounds\]), the transformation (\[BCF1\]) of the partition function is reproduced by (\[axiontop\]). The BCF anomaly in the bulk can be also given a description using the continuum formalism of [@Kapustin:2014gua] describing the gauging of higher form symmetries. This description can also be used at scales longer than the axion Compton wavelength, i.e. applied to the TQFT (\[bulkTQFT\]). We shall find it also useful in Section \[calculabledeformation\] and so we briefly review it next. The continuum formalism of gauging higher form symmetries uses an embedding of the $SU(N_c)$ connection $A^c$ into a $U(N_c)$ connection. We review the construction for $SU(N_c)$; it proceeds similarly for $SU(N_f)$. We take pairs of $U(1)$ $2$-form and $1$-form gauge fields $\left(B^{c(2)},B^{c(1)} \right)$ such that $dB^{c(1)}=N_cB^{c(2)}$. The $1$-form gauge fields satisfy $\oint dB^{c(1)} \in 2 \pi \Z$, where the integrals are taken over closed $2$ surfaces. Thus, we have $\oint B^{c(2)}\in \frac{2 \pi}{N_c}\mathbb Z$. Next, we define the $U(N_c)$ connections $ \tilde A^c\equiv A^c+\frac{B^{c(1)}}{N_c}, $ where the second term is proportional to the $N_c \times N_c$ unit matrix and $A^c$ is the $SU(N_c)$ connection. The gauge field strengths $\tilde F^c=d\tilde A^c+\tilde A^c\wedge\tilde A^c$ satisfy $\mbox{tr}_F \tilde F^c =dB^{c(1)}=N_cB^{c(2)}$. Going from $SU(N_c)$ to $U(N_c)$ introduces extra degrees of freedom. In order to eliminate them, we postulate an invariance under $U(1)$ $1$-from gauge symmetries: $\tilde A^c\rightarrow \tilde A^c+\lambda^{c(1)}$, which translates into $\tilde F^c\rightarrow \tilde F^c+d\lambda^{c(1)}$. The fields $\left(B^{c(2)},B^{c(1)} \right)$ transform as $B^{c(2)}\rightarrow B^{c(2)}+d\lambda^{c(1)}$, $B^{c(1)}\rightarrow B^{c(1)}+ N_c \lambda^{c(1)}$, so that the constraints $dB^{c(1)}=N_cB^{c(2)}$ are invariant (the $1$-form transformation parameter obeys $\oint d \lambda^{c(1)} \in 2 \pi \Z$). Introducing the background fields $(B^{c(1)}, B^{c(2)})$ into the Lagrangian of our gauge theory, as described above, is equivalent to turning on ’t Hooft fluxes for $SU(N_c)$, with topological charge $Q^c$ from (\[backgrounds\]). Similarly, we introduce fields $(B^{f(1)}, B^{f(2)})$ for $SU(N_f)$, along with the corresponding $1$-form symmetry with parameter $\lambda^{f(1)}$, coupled to the $SU(N_f)$ theory via the $U(N_f)$ connection constructed as outlined above. The $(B^{f(1)}, B^{f(2)})$ background corresponds to the introduction of the ’t Hooft flux for $SU(N_f)$, whose topological charge $Q^f$ is given in (\[backgrounds\]). Up to this stage, we have not yet said anything about the baryon background gauge field $A^B$. Invariance of the matter covariant derivatives of ($\psi, \tilde\psi$) under the $U(1)$ $1$-from gauge symmetries demands[^7] that $A^B\rightarrow A^B+n^c\lambda^{c(1)}+ \lambda^{f(1)}$. Using the field strength $F^B=dA^B$ we find $F^B\rightarrow F^B +n^c d\lambda^{c (1)}+ \lambda^{f(1)}$. Thus, we have the following $2$-form combinations $ \tilde F^c-B^{c(2)}$, $\tilde F^f - B^{f(2)}$, $F^B-n^cB^{c(2)}- B^{f(2)}, $ invariant under the $1$-form gauge transformations with parameters $\lambda^{c(1)}$ and $\lambda^{f(1)}$. In terms of these, the topological charge densities appearing in (\[axiontop\]) are$$\begin{aligned} \nonumber q^c &=&\frac{1}{8\pi^2}\left[ \mbox{tr}_F\left(\tilde F^c\wedge \tilde F^c\right)-N_c B^{c(2)}\wedge B^{c(2)} \right],~ q^f = \frac{1}{8\pi^2} \left[ \mbox{tr}_F\left(\tilde F^f\wedge \tilde F^f\right)-N_f B^{f(2)}\wedge B^{f(2)} \right], \\ q^B&=&\frac{1}{8\pi^2} \left[F^B-n^cB^{c(2)}-B^{f(2)} \right]\wedge \left[F^B-n^cB^{c(2)}- B^{f(2)} \right].\label{continuumQ} \end{aligned}$$ Integrating the above $q^{c,f,B}$ over the four dimensional spacetime gives rise to the topological charges in (\[backgrounds\]). One notes that the $U(N_c)$, $U(N_f)$, and $U(1)_B$ topological charges (the terms proportional to $\int\mbox{tr}_F \tilde F^c\wedge \tilde F^c$, $\int\mbox{tr}_F \tilde F^f\wedge \tilde F^f$, $\int F_B \wedge F_B$, respectively) are integer on spin manifolds, while the terms containing $B^{c(2)}$ and $B^{f(2)}$ give rise to the fractional terms in (\[backgrounds\]), once the conditions $\oint B^{c(2)} = {2 \pi \Z \over N_{c}}$, and similar for $c \rightarrow f$, are taken into account. It is now clear that using (\[continuumQ\]) in (\[axiontop\]) reproduces the BCF anomaly (\[BCF1\]) in the effective theory (\[effective1\]). Employing the above formalism, the TQFT action (\[bulkTQFT\]) can also be coupled to the $U(N_f)\over \Z_N$ background $$\label{bulkTQFT1} L_{TQFT} = {N_f T_{\cal{R}} \over 2 \pi}\; \phi^{(0)} \wedge \left(d a^{(3)} -{N_c \over 4 \pi} B^{c(2)}\wedge B^{c(2)} + 2 \pi {d_{\cal{R}} \over N_f T_{\cal{R}}} \; q^f +2\pi {d_{\cal{R}} \over T_{\cal{R}}} \;q^B\right)~,$$ and is easily seen to reproduce the BCF anomaly (\[BCF1\]) upon a $\Z_{N_f T_{\cal{R}}}$ shift of $\phi^{(0)}$ (recalling that $\oint da^{(3)}\in 2 \pi \Z$). Notice that making (\[bulkTQFT1\]) invariant under the same $1$-form transformations that leave (\[axiontop\], \[continuumQ\]) intact, the $3$-form field $a^{(3)}$ acquires a shift under $\Z_{N_c}^{(1)}$ center transformations with parameters $\lambda^{c(1)}$ [@Gaiotto:2014kfa; @Seiberg:2018ntt; @Cordova:2019uob]: $$\label{a3shift} a^{(3)} \rightarrow a^{(3)} + {N_c \over 2 \pi} B^{c(2)} \wedge \lambda^{c(1)} + {N_c \over 4\pi} \lambda^{c(1)} \wedge d \lambda^{c(1)}.$$ Thus, with (\[a3shift\]), the theory (\[bulkTQFT1\]) is invariant under the $1$-form transformations with parameters $\lambda^{c(1)}, \lambda^{f(1)}$ (acting on dynamical and background fields). As already argued, it also reproduces the BCF anomaly (\[BCF1\]) upon a $\Z_{N_f T_{\cal{R}}}$ shift of $\phi^{(0)}$. In the limit we study, $v \gg \Lambda$, the IR theories (\[effective1\], \[effective2\], \[bulkTQFT\]) acquire an emergent $1$-form $\Z_{N_c}^{(1)}$ symmetry,[^8] as the fermions (whose coupling to the gauge field is the only source breaking the center symmetry) decouple. In the case of noncompact $\R^4$, integrating out fermions generates local terms only.[^9] On the other hand, there exist no local terms involving the light dynamical fields—the axion and the $SU(N_c)$ gauge fields—that violate the $\Z_{N_c}^{(1)}$ center symmetry, as the latter only acts on topologically nontrivial line operators. Thus, the long distance theory has an emergent $\Z_{N_c}^{(1)}$ symmetry. In fact, this symmetry is manifest in the TQFT (\[bulkTQFT1\]), where the transformation (\[a3shift\]) is a symmetry (in the absence of the nondynamical baryon and flavor backgrounds). Similar to the case of DW between chirally broken vacua in super-Yang-Mills theory, or DW between CP breaking vacua in $\theta=\pi$ Yang-Mills theory, one expects that this emergent $1$-form symmetry is also broken on the axion DW, hence quarks of all $N$-alities are deconfined. Formally, one argues that the DW between neighboring vacua carries an $SU(N_c)_1$ Chern-Simons theory which has a $\Z_{N_c}^{(1)}$ center symmetry with a ’t Hooft anomaly, for details we refer the reader to [@Gaiotto:2017yup; @Gaiotto:2017tne; @Hsin:2018vcg]. Heuristically, one can argue that this Chern-Simons theory arises because crossing the axion DW implements a $2\pi$ shift of the $\theta$ parameter [@Komargodski:2018odf]. Further, it is natural to identify Wilson loops in the worldvolume Chern-Simons theory with Wilson loops of the gauge theory, taken to lie in the DW worldvolume. Since Wilson loops in Chern-Simons theory are known to obey perimeter law, one concludes that quarks are deconfined on the DW worldvolume. The arguments in favor of deconfinement on axion DW have a somewhat formal flavor and it would be desirable to have a setup where they can be explained more physically. Naturally, this is difficult in nonsupersymmetric theories on $\R^4$, since confinement occurs at strong coupling and a tractable theory thereof is lacking.[^10] However, as we shall show in the next section, using a compactification of the theory on $\R^3\times \S^1$ (but keeping the locally four dimensional nature and all relevant symmetries intact!) the physical mechanism underlying deconfinement on axion DW can be made explicit. On the one hand, this gives us further confidence in arguments based on anomaly inflow, and, on the other, it points towards a more complicated structure of axion DW on $\R^4$ than the one expected from the naive axion effective theory (\[effective2\]). In the next Section, we take this more explicit approach and discuss anomaly matching and the deconfinement of quarks on the axion DW, in a setup where the properties of the ground state of the theory and the associated DW can be studied semiclassically. The BCF anomaly and deconfinement on DW on $\R^3 \times \S^1$ {#calculabledeformation} ============================================================= The calculable setup we discuss here is one where our theory is compactified on a small circle $\S^1$ of circumference $L$. We consider the limit where $v \gg {1\over N_c L} \gg \Lambda$. In addition, the theory is “deformed" by adding massive adjoint fermions whose presence is needed to ensure center stability in the $N_c L\Lambda \ll 1$ semiclassical limit.[^11] We stress that adding massive adjoints does not affect any of the axion couplings and discrete chiral symmetries discussed so far. The effective potential for the $\S^1$-holonomy, to leading order in the $v \gg {1\over N_c L} \gg \Lambda$ limit, is stabilized at the center symmetric value. The theory abelianizes, with broken $SU(N_c) \rightarrow U(1)^{N_c - 1}$ at a high scale $1\over N_c L$, and the long distance $\R^3$ physics is described in terms of the $N_c-1$ dual photons in the Cartan subalgebra of $SU(N_c)$. In the heavy-fermion theory on $\R^3 \times \S^1$, the $(\psi, \tilde\psi)$ fermions have mass $yv \gg 1/L$ and, as before, can be integrated out. When the theory is considered at distances $\gg N_c L$, it has a $0$-form center symmetry $\Z_{N_c}^{(0)}$, the “component" of the emergent $1$-form center in $\R^4$ along the compact direction, as well as a $\Z_{N_c}^{(1)}$ $\R^3$ $1$-form center. There are exponentially suppressed terms, $\sim e^{- y v L}$, local in $\R^3$ but winding around $\S^1$, that break the $0$-form $\Z_{N_c}^{(0)}$ center symmetry[^12], which we ignore. As is the case on $\R^4$, no local terms breaking the $\Z_{N_c}^{(1)}$ $\R^3$ $1$-form emergent center symmetry appear. Thus, at the center-stabilized value of the holonomy, the theory is essentially deformed Yang-Mills theory coupled to an axion field[^13] $a$. The axion appears as a dynamical $\theta$ parameter and has a kinetic term given by the reduction of (\[effective2\]) to $\R^3$, ${v^2 L \over 2} (\partial_\mu a)^2$. To further discuss the long-distance dynamics, we now borrow the results of the recent study of deformed Yang-Mills theory with $\theta$ parameter [@Tanizaki:2019rbk], adding the dynamical axion field and its topological coupling to the background fields from (\[axiontop\]). The kinetic and topological ($\tilde{q}^{f,B}$ denote appropriate $\R^3$ reductions of $q^{f,B}$ from (\[continuumQ\]), given below in (\[r3topological\])) terms in the Euclidean $\R^3$ Lagrangian are:$$\begin{aligned} \label{r3kinetic} L_{kin.} &=& {v^2 L \over 2} |d a|^2 + {1 \over 2 g^2 L}|d\vec{\phi} - N_c A^{c (1)} \vec{\nu}_1|^2 +{g^2 \over 8 \pi^2 L}\left|d \vec{\sigma} + {N_f T_{\cal{R}} a \over 2 \pi}(d \vec{\phi} - N_c A^{c(1)} \vec{\nu}_1)\right|^2 \nonumber \\ &&- i \; {N_c \over 2\pi} \;\vec{\nu}_1 \cdot d \vec{\sigma} \wedge B^{c (2)} + i a (d_{\cal{R}} \tilde{q}^f + N_f d_{\cal{R}} \tilde{q}^B)~.\end{aligned}$$ Here, $L$ is the $\S^1$ circumference and $g$ is the $SU(N_c)$ gauge coupling, frozen at a scale of order $1\over N_c L$. The axion field is the one we already introduced and the scale $v$ is the one from the Higgs potential. The arrows denote vectors in the Cartan subalgebra of $SU(N_c)$: $\vec{\sigma}$ represents the $N_c-1$ Cartan dual photons, which are compact scalars taking values in the weight lattice of $SU(N_c)$ ($\vec\sigma \equiv \vec\sigma + 2 \pi \vec{w}_p$, where $\vec{w}_p$ denote the $N_c-1$ fundamental weight vectors) and $\vec{\phi}$ are the eigenvalues of the $\S^1$-holonomy. The latter is defined so that a fundamental Wilson loop along the $\S^1$ has eigenvalues[^14] $e^{- i \vec\phi \cdot \vec\nu_A}$, $A = 1, \ldots N_c$, where $\vec\nu_A$ are the weights of the fundamental representation, normalized so that $\vec\nu_A \cdot \vec\nu_B = \delta_{AB} - 1/N_c$. Under global $0$-form center symmetry transformations, the holonomy eigenvalues shift as $\vec\phi \rightarrow \vec\phi + 2 \pi \vec\nu_1$, transforming all Wilson loop eigenvalues by a $\Z_{N_c}$ phase, as appropriate. Further, we have dimensionally reduced the four dimensional 2-form gauge field $B^{c(2), 4d} = A^{c(1)} \wedge {d x^3 \over L} + B^{c(2)}$ into a 3d $1$-form $A^{c(1)}$ and a 3d $2$-form $B^{c(2)}$ gauge fields.[^15] The Lagrangian (\[r3kinetic\]) is clearly invariant under the local $0$-form center symmetry, acting as $\vec\phi \rightarrow \vec\phi + N_c \lambda^{(0)} \vec{\nu}_1$ and $A^{c(1)} \rightarrow A^{c(1)} + d \lambda^{(0)}$. The global $0$-form center is recovered upon taking $\lambda^{(0)} = {2 \pi \over N_c}$. Invariance of $e^{- \int_{\R^3} L_{kin}}$ under $1$-form center transformations, $B^{c(2)} \rightarrow B^{c(2)} + d \lambda^{c(1)}$, is ensured by the fact that $\oint d \lambda^{c(1)} \in 2 \pi \Z$ and that the monodromies of the dual photon lie in the weight lattice, i.e. $\oint d \vec{\sigma} \in 2 \pi \Z \vec{w}$, where $\vec{w}$ is any fundamental weight. Most importantly for us, under a $\Z_{N_f T_{\cal{R}}}$ shift (\[axionshift\]) of the axion field, we have that $d \vec{\sigma} \rightarrow d \vec{\sigma} - d \vec{\phi} + N_c A^{c(1)} \vec{\nu}_1$, leading to $\delta L_{kin.} = i {N_c \over 2 \pi} \vec\nu_1 \cdot d \vec{\phi} \;B^{c(2)} - i{N_c \over 2 \pi} (N_c -1) A^{c(1)}\wedge B^{c(2)}$. The first term in the variation of $L_{kin.}$ gives no contribution to $e^{- \int_{\R^3} L_{kin}}$, since $\oint {d\vec{\phi}\over 2 \pi}$ lies in the root lattice [@Argyres:2012ka; @Anber:2015wha] and $\oint N_c B^{c(2)} \in 2 \pi \Z$, while the last term, using $\oint A^{c(1)} \in {2 \pi \Z \over N_c}$, $\oint B^{c(2)} \in {2 \pi \Z \over N_c}$, reproduces the color background contribution to the BCF anomaly (\[BCF1\]). Finally, the other (baryon and flavor) contributions to the BCF anomaly (\[BCF1\]) are due to the last two terms in (\[r3kinetic\]), where $\tilde{q}^f$ and $\tilde{q}^B$ are $\R^3$-reductions of $q^f$ and $q^B$ from (\[continuumQ\]): $$\begin{aligned} \tilde{q}^B &=&\frac{1}{8\pi^2} (d \phi^B - n_c A^{c (1)} - A^{f (1)}) \wedge (F^B - n_c B^{c (2)} - B^{f(2)}),~ \nonumber \\ \tilde{q}^f &=& \frac{1}{8\pi^2} (d \vec\phi^f - N_f A^{f (1)} \vec\nu_1^{(f)})\cdot \wedge (\vec{F}^f - N_f B^{f(2)} \vec\nu_1^{(f)}). \label{r3topological} \end{aligned}$$ Here $\phi^B$ is the $\S^1$ holonomy of the $U(1)_B$ background (and $\oint d \phi^B \in 2 \pi \Z$) and $\vec\phi^f$ are the Cartan components of the $\S^1$ holonomy of the $SU(N_f)$ background fields ($\oint d \vec{\phi}^f/2 \pi$ is in the root lattice of $SU(N_f)$). Likewise $F^B$ ($\oint F^B/2\pi \in \Z)$ is the $\R^3$ $2$-form field strength of $U(1)_B$ and $\vec{F}^f$ denotes the Cartan components of the $\R^3$ $2$-form field strength of $SU(N_f)$ ($\oint \vec{F}^f/2 \pi$ is in the root lattice of $SU(N_f)$). We note that to describe the BCF anomaly, it suffices to turn on Cartan components only; correspondingly, $\vec\nu_1^{(f)}$ denotes the weight of the fundamental of $SU(N_f)$. Invariance under the $1$-form transformations on $\R^4$ (with parameters $\lambda^{c(1)}, \lambda^{f(1)}$, described above (\[continuumQ\])) reduces to invariance under $0$-form transformations on $\R^3$ with parameters $\lambda^{c (0)}, \lambda^{f (0)}$, under which: $\delta A^{c(1)} = d \lambda^{c(0)}$, $\delta A^{f(1)} = d \lambda^{f(0)}$, $\delta \phi^B = n_c \lambda^{c(0)} + \lambda^{f(0)}$, $\delta \vec\phi^f = N_f \vec\nu_1^{(f)} \lambda^{f(0)}$ and $1$-form transformations on $\R^3$ with parameters $\lambda^{c(1)}, \lambda^{f(1)}$, under which $\delta B^{c(2)} = d \lambda^{c(1)}$, $\delta B^{f(2)} = d \lambda^{f(1)}$, $\delta \vec{F}^f = N_f d \lambda^{f(1)} \vec{\nu}_1^{(f)}$, $\delta F_B = n_c d \lambda^{c(1)} + d \lambda^{f(1)}$. It is clear that with the above transformations and the definitions (\[r3topological\]), the $\R^3$ effective lagrangian (\[r3kinetic\]) is invariant under the $\Z_{N_c}$ and $\Z_{N_f}$ transformations ($0$-form and $1$-form on $\R^3$) acting on the dynamical and background fields, as is (\[continuumQ\]) on $\R^4$. Most importantly, it is easy to see that the flavor and baryon contribution to the BCF anomaly is reproduced by the topological couplings in (\[r3kinetic\]); the normalizations $\oint N_{f} A^{f(1)} \in 2 \pi \Z$, $\oint B^{f(2)} N_{f} \in 2 \pi \Z$, along with the corresponding ones for the $A^{c(1)}, B^{c(2)}$ fields given earlier, are important in showing this. After the somewhat lengthy exposition of symmetries and the matching of the BCF anomaly, we now come to the dynamics of the $\R^3 \times \S^1$ theory. Potential terms in the $\R^3$ effective Lagrangian reflect both the $\S^1$-center stabilization and the semiclassical nonperturbative dynamics. The center stabilization is due to the massive adjoint fermions added to stabilize the center. The field $\vec\phi$ is stabilized at the center symmetric value, $\vec\phi = {2 \pi \vec{\rho} \over N_c}$ ($\vec\rho$ is the Weyl vector, $\vec\rho = \sum_{k=1}^{N_c-1} \vec{w}_k$), acquires mass $\sim {g \sqrt{ N_c}\over L}$, and does not participate in the low-energy dynamics [@Unsal:2008ch]. Thus, we ignore $\vec\phi$ and consider the nonperturbative potential, which depends only on the lighter axion field and the dual photons. To leading exponential accuracy in the $v \gg {1\over N_c L} \gg \Lambda$ limit, the potential is given by the deformed Yang-Mills theory potential with a dynamical $\theta$-parameter. The nonperturbative potential, generated by monopole-instantons, is $$\begin{aligned} \label{r3potential} V(\vec\sigma, a) = L^{-3} e^{- {8 \pi^2 \over g^2 N_c}} \sum\limits_{k=1}^{N_c} \left(1 - \cos\left(\vec\alpha_k \cdot \vec\sigma + {a N_f T_{\cal{R}} \over N_c}\right)\right)~, \end{aligned}$$ where $\vec\alpha_{k}$, $k=1,...N_c$, are the simple and affine roots of the $SU(N_c)$ algebra (the affine, or lowest, root is $\vec\alpha_{N_c} = - (\vec\alpha_1 + \ldots \vec\alpha_{N_c-1})$). The nonperturbative factor in front of the potential is associated with the monopole instanton action, $e^{-S_0} = e^{-{8 \pi^2 \over g^2 N_c}}$. The potential (\[r3potential\]) is given up to an inessential multiplicative factor and a constant, ensuring $V \ge 0$, is added. From the discussion after (\[r3kinetic\]), the $\Z_{N_f T_{\cal{R}}}$ shift symmetry $a \rightarrow a + \frac{2 \pi}{N_f T_{\cal{R}}}$ also acts on $\vec\sigma$; in the center-stabilized vacuum $\vec\sigma \rightarrow \vec\sigma - {2 \pi \over N_c} \vec\rho$. We note that this symmetry action is that of an enhanced symmetry,[^16] $\Z_{\cal{Q}}$, with ${\cal{Q}} \equiv {\rm lcm} (N_c, N_f T_{\cal{R}})$. As already discussed, in the limit we study, the potential (\[r3potential\]) has a $0$-form emergent center symmetry, in addition to the $\R^3$ $1$-form center. Its action is such that it cyclically permutes the various monopole-instanton factors, i.e. $\vec\alpha_k \cdot \vec\sigma \rightarrow \vec\alpha_{k+1({\rm mod} N_c)} \cdot \vec\sigma$. In group theoretic terms, this transformation is $\vec\sigma \rightarrow {\cal P} \vec{\sigma}$, where ${\cal{P}}$ denotes an ordered product of Weyl reflections w.r.t. all simple roots [@Anber:2015wha]. This action of the $0$-form center symmetry transformation on the dual photons follows after restricting $\vec\phi$ to the Weyl chamber, see also [@Aitken:2017ayq]. For use below, we note the action of ${\cal P}$ on the Weyl vector $\vec\rho$ and fundamental weights $\vec{w}_k$, $k=1,\ldots N_c -1$ [@Cox:2019aji]: $$\begin{aligned} \label{zeroform2} \Z_{N_c}^{(0)}: {\cal P}^l \; {k \over N_c} \vec\rho &=& {k\over N_c} \vec\rho - k \;\vec{w}_l~, ~l =1, \ldots N_c,~ {\rm with} ~\vec{w}_{N_c} \equiv 0.\end{aligned}$$ To find the minima of the potential (\[r3potential\]), we first extremize $V$ w.r.t. $\vec\sigma$. For any value of $a$, $V$ has $N_c$ extrema with respect to $\vec\sigma$ which lie in the unit cell of the weight lattice. These are at $\vec\sigma_q= {2 \pi q\over N_c} \vec\rho$, $q=0,...N_c-1$. That these are extrema can be easily verified, using $\vec\alpha_{k} \cdot \vec\rho = 1$, $k=1,...,N_c-1$ and $\vec\alpha_{N_c} \cdot \vec\rho = 1- N_c$ (it is known [@Thomas:2011ee] that these are all extrema within the unit cell, see also [@Aitken:2018mbb]). Further, extremizing with respect to $a$, we solve ${d V\over da}(\vec\sigma_q, a) = N \sin\left( {2 \pi q + N_f T_{\cal{R}} a \over N_c}\right) =0$ to obtain $N_f T_{\cal{R}} a_{k,q} = \pi N_c k - 2 \pi q $, $k \in \Z$, $q =0,...N_c-1$. The potential (\[r3potential\]) is nonnegative, hence the minima all have $V(\vec\sigma_q, a_{k,q})=0$, implying that $1 - \cos \pi k = 0$, so that $k=2 r$ is an even integer. Thus, the physically distinct minima of the potential for $a$ occur at $a = {2 \pi (N_c r - q) \over N_f T_{\cal{R}} }$, where $q$ takes the values from $0$ to $N_c-1$ and $r \in \Z$. Counting the distinct ground states of (\[r3potential\]) then reduces to finding pairs of integers $q \in [0,\ldots, N_c -1]$ and $r$ such that the pairs of expectation values $(\langle e^{i a(x)}\rangle, \langle e^{i {\vec\alpha_k \cdot \vec\sigma(x)}}\rangle)= (e^{i {2 \pi (N_c r - q) \over N_f T_{\cal{R}} }}, e^{i {2 \pi q \over N_c}})$ are distinct. ![Axion potential in $SU(5)$ QCD(F) with $N_f=3$: the five branches $V(\vec\sigma_q, a)$, Eq. (\[r3potential\]) evaluated at $\vec\sigma_q = {2 \pi q \over 5} \vec\rho$, $q=0,1,2,3,4$. Shown are three of the $15$ $[{\cal{Q}}={\rm lcm}(5,3)]$ minima, determined as described in the text, related by the broken axion shift symmetry. It is clear that a change of the light field (axion) vacuum entails a rearrangement of the heavy confining degrees of freedom (the dual photons). In an axion EFT this would be described by a transition from one branch of the axion potential to another one. This “branch-hopping" on axion DW is crucial in explaining the deconfinement of quarks.[]{data-label="fig:su5F"}](AxionPotSU5F3.pdf){width=".60\linewidth"} In all cases we have studied, there are ${\cal{Q}} \equiv {\rm lcm} (N_c, N_f T_{\cal{R}})$ values of $q$ and $r$ obeying the above conditions, corresponding to the spontaneous breaking of the $\Z_{N_f T_{\cal{R}}}$ (enhanced to $\Z_{\cal{Q}}$) axion shift symmetry of interest to us. We stress that, for our discussion of deconfinement below, it is important to note that the $0$-form symmetry $\Z_{N_c}^{(0)}$ is unbroken in the $N_f T_{\cal{R}}$ vacua of the axion theory: the first relation in (\[zeroform2\]) implies that the $\Z_{N_c}^{(0)}$ action maps $\vec\sigma_q$ to $\vec\sigma_q + 2 \pi \Z \vec{w}$, i.e. to itself modulo weight-vector shifts, which are identifications of the compact dual photon fields (equivalently, the expectation values of the “covering space" coordinates $e^{i \vec\alpha_k \cdot \vec\sigma}$, invariant under weight-vector shifts, are invariant under $0$-form center transformations). The masses of the dual photons in the minima of (\[r3potential\]) are given by the usual exponentially small nonperturbative scale, $m_\sigma^2 \sim L^{-2} {e^{- {8 \pi^2 \over g^2 N_c}}}$. The axion is significantly lighter due to the high axion scale, $m_a^2 \sim m_\sigma^2 \left({N_f T_{\cal{R}} \over N_c}\right)^2 {1 \over v^2 L^2}$. Thus, one would expect that the dual photons can be integrated out and the long distance dynamics and vacua be described in terms of a potential involving the axion field only. However, as is clear from the discussion of the vacua, different vacua for the axion require a rearrangement of the heavy degrees of freedom, the dual photons. As we shall see, this is crucial to explaining deconfinement on DW. It is useful to plot the axion potential for one of the benchmark theories discussed in the paragraph after (\[globalchiral\]). We consider QCD(F) with $N_c = 5$, $N_f=3$ ($T_{\cal{R}}=1$). On Fig. \[fig:su5F\], we show the function $V(\vec\sigma_q, a)$, i.e. the potential (\[r3potential\]), evaluated at the $q$-th extremum w.r.t. $\vec\sigma$, as a function of the axion field $a$. Following the procedure for finding minima described earlier, we focus on the three minima of the potential shown, all of which lie on different “branches," i.e. have different values of $\vec\sigma_q$. Consider the two neighboring minima at $a={2 \pi \over 3}$ and $a = {4 \pi \over 3}$. They correspond to $\vec\sigma_q$ with $q=4$ and $q=3$, respectively. Thus, a DW interpolating between these two minima must necessarily involve a change not only of $a$, but of $\vec\sigma$ as well, $\Delta \vec\sigma = {2 \pi \over 5}(4 - 3)\vec\rho = {2 \pi \over 5}\vec\rho$. This example shows that a change of the vacuum of the light degree of freedom (the axion) entails a rearrangement of the heavy degrees of freedom (the dual photons). We next explain, following [@Anber:2015kea], how this rearrangement implies that quarks are deconfined on the axion DW. ![The axion DW between two neighboring vacua $(a=0, \vec\sigma=0)$ and $(a = {2 \pi \over 4}, \vec\sigma = {4 \pi \over 3} \vec\rho)$ for $SU(3)$ QCD(F) with $N_f=4, N_f T_{\cal{R}}=4$. On the top two panels, we show the wall profiles for $a$ and $\vec\sigma$ for $v L = 4$ and on the lower two panels, for $vL = 16$. Distances are measured in terms of the inverse dual photon wavelength, the “confinement scale". The multi-scale structure of the DW is evident in the much longer scale of variation of the axion profile for larger $vL$. The overall width of the wall is set by the axion Compton wavelength, while the electric flux—the region with nonzero gradient of $\vec\sigma$—carried by the DW is squeezed into a much narrower region, of order the confinement scale. (This figure was provided to us by Andrew Cox and Samuel Wong. The methods used to obtain the figure are described in [@Cox:2019aji].) []{data-label="fig:axiondw"}](axiondw1.pdf){width="0.9\linewidth"} A consideration of the mass scales leads one to expect that the main contribution to the DW dynamics is given by the axion field and that its Compton wavelength alone determines the DW properties.We notice that this feature is shared by the naive axion effective theory (\[effective2\]) on $\R^4$ where the color degrees of freedom are integrated out. However, it is also clear that if the DW on $\R^4$ are supposed to deconfine quarks, the long-distance axion approximation cannot be the whole story, as it knows nothing about color flux and confinement. The expectation of a multi-scale structure of the axion DW can be clearly seen in our calculable $\R^3 \times \S^1$ set up. The axion DW are more complex objects and also involve scales set by $m_\sigma$, the nonperturbative scale determining the mass gap for gauge fluctuations—this nonperturbative scale plays the role of $\Lambda$ in the $\R^4$ theory. A numerical solution for a DW profile showing the multiscale structure is on Fig. \[fig:axiondw\]. This multi-scale structure of DW is related to the quark deconfinement on DW. It is natural to expect that different scales associated with the axion DW are also important to the physics of quark deconfinement on axion DW on $\R^4$. ![The mechanism of quark deconfinement on axion DW. The quark/antiquark pair shown is suspended on a DW between two vacua of the axion theory (\[r3potential\]). The monodromy of the dual photon $\oint_C d \vec\sigma$ around the junction of two degenerate (due to the $0$-form $\Z_{N_c}^{(0)}$ center symmetry) DW equals the electric charge of the quark, or its weight $2 \pi \vec\lambda$ (the dashed lines are where weight-lattice discontinuities of $\vec\sigma$ occur, but no physical discontinuity). The electric fluxes of the quark and antiquark are absorbed by the DW, as indicated by the arrows. The quarks experience no force, due to the equal tensions of the DW to the left and right of each quark. See [@Cox:2019aji] for detailed explanations of a similar mechanism in super-Yang-Mills theory and for results of actual numerical simulations of DW with suspended quark-antiquark pairs.[]{data-label="fig:deconfwall"}](deconfWall1.pdf){width=".80\linewidth"} To see how deconfinement works in our calculable setup (illustrated on Fig. \[fig:deconfwall\]) recall that the monodromy (i.e. change) of $\vec\sigma$ across the DW is equal to the electric flux carried along the DW worldvolume, as implied by the duality $\partial_x \vec{\sigma} \sim \vec{E}_y, \partial_y \vec{\sigma} \sim -\vec{E}_x$ ($x,y$ are the spatial coordinates). Further, we recall that Cartan subalgebra electric fluxes of quarks of $N$-ality $k$ are $2 \pi \vec{w}_k + 2 \pi \times ({\rm root \;vectors})$. On the other hand, as we saw above, the fluxes carried by axion DW are ${2 \pi \over N_c}\vec{\rho}$ (for simplicity, we focus on DW where $\vec\sigma$ jumps from one branch to the neighboring one, i.e. $|\Delta q| = 1$, as for neighboring vacua on Fig. \[fig:su5F\]; the discussion for non-neighboring DW proceeds similarly). The ${2 \pi \over N_c}\vec{\rho}$ electric flux carried by the DW is, however, only a fraction of the flux carried by quarks. However, we now recall that our theory has a $\Z_{N_c}^{(0)}$ center symmetry (\[zeroform2\]), unbroken in the $N_f T_{\cal{R}}$ vacua. This symmetry maps DW solutions to DW solutions, and implies that there are $N_c$ DW solutions, with equal tensions, between any of the two axion vacua. However, these walls carry different electric fluxes: the first relation in (\[zeroform2\]), with $k=1$, implies that the $N_c$ different domain walls of the same tension carry electric fluxes ${2 \pi \over N_c} \vec\rho + 2 \pi \vec{w}_r$, with $r=0,...N_c-1$ (recall that $\vec w_0 \equiv 0$). Thus the difference between the electric fluxes carried by two walls, labelled by $r, r'$, is $2 \pi (\vec{w}_r - \vec{w}_{r'})$. Thus, as illustrated on Figure \[fig:deconfwall\], the junction of two such DW supports quarks of weights $2 \pi \vec\lambda = 2 \pi (\vec{w}_r - \vec{w}_{r'})$. A quark/antiquark pair suspended on two consecutive junctions experiences no force, due to the equal tensions of the DW. It is clear that any weight of the fundamental representation is deconfined on this $\Delta q=1$ wall. Similar to the discussion of [@Cox:2019aji], this result for the weights of quarks suspended on $|\Delta q| = 1$ DW also implies that Wilson loops for any nonzero $N$-ality representation also show perimeter law on the wall. [****]{} MA acknowledges the hospitality at the University of Toronto, where this work was completed. MA is supported by the NSF grant PHY-1720135. EP is supported by a NSERC Discovery Grant. We are grateful to Andrew Cox and Samuel Wong for their numerical solution of the multi-scale axion DW. [^1]: In terms of $N_f$ 4-component Dirac fermions $\Psi$ in ${\cal{R}}$ of $SU(N_c)$, the Yukawa coupling is $y \Phi\; \bar\Psi_L \cdot \Psi_R + {\rm h.c.}$. [^2]: For $N_f=1$ QCD(F), only fermion number $\Z_2$ is anomaly free; the Yukawa coupling and a fermion mass term then have the same symmetries. [^3]: The topological charge is $Q^c = \int d^4 x \;q^c \in \Z$ for the dynamical fields of the $SU(N_c)$ gauge theory. [^4]: The expectation that axion DW have a more complicated structure than implied by a naive deduction from (\[effective2\]) is not new—see [@Gabadadze:2002ff] for a review of QCD with light quarks, large-$N_c$ limit, super-Yang-Mills, and $D$-branes, and [@Komargodski:2018odf] for a related recent study. The new elements in our discussion are the use of generalized ’t Hooft anomalies and the explicit demonstration, see Section \[calculabledeformation\], of the DW structure and the deconfinement of quarks. [^5]: This can be seen upon canonical quantization, requiring fixing the gauge under the $2$-form gauge transformations of $a^{(3)}$ (see e.g. [@Kapustin:2014gua] for details). [^6]: For a single flavor of the two-index S or AS representation, the phase in (\[BCF1\]) is $\mathbb Z_{N\pm2}$-valued. [^7]: This is equivalent to requiring that the transition functions for the matter fields in the ’t Hooft flux backgrounds (\[backgrounds\]) satisfy the cocycle conditions. [^8]: In addition to the possible $\Z_{p}^{(1)}$ center present in the UV theory when gcd($N_c, n_c) = p >1$. [^9]: In contrast, on $\R^3 \times \S^1$, operators winding the $\S^1$ can be important in the regime of interest. [^10]: Arguments employing monopole-/dyon-condensation confinement (as in Seiberg-Witten theory) have been used to give a physical picture of quark deconfinement on DW (to the best of our knowledge, beginning with [@SJRey:1998], as cited in [@Witten:1997ep]). However, in the nonsupersymmetric theories at hand these excitations are “somewhat elusive"—quoting [@Witten:1997ep]; see [@Greensite:2011zz] for a review—hence these arguments are heuristic at best. The beauty of the $\R^3 \times \S^1$ explanation given in Section \[calculabledeformation\] is that it only involves controllable semiclassical physics. [^11]: To ensure center stability, one can also add nonlocal “double-trace deformations" along $\S^1$ to the compactified theory, but these can be seen to be generated by massive adjoint fermions, with mass ${\cal O}(1/N_c L)$. We take the view that a dynamical explanation of the double-trace deformations is needed to ensure renormalizability of the theory. The setup described above is known as “deformed Yang-Mills theory" and we refer the reader to the original paper [@Unsal:2008ch] for details. See [@Anber:2017pak] for interesting variations and a large list of references. [^12]: Generically, they will lead to an exponentially small shift of the holonomy $\vec\phi$ away from the $Z_{N_c}^{(0)}$ center symmetric point $\vec\phi = 2 \pi \vec\rho/N_c$, which will then only respect the $Z_{p={\rm gcd}(N_c, n_c)}^{(0)}$ center symmetry. We ignore these effects here and leave their study for the future (in particular only Wilson loops which can not be screened by $N$-ality $n_c$ quarks will be deconfined on the wall). [^13]: See [@Anber:2015bba] for an earlier study of axions coupled to deformed Yang-Mills theory. [^14]: The careful reader might notice a slight difference in (\[r3kinetic\]) from the formulae of [@Tanizaki:2019rbk] stemming from the different form of the fundamental $\S^1$ Wilson loop we use; no aspects of the physics are changed. [^15]: For brevity and to reduce clutter, we do not explicitly indicate the three dimensional nature of all forms that appear below. We also warn the reader against confusing the dynamical $4$d color field $A^{c}$ with $A^{c(1)}$, the 3d $1$-form part of $B^{c(2)}$. [^16]: \[enhancement\]This symmetry enhancement is specific to the calculable deformed-Yang-Mills theory setup that we consider here. Consider our $SU(N_f)$ theory with $N_f$ Dirac flavors in ${\cal{R}}$, with $n_f$ Weyl flavors of adjoint fields with a flavor-diagonal mass $m e^{i {\beta \over n_f}}$ added to ensure center stability. Here, we think of $\beta$ as another nondynamical (“spurion") axion field (recall that $n_f \ge2$ with $m \sim 1/L$ is needed for center stability). A monopole-instanton vertex on $\R^3 \times \S^1$, in the limit $v \gg L$, i.e. with the $\cal{R}$ fermions integrated out, is $\sim e^{- {8 \pi^2 \over N_c g^2}} m^{n_f} e^{i \vec\alpha_p \cdot \vec{\sigma}} e^{i \beta} e^{i {a N_f T_{\cal{R}} \over N_c}}$, $p=1, \ldots N_c$. The $m^{n_f}$ factor is due to the lifting of zero-modes in the monopole-instanton background and the other factors follow from [@Unsal:2007jx]. This nonperturbative vertex is invariant under an exact $U(1)$ symmetry shifting both $\beta$ and $a$, as well as the $\Z_{2 N_c n_f}$ anomaly-free discrete chiral symmetry of the adjoint fields, acting as $\beta \rightarrow \beta + {2 \pi \over N_c}$ and $\vec\sigma \rightarrow \vec\sigma - {2 \pi \over N_c }\vec\rho$ (notice that these are the only exact chiral symmetries of the mixed adjoint-$\cal{R}$ theories coupled to axions). Giving an expectation value to the spurion $\beta$ breaks the $U(1)$ but leaves the $\Z_{\cal{Q}}$, ${\cal{Q}} = {\rm lcm}(N_c, N_f T_{\cal{R}})$, symmetry, acting on $\vec\sigma$ by the above $2\pi\vec\rho/N_c$ shifts and on $a$ by $2\pi/(N_f T_{\cal{R}})$ shifts. This is an emergent $0$-form symmetry on $\R^3\times \S^1$. It acts on ’t Hooft loop operators winding around the circle, $e^{i \vec\alpha_p \cdot \vec{\sigma}}$ [@Argyres:2012ka; @Anber:2015wha], and is not detectable by local physics on $\R^4$; for example, an instanton vertex on $\R^4$ does not exhibit an enhancement of the $\Z_{N_f T_{\cal{R}}}$ axion shift symmetry.

--- author: - | , , , , , , , , , ,\ DPNC, University of Geneva, 24 quai Ernest Ansermet, 1211 Genève 4, Switzerland,\ European Organization for Nuclear Research (CERN), 385 Route de Meyrin, 1217 Meyrin, Switzerland\ SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA\ E-mail: title: 'The FE-I4 Telescope for particle tracking in testbeam experiments' --- =1 Introduction {#sec:intro} ============ Telescope description {#sec:desc} ===================== Mechanics and services {#sec:mec} ---------------------- DUT integration {#sec:dut} --------------- Reconstruction {#sec:rec} ============== Performance {#sec:per} =========== Conclusions {#sec:sum} =========== [99]{}

[ **Emergence of quasiparticle Bloch states in artificial crystals crafted atom-by-atom** ]{} J. Girovsky^1^, J. L. Lado^2^, F. E. Kalff^1^, E. Fahrenfort^1^, L. J. J. M. Peters^1^, J. Fernández-Rossier^2,3^ and A. F. Otte^1\*^ [**1**]{} Department of Quantum Nanoscience, Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands.\ [**2**]{} QuantaLab, International Iberian Nanotechnology Laboratory (INL), Avenida Mestre José Veiga, 4715-310 Braga, Portugal.\ [**3**]{} Departamento de Física Aplicada, Universidad de Alicante, San Vicente del Raspeig, 03690 Spain.\ \*Corresponding author: a.f.otte@tudelft.nl Abstract {#abstract .unnumbered} ======== [ **The interaction of electrons with a periodic potential of atoms in crystalline solids gives rise to band structure. The band structure of existing materials can be measured by photoemission spectroscopy and accurately understood in terms of the tight-binding model, however not many experimental approaches exist that allow to tailor artificial crystal lattices using a bottom-up approach. The ability to engineer and study atomically crafted designer materials by scanning tunnelling microscopy and spectroscopy (STM/STS) helps to understand the emergence of material properties. Here, we use atom manipulation of individual vacancies in a chlorine monolayer on Cu(100) to construct one- and two-dimensional structures of various densities and sizes. Local STS measurements reveal the emergence of quasiparticle bands, evidenced by standing Bloch waves, with tuneable dispersion. The experimental data are understood in terms of a tight-binding model combined with an additional broadening term that allows an estimation of the coupling to the underlying substrate.** ]{} Atom manipulation by means of STM is a viable way of constructing atomically precise artificial structures [@1]. Among others, the technique can be use to engineer atomic scale logic devices [@2; @3], low dimensional magnetic systems [@4; @5; @6] or atomic data storages [@7; @8; @9]. As our abilities to manipulate atoms on a large scale are improving, the formation of atomically designed artificial crystals becomes of particular interest driven by a demand for new materials where the properties are defined by emerging quasiparticle states [@10]. Common approaches to build low-dimensional artificial materials by STM include confinement of electronic surface states through precise assembly of individual atoms and/or molecules [@11; @12; @13; @14], self-assembly of molecular networks [@15; @16] and manipulation of dangling bonds [@17] or surface vacancies [@18]. The recent development of large-scale fully automated placement of atomic vacancies on a chlorinated copper crystal surface [@9] provides an excellent platform to explore various lattice compositions. These vacancies were found to host a localized vacancy state in the surface band gap, similar to dopants in semiconductors, allowing their combined electronic states to be modelled by means of tight-binding approximation [@19].\ Here, we present a study of artificial one- and two-dimensional structures built from Cl vacancies in an otherwise perfect monolayer square lattice formed by chlorine atoms on a Cu(100) surface. Using local electron tunnelling spectroscopy, we demonstrate that we are able to reach system scales where the spectral properties no longer depend on size and which we therefore consider to be in the limit of infinite lattice size. For structures with a sufficiently large vacancy density, we observe quasiparticle Bloch waves that can be simulated by using a tight-binding model. Similar wave patterns were reported previously in assembled chains of Au atoms [@14], which were best described in terms of a free electron model. Analysis of the Bloch wave dispersion allows us to extract quasiparticle effective masses, which are found to depend strongly on the chosen lattice structure.\ A monolayer of chlorine atoms on Cu(100) exhibits a surface band gap *$E_{\rm g}$* of about 7 eV (see inset of Fig. 1a) as well as a shift in the substrate’s work function by 1.25 eV [@20], suggesting a significant charge transfer between the substrate and chlorine atoms and formation of the interface dipole moment [@21]. Theoretical calculations predict a charge of 0.5 electron accumulated on chlorine atoms and depletion of the density of states (DOS) at the top-most layer of the copper substrate [@22]. Other materials with a similarly large surface band gap, e.g. ${\rm Cu_2N}$ on Cu(100) (*$E_{\rm g}$* $\sim$ 4 eV) [@23], NaCl bilayers on copper substrates (*$E_{\rm g}$* $\sim$ 8.5 eV) [@24], and non-polar MgO films on Ag(100) (*$E_{\rm g}$* $\sim$ 6 eV) [@25], have found applications in studies of elementary excitations in individual molecules and/or adatoms [@3; @26; @27; @28]. The insulating monolayers formed on the metal substrates have been shown to have a little effect on the valence band maximum, however significantly affect the conduction band minimum, which was found as high as $\sim$ 4 eV for NaCl bi- and tri-layers on copper [@24]. In our case, a sharp step in the differential conductance at $\sim$ 3.5 V denotes the conduction band minimum (Fig. 1a, black curve). The precise onset of the band was determined as the maximum in the normalized differential conductance ${\rm d}I/{\rm d}V\times V/I$ (see Fig. 5).\ As previously reported by Drost et al. [@19], when the Cl/Cu(100) interface possesses defects in the form of missing chlorine atoms (dark square in the inset of Fig. 1a), a localized electronic vacancy state is resolved at lower voltages $\sim$ 3.4 V (green curve Fig. 1a). The vacancy state exhibits similarities to localized states observed on gold atoms adsorbed on NiAl(110) [@14], in the gap region of hydrogen-doped Si(100) surface [@17], and on chlorine vacancies in the NaCl/Cu(111) [@24]. When two vacancies are brought close to each other by means of atom manipulation [@9], the spatial overlap of the wave functions leads to the formation of bonding and anti-bonding orbitals [@14; @17; @24]. These molecular orbitals can be effectively described within the tight-binding model with their energy depends on the hopping term [*t*]{} – a measure of the overlap of the two vacancy states.\ We built one-dimensional lattices of the Cl/Cu(100) vacancies of various lengths and lattice spacing ($\left\{3,0\right\}$, $\left\{2,0\right\}$ and $\left\{1,1\right\}$) as shown in Fig. 1. The notation $\left\{x,y\right\}$ used here, describes spacing between adjacent vacancies in the horizontal and vertical directions, respectively, in multiplies of the lattice constant $a = 3.55~{\rm \AA}$ . Differential conductance ([*[d]{}I[/]{}[d]{}V*]{}) spectra presented in Fig. 1, were acquired along chains of length 16, for all three spacing parameters. The spectra reveal a shift of the band onset towards lower voltages and broadening of the spectral features for the lattice of denser spacing. Both, the shift and the band broadening, result from the increased overlap between neighbouring sites. The observed spectral features show a correlation with the position within the chains, i.e. the band minimum measured on outer vacancies is found at higher energies compared to that resolved on inner ones. The correlation of the band minimum with the position within the chain is related to the broken translational symmetry at the outer positions and leads to the appearance of zero-dimensional states [@14]. This dependence is further corroborated for edge vacancies within denser lattices where the effect is more pronounced (e.g. Fig. 1e). Spectra acquired on the chlorine atoms within the chains show a similar spatial dependence of the band onset.\ In Fig. 1f we plot the dependence of the band onset as a function of the chain length, measured at the centre of each chain. For each lattice spacing, the band onset is found to saturate, however at different lengths: the $\left\{3,0\right\}$ chains saturate at 3.35 eV already for length 3, the $\left\{2,0\right\}$ chains at 3.18 eV for length 5 and the $\left\{1,1\right\}$ chains at 3.1 eV for length 8. The saturation of the band minimum implies an approach of the limit where edge effects no longer play a role for the inner vacancies and the chains can be effectively treated in the limit of infinite lattice size. Furthermore, the observed shift relates to the size of the hopping parameter [*t*]{} which increases for shorter lattice spacing [@19]. ![\[Figure 1 :\]Differential conductance spectra acquired on chlorine vacancy in Cl/Cu(100) substrate and in artificial 1D lattices crafted from the vacancies. ([**a**]{}) [*[d]{}I[/]{}[d]{}V*]{} measurement taken inside a chlorine vacancy (green) and on the bare Cl/Cu(100) substrate (black). Dots depict positions where spectra were taken. The inset shows an [*I*]{}-[*V*]{} curve acquired from -4.5 to 4.0 V where the surface band gap of $E_{\rm g}$ $\sim$ 7 eV is clearly visible. ([**b**]{}) Sketch denoting the energy level of the vacancy state with respect to the band continuum. ([**c**]{}, [**d**]{}, [**e**]{}) [*[d]{}I[/]{}[d]{}V*]{} spectra taken on vacancy sites and/or chlorine interstices (locations indicated by coloured dots in the insets) on lattices with 16 vacancy sites for spacing configurations $\left\{3,0\right\}$ ([**c**]{}), $\left\{2,0\right\}$ ([**d**]{}) and $\left\{1,1\right\}$ ([**e**]{}). ([**f**]{}) Evolution of the band onset as a function of lattice spacing and lattice size. Data points indicate the position of the band onset extracted from spectra taken on the middle vacancy of each lattice. In the case of an even-length chain, the averaged value measured on the two centre vacancies is shown. STM images were acquired in constant current mode [*I*]{} = 2 nA and [*V*]{} = 500 mV. All scale bars are 2 nm.](Figure1.pdf){width="1\linewidth"} \ To further investigate the band formation, we built two-dimensional structures with varying lattice spacing, ($\left\{3,3\right\}$, $\left\{2,3\right\}$, $\left\{2,2\right\}$) as well as ’stripes’ and ’checkerboard’ arrays, all of varying lattice size (Fig. 2). For 2D lattices, the notation $\left\{x,y\right\}$ denotes the lattice spacing in the $x$ and $y$ directions in units of the lattice constant $a$. Moving inward along the diagonal of each structure, the position of the band onset shifts towards lower energies for denser and larger lattices, similar to the 1D lattices. In the case of the stripes lattice we observe two band onsets, $E_{\rm 1}$ = 2.8 eV and $E_{\rm 2}$ = 3.1 eV, measured at the centre vacancy (see Fig. 2g). We attribute these to the two different lattice constants along the lattice diagonals, $a_{\rm 1}$ = 0.51 nm and $a_{\rm 2}$ = 0.69 nm. Assuming that the hopping parameter is exponentially dependent on the distance [@19] and the bandwidth is linearly proportional to the hopping parameter [*t*]{}, the band is expected to be symmetric around the energy [*E*]{} = 3.4 eV of a single vacancy. We estimate the widths of the respective bands to be $W_{\rm 1}$ $\sim$ 1.2 eV and $W_{\rm 2}$ $\sim$ 0.6 eV, leading to a ratio $W_{\rm 1}$/ $W_{\rm 2}$ $\sim$ 2. This ratio is somewhat higher than the ratio between the hopping parameters $t{\rm (}{\it a}_{\rm 1})$/$t{\rm (}{\it a}_{\rm 2})$ $\sim$ 1.2, suggesting that another effect may play a role, affecting the width and/or position of the band, e.g. an electric field due to positively charged neighbouring vacancies observed at polar insulating surfaces. Such an electric field can cause a shift of the band onset towards lower energies, which is expected to be larger for denser lattices [@18].\ The checkerboard lattice (Fig. 2e) was found to be more sensitive to relatively high tunnelling currents than lattices with lower vacancy coverage. For large tunnelling current and voltage values (e.g. $>$ 2 nA at $\sim$ 4.7 V), we observed chlorine atoms changing their position, rendering the structure unstable. For this reason, instead of acquiring d[*I*]{}/d[*V*]{} spectra, we measured the dependence of the tip-sample distance [*z*]{} as a function of sample voltage in constant current mode, i.e. d[*z*]{}/d[*V*]{} curves, that qualitatively resemble the normalized differential conductance d[*I*]{}/d[*V*]{}$\times$[*V*]{}/[*I*]{} (Fig. 5). ![\[Figure 2 :\]Analysis of artificial 2D lattices by chlorine vacancies of Cl/Cu(100). ([**a**]{}, [**b**]{}, [**c**]{}, [**d**]{}, [**e**]{}) Differential conductance spectra measured along diagonals of the lattices with spacing $\left\{3,3\right\}$ ([**a**]{}), $\left\{2,3\right\}$ ([**b**]{}), $\left\{2,2\right\}$ ([**c**]{}) and with stripes ([**d**]{}) and checkerboard ([**e**]{}) patterns. Coloured dots denote the positions where the spectra were acquired. ([**f**]{}) Evolution of band onset as function of lattice spacing and lattice size. Band onset was extracted from the spectra taken in the middle of the lattices. ([**g**]{}) Band onset as a function of lattice density, i.e. number of vacancies divided by number of total positions in the unit cell. STM images were acquired in constant current mode [*I*]{} = 2 nA and [*V*]{} = 500 mV. All scale bars 2 nm.](Figure2.pdf){width="1\linewidth"} \ Apart from preserving the lattice integrity, the d[*z*]{}/d[*V*]{} measurement mode also provides sufficient sensitivity to detect standing wave modes in some of the lattices that are not visible in d[*I*]{}/d[*V*]{} mode (see Methods for details). Fig. 3 shows d[*z*]{}/d[*V*]{} maps acquired on the checkerboard (panels a-d) and stripes (panel j-k) lattices. Interestingly, the modes are resolved very symmetric in the [*x*]{} and [*y*]{} directions, i.e. the number of protrusions in both directions is equivalent, even though the unit cell of the stripes lattice is highly asymmetric.\ To shine more light onto the standing wave pattern, we performed numerical calculations of artificial lattices of size $8 \times 8$ using tight-binding approach that effectively simulate d[*z*]{}/d[*V*]{} maps (Figs. 3e-h, l-n), which are proportional to the density of states (see Appendix for details). The observed modes can be described in terms of two-dimensional confinement modes with [*k*]{}-vectors $k_{\it x} = N\pi/L$ and $k_{\it y} = M\pi/L$, where $L$ is the width of the lattice. The experimentally observed modes resemble some of the calculated modes with [*N*]{} = [*M*]{} (Fig. 6). However, the experimental data show a richer structure with the links connecting the very bright protrusions in [*x*]{} and [*y*]{} direction. Furthermore, the experimentally observed modes are gradually transforming from one mode to another with an increasing number of lobes. At certain energies some of the lobes are not spherical, but rather have an elongated shape, that splits into two with increasing voltage. ![\[Figure 3 :\]d[*z*]{}/d[*V*]{} maps acquired on checkerboard and stripe lattices.([**a**]{}, [**b**]{}, [**c**]{}, [**d**]{}) d[*z*]{}/d[*V*]{} maps taken on an 8$\times$8 checkerboard lattice at different energies. ([**e**]{}, [**f**]{}, [**g**]{}, [**h**]{}) Corresponding numerical calculations using a tight-binding model including an additional hybridization term. ([**i**]{}, [**j**]{}, [**k**]{}) and ([**l**]{}, [**m**]{}, [**n**]{}). Similar as ([**a**]{}–[**d**]{}) and ([**e**]{}–[**h**]{}) for an 8$\times$8 stripes lattice. All scale bars 2 nm. ](Figure3.pdf){width="1.0\linewidth"} \ This smooth crossover can be reproduced by including the coupling of the confined modes to electronic bath underneath. This interaction, which is mathematically represented by a self-energy with finite imaginary part, leads to a finite lifetime of the states and a broadening of the spectral function, which consequently overlap neighbouring energy states and alter the appearance of the modes (see Methods for details on the numerical calculations). The addition of this hybridization term yields a DOS profile composed from the mixture of many individual modes that can no longer be resolved, and resembles up to very fine details the experimental STM maps, including the smooth crossover. Similar broadening of the energy modes was also attributed to strong electron-phonon coupling [@24]. As such, the experimental modes have to be understood as coming from the mixture of many individual modes due to the finite coupling of the lattice to the underlying copper, so that at a particular energy we do not observe a single confined mode, but a weighted mixture of the neighbouring modes. ![\[Figure 4 :\]Fourier analysis of the d[*z*]{}/d[*V*]{} maps. ([**a**]{}, [**b**]{}, [**c**]{}, [**d**]{}, [**e**]{}, [**f**]{}) Evolution of the Fourier map for the checkerboard lattice at different energies, corresponding to d[*z*]{}/d[*V*]{} maps presented in Figs. 3b, c, d and i, j, k, respectively. Increasing energy leads to a shift of the maximal intensity towards higher [*k*]{}-vector values. ([**g**]{}) Evolution of the expectation value of the square of the momentum vs. the energy for checkerboard (purple) and stripes lattice (green) in a wide energy range. Dashed lines define the energy intervals within which the dispersive modes are observed in each lattice. ([**h**]{}) Dispersion plots [*E*]{} vs. $\langle k^{\rm 2}\rangle$ for the energy intervals marked in ([**g**]{}). The experimental and theory data points are represented by full and open circles, respectively. Solid lines show linear fits, from which the effective masses of experimental observed modes are extracted.](Figure4.pdf){width="1.0\linewidth"} \ In a similar manner, we calculated the DOS on the stripes lattice, where the distance between vacancies along two lattice directions is not equal. We included two different hopping terms in our calculations in order to properly reproduce the experimental data. If only the hopping term along the direction of the stripes is considered, the numerically resolved modes within the chains are decoupled from each other and do not reflect the experimentally observed patterns (Figure 7). Comparison of numerical results with the experimental images allows to directly extract the effective hopping parameters for the effective square lattices. In our numerical simulations we used a model with only first neighbour hopping term of $-215$ meV, that reproduces the features of checkerboard lattice very well, but fails to capture the features of stripes lattice. However, the tight binding model with first and second neighbour hopping parameters $-139$ meV and $-38$ meV, respectively, can reproduce both the checkerboard and stripes lattice patterns to a great extent.\ The previous discussion requires investigation of the numerical and experimental d[*z*]{}/d[*V*]{} maps one by one and identify the interference patterns. The emergence of dispersive modes within the maps can be explored systematically using the quantitative fast Fourier transform (FFT) analysis of these d[*z*]{}/d[*V*]{} maps images (Figs. 4a-f). In the following, we calculate the expectation value of the square of the [*k*]{}-vector $\langle k^{\rm 2}\rangle$ of each FFT image, thus assigning a single value to a complete ${\rm d}{\it z}{\rm /d}{\it V}$ map (see Methods for details on how the $\langle k^{\rm 2}\rangle$ values were calculated). Plotting the energy [*E*]{} (i.e. the applied bias voltage) as a function of $\langle k^{\rm 2}\rangle$ we provide a dispersion curve that allows to systematically identify the energy regions where an interference pattern is visible. Fig. 4h shows the obtained dispersion diagrams for the checkerboard in the energy interval 2750 meV to 3140 meV, and for the stripes lattice in the energy interval 2775 meV to 3175 meV. The full [*E*]{} vs. $\langle k^{\rm 2}\rangle$ plots are shown in Fig. 4g.\ We performed linear fits to the *E* vs. $\langle k^{\rm 2}\rangle$ plots in order to extract effective electron masses for the checkerboard *$m_{\rm eff}$* = 1.470 $\pm$ 0.034 $m_e$ and for the stripes lattice *$m_{\rm eff}$* = 0.131 $\pm$ 0.025 $m_e$, where $m_e$ is the free electron mass. One would expect the stripes lattice, being highly anisotropic, to yield different effective masses for the directions parallel and perpendicular to the stripes. However, the weight in the FFT maps is found predominantly along the $k_{\it x}$ and $k_{\it y}$ axis, which are rotated 45 degree with respect to the stripes. Therefore, a single value for the effective mass suffices to describe the observed standing wave patterns. The obtained values suggest that quasiparticle waves in the checkerboard lattice are heavier than those in the stripes lattice. While the calculations confirm this observation (theory: checkerboard: *$m_{\rm eff}$* = 0.98 $\pm$ 0.06 $m_e$ and stripes: *$m_{\rm eff}$* = 0.22 $\pm$ 0.016 $m_e$), intuitively, one might expect the checkerboard lattice, being denser than the stripes lattice, to provide greater band width due to larger hopping parameters, and therefore to yield a lower effective mass. The dispersive properties found from the analysis in Fig. 4 should therefore be considered as phenomenological only. Conclusion {#conclusion .unnumbered} ========== Engineering artificial lattices by means of atom manipulation of chlorine vacancies in the Cl/Cu(100) substrate demonstrate a way to craft artificial one- and two-dimensional materials with tuneable electronic properties. We explore the emergent band formation as we build lattices of varying structure, density and size. For all lattices studied, the bottom of the emerging band is found to shift towards lower energies, in accordance to the tight-binding model, as the lattice size or density is increased. Furthermore, we find that the band onset saturates for larger structures, implying that the effect of finite size can be neglected. In the case of two-dimensional checkerboard- and stripe shaped lattices, we observe standing Bloch waves. These patterns are well explained using a tight-binding model that includes coupling to the electron bath. Surprisingly, the effective mass of the observed Bloch waves is found to depend strongly on the lattice geometry. Our work provides a testing ground for future designer materials where the electronic properties can be defined a priori. Acknowledgements {#acknowledgements .unnumbered} ================ J. G., F. E. K. and A. F. O. acknowledge support from the Netherlands Organisation for Scientific Research (NWO/OCW), NWO VIDI grant no. 680-47-514, and as part of the Frontiers of Nanoscience (NanoFront) program. J. L. L. and J. F.-R. acknowledge financial support by Marie-Curie-ITN 607904-SPINOGRAPH. J. F.-R. acknowledges financial support from MEC-Spain (MAT2016-78625-C2). [99]{} J. A. Stroscio and D. M. Eigler, [*Atomic and Molecular Manipulation with the Scanning Tunnelling Microscope*]{}, Science [**254**]{}, 1319 (1991), . A. J. Heinrich, et al. [*Molecule Cascades*]{}, Science [**298**]{}, 1381 (2002), A. A. Khajetoorians, J. Wiebe, B. Chilian and R. Wiesendanger, [*Realizing All-Spin-Based Logic Operations Atom by Atom*]{}, Science [**332**]{}, 1062 (2011), C. F. Hirjibehedin, et al. [*Spin Coupling in Engineered Atomic Structures*]{}, Science [**312**]{}, 1021 (2006), R. Toskovic, et al. [*Atomic spin-chain realization of a model for quantum criticality*]{}, Nat. Phys. [**12**]{}, 656 (2016), A. Spinelli, B. Bryant, F. Delgado, J. Fernández-Rossier and A. F. Otte, [*Imaging of spin waves in atomically designed nanomagnets*]{}, Nat. Mater. [**13**]{}, 782 (2014), S. Loth, S. Baumann, C. P. Lutz, D. M. Eigler and A. J. Heinrich, [*Bistability in Atomic-Scale Antiferromagnets*]{}, Science [**335**]{}, 196 (2012), A. A. Khajetoorians, et al. [*Current-Driven Spin Dynamics of Artificially Constructed Quantum Magnets*]{}, Science [ **339**]{}, 55 (2013), F. E. Kalff, et al. [*A kilobyte rewritable atomic memory*]{}, Nat. Nanotechnol. [**11**]{}, 926 (2016), L. Venema, et al. [*The quasiparticle zoo*]{}, Nat. Phys. [**12**]{}, 1085 (2016), K. K. Gomes, W. Mar, W. Ko, F. Guinea and H. C. Manoharan, [*Designer Dirac fermions and topological phases in molecular graphene*]{}, Nature [**483**]{}, 306 (2012), Y. Pennec, et al. [*Supramolecular gratings for tuneable confinement of electrons on metal surfaces*]{}, Nat. Nanotechnol. [**2**]{}, 99 (2007), M. R. Slot, et al. [*Experimental realization and characterization of an electronic Lieb lattice.*]{}, Nat. Phys. AOP (2017) Nilius, N., Wallis, T. M. and Ho, W. [*Development of One-Dimensional Band Structure in Artificial Gold Chains*]{}, Science [**297**]{}, 1853 (2002), J. Lobo-Checa, et al. [*Band Formation from Coupled Quantum Dots Formed by a Nanoporous Network on a Copper Surface*]{}, Science [**325**]{}, 300 (2009), F. Klappenberger, et al. [*Dichotomous Array of Chiral Quantum Corrals by a Self-Assembled Nanoporous Kagomé Network*]{}, Nano Lett. [**9**]{}, 3509 (2009), S. R. Schofield, et al. [*Quantum engineering at the silicon surface using dangling bonds*]{}, Nat. Commun. [**4**]{}, 1649 (2013), B. Schuler, et al. [*Effect of electron-phonon interaction on the formation of one-dimensional electronic states in coupled Cl vacancies*]{}. Phys. Rev. B [**91**]{}, 235443 (2015), R. Drost, T. Ojanen, A. Harju and P. Liljeroth, [*Topological states in engineered atomic lattices*]{}, Nat. Phys. AOP (2017), J. Nowakowski, et al. [*Probing the Reactivity of Functionalized Surfaces by Porphyrin Metalation*]{}, ChemistrySelect [**1**]{}, 891 (2016), T. Roman and A. Gro[ß]{}, [*Periodic Density-Functional Calculations on Work-Function Change Induced by Adsorption of Halogens on Cu(111)*]{}, Phys. Rev. Lett. [**110**]{}, 156804 (2013), A. Migani and F. Illas, [*A Systematic Study of the Structure and Bonding of Halogens on Low-Index Transition Metal Surfaces*]{}, J. Phys. Chem. B [**110**]{}, 11894 (2006), C. D. Ruggiero, T. Choi and J. A. Gupta, [*Tunneling spectroscopy of ultrathin insulating films: CuN on Cu(100)*]{}, Appl. Phys. Lett. [**91**]{}, 253106 (2007), J. Repp, G. Meyer, S. Paavilainen, F. E. Olsson and M. Persson, [*Scanning Tunneling Spectroscopy of Cl Vacancies in NaCl Films: Strong Electron-Phonon Coupling in Double-Barrier Tunneling Junctions*]{}, Phys. Rev. Lett. [**95**]{}, 225503 (2005), S. Schintke and W.-D. Schneider, [*Insulators at the ultrathin limit: electronic structure studied by scanning tunnelling microscopy and scanning tunnelling spectroscopy*]{}, J. Phys.: Cond. Matter [**16**]{}, 49 (2004), J. Repp, et al. [*Controlling the Charge State of Individual Gold Adatoms*]{}, Science [**305**]{}, 493 (2004), J. Repp, G. Meyer, S. M. Stojković, A. Gourdon and C. Joachim, [*Molecules on Insulating Films: Scanning-Tunneling Microscopy Imaging of Individual Molecular Orbitals*]{}, Phys. Rev. Lett. [**94**]{}, 026803 (2005), I. G. Rau, et al. [*Reaching the Magnetic Anisotropy Limit of a 3d Metal Atom*]{}, Science [**344**]{}, 988 (2014), Methods {#methods .unnumbered} ======= #### Preparation of chlorine terminated Cu(100): Cu(100) crystals were cleaned by repeated cycles of Argon sputtering and subsequent annealing at 550 $^{\circ}$C. The chlorine terminated Cu(100) substrate was prepared by thermal evaporation of anhydrous CuCl${\rm _2}$ powder from a quartz crucible heated to 300 $^{\circ}$C. Clean Cu(100) crystals are heated to 150 $^{\circ}$C before, during and after the deposition for 10 minutes at each step [@9]. The quality of the surface was verified with LEED and STM. #### Acquisition of d[*z*]{}/d[*V*]{} maps: The arrangement of chlorine atoms in the checkerboard lattice is very sensitive to large tunnelling currents; a current exceeding [*I*]{} $>$ 1 $\mu$A frequently caused unintended displacement of the atoms. Such high currents are reached whilst acquiring differential conductance spectra (d[*I*]{}/d[*V*]{}), and causing the entire structure to collapse. In order to qualitatively extract the local density of states (DOS) in the checkerboard and stripes lattices, we used a method where, instead of acquiring d[*I*]{}/d[*V*]{} spectra, we recorded the tip-sample distance [*z*]{} as a function of applied bias voltage [*V*]{}. The time constant of the feedback-loop was much smaller (t = 25 $\mu$s) than the time set to measure a single data-point (t $\sim$ 1 s), ensuring thus that the tip had enough time to stabilize. In this mode the tunnelling current was kept constant at [*I*]{} = 500 pA. In the next step a numerical derivation of a [*z*]{} vs.[*V*]{} curve, i.e. the d[*z*]{}/d[*V*]{} curve, has been extracted (black, red and blue line in Fig. 5). As the tunnel current [*I*]{} is exponentially proportional to the tip-sample distance [*z*]{},\ $$I(z) = AVe^{-2{\frac{\sqrt{2m{\phi}}}{\hbar}z}}$$ ,where [*A*]{} is a constant, [*V*]{} the bias voltage, [*m*]{} the mass of the tunnelling electron, $\phi$ the height of the tunnelling barrier and $\hbar$ the reduced Planck constant. Extracting [*z*]{} as a function of the tunnelling current and the applied bias voltage results in $$z = \frac{{\rm ln}(\frac{I}{V})-{\rm ln}(A)}{-2{\frac{\sqrt{2m{\phi}}}{\hbar}}}$$ Derivation of the distance [*z*]{} to voltage gives $$\label{eq:Eq3} \frac{{\rm d}z}{{\rm d}V} \propto \frac{{\rm d}I}{{\rm d}V}\frac{V}{I}$$ As can be seen from , the d[*z*]{}/d[*V*]{} is linearly proportional to the normalized differential conductance spectra (d[*I*]{}/d[*V*]{}$\times$[*V*]{}/[*I*]{}), which in turn is proportional to local DOS.\ d[*z*]{}/d[*V*]{} maps have been acquired in constant current mode, where consecutive topography images have been taken on the same area at different bias voltages in 10 mV intervals for checkerboard lattices and in 50 meV intervals for stripes lattices. The consecutive images have been subtracted, thus providing the height difference d[*z*]{} for each point of the topography images for a respective voltage difference. #### Extracting $\langle k^{\rm 2}\rangle$ values: The acquired d[*z*]{}/d[*V*]{} maps were transformed into FFT maps as those shown in Figure 4. Each pixel of the FFT image carries information about the intensity *I*, i.e. weight, of the corresponding *k*-vector value. In the next step the intensity and the corresponding *k*-vector value are squared, i.e. *$I^{\rm2}$* and *$k^{\rm2}$*, respectively and multiplied with each other. The profiles along *$k_x^{\rm 2}$* axis, i.e. *$k_y^{\rm 2}$* = 0 and along *$k_y^{\rm 2}$* axis, i.e. *$k_x^{\rm 2}$* = 0 are normalized by the sum of *$I^{\rm 2}$* leading to a expectation values $\langle k_x^{\rm 2}\rangle$ and $\langle k_y^{\rm 2}\rangle$, respectively. The d[*z*]{}/d[*V*]{} maps exhibit noise signal with very small real-space wavelength, corresponding to a large *k*-vector, thus we calculate the expectation values considering only 3 points left and 3 points right from the maximal *$I^{\rm 2}$*. Profiles along *$k_y^{\rm 2}$* and *$k_y^{\rm 2}$* axis appear identical and we thus calculate the expectation value $\langle k^{\rm 2} \rangle$ = ($\langle k_x^{\rm 2}\rangle$+$\langle k_y^{\rm 2} \rangle$)/2. ![\[Figure 5 :\]Comparing d[*z*]{}/d[*V*]{} and d[*I*]{}/d[*V*]{}$\times$[*V*]{}/[*I*]{} spectra acquired on Cl/Cu(100). Dashed curves represent normalized d[*I*]{}/d[*V*]{}$\times$[*V*]{}/[*I*]{} spectra. The corresponding black, red and blue curves were taken at the same locations as the dashed ones, i.e. on the bare Cl/Cu(100) substrate (black) and on the vacancies within the stripes lattice (cyan and red).](Figure5.pdf){width="1\linewidth"} ![\[Figure 6 :\]Numerical calculations on checkerboard lattice using a tight-binding model. Standing wave patterns acquired for the numerical calculations with the hybridization with the environment term set to zero. The integers [*N*]{} and [*M*]{} denote number of modes for $k_x$ and $k_y$ axis.](Figure6.pdf){width="1\linewidth"} ![\[Figure 7 :\]Numerical calculations on stripes lattice using a tight-binding model. Standing wave pattern within the stripes lattice has been simulated using nearest neighbour and next nearest neighbour hoping terms (top row) or nearest neighbour hopping term only (bottom row). Experimental data are better reproduced using both the nearest and next-nearest hopping term.](Figure7.pdf){width="1\linewidth"} #### Numerical calculations: We performed numerical calculations for a model Hamiltonian defined on two different geometries, for the checkerboard lattice and the second one the stripe lattice. In both situations the size of the lattice we took is exactly the same as in the experiment. The calculations are performed in the tight binding approximation using a Hamiltonian with local orbitals in the form $$\mathcal{H} =\sum_{ij}{t_{ij}c_{j}^{\dagger}c_{i}}$$ where the parameters $t_{ij}$ are the elements of the overlap matrix between states localized within the chlorine vacancies and defined as $$t_{ij} = \langle\psi_{i}|\mathcal{H}|\psi_{j}\rangle$$ and $c_{j}^{\dagger}$ and $c_{i}$ are creation and annihilation parameters at site j and i, respectively.\ In our calculations, we use a value of -139 meV for first neighbour hopping term $t_{\rm 1}$ and a value of -38 meV for second neighbour hopping term $t_{\rm 2}$. Furthermore, to simulate the potential well we use the edge potential of 38 meV for the checkerboard lattice and 80 meV, for the stripes lattice.\ The effect of the hybridization with the metallic bath is taking into account by means of a self-energy parameter with finite imaginary part, that enters the Dyson equation of the Green function. For simplicity we assume the self-energy to be site-independent and diagonal, which allows to precisely reproduce the experimental features in a wide energy range. Green function is thus defined as follow $$G(E) = (E-\mathcal{H}-\Sigma)^{-1}$$ with self-energy term defined as $$\Sigma = i\delta$$ where $\delta$ = 40 meV. Within the previous Green function, the density of states at the site i is given by $$\rho_{i}(E) = {\rm Im}(G_{ii}(E))$$ The spatially resolved DOS is calculated assuming that the local states $\psi_{i}$ is centered in ${\bf r}_i$ have the form $$\psi_{i}({\bf r}) = Ne^{-({\bf r}-{\bf r}_{i}})^{2}/\sigma^{2}$$ with $\sigma$ = 0.9[*a*]{}, where [*a*]{} is the first neighbour vacancy-vacancy distance.\ \ Theoretical calculations of the artificial checkerboard lattice of size 8x8 using tight-binding approach with [*t*]{} = –215 meV without hybridization term are shown in Figure 6, showing the individual modes. In two dimensions, the patterns are characterized by two vectors $k_x$ and $k_y$, that are independent of each other. The vectors are defined as $k_{\it x} = N\pi/L$ and $k_{\it y} = M\pi/L$ where [*N*]{}, [*M*]{} = (1, 2, 3, … , 8) are the mode numbers and [*L*]{} is the size of the lattice ([*L*]{} = 8 in our case). In order to get agreement with the experiment, a finite hybridization with the metal is needed. Furthermore, in the case of the stripes lattice, an additional next nearest neighbour hopping term is needed. In Figure 7 we show calculations for the stripes lattice using the tight-binding model without hybridization term, (i) with nearest neighbour and next nearest neighbour hopping term and (ii) with nearest neighbour hopping term only. The best agreement with the experimental results is found when both terms are included.\

--- author: - 'Haydeé Herrera[^1] and Rafael Herrera[^2] [^3]' title: 'Higher -genera on certain non-spin $S^1$-manifolds' --- [ ]{} Introduction ============ The classical result of Atiyah and Hirzebruch AH about the vanishing of the -genus on Spin manifolds with $S^1$ actions was generalized by Browder and Hsiang Browder to higher -genera in the following form. [[@Browder Theorem 1.8]]{} Let $M$ be a closed Spin manifold with a smooth effective action of a compact, connected, positive-dimensional Lie group $G$. Then $$p_*([M]\cap {\mbox{$\widehat A$}})=0,$$ where $p\colon M{\longrightarrow}M/G$, and ${\mbox{$\widehat A$}}\in H^{4*}(M;\mathbb{Q})$ is the ${\mbox{$\widehat A$}}$ polynomial in the Pontrjagin classes. Furthermore, from this theorem they also deduced a higher -genus theorem analogous to Novikov’s “higher signature".\ By a closed manifold $M$, we mean a compact manifold without boundary. Notice that if $G$ is a compact Lie group not necessarily connected then we restrict our attention to the connected component of the identity element. In this paper, we prove two theorems (Theorems \[theo1\] and \[theo2\]) for non-Spin $G$-manifolds with finite $\pi_2$ and $\pi_4$. They are analogous to those of Browder and Hsiang [@Browder] for Spin manifolds. \[theo1\] Let $M$ be a smooth, closed, connected, oriented (even-dimensional) $G$-manifold with finite $\pi_2(M)$ and $\pi_4(M)$, where $G$ is a compact, connected, positive-dimensional Lie group. Then for any $y\in H^*(M/G, \mathbb{Q})$ $$({\mbox{$\widehat A$}}\cup p^*(y))[M]=0,$$ which implies $$p_*([M]\cap {\mbox{$\widehat A$}})=0,$$ where $p\colon M{\longrightarrow}M/G$ is the projection map, ${\mbox{$\widehat A$}}\in H^{4*}(M;\mathbb{Q})$ is the ${\mbox{$\widehat A$}}$ polynomial. The proof will make use of the $G$-transversality approach of Browder and Quinn [@Browder-Quinn], properties of $G$-transverse submanifolds, and the rigidity of the elliptic genus on manifolds admitting 2-balanced $S^1$ actions (see below). [*Acknowledgements*]{}. The authors wish to thank Anand Dessai for fruitful conversations. The first named author wishes to thank the Centro de Investigación en Matemáticas and the Instituto de Matemáticas of UNAM (México) for their hospitality and support. The second named author wishes to thank the Max Planck Institute for Mathematics (Bonn) and the Institut des Hautes Études Scientifiques (France) for their hospitality and support. $S^1$-transverse submanifolds of manifolds with finite $\pi_2$ and $\pi_4$ ========================================================================== Let $G$ be a connected Lie group acting smoothly on a manifold $M$. Let $H$ be a subgroup of $G$. We denote by $M^H$ the fixed point set of $H$ on $M$. A $G$-invariant submanifold $N$ of $M$ is called [*transverse*]{} if $N$ intersects $M^H$ transversely for every subgroup $H$ of $G$. In order to prove Theorem \[theo1\], we only need to consider a circle action. Thus, we can choose any circle subgroup $S^1 \subseteq G$. We denote by $M^{S^1}$ the fixed point set of the circle action. At a fixed point $p\in M^{S^1}$, the tangent space of $M$ becomes a real representation of $S^1$, whose complexification can be written as $$T_pM\otimes \mathbb{C}= (t^{m_1}+ t^{-m_1}) + \cdots + (t^{m_d} + t^{-m_d})$$ where $t^a$ denotes the representation on which $\lambda\in S^1$ acts by multiplication by $\lambda^a$, and $d$ is half the dimension of $M$. The term $(t^{n} + t^{-n})$ corresponds to the representation $$\lambda=e^{i\theta}\in S^1 \mapsto \left(\begin{array}{cc} \cos(n\theta) & -\sin(n\theta) \\ \sin(n\theta) & \cos(n\theta) \end{array}\right).$$ The numbers $\pm m_1,\ldots, \pm m_d$ are called the [exponents (or weights)]{} of the $S^1$-action at the point $p$. A circle action is called [*$2$-balanced*]{} if the parity of $\sum_{i=1}^dm_i$ does not depend on the connected component of $M^{S^1}$ (cf. ḨiBJ). Since we are only interested in the parity of $\sum_{i=1}^dm_i$, we do not worry about the choice of signs. \[2-balanced\] Let $M$ be a $S^1$-manifold with finite $\pi_2(M)$ and $\pi_4(M)$. Let $N$ be an $S^1$-transverse submanifold of $M$. Then the $S^1$ action on $N$ is $2$-balanced. [*Proof*]{}. Since $N$ meets $M^{S^1}$ transversely, for $p\in N\cap M^{S^1}$ $$T_pM = T_pN + T_pM^{S^1}.$$ Notice that $N^{S^1}= N\cap M^{S^1}$. Let $p, p'\in N^{S^1}$ lie in two different components of $N^{S^1}$. The tangent spaces to $N$ at $p$ and $p'$ become $S^1$ representations so that $$\begin{aligned} T_pN\otimes \mathbb{C} &=& (t^{n_1(p)} + t^{-n_1(p)}) +\ldots + (t^{n_k(p)} + t^{-n_k(p)}),\nonumber \\ T_{p'}N\otimes \mathbb{C} &=& (t^{n_1(p')} + t^{-n_1(p')}) +\ldots + (t^{n_k(p')} + t^{-n_k(p')}) ,\nonumber\end{aligned}$$ and we have to verify that $$(n_1(p)+\ldots +n_k(p))-(n_1(p')+\ldots +n_{k'}(p'))\equiv 0 \quad ({\mbox{{\rm}mod{\kern1pt}{\kern1pt}{\kern1pt}}}2).\label{difference}$$ Observe that the numbers $n_i(p)$ and $n_i(p')$ are, in fact, exponents of the action of $S^1$ on the manifold $M$, and that [(\[difference\])]{} is the difference of exponents of the manifold $M$, since the only missing directions of the tangent space of $M$ are trivial representations (as $N$ and $M^{S^1}$ meet transversely). Since $M$ has finite $\pi_2(M)$ and $\pi_4(M)$, by [@Bredon Theorem V] $$f(t)=T_pM_c - T_{p'}M_c = (1-t)^3 P(t),$$ where $P(t)=\sum b_i t^i$ with only finitely many $b_i$’s different from zero. Since real representations are invariant under the automorphism $t\mapsto t^{-1}$ $$f(t)=f(t^{-1}),$$ i.e. $$(1-t)^3P(t) = \left( 1-{1\over t}\right)^3 P(t^{-1}).$$ Thus, $$t^3P(t) = -P(t^{-1}),$$ $$t^3P(t) +P(t^{-1})=0,$$ $$t^3\sum b_i t^i + \sum b_i t^{-i}=0,$$ $$t^{3/2}\sum b_i (t^{i+3/2} + t^{-i-3/2})=0.$$ Since $b_i\not=0$, for every term of the form $b_i (t^{i+3/2} + t^{-i-3/2})$ there must be another one that cancels it out, i.e. there must be a $j\not =i$ such that $b_i=-b_j$ so that either $j+3/2=i+3/2$ which cannot happen because it contradicts $i\not=j$, or $-j-3/2=i+3/2$, and $i=-3-j$. Then, all the terms of $P(t)$ can be grouped according to the corresponding pairs $$b_it^i + b_jt^j = b_it^i - b_it^{-3-i},$$ which multiplied by $(1-t)^3$ give $$\begin{aligned} (b_it^i - b_it^{-3-i})(1-t)^3&=& b_i(t^i+t^{-i}) -3b_i(t^{i+1}+t^{-(i+1)}) \nonumber\\ &+&3b_i(t^{i+2}+t^{-(i+2)}) -b_i(t^{i+3}+t^{-(i+3)}). \nonumber\end{aligned}$$ Taking the sum of the exponents (with any choice of signs) with multiplicity gives zero (mod 2), $$b_i(i) -3b_i(i+1) + 3b_i(i+2) - b_i(i+3) \equiv 0 \quad({\mbox{{\rm}mod{\kern1pt}{\kern1pt}{\kern1pt}}}2).$$ [Note that the lemma is still valid if we only require the $S^1$ action on $M$ to be $2$-balanced instead of $M$ having finite $\pi_2(M)$ and $\pi_4(M)$. ]{} Elliptic genus on manifolds with $2$-balanced $S^1$-actions =========================================================== Let ${\raise1pt\hbox{${\textstyle}\bigwedge$}}_c^{\pm}$ be the even and odd complex differential forms on the oriented, closed, smooth manifold $X$ under the Hodge $*$-operator, respectively. The signature operator $$d_s^X =d-*d*{\kern1pt}{\kern1pt}{\kern1pt}{\kern1pt}\colon{\kern1pt}{\kern1pt}{\kern1pt}{\kern1pt}{\raise1pt\hbox{${\textstyle}\bigwedge$}}_c^+ {\longrightarrow}{\raise1pt\hbox{${\textstyle}\bigwedge$}}_c^-$$ is elliptic and the virtual dimension of its index equals the signature of $X$, ${\mbox{${\rm sign}$}}(X)$. If $W$ is a complex vector bundle on $X$ endowed with a connection, we can [*twist*]{} the signature operator to forms with values in $W$ $$d_s^X{\otimes}W \colon {\raise1pt\hbox{${\textstyle}\bigwedge$}}_c^+(W) {\longrightarrow}{\raise1pt\hbox{${\textstyle}\bigwedge$}}_c^-(W).$$ This operator is also elliptic and the virtual dimension of its index is denoted by ${\mbox{${\rm sign}$}}(X,W)$. Let $T=TX\otimes{\mbox{${\mathbb}C$}}$ denote the complexified tangent bundle of $X$ and let $R_i$ be the sequence of bundles defined by the formal series $$R(q,T)=\sum_{i=0}^{\infty} {\kern1pt}R_i {\kern1pt}{\kern1pt}q^i= \bigotimes_{i=1}^{\infty}{\raise1pt\hbox{${\textstyle}\bigwedge$}}_{q^i}T {\otimes}\bigotimes_{j=1}^{\infty}{S}_{q^j}T,$$ where ${S}_tT=\sum_{k=0}^{\infty}{\kern1pt}{S}^kT{\kern1pt}{\kern1pt}t^k$, ${\raise1pt\hbox{${\textstyle}\bigwedge$}}_tT=\sum_{k=0}^{\infty} {\kern1pt}{\raise1pt\hbox{${\textstyle}\bigwedge$}}^kT{\kern1pt}{\kern1pt}t^k$, and ${S}^kT$, ${\raise1pt\hbox{${\textstyle}\bigwedge$}}^kT$ denote the $k$-th symmetric and exterior tensor powers of $T$, respectively. The [*elliptic genus*]{} of $X$ is defined as $$\Phi(X) = {\mbox{{\rm}ind}}(d_s^X\otimes R(q,T)) =\sum_{i=0}^{\infty} {\mbox{${\rm sign}$}}(X,R_i) \cdot q^i .\label{definition-EllipticGenus}$$ Note that the first few terms of the sequence $R(q,T)$ are $R_0 = 1$, $R_1 = 2 T$, $R_2 = 2 (T^{\otimes 2} + T)$. In particular, the constant term of $\Phi(X)$ is ${\mbox{${\rm sign}$}}(X)$. If we assume that $G$ is a group acting on $M$ and commuting with the elliptic operator, then for $g\in G$ the equivariant index of $D$ can be defined as $$\mbox{index}(D)_{G}(g) = \mbox{trace}(g,\mbox{Ker}D)-\mbox{trace}(g,\mbox{Coker}D).$$ In an analogous way to the definition of the elliptic genus, now we define the equivariant elliptic genus with respect to the $S^1$ action by $$\Phi(X)_{S^1}({\lambda}) =\sum_{i=0}^{\infty} {\mbox{${\rm sign}$}}(X,R_i)_{S^1}(\lambda){\kern1pt}\cdot q^i,\label{equivariant-elliptic-genus}$$ where $\lambda\in S^1$. \[main\] Let $X$ be an $2n$-dimensional, oriented, closed, smooth manifold admitting a smooth $2$-balanced $S^1$-action. Then $$\Phi(X) = \Phi(X)_{S^1}(\lambda) \label{rig0}$$ for every $\lambda\in S^1$. [*Sketch of proof*]{}. The proof of Theorem \[main\] is along the lines of BT. The equivariant elliptic genus $\Phi(X)_{S^1}(\lambda)$ turns out to be a meromorphic function on $T_{q^2}={\mbox{${\mathbb}C$}}^*/q^2$ (the non-zero complex numbers modulo the multiplicative group generated by $q^2\not =0$). Thus, the proof of the theorem reduces to proving that $\Phi(X)_{S^1}(\lambda)$ has no poles at all on $T_{q^2}$, thus implying that $\Phi(X)_{S^1}(\lambda)$ is constant in $\lambda$. This follows from applying the Atiyah-Segal equivariant index theorem and localizing to the $S^1$-fixed point set and other auxiliary submanifolds. More precisely, one can define the translate $t_a\Phi(M)_{S^1}(\lambda)$ of $\Phi(M)_{S^1}(\lambda)$ by $a\in \mathbb{C}^*$, to be given by the map at the character level ${\lambda}\mapsto a{\lambda}$. In order to prove the rigidity theorem of $\Phi(M)$, we shall show that none of the translates $t_a\Phi(M)$, $a\in T_{q^2}$, by points of finite order on $T_{q^2}$, has a pole on the circle $|{\lambda}|=1$. The translates $t_a\Phi(M)$ can be expressed as [*twists*]{} of the elliptic genus on some auxiliary manifolds. The auxiliary submanifolds are the fixed point sets $X_k$ of the subgroups ${\mbox{${\mathbb}Z$}}_k\subset S^1$, $k\in \mathbb{Z}$. In doing so, the corresponding expressions have no poles at 1, and thus $\Phi(X)_{S^1}(\lambda)$ has no poles at points of finite order in $T_{q^2}$. This argument is valid as long as: - the submanifolds $X_k$ containing $S^1$-fixed points are orientable; - it is possible to choose an orientation of $X_k$ compatible with $X$ and all the components $P$ contained in $X_k$. \(i) is proved in [@HH Lemma 1]. (ii) follows as in [@BT Lemma 9.3] but using the fact that the action is $2$-balanced. \[A=0\] Let $X$ be a even-dimensional, oriented, closed, connected, smooth manifold admitting a $2$-balanced $S^1$ action. If the $S^1$ action is non-trivial then $${\mbox{$\widehat A$}}(X) = 0.$$ The proof follows in the same way as in [@HiSl Theorem, Section 1.5]. [$\Box$]{} \[vanishing-N\] Let $G$ be a compact positive-dimensional Lie group. Let $N$ be a compact $G$-transverse submanifold of a connected, oriented, smooth $G$-manifold with finite $\pi_2(M)$ and $\pi_4(M)$. Then the -genus of $N$ vanishes $${\mbox{$\widehat A$}}(N) =0.$$ This follows from Lemma \[2-balanced\] and Corollary \[A=0\]. [$\Box$]{} [Note that $N$ is not necessarily a Spin manifold since $M$ is not required to be so. Thus, the vanishing of ${\mbox{$\widehat A$}}(N)$ is not a consequence of the Atiyah-Hirzebruch vanishing theorem. ]{} [Note that $M$ does not need to be compact for Corollary \[vanishing-N\] to hold. ]{} Vanishing of higher -genera =========================== In this section we provide the proof of Theorem \[theo1\]. The proof follows that of Theorem 1.8 in [@Browder], and uses the rigidity of the elliptic genus on manifolds admitting 2-balanced $S^1$ actions. We also prove Theorem \[theo2\], which can be thought of as a higher $\hat{A}$-genus theorem for $G$-manifolds with finite $\pi_2(M)$ and $\pi_4(M)$.\ [*Proof of Theorem \[theo1\]*]{}. We use the $G$-transversality approach of Browder and Quinn [@Browder-Quinn]. In it, given a manifold $X$ (not necessarily compact) endowed with an action of $G$, they establish a 1-1 correspondence between transverse bordism classes of compact framed $G$-submanifolds of $X$ of codimension $k$ with homotopy classes of maps from $X/G^*$ to the sphere ${S}^k$, $[X/G^*, S^k]$. Here $X/G^*$ is the 1-point compactification of $X/G$. Given a space $Y$, we denote by $\Sigma^t Y$ the $t$-fold reduced suspension of $Y$, which is homeomorphic to the smash product of $Y$ and $S^t$, $\Sigma^t Y = Y\wedge S^{t} $. To apply [@Browder Theorem 4.2], let $y\in H^l(M/G)$. Since rational stable cohomology and rational stable cohomotopy are isomorphic, we can find $t\in \mathbb{N}$ and a map $$\rho:\Sigma^t(M/G_{+})\longrightarrow S^{l+t}$$ such that $$\rho^*(g)=\Sigma^t y,$$ where $g$ generates $H^{l+t}(S^{l+t})$, and $M/G_+$ is the disjoint union of $M/G$ with a base point. Notice that $\Sigma^t(M/G_+)=(M/G\times \mathbb{R}^t)^*$, the one point compactification of $M/G\times \mathbb{R}^t$. One can consider $M\times \mathbb{R}^t$ as a $G$-manifold, by extending the action of $G$ to the $\mathbb{R}^t$ factor by a trivial action. By [@Browder Theorem 4.2] there is a compact transverse framed $G$-submanifold $i:N\hookrightarrow M\times \mathbb{R}^t$, such that $$p^*\rho^*(g)\cap [(M\times \mathbb{R}^t)^*] =i_*[N],$$ i.e. the Poincaré dual of $i_*[N]$ is $p^*\rho^*(g)$, which follows from the construction of the submanifold $N$ in the proof of Lemma 4.4 in [@Browder]. Note that $${\mbox{$\widehat A$}}(N) = i^*{\mbox{$\widehat A$}}(M),$$ where ${\mbox{$\widehat A$}}(M)\in H^{4*}(M;\mathbb{Q})$. Since $M\times \mathbb{R}^t$ has finite $\pi_2(M\times \mathbb{R}^t)$ and $\pi_4(M\times \mathbb{R}^t)$, and $N$ is a $G$-transverse submanifold of $M\times \mathbb{R}^t$, the $G$ action on $N$ is non-trivial and ${\mbox{$\widehat A$}}(N)[N]=0$. Hence, by Corollary \[vanishing-N\] $$\begin{aligned} 0&=& {\mbox{$\widehat A$}}(N)[N]\nonumber\\ &=& (i^*{\mbox{$\widehat A$}}(M))[M]\nonumber\\ &=& {\mbox{$\widehat A$}}(M)(i_*[N])\nonumber\\ &=& {\mbox{$\widehat A$}}(M)(p^*\rho^*\cap [M\times \mathbb{R}^t])\nonumber\\ &=& {\mbox{$\widehat A$}}(M)(p^*\Sigma^ty\cap[M\times \mathbb{R}^t])\nonumber\\ &=& {\mbox{$\widehat A$}}(M)(p^*y\cap [M])\nonumber\\ &=& ({\mbox{$\widehat A$}}(M)\cup p^*y)[M].\nonumber\end{aligned}$$ [$\Box$]{} Let $f: M{\longrightarrow}K(\pi_1(M),1)$ be a map, assume that $f_*: \pi_1(M) \to \pi_1(M)$ is onto, one can define $\pi'$ to be $\pi_1(M)/f_*i_*(\pi_1(G))$, where $i: G\rightarrow M$ is induced by the action of G on the base point of $M$. Notice that $i_*(\pi_1(G))$ is contained in the center of $\pi_1(M)$ [@Hatcher page 40]. Let $\alpha: \pi_1(M) \to \pi'$ be the projection. \[theo2\] Let $M$ be a closed, connected, smooth manifold with finite $\pi_2(M)$ and $\pi_4(M)$, and let $G$ be a compact, connected, positive-dimensional Lie group acting smoothly and effectively on $M$. Let $f:M{\longrightarrow}K(\pi_1(M),1)$, and $x\in H^*(K(\pi',1);\mathbb{Q})$. Then $(f^*\alpha^*(x)\cup {\mbox{$\widehat A$}})[M]=0$, where ${\mbox{$\widehat A$}}\in H^*(M;\mathbb{Q})$ is the polynomial in the Pontrjagin classes. [*Proof*]{}. By Theorem 1.1 in Browder, there is a map $\phi:H_*(M/S^1,\mathbb{Q})\longrightarrow H_*(K(\pi',1),\mathbb{Q})$ such that the following diagram commutes, $$\begin{array}{ccc} H_*(M,\mathbb{Q})&\longrightarrow \kern-15pt\raise8pt\hbox{\footnotesize{${f_*}$}} &H_*(K(\pi_1(M),1),\mathbb{Q})\\ \downarrow\hbox{\footnotesize{$p_*$}}&&\downarrow\hbox{\footnotesize{$\alpha_*$}}\\ H_*(M/S^1,\mathbb{Q})&\longrightarrow\kern-15pt\raise8pt\hbox{\footnotesize{${\phi}$}} & H_*(K(\pi',1),\mathbb{Q}),\\ \end{array}$$ so that $({\mbox{$\widehat A$}}\cup f^*\alpha^*(x))[M] = ({\mbox{$\widehat A$}}\cup p^*\phi^*(x) ) [M] = 0$, for every $x\in H^*(K(\pi',1),\mathbb{Q})$. [$\Box$]{} [10]{} [AH]{} Atiyah, M.F., Hirzebruch, F.: Spin manifolds and group actions. Essays in Topology and Related Subjects, Springer-Verlag, Berlin, pp. 18–28 (1970) [BT]{} Bott, R., Taubes, T.: On the rigidity theorems of Witten. J. AMS, [**2**]{}, No. 1, 137–186 (1989) [Bredon]{} Bredon, G. E.: Representations at fixed points of smooth actions of compact groups. Ann. of Math. [**89**]{}, 515–532 (1969) [Browder]{} Browder, W., Hsiang, W. C.: $G$-actions and the fundamental group. Invent. Math. [**65**]{} (1981/82), no. 3, 411–424 [Browder-Quinn]{} Browder, W., Quinn, F.: A surgery theory for $G$-manifolds and stratified sets. Manifolds -Tokyo 1973 (Proc. Internat. Conf., Tokyo, 1973), pp. 27–36. Univ. Tokyo Press, Tokyo, 1975. [HH]{} Herrera, H., Herrera, R.: -genus on non-spin manifolds with $S^1$ actions and the classification of positive quaternion-Kähler 12-manifolds. J. Differential Geom. [**61**]{}, no. 3, 341–364 (2002) [HiBJ]{} Hirzebruch, F., Berger, T., Jung, R.: Manifolds and Modular Forms. Aspects of Mathematics, VIEWEG, 1992. [HiSl]{} Hirzebruch, F., Slodowy, P.: Elliptic genera, involutions, and homogeneous spin manifolds. Geometriae Dedicata [**35**]{}, 309–343 (1990) [Hatcher]{} Hatcher, A.: Algebraic Topology. Cambridge University Press, 2002. [^1]: Department of Mathematical Sciences, Rutgers University, Camden, NJ 08102, USA.E-mail: haydeeh@camden.rutgers.edu [^2]: Centro de Investigación en Matemáticas, A. P. 402, Guanajuato, Gto., C.P. 36000, México. E-mail: rherrera@cimat.mx [^3]: Partially supported by CONACYT grants: J48320-F, J110.387/2006.

--- abstract: 'We report on the transport measurements of two-dimensional holes in GaAs field effect transistors with record low densities down to $7\times10^{8}$ cm$^{-2}$. Remarkably, such a dilute system (with Fermi wavelength approaching $1\,\mu$m) exhibits a non-activated conductivity that grows with temperature approximately as a power law at sufficiently low temperatures. We contrast it with the activated transport found in more disordered samples and discuss possible transport mechanisms in this strongly-interacting regime.' author: - Jian Huang - 'D. S. Novikov' - 'D. C. Tsui' - 'L. N. Pfeiffer' - 'K. W. West' title: 'Non-activated transport of strongly interacting two-dimensional holes in GaAs' --- The question of how a strong Coulomb interaction can qualitatively alter an electronic system is fundamentally important. It has generated great interest in studying the transport in two-dimensional (2D) electron systems.[@AFS] Since the interaction becomes effectively stronger with lower 2D electron density, samples with most dilute carriers are desirable to probe the interaction effects. As the density is lowered, strong enough disorder can localize the carriers so that the interaction effect is smeared by the insulating behavior. Therefore, a clean 2D environment is vital to uncover the underlying interaction phenomena. For a long time, the dilute 2D carriers have been known as insulators characterized by activated conductivity. Specifically, in an Anderson insulator,[@Anderson'58; @stl] the conductivity follows the Arrhenius temperature dependence $\sigma \sim e^{-E_g/k_{B}T}$, where $E_g$ is the mobility edge with respect to the Fermi level. The energy relaxation due to phonons in the impurity band results in a softer exponential dependence $\sigma\sim e^{-(T^*/T)^\nu}$, realized via the variable-range hopping (VRH) process.[@Mott-VRH; @ES] Here, the exponent $\nu=1/3$ for non-interacting electrons,[@Mott-VRH] while $\nu=1/2$ if the Coulomb gap opens up at the Fermi level.[@ES] Finally, strong Coulomb interactions are believed to crystallize the 2D system [@wc] which then can become pinned by arbitrarily small disorder. A relation $d\sigma/dT>0$, being a natural consequence of the activated transport, eventually became a colloquial criterion of distinguishing an insulator from a metal.[@AKS-review] The experimental results in the dilute carrier regime are known to be greatly influenced by the sample quality, which has much improved over time. The phonon-assisted hopping transport was observed in early experiments.[@AKS-review] As the sample quality improved, the later experiments performed on 2D electrons in cleaner Si-MOSFETs demonstrated that the temperature-dependence of the resistivity $\rho=\sigma^{-1}$ can be either metal-like ($d\rho/dT>0$), or insulator-like ($d\rho/dT<0$), depending on whether the carrier density $n$ is above or below a critical value $n_c$.[@mit] On the insulating side, where $n<n_{c}$, $\rho(T)$ grows exponentially with cooling.[@Mason] Similar results have since been observed in various low disorder 2D systems, and the resistivity on the insulating side has been consistently found to follow an activated pattern $\rho\sim e^{(T^*/T)^\nu}$, with $\nu$ varying between $1/3$ and $1$. In this work, we focus on the transport properties of clean 2D holes in the dilute carrier regime where the insulating behavior is anticipated. To achieve high quality and low density, we adopt the GaAs/AlGaAs heterojunction insulated-gate field-effect transistor (HIGFET) where the carriers are only capacitively induced by a metal gate.[@kane; @lilly; @noh] Because there is no intentional doping, the amount of disorder is likely to be less, and the nature of the disorder is different from that of the modulation-doped samples. Previous experiments on similar 2D hole[@noh; @noh1] HIGFET devices have demonstrated a non-activated transport. The temperature dependence $\sigma(T)$ of the conductivity becomes approximately linear, $\sigma \propto T$, [@noh1] when the density is lowered to a minimum value of $1.6\times10^{9}$cm$^{-2}$. However, it is unclear whether the linear $T$-dependence will persist for lower densities or it is a crossover to a different transport regime. We have measured several high quality $p$-channel HIGFET samples. The hole density $p$ in our devices can be continuously tuned to as low as $p=7\times10^{8}$ cm$^{-2}$, in which case the nominal Fermi wavelength $\lambda_F = (2\pi/p)^{1/2} \simeq 0.95\,\mu$m. Our main finding is that the conductivity $\sigma(T)$ of the cleanest samples decreases with cooling in a non-activated fashion for densities down to $7\times10^{8}$ cm$^{-2}$. The temperature dependence of the conductivity appears to be best approximated by a non-universal power-law $\sigma \propto T^{\alpha}$ with $1\lesssim\alpha\lesssim2$. The systematic analysis of this dependence will be published elsewhere.[@jian] At base temperature, the magnitude of $\sigma$ is much greater than that of a typical insulator with similar carrier density. Our results point at the presence of the delocalized states in the system with a record low carrier density. Thus our system is not an insulator even though $d\sigma/dT>0$. The device geometry is a standard 3mm$\times 0.8$mm Hall bar. The measurements were performed in a dilution refrigerator with a base temperature of 35 mK. At each value of the gate voltage, the mobility and density were determined through measuring the longitudinal resistivity $\rho$ and its quantum oscillations in the magnetic field. The temperature dependence of the resistivity was measured with an ac four-terminal setup at high carrier density, while both ac and dc setups were used for the low-density high-impedance cases. To ensure linear response, current drive as small as 1pA was used during the measurements at the lowest carrier density. Driving currents of different amplitudes were used for the low density cases and the measured resistivity did not change with varied current drive. The temperature dependence of the resistivity $\rho(T)$ for a number of hole densities from sample \#3 is shown in Fig. \[fig:rt\], with lowest temperature of $80$ mK. At first glance, the density dependence of the $\rho(T)$ curves is similar to that found around the metal-to-insulator transition, [@AKS-review] with $p_c=4\times10^{9}$cm$^{-2}$ being the critical density. For $p>p_c$, the system exhibits the apparent metallic behavior ($d\rho/dT>0$) at sufficiently low temperatures. The downward bending of $\rho(T)$ becomes weaker as $p$ approaches $p_c$, and disappears for lower $p$. The derivative $d\rho/dT$ at low $T$ then becomes negative, a conventional characteristic of an insulator, [@AKS-review] for the whole temperature range. At the transition, the resistivity is of the order of $h/e^{2}$. The value of $p_{c}$ is very close to that obtained in a similar device in Ref. . Fig. \[fig:ct\] shows the conductivities $\sigma(T)$ of the same sample (\#3) for the corresponding densities. For $1.8\times10^{9}$cm$^{-2}<p<3.8\times10^{9}$cm$^{-2}$, the conductivity increases approximately linearly with $T$ at high temperatures (above $\sim 200$mK). The linear regions are almost parallel for different densities, similar to that observed previously.[@noh; @noh1] This linear dependence occurs at temperatures above the nominal Fermi temperature and will be studied in detail elsewhere.[@jian] However, for lower densities, from $8\times10^{8}$cm$^{-2}$ to $1.8\times10^{9}$cm$^{-2}$, the conductivity deviates from the linear decrease at low temperatures. $\sigma(T)$ exhibits a slower change with $T$ as the density is reduced, with weaker $T$-dependence close to the base temperature. The conductivity values ($\sim 0.1e^{2}/h$) are considerably larger than those found in the more disordered sample which will be described later. ![\[fig:VRH\] Comparison of the conductivities $\sigma(T)$ with VRH transport models: (a), (c) Mott; (b), (d) Efros-Shklovskii. Panels (a) and (b) are results from sample \#3, and (c) and (d) are from sample \#4.](po-hop.eps){width="3.4in"} In Fig. \[fig:VRH\], we compare the measured conductivities with the VRH predictions according to Mott [@Mott-VRH] and to Efros and Shklovskii [@ES] for sample \#3 \[panels (a) and (b)\], and sample \#4 \[panels (c) and (d)\]. The hopping conductivity $\sigma\sim e^{-(T^*/T)^\nu}$, repeatedly observed in previous experiments on the insulators, is expected to occur at low temperatures. However, in both of our samples, the conductivity is approximately linear (in the semi-log scale) at high temperatures but nonlinear at low temperatures. It clearly deviates from the VRH law (dotted lines), for both $\nu=1/3$ \[panels (a) and (c)\] and $\nu=1/2$ \[panels (b) and (d)\]. The increasing deviation with cooling indicates that the temperature dependence is weaker than activated. The deviation is slightly larger for the E-S ($\nu=1/2$) case. The qualitative difference in the temperature dependence between our clean samples from more disordered samples (previously measured) is apparent in the log-log scale plot in Fig. \[fig:log\]. Here, the conductivity from a more disordered sample (\#6) is also included for comparison. For about the same densities ($7-8 \times 10^8\,$cm$^{-2}$), the dependence $\log\sigma$ versus $\log T$ for the two cleanest samples appears to be approximately linear below 150mK, indicating a power-law-like relationship $\sigma \propto T^{\alpha}$. The exponent $\alpha$, which corresponds to the slope in the plot, is $\alpha \simeq 2.2$ for $p=7 \times 10^8$cm$^{-2}$ and $\alpha \simeq 1.6$ for $p=8 \times 10^8$cm$^{-2}$. Both numbers differ from the much lower values previously observed in carrier densities around $1.6\times 10^9$cm$^{-2}$. [@lilly; @noh1] The trend, larger $\alpha$ for lower density, is consistent with the results in Ref. . On the other hand, the conductivity for the more disordered sample (\#6) exhibits a clear downward diving, consistent with the activated behavior. Note that the low-$T$ conductivity in the more disordered sample is at least three orders of magnitude smaller than that in the clean ones for the same carrier density. The insulating character of our more disordered sample is consistent with previously observed Anderson insulators in lower quality 2D systems. However, our clean HIGFET results show that, with less disorder, the transport becomes non-activated, indicating the presence of delocalized states. What is responsible for the apparent delocalization? The Anderson localization, being an interference effect, can occur only when the system is sufficiently phase-coherent. At finite temperature, strong electron-electron interaction can dramatically reduce the coherence length $l_\phi$. As the carrier temperatures in our case are of the order of the Fermi temperature $T_F$, there is no suppression of the interaction between the quasiparticles associated with the filled Fermi sea. In this situation it is plausible to assume that $l_\phi \sim \lambda_F$.[@NZA] On the other hand, the 2D localization length $\xi$ is exponentially sensitive to the amount of disorder.[@loc-length] Thus it seems likely for the phase coherence to be broken on the scale $l_\phi < \xi$ in our clean samples, while the opposite is true for the more disordered one. The transport mechanism that leads to the observed temperature dependence $\sigma(T)$ remains unknown. Moreover, even the nature of the ground state of such a system is unsettled.[@spivak2; @spivak; @spivak1; @Das] Below we show that the electron-electron interactions are extremely strong at short distances, and decay relatively fast at larger distances due to the screening by the metallic gate. This nature of interaction between the delocalized carriers suggests that the holes form a strongly-correlated liquid. We now consider the electron-electron interaction in more details. In HIGFETs, the metallic gate at distance $d$ from the 2D hole layer screens the $1/r$ interaction down to $1/r^3$ when $r\gtrsim 2d$. In our case, $d=600\,$nm for sample \#3 and $d=250\,$nm for sample \#4. The short-distance $1/r$-interaction is indeed very strong. If treated classically as a one-component plasma, the interaction parameter $\Gamma = E_C/k_B T \sim 100$, corresponding to an enormous Coulomb energy $E_C=e^2/\epsilon a\sim 10$K, with $\epsilon = 13$. Since the temperature in our system is of the order of the Fermi energy, $E_F=\hbar^2/ma^2 \sim 100\,$mK, quantum effects may also be important. A standard estimate of the strength of interaction is the quantum-mechanical parameter $r_s=a/a_B$, where $a_B=\hbar^2\epsilon/me^2$ is the Bohr radius. It requires the knowledge of the band mass $m$ that has never been measured in such a dilute regime. Higher density cyclotron resonance measurements give $m\simeq 0.2-0.4 \,m_e$, [@pan] whereas low-density theoretical estimates (based on the Luttinger parameters) [@Winkler] give $m\simeq 0.1 m_e$. The $r_s$ value for $p=1\times 10^9\,$cm$^{-2}$ is in the range of $25 - 100$ for the mass range of $m=0.1 m_e - 0.4 m_e$. The long-distance dipolar interaction is relatively weak for such low densities since the $1/r^3$ potential is short-ranged in two dimensions. Therefore, the liquid is favored over the Wigner crystal (WC),[@spivak1] as the quantum fluctuations ($\sim 1/r^2$) overcome the $1/r^3$ interaction. Meanwhile, even for a classical system, the 2D WC melting temperature $T_m\simeq E_C/130k_{B}$ is already low [@Morf]: $T_m=56\,$mK for $p=1\times10^9\,$cm$^{-2}$. The screening further reduces[@Tm-screened] $T_m$ to make the WC even harder to access. The absence of the pinned WC in our samples is corroborated by the non-activated transport in the linear response regime and the absence of singularity in $\sigma(T)$. In summary, we have observed a non-insulating behavior in the putatively insulating regime for the hole densities down to $7\times10^8\,$cm$^{-2}$. Our results suggest that the 2D holes form a strongly-correlated liquid whose properties require further investigation. This work has benefited from valuable discussions with I.L. Aleiner, B.L. Altshuler, P.W. Anderson, R.N. Bhatt, and M.I. Dykman. We also thank Stephen Chou for the use of the fabrication facilities. The work at Princeton University is supported by US DOE grant DEFG02-98ER45683, NSF grant DMR-0352533, and NSF MRSEC grant DMR-0213706. [99]{} T. Ando, A.B. Fowler, and F. Stern, Rev. Mod. Phys. [**54**]{}, 437 (1982). P.W. Anderson, Phys. Rev. [**109**]{}, 1492 (1958). E. Abrahams, P.W. Anderson, D.C. Licciardello, and T.V. Ramakrishnan, Phys. Rev. Lett. [**42**]{}, 673 (1979) N.F. Mott, J. Non-Cryst. Solids [**1**]{}, 1 (1968). B.I. Shklovskii and A.L. Efros, [*Electronic Properties of Doped Semiconductors*]{}, Springer-Verlag, Berlin (1984). B. Tanatar and D. M. Ceperley, Phys. Rev. B [**39**]{}, 5005 (1989). E. Abrahams, S.V. Kravchenko, and M.P. Sarachik, Rev. Mod. Phys. [**73**]{}, 251 (2001). Kravchenko, S. V., G. V. Kravchenko, J. E. Furneaux, V. M. Pudalov, and M. D’Iorio, Phys. Rev. B [**50**]{}, 8039 (1994). W. Mason, S. V. Kravchenko, G. E. Bowker, and J. E. Furneaux, 1995, Phys. Rev. B [**52**]{}, 7857 B. E. Kane, L. N. Pfeiffer, and K. W. West, Appl. Phys. Lett. [**67**]{}, 1262 (1995). M. P. Lilly, J. L. Reno, J. A. Simmons, I. B. Spielman, J. P. Eisenstein, L. N. Pfeiffer, K. W. West, E. H. Hwang, and S. Das Sarma, Phys. Rev. Lett. [**90**]{}, 056806 (2003). Hwayong Noh, M. P. Lilly, D. C. Tsui, J. A. Simmons, E.H.Hwang, S.Das Sarma, L. N. Pfeiffer, and K. W. West, Phys. Rev. B [**68**]{}, 165308 (R) (2003). Hwayong Noh, M. P. Lilly, D. C. Tsui, J. A. Simmons, L. N. Pfeiffer, and K. W. West, Phys. Rev. B [**68**]{}, 241308 (R) (2003). Jian Huang [*et al.*]{} cond-mat/0610320 (2006). B.N. Narozhny, G. Zala, and I.L. Aleiner, Phys. Rev. B [**65**]{}, R180202 (2002) A. MacKinnon and B. Kramer, Phys. Rev. Lett. [**47**]{}, 1546 (1981); D. Vollhardt and P. Wölfle, Phys. Rev. Lett. [**48**]{}, 699 (1982). S. Das Sarma and E. H. Hwang, Phys Rev. B [**68**]{}, 195315 (2003). B. Spivak, S. Kivelson, cond-mat/0510422 (2005). B. Spivak and S. A. Kivelson Phys. Rev. B [**70**]{}, 155114 (2004). B. Spivak, Phys. Rev. B [**67**]{}, 125205 (2003). Pan W, Lai K, Bayrakci SP, Ong NP, Tsui DC, Pfeiffer LN, West KW, Appl. Phys. Lett. [**83**]{}, 3519 (2003). R. Winkler, [*Spin-orbit coupling effects in two-dimensional electron and hole systems*]{} (Springer, New York, 2003). D.S. Fisher, B.I. Halperin, and R. Morf, Phys. Rev. B [**20**]{}, 4692 (1979). P. Hartmann, G.J. Kalman, Z. Donkó, and K. Kutasi, Phys. Rev. E [**72**]{}, 026409 (2005).

--- abstract: 'We consider elections where the voters come one at a time, in a streaming fashion, and devise space-efficient algorithms which identify an approximate winning committee with respect to common multiwinner proportional representation voting rules; specifically, we consider the Approval-based and the Borda-based variants of both the Chamberlin– Courant rule and the Monroe rule. We complement our algorithms with lower bounds. Somewhat surprisingly, our results imply that, using space which does not depend on the number of voters it is possible to efficiently identify an approximate representative committee of fixed size over vote streams with huge number of voters.' author: - | Palash Dey\ \ \ Nimrod Talmon\ \ \ Otniel van Handel\ \ \ bibliography: - 'bib.bib' title: '[Proportional Representation in Vote Streams]{}' --- &lt;ccs2012&gt; &lt;concept&gt; &lt;concept\_id&gt;10003752.10003809.10010055&lt;/concept\_id&gt; &lt;concept\_desc&gt;Theory of computation Streaming, sublinear and near linear time algorithms&lt;/concept\_desc&gt; &lt;concept\_significance&gt;500&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10010147.10010178.10010219.10010220&lt;/concept\_id&gt; &lt;concept\_desc&gt;Computing methodologies Multi-agent systems&lt;/concept\_desc&gt; &lt;concept\_significance&gt;500&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;/ccs2012&gt; Introduction ============ The voting rule suggested by Chamberlin–Courant [@cha-cou:j:cc] and the voting rule suggested by Monroe [@mon:j:monroe], are multiwinner voting rules concentrated on proportional representation. Such proportional representation multiwinner rules aim at selecting a committee of fixed size which represents the society best. Informally, most voters shall be somewhat satisfied by the committees selected by such proportional representation rules, which, roughly speaking, try to best represent the spectrum of different views of the society. This stands in contrast, for example, to $k$-best multiwinner voting rules such as $k$-Borda. Proportional representation multiwinner voting rules have several good axiomatic properties [@elkind2014properties]. Winner determination for these rules, however, is NP-hard [@pro-ros-zoh:j:proportional-representation], though it is possible to compute the winner when some parameters are small [@bet-sli-uhl:j:mon-cc]; that is, winner determination for these rules is fixed-parameter tractable with respect to either the number of voters or the number of candidates. Further, efficient approximation algorithms are known [@sko-fal-sli:c:multiwinner] for these rules as well as heuristic algorithms based on clustering [@clusteringpaper]. Proportional representation multiwinner voting rules have several other applications, besides their original, political application. Specifically, these rules are used for resource allocation [@mon:j:monroe; @sko-fal-sli:c:multiwinner], facility location [@bet-sli-uhl:j:mon-cc], and recommender systems [@bou-lu:c:chamberlin-courant; @skowron2015finding]. In such situations, it is indeed desirable to select a set of $k$ “representative” elements out of a larger set. While the number of voters in some elections is modest, there are situations where the number of voters is huge, making it impossible to store the whole election in order to operate upon it (specifically, to identify a winning committee). Consider, e.g., the preferences of users of an online shopping website: there are lots of potential buyers (corresponding to the voters), each with her own preferences over the items being sold on the website (corresponding to the candidates). The owners of the shopping website might wish to identify a set of, say, $k$ items to advertise on their landing page, with the intent of maximizing the number of users which would be interested in at least one of those displayed items. More generally, as certain tasks which are concerned with the creation of various kinds of product portfolios can be modeled as equivalents of solving winner determination for proportional representation, it is of interest to devise efficient algorithms for such situations which naturally correspond to elections with huge number of voters. Thus, in this paper we are interested in designing algorithms which identify good representative committees, but without being able to store the whole electorate in order to process it; concretely, we are aiming at algorithms whose space complexity does not depend on the number of voters, since this number might be huge. To study space-efficient algorithms for such situations, we consider streaming algorithms which solve the winner determination problem for proportional representation multiwinner voting rules. Specifically, while we consider the set of alternatives as being fixed, we assume that the voters are arriving (that is, voting) one at a time, in what we refer to as a *vote stream*. Concretely, we assume that each voter is arriving only once (such that it is possible to process each voter only once), and we are interested in space-efficient streaming algorithms for finding a winning committee of fixed size $k$ in such vote streams. As it is customary in studying streaming algorithms, we allow our algorithms to be randomized and to find approximate solutions. That is, in order to have algorithms which use only small amounts of space, we will be satisfied with algorithms which find an approximate winning committee; specifically, we will be satisfied with finding a committee whose score under the given voting rule is close to being the optimum score possible for a committee with respect to the given election. Further, we will be satisfied with randomized algorithms, which might not always find such approximate winning committees, but nevertheless are guaranteed to find such approximate winning committees with arbitrarily high probability. A more formal description of our setting is given in Section \[section:preliminaries\]. Our results, which are summarized in Table \[table:results\], imply that it is possible to process huge amount of preferences data (that is, huge amount of voters), using only small space, and still, with high probability, find an almost-optimal winning committee. Since, as briefly mentioned above, the voting rules we consider in this paper have applications not only in political settings, but also in commercial and business settings, our results naturally have implications to those scenarios as well. We further discuss the applicability of our results in Section \[section:outlook\]. Related Work ------------ The two most-related papers to our paper are two papers by Bhattacharyya and Dey [@bhattacharyya2015fishing; @dey2015sample]. The first paper [@bhattacharyya2015fishing] provides an analysis of the space complexity of streaming algorithms for some single-winner voting rules. The second paper [@dey2015sample] provides an analysis concerning the number of samples which are sufficient in order to approximately compute the winner under various single-winner voting rules, and is of relevance to our paper since our algorithms are based on sampling. Another related paper is that of Filtser and Talmon [@distributedmonitoring] which provides efficient protocols for winner determination in distributed streams. We stress that, while the above-mentioned papers deal with single-winner voting rules, our paper deals with multiwinner voting rules which select a committee of fixed size. Another line of work worth mentioning is concerned with developing streaming algorithms for the <span style="font-variant:small-caps;">Max Cover</span> problem. In the <span style="font-variant:small-caps;">Max Cover</span> problem, we are given a collection of sets over some universe and a budget $k$, and the task is to find $k$ sets which cover the largest number of elements. Approval-CC (see Section \[section:preliminaries\]) is equivalent to <span style="font-variant:small-caps;">Max Cover</span> (to see this, interchange voters by elements and candidates by sets; see also, e.g., [@skowron2015fully]). Thus, the very recent paper by McGregor and Vu [@mcgregor2016better] is of relevance to us; specifically, they give an upper bound [@mcgregor2016better Theorem 10] which has some similarities with our Theorem \[theorem:Approval-CC\_UB\], and they give a lower bound [@mcgregor2016better Theorem 20]. However, their model of a stream is different than ours, since the items in their streams are the sets (corresponding to the candidates), while for us the items are the voters (corresponding to the elements). In the context of social choice, there are some further interesting papers to mention. Conitzer and Sandholm [@conitzer2005communication] study communication complexity of various voting rules; they do not consider approximations and therefore the communication complexity of their protocols is generally quite high. Along similar lines, Chevaleyre et al. [@chevaleyre2011compilation] design communication protocols for situations where the set of candidates might change over time. Chevaleyre et al. [@chevaleyre2009compiling] study compilation complexity of various voting rules; roughly speaking, they divide the electorate into two parts, and are concerned with the amount of information which one part shall transmit to the other in order to correctly identify a winner. Xia and Conitzer [@xia2010compilation] extend upon this previously-mentioned paper by considering some further variants as well as some other voting rules not previously studied. Finally, we mention the paper by Conitzer and Sandholm [@con-san:c:strategy-proofness] which is concerned with vote elicitation. Preliminaries {#section:preliminaries} ============= We provide preliminaries regarding elections and proportional representation voting rules, streaming algorithms and vote streams, and mention some useful results from probability theory. We denote the set $\{1, \ldots, n\}$ by $[n]$. Proportional Representation --------------------------- An *election* $E = (C, V)$ consists of a set of *candidates* $C = \{c_1, \ldots , c_m\}$ and a collection of *voters* $V = (v_1, \ldots , v_n)$, where each voter is associated with her *vote*. (For ease of presentation, we refer to the voters as females while the candidates are males.) In this paper we consider two kinds of elections: in *Approval-based* elections, the vote of voter $v_i \in V$ is a subset of $C$, corresponding to the candidates which this voter *approves*; in *Borda-based* elections, the vote of voter $v_i \in V$ is a total order ${\ensuremath{\succ}}_{v_i}$ over $C$. For Borda-based elections, we write ${\mathrm{pos}}_v(c)$ to denote the *position* of candidate $c$ in $v$’s preference order (e.g., if $v$ ranks $c$ on the top position, then ${\mathrm{pos}}_v(c) = 1$). Given an election $E = (C, V)$ and an integer $k$, $k \leq |C|$, a *committee* $S \subseteq C$ consists of $k$ candidates from $C$. A *multiwinner voting rule* ${\mathcal{R}}$ is a function that returns a set ${\mathcal{R}}(E, k)$ of *winning committees* of size $k$ each, and we say that the committees in ${\mathcal{R}}(E, k)$ tie as winners of the election. To formally define the specific voting rules which we consider in this paper, namely Chamberlin–Courant and Monroe, we first discuss assignment functions and satisfaction functions. **Assignment functions.**Given an election $E = (C, V)$ and a committee $S \subseteq C$ of size $k$, a *CC-assignment* function is a function $\Phi \colon V \rightarrow S$. We say that $\Phi(v)$ is the *representative* of voter $v \in V$ and that $v$ is represented by $\Phi(v)$. An *M-assignment* function is a CC-assignment function where $\lfloor \frac{n}{k} \rfloor \leq |\Phi^{-1}(c)| \leq \lceil \frac{n}{k} \rceil$ holds for each $c \in S$. That is, in an M-assignment, each committee member represents roughly (i.e., up to rounding) the same number of voters. **Satisfaction functions.**Intuitively, a *satisfaction function* $\gamma : V \times C \to \mathbb{N}$ is a function measuring the satisfaction of a voter $v$ when she is represented by a certain candidate $c$. For Approval-based elections, we use the satisfaction function $\gamma \equiv \alpha$ where $\alpha(v, c) = 1$ iff $c$ is approved by $v$, and $0$ otherwise (that is, $1$ if $c$ is contained in $v's$ vote; informally, a voter is satisfied only by her approved candidates). For Borda-based elections, we use the satisfaction function $\gamma \equiv \beta$ where $\beta(v, c) = m - {\mathrm{pos}}_v(c)$. **Chamberlin–Courant and Monroe.**Given an election $E = (C, V)$, a size-$k$ committee $S$, and a CC-assignment function $\Phi$, we define the *total satisfaction* of the voters in $V$ from the committee $S$ and the CC-assignment $\Phi$ to be: $$\sum_{v \in V}\gamma(\Phi(v)),$$ where, for Approval-based elections, $\gamma$ equals the $\alpha$ satisfaction function described above, while for Borda-based elections, $\gamma$ equals the $\beta$ satisfaction function described above. For the Chamberlin–Courant rule, the *total satisfaction* of the voters in $V$ from a committee $S$ is defined as the maximum total satisfaction of the voters $V$ from the committee $S$ over all possible CC-assignment functions. The Chamberlin–Courant rule outputs all size-$k$ committees $W$ with the highest total satisfaction. The Monroe rule is defined similarly, but where we consider only M-assignment functions; that is, the total satisfaction of the voters in $V$ from a committee $S$ is defined as the maximum total satisfaction of the voters $V$ from the committee $S$ over all possible M-assignment functions. We denote by *Approval-CC* (*Borda-CC*) the Chamberlin–Courant rule for Approval-based (Borda-based) elections, and by *Approval-M* (*Borda-M*) the Monroe rule for Approval-based (Borda-based) elections. Voting rule Space complexity ------------- ----------------------------------- Approval-CC $O(\epsilon^{-2} k m \log m)$ Borda-CC $O(\epsilon^{-2} k^3 m^3 \log m)$ Approval-M $O(\epsilon^{-2} k^3 m \log m)$ Borda-M $O(\epsilon^{-2}k^3m^5 \log m)$ : Summary of our upper bounds. We list our upper bounds for randomized streaming algorithms which identify $\epsilon$-approximate winning committees under several proportional representation voting rules. $k$ denotes the size of the committee while $m$ denotes the number of candidates which participate in the election.[]{data-label="table:results"} Vote Streams ------------ We assume that the set of candidates $C$ is known, and that the voters $v_1, \ldots, v_n$ arrive (that is, vote) one at a time. More formally we might say that at time $t \in [n]$, voter $v_t$ arrives; importantly, each voter arrives only once. We are interested in randomized algorithms which operate on such vote streams, and find approximate solutions. The following definition is crucial for our notion of approximation. A committee of size $k$ is *$\epsilon$-winning* if it is either a winning committee, or it can become a winning committee by changing at most $\epsilon n$ votes. Specifically, we require that the committees computed by our streaming algorithm shall be, with high probability, $\epsilon$-winning. Such a streaming algorithm, which identify, with high probability, an $\epsilon$-winning committee, is said to be an *$\epsilon$-approximate streaming algorithm*. A streaming algorithm is an *$\epsilon$-approximate* streaming algorithm if it returns, with high probablty, an $\epsilon$-winning committee. Throughout the paper, when we say “with high probability” we mean with probability $1 - O(1/n)$. Such a success probability should be sufficient; as usual in streaming algorithms, can be further tweaked by repetitions. Assuming that the number $n$ of voters is huge, our goal is to devise streaming algorithms whose space complexity do not depend on the number $n$ of voters. Our algorithms are based on sampling voters; by a *subset* of an election we mean a subset of the voters. Let us explain how exactly we sample voters. Let $n$ be the length of the stream (i.e., the total number of voters), and suppose that we want to sample $z$ votes from the stream. Then, we pick each vote with probability $z / (n \delta)$ for some constant $0 \leq \delta \leq 1$. By Markov’s inequality, with probability at least $1 - \delta$ is holds that the sample size is at least $z$ (and not much larger). Hence, every vote belongs to our sample with probability $z / (n \delta)$ independently of other items. Useful Results from Probability Theory -------------------------------------- Since our algorithms are randomized, specifically based on sampling a small number of voters, we make extensive use of the following variant of Hoeffding’s inequality, which upper bounds the probability that the sum of a given set of random variables deviates from its expectation. \[theorem:Hoeffding\] Let $X_1,..., X_t$ be independent random variables such that $0 \leq X_i \leq m$ for each $i \in [t]$. Let $X$ be a random variable such that $X = \sum_{i \in [t]} X_i$. Then, the following two statements hold. $$(1) \quad\quad\quad \Pr [X -\mathbb{E}[X] < \epsilon] \le \exp\left(-\frac{2\epsilon^2 t}{m^2}\right)$$ $$(2) \quad\quad\quad \Pr [\mathbb{E}[X] - X < \epsilon] \le \exp\left(-\frac{2\epsilon^2 t}{m^2}\right)$$ For the special case when $m = 1$, Hoeffding’s inequality simplifies as follows. $$(1) \quad\quad\quad \Pr [\mathbb{E}[X] - X < \epsilon] \le \exp\left(-{2\epsilon^2 t}\right)$$ $$(2) \quad\quad\quad \Pr [X - \mathbb{E}[X] < \epsilon] \le \exp\left(-{2\epsilon^2 t}\right)$$ Results {#section:results} ======= Our main results are summarized in Table \[table:results\]. In Section \[section:upperbounds\] we describe our upper bounds while in Section \[section:lowerbounds\] we describe our lower bounds. Upper Bounds {#section:upperbounds} ------------ We first consider the Approval-CC voting rule, which is arguably the simplest voting rule we consider in this paper. The following algorithm is based on sampling a small number of voters. The proof shows that, with high probability, a winning committee for the election corresponding to the sample has fairly high score in the whole election; specifically, it constitutes an $\epsilon$-approximate winning committee of the whole election. \[theorem:one\]\[theorem:Approval-CC\_UB\] There is an $\epsilon$-approximate streaming algorithm for Approval-CC which uses $O(\epsilon^{-2} k m \log m)$ space. The algorithm operates as follows. We select a sample of $t = 6\epsilon^{-2} k \log m$ voters, uniformly at random. Then, we find a winning committee of the sampled voters (with respect to Approval-CC) and return it as a winning committee for the whole election. We show that a winning committee of the sampled voters is, with high probability, an $\epsilon$-winning committee for the whole election. Notice that in order to store the votes of $t$ voters, our algorithm uses $mt$ space, as claimed. Next we prove that, with high probability, our algorithm returns an $\epsilon$-winning committee. Let $E = (C, V)$ denote the whole election and let $E_R = (C, V_R)$ denote the sampled election, where $V_R$ denotes the set of $t$ sampled voters. Let $S$ be a winning committee in the whole election. Let ${\mathrm{score}}_E(S)$ (${\mathrm{score}}_{E_R}(S)$) denote the score of $S$ in the whole election (in the sampled election, respectively). Let us first consider the case where ${\mathrm{score}}_E(S) < \epsilon n$, that is, where there are less than $\epsilon n$ voters being satisfied by $S$. In this case any committee is $\epsilon$-winning, thus our algorithm is always correct. Therefore, from now on we assume that there are at least $\epsilon n$ voters satisfied by $S$. The next claim concentrates on the winning committee $S$, which, since it is winning in $E$, has high score in $E$; the claim shows that, with high probability, $S$ also has high score in $E_R$. The factor $\frac{n}{t}$ is a normalization factor. \[claim:theorem1claim1\] $\frac{n}{t}\cdot {\mathrm{score}}_{E_R}(S) \geq {\mathrm{score}}_{E}(S) - \frac{\epsilon}{2}n$ holds with probability at least $1-m^{-k}$. For $i \in [t]$, let $X_i$ be an indicator random variable such that $X_i = 1$ if the $i$th sampled voter is satisfied by $S$, and $X_i = 0$ otherwise. Let $X = \sum_{i \in [t]} X_i$. Since ${\mathrm{score}}_{E}(S)$ equals the number of voters in the whole election which are satisfied by $S$, it holds that $$\mathbb{P}[X_i = 1] = {\mathrm{score}}_{E}(S) / n$$ for each $i \in [t]$. Then, from linearity of expectation, we conclude that $$\mathbb{E}[X] = \frac{t}{n} \cdot {\mathrm{score}}_{E}(S).$$ This means that, in expectation, the score of $S$ in $E_R$ is as claimed; we use Hoeffding’s inequality (see Theorem \[theorem:Hoeffding\]) to show concentration, as follows. $$\begin{aligned} \mathbb{P}\left[\frac{n}{t}X < \frac{n}{t}\mathbb{E}[X] - \frac{\epsilon}{2}n\right] &= \mathbb{P}\left[X < \mathbb{E}[X] - \frac{\epsilon}{2}t \right] \\ &\leq e^{-2(\frac{\epsilon}{2})^2 6 \epsilon^{-2} k \log m} \\ &\leq m^{-k}.\end{aligned}$$ (proof of claim \[claim:theorem1claim1\]) Claim \[claim:theorem1claim1\] shows that, with high probability, a committee with high score in the whole election also gets a relatively high score in the sampled election. Next we show that, with high probability, a committee with low score in the whole election also gets a low score in the sampled election. \[claim:theorem1claim2\] Let $S'$ be a committee for which it holds that ${\mathrm{score}}_{E}(S') \leq (1 - \epsilon) \cdot {\mathrm{score}}_{E}(S)$. Then, with probability at least $1 - m^{-2k}$, it holds that $\frac{n}{t}\cdot {\mathrm{score}}_{E_R}(S') \leq (1 - \epsilon) \cdot$ ${\mathrm{score}}_{E}(S) + \frac{\epsilon}{2}n$. Let $S'$ be such that ${\mathrm{score}}_{E}(S') \leq (1 - \epsilon) \cdot {\mathrm{score}}_{E}(S)$. For $i \in [t]$, let $X_i$ be an indicator random variable such that $X_i = 1$ if the $i$th sampled voter is satisfied by $S'$, and $X_i = 0$ otherwise. Let $X = \sum_{i \in [t]} X_i$. Since ${\mathrm{score}}_{E}(S')$ equals the number of voters in the whole election which are satisfied by $S'$, it holds that $$\mathbb{P}[X_i = 1] = \frac{{\mathrm{score}}_{E}(S')}{n}$$ for each $i \in [t]$. Then, from linearity of expectation, we conclude that $$\mathbb{E}[X] = \frac{t}{n} \cdot {\mathrm{score}}_{E}(S') \leq \frac{t}{n} \cdot (1 - \epsilon) \cdot {\mathrm{score}}_E(S).$$ This means that, in expectation, the score of $S$ in $E_R$ is as claimed. Since the $X_i$’s are independent and all of them are bounded, we use Hoeffding’s inequality (see Theorem \[theorem:Hoeffding\]) to show concentration, as follows. $$\begin{aligned} \mathbb{P}\left[\frac{n}{t}\cdot X > \frac{n}{t}\cdot \mathbb{E}[X] + \frac{\epsilon}{2}n\right] &= \mathbb{P}\left[X - \mathbb{E}[X] > \frac{\epsilon }{2}t\right] \\ &< e^{-2 (\epsilon/2)^2 6 \epsilon^{-2} k \log m} \\ &\leq m^{-2k}.\end{aligned}$$ (proof of claim \[claim:theorem1claim2\]) Since there are at most ${m \choose k}\le m^k$ committees, and therefore at most $m^k$ committees $S'$ for which ${\mathrm{score}}_{E}(S') \leq (1 - \epsilon) \cdot {\mathrm{score}}_{E}(S)$ holds (and these are exactly the committees which are not $\epsilon$-winning), we can apply union bound on the result proved in Claim \[claim:theorem1claim2\], to get that with high probability, the score of $S$ in $E_R$ is strictly higher than the score of any committee $S'$ which is not $\epsilon$-winning. Thus, our algorithm returns, with high probability, an $\epsilon$-winning committee. It turns out that it is possible to extend the sampling-based streaming algorithm described in the proof of Theorem \[theorem:Approval-CC\_UB\] to work also for Borda-CC, albeit with some increase of the space complexity. Informally, the increase of the space complexity is because the proof needs to take care for the fact that the score difference induced by a single voter is greater in Borda-CC than it is in Approval-CC: while in Approval-CC, the satisfaction of a voter from a committee is either $0$ or $1$, in Borda-CC it is anything between $0$ to $m - 1$. \[theorem:Borda-CC-UB\] There is an $\epsilon$-approximate streaming algorithm for Borda-CC which uses $O(\epsilon^{-2} k^3 m^3 \log m)$ space. Let $t = 10 \epsilon^{-2} k m^2$. Similarly in spirit to the algorithm presented in the proof of Theorem \[theorem:one\], our algorithm samples $k^2 t$ voters, select a winning committee in the sampled election, and declares it as an $\epsilon$-winning committee for the whole election. Since storing the vote of each sampled voter takes $m \log m$ space, we get the claimed space complexity. Next we prove the correctness of the algorithm. Fix an election $E$, a committee $S$, a committee member $c$, and consider a voter $v$. We define the *score given to $c$ by $v$ with respect to $S$*, denoted by ${\mathrm{score}}_E^{v, S}(c)$ to be the Borda-score of $c$ in the preference order of $v$, if, among the candidates of $S$, $c$ is the representative of $v$; that is, if, among the candidates of $S$, $v$ ranks $c$ the highest. We define it to be $0$ otherwise. Further, we define the *score of $c$ with respect to $S$*, denoted by ${\mathrm{score}}_E^{S}(c)$ to be the sum over all voters, that is, ${\mathrm{score}}_E^{S}(c) = \sum_{i \in [n]} {\mathrm{score}}_E^{{v_i}, S}(c)$. Further, as before, we define ${\mathrm{score}}_E(S)$ to be the score of $S$, and, indeed, it holds that ${\mathrm{score}}_E(S) = \sum_{c \in S} {\mathrm{score}}_E^{S}(c)$. We begin by showing that, fixing a committee $S$ and a committee member $c$, it is possible to estimate the score of $c$ with respect to $S$ by sampling $t$ voters. Let $E$ denote the whole election, and let $E_R$ denote the sampled election, containing $t$ voters (where $t$ is as defined in the beginning of the current theorem’s proof) chosen uniformly at random from $E$. The following claim shows that with high probability the sampled election roughly preserves the score of any committee. \[claim:theorem2claim1\] Let $S$ be a committee and $c$ a committee member. Then, $|\frac{n}{t} \cdot{\mathrm{score}}_{E_R}^{S}(c) - {\mathrm{score}}_{E}^{S}(c)| \leq \epsilon n / 2$ holds with probability at least $1 - 1/m^{3k}$, where $E_R$ is obtained by sampling each voter in $E$ independently with probability $t/n$. For the committee $S$ and the committee member $c$, we define a random variable $X_i$, $i \in [t]$, such that $X_i = {\mathrm{score}}_{E_R}^{v_i, S}(c)$, where $v_i$ is the $i$th sampled voter. It holds that $$\mathbb{E}[X_i] = \frac{1}{n} {\mathrm{score}}_{E}^{S}(c).$$ Letting $X = \sum_{i \in [t]} X_i$, we have the following (from linearity of expectation): $$\mathbb{E}[X] = \frac{t}{n} {\mathrm{score}}_{E}^{S}(c).$$ Importantly, note that the variables $X_i$ have the following properties: - They are independent; this follows since we consider each committee member separately. - They are bounded; specifically, $0 \leq X_i \leq m$ holds for each $i \in [t]$. Utilizing the above two properties, we can apply a variation of Hoeffding’s inequality (see Theorem \[theorem:Hoeffding\]) and conclude that: $$\begin{aligned} \mathbb{P}\left[ | \frac{n}{t}\cdot X - {\mathrm{score}}_{E}^{S}(c) | \geq n\epsilon / 2 \right] &= \mathbb{P}[ | X - \mathbb{E}[X] | \geq t\epsilon / 2 ] \\ &\leq 2 e^{-\frac{ 2(\epsilon/2)^2 t}{ (m + 1)^2}} \\ &\leq \frac{1}{m^{3k}},\end{aligned}$$ where the first inequality follows from Hoeffding’s inequality (see Theorem \[theorem:Hoeffding\]) and last inequality follows from our definition of the sample size $t$. (of claim \[claim:theorem2claim1\]) Claim \[claim:theorem2claim1\] shows that by sampling $t$ voters, we get a good estimation for the score of a candidate with respect to some committee. Let $E$ denote the whole election, and let $E_R$ denote the sampled election, containing $k^2 t$ voters (where $t$ is as defined in the beginning of the current theorem’s proof) chosen uniformly at random from $E$. Next we show that, by sampling $k^2 t$ voters, we get a good estimation for the score of a committee. \[claim:theorem2claim2\] Let $S$ be a committee. Then, $| \frac{n}{t}\cdot{\mathrm{score}}_{E_R}(S) - {\mathrm{score}}_{E}(S) | \leq n \epsilon / 2$ holds with probability at least $1 - 1/m^{k}$, where $E_R$ is obtained by sampling each voter in $E$ independently with probability $k^2 t/n$. Let $S$ be a committee containing the committee members $c_1, \ldots, c_k$. For each $j \in [k]$, we apply Claim \[claim:theorem2claim1\] on the committee $S$ and the committee member $c_j$ with $\epsilon' = \epsilon / k$. Let us denote the random variable containing the estimated score of committee member $c_j$ with respect to committee $S$ by $Y_j$; that is, $Y_j$ is the estimated value of ${\mathrm{score}}_{E}^{S}(c_j)$, therefore, $Y_j = {\mathrm{score}}_{E_R}^{S}(c_j)$ using Claim \[claim:theorem2claim1\]. Let $Y = \sum_{j \in [k]} Y_j$. Since ${\mathrm{score}}_{E}(S) = \sum_{j \in [k]} {\mathrm{score}}_{E}^{S}(c_j)$, and from linearity of expectation, it follows that $$\mathbb{E}[Y] = \frac{t}{n}\cdot{\mathrm{score}}_{E}(S).$$ Further, we have that: $$\begin{aligned} \mathbb{P}\big[ | \frac{n}{t}\cdot Y &- {\mathrm{score}}_{E}(S) | \geq n \epsilon / 2 \big] \\ &\leq \mathbb{P}\left[ \Sigma_{j \in [k]} | Y_j - \mathbb{E}[ Y_j ] | \geq n k \epsilon' / 2 \right] \\ &\leq \sum_{j \in [k]}\left( \mathbb{P}\left[ | Y_j - \mathbb{E}[ Y_j ] | \geq n \epsilon' / 2 \right] \right) \\ &\leq \frac{k}{m^{2k}} \\ &\leq \frac{1}{m^k},\end{aligned}$$ where the first inequality follows from the definitions of $Y$ and $\epsilon'$, the second inequality follows from applying a union bound over the committee members $c_1, \ldots, c_k$, and the third inequality follows from Claim \[claim:theorem2claim1\]. (of claim \[claim:theorem2claim2\]) Finally, building upon Claim \[claim:theorem2claim2\], we apply union bound on all ${m \choose k}$ committees of size $k$. Following this union bound, we conclude that, with high probability, the algorithm returns an $\epsilon$-winning committee. We mention that the result described in Theorem \[theorem:Borda-CC-UB\] transfers to all scoring rules, albeit with some increase of the space complexity. That is, careful analysis of the proof of Theorem \[theorem:Borda-CC-UB\] reveals that, since we can upper bound the values of the random variables $X_j$ by $m$, it follows that we can apply Hoeffding’s inequality (see Theorem \[theorem:Hoeffding\]), which causes an increase of the space complexity by a multiplicative factor of $m^2$, compared to the space complexity that we get for Approval-CC. Considering any normalized scoring vector $(\alpha_1, \alpha_2, \ldots, \alpha_m)$ with $\alpha_1 \geq \ldots \geq \alpha_m$ such that $\alpha_1$ is the value given by a voter to her first-choice candidate, and following the same reasoning as described above, we see that applying Hoeffding’s inequality (see Theorem \[theorem:Hoeffding\]) causes an increase of the space complexity by a multiplicative factor of $\alpha_1^2$, compared to the space complexity we get for Approval-CC. Specifically, the resulting space complexity is $O(\epsilon^{-2}k^3 \alpha_1^2 m \log m)$. We know that scoring rules remain unchanged if we multiply every $\alpha_i$ by any constant $\lambda>0$ and/or add any constant $\mu$. Hence, we can assume without loss of generality that for any score vector $\alpha$, there exists a $j$ such that $\alpha_j - \alpha_{j+1}=1$ and $\alpha_k = 0$ for all $k>j$. We call such an $\alpha$ a normalized score vector. Next we move on to consider Monroe (M), beginning with the arguably simpler case of Approval-M. Our algorithm is again based on sampling a small number of voters and computing a winning committee for them. The analysis is more involved, since we cannot consider all assignments, but only M-assignments. A naive analysis would apply union bound on all M-assignments, but since there are $O(k^n)$ such assignments, we would get linear space in the number of voters, which would be too much. Fortunately, we can do better, building upon some structural observations, as we show next. \[theorem:approval-m\] There is an $\epsilon$-approximate streaming algorithm for Approval-M which uses $O(\epsilon^{-2} k^3 m \log m)$ space. The overall idea is to consider any committee $S$ with its optimal assignment $A^*$. We will show that, with high probability, with respect to $S$, the score of the assignment $A^*$ on a sampled election is close to being the actual score of the committee $S$ on the sampled election. The theorem would then follows by union bound over all ${m \choose k}$ committees. More specifically, for each committee $S$ together with its optimal assignment $A^*$, we define a *preserving subset* to be a subset $E_P$ of the election $E$ such that, for each committee member $c \in S$, the fraction of voters assigned to $c$ which are satisfied by $c$, as well as the fraction of voters assigned to $c$ which are not satisfied by $c$, is preserved. Formally, we define a preserving subset as follows. Let $S$ be a committee, let $A^*$ be its optimal assignment, and let $E_P$ be a subset of the election $E$. Let $\smiley_{E_P}^{A^*}(c_i)$ denote the set of voters in $E_P$ which are assigned to $c_i$ by $A^*$ and are satisfied by $c_i$ (that is, it holds that $c_i \in v$), and let $\frownie_{E_P}^{A^*}(c_i)$ denote the set of voters in $E_P$ which are assigned to $c_i$ by $A^*$ and are not satisfied by $c_i$ (that is, it holds that $c_i \notin v$). Then, a subset $E_P$ of the election $E$ is a *preserving subset* if for each $c_i \in S$ it holds that $$(1)\quad |\smiley_{E_P}^{A^*}(c_i)| = \frac{|E_P|}{|E|} \cdot |\smiley_{E}^{A^*}(c_i)|$$ and that $$(2)\quad |\frownie_{E_P}^{A^*}(c_i)| = \frac{|E_P|}{|E|} \cdot |\frownie_{E}^{A^*}(c_i)|,$$ That is, a preserving subset is a subset of the voters of some given election which, with respect to the optimal assignment of a given committee, preserves the (normalized) number of voters assigned to each candidate and are satisfied (unsatisfied) by it. Next we show that, for each committee $S$, with high probability a random subset containing $t = O(\epsilon^{-2} k^3 \log m)$ is close to being a preserving subset. \[lemma:am2\] Let $E_R$ be a random subset of voters from $E$ obtained by sampling each voter independently at random with probability $t / n$. Then, for each committee $S$, with probability at least $1 - m^{-2k}$, it holds that there exists a preserving subset $E_P$ which can be obtained from $E_R$ by changing the vote of at most $\epsilon t$ voters. It suffices to show that, for each $c_i \in S$, it holds that $\smiley_{E}^{A^*}(c_i) = \frac{n}{t} \smiley_{E_R}^{A^*}(c_i) \pm \frac{\epsilon n}{2k}$ and also it holds that $\frownie_{E}^{A^*}(c_i) = \frac{n}{t} \frownie_{E_R}^{A^*}(c_i) \pm \frac{\epsilon n}{2k}$, since then, the fraction of each of the $k$ sets $\smiley_{E}^{A^*}(c_i)$ and each of the $k$ sets $\frownie_{E}^{A^*}(c_i)$ can preserve its respective fraction by changing the votes of at most $\frac{\epsilon n}{2k}$ voters. Since each voter is sampled with probability $t / n$, we have that $$\mathbb{E}[\smiley_{E_R}^{A^*}(c_i)] = \frac{t}{n} \smiley_{E}^{A^*}(c_i).$$ Since each voter is sampled independently, we can apply Hoeffding’s inequality (see Theorem \[theorem:Hoeffding\]), to have the following. $$\mathbb{P} \left[ | \smiley_{E_R}^{A^*}(c_i) - \mathbb{E}[ \smiley_{E_R}^{A^*}(c_i) ]| \geq \frac{\epsilon t}{2k} \right] \leq 2 e^{-\frac{2t \epsilon^2}{4k^2}} = O(m^{-2k})$$ and $$\mathbb{P} \left[ | \frownie_{E_R}^{A^*}(c_i) - \mathbb{E}[ \frownie_{E_R}^{A^*}(c_i) ]| \geq \frac{\epsilon t}{2k} \right] \leq 2 e^{-\frac{2t \epsilon^2}{4k^2}} = O(m^{-2k}).$$ Thus, we are done. (of claim \[lemma:am2\]) Next we show that, for each committee $S$, its optimal assignment $A^*$ in $E$ is also an optimal assignment in any preserving subset $E_P$ of $E$. Notice that the following claim is not probabilistic but combinatorial. \[lemma:am1\] Let $S$ be a committee, $A^*$ be its optimal assignment, and $E_P$ be a preserving subset of $E$. Then, the restriction of $A^*$ to $E_P$ is an optimal assignment for $S$ in $E_P$. Intuitively, if there was a better assignment $A_P$ than $A^*$ for $S$ in $E_P$, then we could change $A^*$ accordingly and get a better assignment for $S$ in $E$, contradicting the optimality of $A^*$ for $S$ in $E$. More formally, let $S$ be a committee, $A^*$ be its optimal assignment, and $E_P$ be a preserving subset of $E$. Towards a contradiction, assume that there is an assignment $A_P \neq A^*$ such that ${\mathrm{score}}_{E_P}^{A_P}(S) > {\mathrm{score}}_{E_P}^{A^*}(S)$. Consider $\bar{E}_P = E \setminus E_P$ and notice that, since $E_P$ is a preserving subset of $E$, it also holds that $\bar{E}_P$ is a preserving subset of $E$, and we have that $$\begin{aligned} {\mathrm{score}}_{E}^{A^*}(S) &= \frac{|E_P|}{|E|} \cdot {\mathrm{score}}_{E}^{ A^*}(S) + \frac{|\bar{E}_P|}{|E|} \cdot {\mathrm{score}}_{E}^{A^*}(S) \\ &= {\mathrm{score}}_{E_P}^{A^*}(S) + {\mathrm{score}}_{\bar{E}_P}^{A^*}(S) \\ &< {\mathrm{score}}_{E_P}^{A_P} (S) + {\mathrm{score}}_{\bar{E}_P}^{A^*}(S).\end{aligned}$$ Since $A_P$ does not violate the Monroe property, we have constructed a better assignment for $S$ in $E$, contradicting the optimality of $A^*$ for $S$ in $E$. (of claim \[lemma:am1\]) Building upon the last two claims proven above, the following claim shows that, for each committee $S$, with high probability, the score of its optimal assignment $A^*$ on $E$ is a good estimator for its score on the sampled election $E_R$. \[lemma:am0\] For each committee $S$ and its optimal assignment $A^*$, with probability at least $1 - m^{-2k}$ it holds that: $${\mathrm{score}}_{E_R}(S) + \epsilon t \geq {\mathrm{score}}_{E_R}^{A^*}(S) \geq {\mathrm{score}}_{E_R}(S) - \epsilon t.$$ Combining the last two claims, we have that, with high probability, there exists a preserving subset $E_P$, obtained from the sampled election by changing at most $\epsilon t$ voters. Consider the preserving subset $E_P$ which is obtained from the sampled election $E_R$ by changing at most $\epsilon t$ voters. By the first claim, we have that the assignment $A^*$ is optimal for $S$ on $E_P$. Consider any other assignment. Since $A^*$ is optimal for $S$ on $E_P$ and $E_P$ is $\epsilon$-close to $E_R$, the two inequalities hold, since $\epsilon$ bounds the score difference between $A^*$ on $E_R$ and any other assignment. (of claim \[lemma:am0\]) Following the last claim, we have that, for each committee $S$, a random sample is indeed a good estimator for the score of $S$. Then, the claim follows by union bound over all possible ${m \choose k}$ committees. There is an $\epsilon$-approximate streaming algorithm for Borda-M which uses $O(\epsilon^{-2} k^3 m^5 \log m)$ space. The idea of the proof is very similar to Approval-M, when we take into account the following two differences. The first difference is that, instead of only two blocks for each committee member, namely the $\smiley$ block and the $\frownie$ block, in Borda-M we shall consider $m$ blocks for each committee member, where a voter $v$ is assigned to the $l$th block (for $l \in [m]$) of committee member $c$ if $v$ is represented by $c$ and the satisfaction of $v$ from $c$ is $j$. The second difference is that we shall bound the difference between the actual score of a committee and its score in the sampled election differently; specifically, we have that $${\mathrm{score}}_{E_R}(S) + \epsilon m t \geq {\mathrm{score}}_{E_R}^{A^*}(S) \geq {\mathrm{score}}_{E_R}(S) - \epsilon m t,$$ since each voter whose vote is changed can increase or decrease the score of each committee by $O(m)$ and not only by $O(1)$ as for Approval-M. The proof then follows similar lines as the proof given for Approval-M (see Theorem \[theorem:approval-m\]), but the space complexity increases. Specifically, the first difference described above causes the space complexity to multiply by a factor of $O(m^2)$, since we shall consider those $m$ blocks (instead of only $2$) and take into account that the error can multiply by $m$. Similarly, the second difference described above causes the space complexity to multiply by another factor of $O(m^2)$, since we shall increase the size of the sample to account for the increased score difference. Lower Bounds {#section:lowerbounds} ------------ In this section we prove two types of lower bounds which complement our algorithms. We begin by showing that any streaming algorithm shall use space which is linear in the number $m$ of candidates. \[theorem:lowerbound1\] There is an $\epsilon > 0$ such that any $\epsilon$-approximate streaming algorithm for Approval-CC or Approval-M needs $\Omega(m)$ space. We reduce from the <span style="font-variant:small-caps;">Set Disjointness</span> problem in communication complexity. In the <span style="font-variant:small-caps;">Set Disjointness</span> problem, there is a set of elements $U = x_1, \ldots, x_u$, and two players, Alice and Bob. Alice is given a subset $A \subseteq U$ and Bob is given a subset $B \subseteq U$. Then, Alice sends a message to Bob, and Bob has to decide whether $A \cap B = \emptyset$, in which case Bob shall accept; otherwise, that is, if there is some index $i \in [u]$ such that $x_i \in A \cap B$, then Bob shall reject. It is known that Alice shall send $\Omega(u)$ bits in order for Bob to be correct with high probability [@kalyanasundaram1992probabilistic]. We first describe the reduction for <span style="font-variant:small-caps;">Approval-CC</span>; that is, given an instance of <span style="font-variant:small-caps;">Set Disjointness</span>, we construct a vote stream for <span style="font-variant:small-caps;">Approval-CC</span>, as follows. we create an election with $u + 1$ candidates, where for each $x_i$ ($i \in [u]$) we create a corresponding candidate $c_i$, and we have another candidate $d$. Then, Alice inserts two voters, $v_1$ and $v_1'$, to the vote stream, where both $v_1$ and $v_1'$ are approving the candidates corresponding to the elements in $A$ (that is, $v_1 = v_1' = \{c_i : x_i \in A\}$. Then, Bob inserts two voters, $v_2$ and $v_2'$, to the vote stream, where, similarly, $v_2 = v_2' = \{c_i : x_i \in B\}$. Finally, Bob inserts three voters, $v_3, v_4, v_5$, all of which approve only the candidate $d$. This finishes the description of the reduction. For example, letting $U = \{x_1, x_2, x_3\}$ (thus, $u = 3$), $A = \{x_2\}$, and $B = \{x_1, x_2\}$, we will have that $v_1$ and $v_1'$ both approve $c_2$, $v_2$ and $v_2'$ both approve $c_1$ and $c_2$, and $v_3$, $v_4$, and $v_5$ all approve $d$. We assume, towards a contradiction, that there is a streaming algorithm for Approval-CC which uses $o(m)$ space. We use that algorithm with $k = 1$ and $\epsilon = 1/7$. Notice that if $A \cap B = \emptyset$, then each candidate $c_i$ covers at most $2$ voters, while if there is some index $i \in [u]$ such that $x_i \in A \cap B$, then the candidate $c_i$ covers $4$ voters. Irrespectively, the candidate $d$ covers $3$ voters. Thus, the streaming algorithm would declare $d$ as the winner if and only if $A$ and $B$ are disjoint, contradicting the lower bound for <span style="font-variant:small-caps;">Set Disjointness</span>. As for Approval-M, notice that in the reduction described above the size $k$ of the committee is $1$. In this case, Approval-CC and Approval-M are equivalent, thus the reduction transfers to Approval-M as it is. It turns out that with some modifications, the reduction described in the proof of Theorem \[theorem:lowerbound1\] can be made to work also for Borda-CC and Borda-M. \[theorem:lowerbound1\] There is an $\epsilon > 0$ such that any $\epsilon$-approximate streaming algorithm for Borda-CC or Borda-M needs $\Omega(m)$ space. We again reduce from <span style="font-variant:small-caps;">Set Disjointness</span> where Alice (Bob) is given a subset $A \subseteq U$ ($B \subseteq U$), for $U = \{x_1, \ldots, x_u$}, and Alice and Bob shall decide together whether $A \cap B = \emptyset$ (see the proof of Theorem \[theorem:lowerbound1\] for a more detailed description of <span style="font-variant:small-caps;">Set Disjointness</span>). We describe first the reduction for <span style="font-variant:small-caps;">Borda-CC</span>; that is, given an instance of <span style="font-variant:small-caps;">Set Disjointness</span>, we construct a vote stream for <span style="font-variant:small-caps;">Borda-CC</span>, as follows. We create an election with $4u + 1$ candidates, where for each $x_i$ ($i \in [u]$) we create a corresponding candidate $c_i$; we have another candidate $d$; and another $3u$ dummy candidates $d_1, \ldots, d_{3u}$. Corresponding to her set $A$, Alice inserts one voter $v_1$ to the vote stream, ranking first those $|A|$ candidates $c_i$ which correspond to the elements $x_i$ in $A$, then $u - |A|$ dummy candidates $d_1, \ldots, d_{u - |A|}$, then $d$, then the remaining $2u + |A|$ dummy candidates $d_{u - |A| + 1}, \ldots, d_{3u}$, and ranking last those $u - |A|$ candidates $c_i$ which correspond to the elements $x_i$ not in $A$. Bob behaves quite similarly, by inserting one voter $v_2$ to the vote stream, ranking first those $|B|$ candidates $c_i$ which correspond to the elements $x_i$ in $B$, then $u - |A|$ dummy candidates $d_{3u}, \ldots, d_{3u - |B| + 1}$ (notice the change of order of the dummy candidates with respect to $v_1$), then $d$, then the remaining $2u + |B|$ dummy candidates $d_{2u - |B|}, \ldots, d_{1}$ (notice again the change of order), and ranking last those $u - |B|$ candidates $c_i$ which correspond to the elements $x_i$ not in $B$. This finishes the description of the reduction. For example, letting $U = \{x_1, x_2, x_3\}$ (thus, $u = 3$), $A = \{x_2\}$, and $B = \{x_1, x_2\}$, we will have that $v_1 : c_2 {\ensuremath{\succ}}d_1 {\ensuremath{\succ}}d_2 {\ensuremath{\succ}}d {\ensuremath{\succ}}d_3 {\ensuremath{\succ}}d_4 {\ensuremath{\succ}}d_5 {\ensuremath{\succ}}d_6 {\ensuremath{\succ}}d_7 {\ensuremath{\succ}}d_8 {\ensuremath{\succ}}d_9 {\ensuremath{\succ}}c_1 {\ensuremath{\succ}}c_3$ and $v_2 : c_1 {\ensuremath{\succ}}c_2 {\ensuremath{\succ}}d_9 {\ensuremath{\succ}}d {\ensuremath{\succ}}d_8 {\ensuremath{\succ}}d_7 {\ensuremath{\succ}}d_6 {\ensuremath{\succ}}d_5 {\ensuremath{\succ}}d_4 {\ensuremath{\succ}}d_3 {\ensuremath{\succ}}d_2 {\ensuremath{\succ}}c_1 {\ensuremath{\succ}}c_3$. We argue that $d$ is a Borda winner in the reduced election if and only if $A \cap B = \emptyset$. Let us denote the Borda score of a candidate $c$ in the election containing the voters $v_1$ and $v_2$ by $s(c)$. For the dummy candidates we have that $s(d_i) \leq 5u$ (for any $i \in [3u]$); this can be seen by observing that the dummy candidates achieve maximum score in the extreme case where $A = B = \emptyset$, in which $d_i$ is getting $4u - i$ points from $v_1$ and another $u + i$ points from $v_2$. Now, consider a candidate $c_i$ corresponding to an element $x_i$ which appears only in one of the sets, either $A$ or $B$; without loss of generality, let $c_i$ be a candidate corresponding to an element $x_i$ such that $x_i \in A$ and $x_i \notin B$. Then, we have that $c_i$ gets at most $4u$ points from $v_1$ and at most $u - 1$ points from $v_2$. Thus, we conclude that $s(c_i) \leq 5u - 1$. Similarly, consider a candidate $c_i$ corresponding to an element $x_i$ which appears both in $A$ and $B$. Then, we have that $c_i$ gets at least $3u + 1$ points from each of $v_1$ and $v_2$. Thus, we conclude that $s(c_i) \geq 6u + 2$. Finally, notice that, irrespective of the contents of $A$ and $B$, it holds that $s(d) = 6u$ . Therefore, following the computation described in the last paragraph, we conclude that $d$ is a Borda winner if and only if $A$ and $B$ are disjoint. So, assuming, towards a contradiction, that there is a streaming algorithm for Borda-CC which uses $o(m)$ space, we use that algorithm with $k = 1$ and $\epsilon = 1/3$. Since the streaming algorithm would declare $d$ as the winner if and only if $A$ and $B$ are disjoint, it would contradict the lower bound for <span style="font-variant:small-caps;">Set Disjointness</span>. As for Borda-M, notice that in the reduction described above the size $k$ of the committee is $1$. In this case, Borda-CC and Borda-M are equivalent, thus the reduction transfers to Borda-M as it is. We continue by observing the following lower bound, with respect to the required approximation $\epsilon$ (notice that the following theorem is also a corollary of [@bhattacharyya2016optimal Theorem 10]). For any $\epsilon > 0$, any $\epsilon$-approximate streaming algorithm for Approval-CC needs $\Omega(\epsilon^{-1})$ space. We reduce from the <span style="font-variant:small-caps;">$\ell_1$-Heavy Hitters</span> problem, which, given a stream containing $n$ items, each item is of one type out of $m$ item types, an approximation parameter $\epsilon$, and a further parameter $\phi$, asks for returning all items which occur at least $\phi n$ times, while not returning any item which occurs less than $(\phi - \epsilon)n$ times. A lower bound of $O(\epsilon^{-1})$ is known for <span style="font-variant:small-caps;">$\ell_1$-Heavy Hitters</span> [@bhattacharyya2016optimal]. Given an instance of <span style="font-variant:small-caps;">$\ell_1$-Heavy Hitters</span>, we create an instance for Approval-CC, as follows. For each item type, we create a candidate. For each item in the stream, we create a voter approving only the candidate corresponding to its item type. We set $k = 1$, keep the same $\epsilon$, and set $\phi = 1/2$. This finishes the description of the reduction. Correctness and space complexity follows immediately. The reader might notice that the lower bounds presented in this section are not tight. We leave the task of closing the gap between our upper bounds and lower bounds to future research. Discussion and Outlook {#section:outlook} ====================== We have described streaming algorithms which find approximate winners for several well-known proportional representation multiwinner voting rules. Below we mention some extensions to our model, discuss the usefulness of our results, and mention several avenues for future research. [**More general models.**]{} In this paper we concentrated on a simple streaming model where (1) each item in the stream is a voter, (2) there are no assumptions on the order by which the voters arrive to the stream, and (3) the goal is to compute an approximate winner at the end of the stream. There are other relevant models, which we mention below. - In the *sliding windows* model, the goal is to compute an approximate winner with respect to the last $t$ elements in the stream, for some given $t$. Since our streaming algorithms are based on sampling, and sampling from a sliding window can be done efficiently [@babcock2002sampling], our streaming algorithms extend to this model as well. This model is useful for identifying emerging trends. - It is possible to use our streaming algorithms not only to compute an approximate winner at the end of the stream, but, since they are based on sampling, they can be used to compute an approximate winner at any time during the stream. - Our streaming algorithms extend also to situations where we do not know the number $n$ of the voters a-priori, as is apparent by a recent result [@bhattacharyya2016optimal], and since our streaming algorithms are based on sampling. - Consider situations where a voter might gradually approve more candidates. A corresponding stream model might be that each item in the stream is a tuple $(v_i, c_j)$, where an item $(v_i, c_j)$ means that voter $v_i$ have just decided to approve candidate $c_j$. Such a stream model might model online shopping websites, where an item $(v_i, c_j)$ would arrive to the stream whenever the person $v_i$ decided to search for the product $c_j$. Importantly, since we can decide at the beginning of the stream which voters to sample, it follows that our upper bounds also extend to this, more general model. [**Less general models.**]{} It might be interesting to study models where we assume some structure in the stream. Specifically, one might consider *uniform streams*, where the voters are not arriving in an arbitrary (possibly adversarial) order, but in a random order, by choosing a random permutation uniformly at random. The hope is that for such uniform streams it might be possible to design streaming algorithms with better space complexity. Indeed, we believe that, at least for uniform streams, there are streaming algorithms with better space complexity for *round-based* voting rules, such as the greedy versions of Chamberlin–Courant and Monroe [@sko-fal-sli:c:multiwinner] (in short, one might sample several subelections, and use each subelection for a different round). Such results would be relevant also for situations without huge number of voters, but with time constraints; consider the following example (which we thank an anonymous reviewer for suggesting it). A distinguished speaker is to give the same talk at $k$ different dates, and, in order to maximize the total number of attendees, an online scheduling poll is created in order to decide upon the dates. The problem is that we have to decide upon the dates very soon, so we cannot wait for everybody to answer; our sampling-based streaming algorithms (and possibly even better algorithms assuming stream uniformity) could tell us how many voters we need in the scheduling poll. Another restricted model might be to consider restricted domains, thus not considering all possible elections, but only those elections which adhere to some restricted domains, such as single peaked domains and single crossing domains. It is not clear whether imposing structural constraints on the elections would lower the needed space complexity. [**Other multiwinner voting rules.**]{} Indeed, streaming algorithms for other multiwinner voting rules deserve to be studied as well. We specifically mention Single Transferable Vote (STV) which also aims at proportional representation. Naturally, there are other multiwinner voting rules which do not aim at proportional representation; we mention $k$-best rules, committee scoring rules, and various extensions to Condorcet consistent voting rules, as some important families of multiwinner voting rules.

--- abstract: 'We report experimental observations of the convection-driven fingering instability of an iodate-arsenous acid chemical reaction front. The front propagated upward in a vertical slab; the thickness of the slab was varied to control the degree of instability. We observed the onset and subsequent nonlinear evolution of the fingers, which were made visible by a [*p*]{}H indicator. We measured the spacing of the fingers during their initial stages and compared this to the wavelength of the fastest growing linear mode predicted by the stability analysis of Huang [*et. al.*]{} \[[*Phys. Rev. E*]{}, [**48**]{}, 4378 (1993), and unpublished\]. We find agreement with the thickness dependence predicted by the theory.' address: - | Department of Physics, University of Toronto,\ 60 St. George St., Toronto, Ontario, Canada M5S 1A7 - | Department of Physics and Erindale College, University of Toronto,\ 60 St. George St., Toronto, Ontario, Canada M5S 1A7 - 'AT & T Bell Laboratories, Murray Hill, New Jersey 07974-0636' author: - 'Michael R. Carey[@presentaddress]' - 'Stephen W. Morris' - Paul Kolodner title: Convective Fingering of an Autocatalytic Reaction Front --- Most studies of pattern-forming chemical reactions are carried out in thin horizontal layers, or in gels, under conditions where the coupling between chemical concentrations and hydrodynamic effects is suppressed. This coupling arises because reactions usually modify the density of the solution, giving rise to buoyancy forces and hydrodynamic flow. This flow then advects the reacting solution, adding new transport terms to the already complex reaction-diffusion problem. In this paper, we study chemical reaction driven convection in a very simple context, in the autocatalytic reaction of iodate and arsenous acid in a narrow vertical slot[@huang]. This system may be regarded as a simplified prototype for the study of hydrodynamic instabilities in more complex pattern-forming chemical systems, such as the Belousov-Zhabotinsky reaction[@BZ]. In the absence of hydrodynamic effects, the iodate-arsenous acid reaction[@wellknownRx] produces a single, sharply defined reaction front which propagates at a speed governed by the diffusion of the iodide autocatalyst. When such a front propagates upward in a circular tube, it has been shown that the solution near the front is unstable to convection when the tube diameter exceeds a critical value[@pojman]. In this direction of propagation, the unreacted solution, which is denser, lies above the lighter reacted solution in a potentially unstable arrangement. For the vertical slot geometry, which is analytically simpler, there exists a well developed theory[@huang; @unboundedtheory; @moretheory] of the linear instability of a flat front. The theory predicts that the initial instability takes the form of convective rolls which corrugate the front into a series of fingers. In the limit of sharp fronts, the reaction-diffusion part of the problem can be reduced to an eikonal relation in which the local front speed depends on the front curvature[@eikonal]. The relative strength of buoyancy is measured by a dimensionless parameter $S$, described below, which is proportional to the cube of the slot thickness. The spacing of the finger pattern results from the competition between convection, which tends to extend the fingers vertically, and diffusion, acting via the eikonal relation, which tends to smooth the front. More realistic models, which include the full reaction-diffusion equations needed to describe a finite front thickness, have recently been proposed[@huangprivate]. A flat front in a laterally extended slot is expected to be unstable to a periodic chain of convection cells which are analogous to those found in other simple fluid systems, such as Rayleigh-Bénard convection[@bigreview]. This instability also bears some analogy to that of a flame front[@flames]. The objective of the present study is to examine this instability as it grows out of a flat front, and to compare the spacing of the pattern of fingers to what is predicted from the linear theory in the thin-front limit. We also qualitatively observed the instability after it had become fully developed, in which the front dynamics is dominated by interactions between fingers. This highly nonlinear regime is the subject of some recent simulations[@huangprivate]. Fig. \[apparatus\] shows the experimental arrangement. The slot was formed of two glass plates held apart by a rubber gasket. The spacing $a~\sim~1mm$ of the cell was determined by steel shims outside the gasket, which were not in contact with the solution. The width of the slot $w$ was $32 \pm 0.5 mm$. The back face of the cell, which was painted white to make the reaction more visible, was in good thermal contact with a copper block at room temperature, $23^\circ C$. The reaction front was initiated by an electrical trigger which consisted of two thin steel electrodes on opposite sides of the slot which were in contact with the solution. A DC voltage $\approx 10V$ was applied to the electrodes for a few seconds to initiate the reaction. The reaction front, made visible by an indicator dye, was photographed through the front face of the cell. The camera and cell apparatus were mounted on a common stand which could be oriented with respect to vertical. The working solution was prepared as follows. Solid $As_2O_3$ powder was dissolved in distilled water which had been made strongly basic by the addition of $KOH$. When the powder had dissolved, the resulting solution was filtered and neutralized with concentrated $HCl$. This was mixed with a solution of $KIO_3$ and a solution of Congo Red indicator, so that the final concentrations were $[H_3AsO_3]$ = 0.005M, $[KIO_3]$ = 0.0025M, with 0.0001M of indicator. In some thinner cells, a higher concentration of indicator was used. The unreacted solution was adjusted to have a ${\it p}$H $\approx 6$. Congo Red changes from red to blue near ${\it p}$H $\approx 5.2$. With this high initial ${\it p}$H value, the reaction does not start spontaneously, but can be initiated quickly on an electrode as described above. As the reaction front passes a point, a sharp transition in color is seen; additional color gradations in the blue region behind the front served to delineate some flow structures within the fingers. The front speed in the absence of convection, measured for downward propagation in narrow capillary tubes, was $c_0 = (3.47 \pm 0.03) \times 10^{-3} mm/s$ for this solution. This is an order of magnitude smaller than that observed in previous studies[@pojman], which used somewhat different concentrations and ${\it p}$H. We performed runs in cells with various thicknesses $a$ and did not vary the width $w$, the temperature, or the chemical concentrations. In runs in which the front was initiated at the top of the slot and propagated downward, the initial front irregularities grew only slightly during propagation. Fronts which were initiated at the bottom and propagated upward showed strong fingering in all but the thinnest cells. The fingers that formed soon became nonsinusoidal, taking on a scalloped shape, with broad upper ends and narrow cusps between fingers on the lower side. One could clearly see a plume-like structure within each finger in color gradations of the blue state of the indicator. The precise relation of these to the flow is not clear, but they presumably map varying ${\it p}$H regions which persisted behind the front. The fingers were reasonably regular initially, but soon after initiation, some fingers are overtaken and suppressed by their neighbors. There also developed a tendency for fingers to avoid the lateral edges of the cell. We did not generally observe tip splitting or mechanisms which nucleated new fingers. In the late stages, the front advanced farthest, still carrying interacting fingers, near the lateral midpoint of the cell. Eventually, a large central plume of width $\approx w/2$ developed which was accompanied by a strong suppression, or even a reversal, of the front advancement near the ends of the cell. In the thinnest cells, as discussed below, no fingering was observed in the early stages, and the only distortion of the front was the development of this large plume. We measured the spacing of the fingers directly from photos, at the earliest stages of their development for cells of various thickness. Fig. \[lambda\_vs\_a\] shows a plot of the wavelength $\lambda$ of the fingers, defined as the mean trough-to-trough spacing, exclusive of end effects, as a function of the thickness of the cell. For cells thinner than $0.4$ mm, no clear fingers were observed before the broad plume reached the top of the cell. The main uncertainties in the finger spacing came from the small number of fingers across the cell (3 - 7), imperfections of the starting conditions, which tended to be amplified by the instability, and, for thicker cells, the tendency of the fingers to interact and overtake one another at an early stage. However, a clear trend to narrower fingers for thicker cells was evident. The solid line in Fig. \[lambda\_vs\_a\] is a fit discussed below. According to the linear analysis[@huang; @unboundedtheory; @moretheory], the dimensionless parameter that controls the instability is given by $$S = \frac{\delta g a^3 }{\nu D_C}, \label{defineS}$$ where $g$ is the acceleration of gravity, $a$ is the cell thickness, $\nu$ is the kinematic viscosity and $D_C$ the diffusion constant of the autocatalyst. The density change across the front is parameterized by the dimensionless density jump $\delta=(\rho_u/\rho_r)-1$, with $\rho_r$ and $\rho_u$ the densities of the reacted and unreacted solutions. The density change is almost completely due to concentration change, so that the thermal expansion of the fluid due to the heat released by the reaction can be neglected[@neglectheat]. For a laterally unbounded slot[@huang], linear stability analysis predicts that fronts are unstable to convection rolls within a band of wavenumbers $q$, with $0 \leq q \leq q_c(S)$ for [*any*]{} $S > 0$. The critical wavevector $q_c \rightarrow 0$ as $S \rightarrow 0$. The predicted neutral stability curve $S_c=S(q_c)$, given by Eqn. 40 of Ref. [@huang], is shown in Fig. \[S\_vs\_q\]. Physically, this limit means that fronts in wide slots will be unstable to long wavelength convection, even when the slot is made very thin ([*i.e.*]{}, even for small $S$). For slots of finite width $w$, one naturally expects a cutoff when the unstable band implies pattern wavelengths $\lambda \approx w$. The data shown in Fig. \[lambda\_vs\_a\], for which $w = 32 \pm 0.5 mm$, are consistent with this expectation; no fingers are seen in cells with $a < 0.40mm$, a thickness for which only two or three fingers would fit across the width of the cell. The theory also predicts the fastest growing linear mode $q_{max}$, which lies near the midpoint of the unstable band. Fig.\[S\_vs\_q\] shows the finger spacing data on a dimensionless plot of $S$ [*vs.*]{} $q$, with lengths scaled by the cell thickness $a$. We fit the data to the locus of maximum growth rate[@huangprivate] using a one parameter fit of the form $S = k a^3$ [*vs.*]{} $q = 2 \pi a/\lambda$, with $k$ adjustable. The fit minimized the sum of the absolute deviations, a criterion which is more robust to outliers than the usual least squares. We found $k = (2.05 \pm 0.30) \times 10^{11} m^{-3}$. The fit parameter is simply related to the various parameters in Eqn. \[defineS\]. Using the measured value[@pojman; @unboundedtheory] for $D_C = 2.0 \times 10^{-9} m^2/s$, and taking $\nu$ as the viscosity of water, we find that the density jump required by Eqn. \[defineS\], is $\delta \approx 4 \times 10^{-5}$. This is about 50% smaller than the value[@pojman; @unboundedtheory] previously measured for the isothermal density change. The smaller value of this parameter, and our smaller front speed $c_0$, are probably the result of the different chemical concentrations and ${\it p}$H of our solution. The wavelength $\lambda_{max} = 2 \pi a / q_{max}$ corresponding to the fastest growing linear mode is shown as the solid line in Fig. \[lambda\_vs\_a\]. It is interesting to note that the linear theory for a laterally unbounded slot predicts that $\lambda_{max}$ tends to infinity as $a \rightarrow 0$ and to a constant $\lambda_{\infty} \approx 15(\nu^2 / g \delta)^{1/3}$ as $a \rightarrow \infty$, and passes through a minimum at intermediate values of $a$[@huangprivate]. For our solution, this minimum is expected near $a \approx~1.1~mm$, for which $\lambda_{max} \approx 4.6~mm$, which is unfortunately just beyond the range of our data. Observation of this curious minimum would be a very interesting confirmation of the theory. In a slot of finite width, the $\lambda_{max}$ approaches $\lambda_w \sim w$, as $a \rightarrow 0$, due the constraint that at least one convection roll must fit across the slot width. In the late stages of the nonlinear evolution of our fronts, it is evident that flow eventually develops on the scale of $\lambda_w$, in the form of a large upflow at the midpoint of the cell, with downflows near each end. We have compared our data to the existing theory which is valid in the limit of zero front thickness[@huang]. In fact, the front has a finite thickness $d_r \sim D_c/c_0$, which is approximately 0.6 mm for our solution. Thus $d_r \sim a$ for most if not all of our data. While our data do behave as predicted by the simple theory, we expect that a more accurate model which includes the effects of finite front thickness will be required for more detailed comparison to future experiments. In conclusion, we have observed the fingering instability of a vertically propagating chemical reaction front in a cell with a slot geometry. We find that the spacing of the fingers in the early stages of their development is consistent with the fastest growing mode according to the linear theory of convection driven by concentration-induced buoyancy. We find a long wavelength cutoff consistent with the effect of the finite lateral extent of the cell. As the instability develops, we observed interactions of fingers which tended to reduce their number, eventually leading to broad flows on the scale of the cell width. In future experiments, we plan to extend this study to wider cells, with better control over the cell geometry, and to cells which are continuously fed with reactants. With these arrangements, it should be possible to precisely study the onset of the instability and the dynamical evolution of the fingers in the nonlinear regime. We wish to thank Jie Huang and Boyd F. Edwards for communicating the results of their analysis prior to publication. Some preliminary experiments were performed by one of us (P.K.) in collaboration with Sibel Bayrakci. M.R.C and S.W.M. gratefully acknowledge support from the Natural Science and Engineering Research Council of Canada. Present address: Department of Physics, Duke University, PO Box 90305, Durham, NC 27708-0305. J. Huang, D. A. Vasquez, B. F. Edwards and P. Kolodner, [*Phys. Rev. E*]{}, [**48**]{}, 4378 (1993). M. Menzinger, A. Tzalmona, R. L. Armstrong, A Cross and C. Lemaire, [*J. Phys. Chem.*]{}, [**96**]{}, 4725 (1992), K. Matthiessen, S.C. Mueller, [*Phys. Rev. E*]{}, [**52**]{}, 492 (1995). A. Saul and K. Showalter, in [*Oscillations and Traveling Waves in Chemical Systems*]{}, edited by R. J. Field and M. Burger (Wiley, New York, 1985), p. 419. J. A. Pojman, I. R. Epstein, T. J. McManus and K. Showalter, [*J. Phys. Chem.*]{}, [**95**]{}, 1299 (1991). B. F. Edwards, J. W. Wilder and K. Showalter,[*Phys. Rev. A*]{}, [**43**]{}, 749 (1991). D. A. Vasquez, B. F. Edwards and J. W. Wilder, [*Phys. Rev. A*]{}, [**43**]{}, 6694 (1991), D. A. Vasquez, J. W. Wilder and B. F. Edwards, [*Phys. Fluids A*]{}, [**4**]{}, 2410 (1992), J. W. Wilder, D. A. Vasquez and B. F. Edwards, [*Phys. Rev. E*]{}, [**47**]{}, 3761 (1993). J. W. Wilder, B. F. Edwards and D. A. Vasquez, [*Phys. Rev. A*]{}, [**45**]{}, 2320 (1992). J. P. Keener and J. J. Tyson, [*Physica D*]{}, [**32**]{}, 307 (1988). M. C. Cross and P. C. Hohenberg, [*Rev. Mod. Phys.*]{}, [**65**]{}, 851 (1993). M. Gorman, M. el-Hamdi, and K. A. Robbins, [*Combust. Sci. and Tech.*]{}, [**98**]{}, 37 (1994), G. Sivashinsky, [*Annu. Rev. Fluid Mech.*]{}, [**15**]{}, 179 (1983). J. Huang and B. F. Edwards, private communication and unpublished. 0.1in 0.1in 0.1in

Introduction ============ For cultural anthropologists, understanding fads, trends, or, generally, cultural similarity, essentially comes to explaining “the capacity of some representations to propagate until becoming precisely cultural, that is, revealing the reasons of their contagiosity” [@lenc:cult]. This type of research programme admittedly assumes the possibility of, on one hand, describing representations in a consistent manner, and, on the other hand, apprehending processes of social mediation. Defining consistent cultural items is indeed crucial to describe adoption of similar ideas, behaviors, opinions, topics, etc. — the literature proposes here a large variety of concepts, such as using same bags of terms, having identical opinion vectors, duplicating references (for instance to digital content such as online video or news articles, tagged by the same URL) or, more loosely, being “infected” by spreading “memes”. Second, describing social mediation requires to understand jointly how some types of social network configurations and some types of interactions may or may not favor the transmission, reproduction or adoption of behaviors, ideas, etc. Again, a vast amount of research has been concerned with normative models or descriptive protocols aimed at understanding which kind of individuals were more or less likely to pass on some pieces of information, and which type of network positions could favor the diffusion of some items. By relying on large-scale datasets on which individuals talk about what and when, specifically in online communities, social computing has recently contributed to this broad research programme by intensively developing two pragmatic streams of study: detection of “topics”, and characterization of “informational cascades”. Studies focused on topic detection explore bursts and regularities of behavior or term use [e.g., @klei:burs], sometimes in order to infer trends in the general population [@gins:dete; @asur:pred]. In all these studies, cultural representations are assumed to be extremely atomic, based on a single behavior (a vote), item (a reference, a URL), apprehending cultural contagion pretty much similarly to disease contagion — to the notable exception of [@lesk:meme] who gather similar sentences into clusters of quotes, getting closer to the polymorphism of cultural representations emphasized by anthropologists. On the other hand, studies on informational cascades currently adopt a structural stance, migrating from the “two-step-model” to more recent arguments underlining the importance of more horizontal, less hierarchical patterns [@watt:infl; @Cha:2010p2742]. Importantly, in this persective, information flows and diffusion paths are characterized along a given social network, available *a priori*. In many cases however, and certainly in blogs in particular, much of the information regarding the whole underlying interaction infrastructure is simply missing (be it in terms of news media readership, email exchanges and broadly any type of non-blog-based online conversation, phone calls, etc.). In this paper, we aim at bridging these rather separate streams by adopting (i) a looser view on representations, as stories or cultural attractors [@sper:expl; @sper:mode] rather than atomic items and, (ii) by considering information sources, in our case bloggers, as sensors in a social system – in particular as representatives of topics discussed in the society – so as to suggest possible/implicit information diffusion flows or, at least, precedence relationships. As an aside, the current contribution also considers *observed* social networks as effects rather than just causes of information diffusion. We thus propose to identify topic classes, exhibit temporal precedence relations between sources based on *significant plausibility* for an individual to address a topic before others do, and eventually compare this structure with the partial network of interactions constituted by explicit links among bloggers. Classical authority measures are found to have only a weak correlation with our approach, which rather exhibits potential online whistleblowers. The next section presents an overview of the relevant literature, while Sec. \[sec3\] details the empirical protocol used to identify topics. Sec. \[sec4\] then describes our approach to compute probable precedence relationships; results are discussed and reframed in Sec. \[sec5\]. Related work {#sec:related} ============ Temporal detection of topics/bursts. ------------------------------------ Topic characterization from (online) text corpora generally relies on *terms*, *n-grams* ( a basic linguistic unit of $n$ terms) or sentence segments. Once basic text units have been defined and extracted, topics are appraised both quantitatively and temporally, essentially by describing “how much on which period of time they are being discussed”. This led to distinguishing bursts of interest (“spikes”) [@klei:burs], as opposed to continuous discussions (“chatters”) around topics [@gruh:info]. Models of the temporal [@Balog:2006p2268] or spatial [@lloy:news] regularities in the usage of topics have been subsequently developed, up to infering and predicting accurate information regarding the whole population behavior [@gins:dete; @asur:pred]. Another stream of research has focused on improving the qualification of topics: for instance, by detecting whether issues are addressed in a positive light or not [the so-called field of “sentiment analysis”, see @Mishne:2006p2282 among others]; or, closer to our issues, by managing to group portions of text into classes of similar content [@lesk:meme] — thereby implicitly addressing one common critique among social scientists regarding the atomism of “memes” as cultural items. Precedence and influence ------------------------ Empirical studies of influence generally rely on interaction networks, using relational information to characterize contagion paths, and following a long tradition in mathematical sociology of social network-based models of information diffusion. As regards blogspace in particular, after initial descriptions of the underlying social network structure [e.g. @kuma:burs who also discuss bursty behavior in link creation], [@Leskovec:sdm] has been one of the first studies to specifically focus on the structure of link cascades. In a previous study, [@coin:soci] describe more precisely local influence patterns such as the relationship between holistic patterns and the weakness of links, in Granovetter’s sense. [@Java:2006p1951], on the other hand, use various social network structures to show that possible influence of a given blog is best described by strictly structural page-rank-style measures. Since influence is obviously related to precedence relationships, several papers focus rather on temporal behavioral precedence. For instance, the authors of [@koss:stru] exhibit explicit temporal dependencies on a email transmission network by characterizing possible shortcuts in information paths, because a dyad (A,B) could communicate less quickly than (A,C) and (C,B) separately do. In terms of intertwining social network structure and precedence/influence, the relationship between topology and precursors or laggards had also been explored in [@vale:soci], but with the assumption that the social network is known a priori, and by monitoring the adoption of a unique yes-or-no behavior. As said before, it is likely that a lot of information about the social structure is missing in most of the above studies, which consider the (given) social network as the substrate of information propagation. By assuming that the social structure describes only a non-significant fraction of all possible interaction links and contagion paths in the context of (for instance) political discussions, we basically wish to suggest that, here, the social network could just be a secondary material in the study of contagion. Some studies do exactly so and exhibit influence relationships from usage information only: for instance in [@zhou:topi] a Markov Chain Model is used to characterize which topics are most likely to transition into others, using data extracted from scientific bibliographic databases. Back to blogs, “probable” content diffusion paths could be exhibited in [@adar:impl] by using classifiers based upon blog features: for instance, having similar citing and content posting patterns; however, the analysis does not seem to make use of topic dynamics *per se*. Another reference [@Java:2006p1948] introduces an analysis which integrates more semantics, essentially in order to design automatic feed recommenders — which appears nonetheless to be still based on structural features (in-degree statistics) even if a filter is applied over general topics (politics vs. IT, etc.). On the whole, and in the context of partial social network information, the issue of the detection of implicit, non-structural influence flows using *temporal* precedence in addressing topics remains a pending question.

--- abstract: 'We address the uniqueness of the minimal couplings between higher-spin fields and gravity. These couplings are cubic vertices built from gauge non-invariant connections that induce non-abelian deformations of the gauge algebra. We show that Fradkin-Vasiliev’s cubic $2-s-s$ vertex, which contains up to $2s-2$ derivatives dressed by a cosmological constant $\L$, has a limit where: [(i)]{} $\L\rightarrow 0$; [(ii)]{} the spin-2 Weyl tensor scales *non-uniformly* with $s$; and [(iii)]{} all lower-derivative couplings are scaled away. For $s=3$ the limit yields the unique non-abelian spin $2-3-3$ vertex found recently by two of the authors, thereby proving the *uniqueness* of the corresponding FV vertex. We extend the analysis to $s=4$ and a class of spin $1-s-s$ vertices. The non-universality of the flat limit high-lightens not only the problematic aspects of higher-spin interactions with $\L=0$ but also the strongly coupled nature of the derivative expansion of the fully nonlinear higher-spin field equations with $\L\neq 0$, wherein the standard minimal couplings mediated via the Lorentz connection are *subleading* at energy scales $\sqrt{|\L|}<\!\!\!< E<\!\!\!< M_{\rm p}$. Finally, combining our results with those obtained by Metsaev, we give the complete list of *all* the manifestly covariant cubic couplings of the form $1-s-s\,$ and $2-s-s\,$, in Minkowski background.' --- [Nicolas Boulanger$^{*,}$[^1], Serge Leclercq$^{\dagger,}$[^2] and Per Sundell$^{*,}$[^3] ]{} Introduction and overview ========================= No-go and yes-go results for $\L=0$ ----------------------------------- From a general perspective it is a remarkable fact that the full gravitational couplings of lower-spin fields involve at most two derivatives in the Lagrangian. For spin $s\leqslant 1$ the *standard* covariantization scheme, wherein $\partial \rightarrow \nabla = \partial + \o \,$ with $\o$ being a torsion-constrained Lorentz connection, induces the “minimal coupling” $\int d^Dx\, h_{\mu\nu} T^{\mu\nu}$ where $T^{\mu\nu}$ is the Belifante-Rosenfeld stress-tensor which is quadratic and contains up to two derivatives. Actually, for scalars, Maxwell fields and other Lorentz-invariant differential forms, the Lorentz covariantization is trivial and the coupling therefore involves no derivatives of the metric. It is also remarkable that the non-abelian cubic self-coupling of a spin-2 field contains only two derivatives. Turning to gauge fields with $s>2$ and considering $2-s-s\,$ couplings in an expansion around flat spacetime, the standard scheme breaks down as has been known for a long time [@Aragone:1979bm; @Berends:1979wu; @Aragone:1981yn]. These no-go results have recently been strengthened in [@Metsaev:2005ar; @Metsaev:2007rn] following a light-cone method and in [@Porrati:2008rm] with $S$-matrix tools. More interestingly, in the works [@Metsaev:2005ar; @Boulanger:2006gr; @Metsaev:2007rn] some *yes-go* results have been obtained. In the specific case of $s=3\,$, the work [@Boulanger:2006gr] provides a manifestly covariant *non-standard* four-derivative vertex associated with a *nonabelian* deformation of the gauge algebra. These yes-go results suggest a class of minimal nonabelian[^4] non-standard vertices containing $2s-2$ derivatives. We wish to emphasize that the existence of cubic couplings containing $2s-2$ derivatives was explicitly shown in [@Metsaev:2005ar; @Metsaev:2007rn] although the light-cone gauge method used therein does not exhibit the nature of the gauge algebra and does not readily allow for the explicit construction of the corresponding covariant vertices. The results are nonetheless remarkable in that they show the existence of only a few non-trivial cubic vertices of the general form $s-s'-s''\,$ for massive and massless fields (bosonic and fermionic) in flat space of arbitrary dimension $D>3\,$. In the case of integer spins, the possible vertices have $s+s'+s''-p$ derivatives where $p=0,2,\dots,2\min(s,s',s'')$. In the specific massless $2-3-3\,$ case, using the BRST-BV cohomological methods of [@Barnich:1993vg; @Henneaux:1997bm], the vertex of [@Boulanger:2006gr] was shown to be *unique* among the class of vertices that: [(i)]{} contain a *finite* number of derivatives; (ii) manifestly preserve Poincaré invariance and [(iii)]{} induce a *nonabelian* deformation of the gauge algebra. This uniqueness result relies on the fact that other candidate nonabelian deformations cannot be “integrated” cohomologically to gauge transformations and vertices. We have managed to push the uniqueness analysis to the case of $s=4$ and the unique $2-4-4\;$ nonabelian vertex is presented in Section \[sec:244\] together with its corresponding gauge algebra and transformations. We also extend the results of [@Boulanger:2006gr] with the cohomological proof in Section \[sec:2ss\] that the standard two-derivative minimal couplings $2-s-s\,$ are inconsistent, thereby providing an alternative proof for the results recently obtained in [@Metsaev:2005ar; @Metsaev:2007rn] following light-cone methods and in [@Porrati:2008rm] following $S$-matrix methods. In the same section \[sec:2ss\], combining the cohomological approach with the light-cone results of Metsaev [@Metsaev:2005ar; @Metsaev:2007rn], we show that there exists *only one* nonabelian $2-s-s\,$ coupling, which contains $2s-2$ derivatives and must be the flat limit of the well-known nonabelian Fradkin–Vasiliev vertex [@Fradkin:1986qy; @Fradkin:1987ks] in $AdS\,$, as we verify explicitly for $s=3$. There also exist two abelian covariant $2-s-s\,$ vertices containing $2s+2$ and $2s$ derivatives. Their existence was first found in [@Metsaev:2005ar], and we exhibit them here explicitly in their covariant form. The $(2s+2)$-derivative vertex is of the Born-Infeld type, whereas the $2s$-derivative vertex exists only for $D\geqslant 5$ and is gauge invariant up to a total derivative. These three vertices, with $2s-2$, $2s$ and $2s+2$ derivatives, thus exhaust the possibilities of manifestly Lorentz-covariant $2-s-s\,$ couplings in flat space. We begin in Section \[sec:1ss\] by examining the simpler case of $1-s-s\,$ vertices. We build explicitly the unique, nonabelian $1-s-s\,$ coupling, which has $2s-1$ derivatives, together with the only abelian $1-s-s\,$ vertex, which as $2s+1$ derivatives, thereby completing the list of all possible nontrivial, manifestly covariant, $1-s-s\,$ couplings. Again, by the uniqueness of the nonabelian vertex, we know that it is the flat limit of the corresponding $AdS$ Fradkin–Vasiliev (FV) vertex [@Fradkin:1986qy; @Fradkin:1987ks]. The Fradkin–Vasiliev cancelation mechanism for $\L\neq 0$ --------------------------------------------------------- Under the assumptions that the cosmological constant vanishes and that the Lagrangian contains at most two derivatives, the standard covariantization of Fronsdal’s action leads to an inconsistent cubic action of the form[^5] $$\begin{aligned} S^{\L=0}_{2ss}[g,\phi] &=& {1\over \ell_p^{D-2}}\int \left(R+G+\frac12 W_{\m\n\r\s}\b_{(2)}^{\m\n,\r\s}(\phi^{\otimes 2})\right)\ ,\end{aligned}$$ where $\ell_p$ is the Planck length, $\int=\int d^Dx \sqrt{-g}$, the spin-$s$ kinetic term[^6] $G=\frac12 \phi^{\mu(s)}G_{\mu(s)}(g;\nabla\nabla\phi)$ with the Einstein-like self-adjoint operator[^7] $$\begin{aligned} G_{\mu(s)}&=&F_{\mu(s)}-\frac{s(s-1)}4 g_{\m(2)} F'_{\m(s-2)}\ ,\\[5pt] F_{\m(s)}&=& \nabla^2\phi_{\m(s)}-s\nabla_{\m_1}\nabla\cdot \phi_{\m(s-1)}+\frac{s(s-1)}2\nabla_{\m_1} \nabla_{\m_2}\phi'_{\m(s-2)}\ ,\end{aligned}$$ the covariantized Fronsdal field strength. The symbol $\b_{(2)}$ denotes a dimensionless symmetric bilinear form, $W_{\m\n\r\s}$ is the spin-2 Weyl tensor, and $(\ell_p)^2W$ and $\phi$ are assumed to be weak fields. A quantity ${\cal O}$ has a regular weak-field expansion if ${\cal O}=\sum_{n=n({\cal O})}^\infty \stackrel{(n)}{{\cal O}}$ where $\stackrel{(n)}{{\cal O}}$ scales like $g^{n}$ if the weak fields are rescaled by a constant factor $g$, and we shall refer to $\stackrel{(n)}{{\cal O}}$ as being of $n$th in weak fields, or equivalently, as being of order $n-n({\cal O})$ in the $g$ expansion. Under the spin-$s$ gauge transformation $\d_\e\phi_{\mu(s)} =s\,\nabla_{\m_1}\e_{\mu(s-1)}+R_{\m(s)}[g_{\a\b},\phi, \e]$ and $\d_\e g_{\m\n}=R_{\m\n}[g_{\a\b},\phi,\e]$, where $\e$ is a weak traceless parameter and $R_{\m(s)}$ and $R_{\m(2)}$ are quadratic in weak fields, the variation of the action picks up the first-order contribution $$\begin{aligned} \delta_\e \int G&=& \int W^{\m\n\r\s}{\cal A}_{\m\n,\r\s}(g_{\a\b}; \nabla\phi\otimes\e)\ ,\end{aligned}$$ where the bilinear form $$\begin{aligned} {\cal A}_{\m\n,\r\s}&=& 2s(s-1){\bf P}^W \Big[\nabla_\m \phi_{\n\s\t(s-2)}\e_{\r}{}^{\t(s-2)} \nonumber\\[5pt]&& +(s-2)\big(\nabla_\s\phi'_{\n\t(s-3)}-\frac12 \, \nabla\cdot \phi_{\n\s\t(s-3)}+\frac{(s-3)}4 \nabla_{\t_1}\phi'_{\n\s\t(s-4)}\big)\e_{\m\r}{}^{\t(s-3)}\Big]\ ,\end{aligned}$$ that has been shown to be anomalous for $s=3$ [@Aragone:1981yn] (for recent re-analysis see [@Boulanger:2006gr] and also [@Porrati:2008rm] for an $S$-matrix argument) in the sense that it cannot be canceled by any choice of $\b_{(2)}$ nor by abandoning the assumption that the Lagrangian contains at most two derivatives. However, as first realized by Fradkin and Vasiliev [@Fradkin:1986qy], if both $\L\neq 0$ and higher-derivative terms are added to the cubic part of the action, the analogous obstruction can be bypassed. In the weak-field expansion the resulting *minimal cubic action* reads $$\begin{aligned} S^{\L}_{2ss}[g,\phi] &=& \frac1{\ell_p^{D-2}}\int \left(R(g)-\L+G_\L\right)+ \sum_{ \substack{n=2 \\ {n~even}}}^{n_{\rm min}(s)} \frac1{\ell_p^{D-2}}\int V^{(n)}_\L(2,s,s)\ ,\\[5pt] V^{(n)}_\L(2,s,s)&=& \frac1{2\l^{n-2}} \sum_{p+q=n-2} W_{\m\n\r\s}\b^{\m\n,\r\s}_{(n);p,q}(\nabla^p\phi\otimes \nabla^q\phi)\ ,\end{aligned}$$ with $\l^2\equiv -\frac{\L}{(D-1)(D-2)}$ and $G_\L$ is the Einstein-like kinetic term built from $F_\L=F-\frac12 \l^2 M_s^2(\phi^{\otimes 2})$ (see (\[Fs\]) below). The spin-$s$ gauge invariance up to first order uses that at zeroth order $$\begin{aligned} &&[\nabla_\mu,\nabla_\nu] V_\r \ =\ R_{\mu\nu\r}{}^\s V_\s\ \approx\ 2\l^2g_{\r[\n} V_{\m]}+W_{\m\n\r}{}^\s V_\s\ ,\qquad \nabla^\mu W_{\m\n,\r\s}\ \approx 0\ ,\label{nablanabla}\\[5pt] &&R_{\m\n}-\frac12(R-\L) g_{\m\nu}\ \approx\ 0\ ,\qquad F_{\m(s)}- \frac12\, \l^2 M_s^2(\phi^{\otimes 2})\ \approx\ 0\ ,\end{aligned}$$ where $\approx$ is used for equalities that hold on-shell. At zeroth order, the invariance requires the critical mass matrix [@Fronsdal:1978vb] $$\begin{aligned} M_s^2(\phi^{\otimes 2})&=& m_s^2\phi^2+m^{\prime 2}_s\phi^{\prime 2}\ ,\quad m_s^2\ =\ s^2+(D-6)s-2D+6\ ,\quad m^{\prime 2}_s\ =\ s(s-1)\ .\end{aligned}$$ At first order, the classical anomaly $\int W{\cal A}$, which is independent of $\l$, is accompanied by two types of $\l$-independent counter terms, namely $\d_\e \int V^{(2)}_\L$ plus the contributions to $\delta_\e \int V^{(4)}_\L$ from the constant-curvature part of $[\nabla,\nabla]$, that can be arranged to cancel the anomaly at order $\l^0$. At order $\l^{-2}$, the remaining terms in $\delta_\e \int V^{(4)}_\L$ can be canceled against order $\l^{-2}$ contributions from $\delta_\e \int V^{(6)}_\L$ and so on, until the procedure terminates at the *top vertex* $V^{{\rm top}}_\L(2,s,s)=V^{(n_{\rm min}(s))}_\L$ that: [(i)]{} is *weakly gauge invariant* up to total derivatives and terms that are of lower order in $\l$; and [(ii)]{} contains a total number of derivatives given by $$\begin{aligned} n_{\rm min}(s)=2s-2\ .\end{aligned}$$ Counting numbers of derivatives, there is a *gap* between the top vertex and the *tail* of Born-Infeld-like non-minimal cubic vertices, which is *a priori* of the form $$\begin{aligned} S^{\rm nm}_{2ss;\L}&=& \sum_{n=0}^\infty \frac1{(\ell_p)^{D-2}2\l^{2(n+s)}}\sum_{p+q=2n}\int W_{\m\n\r\s} \c^{\m\b,\r\s}_{(n);p,q}(\nabla^pC\otimes \nabla^q C)\ ,\label{BItail}\end{aligned}$$ where $C_{\m(s),\n(s)}$ is the linearized spin-$s$ Weyl tensor and $\c_{(n);p,q}$ are dimensionless bilinear forms. Adapting the flat-space result of [@Metsaev:2005ar] to constantly curved backgrounds suggests that, if the $\c_{(n);p,q}$ fall off with $n$ sufficiently fast, then the couplings with $n\geqslant 1$ can be removed by a suitable, possibly non-local, field redefinition. More generally, turning to higher orders in the weak-field expansion, one may adopt the *canonical* frame of standard fields that by definition minimizes the maximal numbers of derivatives at each order. The *existence* of at least one cancelation procedure has sofar been shown in the literature only for $D=4,5$ [@Fradkin:1986qy; @Vasiliev:2001wa; @Alkalaev:2002rq], following the existence of a more general minimal cubic action given within the frame-like formulation based on a nonabelian higher-spin Lie algebra extension $\mathfrak{h}$ of $\mathfrak{so}(D+1;\mathbb{C})$. The 4D action is a natural generalization of the MacDowell-Mansouri action for $\L$-gravity. It is given by a four-form Lagrangian based on a bilinear form $<\cdot,\cdot>_{\mathfrak{h}}$ such that the resulting action: [(i)]{} contains at most $2$ derivatives at second order in weak fields; [(ii)]{} propagates symmetric rank-$s$ tensor gauge fields with $s\geqslant 1$ and critical mass; [(iii)]{} contains nonabelian $V^{(n)}_{\L}(s,s',s'')$ vertices with $s,s',s''\geqslant 1$ and $n\leqslant n_{\rm min}(s,s',s'')$. The 5D action shares the same basic features [@Vasiliev:2001wa; @Alkalaev:2002rq]. The existence issue in $D>5$ is open at present though all indications sofar hint at that the lower-dimensional cases do actually have a generalization to arbitrary $D$. Recovering the metric-like FV 2-3-3 vertex {#FVsection} ------------------------------------------ Apparently Fradkin and Vasiliev first found the gravitational coupling of the spin-3 field using the metric-like formalism without publishing their result (see [@Vasiliev:2001ur] for an account). Later they obtained and published their (by now famous) result in the frame-like formalism in the general $2-s-s$ case in $D=4$ [@Fradkin:1986qy]. For the purpose of discussing the uniqueness of their result and its extension to $D$ dimensions, we need the explicit form of the $D$-dimensional $2-3-3\,$  FV vertex. To this end, we work within the metric-like formulation and start from the free Lagrangian ${\cal L}_{2}+{\cal L}_{3}\,$ where Fronsdal’s Lagrangian for a symmetric rank-$s$ tensor gauge field in $AdS_D$ reads [@Buchbinder:2001bs] $$\begin{aligned} -\frac{\cl_{s}}{\sqrt{-\bar g}} &=&\frac{1}{2}\, \overline\N_\m \phi_{\a_1...\a_s}\overline\N^\m \phi^{\a_1...\a_s} - \frac{1}{2}\,s\overline\N^\m \phi_{\m\a_1...\a_{s-1}}\overline\N_\n \phi^{\n\a_1...\a_{s-1}}\nonumber\\&& + \frac{1}{2}\,s(s-1)\overline\N_\a \phi'_{\b_1...\b_{s-2}}\overline\N_\m \phi^{\m\a\b_1...\b_{s-2}} - \frac{1}{4}\,s(s-1)\overline\N_\m \phi'_{\a_1...\a_{s-2}}\overline\N^\m \phi'^{\a_1...\a_{s-2}}\nonumber\\&& -\frac{1}{8}s(s-1)(s-2)\overline\N^\m \phi'_{\m\a_1...\a_{s-3}}\overline\N_\n \phi'^{\n\a_1...\a_{s-3}} \nonumber\\&& +\frac12 \,\l^2\left[s^2+(D-6)s-2D+6\right]\phi_{\a_1...\a_s} \phi^{\a_1...\a_s}\nonumber\\&&-\frac14 \l^2 s(s-1) \left[s^2+(D-4)s-D+1\right]\phi'_{\a_1...\a_{s-2}}\phi'^{\a_1...\a_{s-2}}\ , \label{Fs}\end{aligned}$$ given that $\overline R_{\a\b\c\d}=-\l^2(\bar g_{\a\g}\bar g_{\b\d}-\bar g_{\b\g}\bar g_{\a\d})$. We find, using the Mathematica package Ricci [@Lee], that the 2-3-3 FV vertex is given by[^8] $$\begin{aligned} -{\stackrel{(3)}{\cl}_{FV}\over \sqrt{-\bar g}}&\approx&-\frac{11}{2}~ w_{\a\b\c\d}~\phi^{\a\c}_{\phantom{\a\c}\m}\phi^{\b\d\m} + \frac{1}{(D-1)\l^2} ~w_{\a\b\c\d}~\Big[2\ \phi'_\m \overline \N^{(\b}\overline \N^{\d)}\phi^{\a\c\m} + \phi^{\a\c}_{\phantom{\a\c}\m}\overline \N^{(\d}\overline \N^{\m)}\phi'^{\b}\nonumber\\[5pt] &&- 3\ \phi'^{\a}\overline \N^{(\d}\overline \N^{\m)}\phi^{\b\c\phantom{\m}}_{\phantom{\b\c}\m} + 2\ \phi^{\a}_{\phantom{\a}\m\n}\overline \N^{(\d}\overline \N^{\n)}\phi^{\b\g\m} + \overline \N_\m\phi^{\a\c\m}_{\phantom{\a\c\m}}\overline \N_\n\phi^{\b\d\n}_{\phantom{\b\d\n}} - \phi^{\a\c\m}\overline \N_{(\m}\overline \N_{\n)}\phi^{\b\d\n}_{\phantom{\b\d\n}}\nonumber\\[5pt]&& - 2\ \overline \N^{(\m}\phi^{\n)\a\c}\overline \N^{\phantom{\m}}_\m\phi^{\b\d}_{\phantom{\b\d}\n} - 2\ \phi^{\a\c}_{\phantom{\a\c}\m}\overline \N^{(\d}\overline \N^{\n)} \phi^{\b\m\phantom{\n}}_{\phantom{\b\m}\n} + \phi'^{\a}\overline \N^{(\b}\overline \N^{\d)}\phi'^{\g} - \phi^{\a}_{\phantom{\a}\m\n}\overline \N^{(\b}\overline \N^{\d)}\phi^{\c\m\n}\Big]\;.\qquad \label{FVvertex}\end{aligned}$$ The first term corrects the obstruction to the standard minimal scheme at the expense of introducing a new one that can be removed, however, by adding the above particular combination of two-derivative terms (involving only a subset of all possible tensorial structures as expected from the frame- like formulation). We stress again that the top vertex does not introduce any further obstructions, and that the vertex indeed exhibits the gap. Non-Uniform $\L\rightarrow 0$ Limits ------------------------------------ Since for given $s$ the derivative expansion of the minimal $2-s-s$ coupling terminates at the top vertex $V^{(2s-2)}_\L(2,s,s)$, the cubic action $S^{\L}_{2ss}$ admits the *scaling limit* $$\begin{aligned} \l&=&\e (\ell_p)^{-1} \ ,\quad W\ =\ \e^{2s-4}\widetilde{W}\ ,\qquad \e\ \rightarrow\ 0\ ,\end{aligned}$$ with evanescent piece $\widetilde W_{\m\n\r\s}$ held fixed, so that $\widetilde W_{\m\n\r\s}$ can be replaced by the linearized Weyl tensor $\widetilde w_{\m\n\r\s}$ in the cubic vertices, resulting in the action $$\begin{aligned} \widetilde S^{\L=0}_{2ss}[g,\phi]&=&{1\over \ell_p^{D-2}}\int d^Dx \sqrt{-g} \left(R(g)+G_0+\widetilde V^{(2s-2)}_0(2,s,s)\right)\ ,\\[5pt] \widetilde V^{(2s-2)}_0(2,s,s)&=&\frac12 \sum_{p+q=2s-4}\int \widetilde w_{\m\n\r\s}\b^{\m\n,\r\s}_{(n);p,q}(\nabla^p\phi\otimes \nabla^q\phi)\ ,\end{aligned}$$ that is faithful up to cubic order in weak graviton and spin-s fields, and $G_0$ contains the connection $\nabla_0$ obeying the flatness condition $[\nabla_0,\nabla_0]=0$. Alternatively, one may first perturbatively expand the FV action around AdS and then take the $\L\rightarrow 0$ limit as follows: $$\begin{aligned} \l&=&\e\widetilde\ell_p^{-1}\ ,\qquad \ell_p\ =\ \e^{\D_p}\widetilde \ell_p\ , \\[5pt] h_{\m\n}&=&\e^{\D_h}\widetilde h_{\m\n}\ ,\qquad\phi_{\m\n\r}\ =\ \e^{\D_\phi}\widetilde\phi_{\m\n\r}\ ,\qquad \e\ \rightarrow\ 0\ ,\end{aligned}$$ with $\widetilde\ell_p$, $\widetilde h$ and $\widetilde \phi$ kept fixed and $\D_h=\D_\phi=2(s-2)$ and $\D_p=\frac{4(s-2)}{D-2}$. The resulting flat-space 2-3-3 vertex reads $$\begin{aligned} -\stackrel{(3)}{\cl}&=&\frac{1} {D-1}\tilde{w}_{\a\b\c\d}\Big[2\ \widetilde\phi'_\m \6^{\b}\6^{\d}\widetilde\phi^{\a\c\m} + \widetilde\phi^{\a\c}_{\phantom{\a\c}\m}\6^{\d}\6^{\m}\widetilde\phi'^{\b} - 3\ \widetilde\phi'^{\a}\6^{\d}\6^{\m}\widetilde \phi^{\b\c\phantom{\m}}_{\phantom{\b\c}\m} \nonumber\\[5pt]&& + 2\ \widetilde\phi^{\a}_{\phantom{\a}\m\n}\6^{(\d}\6^{\n)}\widetilde \phi^{\b\g\m}+ \6_\m\widetilde\phi^{\a\c\m}_{\phantom{\a\c\m}} \6_\n\widetilde\phi^{\b\d\n}_{\phantom{\b\d\n}} - \widetilde\phi^{\a\c\m}\6_{\m}\6_{\n}\widetilde \phi^{\b\d\n}_{\phantom{\b\d\n}}\nonumber\\[5pt]&& - 2\ \6^{(\m}\widetilde\phi^{\n)\a\c}\6^{\phantom{\m}}_\m \widetilde\phi^{\b\d}_{\phantom{\b\d}\n}- 2\ \widetilde\phi^{\a\c}_{\phantom{\a\c}\m}\6^{\d}\6^{\n} \widetilde\phi^{\b\m\phantom{\n}}_{\phantom{\b\m}\n} + \widetilde\phi'^{\a}\6^{\b}\6^{\d}\widetilde\phi'^{\g} - \widetilde\phi^{\a}_{\phantom{\a}\m\n}\6^{\b}\6^{\d} \widetilde\phi^{\c\m\n}\Big ]\end{aligned}$$ where $\tilde{w}_{\a\b\c\d}=\tilde{K}_{\a\b\c\d}-\frac{2}{D-2}\, (\eta_{\a[\g}\tilde{K}_{\d]\b}-\eta_{\b[\g}\tilde{K}_{\d]\a})+ \frac{2}{(D-1)(D-2)}\,\eta_{\a[\g}\eta_{\d]\b}\tilde{K}$ with $\tilde{K}_{\a\b\c\d}=-\6_{\g}\6_{[\a}\tilde{h}_{\b]\d} +\6_{\d}\6_{[\a}\tilde{h}_{\b]\d}$. As discussed above, the top 2-3-3 vertex must be equivalent modulo total derivatives and linearized equations of motion to the nonabelian 2-3-3 vertex presented in Appendix B of [@Boulanger:2006gr] which we have verified explicitly[^9]. Uniqueness of the $2-3-3\,$ FV vertex ------------------------------------- The uniqueness of the FV cancelation procedure in the case of spin $s=3$ can be now be established for any $D$ as follows. We obtained the $AdS_D$ covariantization $$S^{\L}[h,\phi]=S^{\Lambda}_{free} + g\,S^{\Lambda}_{cubic}$$ of the nonabelian flat spacetime action $$S^{\L=0}[h,\phi]=S^{Flat}_{free} + g\,S^{Flat}_{cubic}$$ obtained in [@Boulanger:2006gr], with $S^{\L=0}_{cubic}= \int d^Dx\, V^{(4)}_0(2,3,3)\,$ and $g$ the deformation parameter. The cubic part $S^{\Lambda}_{cubic}=\int d^Dx \sqrt{-g}\; V_{\Lambda}(2,3,3)$ possesses an expansion in powers of the $AdS$ radius, where the contribution to $V_{\Lambda}(2,3,3)$ with the maximum number of derivatives is called $V^{top}_{\Lambda}(2,3,3)\,$. We recall that, using the power of the BRST-BV cohomological method [@Barnich:1993vg], the first-order deformation $S^{\L=0}[h,\phi]$ has been proved [@Boulanger:2006gr] to be *unique* under the sole assumptions of - Locality, - Manifest Poincaré invariance, - Nonabelian nature of the deformed gauge algebra. The last assumption allows the addition of Born-Infeld-like cubic vertices of the form $V^{BI}_0(2,3,3)=C(h)C(\phi)C(\phi)\,$ where $C(h)$ and $C(\phi)$ denote linearized Weyl tensors and we note that $C(\phi)$ contains $3$ derivatives [@Damour:1987vm]. Such vertices are strictly gauge invariant and do not deform the gauge algebra nor the transformations. We also disregard deformations of the transformations that do not induce nonabelian gauge algebras, as is the case for such deformations involving the curvature tensors. In the following, when we refer to a deformation as unique it should be understood to be up to the addition of other deformations that do not deform the gauge algebra. The uniqueness of $S^{\L=0}[h,\phi]$ is instrumental in showing the uniqueness of its $AdS_D$ completion $S^{\L}[h,\phi]\,$, due to the linearity of the perturbative deformation scheme and the smoothness of the flat limit at the level of cubic actions. The proof goes as follows. First suppose that there exists another action $S^{\prime\L}[h,\phi]=S^{\Lambda}_{free} + g\,S^{\prime\Lambda}_{cubic}$ that admits a nonabelian gauge algebra and whose top vertex $V_{\Lambda}^{\prime {\rm top}}(2,3,3)$ involves $n_{\rm top}$ derivatives with $n_{\rm top}\neq 4$. Then, this action would scale to a nonabelian flat-space action whose cubic vertex would involve $n_{\rm top}$ derivatives. This is impossible, however, because the *only* nonabelian cubic vertex in flat space is $V^{(4)}_0(2,3,3)\,$. Secondly, suppose there exists a nonabelian action $S^{\prime\prime\L}[h,\phi]=S^{\Lambda}_{free} +g\,S^{\prime\prime\Lambda}_{cubic}$ whose top vertex contains $4$ derivatives but is otherwise different from $V_{\Lambda}^{\rm top}(2,3,3)$. Then its flat limit would yield a theory with a cubic vertex, involving 4 derivatives, but different from $V^{\rm top}_{0}(2,3,3)\,$, which is impossible due to the uniqueness of the latter deformation. Thirdly, and finally, suppose there exists a cubic action with top vertex $V^{\rm top}_{\Lambda}(2,3,3)\,$ but differing from $S^{\L}_{cubic}$ in the vertices with lesser numbers of derivatives. By the linearity of the BRST-BV deformation scheme, the difference between this coupling and $S^{\L}_{cubic}$ would lead to a nonabelian theory in $AdS$ with top vertex involving less than 4 derivatives. Its flat-space limit would therefore yield a nonabelian action whose top vertex would possess less than 4 derivatives, which is impossible due to the uniqueness of $S^{\L=0}[h,\phi]\,$. A more rigorous proof can be stated entirely in terms of master actions within the BRST-BV framework. Then all ambiguities resulting from trivial field and gauge parameter redefintions are automatically dealt with cohomologically. Moreover, the possibility of scaling away the nonabelianess while at the same time retaining the vertex is ruled out[^10]. On Separation of Scales in Higher-Spin Gauge Theory --------------------------------------------------- Thanks to Vasiliev’s oscillator constructions [@Vasiliev:1990en; @Vasiliev:2003ev] it has been established that fully nonlinear nonabelian higher-spin gauge field equations exist in arbitrary dimensions in the case of symmetric rank-$s$ tensor gauge fields. Compared to the cubic actions, the full equations exhibit two additional essential features: [(i)]{} a precise spectrum $\mathfrak{D}$ given by an *infinite tower* of $\mathfrak{so}(D+1;\mathbb{C})$ representations forming a unitary representation of (a real form of) the higher-spin algebra $\mathfrak{h}$ (see *e.g.* [@Konstein:1989ij]); [(ii)]{} nonlocal, potentially *infinite*, Born-Infeld tails. The closed form of Vasiliev’s equations requires the *unfolded formulation* of field theory whereby [@D'Auria:1982my; @D'Auria:1982nx; @van; @Nieuwenhuizen:1982zf; @Vasiliev:1988xc; @Vasiliev:1988sa; @Vasiliev:2001ur; @Skvortsov:2008vs]: [(i)]{} standard physical (gauge) fields are replaced as independent action variables by differential forms taking their values in $\mathfrak{so}(D+1;\mathbb{C})$ modules that are finite-dimensional for $p$-forms with $p>0$ and infinite-dimensional for zero-forms; [(ii)]{} the resulting kinetic terms feature only the exterior derivative $d$; [(iii)]{} the standard interactions are mapped to non-linear structure functions appearing in the unfolded first-order equations obeying *algebraic conditions* assuring $d{}^2=0$. Thus the on-shell content of a spin-$s$ gauge field $\phi_s$ is mapped into an infinite-dimensional collection of zero-forms carrying traceless Lorentz indices filling out the covariant Taylor expansion on-shell of the corresponding Weyl tensor $C(\phi_s)$. Letting $X^\a$ denote the complete unfolded field content, the unfolded equations take the form $dX^\a+f^\a(X^\b)=0$ where $f^\a$ are written entirely using exterior algebra, and subject to the algebraic condition $f^\b{\partial^l\over\partial X^\b} f^\a=0\,$ (defining what is sometimes referred to as a free-differential algebra). The salient feature of the unfolded framework is that any consistent deformation is automatically gauge-invariant in the sense that every $p$-form with $p\geqslant 1$ is accompanied by a $(p-1)$-form gauge parameter, independently of whether the symmetry is manifestly realized or not. Vasiliev’s equations provide one solution to the on-shell deformation problem given a one-form $A$ taking its values in the algebra $\mathfrak{h}$, and a zero-form $\Phi$ containing all Weyl tensors and their on-shell derivatives, which is the unfolded counterpart of the massless representation $\mathfrak{D}$. The embedding of the canonical fields $\{g_{\m\n},\phi,\dots\}$ into $\Phi$ and $A$ requires a non-local field redefinition[^11] to microscopic counterparts $\{\widehat g_{\m\n},\widehat \phi,\dots\}$. In the microscopic frame, the standard field equations are non-canonical and actually contain infinite Born-Infeld tails already at first order in the weak-field expansion (see [@Sezgin:2002ru] for a discussion). For example, the first-order corrections to the stress tensor, defined by $\widehat R_{\m\n}-\frac12 \widehat g_{\m\n}(\widehat R-\L)=\widehat T_{\m\n}$, from a given spin $s$ arise in a derivative expansion of the form $\widehat T^{(1)}_{\mu\nu}=\sum_{n=0}^\infty \sum_{p+q=2n}\l^{-2n}\widehat T^{(n);p,q}_{\mu\nu}(\widehat \nabla^p\widehat \phi_s, \widehat \nabla^q\widehat \phi_s)$ where $\widehat \nabla^p\widehat \phi_s$ is a connection if $p<s$ and $(p-s)$ derivatives of $C(\widehat \phi_s)$ if $p\geqslant s$ (see *e.g.* [@Kristiansson:2003xx] for the case of $s=0$). As discussed below (\[BItail\]), the microscopic tails should be related to the canonical vertices via non-local, potentially divergent, field redefinitions. Thus one has the following scheme: $$\begin{aligned} \begin{array}{c}\mbox{Unfolded}\\[-3pt]\mbox{master-field}\\[-3pt] \mbox{equations}\end{array}\quad& \stackrel{\tiny\begin{array}{c}{\rm weak}\\{\rm fields}\end{array}}{\leftrightarrows} &\quad \begin{array}{c}\mbox{Standard-exotic}\\ [-3pt]\mbox{microscopic}\\[-3pt]\mbox{field equations}\end{array}\qquad \stackrel{\tiny\begin{array}{c}{\rm non-local}\\ {\rm field\, redef.}\end{array}}{\rightleftarrows}\qquad \begin{array}{c}\mbox{Standard-exotic}\\ [-3pt]\mbox{canonical}\\[-3pt]\mbox{field equations}\end{array}\end{aligned}$$ Thus, the semi-classical weak-field expansion, whether performed in the microscopic or canonical frames, leads to amplitudes depending on the following three quantities: (i) a dimensionless AdS-Planck constant $g^2\equiv (\l \ell_p)^{D-2}$ that can always be taken to obey $g <\!\!\!< 1$ and that counts the order in the perturbative weak-field expansion, where $\ell_p$ enters via the normalization of the effective standard action and we are working with dimensionless physical fields; and (ii) a massive parameter $\l$ that simultaneously (iia) sets the infrared cutoff via $\L\sim \l^2$ and critical masses $M^2 \sim\l^2$ for the dynamical fields; and (iib) dresses the derivatives in the interaction vertices thus enabling the Fradkin-Vasiliev (FV) mechanism; and [(iii)]{} the weak-field fluctuation amplitudes[^12] $|\nabla^n C(\phi)|\sim (\l \ell)^{-s-n} |\phi|$ where $\ell$ is the characteristic wavelength of the bulk fields. We stress that what makes higher-spin theory exotic is the dual purpose served by $\l$ within the FV mechanism whereby positive and negative powers of $\l$ appear in mass terms and vertices, respectively. Thus, at each order of the canonical weak-field expansion scheme, the local bulk interactions – and in particular the standard minimal gravitational two-derivative couplings – are dominated by strongly coupled top vertices going like finite positive powers of (energy scale)/(IR cutoff), *i.e.* $(\ell\l)^{-1}$. On the other hand, in the microscopic weak-field expansion scheme, each order is given by a potentially divergent Born-Infeld tail, suggesting that classical solutions as well as amplitudes should be evaluated directly within the master-field formalism which offers transparent methods based on requiring associativity of the operator algebra for setting up and assessing regularized calculational schemes. More precisely, the tails are strongly coupled for fluctuations around curved backgrounds that are close to the $AdS_D$ solution, where $\ell\l<\!\!\!< 1$, although it is in principle also possible to expand around backgrounds that are “far” from $AdS_D$, and that might bring in additional new scales altering the nature of the tails. Sticking to the first background scenario, and remaining with the microscopic frame of fields, Vasiliev’s oscillator formalism may offer a natural remedy amounting to augmenting specific classes of composite operators by associative operator products. As a result the tails, which are power-series expansions in $z=(\ell\l)^{-1}$ that define special functions in the unphysical region $|z|<\!\!\!< 1$, would be given physically meaningful continuations into the physical region $|z|>\!\!\!> 1$, leading to microscopic amplitudes ${1\over \ell_p^{D-2}}\int \widehat V_\L(s_1,\dots,s_N)\sim g^{N-2} \widehat A(s_1,\dots,s_N|\ell\l)$, where $\widehat A(s_1,\dots,s_N|\ell\l)$ are analytically continued amplitudes. If these are bounded uniformly in $\{s_i\}$ for $\ell\l<\!\!\!< 1$, then a semi-classical expansion would be possible if $g<\!\!\!< 1$. If on the other hand one redefines away the non-minimal tails, and if the higher-derivative nature of the minimal cubic vertices generalizes to $N>3$, then the remaining top vertices ${V}{}^{({\rm top})}_\L(s_1,\dots,s_n)$ would necessarily contain total numbers of derivatives $n_{\rm top}(\{s_i\})$ growing at least linearly with $\sum_i s_i$, so that the resulting canonical amplitudes $A(s_1,\dots,s_N|\ell\l;g)\sim g^{N-2}(\ell\l)^{-n_{\rm top}(\{s_i\})}$, leaving no room for a uniform semi-classical expansion. Antifield formulation {#AppBV} ===================== Definitions {#sec:Def} ----------- In this section we briefly recall the BRST deformation scheme [@Barnich:1993vg] in the case of spin-$s$ Fronsdal theory, that is irreducible and abelian. The containt of the present section is mainly based on the works [@Bekaert:2005ka; @Boulanger:2000rq; @Bekaert:2005jf]. According to the general rules of the BRST-antifield formalism, a grassmann-odd ghost is introduced, which accompanies each grassmann-even gauge parameter of the gauge theory. It possesses the same algebraic symmetries as the corresponding gauge parameter. In the cases at hand, it is symmetric and traceless in its spacetime indices. Then, to each field and ghost of the spectrum, a corresponding antifield (or antighost) is added, with the same algebraic symmetries but the opposite Grassmann parity. A $\mathbb{Z}$-grading called [*ghost number*]{} ($gh$) is associated with the BRST differential $s$, while the [*antifield number*]{} ($antigh$) of the antifield $Z^*$ associated with the field (or ghost) $Z$ is given by $antigh(Z^*)\equiv gh(Z)+1\,$. It is also named [ *antighost number*]{}. More precisely, in the general class of theories under consideration, the spectrum of fields (including ghosts) and antifields together with their respective ghost and antifield numbers is given by ($s>2$) - the fields $\{ A_{\mu}, h_{\mu\nu}, \phi_{\m_1\ldots\m_s} \}\,$ with ghost number $0$ and antifield number $0$; - the ghosts $\{ C, C_{\mu}, C_{\m_1\ldots\m_{s-1}} \}$ with ghost number $1$ and antifield number $0$; - the antifields $\{ A^{*\mu}, h^{*\mu\nu}, \phi^{*\m_1\ldots\m_s} \}$ , with ghost number $-1$ and antifield number $1$; - the antighosts $\{ C^*, C^{*\mu}, C^{*\m_1\ldots\m_{s-1}} \}$ with ghost number $-2$ and antifield number $2\,$. If the [*pureghost number*]{} ($pgh$) of an expression simply gives the number of ghosts (and derivatives of the ghosts) present in this expression, the ghost number ($gh$) is simply given by $$gh = pgh-antigh\;.$$ The fields and ghosts will sometimes be denoted collectively by $\Phi_I\,$, the antifields by $\Phi^{*I}$. The basic object in the antifield formalism is the BRST generator $W_0\,$. For a spin-$1$ field $A_{\mu}\,$, a spin-$2$ field $h_{\mu\nu}\,$ and a (double-traceless) spin-$s$ Fronsdal field $\phi_{\mu_1\ldots\mu_s}\,$, it reads $$\begin{aligned} W_{0,1}&=&S_{EM}[A_{\mu}] +\int A^{*\mu}\;\6_\mu C\; d^Dx\;, \nonumber \\ W_{0,2}&=&S_{PF}[h_{\mu\nu}]+2\,\int h^{*\mu\nu}\;\6_{(\mu}C_{\nu)} \;d^Dx\ , \nonumber \\ W_{0,s}&=&S_{F}[\phi_{\mu_1...\mu_s}] +s\,\int \phi^{*\mu_1...\mu_s}\;\6_{(\mu_1}C_{\mu_2...\mu_s)}\;d^Dx\ . \nonumber \end{aligned}$$ The functional $W_0$ satisfies the master equation $(W_0,W_0)=0$, where $(\,,)$ is the antibracket given by $$\begin{aligned} (A,B)=\frac{\d^R A}{\d\Phi_I} \frac{\d^L A}{\d\Phi^{*I}} - \frac{\d^R A}{\d\Phi^{*I}} \frac{\d^L A}{\d\Phi_I}\,.\end{aligned}$$ Let us note that this definition is appropriate for both functionals and differentials forms. In the former case, the summation over $I$ also implies an integration over spacetime (de Witt’s condensed notation). See the textbook [@Henneaux:1992ig] for a thorough exposition of the BRST formalism. The action of the BRST differential $s$ is defined by $$sA=(W_0\, , A)\;.$$ The differential $s$ is the sum of the Koszul-Tate differential $\d$ (which reproduces the equations of motion and the Noether identities) and the longitudinal derivative $\g$ (which reproduces the gauge transformations and the gauge algebra). Let us write down explicitely the action of $\d$ and $\g$ (unless it is vanishing): For a spin-1 field: $$\begin{aligned} \d C^* = -\6_\m A^{*\m}\ ,\quad \d A^{*\m}=\6_\r F^{\r\m}\ , \quad \g A_\m=\6_\m C\ .\nonumber\end{aligned}$$ For a spin-2 field: $$\begin{aligned} \d C^{*\n} = -2\,\6_\m h^{*\m\n}\ , \quad \d h^{*\m\n}=-2\, H^{\m\n}\ , \quad \g h_{\m\n}=2\,\6^{}_{(\m} C_{\n)}\ . \nonumber\end{aligned}$$ For a spin-$s$ field: $$\begin{aligned} \d C^{*\mu_1...\mu_{s-1}}=-s\left(\6_{\mu_s}\phi^{*\mu_1...\mu_s}- \frac{(s-1)(s-2)} {2(D+2s-6)}\;\eta^{(\mu_1\mu_2}\6_{\mu_s}\phi'^{*\mu_3...\mu_{s-1})\mu_s} \right)\;, \nonumber\end{aligned}$$ $$\begin{aligned} \d\phi^{*\mu_1...\mu_s}=G^{\mu_1...\mu_s}\ ,\quad \g\phi_{\mu_1...\mu_s}=s\,\6_{(\mu_1}C_{\mu_1...\mu_s)}\nonumber\ ,\end{aligned}$$ where $F_{\m\n}=\6_\m A_\n -\6_\n A_\m\,$ and $K_{\a\b|\m\n}=-\frac{1}{2}(\6^2_{\a\m}h_{\b\n}+\6^2_{\b\n}h_{\a\m} -\6^2_{\a\n}h_{\b\m}-\6^2_{\b\m}h_{\a\n})\,$ are the Maxwell field-strength and the linearized Riemann tensor, respectively[^13]. The linearized Einstein tensor is $H_{\m\n}=K_{\m\n}-\frac{1}{2}\,\eta_{\m\n}K\,$ where $K_{\b\n}=\eta^{\a\m}\,K_{\a\b|\m\n}$ is the linearized Ricci tensor and $K=\eta^{\b\n}\,K_{\b\n}$ the linearized scalar curvature. Finally, the flat-spacetime spin-$s$ Einstein-like and Fronsdal tensors $G_{\mu_1...\mu_s}$ and $F_{\mu_1...\mu_s}$ are given by $$\begin{aligned} G_{\mu_1...\mu_s}&=&F_{\mu_1...\mu_s} -\frac{s(s-1)}{4}\,\eta_{(\mu_1\mu_2}\,F'_{\mu_3...\mu_s)}\;, \nonumber \\ F_{\mu_1...\mu_s}&=&\Box \phi_{\mu_1...\mu_s} -s\,\6^2_{\r(\mu_1}\phi^{\r}_{\phantom{\r}\mu_2...\mu_s)}+ \frac{s(s-1)}{2}\,\6^2_{(\mu_1\mu_2}\phi'_{\mu_3...\mu_s)}\;. \nonumber\end{aligned}$$ For further purposes we also display the spin-$s$ curvature $$\begin{aligned} K_{\mu_1\nu_1|...|\mu_s\nu_s}=2^s\;Y^s(\6^s_{\mu_1...\mu_s} \phi_{\nu_1...\nu_s})\;,\quad s>2\;,\end{aligned}$$ where we have used the permutation operator $$\begin{aligned} Y^s=\frac{1}{2^s}\prod_{i=1}^{s} [e-(\mu_i\nu_i)]\; \nonumber\end{aligned}$$ that performs total antisymmetrization over the pairs of indices $(\mu_i,\nu_i)\,$, $i=1,\ldots,s\,$. Finally, we note that the Fronsdal and curvature tensors are not quite independant. The following relations can be established: $$\begin{aligned} K^{\rho}_{\phantom{\rho}\nu_{s-1}|\rho\nu_s|\mu_1\nu_1| \ldots|\mu_{s-2}\nu_{s-2}} &=& 2^{s-2}\,Y^{s-2}(\6^{s-2}_{\mu_1...\mu_{s-2}}F_{\nu_1...\nu_s})\;, \nonumber \\ \6^{\mu_s} K_{\mu_1\nu_1|...|\mu_s\nu_s}&=& 2^{s-1}\,Y^{s-1}(\6^{s-1}_{\mu_1...\mu_{s-1}}F_{\nu_1...\nu_s})\;. \nonumber\end{aligned}$$ In the following two subsections we give some cohomological results needed for the BRST-BV analysis of the deformation problem. Cohomology $H^*(\g)$ {#Hgamma} -------------------- For a proof of general results, see [@Bekaert:2005ka]. The only gauge-invariant functions for a spin-$s$ gauge field are functions of the field-strength tensor $F_{\m\n}$, the Riemann tensor $K_{\a\b|\m\n}$, the Fronsdal tensor $F_{\mu_1...\mu_s}$ and the curvature tensor $K_{\mu_1\nu_1|\mu_2\nu_2|...|\mu_s\nu_s}\,$. In pureghost number $pgh=0\,$ one has: $H^0(\g)=\{f([F_{\mu\nu}],[K],[F_s],[K_s],[\Phi^{*I}])\}$ where the notation $[\psi]$ indicates the (anti)field $\psi$ as well as all its derivatives up to a finite (but otherwise unspecified) order. In $pgh>0$, it can be shown (along the same lines as in [@Boulanger:2006gr], Appendix A) that one can choose $H^*(\g)$-representatives as the products of an element of $H^0(\g)$ with an appropriate number of non $\g$-exact ghosts. The latter are $\{$ $C$, $C_\m$, $\6_{[\m}C_{\n]}$, $C_{\mu_1...\mu_{s-1}}$ $\}$ together with the traceless part of $Y^j(\6_{\mu_1...\mu_j}C_{\nu_1...\nu_{s-1}})$ for $j\leqslant s-1\,$, that we denote $U^{(j)}_{\mu_1\nu_1|\ldots\mu_j\nu_j|\nu_{j+1}\ldots\nu_{s-1}}\,$. If we denote by $\omega^i_J$ a basis of the products of these objects in $pgh=i$, we get : H\^i(){\^J\^i\_J | \^JH\^0()} . More generally, let $\{\o_I\}$ be a basis of the space of polynomials in these variables (since these variables anticommute, this space is finite-dimensional). If a local form $a$ is $\gamma$-closed, we have $$\begin{aligned} \g a = 0 \quad\Rightarrow\quad a \,=\, \a^J \, \o_J + \g b \,.\end{aligned}$$ If $a$ has a fixed, finite ghost number, then $a$ can only contain a finite number of antifields. Moreover, since the [*local*]{} form $a$ possesses a finite number of derivatives, we find that the $\a^J$ are polynomials. Such a polynomial $\a^J$ will be called an [*invariant polynomial*]{}. We shall need several standard results on the cohomology of $d$ in the space of invariant polynomials. \[2.2\] In form degree less than $D$ and in antifield number strictly greater than $0\,$, the cohomology of $d$ is trivial in the space of invariant polynomials. That is to say, if $\a$ is an invariant polynomial, the equation $d \a = 0$ with $antigh(\a) > 0$ implies $ \a = d \b$ where $\b$ is also an invariant polynomial. The latter property is rather generic for gauge theories (see e.g. Ref. [@Boulanger:2000rq] for a proof), as well as the following: \[csq\] If $a$ has strictly positive antifield number, then the equation $\gamma a + d b = 0$ is equivalent, up to trivial redefinitions, to $\gamma a = 0$. More precisely, one can always add $d$-exact terms to $a$ and get a cocycle $a' := a + d c$ of $\gamma$, such that $\g a'= 0$. Homological groups $H^D_2(\d|d)$ and $H^D_2(\d|d,H^0(\g))$ ---------------------------------------------------------- We first recall a general result (Theorem 9.1 in [@Barnich:1994db]): \[usefll\] For a linear gauge theory of reducibility order $r$, $$\begin{aligned} H_p^D(\d \vert\, d)=0\; for\; p>r+2\,. \nonumber\end{aligned}$$ Since the theory at hand has no reducibility, we are left with the computation of $H_2^D(\d \vert\, d)\,$. Then, as we already claimed in [@Boulanger:2006gr], for a collection of different spins, $H_2^D(\d|d)$ is the direct sum of the homologies of the individual cases. For spin$\,$-$1$: $$H_2^D(\d|d)= \left\{\L C^*\,d^Dx \, |\ \L\in{{\mathbb{R}}}\right\}\,.$$ For spin$\,$-$2$ : H\_2\^D(|d)={\^C\^[\*]{}\_d\^Dx | \_[(]{}\_[)]{}=0} . For spin$\,$-$s$ $(s>2)\,$ ([@Bekaert:2005ka; @Barnich:2005bn]): H\_2\^D(|d)={\^[\_1...\_[s-1]{}]{}C\^\*\_[\_1...\_[s-1]{}]{}d\^Dx  | \_[(\_1]{}\_[\_2...\_s)]{}=0} . BRST deformation {#deformation} ---------------- As shown in [@Barnich:1993vg], the Noether procedure can be reformulated within a BRST-cohomological framework. Any consistent deformation of the gauge theory corresponds to a solution $$W=W_0+g W_1+g^2W_2+\co(g^3)$$ of the deformed master equation $(W,W)=0$. Taking into account field-redefinitions, the first-order nontrivial consistent local deformations $W_1=\int a^{D,\,0}$ are in one-to-one correspondence with elements of the cohomology $H^{D,\,0}(s \vert\, d)$ of the zeroth order BRST differential $s=(W_0\,,\cdot)$ modulo the total derivative $d\,$, in maximum form-degree $D$ and in ghost number $0\,$. That is, one must compute the general solution of the cocycle condition $$\begin{aligned} s a^{D,\,0} + db^{D-1,1} =0\,, \label{coc}\end{aligned}$$ where $a^{D,\,0}$ is a top-form of ghost number zero and $b^{D-1,1}$ a $(D-1)$-form of ghost number one, with the understanding that two solutions of (\[coc\]) that differ by a trivial solution should be identified $$\begin{aligned} a^{D,\,0}\sim a^{D,\,0} + s p^{D,-1} + dq^{D-1,\,0} \nonumber\end{aligned}$$ as they define the same interactions up to field redefinitions. The cocycles and coboundaries $a,b,p,q,\ldots\,$ are local forms of the field variables (including ghosts and antifields). The corresponding second-order interactions $W_2$ must satisfy the consistency condition $$s W_2=-\frac{1}{2} (W_1,W_1)\,.$$ This condition is controlled by the local BRST cohomology group $H^{D,1}(s\vert d)$. Quite generally, one can expand $a$ according to the antifield number, as a=a\_0+a\_1+a\_2+ …a\_k, \[antighdec\]where $a_i$ has antifield number $i$. The expansion stops at some finite value of the antifield number by locality, as was proved in [@Barnich:1994mt]. Let us recall [@Henneaux:1997bm] the meaning of the various components of $a$ in this expansion. The antifield-independent piece $a_0$ is the deformation of the Lagrangian; $a_1$, which is linear in the antifields associated with the gauge fields, contains the information about the deformation of the gauge symmetries; $a_2$ contains the information about the deformation of the gauge algebra (the term $C^{*} C C$ gives the deformation of the structure functions appearing in the commutator of two gauge transformations, while the term $\phi^* \phi^* C C$ gives the on-shell closure terms); and the $a_k$ ($k>2$) give the informations about the deformation of the higher order structure functions and the reducibility conditions. In fact, using standard reasonings (see e.g. [@Boulanger:2000rq]), one can remove all components of $a$ with antifield number greater than 2. The key point, as explained e.g. in [@Bekaert:2005jf], is that the invariant characteristic cohomology $H^{n,inv}_k(\delta \vert d)$ controls the obstructions to the removal of the term $a_k$ from $a$ and that all $H^{n,inv}_k(\delta \vert d)$ vanish for $k>2$ by Proposition \[usefll\] and Theorem \[2.6\] proved in section \[deltamodd4\]. This proves the first part of the following Theorem \[antigh2\], valid up to spin $s=4$: \[antigh2\] Let $a$ be a local top form which is a nontrivial solution of the equation (\[coc\]). Without loss of generality, one can assume that the decomposition (\[antighdec\]) stops at antighost number two, i.e. a=a\_0+a\_1+a\_2. \[defdecomps\] Moreover, the element $a_2$ is cubic: linear in the antighosts and quadratic in the variables $\{C,C_{\mu},\partial_{[\mu}C_{\nu]}, C_{\mu_1\ldots\mu_{s-1}}, U^{(j\leqslant s-1)}_{\mu_1\nu_1|\ldots|\mu_j\nu_j|\nu_{j+1}\ldots\nu_{s-1}} \; \}\vert_{s\leqslant 4}\;$ given in subsection \[Hgamma\]. Similarly to (\[defdecomps\]), one can assume $b=b_0+b_1\,$ in (\[coc\]) (see e.g. [@Boulanger:2000rq]) and insert the expansions of $a$ and $b$ into the latter equation. Decomposing the BRST differential as $s=\d+\g$ yields $$\begin{aligned} \g a_0+\d a_1 +d b_0=0 \,,\label{first}\\ \g a_1+\d a_2 +d b_1=0 \,, \label{second}\\ \g a_2=0\,. \label{minute} \end{aligned}$$ The general solution of (\[minute\]) is given in subsection \[Hgamma\]. **Remark:** Actually, even if the Theorem \[2.6\] cannot be extended to $s>4$ for technical reasons, we can always assume that $a_2$ is cubic as given in the above Theorem \[antigh2\], relax the limitation $s\leqslant 4$ and proceed with the determination of $a_1$ and $a_0$ according to (\[second\]) and (\[first\]). In fact, it is impossible to build a ghost-zero cubic object with $antigh>2$, so a cubic deformation always stops at $antigh\ 2$. Moreover, a cubic element $a_2$ must be proportional to an antighost and quadratic in the ghosts, then, modulo $d$ and $\g$, it is obvious that the only possible cubic deformations are those given in Theorem \[antigh2\]. Finally, combining the cohomological approach with other approaches like the light-cone one [@Metsaev:2005ar; @Metsaev:2007rn] may complete our results, as we actually show in the following. Such a combination of two different methods seems to us the most poweful way to completely solve the first-order deformation problem. Consistent vertices $V^{\L=0}(1,s,s)$ {#sec:1ss} ===================================== In this section we use the antifield formalism reviewed above and apply it to the study of nonabelian interactions between spin-$1$ and spin-$s$ gauge fields. We first examine in detail the interactions of the type $1-2-2\,$, and then move on to the general case $1-s-s\,$. Exotic nonabelian vertex $V^{\L=0}(1,2,2)$ {#sec:Ftheory} ------------------------------------------ In this section we show the existence of a cubic cross-interaction between a spin 1 field and a family of exotic spin 2 fields. The structure constants of this vertex are antisymmetric, which is in contradiction with the result for self-interacting spin 2 fields (see [@Boulanger:2000rq]). In fact, we easily prove that this vertex cannot coexist with the Einstein- Hilbert theory. We consider in the following a set of fields in Minkowski spacetime of dimension $D\,$. First, a single electromagnetic field $A_\mu$ with field strength $F_{\m\n}=\6_\m A_\n -\6_\n A_\m\,$, invariant under the gauge transformations $\stackrel{(0)}{\d_\L} A_\m=\6_\m \L\,$. Then, a family of Pauli-fierz fields $h_{\m\n}^a$ where $a$ is the family index. The linearized Riemann tensor is $K^a_{\a\b|\m\n}=-\frac{1}{2} (\6^2_{\a\m}h^a_{\b\n}+\6^2_{\b\n}h^a_{\a\m} -\6^2_{\a\n}h^a_{\b\m}-\6^2_{\b\m}h^a_{\a\n})\,$. It is invariant under the linearized diffeomorphism gauge transformations $\stackrel{(0)}{\d_\xi} h^a_{\m\n}=2\6^{}_{(\m}\xi^a_{\n)}\,$. The linearized Einstein tensor is $H^a_{\m\n}=K^a_{\m\n}-\frac{1}{2}\eta_{\m\n}K^a\,$, according to the notation given in Section \[sec:Def\]. The free action is the sum of the electromagnetic action and the different Pauli-Fierz actions: S\_0 =( -F\_F\^ - h\_a\^H\^a\_)d\^D x . In order to study the cubic deformation problem efficiently, we have used the antifield formalism of [@Barnich:1993vg], reviewed in the present paper. The antifield formalism allows us to write down every possible nontrivial deformation of the gauge algebra, encoded in the element denoted $a_2$ above. It turns out that only one $a_2$ candidate gives rise to a consistent vertex $a_0\,$. The details of the analysis are relegated to the appendix \[append122\], not to obscure the reading. The cubic vertex, gauge transformations and gauge algebra are $$\begin{aligned} &\bullet &\; \stackrel{(3)}{\cal{L}} = l_{[ab]}\left[- F^{\rho\sigma}\6_{[\mu}h^a_{\nu]\r}\6^{\m}h^{b\nu}_{\phantom{b\nu}\s}+2A^{\s} K^a_{\m\n|\r\s}\6^\m h^{b\n\r}\right]\;, \nonumber \\ &\bullet &\; \stackrel{(1)}{\d_\xi} A_\rho = 2l_{[ab]}\6^{}_{[\m}h^a_{\n]\r}\6^{\m}\xi^{b\n} \nonumber \\ &\bullet &\; \stackrel{(1)}{\d}_{\!\xi,\Lambda}h_{a\nu\rho}= 2l_{[ab]}\left[ \Lambda K^b_{\n\r}+ F^\m_{\phantom{\m}\r} \6_{[\m}\xi^b_{\n]}\right] \displaystyle -\frac{1}{D-2}l_{[ab]}\eta_{\nu\rho} \left[\Lambda K^b+F^{\m\n}\6_\m \xi^b_\n\right] \nonumber \\ &\bullet &\; \left[\stackrel{(0)}{\d}_{\!\xi},\stackrel{(1)} {\d}_{\!\eta}\right]A_\mu+\left[\stackrel{(1)}{\d}_{\!\xi},\stackrel{(0)} {\d}_{\!\eta}\right]A_\mu=\6_\mu\Lambda\quad \textrm{where}\quad \Lambda =2l_{[ab]}\6_{[\m}\xi^a_{\n]}\6^{\m}\eta^{b\n} \;.\end{aligned}$$ Exotic nonabelian vertex $V^{\L=0}(1,s,s)$ ------------------------------------------ The structure that we have found can be easily extended to obtain a set of consistent $1-s-s$ vertices. Using the notation introduced in Section \[sec:Def\], the Fronsdal action reads $$\begin{aligned} S_{Fs}=\frac{1}{2}\,\int \phi^{\mu_1...\mu_s}_a \,G^a_{\mu_1...\mu_s} \,d^Dx\;.\end{aligned}$$ It is gauge invariant thanks to the Noether identites $$\6^{\mu_s}G^a_{\mu_1...\mu_s}-\frac{(s-1)(s-2)} {2(D+2s-6)}\eta_{(\mu_1\mu_2}\6^{\mu_s} G'^a_{\mu_3...\mu_{s-1})\mu_s}\equiv 0$$ and the symmetry of the second-order differential operator defining $G\,$. The deformation analysis is performed exactly along the same lines as for the $1-2-2\,$ vertex. The uniqueness of the solution has not been proved for spin $s>4$, but we show that it is the only cubic solution deforming the gauge algebra. The spin-2 solution can ben extended to spin $s$, which leads us to consider a deformation of the BRST generator stoping at antighost 2, finishing with the following $a_2$ : $$\begin{aligned} a_2=f_{[ab]}C^*Y^{s-1} (\6^{s-1}_{\mu_1...\mu_{s-1}}C^a_{\nu_1...\nu_{s-1}}) Y^{s-1}(\6^{s-1\mu_1...\mu_{s-1}} C^{b\nu_1...\nu_{s-1}})d^Dx\,.\end{aligned}$$ By solving the equation $\d a_2+\g a_1 = d b_1$, we first obtain $$\begin{aligned} a_1=\tilde{a}_1+\bar{a}_1=2f_{[ab]}A^{*\r}Y^{s-1} (\6^{s-1}_{\mu_1...\mu_{s-1}} \phi^a_{\nu_1...\nu_{s-1}\r})Y^{s-1} (\6^{s-1\mu_1...\mu_{s-1}}C^{b\nu_1...\nu_{s-1}})d^Dx+\bar{a}_1\,\end{aligned}$$ with $\bar{a}_1\ |\ \g \bar{a}_1=de_1$. The resolution of $\d a_1+\g a_0=db_0$ provides us with both $\bar{a}_1$ and $a_0\,$: $$\begin{aligned} \bar{a}_1&=& 2f_{[ab]}\6^{(s-1)\mu_2...\mu_{s-1}} \phi^{*a\rho_1\rho_2\nu_3...\nu_{s-1}\t}D^{\nu_1\nu_2\s}_{\rho_1\rho_2\t} \times\nonumber \\ && \left[F^{\mu_1}_{\phantom{\mu_1}\s} Y^{s-1} (\6^{s-1}_{\mu_1...\mu_{s-1}}C^b_{\nu_1...\nu_{s-1}})-\frac{1}{2^{s-1}}C K^{b\mu_1}_{\phantom{b\mu_1}\s|\mu_1\nu_1|...\mu_{s-1}\nu_{s-1}} \right]d^Dx\end{aligned}$$ where $$\begin{aligned} D^{\nu_1\nu_2\s}_{\rho_1\rho_2\t}&=&\d^{\nu_1}_{\rho_1}\d^{\nu_2}_{\rho_2} \d^\s_\t-\frac{1}{2(D+2s-6)}\,\eta_{\rho_1\rho_2} \eta^{\s\nu_1}\d_\t^{\nu_2}-\frac{s-2}{D+2s-6}\,\eta_{\rho_1\rho_2} \eta^{\s\nu_2}\d_\t^{\nu_1}\,, \nonumber \\ a_0 &=& -f_{[ab]}F^{\r\s}Y^{s-1} (\6^{s-1}_{\mu_1...\mu_{s-1}}\phi^a_{\nu_1...\nu_{s-1}\r})Y^{s-1} (\6^{{s-1}\mu_1...\mu_{s-1}} \phi^{b\nu_1...\nu_{s-1}}_{\phantom{b\nu_1...\nu_{s-1}}\s}) d^Dx\nonumber\\&&+f_{[ab]}\frac{1}{2^{s-2}}A^\r K^a_{\mu_1\nu_1|...|\mu_{s-1}\nu_{s-1}|\r\s} Y^{s-1}(\6^{{s-1}\mu_1...\mu_{s-1}} \phi^{b\nu_1...\nu_{s-1}\s})d^Dx\,.\end{aligned}$$ These components of $W_1$ provide the cubic vertex, the gauge transformations and the gauge algebra: $$\begin{aligned} \bullet \stackrel{(3)}{\cl}&=&-f_{[ab]}F^{\r\s}Y^{s-1}(\6^{s-1}_{\mu_1... \mu_{s-1}}\phi^a_{\nu_1... \nu_{s-1}\r})Y^{s-1}(\6^{{s-1}\mu_1...\mu_{s-1}}\phi^{b\nu_1... \nu_{s-1}}_{\phantom{b\nu_1...\nu_{s-1}}\s})\nonumber\\&&+f_{[ab]} \frac{1}{2^{s-2}}\,A^\r K^a_{\mu_1\nu_1|...|\mu_{s-1}\nu_{s-1}|\r\s} Y^{s-1}(\6^{{s-1}\mu_1...\mu_{s-1}}\phi^{b\nu_1...\nu_{s-1}\s})\; ,\end{aligned}$$ $$\begin{aligned} \bullet\stackrel{(1)}{\d}_\xi A_\mu=Y^{s-1}(\6^{s-1}_{\mu_1...\mu_{s-1}} \phi^a_{\nu_1...\nu_{s-1}\r})Y^{s-1}(\6^{s-1\mu_1...\mu_{s-1}} \eta^{b\nu_1...\nu_{s-1}})\end{aligned}$$ $$\begin{aligned} \bullet\stackrel{(1)}{\d}_{\Lambda,\xi} \phi_{a\rho_1\rho_2\nu_3...\nu_{s-1}\t}=2(-1)^{s-1} f_{[ab]}D^{\nu_1\nu_2\s}_{\rho_1\rho_2\t}\times\nonumber \6^{s-1\mu_2...\mu_{s-1}} \Big[ F^{\mu_1}_{\phantom{\mu_1}\s} Y^{s-1}(\6^{s-1}_{\mu_1..\mu_{s-1}} \xi^b_{\nu_1...\nu_{s-1}})\\ -\frac{1}{2^{s-1}}\Lambda K^{b\mu_1}_{\phantom{b\mu_1}\s|\mu_1\nu_1|...|\mu_{s-1}\nu_{s-1}}\Big]\end{aligned}$$ $$\begin{aligned} \bullet \left[\stackrel{(0)}{\d}_{\xi},\stackrel{(1)} {\d}_{\eta}\right]A_\mu+\left[\stackrel{(1)}{\d}_{\xi},\stackrel{(0)} {\d}_{\eta}\right]A_\mu=\6_\mu \Lambda\end{aligned}$$ where $$\begin{aligned} \Lambda=2f_{[ab]}Y^{s-1} (\6^{s-1}_{\mu_1...\mu_{s-1}}\xi^a_{\nu_1...\nu_{s-1}}) Y^{s-1}(\6^{s-1\mu_1...\mu_{s-1}}\eta^{b\nu_1...\nu_{s-1}})\ ,\end{aligned}$$ the other commutators vanishing. Exhaustive list of interactions $V^{\Lambda=0}(1-s-s)\,$ -------------------------------------------------------- The uniqueness of the above cubic nonabelian interactions can be obtained by combining the above results with those obtained in [@Metsaev:2005ar; @Metsaev:2007rn] using a powerful light-cone method. We learn from the work [@Metsaev:2005ar] that there exist only *two* possible cubic couplings between one spin-$1$ and two spin-$s$ fields. The first coupling involves $2s-1$ derivatives in the cubic vertex whereas the other involves $2s+1$ derivatives. Therefore we conclude that the first coupling corresponds to the nonabelian deformation obtained in the previous subsection. The other one simply is the Born-Infeld-like coupling $$\begin{aligned} \stackrel{(3)}{\cl}&=&g_{[ab]}\,F^{\r\s}\, \eta^{\l\t}\,K^a_{\mu_1\nu_1|...|\mu_{s-1}\nu_{s-1}|\r\l} \;K_{\s\t|}^{b~\;\,\mu_1\nu_1|...|\mu_{s-1}\nu_{s-1}} \; ,\end{aligned}$$ which is strictly invariant under the abelian gauge transformations. Uniqueness of the nonabelian $V^{\Lambda=0}(2-4-4)\,$ vertex {#sec:244} ============================================================ The computation of all the possible nonabelian $2-4-4\,$ or $4-2-2\,$ cubic vertices in Minkowski spacetime of arbitrary dimension $D>3$ can be achieved along the same lines as for the $1-s-s$ vertex. We can apply Theorem \[antigh2\] to find a complete list of the possible $a_2$ terms, thanks to the technical result about $H_k^{n,inv}(\d|d)$ that we provide in Appendix \[append244\]. Then by solving equations (\[second\]) and (\[first\]), we find an unique cubic deformation. First, it is easily seen that it is impossible to build a non trivial $a_2$ involving one spin $4$ and two spin $2$. Then, in the $2-4-4$ case, the highest number of derivatives allowed for $a_2$ to be nontrivial is 6, but Poincaré invariance imposes an odd number of derivatives. Here is the only $a_2$ containing 5 derivatives, which gives rise to a consistent cubic vertex: $$\begin{aligned} a_2=f_{AB}C^*_\c U^A_{\a\m|\b\n|\r} V^{B\a\m|\b\n|\c\r}d^Dx \nonumber\end{aligned}$$ where $U^A_{\mu_1\nu_1|\mu_2\nu_2\t}=Y^2(\6^2_{\mu_1\mu_2} C^A_{\nu_1\nu_2\t})$ and $V^A_{\mu_1\nu_1|\mu_2\nu_2|\mu_3\nu_3} =Y^3(\6^2_{\mu_1\mu_2\mu_3}C^A_{\nu_1\nu_2\nu_3})$.\ Then, the inhomogenous solution of $\d a_2+\g a_1=db_1$ can be computed. The structure constants have to be symmetric in order for $a_1$ to exist : $f_{AB}=f_{(AB)}$ $$\begin{aligned} a_1=\tilde{a}_1+\bar{a}_1=f_{(AB)}\left[ h^{*\phantom{\c}\s}_{\phantom{*}\c}\6^2_{\a\b}\phi^A_{\m\n\r\s} V^{B\a\m|\b\n|\c\r} - 2 h^{*\g\s} \6^3_{\a\b[\c}\phi^A_{\r]\m\n\s} U^{B\a\m|\b\n|\r} \right]d^Dx+\bar{a}_1\,. \nonumber\end{aligned}$$ Finally, the last equation is $\d a_0+\g a_1=db_0$. It allows a solution, unique up to redefinitions of the fields and trivial gauge transformations. We have to say that the natural writing of $a_2$ and the vertex written in terms of the Weyl tensor $w_{\a\b|\c\d}$ do not match automatically. In order to get a solution, we first classified the terms of the form $w \6^4(\phi\phi)$. Then we classified the possible terms in $\bar{a}_1$, which can be chosen in $H^1(\g)$. So they are proportional to the field antifields, proportional to a gauge invariant tensor ($K_{PF}$, $F_4$ or $K_4$) and proportional to a non exact ghost. Finally, we had to introduce an arbitrary trivial combination in order for the expressions to match. The computation cannot be made by hand (there are thousands of terms). By using the software FORM [@Form], we managed to solve the heavy system of equations and found a consistent set of coefficients. We obtained the following $\bar{a}_1$: $$\begin{aligned} \bar{a}_1=\frac{4}{D+2}\, f_{AB}\phi^{*A\a}_{\phantom{*A\a}\b}\6^\t K^{\m\n|\a\s} U^B_{\m\n|\b\s|\t}d^Dx-2f_{AB}\phi^{*A\m\r}_{\phantom{*A\m\r}\a\b} \6^\t K^{\a\n|\b\s} U^B_{\m\n|\r\s|\t}d^Dx\nonumber\ .\end{aligned}$$ and the cubic vertex: $$\begin{aligned} a_{0,w}&\approx&f_{AB}w_{\m\n\r\s}\Big[\, \frac{1}{2}\,\6^{\m\r\a}\phi'^{A\n\b}\6_\a\phi'^{B\s}_{\phantom{,B\s}\b} -\frac{1}{3}\,\6^{\m\r\a}\phi^{A\n\b\c\d}\6_\a\phi^{B\s}_{\phantom{B\s}\b\c\d} +\frac{1}{4}\,\6^{\m\r\a}\phi^{A\n\b\c\d}\6_\b\phi^{B\s}_{\phantom{B\n}\a\c\d} \nonumber \\ && +\frac{3}{4}\,\6^{\m\a\b}\phi'^{A\n\r}\6_\a\phi'^{B\s}_{\phantom{,B\s}\b} +\frac{3}{4}\,\6^{\m\a\b}\phi^{A\n\r\c\d}\6_\a\phi^{B\s}_{\phantom{B\s}\b\c\d} -\frac{3}{2}\,\6^{\m\a\b}\phi^{A\n\r}_{\phantom{A\n\r}\b\c}\6_\a\phi'^{B\s\g} \nonumber \\&& -\frac{1}{2}\, \6^{\m}_{\phantom{\m}\b\c}\phi^{A\n\r\a}_{\phantom{A\n\r\a}\d}\6_\a \phi^{B\s\b\c\d}-\frac{3}{4}\, \6^{\m\a\b}\phi^{A\s}_{\phantom{A\s}\b\c\d}\6_\a\phi^{B\n\r\c\d} +\frac{3}{2}\,\6^{\m\a\b}\phi'^{A\s\g}\6_\a\phi^{B\n\r}_{\phantom{B\n\r}\b\c} \nonumber \\&& -\,\6^{\m}_{\phantom{\m}\b\c}\phi'^{A\s\a}\6_\a\phi^{B\n\r\b\c} +\frac{1}{2}\, \6^{\m}_{\phantom{\m}\b\c}\phi^{A\s\a\c}_{\phantom{A\s\a\c}\d}\6_\a \phi^{B\n\r\b\d}-\frac{1}{2}\,\6_{\a\b\c}\phi^{A\m\r\a\d}\6_\d\phi^{B\n\s\b\c} \nonumber \\&& +\frac{1}{2}\, \6_{\a\b\c}\phi^{A\m\r\a\t}\6^\b\phi^{B\n\s\c}_{\phantom{B\n\s\c}\t} +\frac{1}{8}\,\6^{\a\b}\phi'^{A\m\r}\6_{\a\b}\phi'^{B\n\s} +\frac{3}{8}\, \6^{\a\b}\phi^{A\m\r\c\d}\6_{\a\b}\phi^{B\n\s}_{\phantom{B\n\s}\c\d} \nonumber \\&& -\frac{1}{2}\,\6^{\a}_{\phantom{\a}\b}\phi^{A\m\r\b\c}\6_{\a\c}\phi'^{B\n\s} +\frac{1}{2}\, \6_{\a\b}\phi^{A\m\r\b\t}\6^{\a\c}\phi^{B\n\s}_{\phantom{B\n\s}\c\t} -\frac{3}{4}\, \6^{\a\b}\phi^{A\m\r\c\d}\6_{\a\c}\phi^{B\n\s}_{\phantom{B\n\s}\b\d} \nonumber\\&& +\frac{1}{4}\,\6_{\a\b}\phi^{A\m\r\c\d}\6_{\c\d}\phi^{B\n\s\a\b}\; \Big]\;, \label{v244}\end{aligned}$$ where the weak equality means that we omitted terms that are proportional to the free field equations, since they can trivially be absorbed by field redefinitions. The components $a_1$ and $a_2$ correspond to the following deformation of the gauge transformations $$\begin{aligned} \stackrel{(1)}{\d}_\xi h_{\s\t} &=&\frac{1}{2}f_{AB}\Big[\eta_{\t\mu_3}\6^2_{\mu_1\mu_2} \phi^A_{\nu_1\nu_2\nu_3\s}Y^3(\6^{3\mu_1\mu_2\mu_3}\xi^{B\nu_1\nu_2\nu_3}) \nonumber \\ && - 2\6^3_{\mu_1\mu_2[\t}\phi^A_{\r]\nu_1\nu_2\s} Y^2(\6^{2\mu_1\mu_2}\xi^{B\nu_1\nu_2\r})\Big]+ (\s\leftrightarrow\t) \\ \stackrel{(1)}{\d}_\xi \phi_{\a_1\a_2\a_3\a_4} & = & \frac{4}{D+2}\,f_{AB}\eta_{(\a_3\a_4}\d^{\mu_2}_{\a_1}\6^\t K^{\mu_1\nu_1|\phantom{\a_2}\nu_2}_{\phantom{\mu_1\nu_1|}\a_2)} Y^2(\6^2_{\mu_1\mu_2}\xi^B_{\nu_1\nu_2\t}) \nonumber \\ &&-2f_{AB}\d^{\mu_1}_{(\a_1}\d^{\mu_2}_{\a_2}\6^\t K^{\phantom{\a_3}\n_1\phantom{|\a_4)}\n_2}_{\a_3\phantom{\n_1}|\a_4)} Y^2(\6^2_{\mu_1\mu_2}\xi^B_{\nu_1\nu_2\t})\;.\end{aligned}$$ and to the following deformation of the gauge algebra: $$\begin{aligned} \nonumber\left[\stackrel{(0)}{\d}_{\xi},\stackrel{(1)} {\d}_{\eta}\right]h_{\mu\nu}+\left[\stackrel{(1)}{\d}_{\xi},\stackrel{(0)} {\d}_{\eta}\right]h_{\mu\nu}=2\6_{(\mu}j_{\nu)}\end{aligned}$$ where $$\begin{aligned} j_{\mu_3}=f_{(AB)} \6^{2\mu_1\mu_2}\xi^{A\nu_1\nu_2\nu_3}Y^3(\6^3_{\mu_1\mu_2\mu_3} \eta^B_{\nu_1\nu_2\nu_3}) - (\xi\leftrightarrow\eta)\nonumber\end{aligned}$$ Let us now consider the other possible cases for $a_2$, containing 3 or 1 derivatives. The only possibility with three derivatives is $a_{2,3}=g_{AB}C^*_\b \6_{[\a}C^A_{\m]\n\r}U^{B\a\m|\b\n|\r}d^Dx$. Its variation under $\delta$ should be $\g$-closed modulo $d$ but some nontrivial terms remain, of the types $g_{AB}h^*U^A U^B$ and $g_{AB}h^*\6_{[.}C^A_{.]..}V^B$. The first one can be set to zero by imposing symmetric structure constants, but the second cannot be eliminated. The same occurs for one of the candidates with 1 derivative: $a_{2,1,1}=k_{AB} C^{*\b} C^{A\m\n\r}\6_{[\b} C^B_{\m]\n\r}d^Dx$. We are then left with 2 candidates involving the spin 4 antifield. We have found that $\d a_2+\g a_1=d b_1$ can have a solution only if their structure constants are proportional : $$\begin{aligned} a_{2,1}&=&l_{AB}C^{*A\m\n\r}\left[ C^\a \6_{[\a} C^B_{\m]\n\r}+2\6_{[\m}C_{\a]}C^{B\a}_{\phantom{B\a}\n\r}\right]d^Dx \nonumber\\ a_{1,1}&=&l_{AB}\phi^{*\m\n\r\s}\left[2h_\s^{\phantom{\s}\a}\6_{[\a} C^B_{\m]\n\r} -\frac{4}{3}\,C^\a\6_{[\a}\phi^B_{\m]\n\r\s}+8\6_{[\m}h_{\a]\s}C^{B\a}_{\phantom{B\a}\n\r} -2\6_{[\m} C_{\a]}\phi^{B\a}_{\phantom{B\a}\n\r\s}\right] \nonumber\\ && +\frac{2}{D+2}l_{AB}\6_\s\phi'^{*A\r\s}C^\a\phi'^B_{\a\r}\;.\nonumber\end{aligned}$$ There is no homogenous part $\bar{a}_1$ (because the $\g$-invariant tensors contain at least 2 derivatives). Then, we have considered the most general expression for $a_0$, which is a linear combination of 55 terms of the types $h\phi\6^2\phi$ and $h\6\phi\6\phi\,$. We have found that the equation $\d a_1+\g a_0=db_0$ does not admit any solution. We can conclude that the vertex found with 6 derivatives is the *unique* nonabelian $2-4-4$ cubic deformation. This $2-4-4$ vertex, setting $D=4$, should correspond to the flat limit of the corresponding Fradkin-Vasiliev vertex. The uniqueness of the former can be used to prove the uniqueness of the latter, as we did explicitly in the $2-3-3$ case. Consistent vertices $V^{\L=0}(2,s,s)$ {#sec:2ss} ===================================== Nonabelian coupling with $2s-2$ derivatives ------------------------------------------- Our classification of gauge algebra deformations relies on the theorem concerning $H^D_k(\d|d,H^0(\g))$. While apparently obviously true, it actually becomes increasingly harder to prove with increasing spin. If $H^D_k(\d|d,H^0(\g))=H^D_k(\d|d)\cap H^0(\g)$ holds for spin $s>4$ then there is only one candidates for a nonabelian type $2-s-s$ deformation, involving $2s-3$ derivatives in $a_2$. Let us recall that due to the simple expression of $H^D_2(\d|d)$ we are only left with a few traceless building blocks for $a_2$: the antighosts $C^{*\m}$ and $C^{*A\mu_1...\mu_{s-1}}$, and a collection of ghosts and their anti-symmetrized derivatives, namely $C_\m$, $\6_{[\m}C_{\n]}$ and tensors $U^{(j)A}_{\m_1\n_1|...|\m_j\n_j|\n_{j+1}...\n_{s-1}}$ for $j\leq s-1$, that we have defined in section \[Hgamma\] [@Bekaert:2005ka]. Given this, we can divide the $a_2$ candidates into two categories: those proportional to $C^{*A\mu_1...\mu_{s-1}}$ and those proportional to $C^{*\mu}$. The first category is simple to study: $C^{*A\mu_1...\mu_{s-1}}$ carries $s-1$ indices, and the spin 2 ghost can carry at most 2, namely $\6_{[\a}C_{\b]}$. As no traces can be made, the spin 4 ghost can carry at most $s+1$ indices. But $U^{(2)A}_{\mu_1\nu1|\mu_2\nu_2|\nu_3...\nu_{s-1}}$ contains two antisymmetric pairs which cannot be contracted with $C^{*A}$. The only possible combination involving $\6_{[\a}C_{\b]}$ is thus $$f_{AB}C^{*A\mu_1...\mu_{s-1}} \6_{[\mu_1}C_{\a]}C^{B\a}_{\phantom{B\a}\mu_2...\mu_{s-1}}d^Dx\ .$$ If we consider the underivated $C_\a$, the only possibility is obviously: $$g_{AB}C^{*A\mu_1...\mu_{s-1}}C^\a U^{(1)A}_{\a\mu_1|\mu_2...\mu_{s-1}} d^Dx\ .$$ Those two terms contain only one derivative. Just as for the spin 4 case, we can show that they are related to an $a_1$ if $f_{AB}=\frac{s}{2}g_{AB}\,$: $$\begin{aligned} a_{2,1}=g_{AB}C^{*A\mu_1...\mu_{s-1}}\left[C^\a \6_{[\a}C^B_{\mu_1]\mu_2...\mu_{s-1}} +\frac{s}{2}\6_{[\mu_1}C_{\a]}C^{B\a}_{\phantom{B\a}\mu_2...\mu_{s-1}}\right]d^n x\ \end{aligned}$$ and $$\begin{aligned} a_{1,1}&=&l_{AB}\phi^{*A\m_1...\mu_s}\left[\frac{s}{2}h_{\m_s}^{\phantom{\m_s}\a}\6_{[\a} C^B_{\m_1]\m_2...\m_{s-1}}-\frac{s}{s-1}C^\a\6_{[\a}\phi^B_{\m_1]\m_2...\m_s}\right.\nonumber\\&&\left.\qquad\qquad\quad\;\;\;+\frac{s^2}{2}\6_{[\m_1}h_{\a]\m_s}C^{B\a}_{\phantom{B\a}\m_2...\m_s}-\frac{s}{2}\6_{[\m_1} C_{\a]}\phi^{B\a}_{\phantom{B\a}\m_2...\m_s}\right]\nonumber\\&&+\frac{s(s-2)}{4(n+2s-6)}l_{AB}\6_\s\phi'^{*A\m_3...\m_s}C^\a\phi'^B_{\a\m_3...\m_{s-1}}\end{aligned}$$ Then, the proof of the inconsistency of this candidate is exactly the same as for spin 4. In fact, for every spin $s\geq 4$, there are only 55 possible terms in the vertex. We have thus managed to adapt the proof to spin $s$, this deformation is obstructed. For the second category, the structure has to be $C^* U^{(i)} U^{(j)}\ ,\ i<j$ But $C^*$ carries one index and $U^{(i)}$ carries $i+s-1$. As no traces can be taken, it is obvious that $i=j-1$, which leaves us with a family of candidates: $$a_{2,2j-1}=l_{AB}C^{*\a}U^{(j-1)A\mu_1\nu_1|...| \mu_{j-1}\nu_{j-1}|\nu_j...\nu_{s-1}}U^{(j)B}_{\mu_1\nu_1|...| \mu_{j-1}\nu_{j-1}|\a\nu_j|\nu_{j+1}...\nu_{s-1}}d^Dx$$ Let us now check if these candidates satisfy the equation $\d a_2+\g a_1=d b_1$ for some $a_1$. For the second category, we get schematically $\d a_2=d(...)+\g(...)+l_{AB}h^* U^{(j)A}U^{(j)B}+l_{AB}h^* U^{(j-1)A}U^{(j+1)B}$. The first obstruction can be removed by imposing $l_{AB}=l_{(AB)}$ while the second cannot be removed unless $j=s-1$. As the tensor $U^{(s)B}$ does not exist, this term is not present at top number of derivatives, the second candidate $a_2$ that correspond to an $a_1$ is then: $$\begin{aligned} a_{2,2s-3}=l_{AB}C^{*\a}U^{(s-2)A\mu_1\nu_1|...|\mu_{s-2}\nu_{s-2}| \nu_{s-1}}U^{(s-1)B}_{\mu_1\nu_1|...|\mu_{s-2}\nu_{s-2}|\a\nu_{s-1}}d^Dx \,. \label{a22ss}\end{aligned}$$ Exhaustive list of cubic $V^{\Lambda=0}(2,s,s)$ couplings --------------------------------------------------------- Using the results of [@Metsaev:2005ar], we learn that there exist only *three* cubic couplings of the form $V^{\Lambda=0}(2,s,s)\,$. They involve a total number of derivatives in the vertex being respectively $2s+2$, $2s$ and $2s-2\,$. Moreover, it is indicated [@Metsaev:2005ar] that the $\,2s\,$-derivative coupling only exists in dimension $D>4\,$. From our results of the last subsection, we conclude that the last coupling is the nonabelian coupling with $2s-2$ derivatives. The coupling with $2s+2$ derivatives is simply the strictly-invariant Born-Infeld-like vertex $$\begin{aligned} \stackrel{(3)}{\cl}_{BI}&=& t_{(ab)}\,K^{\a\b|\c\d}\, \;K_{\a\b|}^{a~\;\,\mu_1\nu_1|...|\mu_{s-1}\nu_{s-1}} \;K^b_{\c\d|\mu_1\nu_1|...|\mu_{s-1}\nu_{s-1}} \; ,\end{aligned}$$ whereas the vertex with $2s$ derivatives is given by $$\begin{aligned} \stackrel{(3)}{\cl}_{2s}&=& u_{(ab)}\; \delta^{[\mu\n\r\s\l]}_{[\a\b\c\d\e]}\; h^{\a}_{~\m} \;K^{a\; \b\c|~~~\!|\mu_1\nu_1|...|\mu_{s-2}\nu_{s-2}}_{~~~\;\;\;\;\n\r} \;K^{b \;\d\e|}_{~~~\;\;\,\s\l|\mu_1\nu_1|...|\mu_{s-2}\nu_{s-2}}\;.\end{aligned}$$ It is easy to see that this vertex is not identically zero and is gauge-invariant under the abelian transformations, up to a total derivative. Summary and Conclusions ======================= Already for $\L=0$ the notion of minimal coupling needs to be refined to account for nonabelian vertices with more than two derivatives. Using the antifield formulation [@Barnich:1993vg], in order to prove that the first nonabelian vertex involving a set $\{\phi^i\}$ of fields is cubic, one needs a technical cohomological result concerning the nature of $H_k(\d|d)$ in the space of invariant polynomials. This technical result has been obtained previously up to $s=3$ and has been pushed here up to $s=4\,$ (cfr. Appendix \[deltamodd4\]). Supposing that this result holds in the general spin-s case, which is equivalent to supposing that the first nonabelian vertex is cubic, we have shown in Section \[sec:2ss\] that there exist only two possible nonabelian type $2-s-s$ deformations of the gauge algebra that can be integrated to corresponding gauge transformations. One of these two candidates has $2s-3$ derivatives and must therefore give rise to a vertex with $2s-2$ derivatives to be identified with the flat limit of the corresponding FV $2-s-s$ top vertex [@Fradkin:1986qy; @Fradkin:1987ks]. We have shown that the other candidate is obstructed. If liftable to a vertex, it would have given the two-derivatives vertex that corresponds to the minimal Lorentz covariantization. We have thus proved by cohomological methods what has recently been obtained by other methods in [@Metsaev:2005ar; @Metsaev:2007rn; @Porrati:2008rm]. Then, by combining our cohomological results with those of Metsaev [@Metsaev:2005ar], we explicitly built the *exhaustive* list of nontrivial, manifestly covariant vertices $V^{\Lambda=0}(1,s,s)$ and $V^{\Lambda=0}(2,s,s)\,$, notifying the relevant information concerning the nature of the deformed gauge algebra. For $\L\neq 0$, the standard notion of Lorentz covariantization does apply although it only provides the bottom vertex of a finite expansion in derivatives covered by inverse powers of $\L$, whose top vertices therefore dominate amplitudes (unless extra scales are brought in *e.g.* by expansions around non-trivial backgrounds). The top-vertices scale with energy non-uniformly for different spins rendering the standard semi-classical approach ill-defined unless some additional feature shows up beyond the cubic level. Indeed, Vasiliev’s fully non-linear higher-spin field equations may provide such a mechanism whereby infinite tails amenable to re-summation are developed. The two parallel perturbative expansions in $g$ and $(\ell\l)^{-1}$ resembles those in $g_s$ and $\a'\ell^{-2}$ in string theory, suggesting that the strong coupling at $(\ell\l)^{-1}>\!\!\!> 1$ corresponds to a tensionless limit of a microscopic string (or membrane). Indeed, the geometric underpinning of Vasiliev’s equations is that of flat connections and covariantly constant sections over a base-manifold – the “unfold” - taking their values in a fiber. This suggests that the total system is described by a total Lagrangian whose “pull back” to the unfold would be a free differential algebra action (with exterior derivative kinetic terms). On the other hand, its pull-back to the fiber would be a microscopic quantum theory in which the Weyl zero-form is subject to master constraints that are algebraic equations from the unfold point-of-view (thus avoiding the problematic negative powers of $\ell\l$). A candidate for the microscopic theory is the tensionless string/membrane in AdS whose phase-space action has been argued in [@Engquist:2005yt; @Engquist:2007vj] to be equivalent to a topological gauged non-compact WZW model with subcritical level [@Engquist:2007pr]. [**Acknowledgement:**]{} We would like to thank K. Alkalaev, X. Bekaert, C. Iazeolla, R. Metsaev, M. Porrati, A. Sagnotti, Ph. Spindel and M. Vasiliev for discussions. The work of N.B. and P.S. is supported in part by the EU contracts MRTN- CT-2004-503369 and MRTN-CT-2004-512194 and by the NATO grant PST.CLG.978785. During the preparation of this manuscript there appeared the work [@Zinoviev:2008ck] that also addresses the issue of the AdS deformation of the nonabelian 3-3-2 flat-space vertex found in [@Boulanger:2006gr]. The unique $V^{\Lambda=0}(1,2,2)$ vertex {#append122} ======================================== The gauge algebra, transformations and vertex: $a_2$, $a_1$ and $a_0$ --------------------------------------------------------------------- Thanks to he considerations made above, and as Poincaré invariance is required, the only nontrivial $a_2$ terms are linear in the underivated $antigh\ 2$ antifields and quadratic in the non exact ghosts. Family indices can be introduced, which allows to multiply the terms by structure constants. This construction is impossible for 2 spin 1 and 1 spin 2, while there are 3 candidates for 1 spin 1 and 2 spin 2 : - $a_{2,1}=f_{[ab]}C^*C_\m^aC^{b\m} d^Dx$ - $a_{2,2}=g_{ab}C^{*a}_\m C C^{b\m} d^Dx$ - $a_{2,3}=l_{[ab]}C^*\6_{[\m}C^a_{\n]}\6^{\m}C^{b\n} d^Dx$ We must now check if the equation $\d a_2 +\g a_1 = db_1$ admits solutions for the above candidates. Let us note that homogenous solutions for $a_1$ have to be considered : $\g \bar{a}_1=d\bar{b}_1\,$. Thanks to Proposition \[csq\], this equation can be redefined as $\g \bar{a}_1=0\,$. The non trival $\bar{a}_1$ are elements of $H(\g)$, and, as they are linear in the fields, involve at least one derivative. $$\begin{aligned} \bullet~\d a_{2,1} &=&-f_{[ab]}\6_\r A^{*\r}C_\m^aC^{b\m} d^Dx \nonumber \\ &=& 2 f_{[ab]}A^{*\r}\6_\r C^a_\m C^{b\m}d^Dx+d(...) \nonumber \\ &=& -\g (f_{[ab]}A^{*\r}h^a_{\r\m}C^{b\m}d^Dx) +2 f_{[ab]}A^{*\r}\6_{[\r}C^a_{\m]}C^{b\m}d^Dx+d(...)\,.\end{aligned}$$ The second term can not be $\g$-exact, therefore the first candidate has to be discarded. $$\begin{aligned} \bullet~\d a_{2,2}&=&-2g_{ab}\6_{\n} h^{*a\m\n}C C^b_\m d^Dx \nonumber \\ &=& 2g_{ab}h^{*a\m\n}\left[\6_\n C C^b_\m + C \6_\n C^b_\m \right]d^Dx + d(...)\nonumber \\ &=& -\g\left(g_{ab}h^{*a\m\n}\left[2A_\m C^b_\n-C h^b_{\m\n}\right]d^Dx \right) +d(...)\,.\end{aligned}$$ As there is no homogenous solution with no derivatives, we can conclude that a\_[1,2]{}=g\_[ab]{}h\^[\*a]{}d\^Dx +(...). Finally, applying the Koszul-Tate differential on the $a_{2,3}$ gives $$\begin{aligned} \bullet~\d a_{2,3}&=&-l_{[ab]}\6_\r A^{*\r}\6_{[\m}C^a_{\n]}\6^{\m}C^{b\n} d^Dx\nonumber \\ &=& 2 l_{[ab]}A^{*\r}\6^2_{\r[\m}C^a_{\n]}\6^{\m}C^{b\n} d^Dx+d(...) \nonumber \\ &=& -\g\left(2l_{[ab]}A^{*\r}\6^{}_{[\m}h^a_{\n]\r}\6^{\m}C^{b\n} d^Dx\right)+d(...)\,.\end{aligned}$$ Here we may assume the existence of an homogenous solution : a\_[1,3]{}=\_[1,3]{}+|[a]{}\_[1,3]{}=2l\_[\[ab\]]{}A\^[\*]{}\^\_[\[]{} h\^a\_[\]]{}\^C\^[b]{} d\^Dx+|[a]{}\_[1,3]{} | |[a]{}\_[1,3]{}=0. Finally, we can compute the possible vertices $a_0$, that have to be a solution of $\d a_1+\g a_0=d b_0$ where $a_1$ is one of the above candidates. For the candidate $a_{1,2}$, we get $\displaystyle\d a_{1,2}=-2g_{ab}H^{a\m\n}\left[2A_\m C^b_\n-C h^b_{\m\n}\right]d^Dx$. The second term is $\g$-exact modulo $d$ if $g_{ab}=g_{[ab]}$ (thanks to the properties of the Einstein tensor), but the first one does not work (terms of the form $h^a_{\cdot\cdot} \6^2_{\cdot\cdot} A_\cdot C_\cdot$ have non vanishing coefficients and are not $\g$-exact). Let us now compute the solution $a_{0,3}$, given that $\g \6_{[\m}h^a_{\n]\r}=\6^2_{\r[\m}C^a_{\n]}$ and $\6^\r K^a_{\m\n|\r\s}=2\6_{[\m}K^a_{\n]\s}$: $$\begin{aligned} \d \tilde{a}_{1,3}&=&2l_{[ab]}\6_{\s}F^{\s\r}\6_{[\m}h^a_{\n]\r}\6^{\m}C^{b\n}\nonumber\\ &=&2l_{[ab]}\6^{\s}A^{\r}K^a_{\m\n\s\r}\6^{\m} C^{b\n}+\g\left(l_{[ab]}F^{\s\r}\6_{[\m}h^a_{\n]\r}\6^\m h^{b\n\s}\right)+d(...)\nonumber\\ &=&-4 l_{[ab]}A^{\r}\6_{[\m}K^a_{\n]\r}\6^{\m}C^{b\n}+4l_{[ab]}C\6_{[\m} K^a_{\n]\s}\6^{\m} h^{b\n\s}\nonumber\\&&-\g\left(2l_{[ab]}A^{\r} K^a_{\m\n|\s\r}\6^{\m}h^{b\n\s}- l_{[ab]}F^{\s\r}\6_{[\m}h^a_{\n]\r}\6^\m h^{b\n\s}\right)+d(...)\,.\nonumber\end{aligned}$$ The first two terms are $\delta$-exact and correspond to a nontrivial $\bar{a}_{1,3}$. The last two terms are the vertex: $$\begin{aligned} a_{0,3}&=&l_{[ab]}\left[- F^{\rho\sigma}\6_{[\mu}h^a_{\nu]\r}\6^{\m}h^{b\nu}_{\phantom{b\nu}\s} +2A^{\s} K^a_{\m\n|\r\s}\6^\m h^{b\n\r}\right]d^Dx\,, \nonumber \\ \bar{a}_{1,3}&=&2l_{[ab]}h^{*a\n\r}\left[CK^b_{\n\r}+ F^\m_{\phantom{\m}\r} \6_{[\m}C^b_{\n]}\right]-\frac{1}{D-2}l_{[ab]} h^{*a}\,\!'\left[CK^b+F^{\m\n}\6_\m C^b_\n\right]+\g(...)\,.\end{aligned}$$ Inconsistency with Einstein-Hilbert theory ------------------------------------------ Here we show that, as expected, the spin-2 massless fields considered in the previous section cannot be considered as the linearized Einstein-Hilbert graviton. Let us consider the second order in $g$ in the master equation : it can be written $(W_1,W_1)=-2sW_2$. Let us decompose $W_2$ according to the antighost number : $W_2=\int(c_0+c_1+c_2+...)$. We will just check here the highest antighost part : $(a_2,a_2)=-2\g c_2-2\d c_3+d(...)$. But indeed, in any theory in which $a_2$ is linear in the antighost 2 antifields and quadratic in the ghosts, $(a_2,a_2)$ cannot depend on the antighost 1 antifields or on the fields, so that no $\d c_3$ can appear. This indicates that the expansion of $W_2$ stops at antighost 2 for those theories. But in fact, we get here : $$(a_{2,3},a_{2,3})=2\frac{\d a_{2,3}}{\d C^{*\mu}_a} \frac{\d a_{2,3}}{\d C^a_\mu}+2\frac{\d a_{2,3}}{\d C^*} \frac{\d a_{2,3}}{\d C}=0\quad .$$ This just means that the solution that we found is self-consistent at that order. But we also have to check the compatibility with self-interacting spin 2 fields. Let us consider the $a_2$ for a collection of Einstein-Hilbert theories (this can be found in [@Boulanger:2000rq]) : $a_{2,EH}=f_{(abc)}C^{*a\mu}C^{b\nu}\6_{[\mu}C^c_{\nu]}$, in which the coefficients $f_{abc}$ can be chosen diagonal. Let us now compute $$\begin{aligned} (a_{2,EH},a_{2,3})&=&\frac{\d a_{2,EH}}{\d C^{*\r}_e} \frac{\d a_{2,3}}{\d C^e_\r}=-2f^e_{bc}l_{ea}C^{b\n}\6_{[\r}C^c_{\n]}\6_\t\left[C^*\6^{[\t}C^{|a| \r]}\right]\nonumber\\&=&\g(...)+d(...)-2f^e_{bc}l_{ae}C^*\eta^{\s\n}\6_{[\t} C^b_{\s]}\6_{[\r}C^c_{\n]}\6^{[\t}C^{|a|\r]}\,.\end{aligned}$$ This can be consistent only if $f^e_{(bc}l_{a)e}=0$. But if we choose $f_{abc}$ diagonal, we obtain $f^a_{aa}l_{ba}=-2f^a_{ab}l_{aa}=0$, which means that the $f$’s or the $l$’s have to vanish. In other words, the spin 2 particles interacting with the spin 1 in our vertex cannot be Einstein-Hilbert gravitons. The unique $V^{\Lambda=0}(2,4,4)$ nonabelian vertex {#append244} =================================================== Invariant cohomology of $\d$ modulo $d$ for spin 4 {#deltamodd4} -------------------------------------------------- The following theorem is crucial, in the sense that it enables one to prove the uniqueness of the deformations, within the cohomological approach of [@Barnich:1993vg]: \[2.6\] Assume that the invariant polynomial $a_{k}^{p}$ ($p =$ form-degree, $k =$ antifield number) is $\delta$-trivial modulo $d$, $$\begin{aligned} a_{k}^{p} = \delta \mu_{k+1}^{p} + d \mu_{k}^{p-1} ~ ~ (k \geqslant 2). \label{2.37}\end{aligned}$$ Then, one can always choose $\mu_{k+1}^{p}$ and $\mu_{k}^{p-1}$ to be invariant. To prove the theorem, we need the following lemma, proved in [@Boulanger:2000rq]. \[l2.1\] If $a $ is an invariant polynomial that is $\delta$-exact, $a = \d b$, then, $a $ is $\delta$-exact in the space of invariant polynomials. That is, one can take $b$ to be also invariant. The proof of Theorem \[2.6\] for spin-4 gauge field proceeds in essentially the same way as for the spin-3 case presented in detail in [@Bekaert:2005jf], to which we refer for the general lines of reasoning. We only give here the piece of proof where things differ significantly from the spin-3 case. Different situations are considered, depending on the values of $p$ and $k$. In form degree $p<D\,$, the proof goes as in [@Bekaert:2005jf]. In form degree $p=D\,$, two cases must be considered: $k>D$ and $k\leqslant D\,$. In the first case, the proof goes as in [@Bekaert:2005jf], the new features appearing when $p=D\,$ and $k\leqslant D\,$. Rewriting the top equation (i.e. (\[2.37\]) with $p=D$) in dual notation, we have a\_k=b\_[k+1]{}+\_j\^\_[k]{},  (k2). \[2.44\] We will work by induction on the antifield number, showing that if the property expressed in Theorem \[2.6\] is true for $k+1$ (with $k>1$), then it is true for $k$. As we already know that it is true in the case $k>D$, the theorem will be proved. [**[Inductive proof for $k\leqslant D$]{}**]{} : The proof follows the lines of Ref. [@Barnich:1994mt] and decomposes in two parts. First, all Euler-Lagrange derivatives of (\[2.44\]) are computed. Second, the Euler- Lagrange (E.L.) derivative of an invariant quantity is also invariant. This property is used to express the E.L. derivatives of $a_k$ in terms of invariants only. Third, the homotopy formula is used to reconstruct $a_k$ from his E.L. derivatives. This almost ends the proof. [**(i)**]{} Let us take the E.L. derivatives of (\[2.44\]). Since the E.L. derivatives with respect to $C^*_{\a\b\c}$, the antifield associated with the ghost $C^{\a\b\c}$, commute with $\d$, we get first : $$\begin{aligned} \frac{\d^L a_k}{\d C^*_{\a\b\c}} =\d Z^{\a\b\c}_{k-1} \label{2.45}\end{aligned}$$ with $Z^{\a\b\c}_{k-1}=\frac{\d^L b_{k+1}}{\d C^*_{\a\b\c}}$. For the E.L. derivatives of $b_{k+1}$ with respect to $h^*_{\m\n\r\s}$ we obtain, after a direct computation, $$\begin{aligned} \frac{\d^L a_k}{\d h^*_{\m\n\r\s}}=-\d X^{\m\n\r\s}_k + 4\pa^{(\m}Z^{\n\r\s)}_{k-1}. \label{2.46}\end{aligned}$$ where $X^{\m\n\r\s}_{k}=\frac{\d^L b_{k+1}}{\d h^*_{\m\n\r\s} }$. Finally, let us compute the E.L. derivatives of $a_k$ with respect to the fields. We get : $$\begin{aligned} \frac{\d^L a_k}{\d h_{\m\n\r\s}}=\d Y^{\m\n\r\s}_{k+1} + {\cg}^{\m\n\r\s\vert\a\b\g\d} X_{\a\b\g\d\vert k} \label{2.47}\end{aligned}$$ where $Y^{\m\n\r\s}_{k+1}=\frac{\d^L b_{k+1}}{\d h_{\m\n\r\s}}$ and ${\cg}^{\m\n\r\s\vert\a\b\g\d}(\partial)$ is the second-order self-adjoint differential operator appearing in Fronsdal’s equations of motion $0=\frac{\d S^F[h]}{\d h_{\m\n\r\s}}\equiv G^{\m\n\r\s}={\cg}^{\m\n\r\s\vert\a\b\g\d}\,h_{\a\b\g\d}\,.$ The hermiticity of $\cg$ implies ${\cg}^{\m\n\r\s\vert\a\b\g\d}={\cg}^{\a\b\g\d\vert\m\n\r\s}$. [**(ii)**]{} The E.L. derivatives of an invariant object are invariant. Thus, $\frac{\d^L a_k}{\d C^*_{\a\b\c}}$ is invariant. Therefore, by Lemma \[l2.1\] and Eq. (\[2.45\]), we have also $$\begin{aligned} \frac{\d^L a_k}{\d C^*_{\a\b\c}} =\d Z'^{\a\b\c}_{k-1} \label{2.45'}\end{aligned}$$ for some invariant $Z'^{\a\b\c}_{k-1}$. Indeed, let us write the decomposition $Z^{\a\b\c}_{k-1} = Z'^{\a\b\c}_{k-1} + {\tilde{Z}}^{\a\b\c}_{k-1}$, where ${\tilde{Z}}^{\a\b\c}_{k-1}$ is obtained from ${Z}^{\a\b\c}_{k-1}$ by setting to zero all the terms that belong only to $H(\g)$. The latter operation clearly commutes with taking the $\d$ of something, so that Eq. (\[2.45\]) gives $0 = \d {\tilde{Z}}^{\a\b\c}_{k-1}$ which, by the acyclicity of $\d$, yields ${\tilde{Z}}^{\a\b\c}_{k-1}=\d \s_k^{\a\b\c}$ where $\s_k^{\a\b\c}$ can be chosen to be traceless. Substituting $\d \s_k^{\a\b\c} + Z'^{\a\b\c}_{k-1}$ for $Z^{\a\b\c}_{k-1}$ in Eq. (\[2.45\]) gives Eq. (\[2.45’\]). Similarly, one easily verifies that $$\begin{aligned} \frac{\d^L a_k}{\d h^*_{\m\n\r\s}}=-\d X'^{\m\n\r\s}_k + 4\pa^{(\m}Z'^{\n\r\s)}_{k-1}\,, \label{2.46'}\end{aligned}$$ where $X^{\m\n\r\s}_k = X'^{\m\n\r\s}_k + 4\pa^{(\m}\s^{\n\r\s)}_{k} + \d \r_{k+1}^{\m\n\r\s}$. Finally, using ${\cg}^{\m\n\r\s}{}_{\a\b\g\d}\,\pa^{(\a}\s^{\b\g\d)}{}_k=0$ due to the gauge invariance of the equations of motion ($\s_{\a\b\d}$ has been taken traceless), we find $$\begin{aligned} \frac{\d^L a_k}{\d h_{\m\n\r\s}} = \d Y'^{\m\n\r\s}_{k+1} +{\cg}^{\m\n\r\s}{}_{\a\b\g\d}{X'}^{\a\b\g\d}_k \label{2.47'}\end{aligned}$$ for the invariants $X'^{\m\n\r\s}_k$ and $Y'^{\m\n\r\s}_{k+1}$. Before ending the argument by making use of the homotopy formula, it is necessary to know more about the invariant $Y'^{\m\n\r\s}_{k+1}$. Since $a_k$ is invariant, it depends on the fields only through the curvature $K$, the Fronsdal tensor and their derivatives. (We substitute $4\,\pa^{[\d}\pa_{[\g}F^{~\,\s]}_{\r]~~\m\n}$ for $\eta^{\a\b}K^{\d\s}_{~~\,|\a\m|\b\n|\g\r}$ everywhere.) We then express the Fronsdal tensor in terms of the Einstein tensor: $F_{\m\n\r\s} = G_{\m\n\r\s} - \frac{6}{n+2}\,\eta_{(\m\n}G_{\r\s)}$, so that we can write $a_k = a_k([\Phi^{*i}],[K],[G])$, where $[G]$ denotes the Einstein tensor and its derivatives. We can thus write $$\begin{aligned} \frac{\d^L a_k}{\d h_{\m\n\r\s}} = {\cg}^{\m\n\r\s}{}_{\a\b\g\d} {A'}^{\a\b\g\d}_k + \pa_{\a}\pa_{\b}\pa_{\g}\pa_{\d} {M'}^{\a\m\vert\b\n\vert\g\r\vert\d\s}_k \label{2.49}\end{aligned}$$ where $${A'}^{\a\b\g\d}_k\propto\frac{\d a_k}{\d G_{\a\b\g\d}}$$ and $${M'}_k^{\a\m\vert\b\n\vert\g\r\vert\d\s}\propto{\frac{\d a_k} {\d K_{\a\m\vert\b\n\vert\g\r\vert\d\s}}}$$ are both invariant and respectively have the same symmetry properties as the “Einstein" and “Riemann" tensors. Combining Eq. (\[2.47’\]) with Eq. (\[2.49\]) gives $$\begin{aligned} \d Y'^{\m\n\r\s}_{k+1} = \pa_\a\pa_\b\pa_\g\pa_\d{M'}_k^{\a\m\vert\b\n\vert\g\r\vert\d\s} + {\cg}^{\m\n\r\s}{}_{\a\b\g\d} {B'}^{\a\b\g\d}_k \label{2.50}\end{aligned}$$ with ${B'}^{\a\b\g\d}_k:={A'}^{\a\b\g\d}_k-{X'}^{\a\b\g\d}_k$. Now, only the first term on the right-hand-side of Eq. (\[2.50\]) is divergence-free, $\pa_{\m}(\pa_{\a\b\g}{M'}_k^{\a\m\vert\b\n\vert\g\r})\equiv 0$, not the second one which instead obeys a relation analogous to the Noether identities $$\begin{aligned} \partial^{\tau}G_{\m\n\r\tau}-\frac{3}{(n+2)}\, \eta_{(\m\n}\partial^{\tau}G'_{\r)\tau}=0\,. \nonumber\end{aligned}$$ As a result, we have $\d\Big[\pa_{\m}({Y'}^{\m\n\r\s}_{k+1}-\frac{3}{D+2}\,\eta^{(\n\r} {Y'}_{k+1}^{\s)\m})\Big]=0\,$, where ${Y'}_{k+1}^{\m\s}\equiv \eta_{\n\r}{Y'}_{k+1}^{\m\n\r\s}\,$. By Lemma \[l2.1\], we deduce $$\begin{aligned} \pa_{\m}\Big({Y'}^{\m\n\r\s}_{k+1}-\frac{3}{D+2}\,\eta^{(\n\r} {Y'}_{k+1}^{\s)\m}\Big) +\d {F'}_{k+2}^{\n\r\s}=0 \,, \label{truc}\end{aligned}$$ where ${F'}_{k+2}^{\n\r\s}$ is invariant and can be chosen symmetric and traceless. Eq. (\[truc\]) determines a cocycle of $H^{D-1}_{k+1}(d\vert\d)$, for given $\n$, $\r$ and $\s\,$. Using the general isomorphisms $H^{D-1}_{k+1}(d\vert\d)\cong H^{D}_{k+2}(\d\vert d)\cong 0$ ($k\geqslant 1$) [@Barnich:1994db] we deduce $$\begin{aligned} {Y'}^{\m\n\r\s}_{k+1}-\frac{3}{D+2}\,\eta^{(\n\r}{Y'}_{k+1}^{\s)\m}= \pa_{\a}T_{k+1}^{\a\m\vert\n\r\s} + \delta P^{\m\n\r\s}_{k+2} \,, \label{truc2}\end{aligned}$$ where both $T^{\a\m\vert\n\r\s}_{k+1}$ and $P^{\m\n\r\s}_{k+2}$ are invariant by the induction hypothesis. Moreover, $T^{\a\m\vert\n\r\s}_{k+1}$ is antisymmetric in its first two indices. The tensors $T^{\a\m\vert\n\r\s}_{k+1}$ and $P^{\m\n\r\s}_{k+2}$ are both symmetric and traceless in $(\n,\r,\s)$. This results easily from taking the trace of Eq. (\[truc2\]) with $\eta_{\n\r}$ and using the general isomorphisms $H^{D-2}_{k+1}(d\vert\d)\cong H^{D-1}_{k+2}(\d\vert d)\cong H^{D}_{k+3}(\d\vert d)\cong 0$ [@Barnich:1994db] which hold since $k$ is positive. From Eq. (\[truc2\]) we obtain $$\begin{aligned} {Y'}^{\m\n\r\s}_{k+1} = \pa_{\a} [ T_{k+1}^{\a\m\vert\n\r\s}+ \frac{3}{D}\,T_{k+1}^{\a\vert\m(\n}\eta^{\r\s)} ] + \delta (...) \label{truc3}\end{aligned}$$ where $T_{k+1}^{\a\vert\m\n}\equiv \eta_{\t\r}T_{k+1}^{\a\t\vert\r\m\n}\,$. We do not explicit the $\delta$-exact term since it plays no role in the following. Since $Y'^{\m\n\r\s}_{k+1}$ is symmetric in $\m$ and $\n$, we have also $$\pa_{\a}\Big(T_{k+1~~\r\s}^{\a[\m\vert\n]}+ \frac{2}{D}\,T_{k+1~(\s}^{\a\vert[\m}\d^{\n]}_{\r)}\Big) +\;\delta (...)=0\,.$$ The triviality of $H^{D}_{k+2}(d \vert \d)$ ($k>0$) implies again that $T_{k+1~~\r\s}^{\a[\m\vert\n]}+\frac{2} {D}\,T_{k+1~(\s}^{\a\vert[\m}\d^{\n]}_{\r)}$ is trivial, in particular, $$\begin{aligned} \partial_{\beta}S^{'\beta\a|\mu\nu|}_{\qquad~\r\s} + \d(...) = T_{k+1~~\r\s}^{\a[\m\vert\n]}+\frac{2}{D}\, T_{k+1~(\s}^{\a\vert[\m}\d^{\n]}_{\r)} \label{derStoT}\end{aligned}$$ where $S^{'\beta\a|\mu\nu|}_{\qquad~\r\s}$ is antisymmetric in the pairs of indices ($ \b, \a$) and ($\m,\n$), while it is symmetric and traceless in ($ \r, \s$). Actually, it is traceless in $\m, \n, \r\,\s$ as the right-hand side of the above equation shows. The induction assumption allows us to choose $S^{'\beta\a|\mu\nu|}_{\qquad~\r\s}$, as well as the quantity under the Koszul-Tate differential $\d\,$. We now project both sides of Eq. (\[derStoT\]) on the following irreducible representation of the orthogonal group (35,16)(0,0) (1,4)(10.5,0)[3]{}(10,10)[$\a$]{}[$\m$]{}[$\s$]{} (1,-6.5)(10.5,0)[2]{}(10,10)[$\r$]{}[$\n$]{} and obtain $$\begin{aligned} \partial_{\b}W^{'\b|\a\r|\m\n|\s}_{k+1}+ \d (\dots) = 0 \label{derW}\end{aligned}$$ where $W^{'\b|\a\r|\m\n|\s}_{k+1}$ denotes the corresponding projection of $S^{'\beta\a|\mu\nu|\r\s}\,$. Eq. (\[derW\]) determines, for given $(\m, \n, \a, \r,\s)\,$, a cocycle of $H^{D-1}_{k+1}(d\vert\d,H(\gamma))$. Using again the isomorphisms [@Barnich:1994db] $H^{D-1}_{k+1}(d\vert\d)\cong H^{D}_{k+2}(\d\vert d)\cong 0$ ($k\geqslant 1$) and the induction hypothesis, we find $$\begin{aligned} W^{'\b|\a\r|\m\n|\s}_{k+1} = \pa_{\l}\phi^{\l\b\vert\a\r|\m\n|\s}_{k+1} + \d (\dots) \label{Wintermsofphi}\end{aligned}$$ where $\phi^{\l\b\vert\a\r|\m\n|\s}_{k+1}$ is invariant, antisymmetric in $(\l, \b)$ and possesses the irreducible, totally traceless symmetry (35,16)(0,0) (1,4)(10.5,0)[3]{}(10,10)[$\a$]{}[$\m$]{}[$\s$]{} (1,-6.5)(10.5,0)[2]{}(10,10)[$\r$]{}[$\n$]{} in its last five indices. The $\d$-exact term is invariant as well. Then, projecting the equation (\[Wintermsofphi\]) on the totally traceless irreducible representation (35,16)(0,0) (1,4)(10.5,0)[3]{}(10,10)[$\a$]{}[$\m$]{}[$\s$]{} (1,-6.5)(10.5,0)[3]{}(10,10)[$\r$]{}[$\n$]{}[$\b$]{} and taking into account that $W^{'\b|\a\r|\m\n|\s}_{k+1}$ is built out from $S^{'\beta\a|\mu\nu|\r\s}\,$, we find $$\begin{aligned} \partial_{\l}\Psi^{'\l|\a\r|\m\n|\s\b}_{k+1} + \d (\dots) = 0\end{aligned}$$ where $\Psi^{'\l|\a\r|\m\n|\s\b}_{k+1}$ denotes the corresponding projection of $\phi^{\l\b\vert\a\r|\m\n|\s}_{k+1}\,$. The same arguments used before imply $$\begin{aligned} \Psi^{'\l|\a\r|\m\n|\s\b}_{k+1} = \partial_{\t} \Xi^{'\t\l|\a\r|\m\n|\s\b} + \delta(...) \label{PsitoXi}\end{aligned}$$ where the symmetries of $\Xi^{'\t\l|\a\r|\m\n|\s\b}$ on its last 6 indices can be read off from the left-hand side and where the first pair of indices is antisymmetric. Again, $\Xi^{'\t\l|\a\r|\m\n|\s\b}$ can be taken to be invariant. Then, we take the projection of $\Xi^{'\t\l|\a\r|\m\n|\s\b}$ on the irreducible representation (45,16)(0,0) (1,4)(10.5,0)[4]{}(10,10)[$\t$]{}[$\a$]{}[$\m$]{}[$\s$]{} (1,-6.5)(10.5,0)[4]{}(10,10)[$\l$]{}[$\r$]{}[$\n$]{}[$\b$]{} of $GL(D)$ (here we do not impose tracelessness) and denote the result by $\Theta^{'\t\l|\a\r|\m\n|\s\b}\,$. This invariant tensor possesses the algebraic symmetries of the invariant spin-$4$ curvature tensor. Finally, putting all the previous results together, we obtain the following relation, using the symbolic manipulation program *Ricci* [@Lee]: 6[Y’]{}\^\_[k+1]{} = \_\_\_[c]{}\_\^[’c|]{}\_[k+1]{} + \^\_ \_[k+1]{}\^[’]{}+(…), \[result1\] with $$\begin{aligned} \widehat{X}^{'}_{\a\b\g\d\vert k+1} &:=& \frac{\cy^{\m\n\r\s}_{\a\b\c\d}}{D-2}\, \Big[-\frac{1}{3}\,\eta^{\t\l} S_{\t\m|\l\n|\r\s|k+1} + \frac{1}{3(D+1)}\, \eta_{\m\n}\eta^{\t\l}\eta^{\kappa\z} (S_{\t\kappa|\l\z|\r\s|k+1} + 2\,S_{\t\kappa|\l\r|\z\s|k+1}) \nonumber \\ &&+\;\frac{2(D-2)}{D}\,\eta^{\kappa\t}\partial^{\l} \phi_{\kappa\m|\l\n|\t\r|\s} -\frac{4(D-2)}{D(D+2)}\,\eta_{\m\n}\eta^{\kappa\t}\eta^{\x\z}\partial^{\l} \phi_{\kappa\x|\t\z|\l\m|\n}\Big]\; \label{result2}\end{aligned}$$ being double-traceless and where $\cy^{\m\n\r\s}_{\a\b\c\d}$ projects on completely symmetric rank-4 tensors. [**(iii)**]{} We can now complete the argument. The homotopy formula $$\begin{aligned} a_k = \int^{1}_{0}dt\,\left[C^*_{\a\b\c}\frac{\d^L a_k}{\d C^*_{\a\b\c}}+ h^*_{\m\n\r\s}\frac{\d^L a_k}{\d h^*_{\m\n\r\s}}+ h^{\m\n\r\s}\frac{\d^L a_k}{\d h^{\m\n\r\s}}\right](th\,,\,th^*\,,\,tC^*) \label{homotopy}\end{aligned}$$ enables one to reconstruct $a_k$ from its Euler-Lagrange derivatives. Inserting the expressions (\[2.45’\])-(\[2.47’\]) for these E.L. derivatives, we get $$\begin{aligned} a_k=\d\Big(\int^{1}_{0}dt\,[C^*_{\a\b\c}Z'^{\a\b\c}_{k-1} +h^*_{\m\n\r\s}X'^{\m\n\r\s}_{k}+h_{\m\n\r\s}Y'^{\m\n\r\s}_{k+1}](t)\,\Big) +\pa_{\r}k^{\r}.\label{invhf}\end{aligned}$$ The first two terms in the argument of $\d$ are manifestly invariant. In order to prove that the third term can be assumed to be invariant in Eq. (\[invhf\]), we use Eq. (\[result1\]) to find that (absorbing the irrelevant factor 6 in a redefintion of $Y'^{\m\n\r\s}$) $$h_{\m\n\r\s}\,Y'^{\m\n\r\s}_{k+1}=\frac{1}{16}\, \Theta^{'\a\m\vert\b\n\vert\c\r\vert\d\s}_{k+1} K_{\a\m\vert\b\n\vert\c\r\vert\d\s} +G_{\a\b\g\d}\widehat{X}^{'\a\b\g\d}{}_{k+1}+\pa_\r \ell^\r+\d(\ldots)\,,$$ where we integrated by part four times in order to get the first term of the r.h.s. while the hermiticity of ${\cg}^{\m\n\r\s\vert\a\b\g\d}$ was used to obtain the second term. We are left with $a_k = \d \m_{k+1} + \pa_{\r}\n^{\r}_k\,$, where $\m_{k+1}$ is invariant. That $\n^{\r}_{k}$ can now be chosen invariant is straightforward. Acting with $\g$ on the last equation yields $\pa_{\r} (\g {\n}^{\r}_{k}) =0\,$. By the Poincaré lemma, $\g {\n}^{\r}_{k} = \pa_{\s} (\t_k^{[\r \s]})\,$. Furthermore, Proposition \[csq\] concerning $H(\g\vert\, d)$ at positive antighost number $k$ implies that one can redefine $\n^{\r}_{k}$ by the addition of trivial $d$-exact terms such that one can assume $\g {\n}^{\r}_{k}=0\,$. As the pureghost number of ${\n}^{\r}_{k}$ vanishes, the last equation implies that $\n^{\r}_{k}$ is an invariant polynomial. Uniqueness of the $V^{\Lambda=0}(2,4,4)\,$ vertex ------------------------------------------------- As the main theorems are now established up to spin 4, we can classify the nontrivial $a_2$ terms, which correspond to the deformations of the gauge algebra. The highest number of derivaties allowed for $a_2$ to be nontrivial is 6, but Poincaré invariance imposes an odd number of derivatives. Here is the only $a_2$ containing 5 derivatives, which gives rise to a consistent cubic vertex: $$\begin{aligned} a_2=f_{AB}C^*_\c U^A_{\a\m|\b\n|\r} V^{B\a\m|\b\n|\c\r}d^Dx \nonumber\end{aligned}$$ where $U^A_{\mu_1\nu_1|\mu_2\nu_2\t}=Y^2(\6^2_{\mu_1\mu_2} C^A_{\nu_1\nu_2\t})$ and $V^A_{\mu_1\nu_1|\mu_2\nu_2|\mu_3\nu_3} =Y^3(\6^2_{\mu_1\mu_2\mu_3}C^A_{\nu_1\nu_2\nu_3})$.\ Then, the inhomogenous solution of $\d a_2+\g a_1=db_1$ can be computed. The structure constants have to be symmetric in order for $a_1$ to exist : $f_{AB}=f_{(AB)}$ $$\begin{aligned} a_1=\tilde{a}_1+\bar{a}_1=f_{(AB)}\left[ h^{*\phantom{\c}\s}_{\phantom{*}\c}\6^2_{\a\b}\phi^A_{\m\n\r\s} V^{B\a\m|\b\n|\c\r} - 2 h^{*\g\s} \6^3_{\a\b[\c}\phi^A_{\r]\m\n\s} U^{B\a\m|\b\n|\r} \right]d^Dx+\bar{a}_1\,. \nonumber\end{aligned}$$ Finally, the last equation is $\d a_0+\g a_1=db_0$. It allows a solution, unique up to redefinitions of the fields and trivial gauge transformations. We have to say that the natural writing of $a_2$ and the vertex written in terms of the Weyl tensor $w_{\a\b|\c\d}$ do not match automatically. In order to get a solution, we first classified the terms of the form $w \6^4(\phi\phi)$. Then we classified the possible terms in $\bar{a}_1$, which can be chosen in $H^1(\g)$. So they are proportional to the field antifields, proportional to a gauge invariant tensor ($K_{PF}$, $F_4$ or $K_4$) and proportional to a non exact ghost. Finally, we had to introduce an arbitrary trivial combination in order for the expressions to match. The computation cannot be made by hand (there are thousands of terms). By using the software FORM [@Form], we managed to solve the heavy system of equations and found a consistent set of coefficients. We obtained the following $\bar{a}_1$ as well as the vertex $a_0$ that we wrote above (\[v244\]): $$\begin{aligned} \bar{a}_1=\frac{4}{D+2}\, f_{AB}\phi^{*A\a}_{\phantom{*A\a}\b}\6^\t K^{\m\n|\a\s} U^B_{\m\n|\b\s|\t}d^Dx-2f_{AB}\phi^{*A\m\r}_{\phantom{*A\m\r}\a\b} \6^\t K^{\a\n|\b\s} U^B_{\m\n|\r\s|\t}d^Dx\nonumber\ .\end{aligned}$$ Let us now consider the other possible cases for $a_2$, containing 3 or 1 derivatives. The only possibility with three derivatives is $a_{2,3}=g_{AB}C^*_\b \6_{[\a}C^A_{\m]\n\r}U^{B\a\m|\b\n|\r}d^Dx$. Its variation under $\delta$ should be $\g$-closed modulo $d$ but some nontrivial terms remain, of the types $g_{AB}h^*U^A U^B$ and $g_{AB}h^*\6_{[.}C^A_{.]..}V^B$. The first one can be set to zero by imposing symmetric structure constants, but the second cannot be eliminated. The same occurs for one of the candidates with 1 derivative: $a_{2,1,1}=k_{AB} C^{*\b} C^{A\m\n\r}\6_{[\b} C^B_{\m]\n\r}d^Dx$. We are then left with 2 candidates involving the spin 4 antifield. We have found that $\d a_2+\g a_1=d b_1$ can have a solution only if their structure constants are proportional : $$\begin{aligned} a_{2,1}&=&l_{AB}C^{*A\m\n\r}\left[ C^\a \6_{[\a} C^B_{\m]\n\r}+2\6_{[\m}C_{\a]}C^{B\a}_{\phantom{B\a}\n\r}\right]d^Dx \nonumber\\ a_{1,1}&=&l_{AB}\phi^{*\m\n\r\s}\left[2h_\s^{\phantom{\s}\a}\6_{[\a} C^B_{\m]\n\r} -\frac{4}{3}\,C^\a\6_{[\a}\phi^B_{\m]\n\r\s}+8\6_{[\m}h_{\a]\s} C^{B\a}_{\phantom{B\a}\n\r} -2\6_{[\m} C_{\a]}\phi^{B\a}_{\phantom{B\a}\n\r\s}\right] \nonumber\\ && +\frac{2}{D+2}l_{AB}\6_\s\phi'^{*A\r\s}C^\a\phi'^B_{\a\r}\;.\nonumber\end{aligned}$$ There is no homogenous part $\bar{a}_1$ (because the $\g$-invariant tensors contain at least 2 derivatives). Then, we have considered the most general expression for $a_0$, which is a linear combination of 55 terms of the types $h\phi\6^2\phi$ and $h\6\phi\6\phi\,$. We have found that the equation $\d a_1+\g a_0=db_0$ does not admit any solution. We can conclude that the vertex found with 6 derivatives is the *unique* nonabelian $2-4-4$ cubic deformation. [10]{} C. Aragone and S. Deser, “[Consistency problems of spin-2 gravity coupling]{},” [*Nuovo Cim.*]{} [**B57**]{} (1980) 33–49. F. A. Berends, J. W. van Holten, P. van Nieuwenhuizen, and B. de Wit, “[On spin 5/2 gauge fields]{},” [[*Phys. Lett.*]{} [ **B83**]{} (1979) 188](http://dx.doi.org/10.1016/0370-2693(79)90682-8). C. Aragone and H. La Roche, “[Massless second order tetradic spin 3 fields and higher helicity bosons]{},” [*Nuovo Cim.*]{} [**A72**]{} (1982) 149. R. R. Metsaev, “[Cubic interaction vertices for massive and massless higher spin fields]{},” [[ *Nucl. Phys.*]{} [**B759**]{} (2006) 147–201](http://dx.doi.org/10.1016/j.nuclphysb.2006.10.002), [[arXiv:hep-th/0512342]{}](http://arxiv.org/abs/hep-th/0512342). R. R. Metsaev, “[Cubic interaction vertices for fermionic and bosonic arbitrary spin fields]{},” [[arXiv:0712.3526 \[hep-th\]]{}](http://arxiv.org/abs/0712.3526). M. Porrati, “[Universal Limits on Massless High-Spin Particles]{},” [[arXiv:0804.4672 \[hep-th\]]{}](http://arxiv.org/abs/0804.4672). N. Boulanger and S. Leclercq, “[Consistent couplings between spin-2 and spin-3 massless fields]{},” [*JHEP*]{} [**11**]{} (2006) 034, [[arXiv:hep-th/0609221]{}](http://arxiv.org/abs/hep-th/0609221). A. Fotopoulos and M. Tsulaia, “[Gauge Invariant Lagrangians for Free and Interacting Higher Spin Fields. A Review of the BRST formulation]{},” [[arXiv:0805.1346 \[hep-th\]]{}](http://arxiv.org/abs/0805.1346). G. Barnich and M. Henneaux, “[Consistent couplings between fields with a gauge freedom and deformations of the master equation]{},” [[*Phys. Lett.*]{} [ **B311**]{} (1993) 123–129](http://dx.doi.org/10.1016/0370-2693(93)90544-R), [[arXiv:hep-th/9304057]{}](http://arxiv.org/abs/hep-th/9304057). M. Henneaux, “[Consistent interactions between gauge fields: The cohomological approach]{},” [*Contemp. Math.*]{} [**219**]{} (1998) 93, [[arXiv:hep-th/9712226]{}](http://arxiv.org/abs/hep-th/9712226). E. S. Fradkin and M. A. Vasiliev, “[Cubic Interaction in Extended Theories of Massless Higher Spin Fields]{},” [[*Nucl. Phys.*]{} [ **B291**]{} (1987) 141](http://dx.doi.org/10.1016/0550-3213(87)90469-X). E. S. Fradkin and M. A. Vasiliev, “[On the Gravitational Interaction of Massless Higher Spin Fields]{},” [[*Phys. Lett.*]{} [ **B189**]{} (1987) 89–95](http://dx.doi.org/10.1016/0370-2693(87)91275-5). C. Fronsdal, “[Singletons and Massless, Integral Spin Fields on de Sitter Space (Elementary Particles in a Curved Space. 7]{},” [[*Phys. Rev.*]{} [**D20**]{} (1979) 848–856](http://dx.doi.org/10.1103/PhysRevD.20.848). M. A. Vasiliev, “[Cubic interactions of bosonic higher spin gauge fields in AdS(5)]{},” [[*Nucl. Phys.*]{} [**B616**]{} (2001) 106–162](http://dx.doi.org/10.1016/S0550-3213(01)00433-3), [[arXiv:hep-th/0106200]{}](http://arxiv.org/abs/hep-th/0106200). K. B. Alkalaev and M. A. Vasiliev, “[N = 1 supersymmetric theory of higher spin gauge fields in AdS(5) at the cubic level]{},” [[*Nucl. Phys.*]{} [ **B655**]{} (2003) 57–92](http://dx.doi.org/10.1016/S0550-3213(03)00061-0), [[arXiv:hep-th/0206068]{}](http://arxiv.org/abs/hep-th/0206068). M. A. Vasiliev, “[Progress in higher spin gauge theories]{},” [[arXiv:hep-th/0104246]{}](http://arxiv.org/abs/hep-th/0104246). I. L. Buchbinder, A. Pashnev, and M. Tsulaia, “[Lagrangian formulation of the massless higher integer spin fields in the AdS background]{},” [[*Phys. Lett.*]{} [ **B523**]{} (2001) 338–346](http://dx.doi.org/10.1016/S0370-2693(01)01268-0), [[arXiv:hep-th/0109067]{}](http://arxiv.org/abs/hep-th/0109067). J. M. Lee, “[“Ricci” software package]{},”. T. Damour and S. Deser, “[‘Geometry’ of spin 3 gauge theories]{},” [*Ann. Poincare*]{} [**47**]{} (1987) 277. M. A. Vasiliev, “[Consistent equation for interacting gauge fields of all spins in (3+1)-dimensions]{},” [[*Phys. Lett.*]{} [ **B243**]{} (1990) 378–382](http://dx.doi.org/10.1016/0370-2693(90)91400-6). M. A. Vasiliev, “[Nonlinear equations for symmetric massless higher spin fields in (A)dS(d)]{},” [[*Phys. Lett.*]{} [ **B567**]{} (2003) 139–151](http://dx.doi.org/10.1016/S0370-2693(03)00872-4), [[arXiv:hep-th/0304049]{}](http://arxiv.org/abs/hep-th/0304049). S. E. Konstein and M. A. Vasiliev, “[EXTENDED HIGHER SPIN SUPERALGEBRAS AND THEIR MASSLESS REPRESENTATIONS]{},” [[*Nucl. Phys.*]{} [ **B331**]{} (1990) 475–499](http://dx.doi.org/10.1016/0550-3213(90)90216-Z). R. D’Auria and P. Fre, “[CARTAN INTEGRABLE SYSTEMS, THAT IS DIFFERENTIAL FREE ALGEBRAS, IN SUPERGRAVITY]{},”. Lectrues given at September School on Supergravity and Supersymmetry, Trieste, Italy, Sep 6-18, 1982. R. D’Auria and P. Fre, “[Geometric Supergravity in d = 11 and Its Hidden Supergroup]{},” [[*Nucl. Phys.*]{} [ **B201**]{} (1982) 101–140](http://dx.doi.org/10.1016/0550-3213(82)90376-5). P. van Nieuwenhuizen, “FREE GRADED DIFFERENTIAL SUPERALGEBRAS,” Invited talk given at 11th Int. Colloq. on Group Theoretical Methods in Physics, Istanbul, Turkey, Aug 23- 28, 1982. M. A. Vasiliev, “[Equations of motion of interacting massless fields of all spins as a free differential algebra]{},” [[*Phys. Lett.*]{} [ **B209**]{} (1988) 491–497](http://dx.doi.org/10.1016/0370-2693(88)91179-3). M. A. Vasiliev, “[CONSISTENT EQUATIONS FOR INTERACTING MASSLESS FIELDS OF ALL SPINS IN THE FIRST ORDER IN CURVATURES]{},” [[*Annals Phys.*]{} [ **190**]{} (1989) 59–106](http://dx.doi.org/10.1016/0003-4916(89)90261-3). E. D. Skvortsov, “[Mixed-Symmetry Massless Fields in Minkowski space Unfolded]{},” [[arXiv:0801.2268 \[hep-th\]]{}](http://arxiv.org/abs/0801.2268). E. Sezgin and P. Sundell, “[On an exact cosmological solution of higher spin gauge theory]{},” [[arXiv:hep-th/0511296]{}](http://arxiv.org/abs/hep-th/0511296). E. Sezgin and P. Sundell, “[Analysis of higher spin field equations in four dimensions]{},” [*JHEP*]{} [**07**]{} (2002) 055, [[arXiv:hep-th/0205132]{}](http://arxiv.org/abs/hep-th/0205132). F. Kristiansson and P. Rajan, “[Scalar field corrections to AdS(4) gravity from higher spin gauge theory]{},” [*JHEP*]{} [**04**]{} (2003) 009, [[arXiv:hep-th/0303202]{}](http://arxiv.org/abs/hep-th/0303202). E. Sezgin and P. Sundell, “[An exact solution of 4D higher-spin gauge theory]{},” [[ *Nucl. Phys.*]{} [**B762**]{} (2007) 1–37](http://dx.doi.org/10.1016/j.nuclphysb.2006.06.038), [[arXiv:hep-th/0508158]{}](http://arxiv.org/abs/hep-th/0508158). X. Bekaert and N. Boulanger, “[Gauge invariants and Killing tensors in higher-spin gauge theories]{},” [[*Nucl. Phys.*]{} [**B722**]{} (2005) 225–248](http://dx.doi.org/10.1016/j.nuclphysb.2005.06.009), [[arXiv:hep-th/0505068]{}](http://arxiv.org/abs/hep-th/0505068). N. Boulanger, T. Damour, L. Gualtieri, and M. Henneaux, “[Inconsistency of interacting, multigraviton theories]{},” [[*Nucl. Phys.*]{} [ **B597**]{} (2001) 127–171](http://dx.doi.org/10.1016/S0550-3213(00)00718-5), [[arXiv:hep-th/0007220]{}](http://arxiv.org/abs/hep-th/0007220). X. Bekaert, N. Boulanger, and S. Cnockaert, “[Spin three gauge theory revisited]{},” [*JHEP*]{} [**01**]{} (2006) 052, [[arXiv:hep-th/0508048]{}](http://arxiv.org/abs/hep-th/0508048). M. Henneaux and C. Teitelboim, “[Quantization of gauge systems]{},”. Princeton, USA: Univ. Pr. (1992) 520 p. G. Barnich, F. Brandt, and M. Henneaux, “[Local BRST cohomology in the antifield formalism. 1. General theorems]{},” [[*Commun. Math. Phys.*]{} [ **174**]{} (1995) 57–92](http://dx.doi.org/10.1007/BF02099464), [[arXiv:hep-th/9405109]{}](http://arxiv.org/abs/hep-th/9405109). G. Barnich, N. Bouatta, and M. Grigoriev, “[Surface charges and dynamical Killing tensors for higher spin gauge fields in constant curvature spaces]{},” [*JHEP*]{} [**10**]{} (2005) 010, [[arXiv:hep-th/0507138]{}](http://arxiv.org/abs/hep-th/0507138). G. Barnich, F. Brandt, and M. Henneaux, “[Local BRST cohomology in the antifield formalism. II. Application to Yang-Mills theory]{},” [[*Commun. Math. Phys.*]{} [ **174**]{} (1995) 93–116](http://dx.doi.org/10.1007/BF02099465), [[arXiv:hep-th/9405194]{}](http://arxiv.org/abs/hep-th/9405194). J. A. M. Vermaseren, “[New features of FORM]{},” [[math-ph/0010025]{}](http://arxiv.org/abs/math-ph/0010025). J. Engquist and P. Sundell, “[Brane partons and singleton strings]{},” [[*Nucl. Phys.*]{} [**B752**]{} (2006) 206–279](http://dx.doi.org/10.1016/j.nuclphysb.2006.06.040), [[arXiv:hep-th/0508124]{}](http://arxiv.org/abs/hep-th/0508124). J. Engquist, P. Sundell, and L. Tamassia, “[Singleton strings]{},” [[*Fortsch. Phys.*]{} [**55**]{} (2007) 711–716](http://dx.doi.org/10.1002/prop.200610367), [[arXiv:hep-th/0701081]{}](http://arxiv.org/abs/hep-th/0701081). J. Engquist, P. Sundell, and L. Tamassia, “[On singleton composites in non-compact WZW models]{},” [*JHEP*]{} [**02**]{} (2007) 097, [[arXiv:hep-th/0701051]{}](http://arxiv.org/abs/hep-th/0701051). Y. M. Zinoviev, “[On spin 3 interacting with gravity]{},” [[arXiv:0805.2226 \[hep-th\]]{}](http://arxiv.org/abs/0805.2226). [^1]: Work supported by a “Progetto Italia” fellowship. E-mail address: [^2]: E-mail address: [^3]: Also affiliated to INFN. E-mail address: [^4]: We consider only couplings that truly deform the initial abelian gauge algebra into a nonabelian one, similarly to what happens when coupling $N^2-1$ Maxwell fields in order to obtain the Yang-Mills $SU(N)$ theory. Interesting results and references on abelian couplings can be found in the review [@Fotopoulos:2008ka]. [^5]: We use mostly positive signature and $R=g^{\m\r} g^{\n\s} R_{\m\n\r\s}$. The Fierz-Pauli action $\int d^Dx(-\frac12 \partial^\m h^{\r\s}\partial_\m h_{\r\s}+\cdots)$ is recuperated modulo boundary terms from $\frac{1}{(\ell_p)^{D-2}}\int d^D x\sqrt{-g}R(g)$ upon substituting $g_{\mu\nu}=\eta_{\mu\nu}+\sqrt{2}(\ell_p)^{\frac{D-2}2}h_{\mu\nu}$. [^6]: The initial choice of free kinetic terms affects the classical anomaly and the final form of anomaly cancelation terms. [^7]: Repeated indices distinguished by sub-indexation are implicitly symmetrized, $\nabla\cdot V_{\m(s-1)}\equiv \nabla^\n V_{\n\m(s-1)}$ and $V'_{\mu(s-2)}\equiv g^{\n\r}V_{\n\r\m(s-2)}$. [^8]: We use conventions where $h_{\a\b}$ and $\phi_{\a\b\c}$ are dimensionless. The linearized spin-2 Weyl tensor $w_{\a\b\c\d}=s_{\a\b\c\d}-\frac{2}{D-2}(\bar g_{\a[\g}s_{\d]\b}-\bar g_{\b[\g}s_{\d]\a}) +\frac{2}{(D-1)(D-2)}\bar g_{\a[\g}\bar g_{\d]\b}s$, where $s_{\a\b\c\d}\equiv-\overline \nabla_{\g}\overline \nabla_{[\a}h_{\b]\d}+\overline \nabla_{\d}\overline \nabla_{[\a}h_{\b]\d}+ \l^2(\bar g_{\g[\a}h_{\b]\d}-\bar g_{\d[\a}h_{\b]\g})\,$ has the property that at zeroth order $\bar g^{\a\b}s_{\a\c\b\d}\approx0$ and $\overline \nabla^{\a}s_{\a\b\c\d}\approx 0$. The form of the 2-3-3 FV vertex given in (\[FVvertex\]) reflects the initial choice of free Lagrangian made in (\[Fs\]). [^9]: Modulo the Bianchi identities of $R_{\mu\nu}$, there are 49 four-derivative terms that are proportional to the spin-2 field equations: 25 terms of the form $R_{..}\6^2_{..}\phi_{...}\phi_{...}$ and 24 terms of the form $R_{..}\6_{.}\phi_{...}\6_{.}\phi_{...}$. Adding an arbitrary linear combination of these to the vertex, lifting the derivatives from $\tilde{h}$, subtracting the “cohomological” vertex, and finally factoring out the spin-3 equations of motion by eliminating $\partial^2\phi$, yields a simple system of equations that allow us to fit the coefficients. [^10]: Consider a master action $W_\l=\stackrel{(0)}{W_\l}+g\stackrel{(1)}{W_\l}+\cdots$ with $\stackrel{(1)}{W_\l}=\int (a^\l_2+a^\l_1+a^\l_0)$ where $a^\l_2$, $a^\l_1$ and $a^\l_0$, respectively, contain the nonabelian deformation of the gauge algebra, the corresponding gauge transformations and vertices. The master equation amounts to $\c^\l a^\l_2=0$, $\c^\l a_1^\l+\d^\l a_2^\l=d c_1^\l$ and $\c^\l a_0^\l+\d^\l a^\l_1=d c_0^\l$ where $\c^\l$ and $\d^\l$ have $\l$ expansions starting at order $\l^0$. Since the system is linear and determines $a_1^\l$ and $a_0^\l$ for given $a_2^\l$ it follows that all $a_i^\l$ scale with $\l$ the same way in the limit $\l\rightarrow 0$. [^11]: The situation in higher-spin gauge theory is analogous to that in string theory: in both cases the microscopic formulation is defined in terms of “vertex operators” living in an associative algebra associated with an “internal” quantum theory. As a result, the graviton vertex receives corrections leading to a microscopic frame that is different from the canonical Einstein frame (see [@Sezgin:2005hf] for a related discussion). [^12]: The gauge- invariant characterization of the amplitudes is provided by on-shell closed forms built from $\Phi$ and $A$. A simple set of such “observables” are the zero-form charges found in [@Sezgin:2005pv]. [^13]: We use the notation $\partial^{N}_{\mu_1\ldots\mu_N}\equiv \partial_{\mu_1}\ldots\partial_{\mu_N}\,$.

--- abstract: 'Studies of the H0.1em[I]{} in galaxies have clearly shown that subtle details of the H0.1em[I]{} distribution and kinematics often harbour key information for understanding the structure and evolution of galaxies. Evidence for the accretion of material has grown over the past many years and clear signatures can be found in H0.1em[I]{} observations of galaxies. We have obtained new detailed and sensitive H0.1em[I]{} synthesis observations of three nearby galaxies which are suspected of capturing small amounts of H0.1em[I]{} and show that indeed accretion of small amounts of gas is taking place in these galaxies. This could be the same kind of phenomenon of material infall as observed in the stellar streams in the halo and outer parts of our galaxy and M 31' author: - Thijs van der Hulst - Renzo Sancisi title: Evidence for gas accretion in galactic disks --- epsf \#1[[*\#1*]{}]{} \#1[[*\#1*]{}]{} = \#1 1.25in .125in .25in Introduction ============ It has long been an important question how galaxies form and evolve into the great variety of morphologies we see in the nearby universe. The common view is that galaxies form hierarchically from small building blocks through a sequence of merger events. This view has been motivated by the cold dark matter (CDM) simulations which suggest that all galaxies form initially as discs (e.g. Baugh, Cole & Frenk 1996, Klypin et al. 1999, Moore et al. 1999, Steinmetz & Navarro 2002, Abadi et al. 2003). The signatures of the disks can subsequently be erased by multiple galaxy mergers (Barnes 1992). In this paper we consider the formation of large disks through the accretion of small companions, the process often indicated as the nurture of galaxies. Evidence in support of the prediction of this scenario by cold dark matter simulations has come from several directions. Zaritsky (1995) and Zaritsky & Rix (1997) determined the star formation rate of several tens of galaxies from stellar population studies. They also determined the shapes of these galaxies from photometry and found that overall the galaxies with asymmetries in their shapes have the youngest stellar populations. They acsribed this to recent accretion events. Kinematic lopsidedness, observed in the 2-dimensional H0.1em[I]{} velocity fields of galaxies (Verheijen 1997, Swaters et al. 1999) has also been considered as a result of recent minor mergers. Finally there are at least some twenty examples of galaxies which in H0.1em[I]{} show either signs of interactions and/or have small companions (Sancisi 1999). This suggests that galaxies often are in an environment where material for accretion is available. Even clearer evidence that accretion events play an important role has come from studies of the distribution and kinematics of stars in the Milky Way halo. The discovery of the Sagittarius dwarf galaxy (Ibata et al. 1994) has been major proof that accretion is still taking place at the present time. Since such minor merger remnants retain information about their origin for a long time (Helmi & White 2000) studies of the distribution and kinematics of “stellar streams” can in principle be used to trace the merger history of the Milky Way (Helmi & de Zeeuw 2001). Such “stellar streams” are not only seen in the Milky Way, but have also been discovered in the Local Group galaxy M 31 (Ibata et al. 2001, Ferguson et al. 2002, McConnachie et al. 2003). The substructure in the halo of M 31 is another piece of clear evidence that minor mergers still take place. The question now is how to trace such events in more distant galaxies, where we can not observe individual stars, but require other means of detecting the signature of accretion. The use of H0.1em[I]{} is very powerful as it can image interactions very effectively by studying the H0.1em[I]{} distributions and kinematics of nearby galaxies. Examples can be found in Sancisi (1999). The improved sensitivity of modern synthesis radio telescopes brings within reach the detection of faint H0.1em[I]{} signatures of accretion events and we expect that new observations of nearby galaxies will reveal these in the coming decade. To further illustrate this point we here present a few examples of such signatures: NGC 3359, NGC 4565 and NGC 6946. The observations ================ All three galaxies have been observed recently with the Westerbork Synthesis Radio Telescope (WSRT) using the new front-end and correlator providing a much improved sensitivity. NGC 3359 and NGC 4565 were each observed for 12 hours providing a sensitivity of 0.85 mJy/beam for spatial and velocity resolutions of 30$^{\prime\prime}$ and 10 km s$^{-1}$. NGC 6946 was observed for 15 $\times$ 12 hours and reaches a sensitivity of 0.5 mJy/beam for spatial and velocity resolutions of 60$^{\prime\prime}$ and 5 km s$^{-1}$. We will discuss each case individually below. NGC 3359 -------- NGC 3359 is a nearby barred spiral galaxy (Hubble type SB(rs)c) which has been observed in H0.1em[I]{} by Broeils (1992) as part of a study of the mass distribution of a sample of nearby spiral galaxies. It has a total mass of $1.2 \times 10^{11}$ M$_{\odot}$ and an H0.1em[I]{} mass of $7.5 \times 10^{9}$ M$_{\odot}$ (Broeils & Rhee, 1997, adjusted for a Hubble constant of 72 km/s/Mpc). It has well developed spiral structure both in the optical and in H0.1em[I]{}. Kamphuis & Sancisi (1994, see also Sancisi 1999) pointed out the presence of an H0.1em[I]{} companion which appears distorted and may connect to the main H0.1em[I]{} disk of NGC 3359. This observation already suggested the possibility of witnessing accretion of gas by a large galaxy. Our new, more sensitive observations are shown in Figure 1 and convincingly display an H0.1em[I]{} connection between the distorted H0.1em[I]{} companion and the main galaxy. The mass of the H0.1em[I]{} companion is $1.8 \times 10^{8}$ M$_{\odot}$ or 2.4% of the H0.1em[I]{} mass of NGC 3359. Also shown in Figure 1 is a blow-up of the H0.1em[I]{} distribution of the companion which is clearly distorted and shows a tail pointing towards and connecting with the outer spiral structure of NGC 3359. No optical counterpart has yet been identified. The velocity structure of the H0.1em[I]{} companion and the connecting H0.1em[I]{} fits in very well with the regular velocity field of NGC 3359. This is shown in Figure 2 where we display the emission in the individual channels superposed on the total H0.1em[I]{} image of NGC 3359. Contours of different shades of grey (low velocities are dark, high velocities are light) denote the outer edge of the H0.1em[I]{} emission in each of the velocity channels and thus display the basic kinematics of the H0.1em[I]{} without any further analysis of individual velocity profiles. The regularity of the velocities suggests that the process has been going on slowly for at least one rotational period which is of the order of 1.7 Gy. NGC 4565 -------- NGC 4565 is a large edge-on galaxy of Hubble type SAb which was first observed in H0.1em[I]{} by Sancisi (1976) in an early search for galaxies with warped H0.1em[I]{} disks. Rupen (1991) observed NGC 4565 with much higher resolution and presented a detailed study of the kinematics and the warp. NGC 4565 has a small optical companion 6$^{\prime}$ to the north of the center of NGC 4565, F378-0021557, which has $7.4 \times 10^{7}$ M$_{\odot}$ of H0.1em[I]{} compared to an H0.1em[I]{} mass of $2.0 \times 10^{10}$ M$_{\odot}$ for NGC 4565. Another companion, NGC 4562, somewhat larger in H0.1em[I]{} ($2.5 \times 10^{8}$ M$_{\odot}$) and brighter optically can be found found 15$^{\prime}$ to the south-west of the center of NGC 4565. The H0.1em[I]{} distribution, superposed on the DSS is shown in Figure 3. The asymmetric warp is clearly visible. The warp sets in at the edge of the optical disk and does exhibit a bit of apparent thickening of the H0.1em[I]{} disk visible to the north-west and south-east of the disk, a result from projection effects along the line of sight. Inspection of individual channel maps brings to light that in addition to the warp the H0.1em[I]{} distribution shows additional, low surface brightness emission to the north of the center, in the direction of the faint companion F378-0021557. This is best shown in Figure 4 where we show the outer contours of the H0.1em[I]{} emission in individual velocity channels superposed on the total H0.1em[I]{} distribution of NGC 4565. In this Figure one clearly sees that there is an additional extraplanar H0.1em[I]{} component pointing into the direction of F378-0021557, suggestive of a connection between the presence of F378-0021557 and disturbances in the H0.1em[I]{} disk of NGC 4565. Whether this is recently accreted material can not easily be verified, but it definitely shows that there is a component in the H0.1em[I]{} disk of NGC 4565 which can not be associated with the warp and is the kind of small (in terms of H0.1em[I]{} mass) asymmetry that could be the result of accretion. NGC 6946 -------- NGC 6946 is a bright, nearby spiral galaxy of Hubble type SAB(rs)cd which has been studied in H0.1em[I]{} numerous times (Rogstad et al. 1973, Tacconi & Young 1986, Kamphuis 1993). It was in this galaxy that Kamphuis and Sancisi (1993) found the first evidence for an anomalous velocity H0.1em[I]{} component which they associated with outflow of gas from the disk into the halo as a result of stellar winds and supernova events. Evidence for such a component is now evident in more galaxies as discussed by Fraternali et al. (2002, 2003, and also this volume). A much more detailed study of the anomalous H0.1em[I]{} and the structure in the H0.1em[I]{} disk is being carried out by Boomsma et al. (this volume, see also Fraternali et al. this volume) on the basis of very sensitive observations with the WSRT. Here we concentrate on a low resolution (60$^{\prime\prime}$) version of these data. Figure 5 shows a total H0.1em[I]{} image of NGC 6946 down to column density levels of $1.3 \times 10^{-19}$ cm$^{-2}$. To the west two small companion galaxies can be seen. The most intruiging feature is the faint whisp to the north-west of the main H0.1em[I]{} disk of NGC 6946. This faint H0.1em[I]{} extension can only be brought out at this resolution and appears to form a faint H0.1em[I]{} filament which blends smoothly (also kinematically) with the H0.1em[I]{} disk of NGC 6946 at a position some 11$^{\prime}$ (or 19 kpc) south of the tip of the filament. There is no detectable connection with the two companion galaxies farther to the west. The spatial and velocity structure of the object are so regular, yet only connected to the main H0.1em[I]{} disk at one side that we prefer an explanation in terms of a tidally stretched, infalling H0.1em[I]{} object. So yet another example of accretion of small amount of gas onto a large H0.1em[I]{} disk. Similar examples, though much more massive in H0.1em[I]{}, are perhaps the filament discovered in NGC 2403 (Fraternali et al. 2002, 2003 and also this volume), a long H0.1em[I]{} filament in M 33 (van der Hulst, unpublished) and the extraplanar filaments in the northern part of the H0.1em[I]{} halo of NGC 891 (Fraternali et al., this volume). Concluding remarks ================== We have shown three cases with strong evidence for the accretion of small amounts of H0.1em[I]{}. These will not be unique. Such faint features can only be seen in sensitive H0.1em[I]{} observations as the H0.1em[I]{} masses involved are rather modest. We therefore expect that with the increased sensitivity of modern synthesis radio telescopes, more examples will be discovered in the coming decade. There probably is a range of H0.1em[I]{} masses for these accretion events as is already apparent from the six cases mentioned here: NGC 891, NGC 2403, NGC 3359, NGC 4565, NGC 6946 and M 33. The next question to ask is what the effect of accretion will be on the disk of the main galaxy. There may very well be a connection with the star formation activity in galaxies such as NGC 6946 and NGC 2403 and the evidence for gas infall. This then in turn can cause the observed phenomenon of gas outflows from the active disks as seen in these galaxies (Boomsma et al. and Fraternali et al. this volume). It is quite clear that future sensitive and detailed studies of the H0.1em[I]{} in nearby galaxies will provide a more complete census of the phenomena discussed in this paper and enable us to address these issues further and obtain more definitive answers. Abadi M.G., Navarro J.F., Steinmetz M., Eke V.R., 2003, , 591, 499 Baugh C.M., Cole S., Frenk C.S., 1996, , 283, 1361 Barnes J.E., 1992, , 393, 484 Broeils, A. H., 1992, PhD thesis, University of Groningen Broeils, A. H., Rhee, M. -H., 1997, , 324, 877 Ferguson, A. M. N., Irwin, M. J., Ibata, R. A., Lewis, G. F., & Tanvir, N. R. 2002, , 124, 1452 Fraternali, F., van Moorsel, G., Sancisi, R., & Oosterloo, T. 2002, , 123, 3124 Fraternali, F., Oosterloo, T., Sancisi, R., & van Moorsel, G. 2001, , 562, L47 Helmi, A. & Tim de Zeeuw, P. 2000, , 319, 657 Helmi, A. & White, S. D. M. 2001, , 323, 529 Ibata, R., Irwin, M., Lewis, G., Ferguson, A. M. N., & Tanvir, N. 2001, Nature, 412, 49 Ibata, R. A., Gilmore, G., & Irwin, M. J. 1994, Nature, 370, 194 Kamphuis, J. 1993, PhD thesis, University of Groningen Kamphuis, J. & Sancisi, R. 1994, Panchromatic View of Galaxies. Their Evolutionary Puzzle, 317 Kamphuis, J. & Sancisi, R. 1993, , 273, L31 Klypin, A., Kravtsov, A. V., Valenzuela, O., & Prada, F.  1999, , 522, 82 Moore, B., Ghigna, S., Governato, F., Lake, G., Quinn, T., Stadel, J., & Tozzi, P. 1999, , 524, L19 McConnachie, A. W., Irwin, M. J., Ibata, R. A., Ferguson, A. M. N., Lewis, G. F., & Tanvir, N. 2003, , 343, 1335 Rogstad, D. H., Shostak, G. S., & Rots, A. H.  1973, , 22, 111 Rupen, M. P. 1991, , 102, 48 Sancisi, R. 1976, , 53, 159 Sancisi, R. 1999, IAU Symp. 186: Galaxy Interactions at Low and High Redshift, 186, 71 Sancisi, R. 1999, , 269, 59 Steinmetz, M. & Navarro, J. F. 2003, New Astronomy, 8, 557 Tacconi, L. J. & Young, J. S. 1986, , 308, 600 Zaritsky, D. 1995, , 448, L17 Zaritsky, D. & Rix, H. 1997, , 477, 118

--- abstract: 'It is known that any bipartite unitary operator of Schmidt rank three is equivalent to a controlled unitary under local unitaries. We propose a standard form of such operators. Using the form we improve the upper bound for the entanglement cost to implement such operators under local operations and classical communications (LOCC), and provide a corresponding protocol. A part of our protocol is based on a recursive-control protocol which is helpful for implementing other unitary operators. We show that any bipartite permutation unitary of Schmidt rank three can be implemented using LOCC and two ebits. We give two protocols for implementing bipartite permutation unitaries of any Schmidt rank $r$, and showed that one of the protocol uses $O(r)$ ebits of entanglement and $O(r)$ bits of classical communication, while these two types of costs for the other protocol scale as $O(r\log r)$ but the actual values are smaller for all $r<1100$. Based on this we obtain upper bounds of the number of nonlocal CNOT gates needed to implement bipartite classical reversible maps using classical circuits under two different conditions. We also quantify the entangling power of bipartite permutation unitaries of Schmidt rank two and three. We show that they are respectively $1$ ebit and some value between $\log_2 9 - 16/9$ and $\log_2 3$ ebits.' author: - Lin Chen - Li Yu bibliography: - 'channelcontrol.bib' title: Entanglement cost and entangling power of bipartite unitary and permutation operators --- Introduction {#sec:intro} ============ The implementation of unitary operations is a key task in quantum information processing. Bipartite unitaries are a particularly important class to study, because they are the base case for studying multipartite unitaries. Many tasks in quantum communication, games and cryptography are restricted to two parties. The evaluation of entanglement cost and/or classical resources for implementing unitary operations belong to a type of communication cost problems in quantum information theory. It has applications in the study of quantum networks and distributed quantum computation, see [@Akibue15; @ick16] for recent progress on implementing nonlocal unitaries or isometries on multiple qubits, using shared entanglement in a network or using a limited set of basic gates. Any bipartite unitary is the product of controlled unitaries [@bry02; @blb05]. The controlled unitary can be implemented with local operations and classical communication (LOCC) and a maximally entangled state [@ygc10]. The entanglement cost scales with the logarithm of the number of terms of control. The number can be as large as the dimension of the controlling system. Bipartite unitaries of Schmidt rank not greater than three are equivalent to controlled unitaries under local unitaries [@cy13; @cy14; @cy14ap]. Every Schmidt-rank-two bipartite unitary can be implemented using one ebit and LOCC [@cy13], but the best upper bound for the entanglement cost of Schmidt-rank-three unitaries appears to depend on the dimensions of the Hilbert spaces: an upper bound on $d_A\times d_B$ system is $\log_2 \min\{d_A^2,d_B\}$ ebits for $d_A\le d_B$ [@cy14ap]. In this paper we show that all Schmidt-rank-three bipartite unitaries can be implemented using $\log_2\min\big{\{}d_A,d_B^2,4\lfloor d_B/2\rfloor+2\big{\}}$ ebits, where $A$ is the controlling side of the unitary. This is presented in Theorem \[thm:sch3\] based on a standard form constructed in Eq. . We present a protocol for implementing some bipartite unitaries using multiple levels of control, and apply it to Schmidt-rank-three unitaries. Reducing the entanglement cost for implementing nonlocal unitary gates is a key problem in computation or communication tasks on networks, because entanglement is often imperfect and costly to produce. A protocol that uses less entanglement would have less error in the implemented unitary gate, giving rise to less error in the final outcome of the computation or communication task. Some tasks may involve multipartite unitaries or non-unitary operations, and studying the entanglement cost of bipartite unitaries may help the study of the entanglement cost of those operations. The classical communication cost of the protocols in this paper is linear in the entanglement cost. Thus our protocols have less classical communication cost than the previous protocols. This is beneficial since classical communication is subject to noise and security concerns. It is known that there is a dimension-independent upper bound for the entanglement cost of bipartite permutation unitaries with the help of a one-qubit ancilla on one side [@cy15]. The ancilla can be dropped from this statement at the cost of using more entanglement, since it can be prepared from another shared entangled pair of qubits. We construct a standard form of bipartite complex permutation unitaries of Schmidt rank $r$, when a “big row” of the unitary contains at least $r-1$ nonzero blocks. (The big row is defined in Sec \[sec:pre\].) We further investigate the maximum number of distinct nonzero diagonal blocks of a controlled permutation unitary of Schmidt rank $r$. The above two results give upper bounds of entanglement cost for implementing the corresponding types of unitaries. This is presented in Lemmas \[le:persch\] and \[lm:diagonal\_blocks\]. When the Schmidt rank is not greater than four, we give tighter upper bounds of entanglement cost in Lemmas \[lm:sch2\] and \[lm:sch3perm\], and Corollary \[cr:sr4perm\]. In particular, any Schmidt-rank-three bipartite permutation unitary needs only $2$ ebits to implement. We give a protocol that implements any bipartite permutation unitary of Schmidt rank $r$ using $O(r \log r)$ ebits of entanglement and $O(r \log r)$ bits of classical communication. Then we present another protocol for the same task with the costs only scaling as $O(r)$, but the actual values are larger for all $r<1100$, as discussed below Theorem \[thm:permutation\]. These results give upper bounds for the number of nonlocal CNOT gates for implementing a bipartite classical reversible map using a classical circuit under two different conditions (A *nonlocal CNOT gate* is a CNOT gate that acts across the two parties, as opposed to acting locally on the bits within each party). The number is larger in the case that ancillas are required to be restored to the initial value, compared to the opposite case, and both results are under the assumption that the initial values of the ancillas are known. These results are an exponential improvement over the corresponding results in [@cy15]. An example of a Schmidt-rank-four permutation unitary is given in Sec. \[subsec:permutation\_example\] with its entanglement cost analyzed. As a byproduct, we point out that the expression of bipartite complex permutation unitaries in is further evidence supporting a recent conjecture on the ranks and marginals of multipartite states [@chl14]. Classical reversible circuits may have lower energy cost compared to the circuits that involve erasures [@Bennett73]. The current paper touches upon the topic of classical reversible circuits, not only because our main result applies to it, but also we find that the design for the classical reversible circuits could provide hints for designing better quantum LOCC protocols or quantum unitary circuits. The results so far are for the upper bound of entanglement cost for implementing bipartite unitaries. Another interesting topic is finding lower bounds for this quantity, such as the entangling power defined in . Any Schmidt-rank-$r$ unitary can have entangling power at most $\log_2 r$ ebits, see the beginning part of Sec. \[ssec:ent\_power\_permutation\]. In the case of $r=3$, it is much smaller than the upper bound in this paper when $d_A$ and $d_B$ are large. Recently, Soeda *et al* [@stm11] proved that $1$ ebit of entanglement is needed for implementing any 2-qubit controlled unitary by LOCC when the resource state is of Schmidt rank two. Stahlke *et al* [@sg11] proved that if the Schmidt rank of the resource state is equal to the Schmidt rank of the bipartite unitary, and the unitary can be implemented by the state using LOCC or separable operations, then the resource state has equal nonzero Schmidt coefficients. In Example \[ex:ex2\] we present a class of Schmidt-rank-three unitaries for which we do not know of a protocol with constant entanglement cost. In fact it is an open problem whether there is a constant upper bound for the entanglement cost of all Schmidt-rank-three bipartite unitaries. Next, we show that the entangling power of any Schmidt-rank-two bipartite permutation unitary is exactly 1 ebit by Lemma \[le:sch2perm\_entpower\]. The counterpart of Schmidt-rank-three permutation unitary is some value between $\log_2 9 - 16/9$ and $\log_2 3$ ebits, as shown in Proposition \[le:sch3perm\_entpower\]. Again, there is a curious gap between the best known entanglement cost and the entangling power, similar to the case of general Schmidt-rank-three unitaries. The rest of this paper is organized as follows. In Sec. \[sec:brief\] we briefly introduce the appendix. In Sec. \[sec:pre\] we introduce the notations and preliminary lemmas used in the paper. In Sec. \[sec:main\] we present the main result on Schmidt-rank-three bipartite unitary operators. In Sec. \[sec:permutation\] we study bipartite complex permutation unitaries. We first present some preliminary lemmas, and then investigate the entanglement cost of bipartite permutation unitaries of Schmidt rank up to three in Sec. \[subsec:permutation\_small\], and study the protocol and entanglement cost for general bipartite permutation unitaries in Sec. \[subsec:permutation\_large\]. An example is given in Sec. \[subsec:permutation\_example\], and the entangling power of bipartite permutation unitaries is studied in Sec. \[ssec:ent\_power\_permutation\]. Finally we conclude in Sec. \[sec:con\]. Summary of technical results {#sec:brief} ============================ To enhance readability we briefly summarize the results of the current work and their relationships in this section. We have introduced Theorem \[thm:sch3\] in the introduction, which reduces the entanglement cost to about half of the previous upper bound in [@cy14ap] for large classes of bipartite Schmidt-rank-three unitaries. To study this theorem, we introduce Lemma \[lm:lm2\] as a hard case among the possible forms of bipartite unitaries of Schmidt rank three. The proof of Theorem \[thm:sch3\] makes use of Protocols \[ptl1b\] and \[ptl\_gp\], which are respectively a new two-level controlled unitary protocol, and a protocol from [@ygc10] for implementing unitaries with group-type expansion. We study some basic properties of the real or complex bipartite permutation unitaries in terms of the Schmidt rank in Lemmas \[le:persch\] and \[lm:diagonal\_blocks\]. The results are used throughout Sec. \[sec:permutation\]. In Lemmas \[lm:sch2\] and \[lm:sch3perm\] we investigate the structure and entanglement cost for (complex) permutation unitaries of Schmidt rank two or three. In Theorem \[thm:permutation\] we show that any bipartite permutation unitary of Schmidt rank $r$ can be implemented using local operations with the help of $\min\{\log_2(B_{r+1})+r+\log_2 r, 8r-8\}$ ebits of entanglement and twice as many bits of classical communication, where $B_j$ is the Bell number defined before Lemma \[le:combinations\_partial\_permutation\]. The two terms in the result arise from Protocol \[ptl2\] and Protocol \[ptl3\], respectively. This significantly improves over the result in Theorem 22 of [@cy15], which states that such unitary can be implemented using LOCC with $3\times 2^r$ ebits. In Theorem \[thm:classical\_bipartite\], we adapt the two methods of implementing bipartite permutation unitaries in the proof of Theorem \[thm:permutation\] to the decomposition of classical bipartite reversible circuits into local gates and nonlocal CNOT gates. In Proposition \[le:sch3perm\_entpower\], we prove that the entangling power \[defined in Eq. \] of bipartite permutation unitaries of Schmidt rank three is in the range of $[\log_2 9 - 16/9, \log_2 3]$ ebits. Preliminaries {#sec:pre} ============= In this section we introduce the notations and preliminary lemmas used in the paper. Let $\sigma_x,\sigma_y,\sigma_z$ be the usual $2\times 2$ Pauli matrices. Denote the computational-basis states of the bipartite Hilbert space $\cH=\cH_A\ox\cH_B$ by ${|i,j\rangle},i=1,\cdots,d_A$, $j=1,\cdots,d_B$. Let $I_A$ and $I_B$ be the identity operators on the spaces $\cH_A$ and $\cH_B$, respectively. We also denote $I_d$ and $0_d$, respectively, as the identity and zero matrix of order $d$. The bipartite unitary gate $U$ acting on $\cH$ has *Schmidt rank* $n$ if there is an expansion $U=\sum^n_{j=1}A_j \ox B_j$ where the $d_A\times d_A$ matrices $A_1,\cdots,A_n$ are linearly independent, and the $d_B\times d_B$ matrices $B_1,\cdots,B_n$ are also linearly independent. An equivalent definition named as the operator-Schmidt rank has been presented in [@Nielsen03; @Tyson03]. The above expansion is called the *Schmidt decomposition*. We name the $A$ $(B)$ space of $U$ as the space spanned by all $A_j$ $(B_j)$ that appear in a Schmidt decomposition of $U$. It is well defined in the sense that the space is independent of the specific choice of the Schmidt decomposition. Next, $U$ is a *controlled unitary gate*, if $U$ is equivalent to $\sum^{d_A}_{j=1}{| j\rangle\!\langle j |}\ox U_j$ or $\sum^{d_B}_{j=1}V_j \ox {| j\rangle\!\langle j |}$ via local unitaries. To be specific, $U$ is a controlled unitary from $A$ or $B$ side, respectively. In particular, $U$ is controlled in the computational basis from $A$ side if $U=\sum^{d_A}_{j=1}{| j\rangle\!\langle j |}\ox U_j$. Bipartite unitary gates of Schmidt rank two or three are equivalent to controlled unitaries via local unitaries [@cy13; @cy14; @cy14ap]. We shall denote $V\op W$ as the ordinary direct sum of two matrices $V$ and $W$, and denote $V\op_B W$ as the direct sum of $V$ and $W$ from the $B$ side. The latter is called the $B$-direct sum, and $V$ and $W$ respectively act on two subspaces $\cH_A\ox\cH'_B$ and $\cH_A\ox\cH''_B$ such that $\cH_B'\perp\cH_B''$. A permutation matrix (or called “permutation unitary” or “real permutation matrix”) is a unitary matrix containing elements $0$ and $1$ only. The partial permutation matrix is a matrix with elements being $0$ and $1$ only, satisfying that any row sum or column sum is not greater than $1$. So the partial permutation matrix may be not unitary. A bipartite controlled-permutation matrix $U$ is a permutation matrix controlled in the computational basis of one system, i.e., $U=\sum_j P_j \ox V_j$, where the projectors $P_jP_k=\d_{jk}P_j$, $V_j$ is a permutation unitary, and each $P_j \ox V_j$ is a *term* of $U$. A complex permutation matrix is a unitary matrix with exactly one nonzero element in each row and column. A “big row” of the $d_A d_B\times d_A d_B$ unitary matrix $U$ refers to a $d_B\times d_Ad_B$ submatrix given by $_A {\langlej|}U$, for some $j\in\{1,\dots,d_A\}$. Similarly, a “big column” of $U$ refers to a $d_A d_B\times d_B$ submatrix given by $U{|j\rangle}_A$, for some $j\in\{1,\dots,d_A\}$. A “block” of $U$ refers to a $d_B\times d_B$ submatrix given by $_A {\langlej|}U{|k\rangle}$, for some $j,k\in\{1,\dots,d_A\}$, and when $j=k$, the block is called a “diagonal block.” In all the protocols in this paper, the computational basis starts from ${|0\rangle}$ instead of ${|1\rangle}$. For an $n$-dimensional system, we respectively define the Fourier gate $F={1\over \sqrt{n}}\sum_{j,k=0}^{n-1} e^{2\pi i jk/n}{|j\rangle\!\langlek|}$, and the $Z$ gate usually as $Z=\sum_{j=0}^{n-1} e^{2\pi i j/n}{|j\rangle\!\langlej|}$ but sometimes generalizing the ${|j\rangle\!\langlej|}$ to a high-rank projector, see Protocol \[ptl\_ct\]. The $Z$ basis is the computational basis. The $Z$-information means the information about which computational basis state that the state of the quantum system is in. In this paper, the “entanglement cost” of a bipartite unitary $U$ is defined as \[eq:def\_cost\] E\_c(U)=\_p E\_c(p), where $p$ is any one-shot exact deterministic LOCC protocol to implement $U$, and $E_c(p)$ is the amount of initial entanglement needed in the protocol. “One-shot” means that only one copy of the unitary is implemented, while the word “exact” excludes the case that some other unitary that might approximate the given unitary is implemented, and “deterministic” means that the unitary is implemented with no chance of failure. The Schmidt rank of initially entangled state and the dimension of ancillary space are finite in each protocol $p$, and there is no constant upper bound for these quantities. In the case that the resource entangled state is mixed, we suggest to use the entanglement of formation [@pv07] as the entanglement measure, although we do not discuss the mixed entangled state in this paper. If there is entanglement left after the protocol, subtraction of the latter from the cost would lead to definitions of assisted entanglement cost. It is beyond the scope of this paper. The unit for entanglement is “ebit.” The entanglement contained in a maximally entangled pure state of Schmidt rank $N$ is regarded as $\log_2 N$ ebits. Also, to simplify the notation, every bit of classical communication used in a protocol is called a “c-bit.” If the classical message is a signal among $N$ equally possible signals, the amount of classical communication is regarded as $\log_2 N$ c-bits. Linear algebra -------------- Here we present a few preliminary results of linear algebra used throughout our paper. \[le:vandermonde\] Let $D$ be a diagonal unitary matrix. The following four statements are equivalent.\ (i) $D$ has at least three distinct eigenvalues;\ (ii) the identity, $D$ and $D^\dg$ are linearly independent;\ (iii) any unitary in the linear span of the identity and $D$ is proportional to one of them;\ (iv) any multiple of unitary in the linear span of the identity and $D$ is proportional to one of them. $(i)\ra(ii)$. Let $x,y,z$ be the three distinct eigenvalues of $D$. Since $x,y,z$ all have modulus one, the matrix $F=\left( \begin{array}{ccc} 1 & x & x^* \\ 1 & y & y^* \\ 1 & z & z^* \\ \end{array} \right)$ is the product of the diagonal matrix $\diag(x^*,y^*,z^*)$ and a Vandermonde matrix with columns permuted, the latter has determinant $(y-x)(z-x)(z-y)$. Since $x,y,z$ are distinct, $F$ is invertible. Since $F$ is a submatrix of the matrix whose columns are the diagonal vectors of the identity, $D$ and $D^\dg$, the latter are linearly independent. We have proved $(i)\ra(ii)$. $(i)\ra(iii)$. Let the unitary be $U=xI+yD$ where $x,y$ are complex numbers. We have $(xI+yD)(x^* I + y^* D^\dg) = I$, hence $xy^* D^\dg + x^*y D =(1-{|x|}^2-{|y|}^2)I$. Then $(i)\ra(iii)$ follows from $(ii)$, because of $(i)\ra(ii)$. Finally the relations $(ii)\ra(i)$, $(iii)\ra(i)$ and $(iii)\lra(iv)$ are trivial. This completes the proof. In the following lemma, a matrix $A$ is said to be “block diagonal” iff there is a permutation matrix $P$ such that $PAP^\dag=\left( \begin{array}{cc} A_1 & 0 \\ 0 & A_2 \\ \end{array} \right)$, where $A_1$ and $A_2$ are square matrices. We regard a $k\times k$ matrix as being of order $k$. \[le:linearcom\] Suppose $U$ is a unitary matrix of order at least two, and there is a nonzero diagonal matrix $D$ such that there is a nontrivial linear combination of $D$ and $\tilde U=\left( \begin{array}{cc} 0 & U \\ U^\dg & 0 \\ \end{array} \right)$ that is unitary, and we denote it as $V$. Then $X^\dag V X$ is block diagonal, where $X=\left( \begin{array}{cc} W & 0 \\ 0 & W \\ \end{array} \right)$, and $W$ is an $n\times n$ unitary matrix. By assumption, for the given $n\times n$ unitary matrix $U$, where $n\ge 2$, there exists a nonzero complex number $c$ and a nonzero diagonal matrix $D$ such that $V:=c D+\tilde U$ is proportional to a unitary matrix of order $2n$ with $n\ge2$, where $\tilde U=\left( \begin{array}{cc} 0 & U \\ U^\dg & 0 \\ \end{array} \right)$. This $V$ differs from the $V$ in the assertion by a constant factor, hence it suffices to prove the assertion for the current $V$. Suppose $D=\diag(x_1,x_2,\dots,x_n,y_1,y_2,\dots,y_n)$, and the matrix elements of $U$ are $(U)_{ij}=u_{ij}$, $i,j\in\{1,\dots,n\}$. The rows of $V$ are mutually orthogonal. From that the $j$’th and $(n+k)$’th rows of $V$ are orthogonal, where $j,k\in\{1,\dots,n\}$, we have $x^{\ast}_j u^{\ast}_{jk} + u^{\ast}_{jk} y_k=0$, hence \[eq:ykxj\] y\_k=-x\_j\^ u\_[jk]{}0, j,k. Therefore, for any $j\in\{1,\dots,n\}$, it must be that those $x_p$ ($1\le p\le n$) that are equal to $x_j$ and those $y_q$ ($1\le q\le n$) that are equal to $-x_j^{\ast}$ satisfy that their row and column coordinates determine a rectangular block in $U$ consisting of elements $u_{pq}$, and any element of $U$ outside of this block that are in the same rows or the same columns of this block must be zero. The last statement is due to the following reason: Suppose such a rectangular block contains $u_{pq}$, then an element $u_{pq'}$ where $q'$ satisfies $y_{q'}\ne -x_p^{\ast}$ is in the row labeled by $p$ and outside of the rectangular block containing $u_{pq}$; and from , we have $u_{pq'}=0$. Now we consider two cases: The first case is that there exist $j,k\in\{1,\dots,n\}$ such that $x_j\ne x_k$. In this case, the $U$ contains some rectangular blocks that do not overlap in the rows and columns that they occupy. Since $U$ is unitary, these rectangular blocks must be square blocks. Hence, $U$ is block-diagonal after suitable permutation matrices are multiplied before and after it. From the form of $V$, this implies that $V$ is block diagonal in the sense defined before the lemma. Thus the assertion holds with $W$ being the identity matrix $I_n$. The second case is that $x_1=x_2=\dots=x_n$. Then it must be that $y_1=y_2=\dots=y_n=-x_1^{\ast}$, since otherwise it can be deduced from that there would be a column of $U$ that is zero, violating that $U$ is unitary. Since $U$ is unitary, there is an $n\times n$ diagonal matrix $E$ and an $n\times n$ unitary matrix $W$ such that $U=WEW^\dag$, then V=( [cc]{} W & 0\ 0 & W\ ) ( [cc]{} I\_n & E\ E\^& -\^ I\_n\ ) ( [cc]{} W\^& 0\ 0 & W\^\ ), where $\gamma=x_1$. Since $E$, $E^\dag$, and $I_n$ are all diagonal, the matrix $\left( \begin{array}{cc} \gamma I_n & E \\ E^\dg & -\gamma^{\ast} I_n \\ \end{array} \right)$ is the direct sum of $n$ $2\times 2$ matrices up to a similarity transform by a permutation matrix. The rows and columns of the $j$’th $2\times 2$ matrix correspond to the $j$’th and the $(n+j)$’th rows, and the $j$’th and the $(n+j)$’th columns of the original matrix, respectively. This completes the proof. \[le:2x2unitary\] Any real linear combination of the three matrices $I_2$, $\left( \begin{array}{cc} w & 0 \\ 0 & w^* \\ \end{array} \right)$, and $\left( \begin{array}{cc} 0 & x \\ -x^* & 0 \\ \end{array} \right)$ is proportional to a unitary matrix. Let $V=aI_2+b\left( \begin{array}{cc} w & 0 \\ 0 & w^* \\ \end{array} \right)+c\left( \begin{array}{cc} 0 & x \\ -x^* & 0 \\ \end{array} \right)$ where $a,b,c$ are real numbers. By direct computation one can show that $V$ is proportional to a unitary matrix. This completes the proof. Tighter upper bound for entanglement cost of implementing Schmidt-rank-3 unitaries {#sec:main} ================================================================================== On the problem of exact implementation of bipartite nonlocal unitaries using LOCC and shared entanglement, we use or discuss the following three known protocols. (1) The two-way teleportation protocol, i.e., teleporting the system of one party to the other party, performing the unitary there, and teleporting the system back to the original party. (2) The protocol for implementing controlled unitaries in Sec. III of [@ygc10], which is briefly reviewed as Protocol \[ptl\_ct\] below, and it will be called “the basic controlled-unitary protocol.” A simple extension of it is Protocol \[ptl\_ct\_ext\], and the latter is the basis for the two-level controlled Protocols \[ptl1\] and \[ptl1b\]. (3) The group-type protocol in Sec. IV of [@ygc10], which is briefly reviewed as Protocol \[ptl\_gp\] below. Protocol \[ptl1\] is used in Sec. \[sec:permutation\], and Protocols \[ptl1b\] and \[ptl\_gp\] are used in the proof of Theorem \[thm:sch3\] (ii). \[ptl\_ct\] (The basic controlled unitary protocol.) The unitary to be implemented by two parties, Alice and Bob, is $$\label{eq:basic_ct} U=\sum_{k=0}^{N-1} P_k \otimes V_k,$$ where $P_k$ are mutually orthogonal projectors on $\cH_A$, and $V_k$ are unitary operators on ${\cal H}_B$. The $P_k$ may be of rank greater than $1$, meaning that the dimension of $\cH_A$ may be larger than $N$. A figure for this protocol is Fig. 5 of [@ygc10]. This figure was originally for the case that $P_k$ are all rank-one, but with suitable interpretation of the gates in the circuit (see Sec. III C of [@ygc10]), it works for the general case of higher rank $P_k$. For the protocols in this section only, the $X$ gate on a $N$-dimensional Hilbert space is defined as $$\label{eqn:x_minus} X:= \frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}{|(k-1)\mod N\rangle\!\langlek|}.$$ The steps of the protocol are as follows. 0\. The two parties initially share the following entangled state on ancillary systems $a$ and $b$, which are with Alice and Bob, respectively: $$\label{eqn:phi} \vert \Phi\rangle_{ab}=\frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}\vert k\rangle\otimes\vert k\rangle.$$ 1\. Alice performs a controlled-$X^j$ gate $\sum_{j=0}^{N-1} P_j\ox X^j$ on systems $A$ and $a$, with $A$ as the control. (The $X^j$ means $X$ to the power $j$.) Then Alice performs a measurement on $a$ in the standard basis, and sends the result $l$ to Bob. 2\. Bob applies the gate $X^l$ to $b$. This is followed by a controlled gate $\sum_{k=0}^{N-1} {|k\rangle\!\langlek|}\ox V_k$ on $b$ and $B$, with $b$ as the control. Then Bob does a Fourier gate on $b$ (defined in Sec \[sec:pre\]), and measures $b$ in the standard basis. The outcome $m$ is sent to Alice. 3\. Alice carries out a $Z_m = Z^{-m}$ correction on $A$, where the $Z$ is defined as $Z=\sum_{j=0}^{n-1} e^{2\pi i j/N} P_j$ (c.f. Sec. III C of [@ygc10]), and this definition of $Z$ reduces to that in Sec \[sec:pre\] in the case that all $P_k$ are rank-one. This completes the protocol. The resource consumption of the protocol is $\log_2 N$ ebits and $2\log_2 N$ c-bits. \[ptl\_ct\_ext\] (The extension of the basic controlled unitary protocol to the case that some projectors in are replaced with zero operators.) [If the unitary to be implemented by Alice and Bob is given by , but only some $P_k$ are projectors, and some others are zero operators (the output is zero for any input), then the steps of Protocol \[ptl\_ct\] can still be carried out. Note that the controlled-$X^j$ gate in step 1 and the $Z$ gate in step 3 could be defined using the same expression as before but with the $P_j$ understood as being projectors or zero operators. The protocol still uses $\log_2 N$ ebits and $2\log_2 N$ c-bits. Suppose there are $N'<N$ operators among the $\{P_k\}$ that are nonzero; then the same unitary could be carried out with only $\log_2 N'$ ebits and $2\log_2 N'$ c-bits using Protocol \[ptl\_ct\]. Nonetheless, the less efficient protocol turns out to be useful in Protocols \[ptl1\] and \[ptl1b\] below. ]{} Next, we introduce a recursive-control protocol for implementing some bipartite unitaries with LOCC and initial entanglement. \[ptl1\] (Protocol for implementing a bipartite unitary with two levels of control — The special case that the lower-level controlled unitaries are controlled from a fixed side.) The bipartite unitary to be implemented on ${\cal H}_A\otimes {\cal H}_B$ is of the following form: $$\label{eq:calu} U=\sum_{k=0}^{M-1} P_k\otimes S^E_k,$$ where ${\cal H}_A={\cal H}_C\otimes {\cal H}_D$, and ${\cal H}_E={\cal H}_D\otimes {\cal H}_B$, and $P_k$ are orthogonal projectors on $\cH_C$, and $$\label{eq:sek1} S^E_k=\sum_{j=0}^{n_k-1} U^D_{kj} \otimes Q^{(k)}_j$$ are controlled unitaries with local unitaries $U^D_{kj}$ on $\cH_D$. The $Q^{(k)}_j$ are projectors on $\cH_B$ and are orthogonal among different $j$ for the same $k$. Let $N:=\max\{n_k: k=0,1,\dots,M-1\}$. By introducing some zero operators to the set of $Q^{(k)}_j$ and calling the new operators $\tilde Q^{(k)}_j$, we may write all $S^E_k$ using $N$ terms: $$\label{eq:sek2} S^E_k=\sum_{j=0}^{N} U^D_{kj} \otimes \tilde Q^{(k)}_j,$$ where $U^D_{kj}$ are still local unitaries and some of them are not present in Eq. . The idea of the protocol can be roughly summarized as follows. The higher level of the protocol is “$k$ controls $S^E_k$,” and the lower level is “$j$ controls $U^D_{kj}$.” The steps are as follows. 0\. Alice and Bob share a maximally entangled state of Schmidt rank $M$ on ${\cal H}_a\otimes{\cal H}_b$, and another maximally entangled state of Schmidt rank $N$ on ${\cal H}_q\otimes{\cal H}_r$. The subsystems $a$ and $q$ are on Alice’s side, while $b$ and $r$ are on Bob’s side. 1\. They perform the first half of the basic controlled-unitary protocol (Protocol \[ptl\_ct\]) on ${\cal H}_C$ and ${\cal H}_a\otimes{\cal H}_b$, until the $X^l_b$ gate in the protocol is done \[the $X$ is defined in Eq. \]. Now they share a maximally entangled state ${1\over\sqrt M}\sum_{k=0}^{M-1} {|k\rangle}_C\otimes {|k\rangle}_b$. 2\. They perform Protocol \[ptl\_ct\_ext\] to implement $S^E_k$ using their information about $k$ stored in the entangled state above, with the help of a maximally entangled state of the form $\frac{1}{\sqrt{N}}\sum_{j=0}^{N-1}{|j\rangle}\ox{|j\rangle}$. More specifically, in the lower-level protocol, every unitary gate is controlled by the ${|k\rangle}_C$ state on Alice’s side or the ${|k\rangle}_b$ on Bob’s side. If there are measurements not in the standard basis in the lower-level protocol, we decompose it as a unitary followed by a measurement in the standard basis, so that all measurements are in the same basis and thus need not be controlled by information about $k$. 3\. They have effectively performed the $V^B_j$ gate from the protocol in Sec. III of [@ygc10], which is the $S^E_k$ gate in the higher-level of the current protocol. Next, the subsystem $b$ is measured in the Fourier basis, and a local unitary correction, i.e., the integer powers of the generalized $Z$ gate defined in the basic controlled-unitary protocol is done on $C$. Note that $C$ is not being measured, since it is a “data” system and not an ancilla. The whole protocol uses $\log_2(MN)$ ebits and $2\log_2(MN)$ c-bits. Note that in step 2, the measurement outcomes in the lower-level protocol are the same for different controlling states labelled by $k$. This is acceptable, since the Protocol \[ptl\_ct\_ext\] (used as the lower-level protocol here) works under any measurement outcome anyway. \[ptl1b\] (Protocol for implementing a bipartite unitary with two levels of control — The general case that the lower level unitaries are controlled from different sides.) In Protocol \[ptl1\], the lower level unitaries are all controlled from the same side (and opposite to the direction of control in the higher level, since the case of same direction is trivial in that the unitary is then a one-level controlled unitary). Here we consider a generalization: the lower-level unitaries can be controlled from different sides. Formally, the target unitary $U$ is of the following form: $$\label{eq:calu2} U=\sum_{k=0}^{M-1} P_k\otimes S^E_k,$$ where ${\cal H}_A={\cal H}_C\otimes {\cal H}_D$, and ${\cal H}_E={\cal H}_D\otimes {\cal H}_B$, and $P_k$ are orthogonal projectors on $\cH_C$. For each $S^E_k$, there exists an integer $n_k\ge 1$, such that at least one of the following two equations hold: \[eq:sek34\] S\^E\_k&=&\_[j=0]{}\^[n\_k-1]{} U\^D\_[kj]{} Q\^[(k)]{}\_j,\ S\^E\_k&=&\_[j=0]{}\^[n\_k-1]{} R\^[(k)]{}\_j U\^B\_[kj]{}, where $U^D_{kj}$ and $U^B_{kj}$ are local unitaries on $\cH_D$ and $\cH_B$, respectively. The $Q^{(k)}_j$ are projectors on $\cH_B$ and are orthogonal among different $j$ for the same $k$. The $R^{(k)}_j$ are projectors on $\cH_D$ and are orthogonal among different $j$ for the same $k$. Let $N:=\max\{n_k: k=0,1,\dots,M-1\}$. By introducing some zero operators to the set of $Q^{(k)}_j$ and $R^{(k)}_j$, and calling the new operators $\tilde Q^{(k)}_j$ or $\tilde R^{(k)}_j$, we have that for each $S^E_k$, at least one of the following two equations hold: \[eq:sek56\] S\^E\_k&=&\_[j=0]{}\^[N]{} U\^D\_[kj]{} Q\^[(k)]{}\_j,\ S\^E\_k&=&\_[j=0]{}\^[N]{} R\^[(k)]{}\_j U\^B\_[kj]{}, where $U^D_{kj}$ and $U^B_{kj}$ are local unitaries on $\cH_D$ and $\cH_B$, respectively, and some of them are not present in Eq. . The steps of the protocol are modified from Protocol \[ptl1\] as follows: The first two steps are the same as the Steps 0 and 1 of Protocol \[ptl1\], after which both sides have a copy of the computational-basis information of the higher-level controlling state. And since the form of the overall unitary is known, each party knows whether he or she is to act as the controlling party in the lower-level protocol, depending on the higher-level controlling state. So in the modified Step 2 of the protocol, each party does what is supposed to be done locally in the lower-level controlled-unitary protocol, with each unitary gate being controlled by the local higher-level controlling state labeled by $k$, but the measurements are all in the standard basis and thus need not be controlled (if there are measurements not in the standard basis, we decompose it as a unitary followed by a measurement in the standard basis). There are two stages of classical communication (in opposite directions) in Step 2, and for each such communication stage, the party that is supposed to send classical messages does exactly the same operations as before, but the opposite party measures in the computational basis on an extra ancilla initially in the $\frac{1}{\sqrt{N}}\sum_{j=0}^{N-1}{|j\rangle}$ state, and sends the outcome to the other party. The choice of measuring a useful system or a *dummy* ancilla introduced above is determined by the higher level controlling state labeled by $k$. However, for actual implementation, the actual measurement should be on a fixed system. This can be resolved by a controlled-swap gate controlled by $k$, which conditionally swaps the system to be measured into a fixed system before doing the measurement. The final step is similar to Step 3 of Protocol \[ptl1\]. The whole protocol requires the same amount of entanglement as in Protocol \[ptl1\], but generally requires more classical communication, since the correct and *dummy* messages are sent in both directions simultaneously in the two stages of classical communication in Step 2, so we allow twice as much classical communication in the lower-level protocol. Thus the overall protocol uses $\log_2 (MN)$ ebits and $2\log_2 (M N^2)$ c-bits. A dummy message is the measurement outcome of a system which was originally (before the controlled-swap gate mentioned in the previous paragraph) an ancilla in a fixed initial state. Note that the dummy classical message is only dummy for some of the higher-level controlling states labeled by $k$, but is the correct message for some others. Such message, even if “correct”, does not carry any information about the input state for the overall unitary, by the design of the basic controlled-unitary protocol. The rationale behind the above technique is as follows: The choice of which lower-level unitary is being implemented should be indistinguishable from an outside observer, since the information about the higher-level controlling state should not be leaked to the outside observer, which is necessary for implementing a unitary operation. The reason is in Theorem 1 of [@ygc10], which says that implementing a unitary successfully is equivalent to that no information about the input state of the unitary is leaked to an “environment” system (the tensor product of the environment system and the output system of the unitary is the entire output system of the protocol). \[ptl\_gp\] (Protocol for implementing a bipartite unitary given its group-type expansion.) This protocol is illustrated in Fig. 8 in [@ygc10] (except for changes in symbols in the description below), and it implements bipartite unitaries of the form \[eq:groupform\] U=\_[fG]{} V\_A(f) W\_B(f), where the $V_A(f)$ are unitaries acting on $\cH_A$, and they form a projective unitary representation of a finite group $G$, and $W_B(f)$ are arbitrary operators acting on $\cH_B$ but they satisfy that $U$ is unitary. This protocol uses a maximally entangled resource state of Schmidt rank $\vert G\vert$ (the order of $G$). Thus the entanglement cost is $\log_2 \vert G\vert$ ebits. The classical communication cost is $2\log_2 \vert G\vert$ c-bits. For any unitary $U$, we may expand it in the form by letting $G$ be the generalized Pauli group (ignoring overall phases) $\{X^j Z^k: j,k\in [0,d_A-1]\}$ which is of order $d_A^2$, since the $d_A^2$ generalized Pauli matrices form a basis for the space of $d_A\times d_A$ matrices. We abbreviate the steps of the protocol here. For our purposes, a good property of the protocol to be utilized for the proof of Theorem \[thm:sch3\] is that when $U$ is the $A$-direct sum of some unitaries, it is often the case that there is a relatively small group $G$ (by “small” we mean smaller than $d_A^2$) such that $U$ can be expanded in the form . This is because of the following reason: Each component in the $A$-direct sum form of $U$ is also expandable using the form ; thus, its size divided by $d_B$ is the dimension of a (projective) unitary representation of the group $G$, where the representation is obtained by restricting $V_A(f)$ to the relevant subspace of $\cH_A$, for all $f\in G$. Denote the dimension of such a projective representation as $n_i$, $i=1,\dots,K$, where $K$ is the total number of components in the $A$-direct sum form of $U$. Assume that there is a group $G$ that has inequivalent irreducible projective unitary representations of sizes $n_i$, $i=1,\dots,K'$, where $K'\ge K$, and the $n_i$ with $i>K$ (in the case $K'>K$) are arbitrary positive integers (this is, of course, a big assumption and does not hold for most bipartite unitaries, but note that we may regard several blocks in an $A$-direct sum form of $U$ as one block to increase the chance that such a group $G$ exists, which is a technique used in the proof of Theorem \[thm:sch3\]), then we may do the following steps: Arbitrarily choose a factor system (see the definition in [@ygc10]) from the set of factor systems of $G$ that admit inequivalent irreducible projective unitary representations of sizes $n_i$, $i=1,\dots,K'$ (the existence of such a factor system is guaranteed by the assumption above). Then choose a projective unitary representation of $G$ that contains all inequivalent irreducible projective unitary representations belonging to this factor system. This would be a linearly independent set of matrices according to [@ygc10 Theorem 4], and they are of a simultaneous block diagonal form. We then remove some diagonal blocks from all these matrices so that the remaining blocks are of sizes $n_i$, $i=1,\dots,K$. Then the resulting matrices would be generally linearly dependent, and from the construction, the resulting set forms a (possibly overcomplete) basis for the space of matrices with the same block structure. Thus, this set of unitary matrices can be used to expand the bipartite unitary $U$ in the form of . In our application in the proof of Theorem \[thm:sch3\] in this paper, we choose the type of group $G$ directly and figure out its suitable size. A different problem has been discussed in [@Cohen10], which is trying to find the smallest group $G$ when the matrix of $U$ is known. However, there is some similarity: Our reason for choosing the dihedral groups as the type of group $G$ in the proof of Theorem \[thm:sch3\] is based on the $B$-direct-sum form of $U$ that we proved. The algorithm for choosing the group $G$ in [@Cohen10] also is based on finding the $A$-direct-sum form of $U$ (which corresponds to the block diagonal structure of the operators on $\cH_A$ that are used to expand $U$). The protocols with two levels of control can be generalized to protocols with multiple levels of control. Some other generalizations are possible (but not used in this paper): The lower-level operators $S^E_k$ in the target unitary of the form need not be a controlled unitary, but could be unitaries with group-type expansion in Protocol \[ptl\_gp\], and thus the inner level of the protocol becomes Protocol \[ptl\_gp\]. For studying Theorem \[thm:sch3\], we introduce the preliminary lemma below. We note that the simplest type of Schmidt-rank-three bipartite unitaries, which are controlled unitaries with three terms, are generally not included in Lemma \[lm:lm2\], due to the restrictions on the coefficients $c_{j1},c_{j2},c_{j3}$ and the matrices $T_2$ and $T_3$ below. \[lm:lm2\] Suppose there are three linearly independent $d\times d$ unitary matrices $I_d$, $T_2$ and $T_3$, where $I_d$ is the identity matrix, and $T_2$ is diagonal, and $T_3$ is not diagonal, and $T_2,T_3$ are not simultaneously diagonalizable under a unitary similarity transform; and $K$ distinct triplets $(c_{j1},c_{j2},c_{j3})$, where $j=1,\dots,K$, $c_{j1}$ are real and nonnegative, $c_{j2}$ and $c_{j3}$ are nonzero complex numbers, such that \[eq:u14new2\] U=\_[j=1]{}\^[K]{} [|j|]{} (c\_[j1]{} I\_d + c\_[j2]{} T\_2 + c\_[j3]{} T\_3) is a bipartite unitary of Schmidt rank $3$ on a $K\times d$ space $\cH_A\ox\cH_B$. Then up to local unitaries, there is a decomposition of $U$ with the following direct sum structure on $\cH_B$: $U=\bigoplus_{k=1}^n U_k$, $I_d=\bigoplus_{k=1}^n I^{(k)}$, $T_2=\bigoplus_{k=1}^n T_2^{(k)}$ and $T_3=\bigoplus_{k=1}^n T_3^{(k)}$, satisfying that each \[eq:ui\] U\_k=\_[j=1]{}\^[K]{} [|j|]{} (c\_[j1]{} I\^[(k)]{} + c\_[j2]{} T\_2\^[(k)]{} + c\_[j3]{} T\_3\^[(k)]{}) is a unitary on the $K\times d_{k}$ subspace $\cH_A\ox\cH_{B_k}$ with $d=\sum_{k=1}^n d_{k}$, and that $T_3^{(1)}$ is diagonal, and for each $k>1$, $T_2^{(k)}=\diag(e^{i\a_k},-e^{-i\a_k})$, $\a_k\in\mathbb{R}$; $T_3^{(k)}$ is a non-scalar $2\times2$ unitary whose non-diagonal entries are equal and positive. The proof of this lemma is given in Appendix \[app:[lm:lm2]{}\]. Lemma \[lm:lm2\] leads to the following result, where assertion (i) is a structure theorem for Schmidt-rank-3 bipartite unitaries. Note that the assumption of the result implies $d_A\ge 3$ and $d_B\ge 2$. \[thm:sch3\] Assume that $U$ is a Schmidt-rank-3 bipartite unitary controlled from the A side. Then the following assertions hold.\ (i) Either $U$ is the $A$-direct sum of at most three unitaries of Schmidt rank at most $2$, or $U$ is locally equivalent to a $B$-direct sum of controlled unitaries of Schmidt rank at most $3$. Each of the controlled unitaries is on a $d_A\times 1$ or $d_A\times 2$ space controlled in the computational basis of $\cH_A$.\ (ii) $U$ can be implemented by local operations and \[eq:log2\] \_2d\_A,d\_B\^2,4d\_B/2+2 ebits of entanglement and \[eq:log2cc\] 2\_2d\_A,d\_B\^2,{12,4d\_B/2+2} c-bits. The proof of this theorem is given in Appendix \[app:[thm:sch3]{}\]. Given that the $A$ side is the control, the result in [@cy14ap] gives an entanglement cost upper bound of $\log_2\min\{d_A,d_B^2\}$ ebits. This old upper bound is always not less than the new upper bound in . When $d_A, d_B$ are both large and $d_A$ is about $d_B^2$, the new upper bound in is about $\log_2 (2d_B)=1+\log_2 d_B$ ebits, which is about half of the old upper bound which is about $\log_2 d_B^2=2\log_2 d_B$ ebits. We show two classes of examples. The first shows that for some $U$, the entanglement cost can be much less than the upper bound in Theorem \[thm:sch3\](ii). \[ex:ex1\] Consider a Schmidt-rank-three unitary $U$ of the form . Let $\cH_B$ be of dimension $2n$, and $T_1=I_B$, $T_2=\oplus_{j=1}^n \sigma_z$, $T_3=\oplus_{j=1}^n [\cos(t_j)\sigma_x + \sin(t_j) \sigma_y]$, where $t_j$ ($1\le j\le n$) are some different real numbers. Then $(T_2)^2=(T_3)^2=I_B$, and $T_2 T_3 = -T_3 T_2$. Actually, by conjugation using a local diagonal unitary on $\cH_B$, we can transform $T_3$ into $\oplus_{j=1}^n \sigma_x$ while keeping $T_1$ and $T_2$ unchanged. The other $T_j$ with $j>3$ are given by $T_j=\cos\theta_j T_1+i\sin\theta_j\cos\phi_j T_2+i\sin\theta_j\sin\phi_j T_3$, where $\theta_j$ and $\phi_j$ are real. The $B$ space of $U$ is spanned by a projective representation of an Abelian group of order $4$ (the Klein-four group), hence Protocol \[ptl\_gp\] implements $U$ using 2 ebits of entanglement and LOCC. This is much less than the upper bound in Theorem \[thm:sch3\](ii) when $d_A$ and $d_B$ are large. The second class of examples is still for unitary $U$ of the form in Lemma \[lm:lm2\], but with essentially different blocks in different subspaces of $\cH_B$. It suggests that there might not be an easy improvement to the upper bound in Theorem \[thm:sch3\](ii) for general Schmidt-rank-three bipartite unitaries. \[ex:ex2\] We use the notations in the proof of Lemma \[lm:lm2\], but assume that the unitary is without the diagonal part, i.e. the subspace $\cH_{B_1}$ is a null space. Assume the diagonal elements of the $2\times 2$ matrices $T_2^{(k)}$ and $D_3^{(k)}$ are $s_k \sqrt{1-b^2}+b i$ and $s_k t b\sqrt{\frac{1-b}{1+b}}+t b i$, respectively, where $b\in(0,1]$ is a variable dependent on $k$, and $t$ is a positive constant less than $1$, e.g. $t=1/2$, and the sign factor $s_k$ for the real part is either $1$ or $-1$. Suppose the diagonal elements with the positive $s_k$ appear first in each $T_2^{(k)}$ and $D_3^{(k)}$, and denote such elements as $T_{2k}$ and $D_{3k}$, respectively. Then ${\rm Im}(T_{2k})$, ${\rm Im}(D_{3k})$, and ${\rm Re}(T_{2k} D^\ast_{3k})$ are $b$, $t b$, and $t b$, respectively, which is useful for checking the result below. Since $\vert D_{3k}\vert\le1$, the two off-diagonal elements of $D_3^{(k)}$ are chosen to be equal real numbers such that $D_3^{(k)}$ is unitary. Let the $(c_{j1},c_{j2},c_{j3})$ satisfy that $c_{j1}=(t y -1)/\sqrt{(1+y^2)(t y-1)^2 + t^2 y^2}$, and $c_{j2}=i c_{j1} t y /(t y -1)$, $c_{j3}=i c_{j1} y$, for $j=1,\dots,M$, where $M$ is an arbitrary positive integer, and $y=y_j>1/t$ is a real positive number independent of $k$ but dependent on $j$. Note that $b=b_k$ is independent of $j$. The diagonal part of Eq.  can be written as \[eq:cjtj4\] && (c\_[j1]{})\^2 + (c\_[j2]{})\^2 + (c\_[j3]{})\^2 - 2 c\_[j1]{} c\_[j2]{} [Im]{}(T\_[2k]{})\ && - 2 c\_[j1]{} c\_[j3]{} [Im]{}(D\_[3k]{}) + 2 c\_[j2]{} c\_[j3]{} [Re]{}(T\_[2k]{} D\^\_[3k]{})=1    for $k=1,2,\dots,d$. Here we have used that $c_{j1}$ is real, and $c_{j2}$ and $c_{j3}$ are pure imaginary, and that $T_2$, $T_3$ are unitary, and we denote $\tilde c_{j2}:={\rm Im}(c_{j2})$, $\tilde c_{j3}:={\rm Im}(c_{j3})$. It is easily verified that there are an infinite number of solutions of $y=y_j$ and $b=b_k$ for when $t$ is fixed, and by choosing some sufficient but finite number of them to be used in the matrix $U$, the $U$ would have Schmidt rank three. The $U$ is unitary because each $2\times 2$ block in each controlled operator on the $B$ side is unitary, and the latter follows from Lemma \[le:2x2unitary\] and our choice of the $T_{2k}$ and $D_{3k}$, and that $c_{j1}$ is real, and $c_{j2}$ and $c_{j3}$ are pure imaginary. The statement about the number of solutions above implies that the dimensions $d_A$ and $d_B$ are arbitrarily large, and we do not know of any simple protocol that implements this class of unitaries with a constant number of ebits and LOCC. This suggests there might not be an easy improvement to the upper bound in Theorem \[thm:sch3\](ii). Entanglement cost and entangling power of bipartite permutation unitaries {#sec:permutation} ========================================================================= This section is motivated by the following question. What is the entanglement cost for Schmidt-rank-three bipartite permutation unitaries? The result in Theorem 22 of [@cy15] gives an upper bound of 24 ebits, with the help of a one-qubit ancilla on one side. Other motivations to study the permutation unitaries are in the first paragraph of Sec. \[subsec:permutation\_example\], and also in [@cy15]. We shall first develop some preliminary results about bipartite (complex) permutation unitaries of general Schmidt rank, and then derive the improved upper bounds for the entanglement cost for bipartite permutation unitaries of small Schmidt rank in Sec. \[subsec:permutation\_small\]. The case of general Schmidt rank is studied in Sec. \[subsec:permutation\_large\]. We give an example in Sec. \[subsec:permutation\_example\], and study the entangling power of bipartite permutation unitaries of Schmidt rank up to three in Sec. \[ssec:ent\_power\_permutation\]. \[le:persch\] Let $U$ be a complex bipartite permutation matrix of Schmidt rank $r$. Then the following assertions hold.\ (i) The nonzero blocks in any big row or big column of $U$ are linearly independent. The number of them is at least $1$ and at most $r$.\ (ii) Suppose a big row of $U$ contains $r$ linearly independent blocks. Then up to local complex permutation matrices the first $r$ blocks in the big row are orthogonal projectors, whose sum is the identity matrix. A similar statement holds when all “row” are replaced with “column”.\ (iii) Under the assumption in (ii), up to local complex permutation matrices $U$ is a complex $r$-term controlled-permutation unitary from the $B$ side. The projectors in the terms are exactly the projectors in (ii). Such unitary can be implemented using $\log_2 r$ ebits and LOCC.\ (iv) If $U$ is a real permutation unitary, then (ii) and (iii) hold with all occurrences of the word “complex” removed.\ (v) Suppose a big row of $U$ contains $r-1$ linearly independent blocks. Then up to local complex permutation matrices the first $r-1$ blocks in the big row are orthogonal projectors, whose sum is the identity matrix. A similar statement holds when all “row” are replaced with “column”.\ (vi) Under the assumption in (v), assume that the projectors and their orders are respectively $P_j$ and $s_j$ for $j=1,\cdots,r-1$. Up to local complex permutation matrices, we have \[eq:ubigg\] U= ( (Q P) \_A \^n\_[j=1]{}(Q\_j P\_j) ) \_B ( (\^[r-1]{}\_[j=n+1]{})\_B U\_j )\ where $n\in\{0\}\cup[2,r-1]$, $P$, $Q$ and $Q_j$ are all complex permutation matrices on their respective subspaces. $P$ is of size $(\sum^{n}_{j=1} s_j) \times (\sum^{n}_{j=1} s_j)$, and the pair of matrices $Q$ and $Q_j$ ($\forall j\le n$) are orthogonal in both the input and output spaces. Furthermore, $U_j$ is a complex permutation matrix of Schmidt rank at most two on the bipartite Hilbert space $\cH_A\times \lin\{{|s_1+\cdots+s_n+1\rangle},\cdots,{|d_B\rangle}\}$. The $B$ space of $U_j$ contains $P_j$. If $n\in[2,r-2]$, then $U$ can be implemented using $\max \{2 + \log_2 n, 2+ \log_2 (r-n-1)\}$ ebits and LOCC. If $n=0$, then $U$ can be implemented using $2+ \log_2 (r-1)$ ebits and LOCC. If $n=r-1$, then $U$ can be implemented using $1+ \log_2 (r-1)$ ebits and LOCC.\ (vii) In (vi), if $U$ is a real permutation unitary, and $n=0$, then under local permutations, either $U$ can be written in the $n=r-1$ case of the form of , or $U$ is a controlled-permutation unitary controlled from the $B$ side with at most $2(r-1)$ terms, thus $U$ can be implemented using $1+ \log_2 (r-1)$ ebits of entanglement. The proof of this lemma is given in Appendix \[app:[le:persch]{}\]. The partial transpose has been used to study the separability problem in entanglement theory [@peres1996; @hhh96]. Recently it has been used to study the ranks of marginals of multipartite quantum states [@chl14], in terms of the following conjectured inequality \[eq:ineq\] (\^k\_[j=1]{} A\_j B\_j ) k (\^k\_[j=1]{} A\_j B\_j\^T ), where $A_j$ (resp. $B_j$) are matrices of the same size and $T$ denotes the transpose. In previous works we have presented a few bipartite unitaries satisfying the inequality [@cy14; @cy14ap]. One can verify that the partial transpose of the complex permutation unitaries in (ii) and are still unitary matrices. When considered as one of the bracket expressions in the lhs or rhs of , they both satisfy . They provide further evidence supporting the conjecture. We do not know whether all bipartite permutation matrices or complex permutation matrices satisfy . Next we describe some simple properties about the $d_B\times d_B$ blocks in bipartite permutation matrices. Let $m(r)$ denote the maximum possible number of distinct diagonal blocks in a Schmidt-rank-$r$ bipartite controlled-permutation unitary. Let $m'(r)$ denote the maximum possible number of distinct permutation matrices in the $B$-space of a Schmidt-rank-$r$ bipartite permutation unitary. Let $n(r)$ denote the maximum possible number of distinct nonzero partial permutation matrices in the $B$-space of a Schmidt-rank-$r$ bipartite permutation unitary. Using these definitions we state the following lemma. \[lm:diagonal\_blocks\] (i) $m(r)$ is equal to the maximum number of distinct permutation matrices in the linear span of $r$ arbitrary permutation matrices of the same size.\ (ii) $m(r)=2^{r-1}$.\ (iii) The entanglement cost of any Schmidt-rank-$r$ controlled-permutation unitary is not more than $r-1$ ebits.\ (iv) $m'(r)$ is not greater than the maximum number of distinct permutation matrices in the linear span of $r$ arbitrary partial permutation matrices of the same size.\ (v) $m'(r)=2^{r-1}$.\ (vi) $n(r)=2^r-1$, and the maximum in the definition of $n(r)$ is achieved only when the bipartite permutation unitary is equivalent to a controlled unitary from the $B$ side under local permutation unitaries. The proof of this lemma is given in Appendix \[app:[lm:diagonal\_blocks]{}\]. Evidently the $m'(r)$ and $n(r)$ would be unaffected if we replace $B$ by $A$ in their definition. Entanglement cost of bipartite permutation unitaries of Schmidt rank two or three {#subsec:permutation_small} --------------------------------------------------------------------------------- We have studied the properties of the complex bipartite permutation unitaries in terms of the Schmidt rank in Lemmas \[le:persch\] and \[lm:diagonal\_blocks\]. In this subsection we study the bipartite permutation unitaries of Schmidt rank two or three. They are locally equivalent to controlled unitaries [@cy13; @cy14; @cy14ap]. So they can be implemented using the basic controlled-unitary protocol by directly using the controlled form, however this might require more than minimal amount of entanglement. The Lemma \[lm:sch2\] (i) below, together with Lemma \[le:sch2perm\_entpower\], imply that the entanglement cost by directly using the controlled form is minimal for the case of Schmidt rank two. \[lm:sch2\] (i) Any Schmidt-rank-two bipartite permutation unitary is equivalent to a two-term controlled-permutation unitary under local permutation unitaries.\ (ii) Any Schmidt-rank-two bipartite complex permutation unitary is equivalent to a two-term controlled-complex-permutation unitary under local complex permutation unitaries. Let us prove (ii) first. Denote the complex unitary as $U$. Its standard matrix form, also denoted by $U$, is a $d_A d_B\times d_A d_B$ matrix. If there is a big row or column of $U$ containing two nonzero blocks, then the assertion follows from Lemma \[le:persch\](ii)(iii). It suffices to consider the case that there is exactly one nonzero block in any big row or column of $U$. Up to local permutation matrices on $\cH_A$ we may assume that $U$ is a block-diagonal complex permutation matrix, and the first two diagonal blocks $D_1,D_2$ are linearly independent. Up to a local complex permutation matrix on $\cH_B$, we may assume $D_1=I_B$. If all diagonal blocks of $U$ are proportional to $D_1$ or $D_2$, then the assertion follows. If there is a diagonal block which is not proportional to any one of $D_1,D_2$, then $D_2$ has to be diagonal and if $D_2$ has only two distinct diagonal entries, then $U$ is equivalent to a controlled complex permutation unitary from the $B$ side with two terms, up to local permutation unitaries. Thus we only need to consider the remaining case, i.e., that $D_2$ is diagonal and has at least three distinct diagonal entries. However in this case $D_2$ cannot be unitary by Lemma \[le:vandermonde\]. This completes the proof of (ii). The proof for (i) is similar. If there is a big row or column of $U$ containing two nonzero blocks, the assertion follows from Lemma \[le:persch\](iv). In the remaining case, the result follows from Lemma \[lm:diagonal\_blocks\](ii). Now we investigate the structure and entanglement cost for complex permutation unitaries of Schmidt rank three. In particular, the real counterpart is completely characterized in (i). \[lm:sch3perm\] (i) Up to local permutation unitaries, any Schmidt-rank-three bipartite permutation unitary is either equivalent to a three-term or four-term controlled-permutation unitary, or equivalent to the direct sum of a product permutation unitary and a two-term controlled-permutation unitary. Therefore such unitary can be implemented using $2$ ebits and $4$ c-bits.\ (ii) Any Schmidt-rank-three bipartite complex permutation unitary that is not equivalent to a diagonal unitary under local permutation unitaries can be implemented using $3$ ebits and LOCC.\ (iii) Any diagonal Schmidt-rank-three bipartite complex permutation unitary, whose diagonal blocks contain the identity matrix and a diagonal matrix of exactly two distinct diagonal elements, can be implemented using $2$ ebits and LOCC. The proof of this lemma is given in Appendix \[app:[lm:sch3perm]{}\]. An example for “the direct sum of a product permutation unitary and a two-term controlled-permutation unitary” is given by the following unitary on $3\times 2$ system: \[eq:uketbra11\] U&=&\[[|11|]{}([|12|]{}+[|21|]{})\]\ &\_A& \[([|22|]{}+[|33|]{})[|11|]{}+([|23|]{}+[|32|]{})[|22|]{}\].\ Entanglement cost of bipartite permutation unitaries of general Schmidt rank {#subsec:permutation_large} ---------------------------------------------------------------------------- The following Protocol \[ptl2\] implements bipartite permutation unitary $U$ of arbitrary Schmidt rank $r$. The computational basis for each system starts with ${|0\rangle}$. The entanglement and classical communication cost of the protocol in terms of $r$ is analyzed in Theorem \[thm:permutation\]. Before introducing the protocol, we define the so-called effective input and output dimensions for $U$. An example unitary illustrating these definitions is in Example \[ex4\] in Sec. \[subsec:permutation\_example\]. \[def\_state\_types\] (i). The effective input dimension of $A$ is the number of types of input states of $A$. A type of input states of $A$ (or “an input type of $A$”) is a subspace of $\cH_A$ spanned by computational basis states, so that any two big columns of $U$ corresponding to two computational basis states in the subspace have the same collection of blocks in them, ignoring the positions and the relative order of the nonzero blocks in the big column. (2). The effective output dimension of $A$ relative to an input computational basis state of $\cH_A$ is the number of nonzero blocks in the big column of $U$ corresponding to the input computational basis state of $\cH_A$. And the labels for each effective dimension for a given input computational state of $\cH_A$ is determined by the order in which the nonzero block appears in the big column. The output computational basis state of $\cH_A$ corresponding to the big row with a nonzero block in the given big column is called an output type of $A$ relative to the input of $A$, abbreviated as “a relative output type of $A$”. (3). The effective output dimension of $B$ is the number of output types of $B$, where an output type of $B$ is a subspace of $\cH_B$ spanned by computational basis states, so that each computational basis state in such subspace has the same combination of being in or not in the output space of the partial permutation operators in the $B$ space of $U$. It turns out that for this definition of the output type of $B$, it suffices to consider a linearly independent set of $r$ partial permutation operators in the $B$ space of $U$, which form a basis for the $B$ space of $U$, and we call such revised definition the *simplified definition*. Such a basis of $r$ partial permutation operators do exist, and they can be selected from the $d_B\times d_B$ blocks in the matrix $U$. Any other partial permutation operator in the $B$ space of $U$ is a linear combination of these $r$ basis operators. Suppose the simplified definition is inequivalent to the original definition. Then there are two computational basis states in the output space $\cH_B$ so that they are simultaneously in or not in the output space of any of the $r$ basis operators, while one and only one of them is in the output space of another partial permutation operator $Q_B$ in the $B$ space of $U$. The $Q_B$ is a linear combination of the $r$ basis operators, each of which has row sums being equal between the two said output types, hence the row sums of $Q_B$ are equal between the two said output types, and we have arrived at a contradiction. Therefore, the simplified definition is equivalent to the original definition. \[ptl2\] (A protocol that implements a general bipartite permutation unitary $U$.)  The circuit diagram for the protocol is shown in Fig. \[fgr1\]. The steps of the protocol are as follows. 1\. Alice prepares an ancilla $a$ in the state ${|0\rangle}$, and performs a controlled-$X^j$ gate on $A$ and $a$ (with projectors on $\cH_A$ of rank possibly greater than one) so that the system $a$ stores in its $Z$ basis the information about the type of input state on system $A$, which is defined in Def. \[def\_state\_types\](i), and is abbreviated as “the input type of $A$”. The integer $j\in\{0,1,\dots,d-1\}$ labels the type of the input state of $A$, where $d$ is the dimension of system $a$. The $X$ is the cyclic shift gate $\sum_{j=0}^{d-1} {|(j+1)\mod d\rangle\!\langlej|}$ (note it was the minus sign in [@ygc10] and Protocol \[ptl1\] instead of the plus sign). 2\. Alice sends the $Z$-information about $a$ to Bob’s side, so that Alice has a copy $a$ storing the $Z$-information about $a$, and Bob has a copy $e'$. This requires a prior shared maximally entangled pair of $d$-dimensional qudits $ee'$ in the state $\frac{1}{\sqrt{d}}\sum_{j=0}^{d-1} {|jj\rangle}$, and involves a controlled cyclic-shift gate on $ae$ and a measurement of $e$ in the standard basis on Alice’s side, with the outcome sent to Bob using a classical channel, and a cyclic-shift gate on $e'$ on Bob’s side according to the measurement outcome. 3\. Bob has an ancilla system $f_0$ initialized in ${|0\rangle}$. He performs a controlled permutation unitary $W$ on $e'$ (which now stores the input type of $A$), $f_0$ and $B$, with $e'$ being the control, to prepare the output type of $A$ on the output system $f$ relative to the input $f_0$ \[defined in Def. \[def\_state\_types\](ii)\], and at the same time prepare the output state of $B$ (under the action of $U$) on the system $B$. Note that if the input $f_0$ and the corresponding output $f$ for the gate $W$ are removed, the $W$ would not be unitary in general. 4\. Bob measures $e'$ in the Fourier basis and a phase correction (an integer power of $Z=\sum_{j=0}^{d-1} e^{2\pi i j/d}{|j\rangle\!\langlej|}$) is done on the $a$ by Alice according to the measurement outcome sent classically. Bob teleports $f$ to the $A$ side, denoted as $f'$. 5\. Alice performs a controlled permutation unitary gate $V$ on three systems $A$, $a$, and $f'$, with the joint system $af'$ being the control, to get the output of $A$. 6\. The remaining task is to erase the state on $a$ and $f'$. The $a$ stores the input type of $A$, and the $f'$ stores the relative output type of $A$, and both are determined jointly by the output of $A$ together with the output type of $B$. Hence a preparation of a system $h$ containing the output type of $B$ \[defined in Def. \[def\_state\_types\](iii)\] is needed, and the $h$ is teleported to the $A$ side (and denoted $h'$), for Alice to erase $a$ and $f'$ to ${|0\rangle}_a {|0\rangle}_{f'}$ by a controlled permutation unitary gate $T$ acting on $Aaf'h'$, with the joint system $Ah'$ being the control. Finally the $h'$ is measured in the Fourier basis and the outcome is sent to Bob classically, and a phase correction is done on system $B$. The phase correction gate is denoted as an integer power of $\hat Z$ to indicate that it is a diagonal operator with eigenvalues being the $d$-th roots of unity but with some degeneracies, where $d$ is the number of the output types of $B$. This completes the protocol, with the output of $U$ in systems $A$ and $B$. The following lemma gives an upper bound of the maximum number of types of the input state on system $A$ defined in Def. \[def\_state\_types\] (i). A matrix *occupies a column* if and only if it has a nonzero element in that column. Suppose $S$ is a set of nonzero $d\times d$ partial permutation matrices. A subset $S'\subseteq S$ is called a *covering* subset if and only if any two matrices in $S'$ do not occupy the same column, and any column is occupied by some matrix in $S'$. A *basis* of $S$ is a maximal linearly independent set of matrices in $S$. The Bell number $B_r$ is the number of different ways to partition a set of $r$ distinguishable elements, regardless of the order of partitions and the order of elements within each partition. By simple calculation, $B_1=1, B_2=2, B_3=5, B_4=15, B_5=52$, and it is known that $B_r<[0.792 r/\log_e (r+1)]^r$ for any integer $r\ge 1$ [@bt10]. \[le:combinations\_partial\_permutation\] Suppose $S$ is a set of nonzero $d\times d$ partial permutation matrices which include exactly $r$ linearly independent matrices, and each column is occupied by some matrix in $S$. The number of covering subsets of $S$ is not greater than $B_{r+1}$. The assertion apparently holds when $r=1$. In the following we assume $r\ge 2$. From Lemma \[lm:diagonal\_blocks\](vi), the size of $S$ is at most $2^r-1$. A covering subset of $S$ can contain at most $r$ elements, since elements of a covering subset must be linearly independent. Let us fix a basis of $r$ linearly independent matrices in $S$. From the proof of Lemma \[lm:diagonal\_blocks\](vi), there are $r$ positions (matrix elements) of $d\times d$ matrices that determine a partial permutation matrix in the space spanned by the $r$ basis matrices. Let us call these $r$ matrix elements as “key elements”. Some of the key elements may be in the same column. For any two matrices in the same covering subset of $S$, they occupy disjoint sets of columns, hence they cannot both contain $1$’s at the position of the same key element, nor can they contain a “$1$” respectively at one of two different key elements in the same column. Hence any matrix in a covering subset of $S$ is characterized by a set of key elements among the given $r$ key elements, and a covering subset of $S$ is characterized by a partitioning of the key elements, but possibly with some key element(s) not belonging to any matrix in the covering subset, in the latter case we arrange the “extra” key element(s) into a partition, and mark this set with an auxiliary element, i.e. let the auxiliary element and the extra key element(s) be put into the same part in the partition of $r+1$ elements. In the case that no extra key element exists, the auxiliary element is a part of the partition by itself. Therefore, the total number of covering subsets of $S$ is at most the partition number of $r+1$ elements, which is $B_{r+1}$. This completes the proof. Now we introduce a new definition of the number of input types on system $A$ (the definition for system $B$ is similar), which will be used in Protocol \[ptl3\] below. If the sum of all blocks in a big column of $U$ is equal to the corresponding sum for another big column, then these two big columns are regarded as *of the same type in the loose sense*. The reason for this new definition is that any input computational basis state on $B$ is mapped to the same output state on $B$ under the maps represented by the two big columns which satisfy that the sum of blocks in them are equal. \[le:input\_types\_A\] The number of distinct types of big columns of a bipartite permutation matrix of Schmidt rank $r$ in the loose sense is at most $2^{r-1}$. This bound is tight. Denote the matrix as $U$, and denote the maximum value of the quantity in the assertion as $f(r)$, which is a function of $r$ only. The sum of all blocks in a big column of $U$ is in the $B$ space of $U$, and is a matrix with elements being $0$ or $1$ and with sum of elements in each column equal to $1$. By an argument similar to that in the proof of Lemma \[lm:diagonal\_blocks\](ii), there are at most $2^{r-1}$ such matrices in the $B$ space of $U$. By definition, two big columns of different types in the loose sense are different in the sum of their blocks. Hence $f(r)\le 2^{r-1}$. The example of $U$ that reaches the maximum value of $2^{r-1}$ is in the proof of Lemma \[lm:diagonal\_blocks\](ii). \[ptl3\] (Another protocol that implements a general bipartite permutation unitary $U$.)  The circuit diagram for the protocol is shown in Fig. \[fgr2\]. The steps of the protocol are as follows. 1\. Alice prepares an ancilla $a$ in the state ${|0\rangle}$, and performs a controlled-$X^j$ gate on $A$ and $a$ (with projectors on $\cH_A$ of rank possibly greater than one) so that the system $a$ stores in its $Z$ basis the information about the type of input state of $A$ in the loose sense, which is defined before Lemma \[le:input\_types\_A\]. She teleports $a$ to Bob’s side using prior shared entanglement and LOCC. Similarly, Bob prepares an ancilla $b$ storing the information about the type of input state of $B$ in the loose sense, and he teleports $b$ to Alice’s side. 2\. Alice performs a controlled-permutation unitary $W$ on $A'$, $A$ and the teleported $b$, with $A$ and $b$ as the control, and the $A'$ was initialized in ${|0\rangle}$ before such gate. The controlled operator acting on $A'$ in the gate $W$ is a permutation unitary that only swaps the ${|0\rangle}$ state with the output state determined by the state on the control registers, and keeps other $Z$ basis states of $A'$ unchanged (those states are not the actual input state anyway). After the $W$, the $Z$-information about the output of $A$ under the action of $U$ is stored in the $Z$ basis of $A'$. Similarly, Bob performs $\tilde W$ and the $B'$ now contains the $Z$ information about the output of $B$ under $U$. 3\. Alice teleports $b$ back to Bob’s side, and Bob teleports $a$ back to Alice’s side. Each party performs the inverse of the controlled gate in step 1 to erase the $a$ and $b$ to ${|0\rangle}$. 4\. This step is similar to step 1, except that $U^\dag$ instead of $U$ is considered here, and the $A'$ and $B'$ are regarded as the input for the unitary $U^\dag$. An ancillary system $a'$ is initialized in ${|0\rangle}$, and after the controlled gate on $A'$ and $a'$, the $a'$ contains the type of state of $A'$ in the loose sense, and is teleported to the other side. Similarly, the $b'$ containing the type of state of $B'$ in the loose sense is teleported to Alice’s side. 5\. This step is similar to step 2. The controlled permutation gates $T$ and $\tilde T$ are defined similar to the $W$ and $\tilde W$ in step 2, but with $U^\dag$ instead of $U$ and the $A'$ and $B'$ taking the role as the input for $U^\dag$. Because of the form of the $T$ and $\tilde T$ gates and the states of $A$ and $B$ just prior to this step, the $A$ and $B$ are erased to ${|0\rangle}$. 6\. This step is similar to step 3. Alice teleports $b'$ back to Bob’s side, and Bob teleports $a'$ back to Alice’s side. Each party performs the inverse of the controlled gate in step 4 to erase the $a'$ and $b$ to ${|0\rangle}$. This completes the protocol, with the output of $U$ in systems $A'$ and $B'$. In the protocol above, we need to erase the $A$, $B$ (which become ancillary systems in the end) and other ancillas to some fixed state, because no information about the input should be leaked to ancillas in the end; otherwise the protocol does not implement a unitary operator (c.f. [@ygc10], Sec. II C). The above protocol computes the correct output states on $A'B'$ for the input computational states on $AB$ without introducing extra phases, and by linearity, it implements the unitary $U$ on all input quantum states. \[thm:permutation\] Any bipartite permutation unitary of Schmidt rank $r$ can be implemented using local operations with the help of $\min\{\log_2(B_{r+1})+r+\log_2 r, 8r-8\}$ ebits of entanglement and twice as many c-bits. The proof of this theorem is given in Appendix \[app:[thm:permutation]{}\]. This significantly improves over the result in Theorem 22 of [@cy15], which states that such unitary can be implemented using LOCC with $3\times 2^r$ ebits. Since $B_r<[0.792 r/\log_e (r+1)]^r$ for any integer $r\ge 1$ [@bt10], the first term in the result of Theorem \[thm:permutation\] scales as $O(r\log(r))$, while the second term $8r-8$ scales as $O(r)$, but the first term is smaller for many integer values of $r$, at least including all $r<1100$ (note that for very small $r$, the exact value of $B_{r+1}$ is used in the calculation rather than the asymptotic bound above). Also, note that the duration of time of classical communication in Protocol \[ptl2\] (not including the time for entanglement preparation) could be as low as $3L/c$ (since the two teleportations from the $B$ side to the $A$ side can be done simultaneously with the sending of the classical message $m$; there are also two classical messages $l$ and $n$ sent from $A$ to $B$ before and after such step), where $L$ is the distance between the two parties, and $c$ is the speed of light. The communication time required by Protocol \[ptl3\] is also $3L/c$, since the middle two among the four stages of teleportations can be combined into one. Combining the considerations of entanglement cost and communication time, Protocol \[ptl2\] has a definite advantage over Protocol \[ptl3\] for small $r$. In the case $r=4$, an improved bound is provided by the following corollary: \[cr:sr4perm\] Any bipartite permutation unitary of Schmidt rank four can be implemented using LOCC with the help of not more than $10.71$ ebits of entanglement. Denote the bipartite permutation unitary as $U$. If there is a big column of $U$ containing four nonzero blocks, from Lemma \[le:persch\] (iv), $U$ is a controlled permutation unitary with four terms, hence the entanglement cost is at most $\log_2 4=2$ ebits. If there is a big column of $U$ containing three nonzero blocks, from Lemma \[le:persch\] (vi) and (vii), the entanglement needed is not more than $\max\{2+\log_2 2, 1+\log_2 3\}=3$ ebits, under a protocol that may have up to three levels of control depending on $U$. For the remaining cases, there is a formula $\log_2 (B_{r+1} \cdot r \cdot 2^r)$ in the proof of Theorem \[thm:permutation\], and the $r$ term is now replaced with $2$ because any big column of $U$ contains at most $2$ nonzero blocks. This gives $\log_2 (52\times 2\times 16)<10.71$ ebits. Taking into consideration all cases, the entanglement cost of $U$ is not greater than $10.71$ ebits. In Theorem \[thm:classical\_bipartite\](i) below, the two methods for implementing a bipartite permutation unitary in the proof of Theorem \[thm:permutation\] are adapted to the classical bipartite reversible circuits after simple changes. The implementation in Theorem \[thm:classical\_bipartite\](ii) below has some ancillas with final values not equal to initial values. Generally, in a classical computation on one party that uses reversible gates only, if it is required to restore the ancillas to their initial value in the end, we may copy the computation result by CNOT gates (the CNOT is a reversible gate) to some blank register, and the other ancillas can be restored to their initial value by running the inverse of the original reversible circuit. Such process is discussed in [@Bennett73], and a significantly modified method is used in Protocol \[ptl3\] (modification is needed because the initial inputs are still present after the first part of the protocol, and they should be gotten rid of in the end for implementing a quantum unitary operation), helping us obtain the $8r-8$ term in the result about entanglement cost in Theorem \[thm:permutation\]. The Theorem \[thm:classical\_bipartite\] (i) below can be directly adapted for quantum circuits that do not use entanglement but use nonlocal CNOT gates, as stated in (iii). In the following, a *bipartite classical reversible map* is a reversible map from $n+m$ bits to $n+m$ bits, where the $n$ bits are on party $A$, and the $m$ bits are on party $B$. The matrix of such map is a permutation matrix. The *Schmidt rank of a bipartite classical reversible map* is defined as the Schmidt rank of the corresponding quantum map, which is a bipartite permutation unitary and has the same matrix as the bipartite classical reversible map. \[thm:classical\_bipartite\] (i) Any bipartite classical reversible map of Schmidt rank $r$ can be implemented using classical local reversible gates and $\min\{2\lceil\log_2(B_{r+1})\rceil+2r+2\lceil\log_2 r\rceil, 8r-8\}$ classical nonlocal CNOT gates, if ancillas start with some known value and are required to be restored to the same value at the end.\ (ii) Any bipartite classical reversible map of Schmidt rank $r$ can be implemented using classical local reversible gates and $2r-2$ classical nonlocal CNOT gates, with ancillas starting with some known value but without any requirement about their final value.\ (iii) The assertion (i) also holds for quantum circuits, when the terms “classical reversible map”, “classical local reversible gates”, “classical nonlocal CNOT” and “value” are replaced by “permutation unitary”, “local permutation unitaries”, “nonlocal CNOT” and “computational basis state”, respectively. The proof of this theorem is given in Appendix \[app:[thm:classical\_bipartite]{}\]. Note that Theorem \[thm:classical\_bipartite\](ii) does not have a corresponding statement for the quantum permutation unitaries, because to implement a unitary operator, the ancillas at the end of the protocol should not contain information about the input, as mentioned in the proof of Theorem \[thm:permutation\]. Also note that we do not know whether Theorem \[thm:classical\_bipartite\] holds if all ancillas are required to start in some unknown state. This kind of consideration also appears in [@xu2015], which uses the term “borrowed bit” to describe an ancillary bit whose initial value is not known and is returned to the initial value at the end of the computation. On the other hand, in Theorem \[thm:permutation\] there is no specific requirement on the ancillas, so ancillas initialized in fixed quantum states are allowed and are actually used in the protocols in the proof. Examples {#subsec:permutation_example} -------- The simplest examples of Schmidt-rank-four permutation gates are the two-qubit SWAP and DCNOT (double-CNOT [@Collins01]) gates. In the following we show a more nontrivial example of Schmidt-rank-four permutation gate, that can be implemented using Protocol \[ptl2\]. The Example \[ex4\] below is about a unitary which is the product of a few transpositions on the input system, where a transposition is a swap of two states among the computational basis states. Such gates are of interest for quantum computation: In quantum algorithms involving queries such as the Grover’s algorithm [@Grover96; @Grover97], the oracle often acts nontrivially on only one or a few computational basis states, and is either a complex permutation gate or permutation gate, and in the former case it can often be implemented by a permutation gate with the help of ancilla qubit(s), which is illustrated in [@Grover01] in case of Grover’s algorithm. We consider the problem of minimizing the entanglement cost across some bipartite cut of the whole input system. This is not only useful when the two parties are located in separated locations, but is also useful for a local quantum computer where some gates between certain sets of qubits may be harder to implement than other gates due to the design of the layout of the qubits, etc. In the latter case the CNOT-gate cost may be a more relevant measure than entanglement cost, but our protocols can easily be modified to use CNOT gates across a bipartite division of the whole system instead of using entanglement and classical communication (both cases are with the help of local gates), usually with linear overhead. An example for such overhead is in the proof of Theorem \[thm:classical\_bipartite\], which is for classical reversible circuits but can be immediately translated into a result for quantum circuit involving permutation gates. \[ex4\] Suppose $U$ is a Schmidt-rank-four permutation unitary on a $5\times 6$ dimensional system. The matrix form of $U$ expressed using blocks is \[eq:transposition3\] ( [ccccc]{} T\_1 & T\_3 & 0 & 0 & 0\ T\_2 & 0 & T\_3 & 0 & 0\ 0 & T\_2 & 0 & T\_3 & 0\ 0 & 0 & T\_2 & 0 & T\_3\ 0 & 0 & 0 & T\_2 & T\_4\ ) where $T_1=\diag(1,1,1,0,0,0)$, and \[eq:t2t3\] T\_2=( [cccccc]{} 0 & 0 & 0 & 1 & 0 & 0\ 0 & 0 & 0 & 0 & 1 & 0\ 0 & 0 & 0 & 0 & 0 & 1\ 0 & 0 & 0 & 0 & 0 & 0\ 0 & 0 & 0 & 0 & 0 & 0\ 0 & 0 & 0 & 0 & 0 & 0\ ), and $T_3$ is the transpose of $T_2$, and \[eq:t2t3b\] T\_4=( [cccccc]{} 0 & 0 & 0 & 0 & 0 & 0\ 0 & 0 & 0 & 0 & 0 & 0\ 0 & 0 & 0 & 0 & 0 & 0\ 0 & 0 & 0 & 0 & 1 & 0\ 0 & 0 & 0 & 1 & 0 & 0\ 0 & 0 & 0 & 0 & 0 & 1\ ). The $B$ space of $U$ is spanned by $T_1,T_2,T_3,T_4$, hence $U$ is of Schmidt rank four. The $U$ is a symmetric matrix, so it is easy to express the action of $U$ as the swapping of some pairs of computational basis states. The $U$ can be implemented using Protocol \[ptl2\]. The effective input dimension of system $A$ is three, because the second, third and fourth big columns of $U$ all have the same two nonzero blocks $T_2$ and $T_3$ in them, so the corresponding three computational basis states of $\cH_A$ are regarded as the same type of input state of $A$. The effective output dimension of $A$ relative to any of the input computational basis state of $\cH_A$ is two, because there are only two nonzero blocks in each big column of $U$. The effective output dimension of $B$ is two, because the first three computational basis states of $\cH_B$ appear in the output of $T_1$ and $T_2$, but not in $T_3$ or $T_4$, so these three states are counted as one type of output state of $B$, and the same holds for the last three computational basis states of $\cH_B$. Hence the Protocol \[ptl2\] requires $2\log_2 (3\times2\times 2)<3.59$ ebits for this $U$. In contrast, implementing $U$ using two-way teleportation (see the beginning of Sec. \[sec:main\]) would need $2\log_2 5 > 4.64$ ebits. This shows that Protocol \[ptl2\] can sometimes be more efficient than two-way teleportation. Entangling power of bipartite permutation unitaries of small Schmidt rank {#ssec:ent_power_permutation} ------------------------------------------------------------------------- To know how tight our upper bounds for the entanglement cost for bipartite permutation unitaries of small Schmidt rank are, it is helpful to know the entangling power of those unitaries, since the entangling power (the quantity $K_E$ in [@Nielsen03]) gives a lower bound for the entanglement cost under LOCC. Formally for a bipartite unitary $U$ acting on systems $AB$, we have \[eq:K\_e\] K\_E(U) = \_[[|]{},[|]{}]{} E(U([|]{}[|]{})). Here ${|\alpha\rangle}$ and ${|\beta\rangle}$ are pure states on system $A R_A$ and $B R_B$ respectively, $R_A$ and $R_B$ are local ancillas, and the $E$ is the von Neumann entropy of the reduced density matrix on one of the two systems $A R_A$ and $B R_B$. From the definition of $K_E$, we have $K_E(U)\le\log_2 r$ ebits for any $U$ of Schmidt rank $r$. \[le:sch2perm\_entpower\] The entangling power and entanglement cost of any Schmidt-rank-two bipartite permutation unitary are both 1 ebit. From Lemma \[lm:sch2\] (i), up to local permutation unitaries and possibly a relabelling of the $A$ and $B$ sides, we may write the Schmidt-rank-two bipartite permutation unitary as $U=P_1\ox I_B + P_2 \ox V_B$, where $P_1,P_2$ are orthogonal projectors that add up to $I_A$, and $V_B$ is a permutation unitary satisfying that $V_B{|1\rangle}_B={|t\rangle}_B$, where $t\ge 2$ is an integer, and $\{{|j\rangle}_B\}$ is the computational basis of $\cH_B$. Suppose ${|1\rangle}_A$ and ${|s\rangle}_A$ are computational basis states of $\cH_A$ in the support of $P_1$ and $P_2$, respectively, where $s\ge 2$ is an integer. Then for the input product state $\frac{1}{\sqrt{2}}({|0\rangle}_A+{|s\rangle}_A)\ox {|1\rangle}$, the output is $\frac{1}{\sqrt{2}}\left({|1\rangle}_A\ox{|1\rangle}_B+{|s\rangle}_A\ox{|t\rangle}_B\right)$ which contains 1 ebit of entanglement. On the other hand, we have commented previously that the entangling power of any bipartite unitary of Schmidt rank $r$ is at most $\log_2 r$ ebits. This shows the entangling power of any Schmidt-rank-two bipartite permutation unitary is exactly 1 ebit. From the basic controlled-unitary protocol and Lemma \[lm:sch2\](i) (or from [@cy13]), the entanglement cost of any Schmidt-rank-two bipartite permutation unitary is not greater than 1 ebit. Since the entangling power of 1 ebit provides a lower bound for the entanglement cost, the entanglement cost of any Schmidt-rank-two bipartite permutation unitary is exactly 1 ebit. This completes the proof. It should be noted that the “entangling power” in Lemma \[le:sch2perm\_entpower\] can be understood as $K_E$ or $K_{\Delta E}$ (also defined in [@Nielsen03]), or the amortized $K_E$ or $K_{\Delta E}$ over many copies of the unitary, since all four quantities are lower bounds for the entanglement cost which is $1$ ebit in the current case. As a side note, we consider the entangling power of complex bipartite permutation unitaries of Schmidt rank two. Their entangling power $K_E$ can take any value in the interval $(0,1]$ (ebit). A simplest class of examples are locally equivalent to the ones in [@Nielsen03]: $U=\sqrt{1-p} I\ox I + i \sqrt{p} \sigma_z\ox \sigma_z$, where $p\in (0,1]$. When the definition is extended to $p=0$, $U$ is a Schmidt-rank-one unitary, with $K_E(U)=0$. By the continuity of $K_E$ (see [@Nielsen03]), when $p$ is near zero, the $K_E(U)$ is near zero while $U$ is a Schmidt-rank-two diagonal unitary. When $p$ is near $1/2$, the $K_E(U)$ is near $1$. **Entangling power of Schmidt-rank-three bipartite permutation unitaries.** The Schmidt-rank-three bipartite unitary $U$ cannot be on a $2\times2$ system [@Nielsen03]. Hence the maximum of $d_A$ and $d_B$ is at least three and it is indeed reachable. An example acting on $\bC^3\ox\bC^2$ is in . The structure of the Schmidt-rank-three bipartite permutation unitary $U$ has been partially investigated in Lemma \[lm:sch3perm\] (i). The following result gives a range for the entangling power of such unitaries, although we do not know whether the lower bound is optimal. The upper bound of $\log_2 3$ ebits is likely not optimal for some unitaries, see case (I.1) in the proof. \[le:sch3perm\_entpower\] The entangling power of a Schmidt-rank-three bipartite permutation unitary is at least $\log_2 9 - 16/9 \approx 1.392$ ebits and at most $\log_2 3\approx 1.585$ ebits. The proof of this Proposition is in Appendix \[app:[le:sch3perm\_entpower]{}\]. In the proof, the only case where the entangling power may be less than $\log_2 3$ ebits is case (I.1), in which case the entangling power of $U$ is at least $\log_2 9 - 16/9$ ebits, and such $U$ can be implemented using $\log_2 3$ ebits, while in general $U$ can be implemented using $2$ ebits, according to Lemma \[lm:sch3perm\](i). Hence the gap between the entangling power and the entanglement cost of a Schmidt-rank-three bipartite permutation unitary is at most $\max\{2-\log_2 3, \log_2 3-(\log_2 9 - 16/9)\}< 0.42$ ebits. Taking clue from the results above, we present the following conjecture: \[cj:entpower\] (1) The entangling power of any bipartite permutation unitary of Schmidt rank three can only take one of two values: $\log_2 9 - 16/9$ or $\log_2 3$ ebits.\ (2) The entangling power of any bipartite permutation unitary of Schmidt rank $r$ can only be one of $f(r)$ distinct values, where $f(r)$ is a finite integer-valued function of $r$. Numerical calculations suggest that (1) is likely to hold. In the calculations we have assumed the most general form of initial product pure state with ancillas $a$ and $b$, whose sizes are assumed to be equal to those of the corresponding input system $A$ and $B$, respectively. The sizes of $a$ and $b$ need not be larger since it suffices to consider the Schmidt decomposition on $aA$ and $bB$, respectively. Conclusions {#sec:con} =========== We have improved the upper bound for the entanglement cost of bipartite unitary operators of Schmidt rank three under LOCC protocols. Lemma \[lm:lm2\] implies a structure theorem for Schmidt-rank-3 bipartite unitaries, as stated in Theorem \[thm:sch3\]. We have presented a protocol attaining the improved upper bound for the entanglement cost for such unitaries. We have also studied the structure and entanglement cost of bipartite permutation unitaries of Schmidt rank up to three, and presented two protocols for implementing bipartite permutation unitaries of arbitrary Schmidt rank, and analyzed the entanglement and classical communication costs of the protocols. These results are independent of the dimensions of the spaces that the unitary acts on, and they significantly improve over the corresponding results in [@cy15]. The results are applied to classical circuits for implementing bipartite permutation operations, and the protocols we found are such that whether requiring the ancillas to be restored to the fixed initial state makes a difference in the required number of nonlocal CNOT gates. As for the complex permutation unitaries, our progress is mostly restricted to Schmidt rank three (apart from some results for special cases of general Schmidt rank in Lemma \[le:persch\]): Any Schmidt-rank-three bipartite complex permutation unitary that is not equivalent to a diagonal unitary under local permutation unitaries can be implemented with $3$ ebits and LOCC, but it remains open whether there is a constant upper bound of entanglement cost for implementing an arbitrary Schmidt-rank-three bipartite diagonal unitary. We also have quantified the entangling power of bipartite permutation unitaries of Schmidt rank two and three, and in the Schmidt-rank-three case the results suggest that there might be a gap between the entanglement cost and the entangling power. The examples of Schmidt-rank three bipartite permutation unitaries appearing in our proofs may be in some sense the simplest examples of a gap between the entanglement cost and the entangling power, if such gap exists at all: Although there are Schmidt-rank-two unitaries that may have such gap, those are not permutation unitaries and thus may be harder to study. Also, there is some correspondence between the permutation unitaries and the classical reversible circuits. So if the gap exists, there might be some operational implications even classically. Looking at this gap problem from the limit of large Schmidt rank, an apparent open problem is whether the results of Theorems \[thm:permutation\] and \[thm:classical\_bipartite\] can be improved. It is known [@Lovett14] that any total boolean function of rank $r$ can be computed by a deterministic classical communication protocol with $O(\sqrt{r}\cdot\log(r))$ bits of communication. The problem of implementing bipartite permutations might be a harder problem than computing a boolean function on bipartite inputs, but it would be interesting to find out more about the relation between the two problems. Acknowledgments {#acknowledgments .unnumbered} =============== L.Y. thanks Kae Nemoto for helpful discussions. L.C. was supported by the NSF of China (Grant No. 11501024), and the Fundamental Research Funds for the Central Universities (Grant Nos. 30426401 and 30458601). L.Y. was supported by NICT-A (Japan). The proof of Lemma \[lm:lm2\] {#app:{lm:lm2}} ============================= Firstly, note that $T_2$ must have at least two distinct eigenvalues, since otherwise $U$ is of Schmidt rank $2$, violating the assumption that it is of Schmidt rank $3$. Another observation is that the ratio $c_{j2}/\sqrt{\vert c_{j1}\vert^2 + \vert c_{j2}\vert^2}$ (and hence $c_{j2}^\ast/\sqrt{\vert c_{j1}\vert^2 + \vert c_{j2}\vert^2}$) takes at least two different values among different $j$, since otherwise $U$ is expandable using the two operators $c_{j1}I_d+c_{j2}T_2$ and $T_3$ on the second system with any particular $j$, implying that $U$ is of Schmidt rank $2$. Let \[eq:t3\] T\_3=D\_3+E\_3, where $D_3$ is diagonal, and all diagonal elements of $E_3$ are zero. Then $E_3$ is nonzero. Since $U$ is unitary, implies that \[eq:cj\_tj\] (c\_[j1]{} I\_d + c\_[j2]{} T\_2 + c\_[j3]{} T\_3) (c\_[j1]{} I\_d + c\_[j2]{}\^T\_2\^+ c\_[j3]{}\^T\_3\^)=I\_d,  \ (c\_[j1]{} I\_d + c\_[j2]{}\^T\_2\^+ c\_[j3]{}\^T\_3\^) (c\_[j1]{} I\_d + c\_[j2]{} T\_2 + c\_[j3]{} T\_3)=I\_d,   for all $j\in \{1,\dots,K\}$. Given that $T_3 T_3^\dag=T_3^\dag T_3=I_d$, we subtract terms with $T_3 T_3^\dag$ or $T_3^\dag T_3$ from both sides of each equation in . Since any $c_{j1}$ is real, the off-diagonal part of the resulting equations gives that \[eq:cj\_tj\_2\] c\_[j1]{} c\_[j3]{}\^E\_3\^+ c\_[j1]{} c\_[j3]{} E\_3 + c\_[j2]{} c\_[j3]{}\^T\_2 E\_3\^+ c\_[j2]{}\^c\_[j3]{} E\_3 T\_2\^=0,  \ c\_[j1]{} c\_[j3]{}\^E\_3\^+ c\_[j1]{} c\_[j3]{} E\_3 + c\_[j2]{} c\_[j3]{}\^E\_3\^T\_2 + c\_[j2]{}\^c\_[j3]{} T\_2\^E\_3 =0    for all $j\in \{1,\dots,K\}$. Since $c_{j3}$ are nonzero, we may divide both sides of the first equation in by $c_{j3}$, and obtain two independent equations of variables $E_3$ and $E_3 T_2^\dag$ by letting $c_{j2}^\ast/\sqrt{\vert c_{j1}\vert^2 + \vert c_{j2}\vert^2}$ take two different values (the other two terms containing $E_3^\dag$ and $T_2 E_3^\dag$ are viewed as “constants”). Hence $E_3$ and $E_3 T_2^\dag$ are in the space $H:=\lin\{E_3^\dag,T_2 E_3^\dag\}$. If $E_3^\dag\propto T_2 E_3^\dag$, then $T_2$ is proportional to the identity matrix on the rows in which $E_3^\dag$ is nonzero. The remaining diagonal elements of $T_2$ are in the rows in which $E_3^\dag$ is zero. By and the unitarity of $T_3$, the columns of $E_3^\dag$ that contain these diagonal entries (at the same positions in both $T_2$ and $T_3$) are also zero. Hence $T_2$ and $T_3$ are simultaneously block-diagonal under a block structure where the first block of $T_2$ is proportional to the identity matrix. It violates the assumption that $T_2$ and $T_3$ are not simultaneously diagonalizable under a unitary similarity transform. Therefore $H$ has dimension two. We discuss two cases. Case (a). Here $E_3^\dag$ and $E_3$ are not proportional, so they form a basis of $H$. We have $T_2 E_3^\dag=g E_3 + h E_3^\dag$ with complex numbers $g,h$. Since $E_3^\dag$ and $T_2 E_3^\dag$ also form a basis of $H$, we have $g\ne0$. Then \[eq:t2’\] T\_2’ E\_3\^= E\_3 with a diagonal matrix $T_2':=(T_2-h I_d)/g$. Denote $t_j$ as the $j$-th diagonal element of $T_2'$. It follows from and the unitarity of $T_3$ that the row vector and column vector of $E_3$ containing a diagonal entry of the same position have equal norm. Let $e_{jk}$ be the $(j,k)$ element of $E_3$. Let $S:=\{j: \exists k\,\,\mbox{s.t.}\,\,e_{jk}\ne 0\}$. Then it follows from that $t_j$ for those $j\in S$ all have modulus one. It follows from that $t_j e_{kj}^*=e_{jk}$ and $t_k e_{jk}^*=e_{kj}$, $\forall j,k\in\{1,\cdots,d\}$. So if $e_{jk}\ne0$, then $j\in S$ and $t_j=t_k$. Then $t_j\ne t_k$ implies $e_{jk}=0$. The last result, combined with the definition of $T_2'$ and , implies that $T_2$ and $T_3$ are simultaneously block-diagonal, where the blocks are such that each diagonal block of $T_2$ is a scalar matrix. Hence $T_2$ and $T_3$ are simultaneously diagonalizable under a unitary similarity transform. It is a contradiction with the assumption in the lemma. So case (a) has been excluded. Case (b). Hence $E_3^\dag$ and $E_3$ are proportional. By adjusting the phase for $E_3$, while multiplying all $c_{j3}$ by a corresponding phase factor to keep $U$ unchanged, we have $E_3^\dag=E_3$. Applying this equation to the two equations in , we have \[eq:cj1j3\] &&-(c\_[j1]{} c\_[j3]{}\^+ c\_[j1]{} c\_[j3]{}) E\_3\ &=& c\_[j2]{} c\_[j3]{}\^T\_2 E\_3 + c\_[j2]{}\^c\_[j3]{} E\_3 T\_2\^\ &=& c\_[j2]{} c\_[j3]{}\^E\_3 T\_2 + c\_[j2]{}\^c\_[j3]{} T\_2\^E\_3 for all $j\in \{1,\dots,K\}$. Left-multiplying the last line (which is equal to the first line) by $T_2$ and right-multiplying it by $T_2^\dg$, we obtain the second line, which is also equal to the first line, thus we have $(c_{j1} c_{j3}^\ast+ c_{j1} c_{j3}) (E_3 - T_2 E_3 T_2^\dg )=0$. Since the unitaries $T_2$ and $T_3$ are not simultaneously diagonalizable, we have \[eq:cj1\] c\_[j1]{} c\_[j3]{}\^+ c\_[j1]{} c\_[j3]{}=0,  j{1,…,K}.   Hence the first line of is zero, thus the second line of is zero, and since $c_{j2}$ and $c_{j3}$ are nonzero, we have $T_2 E_3\propto E_3 T_2^\dag$. We may adjust the phase of $T_2$ (while multiplying all $c_{j2}$ by a corresponding phase factor) so that \[eq:t2e3\] T\_2 E\_3=- E\_3 T\_2\^. From and the fact that all $c_{j1}$ are positive, we obtain that all $c_{j3}$ are pure imaginary. The last two statements, combined with that the second line of is zero, imply that all adjusted $c_{j2}$ are also pure imaginary. In the rest of the proof we use three assumptions. First, up to a relabeling of the computational basis states of $\cH_B$, $T_2=\bigoplus_{k=1}^n T_2^{(k)}$, and $T_3=\bigoplus_{k=1}^n T_3^{(k)}$, where $T_2^{(k)}$ and $T_3^{(k)}$ both act on the subspace $\cH_{B_k}$ of $\cH_B$, and $T_2^{(k)}$ is diagonal. Second, $T_2^{(1)}$ and $T_3^{(1)}$ commute, and the order of the matrix $T_2^{(1)}$ is the largest possible under this requirement and the first assumption. Of course it may be possible that such order is zero. If the order is nonzero, there is a unitary change of basis in the subspace $\cH_{B_1}$, such that the transformed $T_2^{(1)}$ and $T_3^{(1)}$ are diagonal, while keeping the identity matrix in this subspace \[see the $I_d$ term in \] unchanged. Third, for any $k>1$, $T_2^{(k)}$ and $T_3^{(k)}$ do not commute, and no $T_3^{(k)}$ can be block diagonal in a basis in which $T_2^{(k)}$ is diagonal. So any $T_2^{(k)}$ with $k>1$ has at least two distinct eigenvalues. It can be easily verified that the three assumptions as a whole is always valid, although it is possible that $\cH_{B_1}$ is a null space for some $U$. In the following derivations the $k$ is always greater than $1$ unless otherwise stated. Using , we have \[eq:t3k\] T\_3\^[(k)]{}=D\_3\^[(k)]{}+E\_3\^[(k)]{}, where $D_3^{(k)}$ is diagonal and the diagonals of $E_3^{(k)}$ are zero. Using and , we have \[eq:t2ke3\] T\_2\^[(k)]{} E\_3\^[(k)]{}=- E\_3\^[(k)]{} (T\_2\^[(k)]{})\^. This equation and the assumptions imply that any $T_2^{(k)}$ has exactly two distinct eigenvalues $e^{i\a_k},-e^{-i\a_k}$ with a real number $\a_k$. There exists a permutation matrix $P_k$ such that \[eq:pkt2k\] P\_k T\_2\^[(k)]{} P\_k\^&=& e\^[i\_k]{} I\^[(k)]{}\_[d\_k]{} (-e\^[-i\_k]{}) I\_[e\_k]{}\^[(k)]{} ,\ \[eq:pke3\] P\_k E\_3\^[(k)]{}P\_k\^&=& ( [cc]{} 0 & F\_3\^[(k)]{}\ G\_3\^[(k)]{} & 0\ ), where $d_k$ and $e_k$ are positive integers. Since $E_3^\dag=E_3$, we have $G_3^{(k)}=(F_3^{(k)})^\dg$. Since $T_3^{(i)}$ is unitary, implies that any two row vectors of $F_3^{(k)}$ are orthogonal, and any two column vectors of $F_3^{(k)}$ are also orthogonal. Our assumptions and the unitarity of $T_3^{(k)}$ imply that there is no zero row or column in $F_3^{(k)}$. The last two sentences imply $e_k=d_k\ge 1$. Then the unitary \[eq:s2k\] S\_2\^[(k)]{}:= I\_[d\_k]{} (-I\_[d\_k]{}){I\^[(k)]{},T\_2\^[(k)]{}} satisfies $S_2^{(k)}=(S_2^{(k)})^\dg$. If $d_k>1$, suppose $D_3^{(k)}$ is nonzero. Let the $V$ and $D$ in Lemma \[le:linearcom\] correspond to $T_3^{(k)}$ and $D_3^{(k)}$, respectively. From the form of $T_2^{(k)}$ in , and noting the form of the unitary similarity transform in Lemma \[le:linearcom\], it can be found that Lemma \[le:linearcom\] contradicts with the assumption that “no $T_3^{(k)}$ can be block diagonal in a basis in which $T_2^{(k)}$ is diagonal.” Hence $D^{(k)}_3=0$. Then and imply that \[eq:t3ke3k\] T\_3\^[(k)]{}=E\_3\^[(k)]{} = ( [cc]{} 0 & F\_3\^[(k)]{}\ (F\_3\^[(k)]{})\^& 0\ ), is a unitary matrix. Then $F_3^{(k)}$ is a unitary of order $d_k$. Let the $D$ and $\tilde U$ in Lemma \[le:linearcom\] correspond to $c_{j1}I_{2d_k}+c_{j2}T_2^{(k)}$ and $T_3^{(k)}$, respectively, for some $j\in\{1,\dots,K\}$, where $K$ is from . From , there is a nontrivial linear combination of these two matrices that is a unitary, so it corresponds to $V$ in Lemma \[le:linearcom\]. By noting the form of $T_2^{(k)}$ in , and the form of the unitary similarity transform in Lemma \[le:linearcom\], and the fact that a basis in which $T_2^{(k)}$ is diagonal is also a basis in which $c_{j1}I_{2d_k}+c_{j2}T_2^{(k)}$ is diagonal, and vice versa, it can be found that Lemma \[le:linearcom\] contradicts with the assumption that “no $T_3^{(k)}$ can be block diagonal in a basis in which $T_2^{(k)}$ is diagonal.” The argument above excludes the possibility of $d_k>1$. We have $d_k=1$. The above argument implies that $T_2^{(k)}$ in and $T_3^{(k)}$ in are both $2\times2$ unitary matrices, and $\det T_2^{(k)}=-1$. From , by doing a conjugation by a suitable diagonal $2\times2$ unitary: $T_3^{(k)}\ra Q_k T_3^{(k)} Q_k^\dg$, we may assume that the two non-diagonal entries of $T_3^{(k)}$ are equal and positive. The conjugation by the diagonal unitary $I_{T_2^{(1)}}\op (\op^n_{k=2} Q_k)$ does not change the $I_d$ and $T_2$ in since the latter are both diagonal. This completes the proof. The proof of Theorem \[thm:sch3\] {#app:{thm:sch3}} ================================= \(i) The assertion follows from the following argument which uses Lemma \[lm:lm2\]. The condition that $U$ is a Schmidt-rank-3 bipartite unitary controlled from the $A$ side implies $d_A\ge 3$ and $d_B\ge 2$. We consider the following decomposition of a general Schmidt-rank-three unitary $U$ controlled from the $A$ side: \[eq:u\_sum\_T\_j\] U=\_[j=1]{}\^[d\_A]{} [|j|]{} T\_j, where the unitaries $T_1,T_2$ and $T_3$ are linearly independent, and other $T_j\in\lin\{T_1,T_2,T_3\}$ are unitary. Using a local unitary on $\cH_B$, we assume $T_1=I_B$. We define the set $S_1:=\{T_j: T_j\in\lin\{T_1,T_2\}\}$, thus $T_1$ and $T_2$ are in $S_1$. We refer to $S_2$ as the set of $T_j$ (including $T_3$) that are in $\lin\{T_1,T_3\}$ but not in $S_1$. We also refer to $S_3$ as the set of $T_j$ that are not in $S_1\cup S_2$. Every $T_j$ in $S_3$ is of the form $T_j=\sum_{k=1}^3 h^{(j)}_k T_k$ with nonzero $h^{(j)}_2$ and $h^{(j)}_3$. The set $\{T_j\}_{j=1}^{d_A}$ is the union of the disjoint sets $S_1$, $S_2$ and $S_3$. Let the part of unitary $U$ corresponding to the set $S_{k}$ be denoted by $W_k$, $k=1,2,3$. Using these notions and , we have that up to a relabelling of the computational-basis states on $\cH_A$, \[eq:u1\_1\] U=W\_1 \_A W\_2 \_A W\_3. Evidently each of $W_1$ and $W_2$ has Schmidt rank at most two, and $W_3$ has Schmidt rank at most three. Consider the following two cases. Case (a): $W_3$ has Schmidt rank not greater than two. In this case $U$ is of the first standard form in assertion (i), according to . Case (b): $W_3$ has Schmidt rank exactly three. We may apply suitable local unitaries on $\cH_B$ before and after $U$ so that $T_1=I_B$ and $T_2$ is diagonal, thus in the case that $T_2$ and $T_3$ are not simultaneously diagonal, Lemma \[lm:lm2\] could be applied to $W'_3=(D^{(3)}_A\ox I_B) W_3$, where $D^{(3)}_A$ is a diagonal unitary on the subspace of $\cH_A$ that $W_3$ resides in, so as to let $W'_3$ satisfy the assumption in Lemma \[lm:lm2\] that all $c_{j1}$ are real, thus $W'_3$ is of the second standard form in assertion (i), then so is $W_3$. The case that $T_2$ and $T_3$ are simultaneously diagonalizable is excluded in the assumptions of Lemma \[lm:lm2\], but this case is possible, and $W_3$ is locally equivalent to a diagonal unitary in this case, so the second standard form in assertion (i) still holds for $W_3$. Then since $T_1,T_2,T_3$ span the $B$ space of $U$ as well as the $B$ space of $W_3$, the unitary $U$ also is of the second standard form. \(ii) Since $U$ is controlled from the $A$ side, it can be implemented using the basic controlled-unitary protocol with $\log_2 d_A$ ebits and LOCC. This gives the $d_A$ term inside the $\min\{\}$ symbol in Eq. . The two-way teleportation protocol with the $B$ system being teleported, gives the $d_B^2$ term inside the $\min\{\}$ symbol in Eq. . If $U$ is of the first standard form in assertion (i), the $U$ is a two-level controlled unitary, where the higher level controls which of the (up to) three unitaries $W_1,\dots,W_3$ is to be implemented in the lower level. Each of the three unitaries in the lower level is a controlled unitary of Schmidt rank two, thus there is one side in which it is controlled with two terms [@cy13]. Thus $U$ can be implemented under Protocol \[ptl1b\] with at most $\log_2 3 + 1 = \log_2 6$ ebits and at most $2\log_2 3 + 4=2\log_2 12$ c-bits. Since $d_B\ge 2$, the $\log_2 6$ ebits is not greater than the entanglement cost discussed in the next paragraph, and the relation of the entanglement costs in the current paragraph and the next paragraph is that the maximum is to be taken between these two, thus the $\log_2 6$ term does not appear in Eq. . Now consider the second standard form in assertion (i). We may use Protocol \[ptl\_gp\] with the choice of group being the dihedral group $D_{2n}$ with odd $n$. The group is of order $2n$, using the convention in [@dihedral] (note that the same group is sometimes denoted as $D_n$ in the literature). From the representation theory of dihedral groups [@dihedral], such group $D_{2n}$ has $(n-1)/2$ irreducible two-dimensional representations and two one-dimensional representations. There are $\lfloor d_B/2\rfloor$ $2\times 2$ blocks and possibly a $1\times 1$ block on the $B$ side of the expansion of the bipartite unitary, by viewing (as many as possible) pairs of $1\times 1$ blocks as $2\times 2$ blocks. Thus we have $n=2\lfloor d_B/2\rfloor+1$, and the order of the group is $2n=4\lfloor d_B/2\rfloor+2$. So the group-type protocol needs $\log_2 (4\lfloor d_B/2\rfloor+2)$ ebits. The asserted entanglement-cost upper bound is obtained by combining the results of the cases above. In all the cases mentioned above except the first standard form in assertion (i), the number of bits of classical communication is twice the amount of ebits contained in the resource entangled state, thus the claim of classical communication cost in assertion (ii) holds. The proof of Lemma \[le:persch\] {#app:{le:persch}} ================================ \(i) Since $U$ is a complex permutation matrix, any two nonzero entries in $U$ are in different rows of $U$. So the first assertion holds. The number of the nonzero blocks cannot exceed the Schmidt rank of $U$, which is $r$. On the other hand the number cannot be zero because $U$ is unitary. So the second assertion holds. \(ii) Up to local permutation matrices, we may assume that the first $r$ blocks in the big row are nonzero, and the remaining blocks in the big row are zero. Since $U$ is a complex permutation matrix and each block is of size $d_B\times d_B$, there are exactly $d_B$ nonzero entries in distinct rows of the big row. If the $r$ nonzero blocks in the given big row contains a common zero column vector, then any linear combination of them contains a zero column vector of the same position. And since $U$ is of Schmidt rank $r$, any block of $U$ is zero in that particular column. It is a contradiction with the fact that $U$ is unitary. So these $r$ blocks do not contain any common zero column vector. Since there are exactly $d_B$ nonzero entries in the $r$ blocks, the nonzero entries in the $r$ blocks are in different columns. So the assertion follows. Similarly, the assertion holds when all “row” are replaced with “column”. \(iii) The first two sentences in the claim follow from the fact that any block in $U$ is the linear combination of the $r$ nonzero blocks described in (ii). The last sentence in the claim is reached by using the basic controlled-unitary protocol. \(iv) The argument is exactly similar to the proof of (ii)(iii), so we abbreviate it here. \(v) Up to local permutation unitaries, we may assume that the first big row of $U$ contains exactly $r-1$ nonzero blocks, and the nonzero blocks in it are the first $r-1$ blocks, with the first one being equal to $I_s \oplus 0_{d_B-s}$, where $1\le s\le d_B-r+2$. In the following we prove that up to local permutations all the $r-1$ nonzero blocks in the first big row can be written as orthogonal projectors. Suppose this were not true, then there would be at least one common zero column in these $r-1$ nonzero blocks, and the $r$-th linearly independent block in $U$ must contain a nonzero element in this column. The linear combination of the $r$-th block and the first $r-1$ blocks (with nonzero coefficient for the $r$-th block) can appear at most once in each big row except the first big row, but must appear in each big column. Thus the count of such linear combination is both not more than $d_A-1$ and exactly equal to $d_A$, and this is a contradiction. Hence, up to local permutations all the $r-1$ nonzero blocks in the first big row can be written as orthogonal projectors. So the assertion holds. \(vi) The conditions imply that $\sum^{r-1}_{j=1} P_j = I_B$, $\sum^{r-1}_{j=1} s_j = d_B$, and the sum of orders of $Q$ and $Q_j$ ($\forall j\le n$) is $d_A$. These facts are used in the following proof. Since $U$ has Schmidt rank $r$, there is the $r$’th linearly independent block in $U$. This is a complex partial permutation matrix named as $R$. We regard it as a partitioned matrix \[eq:rsum\] R:=\^[r-1]{}\_[j,k=1]{}[|j|]{}R\_[jk]{}, where the subblock $R_{jk}$ is of size $s_j \times s_k$. In particular, the diagonal subblock $R_{jj}$ is in the same position and of the same size as that of $P_j$ in any diagonal block of $U$. Up to local permutation matrices on $U$, we may use the hypothesis that $n$ is the integer such that for any $j\in[1,n]$, there is a nonzero $R_{j,k_1}$ or $R_{k_2,j}$; and at the same time any $R_{j,k_1}$ and $R_{k_2,j}$ are both zero when $j>n$, $j\ne k_1,k_2$, and $k_1,k_2\in[1,r-1]$. In other words, $R$ is the direct sum of the upper left $(\sum^n_{j=1} s_j)\times(\sum^n_{j=1} s_j)$ submatrix $R'$ and $r-n-1$ subblocks $R_{jj}$ of size $s_j \times s_j$, $j=n+1,\cdots,r-1$, where the integer $n\in\{0\}\cup\{2,\dots,r-1\}$, since $n=1$ implies that there is a nonzero off-diagonal block $R_{1k}$ where $k\ge 2$, meaning that $n\ge 2$, thus the case $n=1$ does not exist. Since $U$ is a complex permutation matrix, any block of $U$ is a complex partial permutation matrix, which is the linear combination of $P_1,\cdots,P_{r-1}$ and $R$. These facts, and the hypothesis imply that $P_1,\cdots,P_n$ do not appear in the linear combination containing $R$ of nonzero coefficient. So any block in $U$ is either the linear combination of $P_1,\cdots,P_{r-1}$, or the direct sum of $R'$ multiplied by a phase and $r-n-1$ subblocks of size $s_j \times s_j$, $j=n+1,\cdots,r-1$. The hypothesis implies that each subblock is the linear combination of $R_{jj}$ and $P_j$. The submatrix of $U$ on the bipartite Hilbert space $\cH_A\times \lin\{{|s_1+\cdots+s_n+1\rangle},\cdots,{|d_B\rangle}\}$ form the second bracket in . The remaining part of unitary $U$, named as $U'$, acts on the bipartite Hilbert subspace $\cH_A\times\lin\{{|1\rangle},\cdots,{|s_1+\cdots+s_n\rangle}\}$. The above argument implies that each block of $U'$ is the linear combination of $P_1,\cdots,P_n$ and $R'$. In particular, the block has to be proportional to $R'$ when $R'$ appears in the linear combination. The hypothesis also implies that the big row or big column of $U'$ containing $R'$ does not contain any other nonzero block. So $R'$ is a complex permutation matrix. By letting $R'=P$, we can decompose $U'$ into the expression in the first bracket of . For the $U_j$ in the last big bracket in , it has Schmidt rank at most two, since $R$ and the term with the specific $P_j$ (where $j>n$) each contributes at most $1$ to the Schmidt rank. So the first paragraph in the claim holds. The last paragraph in the claim is from a multiple-level recursive control protocol generalized from Protocol \[ptl1\]. In the case $n\in[2,r-2]$, the protocol has three levels. The first level is choosing between the two terms in . If the choice is the first term, the second level then chooses between the two terms in the first big bracket in . Otherwise, the second level chooses between the terms in the last big bracket in , and the third level implements a Schmidt-rank-two unitary using the basic controlled-unitary protocol. In the case $n=0$, the protocol similarly has three levels but the first branch in the choices does not have the second or third level. In the case $n=r-1$, the protocol has only two levels since the last term in does not exist. In all cases, the lowest level of the protocol is the basic controlled-unitary protocol. \(vii) Let $U$ be a real permutation matrix and let it be of the form of the $n=0$ case in . We can instead expand the $U$ using orthogonal projectors $P_1, P_2, \dots, P_{r-1}$, and the matrix $R$ on the $B$ side, where $R$ is defined in the proof of (v) and is block-diagonal in the sense that $R_{jk}=0$ for $j\ne k$ in , since $n=0$. If $R$ is a diagonal matrix, then $R$ cannot be the identity matrix since then it would be in $\lin\{P_1, \dots, P_{r-1}\}$, violating that $U$ is of Schmidt rank $r$. But $R$ can be of less than full rank under the assumption that $R$ is diagonal, and in such case $U$ is a controlled permutation matrix controlled from $B$ side with at most $2(r-1)$ terms, which is the second form for $U$ in the assertion. Now suppose $R$ is not diagonal. Any block in $U$ cannot be a linear combination of $R$ and $P_1, \dots, P_{r-1}$ with nonzero coefficient for $R$, since then it would have two nonzero elements in some row. Thus any block of $U$ must be either $R$ or a linear combination of $P_1,\dots\,P_{r-1}$. Thus $U$ is the $A$-direct sum of a unitary whose $B$ space is spanned by $R$ only, and another unitary whose $B$ space is spanned by $P_1,\dots,P_{r-1}$, and the latter is a $(r-1)$-term controlled-permutation unitary controlled from the $B$ side. This is exactly the form for the case $n=r-1$ in . Thus the assertion holds, and the statement about entanglement cost follows from Protocol \[ptl1\]. This completes the proof. The proof of Lemma \[lm:diagonal\_blocks\] {#app:{lm:diagonal_blocks}} ========================================== \(i) The claim holds by definition.\ (ii) The equality obviously holds when $r=1$. In the following we assume $r\ge 2$. Denote the unitary as \[eq:usum\] U=\_[j=1]{}\^[d\_A]{} [|j|]{} V\_j. A class of examples $U$ with $2^{r-1}$ distinct diagonal blocks satisfy $d_A=2^{r-1}$, $d_B=2r-2$, $V_1=I_{2r-2}$, and for $k=2,\dots,r$, $V_k:=I_{2r-2}+{|2k-3\rangle\!\langle2k-2|}+{|2k-2\rangle\!\langle2k-3|}-{|2k-3\rangle\!\langle2k-3|}-{|2k-2\rangle\!\langle2k-2|}$. The $2^{r-1}$ diagonal blocks $V_j$ are of the form $V_r +\sum_{k=2}^r y_k (V_k-V_r)$, where $y_k$ is $0$ or $1$ for each $k\in[2,r]$. Hence \[eq:2r-1\] m(r)2\^[r-1]{}. Now we proceed with the main proof. Up to local permutations on $\cH_A$, we may assume the first $r$ diagonal blocks of $U$ in are linearly independent. We still denote them as $V_1, V_2, \dots, V_r$. Since each $V_h$ is the linear combination of them, we have $V_h=\sum_{k=1}^r x^{(h)}_k V_k$. Since all $V_j$ are permutation matrices, the sum of elements in each row of any $V_j$ is $1$. Thus we have $\sum_{k=1}^r x^{(h)}_k=1$. These two equations imply \[eq:vh\] V\_h-V\_1=\_[k=2]{}\^r x\^[(h)]{}\_k (V\_k-V\_1). For each $k=2,\cdots,r$ we regard the $V_k-V_1$ as a $d_B^2$-dimensional vector. Let the $d_B^2\times (r-1)$ matrix $M$ be consisted of column vectors $V_2-V_1,\cdots,V_r-V_1$. Since $V_2,\cdots,V_r$ are linearly independent, $M$ is of full rank $r-1$. Since the entry sum in each row of the matrix $V_k-V_1$ is zero, we can perform fixed row operations on $M$, to make zero the $d_B$ rows corresponding to the nonzero entries of $V_1$. The resulting matrix $M'$ has the same rank as $M$, since row operations preserve the matrix rank. There is a matrix $M''$ which is a $(r-1)\times (r-1)$ submatrix of $M'$, obtained by deleting the $d_B$ zero rows and some other rows in $M'$, which has the same rank as $M$, namely $r-1$. Then is equivalent to the fact that the vector $M''\cdot [x^{(h)}_2,\cdots,x^{(h)}_r]^T$ has entries one or zero, since all entries of $V_h$ are $0$ or $1$, and the nonzero entries of $V_1$ are excluded by the deletion mentioned above. So there are at most $2^{r-1}$ sets of solutions of $x^{(h)}_2,\cdots,x^{(h)}_r$. It implies $m(r)\le 2^{r-1}$. Combining it with we have $m(r)=2^{r-1}$. \(iii) The claim follows from (ii) and the basic controlled-unitary protocol. \(iv) A set of $B$-side Schmidt operators of $U$ can be chosen to be a set of linearly independent $d_B\times d_B$ blocks in the matrix $U$, hence they are partial permutation matrices (but in general they cannot be an arbitrary set of partial permutation matrices, since they jointly have to have support on every input computational-basis state). Then the assertion follows by definition. \(v) If $m'(r)>2^{r-1}$, by assertions (i), (ii) and that $r\ge 1$, there must be at least $r$ linearly independent ones among these $m'(r)$ distinct permutation matrices. Then assertion (ii) implies $m'(r)=2^{r-1}$, a contradiction. Hence $m'(r)\le 2^{r-1}$. But by definition $m'(r)\ge m(r)$, hence $m'(r)=2^{r-1}$. \(vi) The following argument is almost the same as the last paragraph of the proof of Lemma 21 in [@cy15]. For completeness we include the rewritten argument below. Suppose $\{F_i\}_{i=1}^r$ is a set of $r$ linearly independent matrices among the blocks of $U$. All nonzero partial permutation matrices in the $B$-space of $U$ are linear combinations of $\{F_j\}_{j=1}^r$. This last property still holds if we replace $\{F_i\}_{i=1}^r$ with $\{G_i\}_{i=1}^r$, defined as follows: Each $G_i$ is a linear combination of $\{F_j\}_{j=1}^r$, and satisfies $G_i(t)=\delta_{it}$, $i,t\in\{1,2,\dots,r\}$, where $G_i(t)$ is the $t$-th matrix element of $G_i$ according to some fixed ordering of the matrix elements, and $\delta_{it}$ is the Kronecker delta. Such ordering of the matrix elements must exist but the exact choice depends on the set $\{F_i\}_{i=1}^r$. We do not have extra restrictions on the $G_i(t)$ with $t>r$. Any nonzero partial permutation matrices in the $B$-space of $U$ is a linear combination of $G_i$ ($i=1,2,\dots,r$), and the coefficient for each $G_i$ is either $0$ or $1$, since the resulting matrix is a partial permutation matrix which implies that its first $r$ elements (in the ordering above) must be either $0$ or $1$. Since we only consider the nonzero matrices, the coefficients cannot all be zero, thus there are at most $2^r-1$ nonzero partial permutation matrices in the $B$-space of $U$. This proves $n(r)\le 2^r-1$. The value $2^r-1$ is attained by a $r$-term controlled unitary controlled from the $B$ side. To prove that no other type (up to local permutation equivalence) of bipartite permutation unitaries $U$ can achieve the value $2^r-1$, we make use of the essence of the argument in the last paragraph of the proof of Lemma 21 in [@cy15], that is, there are $r$ positions in the $d_B\times d_B$ matrix such that the value of these elements (each is $0$ or $1$, and is called a “key bit” below) determine the values of other entries of the matrices in the $B$ space of $U$ via fixed linear relations. Since there are $2^r-1$ nonzero partial permutation matrices in the $B$ space of $U$, it must be that every binary combination of the values of the $r$ key bits except the all-zero combination appear in a partial permutation matrix in the $B$ space of $U$. (Note that if the number $2^r-1$ were a smaller number, in general any binary combination of the values of the $r$ key bits does appear in some matrix in the $B$ space of $U$ but such matrix might not be a partial permutation matrix.) Thus no two key bits are located in the same row or column, since otherwise the matrix corresponding to the two key bits being both $1$ cannot be a partial permutation matrix. Suppose one key bit is at position $(r_1,c_1)$, i.e. row $r_1$ and column $c_1$, and another key bit is at position $(r_2,c_2)$. By considering the $(i,j)$ entry of the $d_B\times d_B$ matrix corresponding to both key bits being set to $1$, where $(i,j)\ne (r_1,c_1)$ and $(i,j)\ne (r_2,c_2)$, we find that such $(i,j)$ entry cannot be both $1$ in the two matrices corresponding to the two key bits being set to $1,0$ and $0,1$, respectively, as the latter two matrices add up to the former matrix. This shows that any matrix corresponding to only one key bit set to $1$ must be orthogonal to any other such matrix, where orthogonal means having no common nonzero rows and no common nonzero columns. And since the $U$ is unitary, for any row and column in the $d_B\times d_B$ matrix there has to be at least one nonzero element appearing in a partial permutation matrix with only one key bit set to $1$, thus the bipartite permutation unitary is equivalent to a controlled unitary from the $B$ side under local permutation unitaries. This completes the proof. The proof of Lemma \[lm:sch3perm\] {#app:{lm:sch3perm}} ================================== \(i) We call the first statement the “assertion.” In the following we prove the assertion first, then prove the statement about entanglement cost at the end. We use the same notations as in the proof of Lemma \[lm:sch2\]. First, if there is a big row or column of $U$ containing three nonzero blocks, then from Lemma \[le:persch\] (iv), $U$ is equivalent to a three-term controlled-permutation unitary controlled from the $B$ side, up to local permutation unitaries. Next, if there is exactly one nonzero block in each big row of $U$, then up to local permutation unitaries, $U$ is equivalent to a controlled-permutation unitary controlled from the $A$ side. The number of terms is between the Schmidt rank $r$ and $2^{r-1}$ by Lemma \[lm:diagonal\_blocks\] (ii). So it is either three or four. The remaining case is that there is a big row of $U$ containing exactly two nonzero blocks. From Lemma \[le:persch\] (vi), we have a standard form in , which satisfies the assertion except in the case $n=0$. In the case $n=0$, the assertion follows from Lemma \[le:persch\] (vii). Now we prove that the entanglement cost is at most $2$ ebits. In the first case in the assertion, the result follows from the basic controlled-unitary protocol. In the only remaining case in the assertion, the result follows from Protocol \[ptl1\], where the higher level of this two-level protocol determines which of the product permutation unitary or the two-term controlled-permutation unitary is to be implemented in the lower level. The entanglement cost for the two-level protocol is $\log_2 2+\log_2 2=2$ ebits. For each ebit used in the protocols, two c-bits are used, hence the classical communication cost is not more than $4$ c-bits. So the assertion holds. \(ii) Suppose $U$ is a Schmidt-rank-three bipartite complex permutation unitary that is not equivalent to a diagonal unitary under local permutation unitaries. It follows from Lemma \[le:persch\] (i) that some big column or row of $U$ contains the number of at most three nonzero blocks. If the number is exactly three or two, then the assertion respectively follows from Lemma \[le:persch\] (iii) or (vi). It remains to investigate the case when the number is one. We exchange the $A$ and $B$ systems of $U$ to obtain another matrix $\tilde U$, which is still a Schmidt-rank-three bipartite complex permutation unitary. Since $U$ is not equivalent to a diagonal unitary under local permutation unitaries, the nonzero blocks of $U$ do not have the same nonzero patterns (the pattern about which of the elements are nonzero), hence there are two nonzero blocks of $U$ such that there is nonzero element located in the same row within each block but at different column positions. This means that some big row of $\tilde U$ contains at least two nonzero blocks. The assertion again follows from Lemma \[le:persch\] (iii) and (vi). \(iii) Let the unitary be $U={| 1\rangle\!\langle 1 |}\ox I_B + {| 2\rangle\!\langle 2 |}\ox (x P + y P^\perp) + \sum^{d_A}_{j=3}{| j\rangle\!\langle j |}\ox V_j$ with two different phases $x,y$ and a projector $P$ onto some states in the computational basis of $\cH_B$, which can be assumed to be the first states in the basis, i.e. their labels are before the states in the support of the projector $P^\perp:=I_B-P$. The $V_j$ are diagonal matrices. We have $U=U_1 \oplus_B U_2$, where \[eq:u1u2\] U\_1 &=& ([| 11 |]{}+x[| 22 |]{})P +\ &&\^[d\_A]{}\_[j=3]{} [| jj |]{}P V\_j P,\ U\_2 &=& ([| 11 |]{}+y[| 22 |]{})P\^+\ &&\^[d\_A]{}\_[j=3]{} [| jj |]{}P\^V\_j P\^, Since $U$ is of Schmidt rank $3$, there is a $V_j$ (denoted $V_3$ without loss of generality) that is not a linear combination of $I_B$ and $x P + y P^\perp$. Every other $V_j$ is in $\lin\{I_B,x P + y P^\perp,V_3\}$. The matrices $P V_3 P$ and $P^\perp V_3 P^\perp$ are diagonal. If $P V_3 P$ has three or more distinct nonzero diagonal elements, then among the matrices $P V_j P$ there cannot be any linear combination of $P$ and $P V_3 P$ with nonzero coefficients for both terms, because of Lemma \[le:vandermonde\](iii) and the fact that the set $\{P V_j P\}\cup \{P\}$ contains exactly two linearly independent matrices, the latter is because the set $\{V_j\}\cup \{I_B,x P + y P^\perp\}$ which span the $B$ space of the Schmidt-rank-three unitary $U$ contains exactly three linearly independent matrices. Thus every $V_j$ is either proportional to $V_3$, or is in $\lin\{I_B,x P + y P^\perp\}$. Thus $U$ can be written as $U=W_1\oplus_A W_2$, where $W_1$ is a Schmidt-rank-two unitary with the $B$ space being $\lin\{I_B,x P + y P^\perp\}$, and the $W_2$ is a product unitary with the $B$ space being spanned by $V_3$. Thus $U$ can be implemented using Protocol \[ptl1\], with the lower level of this two-level protocol using at most $1$ ebit of entanglement, and the higher level (choosing between $W_1$ and $W_2$) using $1$ ebit. Thus $U$ can be implemented by $2$ ebits and LOCC in this case. If $P^\perp V_3 P^\perp$ has three or more distinct nonzero diagonal elements, we similarly have that $U$ can be implemented with $2$ ebits and LOCC. Now suppose $P V_3 P$ and $P^\perp V_3 P^\perp$ each has at most two distinct nonzero diagonal elements. Apparently any $P V_j P$ is in $\lin\{P,P V_3 P\}$, thus the $U_1$ (not $U$) is a unitary of Schmidt rank one or two, and can be written in a form of being controlled from the $B$ side (up to local unitaries) with at most two controlling terms. Similarly, by considering $P^\perp V_3 P^\perp$, we get that $U_2$ is controlled from the $B$ side (up to local unitaries) with at most two controlling terms. And since $U=U_1 \oplus_B U_2$, the $U$ is locally equivalent to a controlled unitary with at most $4$ controlling terms on the $B$ side, hence $U$ can be implemented using at most $2$ ebits and LOCC under the basic controlled-unitary protocol. Hence, in all cases, the $U$ can be implemented by $2$ ebits and LOCC. The proof of Theorem \[thm:permutation\] {#app:{thm:permutation}} ======================================== Denote the unitary as $U$. We first prove for the term $\log_2(B_{r+1})+r+\log_2 r$ in the assertion. In the following we consider the cases that $r\ge 4$, and the method is just to apply Protocol \[ptl2\] to the unitary $U$. The cases of $r\le 3$ will be mentioned later. The dimension of $a$ (and $e$, $e'$) in Protocol \[ptl2\] is the effective input dimension of $A$, i.e., number of different input types of $A$, or the number of different big columns of $A$ characterized by the set of nonzero blocks in the big column regardless of the order of the blocks. The effective input dimension of $A$ is at most $B_{r+1}$, which follows from Lemma \[le:combinations\_partial\_permutation\] by noting the following: All the blocks of $U$ are in the linear span of $r$ linearly independent blocks in $U$, and we may regard the $S$ in Lemma \[le:combinations\_partial\_permutation\] as the set of all blocks in $U$, and each big column of $U$ corresponds to a covering subset of $S$ determined by which nonzero blocks are in the big column. The dimension of $f'$ in Protocol \[ptl2\] is the effective output dimension of $A$ relative to the input computational basis state of $\cH_A$, and it is at most $r$, because there can be at most $r$ nonzero blocks in a big column of $U$. The dimension of $h'$ in Protocol \[ptl2\] is the effective output dimension of $B$. In Def. \[def\_state\_types\](iii) it is shown that the simplified definition is equivalent to the original definition for the effective output dimension of $B$, thus there are at most $2^r$ output types of $B$. It may be worth noting that the definition of such output types of $B$ above is independent of the output of $A$, and this is for the final phase correction $\hat Z^{-n}$ in Fig. \[fgr1\] to be successfully carried out. Thus, when $r\ge 4$, the number of ebits needed in the whole protocol is at most $\log_2 (B_{r+1} \cdot r \cdot 2^r)=\log_2 B_{r+1} + r + \log_2 r <\log_2 [0.792 r/\log_e (r+1)]^r + r + \log_2 r=O(r \log r)$. For each ebit in the protocol, $2$ c-bits are needed. When $r\le 3$, the number of ebits needed are $0$, $1$, and $2$ ebits for $r=1,2,3$, respectively, where the latter two results are from Lemma \[lm:sch2\] and \[lm:sch3perm\], respectively. Again, for each ebit in the protocols, $2$ c-bits are needed. The above shows that $U$ can be implemented using at most $\log_2(B_{r+1})+r+\log_2 r$ ebits and twice as many c-bits. In the following we prove for the term $8r-8$ in the assertion. From Lemma \[le:input\_types\_A\] and the symmetry of the two sides, the number of possible input types in the loose sense on each of the $A$ and $B$ sides is not more than $2^{r-1}$. Consider the Protocol \[ptl3\] shown in Fig. \[fgr2\]. The $a$ contains the input type of system $A$ in the loose sense, so its dimension is at most $2^{r-1}$. Hence the teleportation of $a$ to Bob’s side requires at most $r-1$ ebits and $2r-2$ c-bits. Similarly, the teleportation of $b$ to Alice’s side requires at most $r-1$ ebits and $2r-2$ c-bits. Teleporting these systems back requires the same amount of nonlocal resources. Since $U^\dag$ has the same Schmidt rank as $U$, the entanglement and classical communication cost of the second part of the protocol is bounded above by the same numbers as in the first part of the protocol. Hence, $8r-8$ ebits and $16r-16$ c-bits suffice to implement the $U$. Thus the assertion is proved by combining the upper bounds for the two protocols above. The proof of Theorem \[thm:classical\_bipartite\] {#app:{thm:classical_bipartite}} ================================================= \(i) For $r=2$ and $r=3$, we use the basic controlled-unitary protocol or the recursive controlled protocol \[Protocol \[ptl1\](a)\] which are used in the proof of Lemma \[lm:sch2\](i) and \[lm:sch3perm\](i), respectively, but with modifications to use nonlocal CNOT gates instead of entanglement, similar to those below for the case of general $r$. For $r\ge 4$, we use the adapted versions of the two protocols in the proof of Theorem \[thm:permutation\]. The details would be given in the following paragraphs but the main idea is to use local classical reversible gates instead of the local quantum permutation gates, and replace the entangled state and teleportation and the directly related LOCC operations with the classical nonlocal CNOT gate. According to the definition of ebits in Sec. \[sec:pre\], the non-integer entanglement cost in Theorem \[thm:permutation\] means that a maximally entangled state on $k\times k$ system is used, where $k$ is not a power of $2$. Since we are concerned with the CNOT gate cost, we extend such entangled state to be a maximally entangled state on a $2^n\times 2^n$ system, where $n\in\mathbb{N}$, and this gives the ceiling function in the assertion. In the following we consider the two protocols in the proof of Theorem \[thm:permutation\] respectively. For the first protocol in the proof of Theorem \[thm:permutation\] which is Protocol \[ptl2\], we may use an integer number of nonlocal CNOT gates to prepare $e'$ on the $B$ side, where $e'$ is the input to the $W$ gate in Protocol \[ptl2\], and similarly the same number of nonlocal CNOT gates is needed later to erase the $e'$, so two nonlocal CNOT gates are needed for every ebit in the $ee'$ state in Protocol \[ptl2\]. The teleportation of qubits from the $B$ side to the $A$ side are replaced with an integer number of the classical DCNOT (double-CNOT, see the quantum version in [@Collins01]) gates, where each DCNOT gate includes a CNOT gate controlled from $B$, where the controlled bit on $A$ is an auxiliary bit initially in the fixed value $0$, followed by a CNOT gate controlled from $A$. In other words, two nonlocal CNOT gates are used to transfer each bit from $B$ to $A$ while sending the auxiliary bit initialized in $0$ from $A$ to $B$. The original teleportation needs one ebit to teleport each qubit. Thus each term in the expression for the number of required nonlocal CNOT gates is at most two times the ceiling function of the number of ebits used in the corresponding part of Protocol \[ptl2\]. For the second protocol in the proof of Theorem \[thm:permutation\] which is Protocol \[ptl3\], each ebit can be turned into one nonlocal CNOT gate. For example, the first teleportation of the $a$ ($b$) system can be implemented by at most $r-1$ nonlocal CNOT gates to send the information about the computational basis of the register $a$ ($b$), and the teleportation back later can be implemented by at most $r-1$ CNOT gates to erase the state on one party, and then the remaining local copy of $a$ ($b$) can be locally erased by the inverse circuit of the local circuit used to prepare it. Thus the number of nonlocal CNOT gates needed is equal to the number of ebits used in Protocol \[ptl3\]. This completes the proof of (i).\ (ii) The following is the classical version of the first part of Protocol \[ptl3\]. From Lemma \[le:input\_types\_A\] and the symmetry of the two sides, the number of possible input types in the loose sense on each of the $A$ and $B$ sides is not more than $2^{r-1}$. Consider a classical circuit where Alice sends the input type of system $A$ to the $B$ side using $r-1$ CNOT gates, and Bob sends the input type of $B$ to the $A$ side using $r-1$ CNOT gates. Then each party computes the output of the local system, while keeping a copy of the inputs (both the local input and the received information about input types on the other system), in order to make the local circuit reversible, but this leaves some local ancillas with some value dependent on the inputs. Hence $2r-2$ CNOT gates suffice under the condition in the assertion.\ (iii) The assertion follows from (i) as well as the fact that in the circuits in the proof of (i), the ancillas in the end do not contain information about the input. This last condition about the final state of ancillas is necessary for implementing a quantum unitary operation, and is actually sufficient as long as there are no measurements and all gates are unitary; see Theorem 1 of [@ygc10]. The proof of Proposition \[le:sch3perm\_entpower\] {#app:{le:sch3perm_entpower}} ================================================== The upper bound follows from the definitions of the entangling power and the Schmidt rank of the bipartite unitary. To prove the lower bound, we consider three possible forms of $U$, which are studied in detail below. In all cases except case (I.1), the entangling power is $\log_2 3$ ebits. Case (I). Suppose $U$ is a controlled permutation unitary with three terms, and is controlled from the $A$ side. Up to local permutation matrices, we may assume \[eq:ud1\] U &=& D\_1I\_B\ &+& D\_2 (I\_m I\_n V\_1 V\_2)\ &+& D\_3 (I\_m V\_3 I\_q V\_4), where $D_jD_k=\d_{jk} D_j$, $\sum_j D_j = I_A$, and $V_1,V_2,V_3$ and $V_4$ are permutation matrices. $V_1$ and $V_3$ are respectively of size $q\times q$ and $n\times n$, and $V_2$ and $V_4$ are both of size $p\times p$ where $p=d_B-m-n-q$. If $V_1$ or $V_3$ contains a nonzero diagonal entry, then we can move the entry by local permutation matrices on $\cH_B$ so that $I_m$ is replaced with $I_{m+1}$. So $V_1$ and $V_3$ do not contain any nonzero diagonal entry. Similarly, we may assume that $V_2$ and $V_4$ do not have a nonzero diagonal entry in the same column when $p>0$. For the purpose of studying the entangling power of $U$, we may assume that all $D_j$ in are one-dimensional projectors, since the input state is a product state. In the following we consider three cases. The first case (I.1) is that $p=0$, namely $V_2$ does not exist in . We perform $U$ on the product vector ${|e\rangle}({|a\rangle}+{|b\rangle}+{|c\rangle})$ where ${|a\rangle}$, ${|b\rangle}$, and ${|c\rangle}$ are respectively in the support of $I_m$, $I_n$ and $I_q$ in . If the resulting state is maximally entangled, then the three states ${|a\rangle}+{|b\rangle}+{|c\rangle}$, ${|a\rangle}+{|b\rangle}+V_1{|c\rangle}$, and ${|a\rangle}+V_3{|b\rangle}+{|c\rangle}$ are pairwise orthogonal. The solution is ${|a\rangle}={|b\rangle}={|c\rangle}=0$. It is a contradiction with the resulting maximally entangled state. Hence, if ancillas are not allowed, then $U$ cannot create $\log_2 3$ ebits. The unitary $U$ with $m=p=0$ and $n=q=2$ can generate $\log_2 9 - 16/9$ ebits of entanglement starting from a product state without ancillas. A corresponding choice of such input state is $\frac{1}{\sqrt{3}}(1,1,1)\otimes (g,h,g,h)$, where $g=\frac{\sqrt{3}+\sqrt{6}}{6}$, $h=\frac{\sqrt{3}-\sqrt{6}}{6}$. Numerical evidence suggests that this number of $\log_2 9 - 16/9 \approx 1.392$ ebits is optimal for this $U$, even when ancillas are allowed. Of course, if $m>0$ in the case above, we can still create the same amount of entanglement by letting the input state have zero amplitude in the support of the $I_m$. When $q$ or $n$ is greater than $2$, up to local permutations there is always an $s\times s$ cyclic shift submatrix $V_{11}$ in $V_1$ and a $t\times t$ cyclic shift submatrix $V_{31}$ in $V_3$, respectively. We ignore the $I_m$ and other parts of $V_1$ and $V_3$, which means the $B$-side input state has zero amplitude in the support of those matrices. Under these conventions, we choose the input state to be of the form $\frac{1}{\sqrt{3}}(1,1,1)\otimes (v_1,v_2)$, where $v_1$ and $v_2$ are vectors of length $t$ and $s$, respectively, and the elements in $v_1$ are just two real numbers appearing alternately: $e,f,e,f,\dots$, and thus the last number in $v_1$ is $e$ if $t$ is odd, and is $f$ if $t$ is even. Similarly the elements in $v_2$ are just two real numbers appearing alternately: $g,h,g,h,\dots$, and thus the last number in $v_2$ is $g$ if $s$ is odd, and is $h$ if $s$ is even. With suitable choices of real numbers $e,f,g,h$, this would give rise to $\log_2 9 - 16/9 \approx 1.392$ ebits of entanglement in the output state. A class of choices of the real $4$-tuple $(e,f,g,h)$ for arbitrary $t,s\ge 2$ is given by $e-f=\frac{2}{\sqrt{6\lfloor t/2\rfloor}}$, $g-h=\frac{2}{\sqrt{6\lfloor s/2\rfloor}}$ and $\vert v_1\vert=\vert v_2\vert=\frac{1}{\sqrt{2}}$. When these equations are satisfied, the output reduced density operator on the $A$ side would be determined, and is the same as that corresponding to the optimal output entangled state in the case $t=s=2$. It is not hard to see that there are two solutions for the pair $(e,f)$ and two solutions for the pair $(g,h)$ for the equations above, thus there are four solutions $(e,f,g,h)$ for these equations, for any $t$ and $s$. This shows that $K_E(U)\ge\log_2 9 - 16/9 \approx 1.392$ ebits for all $U$ in case (I.1) . The second case (I.2) is that $p>0$ and $V_2\ne V_4$. Then both of $V_2$ and $V_4$ are nonzero. Up to local permutation matrices on $\cH_B$, we may assume \[eq:v2v4=\] V\_2&=&I\_s,\ V\_4&=&\[V\_[41]{},V\_[42]{}\]I\_t with $s,t\ge0$, where the submatrices $V_{21}$ and $V_{42}$ act on the same subspace $\lin\{{|s+1\rangle},\cdots,{|p-t\rangle}\}$ of dimension $p-s-t$. The moves in the paragraph including imply that $p>s+t$. So $V_{21}$ and $V_{42}$ are both nonzero, and are in the column vectors of the same position of $V_2$ and $V_4$. Furthermore $V_{21}$ and $V_{42}$ respectively have no nonzero diagonal entries of $V_2$ and $V_4$. So $\left( \begin{array}{cc} 0 \\ V_{21} \end{array} \right){|j\rangle}\ne{|j\rangle}$ and $\left( \begin{array}{cc} V_{42} \\ 0 \end{array} \right){|j\rangle}\ne{|j\rangle}$ for all $j\in[s+1,p-t]$. Note that $V_{21}$ and $V_{42}$ are both of full rank. If $\left( \begin{array}{cc} 0 \\ V_{21} \end{array} \right){|j\rangle}=\left( \begin{array}{cc} V_{42} \\ 0 \end{array} \right){|j\rangle}$ for all $j\in[s+1,p-t]$, then $V_{21}=\left( \begin{array}{cc} X \\ 0 \\ \end{array} \right)$ and $V_{42}=\left( \begin{array}{cc} 0 \\ X \\ \end{array} \right)$ with a permutation matrix $X$, and thus from we obtain that $V_2$ and $V_4$ are both equal to $X$ up to the moves in the paragraph including . This is a contradiction with the assumption at the beginning of this paragraph. Hence, we can find out some $j\in[s+1,p-t]$, such that $\left( \begin{array}{cc} 0 \\ V_{21} \end{array} \right){|j\rangle}\ne \left( \begin{array}{cc} V_{42} \\ 0 \end{array} \right){|j\rangle}$. It implies that ${|j\rangle}$, $V_2{|j\rangle}$ and $V_4{|j\rangle}$ are pairwise orthogonal. Let $U$ act on the product state $ {1\over\sqrt3} ( {|a_1\rangle} +{|a_2\rangle} +{|a_3\rangle}) {|j\rangle}, $ where the state ${|a_j\rangle}$ satisfies $D_j{|a_k\rangle}=\d_{jk}{|a_j\rangle}$, i.e., $D_j$ is the stabilizer of ${|a_j\rangle}$. So the resulting state ${1\over\sqrt3}({|a_1\rangle}{|j\rangle}+{|a_2\rangle}V_2{|j\rangle}+{|a_3\rangle}V_4{|j\rangle})$ is a Schmidt-rank-three maximally entangled state, and we have created $\log_2 3$ ebits. The third case (I.3) is that $p>0$ and $V_2=V_4$. So we may assume that $V_2$ does not have nonzero diagonal entries, and thus $p>1$. Since $U$ has Schmidt rank three, $n$ and $q$ are not simultaneously zero. If $n=0$ or $q=0$, by performing the local permutation matrix $I_A\ox (I_{m+n+q}\op V_2^\dg)$ on the lhs of $U$, we obtain a new unitary of the type of case (I.1). Thus we may assume $n>0$ and $q>0$. Since $V_1$ and $V_3$ have no nonzero diagonal entries, we have $n>1$ and $q>1$. Since the identity matrix and any permutation matrix are simultaneously diagonalizable, $U$ is locally equivalent to a Schmidt-rank-three diagonal unitary. The unitary $U$ with $m=0$ and $n=q=p=2$ can generate exactly $\log_2 3$ ebits of entanglement starting from a product state without ancillas. An optimal choice of the input state is $\frac{1}{\sqrt{3}}(1,1,1)\otimes (g,h,g,h,g,h)$, where $g=\frac{1+\sqrt{3}}{2\sqrt{6}}$, $h=\frac{1-\sqrt{3}}{2\sqrt{6}}$. For generic cases in the case (I.3), we may assume $m=0$ for the same reason as in case (I.1) above, and consider $n,q,p$ to be integers not less than two. Up to local permutation unitaries there is a cyclic shift (of length $t,s,u$ respectively) in each of the three permutation unitaries $V_1$, $V_2$ and $V_3$, and we let the input state to have nonzero amplitude on the support of these operators only and let them of the form $\frac{1}{\sqrt{3}}(1,1,1)\otimes (v_1,v_2,v_3)$, where the $v_1,v_2,v_3$ are real vectors of length $t,s,u$, respectively. The elements in $v_1$ are just two real numbers appearing alternately: $e,f,e,f,\dots$, and thus the last number in $v_1$ is $e$ if $t$ is odd, and is $f$ if $t$ is even. Similarly, $v_2=(g,h,g,h,\dots)$, and the last number in $v_2$ is $g$ if $s$ is odd, and is $h$ if $s$ is even. And $v_3=(y,z,y,z,\dots)$, and the last number in $v_3$ is $y$ if $u$ is odd, and is $z$ if $u$ is even. Then the maximal output entanglement of $\log_2 3$ ebits is achievable, by choosing $e,f,g,h,y,z\in\mathbb{R}$ which satisfy that $e-f=\frac{1}{\sqrt{2\lfloor t/2\rfloor}}$, $g-h=\frac{1}{\sqrt{2\lfloor s/2\rfloor}}$, $y-z=\frac{1}{\sqrt{2\lfloor u/2\rfloor}}$, and $\vert v_1\vert=\vert v_2\vert=\vert v_3\vert=\frac{1}{\sqrt{3}}$. It is not hard to see that there are $2^3=8$ real solutions $(e,f,g,h,y,z)$ to the equations above, for any $t,s,u$. And since $K_E(U)\le\log_2 r$ ebits for any $U$ of Schmidt rank $r$, we have that $K_E(U)=\log_2 3$ ebits for all $U$ in case (I.3). Case (II). Suppose $U$ is a Schmidt-rank-three controlled permutation unitary with four terms, and is controlled from the $A$ side. By following similar arguments as in (I) but also noting that the $B$-side operators in all four terms in $U$ are permutation matrices, it can be shown that up to local permutation unitaries, the $U$ is of the form \[eq:perm\_u\_4terms\] U= & D\_1I\_B + D\_2 (I\_m I\_n V\_1) +\ & D\_3 (I\_m V\_2 I\_q)+ D\_4 (I\_m V\_2 V\_1),where $D_j$ ($j=1,\dots,4$) are orthogonal projectors onto the computational basis states that add up to $I_A$, while $V_1$ and $V_2$ are permutation matrices of size $q\times q$ and $n\times n$, respectively, and their diagonal elements are all zero. And $m\ge 0$ is an integer. Again, for the purpose of studying the entangling power of $U$, we may assume that all $D_j$ in are one-dimensional projectors. When $q=n=2$, the entangling power of $U$ is exactly $\log_2 3$ ebits, and this number is achieved by a product input state without ancillas. For example, when $m=0$, there is an input state of the form $\frac{1}{2}(1,1,1,1)\otimes (g,h,g,h)$ which gives the optimal output entanglement, where $g=\frac{\sqrt{3}+\sqrt{6}}{6}$, and $h=\frac{\sqrt{3}-\sqrt{6}}{6}$ are the same numbers as in case (I.1). When $m>0$, we choose the $B$-side input state so that it has zero amplitude in the support of $I_m$ in , then we are back to the $m=0$ case. For other values of $q$ and $n$, and arbitrary $m\ge 0$ (which is treated as $m=0$), we also have that the entangling power of $U$ is exactly $\log_2 3$ ebits. A class of the optimal input states is the same as those in case (I.1), although it is possible that there are other classes of optimal input states as well. Case (III). Now the only remaining case is that $U$ is of the form of the last case in Lemma \[lm:sch3perm\](i). An example of this case is in . When no ancillas are allowed, the $U$ in can generate at most 1 ebit, since it is on a $3\times 2$ dimensional system. When ancillas are allowed, we choose the ancillas $A'$ and $B'$ to be of the same size as the input systems $A$ and $B$, respectively, and let the input state on the two sides be the maximally entangled states $\sum_{j=1}^3 {|jj\rangle}_{AA'}$ and $\sum_{k=1}^2 {|kk\rangle}_{BB'}$, respectively, then the output state contains exactly $\log_2 3$ ebits. For other unitaries $U$ of the type of case (III), up to local permutations and a swap of the two systems we may write $U$ as \[eq:uketbra11b\] U&=&(P\_AV\_B)\ &\_A& \[(I\_A-P\_A)Q\_B + W\_A (I\_B-Q\_B)\], where $P_A$ and $Q_B$ are projectors onto computational basis states of $\cH_A$ and $\cH_B$, respectively, and $W_A$ is a partial permutation matrix which is of full rank in the support of $I_A-P_A$, and $V_B$ is a permutation matrix. We choose the input state on the $A$ side to be of the form $\sum_{j=1}^{d_A} \mu_j{|jj\rangle}$, where the real coefficients $\mu_j$ take at most three different values including zero, and $\mu_j=0$ iff ${\langlej|}W_A{|j\rangle}\ne 0$. The nonzero values of $\mu_j$ are the same for ${|j\rangle}_A$ in the support of $P_A$. And the same statement holds for the support of $I_A-P_A$. And choose the input state on the $B$ side to be $\sum_{k=1}^{d_B} \nu_k {|kk\rangle}_{BB'}$, where the real coefficients $\nu_k$ take at most three different values including zero, and $\nu_k=0$ iff ${\langlek|}V_B{|k\rangle}\ne 0$. The nonzero values of $\nu_k$ are the same for ${|k\rangle}_B$ in the support of $Q_B$. And the same statement holds for the support of $I_B-Q_B$. With a suitable choice of the $\mu_j$ and $\nu_k$ subject to the constraints above, the output entanglement is exactly $\log_2 3$ ebits. Therefore, the entangling power of $U$ in case (III) is always $\log_2 3$ ebits. In summary, we have considered all forms of $U$, and thus the assertion holds.

--- abstract: 'We show that there is a constant $c$ such that any colouring of the cube $[3]^n$ in $c \log \log n$ colours contains a monochromatic combinatorial line.' author: - 'David Conlon[^1]' title: Monochromatic combinatorial lines of length three --- Introduction ============ The Hales–Jewett theorem [@HJ63] is a central result in Ramsey theory, an abstract version of van der Waerden’s theorem [@vdW27], saying that finite colourings of high-dimensional cubes contain monochromatic lines. To state the Hales–Jewett theorem formally, we consider the cube $[m]^n$ with $[m] = \{1,2, \dots, m\}$. A subset $L$ of $[m]^n$ is a combinatorial line if there is a non-empty set $I \subseteq [n]$ and $a_i \in [m]$ for each $i \notin I$ such that $$L = \{(x_1, \dots, x_n) \in [m]^n : x_i = a_i \mbox{ for all } i \notin I \mbox{ and } x_i = x_ j \mbox{ for all } i, j \in I\}.$$ The Hales–Jewett theorem is then as follows. For any $m$ and $r$, there exists an $n$ such that any $r$-colouring of $[m]^n$ contains a monochromatic combinatorial line. If we define $HJ(m, r)$ to be the smallest $n$ such that the Hales–Jewett theorem holds, then the original proof results in bounds of Ackermann type for $HJ(m, r)$. In the late eighties, Shelah [@S88] made a major breakthrough by finding a new way to prove the theorem which yielded primitive recursive bounds. This also gave the first primitive recursive bounds for van der Waerden’s theorem. In this special case, Shelah’s bound has since been drastically improved by Gowers [@G01]. The main result of this note is a reasonable bound for the $m = 3$ case. \[main\] There exists a constant $c$ such that $$HJ(3,r) \leq 2^{2^{cr}}.$$ After proving this theorem, we found that a result of this type was claimed at the end of Shelah’s seminal paper. However, the brief sketch given there is at best incomplete and the bound for $HJ(3,r)$ is usually stated as being of tower type in $r$. For instance, this is the case in a paper of Graham and Solymosi [@GS06] where they prove a result comparable to Theorem \[main\] for the coloured version of the corners theorem, that is, for finding monochromatic $(x,y)$, $(x + d, y)$, $(x, y+d)$ in any $r$-colouring of $[n]^2$. Their result is now a simple corollary of our own. We proceed straight to the details. The main idea, for those who know Shelah’s proof, is to use a one-sided version of his cube lemma. The proof ========= For $j = 1, \dots, t$, let $n_j = r^{6^{t-j}}$ and $s_j = n_1 + \dots + n_j$. Suppose now that $n = s_t = n_1 + \dots + n_t$ and $\chi$ is an $r$-colouring of $[3]^n$. We will show by induction, starting at $j = t$ and working downwards to $j = 0$, that for each word $w$ of length $s_j$, there are functions $f_k : [3]^{s_j} \rightarrow \binom{0 \cup [n_k]}{2}$ for all $k > j$ such that if $f_k(w) = \{p_{k,1}, p_{k,2}\}$ with $p_{k,1} < p_{k,2}$, then the following holds: For any $\ell > j$ and elements $a_{\ell + 1}, \dots, a_t$ of $[3]$, the two words $v_i = v_i(\ell; a_{\ell + 1}, \dots, a_t)$, $i = 1, 2$, have the same colour, where $$v_i = w \underbrace{11\dots1}_{p_{j+1, 2}}\underbrace{22\dots2}_{n_{j+1} - p_{j+1,2}} \dots \underbrace{11\dots1}_{p_{\ell, i}}\underbrace{22\dots2}_{n_{\ell} - p_{\ell,i}} \dots \underbrace{11\dots1}_{p_{t, 1}}\underbrace{a_t a_t \dots a_t}_{p_{t,2}-p_{t,1}}\underbrace{22\dots2}_{n_t - p_{t,2}}.$$ More precisely, $v_i$ is equal to $w$ for the first $s_j$ letters; for $j < k < \ell$, $v_i$ has $p_{k,2}$ ones followed by $n_k - p_{k,2}$ twos in the interval $[s_{k-1} + 1, s_k]$; in $[s_{\ell - 1} + 1, s_\ell]$, $v_i$ has $p_{\ell,i}$ ones followed by $n_\ell - p_{\ell,i}$ twos (this is the only use of the variable $i$); and, for $k > \ell$, the interval $[s_{k-1} + 1, s_k]$ consists of $p_{k, 1}$ ones, followed by $p_{k,2}-p_{k,1}$ copies of $a_k$, then by $n_k - p_{k,2}$ twos. When $j = t$, there is nothing to prove. Suppose now that for any word $w'$ of length $s_{j+1}$, we have $f_{j+2}(w'), \dots, f_t(w')$. Let $w$ be a word of length $s_j$ and, for each $0 \leq q \leq n_{j+1}$, consider the word $$w(q) = w\underbrace{11\dots1}_{q}\underbrace{22\dots2}_{n_{j+1} - q}$$ and write $f_k(w(q)) := \{p_{k,1}(q), p_{k, 2}(q)\}$ for all $k > j+1$. Then, for any $a_{j+2}, \dots, a_t \in [3]$, let $$w(q; a_{j+2}, \dots, a_t) = w(q) \underbrace{11\dots1}_{p_{j+2, 1}(q)}\underbrace{a_{j+2} a_{j+2} \dots a_{j+2}}_{p_{j+2,2}(q)-p_{j+2,1}(q)}\underbrace{22\dots2}_{n_{j+2} - p_{j+2,2}(q)} \dots \underbrace{11\dots1}_{p_{t, 1}(q)}\underbrace{a_t a_t \dots a_t}_{p_{t,2}(q)-p_{t,1}(q)}\underbrace{22\dots2}_{n_t - p_{t,2}(q)}.$$ That is, $w(q; a_{j+2}, \dots, a_t)$ equals $w(q)$ for the first $s_{j+1}$ letters, then, for each $k > j+1$, the interval $[s_{k-1} + 1, s_k]$ has $p_{k,1}(q)$ ones, followed by $p_{k,2}(q) - p_{k,1}(q)$ copies of $a_k$, then by $n_k - p_{k,2}(q)$ twos. Now, to each $w(q)$, we assign a colour $\chi_{j+1}(w(q))$, namely, $$\prod_{k = j+2}^t f_k(w(q)) \times \prod_{a_{j+2}, \dots, a_t \in [3]} \chi(w(q; a_{j+2}, \dots, a_t)).$$ Note that the number of colours is $$\prod_{k=j+2}^t \binom{n_k + 1}{2} r^{3^{t - j - 1}} \leq (n_{j+2} \dots n_t)^{2} r^{3^{t-j-1}}.$$ Therefore, since $n_{j+1} \geq (n_{j+2} \dots n_t)^{2} r^{3^{t-j-1}}$, we see that there must exist two choices $p_{j+1, 1}$ and $p_{j+2, 2}$ for $q$ such that $\chi_{j+1}(w(p_{j+1, 1})) = \chi_{j+1}(w(p_{j+1, 2}))$. We now claim that letting $f_{j+1}(w) = \{p_{j+1, 1}, p_{j+2, 2}\}$ and, for $k > j+1$, $f_k(w) = f_k(w(p_{j+1,1})) = f_k(w(p_{j+1,2}))$ suffices. For $\ell = j + 1$, the $v_i(j+1; a_{j+2}, \dots, a_t)$ receive the same colour by the choice of $p_{j+1, 1}$ and $p_{j+1, 2}$. For $\ell > j + 1$, first note that $v_i(\ell; a_{\ell+1}, \dots, a_t)$ is the same as $v'_i(\ell; a_{\ell+1}, \dots, a_t)$, where $v'_i$ is defined relative to the word $w(p_{j+1,2})$. But, by induction, the $v'_i(\ell; a_{\ell+1}, \dots, a_t)$ receive the same colour for all choices of $\ell > j + 1$ and all $a_{\ell+1}, \dots, a_t \in [3]$. This completes our induction. Continuing our induction all the way to $j = 0$ gives, for all $k = 1, \dots, t$, numbers $p_{k,1}, p_{k,2}$ with $0 \leq p_{k,1} < p_{k,2} \leq n_k$ such that For any $0 \leq \ell \leq t$ and elements $a_{\ell + 1}, \dots, a_t$ of $[3]$, the two words $v_i = v_i(\ell; a_{\ell + 1}, \dots, a_t)$, $i = 1, 2$, have the same colour, where $$v_i = \underbrace{11\dots1}_{p_{1, 2}}\underbrace{22\dots2}_{n_{1} - p_{1,2}} \dots \underbrace{11\dots1}_{p_{\ell, i}}\underbrace{22\dots2}_{n_{\ell} - p_{\ell,i}} \dots \underbrace{11\dots1}_{p_{t, 1}}\underbrace{a_t a_t \dots a_t}_{p_{t,2}-p_{t,1}}\underbrace{22\dots2}_{n_t - p_{t,2}}.$$ In words, for $1 \leq k < \ell$, $v_i$ has $p_{k,2}$ ones followed by $n_k - p_{k,2}$ twos in the interval $[s_{k-1} + 1, s_k]$; in $[s_{\ell - 1} + 1, s_\ell]$, $v_i$ has $p_{\ell,i}$ ones followed by $n_\ell - p_{\ell,i}$ twos; and, for $k > \ell$, the interval $[s_{k-1} + 1, s_k]$ consists of $p_{k, 1}$ ones, followed by $p_{k,2}-p_{k,1}$ copies of $a_k$, then by $n_k - p_{k,2}$ twos. To conclude the proof, take $t = r$ and consider the words $v(q) = v(0; 1, 1, \dots, 1, 3, 3, \dots, 3)$ (the $i$ is redundant when $\ell = 0$) where there are $q$ ones followed by $r - q$ threes for some $0 \leq q \leq r$. By the pigeonhole principle, there must exist $q_1$ and $q_2$ with $q_1 < q_2$ such that $\chi(v(q_1)) = \chi(v(q_2))$. But then $$\begin{aligned} \chi(v(q_2)) & = \chi(v_2(q_2;3,3, \dots, 3))\\ & = \chi(v_1(q_2; 3, 3, \dots, 3))\\ & = \chi(v_2(q_2 - 1; 2, 3, 3, \dots, 3))\\ & = \dots \\ & = \chi(v_2(q_1 + 1; 2, 2, \dots, 2, 3, 3, \dots, 3)\\ & = \chi(v_1(q_1 + 1; 2, 2, \dots, 2, 3, 3, \dots, 3)\end{aligned}$$ where there are always $r - q_2$ threes. Here, alternate lines follow from identification between words and from an application of the conclusion above. But $$v_1(q_1 + 1; 2, 2, \dots, 2, 3, 3, \dots, 3) = v(0; 1, 1, \dots, 1, 2, 2, \dots, 2, 3, 3, \dots, 3),$$ where there are $q_1$ ones, followed by $q_2 - q_1$ twos, then $r- q_2$ threes, so together with $v(q_1)$ and $v(q_2)$, where the twos are replaced with threes and ones, respectively, we get the required monochromatic combinatorial line. As a closing remark, we note that a similar iteration with Theorem \[main\] as a base shows that $HJ(4,r)$ is at most a tower of twos of height $O(r)$, improving also the bound for lines of length four. (2001), 465–588. in Topics in discrete mathematics, 129–132, Algorithms Combin., 26, Springer, Berlin, 2006. (1963), 222–229. (1988), 683–697. (1927), 212–216. [^1]: Mathematical Institute, Oxford OX2 6GG, United Kingdom. E-mail: [david.conlon@maths.ox.ac.uk]{}. Research supported by ERC Starting Grant 676632.

--- abstract: 'Recent numerical relativistic simulations of black hole coalescence suggest that in certain alignments the emission of gravitational radiation can produce a kick of several thousand kilometers per second. This exceeds galactic escape speeds, hence unless there a mechanism to prevent this, one would expect many galaxies that had merged to be without a central black hole. Here we show that in most galactic mergers, torques from accreting gas suffice to align the orbit and spins of both black holes with the large-scale gas flow. Such a configuration has a maximum kick speed $<200$ km s$^{-1}$, safely below galactic escape speeds. We predict, however, that in mergers of galaxies without much gas, the remnant will be kicked out several percent of the time. We also discuss other predictions of our scenario, including implications for jet alignment angles and X-type radio sources.' author: - 'Tamara Bogdanović, Christopher S. Reynolds, and M. Coleman Miller' title: Alignment of the spins of supermassive black holes prior to coalescence --- Introduction ============ When two black holes spiral together and coalesce, they emit gravitational radiation which in general possesses net linear momentum. This accelerates (i.e., “kicks”) the coalescence remnant relative to the initial binary center of mass. Analytical calculations have determined the accumulated kick speed from large separations until when the holes plunge towards each other [@Per62; @Bek73; @Fit83; @FD84; @RR89; @W92; @FHH04; @BQW05; @DG06], but because the majority of the kick is produced between plunge and coalescence, fully general relativistic numerical simulations are necessary to determine the full recoil speed. Fortunately, the last two years have seen rapid developments in numerical relativity. Kick speeds have been reported for non-spinning black holes with different mass ratios [@HSL06; @Baker06; @Gonzalez06] and for binaries with spin axes parallel or antiparallel to the orbital axes [@Herrmann07; @Koppitz07; @Baker07], as well as initial explorations of more general spin orientations [@Gonzalez07; @Campanelli07a; @Campanelli07b]. For mergers with low spin or spins both aligned with the orbital angular momentum, these results indicate maximum kick speeds $<200$ km s$^{-1}$. Remarkably, however, it has recently been shown that when the spin axes are oppositely directed and in the orbital plane, and the spin magnitudes are high (dimensionless angular momentum ${\hat a}\equiv cJ/GM^2\sim 1$), the net kick speed can perhaps be as large as $\sim 4000$ km s$^{-1}$ [@Gonzalez07; @sb07; @Campanelli07b]. The difficulty this poses is that the escape speed from most galaxies is $<1000$ km s$^{-1}$ (see Figure 2 of @Merritt04), and the escape speed from the central bulge is even smaller. Therefore, if large recoil speeds are typical, one might expect that many galaxies that have undergone major mergers would be without a black hole. This is in clear contradiction to the observation that galaxies with bulges all appear to have central supermassive black holes (see @FF05). It therefore seems that there is astrophysical avoidance of the types of supermassive black hole coalescences that would lead to kicks beyond galactic escape speeds. From the numerical relativity results, this could happen if (1) the spins are all small, (2) the mass ratios of coalescing black holes are all much less than unity, or (3) the spins tend to align with each other and the orbital angular momentum. The low-spin solution is not favored observationally. X-ray observations of several active galactic nuclei reveal relativistically broadened Fe K$\alpha$ fluorescence lines indicative of spins ${\hat a}>0.9$ [@Iwasawa96; @Fab02; @RN03; @BR06]. A similarly broad line is seen in the stacked spectra of active galactic nuclei in a long exposure of the Lockman Hole [@Streb05]. More generally, the inferred average radiation efficiency of supermassive black holes suggests that they tend to rotate rapidly (@Sol82 [@YT02]; see @marconi04 for a discussion of uncertainties). This is also consistent with predictions from hierarchical merger models (e.g., @volonteri05). Mass ratios much less than unity may occur in some mergers, and if the masses are different enough then the kick speed can be small. For example, @Baker07, followed by [@Campanelli07b], suggest that the spin kick component scales with mass ratio $q\equiv m_1/m_2\leq 1$ as $q^2({\hat a}_2-q{\hat a}_1)/(1+q)^5$, hence for ${\hat a}_1=-{\hat a_2}=1$ the maximum kick speed is $\propto q^2/(1+q)^4$. For $q<0.1$ this scales roughly as $q^2$ and hence kicks are small. However, for $q>0.2$ the maximum kick is within a factor $\sim 3$ of the kick possible for $q=1$. An unlikely conspiracy would thus seem to be required for the masses always to be different by the required factor of several. Some tens of percent of galaxies appear to have undergone at least one merger with mass ratio $>0.25$ within redshift $z<1$ (for recent observational results with different methods, see @Bell06b [@Lotz06], and for a recent simulation see @Maller06). The well-established tight correlations between central black hole mass and galactic properties such as bulge velocity dispersion (see @FF05 for a review) then suggest strongly that coalescence of comparable-mass black holes should be common. The most likely solution therefore seems to be that astrophysical processes tend to align the spins of supermassive black holes with the orbital axis. This astrophysical alignment is the subject of this [ *Letter*]{}. Here we show that gas-rich mergers tend to lead to strong alignment of the spin axes with the orbital angular momentum and thus to kick speeds much less than the escape speeds of sizeable galaxies. In contrast, gas-poor mergers show no net tendency for alignment, assuming an initially uniform distribution of spin and orbital angular momentum vectors. We demonstrate this aspect of gas-poor mergers in § 2. In § 3 we discuss gas-rich mergers, and show that observations and simulations of nuclear gas in galactic mergers suggest that the black holes will be aligned efficiently. We discuss consequences and predictions of this alignment in § 4. Gas-poor Mergers ================ Several recent models and observations have been proposed as evidence that some galactic mergers occur without a significant influence of gas. Possible signatures include the metal richness of giant ellipticals (e.g., @NO07) and slow rotation and the presence of boxy orbits in the centers of some elliptical galaxies (e.g., @Bell06a [@Naab06]). Consider such a gas-free merger, and assume that we can therefore treat the gradual inspiral of two spinning black holes as an isolated system. As laid out clearly by [@Schnittman04], throughout almost the entire inspiral there is a strong hierarchy of time scales, such that $t_{\rm inspiral}\gg t_{\rm precess}\gg t_{\rm orbit}$. [@Schnittman04] therefore derived orbit-averaged equations for the spin evolution in the presence of adiabatic dissipation. Such effects can lead to relaxation onto favored orientations. The question is then whether, with the uniform distribution of orbital and spin directions that seems expected in galactic mergers, there is a tendency to align in such a way that the net kicks are small. Using equations A8 and A10 from [@Schnittman04], we have evolved the angles between the two spin vectors, and between the spins and the orbital angular momentum. We find that for isotropically distributed initial spins and orbits, the spins and orbits at close separation are also close to isotropically distributed (see Figure \[fig:finalangles\]). Thus, although (as we confirm) [@Schnittman04] showed that for special orientations the spins might align (e.g., for an initial $\cos\theta_1\approx 1$, or as we also discovered, for an initial $\cos\theta_2\approx -1$), the initial conditions resulting in such alignment are special and subtend only a small solid angle. The conclusion is that gas-poor mergers alone cannot align spins sufficiently to avoid large kicks due to gravitational radiation recoil. Indeed, [@sb07] find that for mass ratios $q>0.25$, spin magnitudes ${\hat a}_1={\hat a_2}=0.9$, and isotropic spin directions, $\sim 8$% of coalescences result in kick speeds $>1000$ km s$^{-1}$ and $\sim 30$% yield speeds $>500$ km s$^{-1}$. The high maximum speeds inferred by [@Campanelli07b] are likely to increase these numbers. We now discuss gas-rich mergers, which can naturally reduce the kick speeds by aligning black hole spins with their orbital axis. Gas-rich Mergers \[S\_wet\_mergers\] ==================================== Consider now a gas rich environment, which is common in many galactic mergers. The key new element is that gas accretion can exert torques that change the direction but not the magnitude of the spin of a black hole, and that the lever arm for these torques can be tens of thousands of gravitational radii [@bp75; @np98; @na99]. In particular, @np98 and [@na99] demonstrate that the black hole can align with the larger scale accretion disk on a timescale that is as short as 1% of the accretion time. An important ingredient of this scenario is the realization by @pp83 that the warps are transmitted through the disk on a timescale that is shorter by a factor of $1/2\alpha^2$ compared with the transport of the orbital angular momentum in flat disks, where $\alpha\sim 0.01-0.1$ is the standard viscosity parameter [@ss73]. The question that distinguishes gas-rich from gas-poor mergers is therefore whether the accreted mass is $\sim 0.01-0.1M_{\rm bh}$ during the sinking of the black holes towards the center of the merged galaxy, where $M_{\rm bh}$ is a black hole mass. Numerical simulations show that galactic mergers trigger large gas inflows into the central kiloparsec, which in gas rich galaxies can result in a $\sim 10^9 {\>{\rm M_{\odot}}}$ central gas remnant with a diameter of only few$\times$100 pc [@bh91; @bh96; @mh94; @msh05; @kazantzidis05]. Such mergers are thought to be the progenitors of ultraluminous infrared galaxies. @kazantzidis05 find that the strong gas inflows observed in cooling and star formation simulations always produce a rotationally supported nuclear disk of size $\sim 1-2$ kpc with peak rotational velocities in the range of 250$-$300 ${\rm km\, s^{-1}}$. The results of numerical simulations are in good agreement with observations, which also show that the total mass of the gas accumulated in the central region of merger galaxies can reach $10^9-10^{10}{\>{\rm M_{\odot}}}$ and in some cases account for about half of the enclosed dynamical mass [@tacconi99]. Observations imply that the cold, molecular gas settles into a geometrically thick, rotating structure with velocity gradients similar to these obtained in simulations and with densities in the range $10^2-10^5\,{\rm cm^{-3}}$ [@ds98]. Both observations and simulations of multiphase interstellar matter with stellar feedback show a broad range of gas temperatures, where the largest fraction of gas by mass has a temperature of about $100\,$K [@wn01; @wn02]. We therefore consider an idealized model based on these observations and simulations. In our model, the two black holes are displaced from the center embedded within the galactic-scale gas disk. We are mainly concerned with the phase in which the holes are separated by hundreds of parsecs, hence the enclosed gas and stellar mass greatly exceeds the black hole masses and we can assume that the black holes interact independently with the disk. Based on the results of @escala04 [@escala05], @mayer06, and @dotti06 the time for the black holes to sink from these separations to the center of the disk due to dynamical friction against gaseous and stellar background is $\leq5\times10^7$yr, which is comparable to the starburst timescale, $\sim10^8$yr [@larson87]. The accretion onto the holes is mediated by their nuclear accretion disks fed from the galactic scale gas disk at the Bondi rate, ${\dot M}_{\rm Bondi}$, as long as it does not exceed the Eddington rate, ${\dot M_{\rm Edd}}$ [@GT04]. Locally, one can estimate the Bondi radius $R_{\rm Bondi}=GM_{\rm bh}/v_g^2\approx 40~{\rm pc}\,(M_{\rm bh}/10^8\,M_\odot)(v_g/100~{\rm km~s}^{-1})^{-2}$ that would be appropriate for a total gas speed at infinity, relative to a black hole, of $v_g$ (we use a relatively large scaling of 100 km s$^{-1}$ for this quantity to be conservative and to include random motions of gas clouds as well as the small thermal speed within each cloud). The accretion rate onto the holes will then be $\min({\dot M_{\rm Bondi}},{\dot M_{\rm Edd}})$, where ${\dot M}_{\rm Bondi}\approx 1~M_\odot\,{\rm yr}^{-1}\; (v_g/100\,{\rm km\,s}^{-1})^{-3}(n/100\,{\rm cm}^{-3})(M_{\rm bh}/10^8{\>{\rm M_{\odot}}})^2$. We also note that clearing of a gap requires accretion of enough gas to align the holes with the large-scale gas flow. At this rate, the holes will acquire 1-10% of their mass in a time short compared to the time needed for the holes to spiral in towards the center or the time for a starburst to deplete the supply of gas. The gas has significant angular momentum relative to the black holes: analogous simulations in a planetary formation environment suggest that the circularization radius is some hundredths of the capture radius (e.g., @HB91 [@HB92]). This corresponds to more than $10^5$ gravitational radii, hence alignment of the black hole spin axes is efficient (and not antialignment, since the cumulative angular momentum of the accretion disk is much greater than the angular momentum of the black holes; see @king05 [@lp06]). If the black hole spins have not been aligned by the time their Bondi radii overlap and a hole is produced in the disk, further alignment seems unlikely [for a different interpretation see @liu04]. The reason is twofold: the shrinking of the binary due to circumbinary torques is likely to occur within $<{\rm few}\times 10^7$ yr [@escala04; @escala05], and accretion across the gap only occurs at $\sim 10$% of the rate it would have for a single black hole [@lsa99; @ld06; @mm06], with possibly even smaller rates onto the holes themselves. This therefore leads to the gas-poor merger scenario, suggesting that massive ellipticals or ellipticals with slow rotation or boxy orbits have a several percent chance of having ejected their merged black holes but that other galaxy types will retain their holes securely. Predictions, Discussion and Conclusions ======================================= We propose that when two black hole accrete at least $\sim 1-10$% of their masses during a gas-rich galactic merger, their spins will align with the orbital axis and hence the ultimate gravitational radiation recoil will be $<200$ km s$^{-1}$. In this section we discuss several other observational predictions that follow from this scenario. The best diagnostic of black hole spin orientation is obtained by examining AGN jets. All viable jet formation mechanisms result in a jet that is initially launched along the spin axis of the black hole. This is the case even if the jet is energized by the accretion disk rather than the black hole spin since the orientation of the inner accretion disk will be slaved to the black hole spin axis by the Bardeen-Petterson effect [@bp75]. At first glance, alignment of black hole spin with the large scale angular momentum of the gas would seem to run contrary to the observation that Seyfert galaxies have jets that are randomly oriented relative to their host galaxy disks [@kinney00]. However, Seyfert morphology is not consistent with recent major mergers [@veilleux03], hence randomly-oriented minor mergers or internal processes (e.g., scattering of a giant molecular cloud into the black hole loss cone) are likely the cause of the current jet directions in Seyfert galaxies. Within our scenario, one will never witness dramatic spin orientation changes during the final phase of black hole coalescence following a gas-rich merger. There is a class of radio-loud AGN known as “X-shaped radio galaxies”, however, that possess morphologies interpreted precisely as a rapid ($<10^5$yr) re-alignment of black hole spin during a binary black hole coalescence [@ekers78; @dt02; @wang03; @komossa03b; @lr07; @cheung07]. These sources have relatively normal “active” radio lobes (often displaying jets and hot-spots) but, in addition, have distinct “wings” at a different position angle. The spin realignment hypothesis argues that the wings are old radio lobes associated with jets from the one of the pre-coalescence black holes in which the spin axis possessed an entirely different orientation to the post-coalescence remnant black hole [@me02]. If this hypothesis is confirmed by, for example, catching one of these systems in the small window of time in which both sets of radio lobes have active hot spots, it would contradict our scenario unless it can be demonstrated that all such X-shaped radio-galaxies originate from gas-poor mergers. However, the existence of a viable alternative mechanism currently prevents a compelling case from being made that X-shaped radio sources are a unique signature of mis-aligned black hole coalescences. The collision and subsequent lateral expansion of the radio galaxy backflows can equally well produce the observed wings [@capetti02]. Indeed, there is circumstantial evidence supporting the backflow hypothesis. @kraft05 present a Chandra observation of the X-shaped radio galaxy 3C 403 and find that the hot ISM of the host galaxy is strongly elliptical, with a (projected) eccentricity of $e\sim 0.6$. Furthermore, the wings of the “X” are closely aligned with the minor axis of the gas distribution, supporting a model in which the wings correspond to a colliding backflow that has “blown” out of the ISM along the direction of least resistance. Although 3C 403 is the only X-shaped radio galaxy for which high-resolution X-ray maps of the hot ISM are available, @capetti02 have noted that a number of X-shaped sources have wings that are oriented along the minor axis of the [*optical*]{} host galaxy. This suggests that the conclusion of @kraft05 for 3C 403 may be more generally true. There is one particular system, 0402$+$379, that might provide a direct view of spin alignment in a binary black hole system. Very Long Baseline Array (VLBA) imaging of this radio galaxy by @maness04 discovered two compact flat spectrum radio cores, and follow-up VLBA observations presented by @rodriguez06 showed the cores to be stationary. A binary supermassive black hole is the most satisfactory explanation for this source, with the projected distance between the two black holes being only 7.3pc. Within the context of our gas-rich merger scenario, these two black holes already have aligned spins. Existing VLBA data only show a jet associated with one of the radio cores. We predict that, if a jet is eventually found associated with the other radio core, it will have the same position angle as the existing jet. Evidence already exists for alignment of a coalescence remnant with its galactic scale gas disk. @perlman01 imaged the host galaxies of three compact symmetric objects and discovered nuclear gas disks approximately normal to the jet axis. The presence of such a nuclear gas disk as well as disturbances in the outer isophotes of all three host galaxies suggests that these galaxies had indeed suffered major gas-rich mergers within the past $10^8$yr. In conclusion, we propose that in the majority of galactic mergers, torques from gas accretion align the spins of supermassive black holes and their orbital axis with large-scale gas disks. This scenario helps explain the ubiquity of black holes in galaxies despite the potentially large kicks from gravitational radiation recoil. Further observations, particularly of galaxy mergers that do not involve significant amounts of gas, will test our predictions and may point to a class of large galaxies without central black holes. We thank Doug Hamilton and Eve Ostriker for insightful discussions. TB thanks the UMCP-Astronomy Center for Theory and Computation Prize Fellowship program for support. CSR and MCM gratefully acknowledge support from the National Science Foundation under grants AST0205990 (CSR) and AST0607428 (CSR and MCM). [99]{} , J. G., [Boggs]{}, W. D., [Centrella]{}, J., [Kelly]{}, B. J., [McWilliams]{}, S. T., [Miller]{}, M. C., & [van Meter]{}, J. 2007, ApJ, submitted (astro-ph/0702390) , J. G., [Centrella]{}, J., [Choi]{}, D.-I., [Koppitz]{}, M., [van Meter]{}, J. & [Miller]{}, M. C. 2006, ApJ, 653, L93 Bardeen, J. M., & Petterson, J. A. 1975, , 195, L65 Barnes, J. E., & Hernquist, L. E. 1991, , 370, L65 Barnes, J. E., & Hernquist, L. 1996, , 471, 115 , J. D. 1973, ApJ, 183, 657 , E. F., et al. 2006a, ApJ, 640, 241 , E. F., [Phleps]{}, S., [Somerville]{}, R. S., [Wolf]{}, C., [Borch]{}, A., & [Meisenheimer]{}, K. 2006b, ApJ, 652, 270 , L., [Qusailah]{}, M. S. S., & [Will]{}, C. M. 2005, ApJ, 635, 508 , L. W., & [Reynolds]{}, C. S. 2006, ApJ, 652, 1028 , M., [Lousto]{}, C. O., [Zlochower]{}, Y., & [Merritt]{}, D. 2007a, gr-qc/0701164 , M., [Lousto]{}, C. O., [Zlochower]{}, Y., & [Merritt]{}, D. 2007b, gr-qc/0702133 Capetti, A., Zamfir, S., Rossi, P., Bodo, G., Zanni, C., Massaglia, S., 2002, A&A, 394, 39 Cheung, C. C., 2007, AJ, in press (astroph/0701278) , T., & [Gopakumar]{}, A. 2006, Phys. Rev. D, 73, 124006 Dennett-Thorpe, J., Scheuer, P. A. G., Laing, R. A., Bridle, A H., Pooley, G. G., Reich, W., 2002, MNRAS, 330, 609 Di Matteo, T., Springel, V., & Hernquist, L. 2005, , 433, 604 Dotti, M., Colpi, M., & Haardt, F. 2006, , 367, 103 Downes, D., & Solomon, P. M. 1998, , 507, 615 Ekers, R. D., Fanti, R., Lari, C., Parma, P., 1978, Nature, 276, 588 Escala, A., Larson, R. B., Coppi, P. S., & Mardones, D. 2004, , 607, 765 Escala, A., Larson, R. B., Coppi, P. S., & Mardones, D. 2005, , 630, 152 , A. C. et al. 2002, MNRAS, 335, L1 , M., [Hughes]{}, S. A., & [Holz]{}, D. E. 2004, ApJ, 607, L5 , L., & [Ford]{}, H. 2005, Sp. Sci. Rev., 116, 523 , M. J. 1983, MNRAS, 203, 1049 , M. J., & [Detweiler]{}, S. 1984, MNRAS, 211, 933 , J. A., [Hannam]{}, M. D., [Sperhake]{}, U., [Brügmann]{}, B., [Husa]{}, S. 2007, gr-qc/0702052 , J. A., [Sperhake]{}, U., [Brügmann]{}, B., [Hannam]{}, M. D., [Husa]{}, S. 2006, gr-qc/0610154 , J., & [Tan]{}, J. C. 2004, ApJ, 608, 108 , D. P., & [Burns]{}, J. A. 1991, Icarus, 92, 118 , D. P., & [Burns]{}, J. A. 1992, Icarus, 96, 43 , F., [Hinder]{}, I., [Shoemaker]{}, D., [Laguna]{}, P., & [Matzner]{}, R. A. 2007, gr-qc/0701143 , F., [Shoemaker]{}, D., & [Laguna]{}, P. 2006, gr-qc/0601026 , K. et al. 1996, MNRAS, 282, 1038 Kazantzidis, S., et al. 2005, , 623, L67 King, A. R., Lubow, S. H., Ogilvie, G. I., & Pringle, J. E. 2005, , 363, 49 Kinney, A. L., Schmitt, H. R., Clarke, C. J., Pringle, J. E., Ulvestad, J. S., Antonucci, R. R. J., 2000, , 537, 152 Kraft, R. P., Hardcastle, M. J., Worrall, D. M., Murray, S. S., 2005, , 622, 149 Komossa, S. 2003b, AIP Conf. Proc. 686, The Astrophysics of Gravitational Wave Sources, ed. J. Centrella (New York: API), 161 , M., [Pollney]{}, D., [Reisswig]{}, C., [Rezzolla]{}, L., [Thornburg]{}, J., [Diener]{}, P., & [Schnetter]{}, E. 2007, gr-qc/0701163 Lal,  D. V. & Rao, A. P., 2007, MNRAS, 374, 1085 Larson, R. B. 1987, Starbursts and Galaxy Evolution, 467 Liu, F. K. 2004, , 347, 1357 Lodato, G., & Pringle, J. E. 2006, , 368, 1196 , J. M. et al. 2006, ApJ, submitted (astro-ph/0602088) Lubow, S. H., & D’Angelo, G. 2006, , 641, 526 Lubow, S. H., Seibert, M., & Artymowicz, P. 1999, , 526, 1001 MacFadyen, A. I., & Milosavljevic, M. 2006 (astro-ph/0607467) , A. H., [Katz]{}, N., [Keres]{}, D., [Davé]{}, R., & Weinberg, D. H. 2006, ApJ, 647, 763 Maness, H. L., Taylor, G. B., Zavala, R. T., Peck, A. B., Pollack, L. K., 2004, , 602, 123 Marconi, A., Risaliti, G., Gilli, R., Hunt, L. K., Maiolino, R., & Salvati, M. 2004, , 351, 169 Mayer, L., Kazantzidis, S., Madau, P., Colpi, M., Quinn, T., & Wadsley, J.2006, to appear in Proceedings of “Relativistic Astrophysics and Cosmology - Einstein’s Legacy” (astro-ph/0602029) Merritt, D., & Ekers, R. D., 2002, Science, 297, 1310 , D., [Milosavljevic]{}, M., [Favata]{}, M., [Hughes]{}, S. A., & [Holz]{}, D. E. 2004, ApJ, 607, L9 Mihos, J. C., & Hernquist, L. 1994, , 425, L13 , T., [Jesseit]{}, R., & [Burkert]{}, A. 2006, MNRAS, 372, 839 , T., & [Ostriker]{}, J. P. 2007, astro-ph/0702535 Natarajan, P., & Armitage, P. J. 1999, , 309, 961 Natarajan, P., & Pringle, J. E. 1998, , 506, L97 Papaloizou, J. C. B., & Pringle, J. E., 1983, , 202, 1181 , A. 1962, Phys. Rev., 128, 2471 Perlman, E. S., Stocke, J. T., Conway, J., & Reynolds, C. 2001, , 122, 536 , I. H., & [Rees]{}, M. J. 1989, Commun. Astrophys., 14, 165 , C. S., & [Nowak]{}, M. A. 2003, Phys. Rep., 377, 389 Rodriguez, C., Taylor, G. B., Zavala, R. T., Peck, A. B., Pollack, L. K., Romani, R. W., 2004, , 646,49 , J. D. 2004, PRD, 70, 124020 Schnittman, J. D. & Buonanno, A 2007, submitted to (astro-ph/0702641) Shakura, N. I., & Sunyaev, R. A. 1973, , 24, 337 , A. 1982, MNRAS, 200, 115 , A., [Hasinger]{}, G., [Finoguenov]{}, A., [Barcons]{}, X., [Mateos]{}, S., & [Fabian]{}, A. C. 2005, A&A, 432, 395 Tacconi, L. J., Genzel, R., Tecza, M., Gallimore, J. F., Downes, D., & Scoville, N. Z. 1999, , 524, 732 Veilleux, S. 2003, ASP Conf. Ser. 290, Active Galactic Nuclei: From Central Engine to Host Galaxy, ed. S. Collin, F. Combes & I. Shlosman (San Francisco: ASP), 11 Volonteri, M., Madau, P., Quataert, E., & Rees, M. J. 2005, , 620, 69 Wada, K., & Norman, C. A. 2001, , 547, 172 Wada, K., & Norman, C. A. 2002, , 566, L21 Wang, T. , Zhou, H.,& Dong, X., 2003, , 126, 113 , A. G. 1992, PRD, 46, 1517 , Q., & [Tremaine]{}, S. 2002, MNRAS, 335, 965

--- abstract: 'Results from the Wilkinson Microwave Anisotropy Probe (WMAP), Atacama Cosmology Telescope (ACT) and recently from the South Pole Telescope (SPT) have indicated the possible existence of an extra radiation component in addition to the well known three neutrino species predicted by the Standard Model of particle physics. In this paper, we explore the possibility of the apparent extra *dark* radiation being linked directly to the physics of cold dark matter (CDM). In particular, we consider a generic scenario where dark radiation, as a result of an interaction, is produced directly by a fraction of the dark matter density effectively decaying into dark radiation. At an early epoch when the dark matter density is negligible, as an obvious consequence, the density of dark radiation is also very small. As the Universe approaches matter radiation equality, the dark matter density starts to dominate thereby increasing the content of dark radiation and changing the expansion rate of the Universe. As this increase in dark radiation content happens naturally after Big Bang Nucleosynthesis (BBN), it can relax the possible tension with lower values of radiation degrees of freedom measured from light element abundances compared to that of the CMB. We numerically confront this scenario with WMAP+ACT and WMAP+SPT data and derive an upper limit on the allowed fraction of dark matter decaying into dark radiation.' address: - 'Department of Physics and Astronomy, University of Aarhus, 8000 Aarhus C, Denmark' - | Institut für Theoretische Teilchenphysik und Kosmologie\ RWTH Aachen, D-52056 Aachen, Germany, Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, BC V6T 1Z1, Canada - 'Department of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, UK' author: - Ole Eggers Bjaelde - Subinoy Das - Adam Moss title: 'Origin of $\Delta N_{\rm eff}$ as a Result of an Interaction between Dark Radiation and Dark Matter' --- [ TTK-12-14]{} Introduction ============ Measurements of fluctuations in the Cosmic Microwave Background (CMB) power spectra [@Komatsu:2010fb] and light element abundances from Big Bang Nucleosynthesis (BBN) [@Steigman:2007xt] have been corner stones in the era of precision cosmology. Though standard $\Lambda$CDM cosmology is a very good fit to present data from these measurements, there are also tantalising hints of physics beyond standard $\Lambda$CDM. One such example is the possible presence of an extra *dark* radiation component during the epoch of decoupling. Recent analyses of CMB data point towards the possible existence of one or more radiation component(s) [@Hamann:2007pi; @Archidiacono:2011gq; @Calabrese:2011hg; @Hou:2011ec; @Archidiacono:2012gv] with no standard electromagnetic and electroweak interactions other than those predicted by the standard model of particle physics. This extra radiation needs to be *dark* in the sense that the presence of an extra photon like component would not only spoil the success of BBN, but also generate a chemical potential for the photon – something which is constrained by CMB observations. The indication of excess radiation arises mainly through the precise observation of less power in the smaller scales of CMB anisotropy spectra. It has also been confirmed that these hints for extra radiation are indeed ‘real’, insofar as not being a statistical ambiguity from the choice of confidence interval [@Hamann:2011hu]. Recently, the evidence was bolstered through the possible indication of ultra light sterile neutrino states in the neutrino oscillation experiments [@Kopp:2011qd; @Akhmedov:2010vy; @Agarwalla:2010zu; @Giunti:2011gz; @Hamann:2010bk] and also through the reactor neutrino anomaly [@Mention:2011rk]. The nature of the dark radiation component is a topic of much debate. Current data allow the dark radiation component to be comprised by both sterile neutrinos as well as active neutrinos with a nonthermal distribution (see e.g. [@Cuoco:2005qr]). If dark radiation is comprised by massless sterile neutrinos, we expect them to behave as relativistic particles with effective sound speed $c_{\rm eff}^2$ and viscosity parameter $c_{\rm vis}^2$ satisfying $c_{\rm eff}^2=c_{\rm vis}^2=1/3$[^1]. Possible deviations from these values could indicate nonstandard interactions in the neutrino sector [@Beacom:2004yd; @Hannestad:2004qu; @Cuoco:2005qr; @Basboll:2008fx; @Basboll:2009qz]. Luckily, measurements of CMB anisotropies can help in constraining these parameters [@Hu:1998tk] and most analyses are consistent with the $c_{\rm eff}^2=c_{\rm vis}^2=1/3$ (see e.g. [@Trotta:2004ty; @DeBernardis:2008ys; @Archidiacono:2012gv]) - although [@Smith:2011es] reported on finding $c_{\rm eff}^2<1/3$. The bottom line is that cosmological data is sensitive to the details of dark radiation and can help in predicting its nature. The helium abundance $Y_P$ is very sensitive to the expansion rate of the Universe (and hence the amount of radiation present) at the time when $ T \sim {\rm MeV}$. However, the evidence for extra radiation from BBN data is somewhat ambiguous. In some analyses, it is reported that one can accommodate one extra dark radiation component [@Izotov:2010ca], while in other works it is concluded that there is no need for extra radiation during BBN [@Simha:2008zj; @Aver:2010wq]. The former of these studies, for example, found $N_{\rm eff}^{\rm BBN} = 2.4 \pm 0.4$ [@Simha:2008zj]. The main reason for the confusion is that $N_{\rm eff}$ is highly sensitive to how the helium abundance is treated in the analysis [@Nollett:2011aa]. In general, from many data analyses, it remains a possibility that the central value for the number of relativistic degrees of freedom allowed by CMB data is higher than that of BBN. Taking CMB data alone, the estimate is $N_{\rm eff}^{\rm CMB} = 5.3 \pm 1.3$ from the Wilkinson Microwave Anisotropy Probe (WMAP) 7-year with Atacama Cosmology Telescope (ACT) data [@Dunkley:2010ge]. Combined WMAP and South Pole Telescope (SPT) data give a slightly lower value of $N_{\rm eff}^{\rm CMB} = 3.85 \pm 0.62$ [@Keisler:2011aw]. The addition of baryon acoustic oscillations (BAO) data and the measurement of the Hubble parameter $H_0$ improves these constraints somewhat. It is found that $N_{\rm eff}^{\rm CMB} = 4.56 \pm 0.75$ for WMAP+ACT+BAO+$H_0$ [@Dunkley:2010ge] and $N_{\rm eff}^{\rm CMB} = 3.86 \pm 0.42$ for WMAP+SPT+BAO+$H_0$ [@Keisler:2011aw]. Taken at face value, the latter two results suggest $\sim 2\sigma$ evidence for extra relativistic species. The Planck satellite will dramatically increase the precision of the inferred value of $\Delta N_{\rm eff} \simeq 0.26$ [@Hamann:2007sb] and should be able to find a mismatch (if there is one) between $N_{\rm eff}^{\rm CMB}$ and $N_{\rm eff}^{\rm BBN}$ at the level of $4-5 \sigma$ [@Hamann:2007pi]. To explain the apparent radiation excess, one can, of course, just add a weakly interacting neutrino-like fermion by hand. However, then the question remains of explaining the origin of such a particle. For a recent particle physics model explaining the radiation excess with three flavours of light right-handed neutrinos, though, see [@Anchordoqui:2011nh]. See also [@Sikivie:2009qn; @Sikivie:2011; @Lundgren:2010] for another explanation of excess radiation during decoupling through the Bose-Einstein condensation of a coherently oscillating dark matter axion. We would like to point out that though LSND [@Aguilar:2001ty] and MiNiBooNE [@AguilarArevalo:2010wv] indicate the existence of one or more eV scale sterile neutrinos [@Abazajian:2012ys], which are excellent candidates for the excess radiation hinted by small scale CMB data, it is very hard to reconcile two eV scale sterile neutrinos as dark radiation with the large scale structure and other measurements [@Hamann:2011ge] unless sterile neutrinos have other interactions [@Fan:2012ca; @Barger:2003rt; @Mangano:2010ei; @Fardon:2003eh; @Antusch:2008hj]. So, it is highly possible that the dark radiation may be a result of dark sector physics. For instance, if dark matter decays into dark radiation, that can explain the dark radiation excess and its effect could be found in the structure formation of the Universe. Note that some hidden sector models [@Feng:2011uf; @Blennow:2012de; @Ichikawa:2007jv] motivated by other issues of particle physics and cosmology can also provide extra $\Delta N_{\rm eff}$.\ If it is indeed the case that there is a change (increase) in the number of radiation degrees of freedom between the epoch of BBN and CMB, that will be an extremely interesting and surprising result. From a theoretical point of view, some new physics has to set in at a low energy scale $(T \sim {\rm eV})$. Recently, there have been a few interesting works in this line of thought [@Fischler:2010xz; @Hasenkamp:2011em], where $ \Delta N^{\rm BBN}_{\rm eff} \neq \Delta N^{\rm CMB}_{\rm eff}$. From a particle physics view point, this indicates that a particle (beyond the frame work of the standard model) has to decay [@hep-ph/0703034] into an extra dark radiation component in between the epoch of BBN and photon decoupling. In this paper we propose a very simple mechanism where one naturally generates an extra radiation component when the Universe approaches the era of matter radiation equality and decoupling. The basic idea is to allow dark matter to interact with and decay into dark radiation: As the Universe approaches matter radiation equality (MRE), the density of dark matter starts to dominate the universal energy budget. As a result of the interaction, the densities of dark matter and dark radiation are proportional, hence the density of dark radiation increases as we approach MRE. In this scenario, one would naturally see an increase in the dark radiation component after BBN but before decoupling. One extra advantage of this scenario is the fact that the decay naturally reduces the amount of dark matter in galaxies and clusters. As a consequence it may help to alleviate [@Bell:2010fk; @Abdelqader:2008wa] the well known small scale structure issues in $\Lambda$CDM cosmology - the problems with cuspy cores and overproduction of satellite galaxies in numerical simulations of structure formation [@Moore:1999nt; @Klypin:1999uc; @Navarro:1996gj]. We leave the details of this as possible future work. In this paper we solve for the dark radiation density as a function of redshift numerically and show that we can obtain $\Delta N_{\rm eff}\rightarrow1$ as the Universe approaches the epoch of photon decoupling. We confront this generic scenario with the present cosmological data. We take a model-independent approach, where we use WMAP with either ACT or SPT data[^2] to constrain the fraction of the dark matter density which is allowed to be converted into dark radiation. We show that one can easily find a viable region in parameter space where $\Delta N_{\rm eff}^{\rm CMB}$ can be greater than $\Delta N_{\rm eff}^{\rm BBN}$ by of order unity. The plan of the paper is as follows: In section \[interaction\], we quantify the production of dark radiation and solve for the background solution. We present an analytical expression of $\Delta N_{\rm eff}$ and plot its dependence on scale factor and the coupling between dark matter and dark radiation. In section \[perturbations\], we derive the cosmological perturbation equations for our scenario and, in section \[cosmomc\], we discuss our main numerical results from a COSMOMC analysis using various datasets. We demonstrate that observations are consistent with $\Delta N_{\rm eff}\sim1$ around decoupling and $\Delta N_{\rm eff}\sim0$ around BBN. In section \[origin\], we present a specific model in which the dark radiation production can be realised and constrain the model parameters. Finally we conclude in section \[conclusion\]. Interaction between dark radiation and dark matter {#interaction} ================================================== Background evolution -------------------- If dark radiation belongs to the dark sector along with dark matter, an interaction between the two could be possible. A general coupling (at the background level) can be described by the energy balance equations $$\begin{aligned} \label{eq:contnu} \dot{{\rho}}_{\rm DM}&+&3H{\rho}_{\rm DM}=-Q\,, \nonumber \\ \dot{{\rho}}_{\rm dark}&+&3H \left( 1 + w_{\rm dark} \right) {\rho}_{\rm dark}= Q\,, \label{eq:contDM}\end{aligned}$$ where $\rho_{\rm DM}$ and $\rho_{\rm dark}$ are the dark matter/radiation energy densities and $H=\dot{a}/a$ is the Hubble rate, where $a$ is the scale factor and an overdot denotes the derivative with respect to conformal time $\tau$. For our case of dark matter being converted into dark radiation then $w_{\rm dark}= P_{\rm dark}/\rho_{\rm dark} = 1/3$. The rate of energy transfer is given by $Q$ – a positive $Q$ denotes the direction of energy transfer from dark matter to dark radiation. A non-zero $Q$ means that dark matter no longer redshifts exactly as $1/a^3$ and also that dark radiation does not redshift as $1/a^4$. It is important to remind ourselves that we require a covariant form for the energy momentum transfer $Q$. Several papers over the recent years have studied different forms of the energy transfer rate $Q$ in the context of interacting dark matter-dark energy [@Zimdahl:2001ar; @Wang:2005jx; @Olivares:2006jr; @Koivisto:2005nr; @Ziaeepour:2003qs; @Setare:2007we; @Bjaelde:2007ki; @Bjaelde:2008yd; @Das:2005yj; @Malik:2002jb; @Valiviita:2008iv]. We adopt the covariant form of energy momentum transfer 4-vector introduced in [@Valiviita:2008iv] $$Q_{\rm DM}= \Gamma \rho_{\rm DM} \,, \label{eq:covQ}$$ where the form of interaction rate $\Gamma$ depends on the details of the particle physics of the decay process. Many forms of $\Gamma$ has been studied in literature, we adopt a simple case where $\Gamma= \alpha H $, where $\alpha$ is a constant and $H$ is the Hubble rate. As discussed in [@Valiviita:2008iv], an implicit assumption behind this form of $\Gamma$ is that the interaction rate varies with time but not with space, which explains the presence of $H$ in the place of the interaction rate in Eq. \[eq:covQ\]. This form of $\Gamma$ can arise from different models of dark matter decay. In section \[origin\], we demonstrate a model of dark matter decaying into dark radiation and we show that the above mentioned form of $\Gamma$ can be easily realised in nature. For this form of the coupling it is easy to solve the background energy density equations [@Valiviita:2008iv] $$\begin{aligned} \label{eq:rhodm} \rho_{\rm DM} &=& \rho_{\rm DM,0} a^{-(3+\alpha)}\,,\nonumber\\ \rho_{\rm dark} &=& \rho_{\rm dark,0}a^{-3(1+w_{\rm dark})}+\left( \frac{\alpha}{\alpha-3w_{\rm dark}} \right) \rho_{\rm DM,0} a^{-3} (a^{-3w_{\rm dark}}-a^{-\alpha})\,.\end{aligned}$$ With $w_{\rm dark}= 1/3 $ the equation for $\rho_{\rm dark}$ can be collected into two terms $$\rho_{\rm dark} = \beta a^{-4} + \left( \frac{\alpha}{1-\alpha} \right) \rho_{\rm DM,0} a^{-(3+\alpha)}, \label{eq:rhodark}$$ where $\beta$ is a constant. The first part behaves like a standard radiation density and the second part behaves like a fluid with an equation of state $\alpha/3$. In the case of a weak coupling between dark matter and dark radiation, we require that $\alpha$ is small, which in turn leads to $\beta\sim0$. In the following we only keep the second term in Eq. \[eq:rhodark\]. This is further justified by the fact that the fraction of $\rho_{\rm dark}$, which redshifts like $1/a^4$, will be subdominant to the fraction which redshifts like dark matter due to the expansion of the Universe. With this assumption we obtain $$\frac{\rho_{\rm dark}}{ \rho_{\rm DM}} \to \frac{\alpha}{3w_{\rm dark}-\alpha} = \frac{\alpha}{1-\alpha}\,. \label{eq:assumption}$$ As we will see later from our numerical analysis, $ \alpha \ll 1$, so the ratio reduces to ${\rho_{\rm dark}}/{\rho_{\rm DM}}= \alpha $. In fact, in section 6 we study a model of dark matter decay for a specific interaction and decay mechanism where our assumption is realised. This means that the pure radiation-like component (the term with coefficient $\beta$) is indeed absent. Calculation of $\Delta N_{\rm eff}$ ------------------------------------ In the standard cosmological scenario, it is a standard practice to define $\Delta N_{\rm eff}$ by $$\rho_{\rm rad} = \left[ 1 + \frac{7}{8} N_{\rm eff} \left( \frac{T_{\nu}}{T_{\gamma}} \right)^4 \right] \rho_{\gamma}\,, \label{rhorad}$$ where the radiation density $\rho_{\rm rad}$ is given as a sum of the energy density in photons $\rho_{\gamma}=(\pi^2/15) T_{\gamma}^4$ and standard model neutrinos. In the standard model this predicts $N_{\rm eff}^{\rm SM}=3.046$ with $T_{\nu}/T_{\gamma} = (4/11)^{1/3}$. Any departure from the standard scenario is parameterized as $N_{\rm eff} = N_{\rm eff}^{\rm SM} + \Delta N_{\rm eff}$. One implicit assumption in the above definition is that the dark matter dilutes as $1/a^3$. But in our case neither dark matter nor dark radiation dilutes in the standard way. This means we cannot simply use the definition above – a more appropriate model independent method is needed to compare the expansion rate $H$ to that of the standard model $H_{\rm SM}$ and attribute the difference to $ \Delta N_{\rm eff}$. In our model $$\begin{aligned} 3 H^2 M_{Pl}^2 &=& \rho_{\rm DM,0} / a^{3 + \alpha} + \frac{\alpha}{1-\alpha} \rho_{\rm DM,0}/a^{3 + \alpha} + \, \rho_{\rm rest} \\ \nonumber &=& \frac{1}{1-\alpha} \rho_{\rm DM,0}/a^{3 + \alpha} + \, \rho_{\rm rest} \,,\end{aligned}$$ where $\rho_{\rm rest}$ stands for the normal radiation and dark energy components. We then compare to the standard $H_{\rm SM}$ (with non-zero $\Delta N_{\rm eff}$) to obtain $$\frac{7}{8} \Delta N_{\rm eff} \, \left (\frac{T_{\nu}}{T_{\gamma}} \right)^4 \, \frac{\rho_{\gamma 0}}{a^4} = \frac{1}{1-\alpha} \rho_{\rm DM,0}/a^{3 + \alpha} - \rho_{\rm DM,0}/a^{3}\,.$$ Using the above definition we find that $\Delta N_{\rm eff}$ depends on the decay constant $\alpha$ as well as scale factor $a$. We show $\Delta N_{\rm eff}$ as function of $a$ in the top-left panel of Fig. \[fig:plots\] for $\alpha=0.02$ and 0.04. At the time of decoupling this produces a $\Delta N_{\rm eff}$ of order unity. ![For each panel we show the best-fit vanilla 6-parameter model from WMAP+SPT (black), then models with the [*same*]{} parameters but one extra relativistic species (dotted-red), a lower dark matter density of $\Omega_{\rm DM} = 0.085$ (dashed-blue, as opposed to $\Omega_{\rm DM} = 0.112$) and a decaying dark matter model with $\alpha = 0.02$ (dot-dash green) and 0.04 (dot-dot-dash magenta). (Top-left) $\Delta N_{\rm eff}$ as a function of scale factor. (Top-right) Hubble rate compared to the standard model. (Bottom-left) Effective (total) equation of state. (Bottom-right) Ratio of the gravitational potential $\Phi$ for a Fourier mode with $k=0.02\, {\rm Mpc}^{-1}$ compared to the standard model. Horizon entry for this mode is indicated by the dashed vertical line.[]{data-label="fig:plots"}](plots.eps){width="6.0"} As already discussed, the contribution to $\Delta N_{\rm eff}$ arises mainly due to the faster expansion rate at the time of decoupling and MRE in our model. It is interesting to note that at an early epoch deep in the radiation-dominated era, when the dark matter density is negligible, the deviation from the standard expansion rate is close to zero, as expected. At late times – after $ z \sim 100$ – the expansion rate again approaches the standard model expansion rate, as demonstrated in Fig. \[fig:plots\]. This is because $ a \rightarrow 1$ and $\alpha$ is still small – in other words the interacting dark matter starts to behave in the same way as in standard $\Lambda$CDM. Perturbations of dark radiation {#perturbations} =============================== To determine the perturbations to dark radiation and dark matter we need to consider the Boltzmann equation for its distribution. For simplicity we will consider a massive dark matter particle decaying into a pair of massless daughter particles. Following standard practice (e.g. [@Ma:1995ey]) we expand the distribution function for each species $j$ in terms of a zero-order component $f^0_j$ and a perturbation $\Psi_j$ $$f_j(x^i,q_j,n_i,\tau) = f_j^0(q_j,\tau)[1+\Psi_j(x^i,q_j,n_i,\tau)]\,,$$ which depends on position $x^i$, magnitude of momentum $q_j$, direction $n_i$ and conformal time $\tau$. The phase space of each species obeys the Boltzmann equation $$\frac{Df_j}{d\tau}=\frac{\partial f_j}{ \partial \tau}+\frac{\partial f_j}{\partial x^i}\frac{dx^i}{d\tau} +\frac{\partial f_j}{\partial q_j}\frac{dq_j}{d\tau}+\frac{\partial f_j}{\partial n_i}\frac{dn_i}{d\tau}=\left(\frac{df_j}{d\tau}\right)_C\,,$$ where $\left(\frac{df_j}{d\tau}\right)_C$ is the collision term, which depends on particle interactions. At zeroth-order the Boltzmann equation for the dark matter distribution function can then be written as [@Kawasaki:1992kg; @Kaplinghat:1999xy] $$\dot{f}_{\rm DM}^0=-\alpha H f_{\rm DM}^0\,, \label{eq:collision}$$ under the assumption that Eq. \[eq:assumption\] is fulfilled. Upon multiplying by the proper energy $\epsilon_j = \sqrt{q_j^2 + a^2 m_j^2}$ and integrating over all momenta one obtains the same continuity equation for dark matter as in Eq. \[eq:contDM\]. Working in the synchronous gauge, we can now work out the equations of motion for the perturbations to dark matter and its decay product. For the perturbations to dark matter we write out the Boltzmann equation and the perturbation to the energy density by following the machinery described in Ref. [@Ma:1995ey]. In the end, the equations of motion for the dark matter perturbations reduce to the case of stable dark matter particles - the only difference being the different distribution function $f_{\rm DM}^0$ specified by Eq. \[eq:collision\]. This result was also obtained in Refs. [@Kaplinghat:1999xy; @Wang:2010ma; @Wang:2012ek] Following Ref. [@Ma:1995ey], for the massless decay product we integrate out the $q$ dependence of the distribution function and expand the angular component in terms of Legendre polynomials, $$\label{fsubl} F_j(\vec{k},\hat{n},\tau) \equiv {\int q_j^2 dq_j\,q_j f^0_j(q_j)\Psi \over \int q_j^2 dq_j\,q_j f^0_j(q_j)} \equiv \sum_{l=0}^\infty(-i)^l (2l+1)F_{j\,l}(\vec{k},\tau)P_l(\hat{k}\cdot\hat{n})\,,$$ where $\mu=\hat{k}\cdot\hat{n}$ and $P_n(\mu)$ are the Legendre polynomials of order $n$. The Boltzmann equation can then be worked out to give $$\label{frddot} \dot{F}_{\rm dark} + i k \mu F_{\rm dark} = -\frac{2}{3} \dot{h} - \frac{4}{3} \left(\dot{h} + 6 \dot{\eta} \right) P_2 (\mu) + H(1-\alpha) \left( \delta_{\rm DM} - F_{\rm dark} \right)\,,$$ where the expression for the density perturbation $\delta=\delta \rho/\rho$ in terms of phase-space integrals, and the definition of the synchronous gauge metric perturbations $h$ and $\eta$ can be found in Ref. [@Ma:1995ey]. Eq. \[frddot\] can be translated into a hierarchy of perturbation equations of motion for the dark radiation by inserting the expansion in Eq. \[fsubl\] and collecting terms. The final results are $$\begin{aligned} \label{massless} \dot{\delta}_{\rm dark} &=& -{4\over 3}\theta_{\rm dark} -{2\over 3}\dot{h}-H(1-\alpha)(\delta_{\rm dark}-\delta_{\rm DM}) \,,\\ \dot{\theta}_{\rm dark} &=& k^2 \left(\frac{1}{4}\delta_{\rm dark} - \sigma_{\rm dark} \right)-H(1-\alpha)\theta_{\rm dark}\,,\nonumber\\ \dot{F}_{{\rm dark}\,2} &=& 2\dot\sigma_{\rm dark} = {8\over15}\theta_s - {3\over 5} k F_{{\rm dark}\,3} + {4\over15}\dot{h} + {8\over5} \dot{\eta}-H(1-\alpha)F_{{\rm dark}\,2} \,,\nonumber\\ \dot{F}_{{\rm dark}\,l} &=& {k\over2l+1}\left[ l F_{{\rm dark}\,(l-1)} - (l+1) F_{{\rm dark}\,(l+1)} \right]-H(1-\alpha)F_{{\rm dark}\,l}\,, \quad l \geq 3 \,, \nonumber\end{aligned}$$ where $\delta_{\rm dark} = F_{\rm dark \,0}$, $\theta_{\rm dark} = 3k/4 F_{\rm dark\,1}$ and $\sigma_{\rm dark} = F_{\rm dark\,2}/2$. This is an infinite hierarchy so we need to truncate the hierarchy at some $l_{\rm max}$, for which we choose [@Ma:1995ey] $$\label{truncnu} F_{\nu\,(l_{\rm max}+1)}\approx{(2l_{\rm max}+1)\over k\tau}\,F_{\nu \,l_{\rm max}}-F_{\nu\,(l_{\rm max}-1)}\ .$$ These equations are equivalent to those in Ref. [@Kaplinghat:1999xy] for their choice of decay variables. We implemented these equations in a modified version of [CAMB]{} [@CAMB]. This amounts to: (1) Changing the scaling behaviour of dark matter to $a^{-(3+\alpha)}$; (2) Modifying the background evolution to include an additional component whose energy density scales like dark matter but with $w_{\rm dark} = 1/3$ ; (3) Implementing the hierarchy of perturbation equations, which are similar (with the exception of the final terms in Eq. \[massless\]) to the existing massless neutrino perturbation equations. Results {#cosmomc} ======= ![Temperature power spectrum for the models listed in Fig. \[fig:plots\]. []{data-label="fig:cl"}](cl.eps){width="4.2"} In Fig. \[fig:cl\] we show the temperature power spectrum each of the models listed in Fig. \[fig:plots\]. The general features can be understood qualitatively by the following - For models with an extra relativistic species, lower CDM and decaying dark matter (DDM) the acoustic peaks are shifted to smaller scales. The angular scale of the peaks is set by the ratio of the sound horizon at decoupling to the angular diameter distance to decoupling. For a flat Universe this is approximately the ratio of the conformal time at decoupling to that today, i.e. $\theta_{\rm A} \approx \tau_{\rm dec}/\tau_0$. For extra relativistic species the increased Hubble rate at early times decreases $\tau_{\rm dec}$, while $\tau_0$ remains similar. For lower CDM both $\tau_{\rm dec}$ and $\tau_0$ increase, with the relative increase in $\tau_0$ compared to the standard model greater. For DDM both $\tau_{\rm dec}$ and $\tau_0$ decrease, with the relative decrease in $\tau_{\rm dec}$ greater. - The first two peaks are noticeably enhanced in models with an extra relativistic species and lower CDM, but are suppressed in the DDM model. This arises from the driving effect (modes entering the horizon during the radiation era are enhanced due to the decay of gravitational potentials) and (for the first peak) the early Integrated Sachs-Wolfe (ISW) effect. The total (effective) equation of state of the Universe for each model is shown in Fig. \[fig:plots\]. For extra relativistic species and lower CDM, radiation domination ($w_{\rm eff} \approx 1/3$) is extended, while for the decaying dark matter model the Universe actually departs from radiation domination faster. This is somewhat opposite to what one might expect, since dark matter is decaying into dark radiation, but is a consequence of fixing $\Omega_{\rm DM}$ today to be the same as the standard model and it scaling as $a^{-(3+\alpha)}$. The result of this can be seen in Fig. \[fig:plots\], by plotting the gravitational potential $\Phi$ for a mode with $k=0.02 \, {\rm Mpc}^{-1}$, which enters the horizon around $a \approx 10^{-4}$. Potential decay is suppressed in the DDM model by the time of decoupling. - The first peak is also affected by the early ISW – here the potential can still decay after decoupling as the Universe is not completely matter dominated. For DDM the total equation of state is closer to the standard model at decoupling than in models with an extra relativistic species or lower CDM, so the early ISW contribution is smaller. There is, however, a late time ISW effect, resulting in more power on large scales. This is because the total equation of state $w_{\rm eff} \ne 0$, and also because the radiation decay product (whose energy density is $\alpha \rho_{\rm DM}$) contributes a source of pressure and anisotropic stress. - On small scales there is reduced power in the damping tail of the CMB for both extra relativistic species and DDM. This is due to the increased expansion rate prior to decoupling, resulting in higher diffusion damping. The opposite occurs for lower CDM due to the decreased expansion rate. In order to confront the DDM model with observations we perform parameter estimation using a modified version of the COSMOMC package [@cosmomc]. For our analysis we use data from the 7-year WMAP release [@Komatsu:2010fb], the 148 GHz 2008 ACT data [@Dunkley:2010ge] and the 150 GHz 2008/2009 SPT data [@Keisler:2011aw]. Due to the small overlapping sky coverage between ACT and SPT we consider WMAP + ACT and WMAP + SPT independently. We use software provided by each team to compute the likelihood of cosmological models. ![Marginalized parameter constraints for WMAP + ACT (solid black) and WMAP + SPT (dashed red).[]{data-label="fig:1d"}](SPT_ACT_comp.eps){width="5.2"} To parameterize these models we fit for 7 parameters, imposing the flatness condition $\Omega_{\rm k}=0$[^3]: the baryon density $\Omega_{\rm b} h^2$, cold dark matter density $\Omega_{\rm DM} h^2$, Hubble parameter $H_0 = 100 \, h \, {\rm Mpc}^{-1} \, {\rm km} \, {\rm s}^{-1}$, optical depth to reionization $\tau$, and the amplitude $A_{\rm s}$ and spectral index $n_{\rm s}$ of initial fluctuations. In addition we fit for the dark matter decay constant $\alpha$, imposing a prior that $\alpha \ge 0$. Since ACT and SPT observe much smaller scales than WMAP, marginalisation over foregrounds is also required, since these contribute to the small scale temperature power spectrum. We follow the same procedure as in the ACT and SPT analysis, marginalising over a combined thermal and kinetic Sunyaev-Zeldovich (SZ), and a clustered point source, template, together with a Poisson ($C_{\ell} = {\rm const}$) point source component. The reader is referred to these references for more details on the templates used. This brings the total number of parameters fitted to 10 (7 cosmological and 3 secondary foreground parameters). We used the lensed theoretical CMB spectra from [CAMB]{} in our fits, since lensing is favoured by WMAP + ACT/SPT at the level of several $\sigma$. ![Marginalized constraints on the 7 fitted cosmological parameters for WMAP + ACT (solid black) and WMAP + SPT (dashed red). Likelihood contours show the 68% and 95% confidence levels.[]{data-label="fig:contour"}](contour.eps){width="5.2"} The results of our analysis are shown in Fig. \[fig:1d\] and Fig. \[fig:contour\], where we show marginalised 1-dimensional and 2-dimensional parameter constraints. There is no preference for a non-zero decay constant, with a $2\sigma$ upper-limit of $\alpha< 0.027$ for WMAP + ACT and $\alpha< 0.028$ for WMAP + SPT. This likely stems from the different effect on the CMB spectrum for DDM than for extra relativistic species, in particular the suppression of the first peak instead of an enhancement. There is some degeneracy with other parameters, in particular the spectral index and Hubble parameter. Repeating our analysis for the standard model ($\alpha=0$) we find, for example, WMAP + SPT gives $n_{\rm s} = 0.964 \pm 0.011$ and $h=0.706 \pm 0.021$, while allowing $\alpha$ to vary then $n_{\rm s} = 0.976 \pm 0.015$ and $h=0.765 \pm 0.052$. Origin of interaction: A phenomenological model {#origin} =============================================== An empirical form of energy transfer has been introduced in most studies [@Zimdahl:2001ar; @Wang:2005jx; @Olivares:2006jr; @Koivisto:2005nr; @Ziaeepour:2003qs; @Setare:2007we; @Bjaelde:2007ki; @Bjaelde:2008yd; @Das:2005yj; @Malik:2002jb; @Valiviita:2008iv] where interactions between different cosmological sectors has been considered. Till now we also have chosen an empirical form $\Gamma = \alpha \, H$ for our model and used it to confront with data. For this form of coupling we have shown that the ratio of the energy densities of dark radiation and dark matter $\rho_{\rm dark} / \rho_{DM}= \alpha$ is practically constant in time for small $\alpha$. Hence, putting an upper bound on $\alpha$ basically gives us an estimate of how large a fraction of dark matter is allowed to decay into dark radiation, obeying all cosmological constraints. Though the main goal of this paper is not a dark matter model, we will now discuss a specific phenomenological model where one can analytically derive the energy transfer equations between dark matter and dark radiation. We show that a time independent $\alpha$ can indeed emerge in a phenomenological model. Note, however, that our numerical results are not limited to this specific model. Any dark matter model with a coupling to dark radiation and where the fraction of dark radiation to dark matter does not change much in the course of a Hubble time will be subject to the constraints from section \[cosmomc\].\ We consider a coherently oscillating scalar field which plays the role of CDM [@Das:2006ht; @Bjaelde:2010vt; @Kobayashi:2011hp]. This has a Yukawa type coupling to a nearly massless *dark* fermion $\psi_d$ (${\cal L} \supset \lambda \phi \psi_d \bar{\psi_d}$). This type of CDM can in principle decay parametrically into dark radiation and the situation is very similar to the fermionic preheating scenario in the context of inflation [@Greene:2000ew]. However, the energy scale which we are considering here is much lower compared to that of inflation. We refer our readers to Refs. [@Das:2006ht; @Bjaelde:2010vt; @Kobayashi:2011hp] for details of the dark matter decay process in this scenario. Here we present a brief review about the basic mechanism of *dark* radiation production from CDM and finally we put constaints on the model parameters on the basis of the numerical results obtained in section \[cosmomc\]. By adopting the results of fermionic preheating [@Bjaelde:2010vt; @Greene:2000ew] in an expanding background, the comoving number density of dark fermions can be found by solving the well known Mathieu equations $$X_k^{''} + [\kappa^2 + (\tilde{m} + \sqrt{q} f)^2 -i \sqrt{q} f'] X_k =0,$$ where the resonance parameter $q\equiv \lambda^2 \phi_0^2 / m_{\phi}^2$, $\phi_{0}f(t)$ is the background solution for the time evolution of the oscillating scalar field, $\kappa \equiv k / m_{\phi}$ is the dimensionless fermion mass, and $\tilde{m} \equiv m_{\psi} / m_{\phi}$. These three parameters completely determine the parametric production of fermions. We consider the oscillation of the field with the usual quadratic potential $V= \frac{1}{2} m^2 \phi^2 $, which is a good first order approximation around the minima of any potential. The term $(\tilde{m} + \sqrt{q} f)$ can be thought of as an effective mass of the fermion. As the scalar field oscillates, the effective mass itself will oscillate around zero and the parametric production of fermions is enhanced when the effective mass crosses zero. It can be shown numerically that $n_k(t)$ oscillates and due to Pauli blocking its maximum value never crosses unity.\ The expansion of the Universe has not been taken into account in the discussion above. As the mass of the oscillating scalar is very low, for a complete treatment, however, we have to consider the expansion of the Universe. When the expansion is taken into account, the resonance parameter $q \equiv \lambda^2 \phi (t)^2 / m_{\phi}^2 $ becomes time-dependent and the periodic modulation of the comoving number density does not hold any more. It has been shown [@Bjaelde:2010vt] (though in a different context of neutrino cosmology) that the parametric production of dark radiation happens as long as the time dependent resonance parameter satisfies $q(z) \gg 1$. In this regime, the produced dark radiation density takes a very simple form $\rho_{\rm dark} = 8 \pi \lambda^2 \rho_{\rm DM}$, which gives $\alpha$ in terms of model parameters $\alpha = 8 \pi \lambda^2$. The only requirement for the above relation is $q \gg 1$, which translates into $$2 \, \frac{\lambda^2 \, \rho_{\rm DM, 0}}{m_{\phi}^4} \, (1+z)^3 \gg 1, \label{eq:lambdacon}$$ which we assumes holds true from BBN till the present epoch denoted by the ’0’ superscript. So far the dark matter mass $m_{\phi}$ has not been constrained in our analysis. This is because the dark matter mass does not enter explicitly into the perturbation analysis and thus our numerical results do not constrain the dark matter mass directly. One can, however, place an upper bound on the dark matter mass for this specific model of parametric decay. This is possible from the requirement $m_{\phi} \geq H$ – otherwise the Hubble friction would prevent the field from oscillating coherently and will not allow it to behave as CDM. From the constraints on CDM matter power spectra we know that dark matter has to be present in the Universe at least couple of e-foldings before MRE – otherwise there would be too much suppression in the linear matter power-spectra on small scales [@Das:2006ht]. So to get an estimate we assume the scalar started to oscillate coherently when the temperature of the Universe was around $T=T_{\rm osc} \simeq 100 \,$ eV. This choice keeps us out of the conflict with constraints from linear matter power spectra measurements from Lyman-$\alpha$ and SDSS data [@McDonald:2004eu; @Viel:2005ha]. This, in turn, constrains the dark matter mass $ m_{\phi} \gg H(T_{\rm osc}) $. In Fig. \[fig5\] we show the allowed region in the $(m_{\phi}, \lambda)$ plane which satisfy all of the above three constraints namely: a) To satisfy the parametric production of dark radiation; b) To be consistent with our numerical results (upper bound on $\alpha$) from section \[cosmomc\]; c) To obey the condition for the coherent oscillation setting in before MRE. ![Allowed region in the $( m_{\phi}, \lambda)$ plane for a coherently oscillating scalar dark matter decaying into dark radiation. The blue area is excluded from the requirement that the resonance parameter is not high enough to produce dark radiation through parametric resonance. The purple region is excluded from the upper limit on the fraction of dark matter decaying into dark radiation taken from our numerical result in section \[cosmomc\]. The lower grey area is excluded from the requirement $m_{\phi} \gg H (T_{\rm osc}) $. []{data-label="fig5"}](dark-rad.eps){width="3.2"} Discussion and conclusion {#conclusion} ========================= The evidence for the existence of dark radiation at the CMB epoch is intriguing. If future experiments find a mismatch between the radiation content of the Universe at the epoch of BBN and decoupling, the production of dark radiation may be a late-time phenomenon in the cosmic history which took place some e-foldings after BBN. Future surveys like Planck will measure the effective number of radiation degrees of freedom with an accuracy of $ \Delta N_{eff} = 0.026 $ [@Hamann:2007sb] and will also be able to probe if extra radiation has been produced after BBN at all, so that $ \Delta N^{\rm BBN}_{\rm eff} \neq \Delta N^{\rm CMB}_{\rm eff}$. In this paper, we have shown that if dark radiation is produced from dark matter decays, the Universe naturally gets populated with an extra radiation component after BBN but before photon decoupling. The reason is that as the Universe cools the dark matter density increases and, as a result, so does the dark radiation produced from it. We have constrained the fraction of dark matter which is allowed to be converted into dark radiation using the WMAP7 + ACT and WMAP + SPT data. We find an upper bound on this fraction using a COSMOMC analysis and show that it is possible to get an increase in $N_{\rm eff}$ by of order unity as the Universe approaches the epoch of photon decoupling. However, the effect on the temperature power spectrum is somewhat different than adding in extra relativistic species by hand, most noticeably in the suppression of the first acoustic peak. For this reason, if Planck confirms the mismatch between $\Delta N^{\rm CMB}_{\rm eff}$ and $\Delta N^{\rm BBN}_{\rm eff}$, it remains to be seen how well the decaying dark matter model fits data. As a phenomenological example we have presented a model of dark matter decay and calculated the decay rate as a function of the coupling between CDM and dark radiation.\ Dark matter decaying into dark radiation could also have important implications for cosmological structure formation. Very recently the observations of high redshift massive galaxy clusters [@arXiv:1006.5639] has put $\Lambda$CDM cosmology under stringent constraints [@arXiv:1009.3884; @arXiv:1006.1950; @Mortonson:2010mj]. In fact, the presence of extra dark radiation during the CMB epoch may play a role in resolving these issues [@Bashinsky:2003tk]. Note in this context that in Fig. \[fig:plots\] we have showed that the decay of the gravitational potential during decoupling is suppressed in the decaying dark matter model. This may boost early structure formation as discussed above. The detailed study of this effect in the context of our model is beyond the scope of this paper and we leave it for future work.\ We thank Jim Zibin and Yvonne Wong for useful discussions. OEB acknowledges support from the Villum Foundation. References {#references .unnumbered} ========== [99]{} E. Komatsu [*et al.*]{} \[WMAP Collaboration\], Astrophys. J. Suppl.  [**192**]{}, 18 (2011) \[arXiv:1001.4538 \[astro-ph.CO\]\]. G. Steigman, Ann. Rev. Nucl. Part. Sci.  [**57**]{}, 463 (2007) \[arXiv:0712.1100 \[astro-ph\]\]. J. Hamann [*et al.*]{}, JCAP [**0708**]{}, 021 (2007) \[arXiv:0705.0440 \[astro-ph\]\]. M. Archidiacono, E. Calabrese, A. Melchiorri, \[arXiv:1109.2767 \[astro-ph.CO\]\]. E. Calabrese, D. Huterer, E. V. Linder, A. Melchiorri, L. Pagano, Phys. Rev.  [**D83**]{}, 123504 (2011) \[arXiv:1103.4132 \[astro-ph.CO\]\]. Z. Hou, R. Keisler, L. Knox, M. Millea and C. Reichardt, \[arXiv:1104.2333 \[astro-ph.CO\]\]. M. Archidiacono, E. Giusarma, A. Melchiorri and O. Mena, arXiv:1206.0109 \[astro-ph.CO\]. J. Hamann, JCAP [**1203**]{} 021 (2012) \[arXiv:1110.4271 \[astro-ph.CO\]\]. J. Kopp, M. Maltoni and T. Schwetz, \[arXiv:1103.4570\]. E. Akhmedov, T. Schwetz, JHEP [**1010**]{} (2010) 115. \[arXiv:1007.4171 \[hep-ph\]\]. S. K. Agarwalla, P. Huber, Phys. Lett.  [**B696**]{} (2011) 359 \[arXiv:1007.3228 \[hep-ph\]\]. C. Giunti, M. Laveder, \[arXiv:1107.1452 \[hep-ph\]\]. J. Hamann, S. Hannestad, G. G. Raffelt, I. Tamborra and Y. Y. Y. Wong, Phys. Rev. Lett.  [**105**]{}, 181301 (2010) \[arXiv:1006.5276 \[hep-ph\]\]. G. Mention, M. Fechner, T. Lasserre, T. A. Mueller, D. Lhuillier, M. Cribier and A. Letourneau, Phys. Rev.  D [**83**]{}, 073006 (2011) \[arXiv:1101.2755 \[hep-ex\]\]. A. Cuoco, J. Lesgourgues, G. Mangano and S. Pastor, Phys. Rev. D [**71**]{} (2005) 123501 \[astro-ph/0502465\]. W. Hu, Astrophys. J.  [**506**]{}, 485 (1998) \[astro-ph/9801234\]. J. F. Beacom, N. F. Bell and S. Dodelson, Phys. Rev. Lett.  [**93**]{} (2004) 121302 \[astro-ph/0404585\]. S. Hannestad, JCAP [**0502**]{} (2005) 011 \[astro-ph/0411475\]. A. Basboll, O. E. Bjaelde, S. Hannestad and G. G. Raffelt, Phys. Rev. D [**79**]{} (2009) 043512 \[arXiv:0806.1735 \[astro-ph\]\]. A. Basboll and O. E. Bjaelde, Phys. Rev. D [**81**]{} (2010) 123503 \[arXiv:0911.1771 \[astro-ph.CO\]\]. W. Hu, D. J. Eisenstein, M. Tegmark and M. J. White, Phys. Rev. D [**59**]{} (1999) 023512 \[astro-ph/9806362\]. R. Trotta and A. Melchiorri, Phys. Rev. Lett.  [**95**]{} (2005) 011305 \[astro-ph/0412066\]. F. De Bernardis, L. Pagano, P. Serra, A. Melchiorri and A. Cooray, JCAP [**0806**]{} (2008) 013 \[arXiv:0804.1925 \[astro-ph\]\]. T. L. Smith, S. Das and O. Zahn, Phys. Rev. D [**85**]{} (2012) 023001 \[arXiv:1105.3246 \[astro-ph.CO\]\]. Y. I. Izotov and T. X. Thuan, Astrophys. J.  [**710**]{}, L67 (2010) \[arXiv:1001.4440 \[astro-ph.CO\]\]. V. Simha and G. Steigman, JCAP [**0806**]{}, 016 (2008) \[arXiv:0803.3465 \[astro-ph\]\]. E. Aver, K. A. Olive and E. D. Skillman, JCAP [**1005**]{}, 003 (2010) \[arXiv:1001.5218 \[astro-ph.CO\]\]. K. M. Nollett and G. P. Holder, arXiv:1112.2683 \[astro-ph.CO\]. J. Dunkley [*et al.*]{}, Astrophys. J.  [**739**]{}, 52 (2011) \[arXiv:1009.0866 \[astro-ph.CO\]\]. R. Keisler [*et al.*]{}, Astrophys. J.  [**743**]{}, 28 (2011) \[arXiv:1105.3182 \[astro-ph.CO\]\]. J. Hamann, J. Lesgourgues and G. Mangano, JCAP [**0803**]{}, 004 (2008) \[arXiv:0712.2826 \[astro-ph\]\]. L. A. Anchordoqui and H. Goldberg, Phys. Rev. Lett.  [**108**]{} (2012) 081805 \[arXiv:1111.7264 \[hep-ph\]\]. P. Sikivie and Q. Yang, Phys. Rev. Lett.  [**103**]{}, 111301 (2009) \[arXiv:0901.1106 \[hep-ph\]\]. O. Erken, P. Sikivie, H. Tam and Q. Yang, \[arXiv:1104.4507 \[astro-ph.CO\]\]. A. P. Lundgren, M. Bondarescu, R. Bondarescu and J. Balakrishna, Astrophys. J.  [**715**]{}, L35 (2010) \[arXiv:1001.0051 \[astro-ph.CO\]\]. A. Aguilar-Arevalo [*et al.*]{} \[LSND Collaboration\], Phys. Rev. D [**64**]{} (2001) 112007 \[hep-ex/0104049\]. A. A. Aguilar-Arevalo [*et al.*]{} \[The MiniBooNE Collaboration\], Phys. Rev. Lett.  [**105**]{} (2010) 181801 \[arXiv:1007.1150 \[hep-ex\]\]. K. N. Abazajian, M. A. Acero, S. K. Agarwalla, A. A. Aguilar-Arevalo, C. H. Albright, S. Antusch, C. A. Arguelles and A. B. Balantekin [*et al.*]{}, arXiv:1204.5379 \[hep-ph\]. J. Hamann, S. Hannestad, G. G. Raffelt and Y. Y. Y. Wong, JCAP [**1109**]{}, 034 (2011) \[arXiv:1108.4136 \[astro-ph.CO\]\]. J. Fan and P. Langacker, JHEP [**1204**]{}, 083 (2012) \[arXiv:1201.6662 \[hep-ph\]\]. V. Barger, J. P. Kneller, P. Langacker, D. Marfatia and G. Steigman, Phys. Lett.  B [**569**]{}, 123 (2003) \[arXiv:hep-ph/0306061\]. G. Mangano, G. Miele, S. Pastor, O. Pisanti and S. Sarikas, JCAP [**1103**]{}, 035 (2011) \[arXiv:1011.0916 \[astro-ph.CO\]\]. R. Fardon, A. E. Nelson and N. Weiner, JCAP [**0410**]{}, 005 (2004) \[arXiv:astro-ph/0309800\]. S. Antusch, S. Das and K. Dutta, JCAP [**0810**]{}, 016 (2008) \[arXiv:0807.4930 \[astro-ph\]\]. J. L. Feng, V. Rentala and Z. Surujon, Phys. Rev.  D [**84**]{}, 095033 (2011) \[arXiv:1108.4689 \[hep-ph\]\]. M. Blennow, E. Fernandez-Martinez, O. Mena, J. Redondo and P. Serra, \[arXiv:1203.5803 \[hep-ph\]\]. K. Ichikawa, M. Kawasaki, K. Nakayama, M. Senami and F. Takahashi, JCAP [**0705**]{}, 008 (2007) \[arXiv:hep-ph/0703034\]. W. Fischler, J. Meyers, Phys. Rev.  [**D83**]{}, 063520 (2011) \[arXiv:1011.3501 \[astro-ph.CO\]\]. J. Hasenkamp, \[arXiv:1107.4319 \[hep-ph\]\]. K. Ichikawa, M. Kawasaki, K. Nakayama, M. Senami and F. Takahashi, JCAP [**0705**]{}, 008 (2007) \[hep-ph/0703034 \[HEP-PH\]\]. N. F. Bell, A. J. Galea and K. Petraki, Phys. Rev. D [**82**]{}, 023514 (2010) \[arXiv:1004.1008 \[astro-ph.HE\]\]. M. Abdelqader and F. Melia, Mon. Not. Roy. Astron. Soc.  [**388**]{} (2008) 1869 \[arXiv:0806.0602 \[astro-ph\]\]. B. Moore, S. Ghigna, F. Governato, G. Lake, T. R. Quinn, J. Stadel and P. Tozzi, Astrophys. J.  [**524**]{} (1999) L19 \[astro-ph/9907411\]. A. A. Klypin, A. V. Kravtsov, O. Valenzuela and F. Prada, Astrophys. J.  [**522**]{} (1999) 82 \[astro-ph/9901240\]. J. F. Navarro, C. S. Frenk and S. D. M. White, Astrophys. J.  [**490**]{} (1997) 493 \[astro-ph/9611107\]. J. R. Kristiansen and O. Elgaroy, arXiv:1104.0704 \[astro-ph.CO\]. E. Giusarma, M. Archidiacono, R. de Putter, A. Melchiorri and O. Mena, Phys. Rev. D [**85**]{}, 083522 (2012) \[arXiv:1112.4661 \[astro-ph.CO\]\]. W. Zimdahl and D. Pavon, Phys. Lett. B [**521**]{}, 133 (2001) \[astro-ph/0105479\]. B. Wang, Y. -G. Gong and E. Abdalla, Phys. Lett. B [**624**]{}, 141 (2005) \[hep-th/0506069\]. M. R. Setare and E. C. Vagenas, Int. J. Mod. Phys. D [**18**]{}, 147 (2009) \[arXiv:0704.2070 \[hep-th\]\]. H. Ziaeepour, Phys. Rev. D [**69**]{}, 063512 (2004) \[astro-ph/0308515\]. G. Olivares, F. Atrio-Barandela and D. Pavon, Phys. Rev. D [**74**]{}, 043521 (2006) \[astro-ph/0607604\]. T. Koivisto, Phys. Rev. D [**72**]{}, 043516 (2005) \[astro-ph/0504571\]. O. E. Bjaelde, A. W. Brookfield, C. van de Bruck, S. Hannestad, D. F. Mota, L. Schrempp and D. Tocchini-Valentini, JCAP [**0801**]{}, 026 (2008) \[arXiv:0705.2018 \[astro-ph\]\]. O. E. Bjaelde and S. Hannestad, Phys. Rev. D [**81**]{} (2010) 063001 \[arXiv:0806.2146 \[astro-ph\]\]. S. Das, P. S. Corasaniti and J. Khoury, Phys. Rev. D [**73**]{}, 083509 (2006) \[astro-ph/0510628\]. K. A. Malik, D. Wands and C. Ungarelli, Phys. Rev. D [**67**]{}, 063516 (2003) \[astro-ph/0211602\]. J. Valiviita, E. Majerotto and R. Maartens, JCAP [**0807**]{}, 020 (2008) \[arXiv:0804.0232 \[astro-ph\]\]. C. -P. Ma, E. Bertschinger, Astrophys. J.  [**455**]{}, 7-25 (1995) \[astro-ph/9506072\]. M. Kawasaki, G. Steigman, H. -S. Kang, Nucl. Phys.  [**B403**]{}, 671-706 (1993). M. Kaplinghat, R. E. Lopez, S. Dodelson, R. J. Scherrer, Phys. Rev.  [**D60**]{}, 123508 (1999) \[astro-ph/9907388\]. M. -Y. Wang and A. R. Zentner, Phys. Rev. D [**82**]{} (2010) 123507 \[arXiv:1011.2774 \[astro-ph.CO\]\]. M. -Y. Wang and A. R. Zentner, Phys. Rev. D [**85**]{} (2012) 043514 \[arXiv:1201.2426 \[astro-ph.CO\]\]. A. Lewis, A. Challinor and A. Lasenby, Astrophys. J.  [**538**]{}, 473 (2000) \[astro-ph/9911177\]. A. Lewis and S. L. Bridle, Phys. Rev.  [**D66**]{}, 103511 (2002) \[astro-ph/0205436\]. S. Das and N. Weiner, \[arXiv:astro-ph/0611353\]. O. E. Bjaelde and S. Das, Phys. Rev.  D [**82**]{}, 043504 (2010) \[arXiv:1002.1306 \[astro-ph.CO\]\]. T. Kobayashi, F. Takahashi, T. Takahashi, M. Yamaguchi, \[arXiv:1111.1336 \[astro-ph.CO\]\]. P. B. Greene and L. Kofman, Phys. Rev.  D [**62**]{}, 123516 (2000) \[arXiv:hep-ph/0003018\]. P. McDonald [*et al.*]{} \[SDSS Collaboration\], Astrophys. J. Suppl.  [**163**]{}, 80 (2006) \[arXiv:astro-ph/0405013\]. M. Viel and M. G. Haehnelt, Mon. Not. Roy. Astron. Soc.  [**365**]{}, 231 (2006) \[arXiv:astro-ph/0508177\]. M. Brodwin, J. Ruel, P. A. R. Ade, K. A. Aird, K. Andersson, M. L. N. Ashby, M. Bautz and G. Bazin [*et al.*]{}, Astrophys. J.  [**721**]{}, 90 (2010) \[arXiv:1006.5639 \[astro-ph.CO\]\]. B. Hoyle, R. Jimenez and L. Verde, Phys. Rev. D [**83**]{}, 103502 (2011) \[arXiv:1009.3884 \[astro-ph.CO\]\]. L. Cayon, C. Gordon and J. Silk, \[arXiv:1006.1950 \[astro-ph.CO\]\]. M. J. Mortonson, W. Hu and D. Huterer, Phys. Rev.  D [**83**]{}, 023015 (2011) \[arXiv:1011.0004 \[astro-ph.CO\]\]. S. Bashinsky and U. Seljak, Phys. Rev.  D [**69**]{} 083002 (2004) \[arXiv:astro-ph/0310198\]. [^1]: Check definitions in [@Hu:1998kj]. [^2]: We do not use ACT and SPT data simultaneously. This is because the observational fields slightly overlap and hence require a more detailed analysis of the combined noise properties. We thank Mark Halpern for pointing this out. [^3]: The effect of leaving $\Omega_{\rm k}$ as a free parameter has been investigated in e.g. [@Kristiansen:2011mp; @Giusarma:2011zq]. Interestingly, in a model with two sterile neutrinos the cosmological constant seems to be ruled out at 95 % confidence level. Furthermore, models with sterile neutrinos seem to prefer $w<-1$ for the dark energy equation of state.

--- abstract: 'Nonsymmetric Koornwinder polynomials are multivariable extensions of nonsymmetric Askey-Wilson polynomials. They naturally arise in the representation theory of (double) affine Hecke algebras. In this paper we discuss how nonsymmetric Koornwinder polynomials naturally arise in the theory of the Heisenberg XXZ spin-$\frac{1}{2}$ chain with general reflecting boundary conditions. A central role in this story is played by an explicit two-parameter family of spin representations of the two-boundary Temperley-Lieb algebra. These spin representations have three different appearances. Their original definition relates them directly to the XXZ spin chain, in the form of matchmaker representations they relate to Temperley-Lieb loop models in statistical physics, while their realization as principal series representations leads to the link with nonsymmetric Koornwinder polynomials. The nonsymmetric difference Cherednik-Matsuo correspondence allows to construct for special parameter values Laurent-polynomial solutions of the associated reflection quantum KZ equations in terms of nonsymmetric Koornwinder polynomials. We discuss these aspects in detail by revisiting and extending work of De Gier, Kasatani, Nichols, Cherednik, the first author and many others.' address: - 'J.S.: KdV Institute for Mathematics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands & IMAPP, Radboud University, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands.' - 'B.V.: KdV Institute for Mathematics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands.' author: - 'Jasper Stokman & Bart Vlaar' title: Koornwinder polynomials and the XXZ spin chain --- Dedicated to the 80th birthday of Dick Askey. Introduction ============ Nonsymmetric Koornwinder polynomials ------------------------------------ The four-parameter family of Askey-Wilson polynomials, introduced in 1985 in the famous monograph [@AW], is the most general class of classical $q$-orthogonal polynomials in one variable. Important generalizations have appeared since: the associated Askey-Wilson polynomials [@IR], Rahman’s [@Ra] biorthogonal rational ${}_{10}\phi_9$ functions, nonsymmetric Askey-Wilson polynomials [@NS; @Sa], Askey-Wilson functions [@KS], elliptic analogs of Askey-Wilson polynomials [@Sp; @Rai], etc. There exist two completely different [*multivariable*]{} extensions of the Askey-Wilson polynomials. The first is due to Gasper and Rahman [@GRBook]. In this case the multivariable Askey-Wilson polynomials admit explicit multivariate basic hypergeometric series expansions, allowing the corresponding theory to be developed by classical methods. See [@GI] for the corresponding multivariable generalization of the Askey-Wilson function. The second type of multivariable generalization of the Askey-Wilson polynomials is part of the Macdonald-Cherednik [@Macd; @CBook; @StBook] theory on root system analogs of continuous $q$-ultraspherical, continuous $q$-Jacobi and Askey-Wilson polynomials. In this case the multivariable extension of the Askey-Wilson polynomial, the nonsymmetric Askey-Wilson polynomial and the Askey-Wilson function are the Koornwinder polynomial [@Ko], the nonsymmetric Koornwinder polynomial [@Sa] and the basic hypergeometric function [@St0; @St2], respectively. These multivariable extensions depend on five parameters and arise in harmonic analysis on (quantum) symmetric spaces [@NDS; @Let], representation theory of affine Hecke algebras [@CBook; @Sa] and in quantum relativistic integrable one-dimensional many-body systems of Calogero-Moser type [@Ru; @vD]. Multivariable extensions of Rahman’s [@Ra] biorthogonal rational functions and of the elliptic analogs of the Askey-Wilson functions are due to Rains [@Rai]. Heisenberg spin chains ---------------------- *Spin models* originate in the statistical mechanical study [@Ba; @Be; @He] of magnetism. The *Heisenberg spin chain* is a one-dimensional quantum model of magnetism with a quantum spin particle residing at each site of a one-dimensional lattice. The interaction of the quantum spin particles is given by nearest neighbour spin-spin interaction. Assuming that the number of sites is finite, say $n$, boundary conditions need to be imposed to determine how the quantum spin particles at the boundary sites $1$ and $n$ are treated in the model. The most investigated choice is to impose *periodic (or closed) boundary conditions*, in which case we assume that the sites $1$ and $n$ are also adjacent (the one-dimensional lattice is put on a circle). In this paper we focus on so-called *reflecting (or open) boundary conditions*, in which case the quantum spin particles at the boundary sites $1$ and $n$ are interacting with reflecting boundaries on each side. The quantum Hamiltonian $H_{\text{bdy}}$ for the Heisenberg spin-$\frac{1}{2}$ chain with reflecting boundaries has the following form [@IK; @dVGR; @Ne]. The Pauli [@Pauli] spin matrices are $$\sigma^X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad \sigma^Y = \begin{pmatrix} 0 & -\sqrt{-1} \\ \sqrt{-1} & 0 \end{pmatrix}, \qquad \sigma^Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$$ Then $H_{\text{bdy}}$ is the linear operator on $({\mathbb{C}}^2)^{\otimes n} = \underset{(1)}{{\mathbb{C}}^2} \otimes \cdots \otimes \underset{(n)}{{\mathbb{C}}^2}$ given by $$\begin{aligned} H_{\text{bdy}} &= \sum_{i=1}^{n-1} \left( J_X \sigma^X_i \sigma^X_{i+1} + J_Y \sigma^Y_i \sigma^Y_{i+1} + J_Z \sigma^Z_i \sigma^Z_{i+1} \right) + \\ &+ \left(M^l_X \sigma^X_1 + M^l_Y \sigma^Y_1 + M^l_Z \sigma^Z_1\right) + \left(M^r_X \sigma^X_n + M^r_Y \sigma^Y_n + M^r_Z \sigma^Z_n\right),\end{aligned}$$ where the subindices indicate the tensor leg of $({\mathbb{C}}^2)^{\otimes n}$ on which the Pauli matrix is acting. Besides the three coupling constants $J_X, J_Y$ and $J_Z$ there are three parameters $M^l_X, M^l_Y$ and $M^l_Z$ at the left reflecting boundary and three parameters $M^r_X, M^r_Y$ and $M^r_Z$ at the right reflecting boundary of the one-dimensional lattice. In general, the three coupling constants are distinct, in which case $H_{\text{bdy}}$ corresponds to the XYZ spin-$\frac{1}{2}$ model with general reflecting boundary conditions; in its full generality it was first obtained in [@IK]. In this paper we consider the special case that $J_X=J_Y\not=J_Z$, in which case the spin chain is the XXZ spin-$\frac{1}{2}$ model with general reflecting boundary conditions. We denote the resulting quantum Hamiltonian by $H_{\text{bdy}}^{XXZ}$. Up to rescaling, $H_{\text{bdy}}^{XXZ}$ thus has seven free parameters; also see [@dVGR; @Ne]. The XXZ spin-$\frac{1}{2}$ chain with generic reflecting boundary conditions is quantum integrable, in the sense that the quantum Hamiltonian is part of a large commuting set of linear operators on $({\mathbb{C}}^2)^{\otimes n}$ encoded by a transfer operator (see [@Ne]). For $H_{\text{bdy}}^{XXZ}$ the transfer operator is produced from the standard XXZ spin-$\frac{1}{2}$ solution of the quantum Yang-Baxter equation and three-parameter solutions of the associated left and right reflection equations, see [@Ne] and Subsection \[Hsection\]. It opens the way to study the spectrum and eigenfunctions of $H_{\text{bdy}}^{XXZ}$ using Sklyanin’s [@Sk1988] generalization of the algebraic Bethe ansatz to quantum integrable models with reflecting boundary conditions. Representation theory --------------------- A representation theoretic context for the five parameter family of Koornwinder polynomials is provided by Letzter’s [@Let] notion of quantum symmetric pairs. The quantum symmetric pairs pertinent to Koornwinder polynomials are given by a family of coideal subalgebras of the quantized universal enveloping algebra of $\mathfrak{gl}_N$ naturally associated to quantum complex Grassmannians (see [@NDS; @OS]). For the XXZ spin-$\frac{1}{2}$ spin chain with general reflecting boundary conditions, a representation theoretic context is provided by coideal subalgebras of the quantum affine algebra of $\widehat{\mathfrak{sl}}_2$ known as $q$-Onsager algebras (see [@BK]). $q$-Onsager algebras are examples of quantum [*affine*]{} symmetric pairs [@Kolb]. The five-parameter double affine Hecke algebras of type $\widetilde{C}_n$ [@Sa] provides another representation theoretic context for Koornwinder polynomials. On the other hand, various special cases of the XXZ spin-$\frac{1}{2}$ chain with general reflecting boundary conditions have been related to the three-parameter affine Hecke algebra of type $\widetilde{C}_n$, which is a subalgebra of the double affine Hecke algebra (see [@dGN] and references therein). It is one of the purposes of this paper to show that the double affine Hecke algebra governs the whole seven-parameter family of XXZ spin-$\frac{1}{2}$ chains with reflecting boundary conditions. The double affine Hecke algebra gives rise to a Baxterization procedure for affine Hecke algebra representations, producing for a given representation a two-parameter family of solutions of quantum Yang-Baxter and reflection equations (see Subsection \[Baxt\]). In the case of spin representations, which form a two-parameter family of representations of the affine Hecke algebra of type $\widetilde{C}_n$ on the state space $({\mathbb{C}}^2)^{\otimes n}$ [@dGP1; @dGN], we end up with solutions depending on seven parameters: three affine Hecke algebra parameters denoted by $\underline{\kappa}=(\kappa_0,\kappa,\kappa_n)$, two Baxterization parameters denoted by $\upsilon_0,\upsilon_n$, and two spin representation parameters denoted by $\psi_0,\psi_n$. The resulting transfer operator reproduces $H_{\text{bdy}}^{XXZ}$ as the associated quantum Hamiltonian under an appropriate parameter correspondence. The parameters $\underline{\kappa},\upsilon_0,\upsilon_n$ are the five parameters of the Koornwinder polynomials. The spin representations factor through the two-boundary Temperley-Lieb algebra [@dGN]. The two-boundary Temperley-Lieb algebra has a natural two-parameter family of representations on the formal vector spaces spanned by two-boundary non-crossing perfect matchings (see Definition \[perfectDef\]), in which the generators of the algebra act by “matchmakers”, see, e.g., [@dG; @dGN] and Subsection \[Matchsection\]. It is known [@dGN] that generically the resulting matchmaker representations are isomorphic to the spin representations under a suitable parameter correspondence. We provide a new proof for this in Subsection \[Matchsection\] by constructing an explicit intertwiner. The Baxterization of the matchmaker representation leads to a transfer operator acting on the formal vector space of two-boundary non-crossing perfect matchings. The associated quantum Hamiltonian corresponds to the Temperley-Lieb loop model (also known as dense loop model) with general open boundary conditions (see Subsection \[Hsection\]); special cases are discussed in, e.g., [@MNGB; @dG; @dGN]. It leads to the conclusion that generically the XXZ spin-$\frac{1}{2}$ chain and the Temperley-Lieb loop model for general reflecting boundary conditions are equivalent under a suitable parameter correspondence. Weyl group invariant solutions of the reflection quantum KZ equations --------------------------------------------------------------------- The quantum Knizhnik-Zamolodchikov (KZ) equations are a consistent system of first order linear $q$-difference equations, which were first studied by Smirnov in relation to integrable quantum field theories [@Sm]. They were related to representation theory of quantum affine algebras by Frenkel and Reshetikhin [@FR]. Their solutions entail correlation functions for XXZ spin chains with quasi-periodic boundary conditions, see, e.g., [@JM]. More recently, links to algebraic geometry have been exposed [@dFZJ2]. Generalizations of these equations for arbitrary root systems were constructed by Cherednik [@CQKZ; @CQKZ3]. For type A, the aforementioned Smirnov-Frenkel-Reshetikhin quantum KZ equations are recovered; *reflection* quantum KZ equations are the equations in Cherednik’s framework associated to the other classical types [@CQKZ §5]. In the context of the spin-$\frac{1}{2}$ Heisenberg chain, the reflection quantum KZ equations were first derived in [@JKKKM]. A choice of solutions of the quantum Yang-Baxter equation and the associated reflection equations gives rise to reflection quantum KZ equations [@CQKZ]. In case the solutions are obtained from the Baxterization of the spin representations we obtain reflection quantum KZ equations depending on seven parameters. These are the reflection quantum KZ equations under investigation in the last section of the paper (Section \[Solsection\]). Their solutions are expected to give rise to correlation functions for the spin chain governed by $H_{\text{bdy}}^{XXZ}$. The close link to the spin chain is apparent from the fact that the transport operators of the reflection quantum KZ equations at $q=1$ are higher quantum Hamiltonians for the spin chain (see Subsection \[spsection\]). For special choices of reflecting boundary conditions methods have been developed to construct explicit solutions of the reflection quantum KZ equations, e.g. in [@JKKKM; @JKKMW; @We; @dFZJ; @dFZJ2; @ZJ2; @Pa; @RSV]. In Section \[Solsection\] we construct solutions using nonsymmetric Koornwinder polynomials, following the ideas from [@Ka; @St2]. The solution space of the reflection quantum KZ equations associated to $H_{\text{bdy}}^{XXZ}$ has a natural action of the Weyl group $W_0$ of type $C_n$, which is the hyperoctahedral group $S_n\ltimes\{\pm 1\}^n$. In Section \[Solsection\] we show that generically, the space of Weyl group invariant solutions is in bijective correspondence to a suitable space of eigenfunctions of the Cherednik-Noumi [@N] $Y$-operators. The $Y$-operators are commuting scalar-valued $q$-difference-reflection operators generalizing Dunkl operators. The link is provided by the nonsymmetric Cherednik-Matsuo correspondence [@St1]. This correspondence can be applied in the present context, because the spin representations are examples of principal series representations (see Subsection \[PrincipalSection\]). The nonsymmetric basic hypergeometric function of Koornwinder type [@St0; @St2] provides distinguished examples of eigenfunctions of the $Y$-operators. We show that it produces a nontrivial $W_0$-invariant solution of the reflection quantum KZ equations for generic values of the parameters. It is exactly known under which specialization of the spectral parameters the nonsymmetric basic hypergeometric function reduces to a nonsymmetric Koornwinder polynomial. It follows that if the parameters satisfy one of the following two conditions, $$\begin{split} \psi_0\psi_nq^m&=\kappa_0\kappa_n\kappa^{n-1}\qquad\,\,\,\,\,\,\, \textup{for some } m\in\mathbb{Z}_{\geq 1},\\ \psi_0\psi_nq^m&=\kappa_0^{-1}\kappa_n^{-1}\kappa^{1-n}\qquad \textup{for some } m\in\mathbb{Z}_{\leq 0}, \end{split}$$ then nontrivial $W_0$-invariant Laurent polynomial solutions of the reflection quantum KZ equations exist (see Theorem \[mainthm\]). Note that these conditions do not depend on the two Baxterization parameters $\upsilon_0,\upsilon_n$. $W_0$-invariant Laurent polynomial solutions are expected to play an important role in generalizations of Razumov-Stroganov conjectures [@dGR], cf., e.g., [@dGP2; @ZJ1; @KaPrep]. Outlook ------- It is an open problem how the solutions of the reflection quantum KZ equations in terms of nonsymmetric basic hypergeometric functions and nonsymmetric Koornwinder polynomials are related to other constructions of explicit solutions [@JKKKM; @JKKMW; @We; @dFZJ; @dFZJ2; @ZJ2; @Pa; @RSV]. Wall-crossing formulas for the reflection quantum KZ equations associated to $H_{\text{bdy}}^{XXZ}$ are in reach by [@St1; @St3]. They are expected to lead to elliptic solutions of quantum dynamical Yang-Baxter equations and associated dynamical reflection equations governing the integrability of elliptic solid-on-solid models with reflecting ends, cf. [@Fi] for such models with diagonal reflecting ends. Pursuing the techniques of the present paper at the elliptic level, either for the XYZ spin chain or elliptic solid-on-solid models, is expected to lead to connections to elliptic hypergeometric functions [@Ra; @Sp; @SpSurvey; @SpInt]. Only some first steps have been made here, see, e.g., [@ZJell; @Tak]. Acknowledgements ---------------- We are grateful to P. Baseilhac, J. de Gier, B. Nienhuis, N. Reshetikhin and P. Zinn-Justin for stimulating discussions. The work of B.V. was supported by an NWO Free Competition grant (“Double affine Hecke algebras, Integrable Models and Enumerative Combinatorics”). Notational conventions {#notational-conventions .unnumbered} ---------------------- Throughout the paper $n\in\mathbb{Z}_{\geq 2}$. Linear operators of $\mathbb{C}^2$ are represented as $2\times 2$-matrices with respect to a fixed ordered basis $(v_+,v_-)$ of $\mathbb{C}^2$, and linear operators of $\mathbb{C}^2\otimes\mathbb{C}^2$ are represented as $4\times 4$-matrices with respect to the ordered basis $(v_+\otimes v_+,v_+\otimes v_-,v_-\otimes v_+,v_-\otimes v_-)$ of $\mathbb{C}^2\otimes\mathbb{C}^2$. Groups and algebras of type {#algebrasection} ============================ Affine braid group ------------------ The affine braid group of type $\widetilde{C}_n$ is the group $\mathcal{B}$ with generators $\sigma_0,\sigma_1,\ldots,\sigma_n$ subject to the braid relations $$\begin{aligned} \sigma_0\sigma_1\sigma_0\sigma_1&=\sigma_1\sigma_0\sigma_1\sigma_0,\\ \sigma_{n-1}\sigma_n\sigma_{n-1}\sigma_n&=\sigma_n\sigma_{n-1}\sigma_n\sigma_{n-1},\\ \sigma_i\sigma_{i+1}\sigma_i&=\sigma_{i+1}\sigma_i\sigma_{i+1}, && 1\leq i<n-1, \\ \sigma_i \sigma_j &= \sigma_j \sigma_i, && 1 \leq i,j \leq n, \, |i-j|>1. \end{aligned}$$ It can be topologically realized as follows (see the type $\widetilde{C}_n$ case in [@Al §4]). Consider the line segments $P_l=\{0\}\times\{0\}\times [0,1]$ and $P_r=\{0\}\times\{n+1\}\times [0,1]$ in $\mathbb{R}^3$. Let $\widetilde{\mathcal{B}}$ be the group of $n$-braids in the strip $\bigl(\mathbb{R}^2\times [0,1]\bigr)\setminus (P_l\cup P_r)$, with the $n$-braids attached to the floor at $(0,i,0)$ ($1\leq i\leq n$) and to the ceiling at $(0,i,1)$ ($1\leq i\leq n$). The group operation $\sigma\tau$ is putting the $n$-braid $\tau$ on top of $\sigma$ and shrinking the height by isotopies. The group isomorphism $\mathcal{B}\simeq\widetilde{\mathcal{B}}$ is induced by the identification $$\begin{gathered} \sigma_i = \begin{minipage}[c]{36mm} \begin{tikzpicture}[scale=36/84] \draw[gray,line width = 5pt] (0,0) -- (0,4) node[at start,below] {$P_l$}; \draw[gray,line width = 5pt] (7,0) -- (7,4) node[at start,below] {$P_r$}; \draw[very thick] (1,0) -- (1,4) node[at start, below] {1}; \draw(2,2) node{\ldots}; \draw[very thick] (3,0) .. controls (3,1.6) .. node[at start, below]{$i \vphantom{1}$} (3.5,2) .. controls (4,2.4) .. (4,4) ; \draw[white,line width = 5pt] (4,0) .. controls (4,1.6) .. (3.5,2) .. controls (3,2.4) .. (3,4); \draw[very thick] (4,0) .. controls (4,1.6) .. node[at start, below]{$i\!\!+\!\!1$} (3.5,2) .. controls (3,2.4) .. (3,4); \draw(5,2) node{\ldots}; \draw[very thick] (6,0) -- (6,4) node[at start, below] {$n \vphantom{1}$}; \end{tikzpicture} \end{minipage} \qquad 1 \leq i < n, \\ \sigma_0 = \begin{minipage}[c]{32mm} \begin{tikzpicture}[scale=32/72] \draw[gray,line width = 5pt] (0,0) -- (0,4) node[at start,below] {$P_l$}; \draw[gray,line width = 5pt] (5,0) -- (5,4) node[at start,below] {$P_r$}; \draw[white,line width = 5pt] (1,0) .. controls (1,1.6) .. (0,1.6) .. controls (-1,1.6) .. (-1,2); \draw[very thick] (1,0) .. controls (1,1.6) .. node[at start, below] {1} (0,1.6) .. controls (-1,1.6) .. (-1,2) .. controls (-1,2.4) .. (-.3,2.4); \draw[very thick] (.3,2.4) .. controls (1,2.4) .. (1,4); \draw[very thick] (2,0) -- (2,4) node[at start, below] {2}; \draw(3,2) node{\ldots}; \draw[very thick] (4,0) -- (4,4) node[at start, below] {$n \vphantom{1}$}; \end{tikzpicture} \end{minipage} \qquad \sigma_n = \begin{minipage}[c]{32mm} \begin{tikzpicture}[scale=32/72] \draw[gray,line width = 5pt] (0,0) -- (0,4) node[at start,below] {$P_l$}; \draw[gray,line width = 5pt] (5,0) -- (5,4) node[at start,below] {$P_r$}; \draw[very thick] (1,0) -- (1,4) node[at start, below] {1}; \draw(2,2) node{\ldots}; \draw[very thick] (3,0) -- (3,4) node[at start, below] {$\! n\!\!-\!\!1 \,$}; \draw[very thick] (4,0) .. controls (4,1.6) .. node[at start, below] {$n \vphantom{1}$} (4.7,1.6); \draw[white,line width = 5pt] (6,2) .. controls (6,2.4) .. (5,2.4) .. controls (4,2.4) .. (4,4); \draw[very thick] (5.3,1.6) .. controls (6,1.6) .. (6,2) .. controls (6,2.4) .. (5,2.4) .. controls (4,2.4) .. (4,4); \end{tikzpicture} \end{minipage}\end{gathered}$$ where the right hand sides are the braid diagram projections on the $\mathbb{R}^2\simeq\{0\}\times\mathbb{R}^2$-plane. For $1 \leq i \leq n$, it is easy to check topologically that the elements $$\begin{aligned} g_i &:=\sigma_{i-1}^{-1}\cdots\sigma_1^{-1}\sigma_0\sigma_1\cdots\sigma_{n-1}\sigma_n\sigma_{n-1}\cdots\sigma_i \\ &= \begin{minipage}[c]{60mm} \begin{tikzpicture}[scale=.5] \draw[gray,line width = 5pt] (0,0) -- (0,4) node[at start,below] {$P_l$}; \draw[gray,line width = 5pt] (8,0) -- (8,4) node[at start,below] {$P_r$}; \draw[very thick] (.3,2) -- (1.3,2); \draw[very thick, dashed] (1.3,2) -- (2.7,2); \draw[very thick] (2.7,2) -- (5.3,2); \draw[very thick, dashed] (5.3,2) -- (6.7,2); \draw[very thick] (6.7,2) -- (7.7,2); \draw[white,line width = 5pt] (3,1) -- (0,1) .. controls (-1,1) .. (-1,1.5) ; \draw[very thick] (4,0) .. controls (4,1) .. node[at start, below]{$i \vphantom{1}$} (3,1) -- (2.7,1); \draw[very thick,dashed] (2.7,1) -- (1.3,1); \draw[very thick] (1.3,1) -- (0,1) .. controls (-1,1) .. (-1,1.5) .. controls (-1,2) .. (-.3,2) ; \draw[white,line width = 5pt] (1,0) -- (1,4); \draw[white,line width = 5pt] (3,0) -- (3,4); \draw[white,line width = 5pt] (5,0) -- (5,4); \draw[white,line width = 5pt] (7,0) -- (7,4); \draw[very thick] (1,0) -- (1,4) node[at start, below] {1}; \draw[very thick] (3,0) -- (3,4) node[at start, below] {$i\!\!-\!\!1$}; \draw[very thick] (5,0) -- (5,4) node[at start, below] {$i\!\!+\!\!1$}; \draw[very thick] (7,0) -- (7,4) node[at start, below] {$n \vphantom{1}$}; \draw[white,line width = 5pt] (9,2.5) .. controls (9,3) .. (8,3) -- (5,3) .. controls (4,3) .. (4,4); \draw[very thick] (8.3,2) .. controls (9,2) .. (9,2.5) .. controls (9,3) .. (8,3) -- (6.7,3); \draw[very thick,dashed] (6.7,3) -- (5.3,3); \draw[very thick] (5.3,3) -- (5,3) .. controls (4,3) .. (4,4); \end{tikzpicture} \end{minipage}\end{aligned}$$ of the affine braid group $\mathcal{B}$ pairwise commute. The affine Weyl group --------------------- The affine Weyl group $W$ of type $\widetilde{C}_n$ is the quotient of $\mathcal{B}$ by the relations $\sigma_i^2=1$ ($0\leq i\leq n$). The generators $\sigma_0,\ldots,\sigma_n$ descend to group generators of $W$, which we denote by $s_0,\ldots,s_n$. The affine Weyl group $W$ is a Coxeter group with Coxeter generators $s_0,\ldots,s_n$. Write $\tau_i$ for the commuting elements in $W$ corresponding to $g_i\in\mathcal{B}$ under the canonical surjection $\mathcal{B}\twoheadrightarrow W$, so that $$\label{taui} \tau_i=s_{i-1}\cdots s_1s_0s_1\cdots s_{n-1}s_ns_{n-1}\cdots s_{i}.$$ The affine Weyl group $W$ acts faithfully on $\mathbb{R}^n$ by affine linear transformations via $$\begin{split} s_0(x_1,\ldots,x_n)&=(1-x_1,x_2,\ldots,x_n),\\ s_i(x_1,\ldots,x_n)&=(x_1,\ldots,x_{i-1},x_{i+1},x_i,x_{i+2},\ldots,x_n),\\ s_n(x_1,\ldots,x_n)&=(x_1,\ldots,x_{n-1},-x_n) \end{split}$$ for $1\leq i<n$. Note that the $s_j$ act on $\mathbb{R}^n$ as orthogonal reflections in affine hyperplanes. We sometimes call $s_0,\ldots,s_n$ the simple reflections of $W$. The subgroup $W_0$ of $W$ generated by $s_1,\ldots,s_n$, acting by permutations and sign changes of the coordinates, is isomorphic to $S_n\ltimes (\pm 1)^n$ with $S_n$ the symmetric group in $n$ letters (which, in turn, is isomorphic to the subgroup generated by $s_1,\ldots,s_{n-1}$). Note furthermore that $$\tau_i(x_1,\ldots,x_n)=(x_1,\ldots,x_{i-1},x_i+1,x_{i+2},\ldots,x_n),\qquad 1\leq i\leq n,$$ hence the abelian subgroup of $W$ generated by $\tau_1,\ldots,\tau_n$ is isomorphic to $\mathbb{Z}^n$. We write $\tau( \lambda):=\tau_1^{\lambda_1}\cdots\tau_n^{\lambda_n}\in W$ for $ \lambda=(\lambda_1,\ldots,\lambda_n)\in\mathbb{Z}^n$. We have $W \cong W_0\ltimes \mathbb{Z}^n$, with $W_0$ acting on $\mathbb{Z}^n$ by permutations and sign changes of the coordinates. Affine Hecke algebra -------------------- Fix $\underline{\kappa}=(\kappa_0,\kappa_1,\ldots,\kappa_n)\in\bigl(\mathbb{C}^*\bigr)^{n+1}$ with $\kappa_1=\kappa_2=\cdots=\kappa_{n-1}$. We write $\kappa$ for the value $\kappa_i$ ($1\leq i<n$). The affine Hecke algebra $H(\underline{\kappa})$ of type $\widetilde{C}_n$ is the quotient of the group algebra $\mathbb{C}[\mathcal{B}]$ of the affine braid group $\mathcal{B}$ by the two-sided ideal generated by the elements $(\sigma_j-\kappa_j)(\sigma_j+\kappa_j^{-1})$ for $0\leq j \leq n$. The generators $\sigma_0,\ldots,\sigma_n$ descend to algebraic generators of $H(\underline{\kappa})$, which we denote by $T_0,\ldots,T_n$. The quadratic relation $(T_j-\kappa_j)(T_j+\kappa_j^{-1})=0$ in $H(\underline{\kappa})$ implies that $T_j$ is invertible in $H(\underline{\kappa})$ with inverse $T_j-\kappa_j+\kappa_j^{-1}$. For $\underline{\kappa}=(1,1,\ldots,1)$ the affine Hecke algebra $H(\underline{\kappa})$ is the group algebra $\mathbb{C}[W]$ of the affine Weyl group $W$ of type $C_n$. Let $Y_i$ be the element in $H(\underline{\kappa})$ corresponding to $g_i\in\mathcal{B}$ under the canonical surjection $\mathbb{C}[\mathcal{B}]\twoheadrightarrow H(\underline{\kappa})$. Then $$Y_i=T_{i-1}^{-1}\cdots T_1^{-1}T_0T_1\cdots T_{n-1}T_nT_{n-1}\cdots T_i,\qquad 1\leq i\leq n.$$ The $Y_i$ pairwise commute in $H(\underline{\kappa})$. They are sometimes called Murphy elements (cf., e.g., [@dGN Def. 2.8]). In the context of representation theory of affine Hecke algebras, they naturally arise in the Bernstein-Zelevinsky presentation of the affine Hecke algebra, see [@Lu]. We write $$Y^{ \lambda}:=Y_1^{\lambda_1}Y_2^{\lambda_2}\cdots Y_n^{\lambda_n},\qquad \lambda=(\lambda_1,\ldots,\lambda_n) \in\mathbb{Z}^n.$$ The elements $T_1,\ldots,T_n$ and $Y^{ \lambda}$ ($ \lambda \in \mathbb{Z}^n$) generate $H(\underline{\kappa})$. The two-boundary Temperley-Lieb algebra {#2BTL} --------------------------------------- In analogy to the setup for the affine Hecke algebra, fix $\underline{\delta} = (\delta_0,\delta_1,\cdots, \delta_n) \in\bigl(\mathbb{C}^*\bigr)^{n+1}$ with $\delta_1 = \delta_2 = \cdots = \delta_{n-1} =: \delta$. The *two-boundary Temperley-Lieb algebra* TL$(\underline{\delta})$ [@dGN], also known as the *open Temperley-Lieb algebra* [@dGP1], is the unital associative algebra over ${\mathbb{C}}$ with generators $e_0,\ldots,e_n$ satisfying $$\begin{aligned} && e_i^2 &= \delta_i e_i \label{TLquadraticrelation} \\ && e_i e_{i \pm 1} e_i &= e_i && 1 \leq i < n, \label{TLnoncommutingrelation} \\ && e_i e_j &= e_j e_i,&& |i-j|>1. \label{TLcommutingrelation}\end{aligned}$$ See [@dGN; @dGP1] for a diagrammatic realization of $\textup{TL}(\underline{\delta})$. To generic affine Hecke algebra parameters $\underline{\kappa}$ we associate two-boundary Temperley-Lieb parameters $\underline{\delta}$ by $$\label{eq:deltas} \delta_j = -\frac{\kappa_j+\kappa_j^{-1}}{\kappa \kappa_j^{-1} +\kappa^{-1} \kappa_j}, \qquad\quad \delta = -(\kappa+\kappa^{-1})$$ for $j=0$ and $j=n$. In the remainder of the paper we always assume that the affine Hecke algebra parameters and the two-boundary Temperley-Lieb algebra parameters are matched in this way. Then there exists a unique surjective algebra homomorphism $\phi: H(\underline \kappa) \twoheadrightarrow TL(\underline \delta)$ such that $$\label{phiimages} \phi(T_j) = \begin{cases} \kappa_j + (\kappa \kappa_j^{-1} +\kappa^{-1} \kappa_j) e_j,\qquad & j=0,n, \\ \kappa+e_j,\qquad & 1 \leq j < n, \end{cases}$$ see [@dGN Prop. 2.13]. In particular, $\textup{TL}(\underline{\delta})$ is isomorphic to a quotient of $H(\underline{\kappa})$. The kernel of $\phi$ can be explicitly described, see [@dGN Lem. 2.15]. Representations {#repsection} =============== Spin representation {#Spinrep} ------------------- In the following lemma we give a two-parameter family of representations of the two-boundary Temperley-Lieb algebra $\textup{TL}(\underline{\delta})$ on the state space $\bigl(\mathbb{C}^2\bigr)^{\otimes n}$ of the Heisenberg XXZ spin-$\frac{1}{2}$ chain. It includes the one-parameter family of representations that has been intensively studied in the physics literature (see, e.g., [@dGN §2.3] and references therein). We use the standard tensor leg notations for linear operators on $\bigl(\mathbb{C}^2\bigr)^{\otimes n}$. Note also the convention on the matrix notation for linear operators on $\mathbb{C}^2\otimes\mathbb{C}^2$ as given at the end of the introduction. Let the free parameters $\underline{\delta}$ of the two-boundary Temperley-Lieb algebra be given in terms of the generic affine Hecke algebra parameters $\underline{\kappa}$ by . Let $\psi_0,\psi_n\in\mathbb{C}^*$. There exists a unique algebra homomorphism $$\hat{\rho}=\hat{\rho}^{\underline{\kappa}}_{\psi_0,\psi_n}: \textup{TL}(\underline{\delta})\rightarrow\textup{End}_{\mathbb{C}}\bigl((\mathbb{C}^2)^{\otimes n}\bigr)$$ such that $$\begin{aligned} \hat \rho(e_0) &= \frac{1}{\kappa \kappa_0^{-1} + \kappa^{-1} \kappa_0} \begin{pmatrix} -\kappa_0^{-1} & \psi_0 \\ \psi_0^{-1} & -\kappa_0 \end{pmatrix}_1 \\ \hat \rho(e_i) &= \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & -\kappa & 1 & 0 \\ 0 & 1 & -\kappa^{-1} & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}_{i , i\!+\!1} \\ \hat \rho(e_n) &= \frac{1}{\kappa \kappa_n^{-1} + \kappa^{-1} \kappa_n} \begin{pmatrix} -\kappa_n & \psi_n^{-1} \\ \psi_n & -\kappa_n^{-1} \end{pmatrix}_n. \end{aligned}$$ This is a straightforward verification. We lift $\hat{\rho}$ to a representation $$\rho=\rho^{\underline{\kappa}}_{\psi_0,\psi_n}: H(\underline{\kappa})\rightarrow \textup{End}_{\mathbb{C}}\bigl((\mathbb{C}^2)^{\otimes n}\bigr)$$ of the affine Hecke algebra via the surjection $\phi: H(\underline{\kappa})\twoheadrightarrow \textup{TL}(\underline{\delta})$, so $\rho:=\hat{\rho}\circ\phi$. Then $$\rho(T_0):=\bar K_1,\qquad \rho(T_i):=(\Upsilon\circ P)_{i,i+1},\qquad \rho(T_n):=K_n$$ ($1\leq i<n$) with $$\begin{split} \bar K&:=\left(\begin{matrix} \kappa_0-\kappa_0^{-1} & \psi_0\\ \psi_0^{-1} & 0\end{matrix}\right),\qquad K:=\left(\begin{matrix} 0 &\psi_n^{-1}\\ \psi_n & \kappa_n-\kappa_n^{-1}\end{matrix}\right),\\ \Upsilon&:=\left(\begin{matrix} \kappa & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & \kappa-\kappa^{-1} & 1 & 0\\ 0 & 0 & 0 & \kappa\end{matrix}\right) \end{split}$$ and $P:\mathbb{C}^2\otimes\mathbb{C}^2\rightarrow \mathbb{C}^2\otimes\mathbb{C}^2$ the flip operator $P(v\otimes w)=P(w\otimes v)$. The braid relations of $T_0,\ldots,T_n$ imply that $\Upsilon$ is a solution of the quantum Yang-Baxter equation and that $K$ and $\bar K$ are solutions to associated reflection equations (all equations without spectral parameter). The solution $\Upsilon$ is the well-known solution of the quantum Yang-Baxter equation arising from the universal $R$-matrix of the quantized universal enveloping algebra $\mathcal{U}_{1/\kappa}(\mathfrak{sl}_2)$ acting on $\mathbb{C}^2\otimes\mathbb{C}^2$, with $\mathbb{C}^2$ viewed as the vector representation of $\mathcal{U}_{1/\kappa}(\mathfrak{sl}_2)$. The representations $\rho$ and $\hat{\rho}$ are called spin representations of $H(\underline{\kappa})$ and $\textup{TL}(\underline{\delta})$, respectively. Principal series representation {#PrincipalSection} ------------------------------- One of the main aims of this paper is to relate solutions of reflection quantum Knizhnik-Zamolodchikov equations for Heisenberg XXZ spin-$\frac{1}{2}$ chains with boundaries to Macdonald-Koornwinder type functions. Such a connection is known in the context of so-called principal series representations of affine Hecke algebras; see Subsection \[dCMsection\] and references therein. Principal series representations are a natural class of finite dimensional representations of the affine Hecke algebra, which we define now first. In the second part of this subsection we show that the spin representation $\rho$ is isomorphic to a principal series representation. Set $T:=\bigl(\mathbb{C}^*\bigr)^n$. Note that $W_0\simeq S_n\ltimes (\pm 1)^n$ acts on $T$ by permutations and inversions of the coordinates. \[basicdefprincipal\] Let $I\subseteq\{1,\ldots,n\}$. 1. Let $H_I(\underline{\kappa})$ be the subalgebra generated by $T_i$ ($i\in I$) and $Y^\lambda$ ($\lambda\in\mathbb{Z}^n$). 2. Let $T_I^{\underline{\kappa}}$ be the elements $\gamma\in T$ satisfying $\gamma_n=\kappa_0\kappa_n$ if $n\in I$ and satisfying $\gamma_i/\gamma_{i+1}=\kappa^2$ if $i\in \{1,\ldots,n-1\}\cap I$. Let $\gamma\in T_I^{\underline{\kappa}}$. There exists a unique algebra map $\chi_{I,\gamma}^{\underline{\kappa}}: H_I(\underline{\kappa})\rightarrow\mathbb{C}$ satisfying $$\begin{split} \chi_{I,\gamma}^{\underline{\kappa}}(T_i)&=\kappa_i,\qquad i\in I,\\ \chi_{I,\gamma}^{\underline{\kappa}}(Y^\lambda)&=\gamma^\lambda,\qquad \lambda\in\mathbb{Z}^n. \end{split}$$ The proof is a straightforward adjustment of the proof of [@St1 Lem. 2.5(i)]. We write $\mathbb{C}_{I,\gamma}$ for $\mathbb{C}$ viewed as $H_I(\underline{\kappa})$-module with representation map $\chi_{I,\gamma}^{\underline{\kappa}}$. Let $I\subseteq\{1,\ldots,n\}$ and $\gamma\in T_I^{\underline{\kappa}}$. The associated principal series module is the induced $H(\underline{\kappa})$-module $M_I^{\underline{\kappa}}(\gamma):= \textup{Ind}_{H_I(\underline{\kappa})}^{H(\underline{\kappa})}(\mathbb{C}_{I,\gamma})$. We write $\pi_{I,\gamma}^{\underline{\kappa}}$ for the corresponding representation map. Concretely, $M_I^{\underline{\kappa}}(\gamma)=H(\underline{\kappa})\otimes_{H_I(\underline{\kappa})}\mathbb{C}_{I,\gamma}$ with the $H(\underline{\kappa})$-action given by $$\pi_{I,\gamma}^{\underline{\kappa}}(h)(h^\prime\otimes_{H_I(\underline{\kappa})}1):=(hh^\prime)\otimes_{H_I(\underline{\kappa})}1,\qquad h,h^\prime\in H(\underline{\kappa}).$$ The principal series module $M_I^{\underline{\kappa}}(\gamma)$ is finite dimensional. In fact, $$\textup{Dim}_{\mathbb{C}}\bigl(M_I^{\underline{\kappa}}(\gamma)\bigr)=\#W_0/\#W_{0,I}$$ with $W_{0,I}$ the subgroup of $W_0$ generated by the $s_i$ ($i\in I$). The $W_0$-orbit $W_0\gamma$ of $\gamma$ in $T$ is called the central character of $M_I^{\underline{\kappa}}(\gamma)$. \[rhopiequivalence\] Let $\psi_0,\psi_n\in\mathbb{C}^*$. We have $$\rho_{\psi_0,\psi_n}^{\underline{\kappa}} \simeq \pi_{J, \zeta}^{\underline{\kappa}}$$ with $$\begin{split} J&:=\{1,2,\ldots,n-1\},\\ \zeta&:=(\psi_0\psi_n\kappa^{n-1},\psi_0\psi_n\kappa^{n-3},\ldots,\psi_0\psi_n\kappa^{1-n}). \end{split}$$ Note that $\zeta_i/\zeta_{i+1}=\kappa^2$ for $i\in J$, hence $\pi_{J, \zeta}^{\underline{\kappa}}$ is well defined. Observe that $\rho_{\psi_0,\psi_n}^{\underline{\kappa}}$ is a cyclic $H(\underline{\kappa})$-representation with cyclic vector $v_+^{\otimes n}$. Note furthermore that $$\rho_{\psi_0,\psi_n}^{\underline{\kappa}}(T_i)v_+^{\otimes n}=\kappa v_+^{\otimes n},\qquad i\in J.$$ In addition, for $i\in\{1,\ldots,n\}$, $$\begin{split} \rho_{\psi_0,\psi_n}^{\underline{\kappa}}(Y_i)v_+^{\otimes n}&=\kappa^{n-i}\rho_{\psi_0,\psi_n}^{\underline{\kappa}}(T_{i-1}^{-1}\cdots T_1^{-1}T_0\cdots T_{n-1}T_n)v_+^{\otimes n}\\ &=\psi_n\kappa^{n-i}\rho_{\psi_0,\psi_n}^{\underline{\kappa}}(T_{i-1}^{-1}\cdots T_1^{-1}T_0\cdots T_{n-1})(v_+^{\otimes (n-1)}\otimes v_-)\\ &=\psi_n\kappa^{n-i}\rho_{\psi_0,\psi_n}^{\underline{\kappa}}(T_{i-1}^{-1}\cdots T_1^{-1}T_0)(v_-\otimes v_+^{\otimes (n-1)})\\ &=\psi_0\psi_n\kappa^{n-i}\rho_{\psi_0,\psi_n}^{\underline{\kappa}}(T_{i-1}^{-1}\cdots T_1^{-1})v_+^{\otimes n}\\ &=\psi_0\psi_n\kappa^{n-2i+1}v_+^{\otimes n}. \end{split}$$ Hence $\rho_{\psi_0,\psi_n}^{\underline{\kappa}}(h)v_+^{\otimes n}=\chi_{J, \zeta}^{\underline{\kappa}}(h)v_+^{\otimes n}$ for $h\in H_I(\underline{\kappa})$. We thus have a surjective linear map $M_J^{\underline{\kappa}}( \zeta)\twoheadrightarrow \bigl(\mathbb{C}^2\bigr)^{\otimes n}$ mapping $h\otimes_{H_I(\underline{\kappa})}1$ to $\rho_{\psi_0,\psi_n}^{\underline{\kappa}}(h)v_+^{\otimes n}$ for all $h\in H(\underline{\kappa})$, which intertwines the $\pi_{J,\zeta}^{\underline{\kappa}}$-action on $M_J^{\underline{\kappa}}(\gamma)$ with the $\rho_{\psi_0,\psi_n}^{\underline{\kappa}}$-action on $\bigl(\mathbb{C}^2\bigr)^{\otimes n}$. A dimension count shows that it is an isomorphism. Matchmaker representation {#Matchsection} ------------------------- In this subsection we revisit some of the key results from [@dGN] and reestablish them by different methods. The diagrammatic realization of (boundary) Temperley-Lieb algebras (see, e.g., [@dG; @dGP1; @dGN]) gives rise to natural examples of Temperley-Lieb algebra actions on linear combinations of non-crossing matchings (or equivalently, link patterns) in which the Temperley-Lieb algebra generators act as matchmakers. We give a two-parameter family of such matchmaker representations of $\textup{TL}(\underline{\delta})$. We reestablish the result from [@dGN] that this family is generically isomorphic to the family of spin representations of $\textup{TL}(\underline{\delta})$ by constructing an explicit intertwiner involving a subtle combinatorial expression given in terms of the various orientations of the pertinent non-crossing matchings. The origin of such explicit intertwiners traces back to the explicit link between loop models and the XXZ spin chain from [@MNGB §8]. In the quasi-periodic context, related to the affine Hecke algebra of type $\widetilde{A}$ and the affine Temperley-Lieb algebra, such intertwiners were considered in [@ZJ2 §3.2] and [@MDSA Prop. 3.2]. Let $\mathcal{S}$ be the set of unordered pairs $\{i,j\}$ ($i\not=j$) of $[n+1]:=\{0,\ldots,n+1\}$ and denote by $\mathcal{P}(\mathcal{S})$ the power set of $\mathcal{S}$. If $\mathfrak{p}\in\mathcal{P}(\mathcal{S})$ then $i\in [n+1]$ is said to be matched to $j\in [n+1]$ if $\{i,j\}\in\mathfrak{p}$. \[perfectDef\] $\mathfrak{p}\in\mathcal{P}(\mathcal{S})$ is called a two-boundary non-crossing perfect matching if the following conditions hold: 1. Each $i\in\{1,\ldots n\}$ is matched to exactly one $j\in [n+1]$. 2. There are no pairs $\{i,j\}, \{k,l\}\in\mathfrak{p}$ with $i<k<j<l$. 3. $\{0,n+1\}\not\in\mathfrak{p}$. We write $\mathcal{M}$ for the set of two-boundary non-crossing perfect matchings of $[n+1]$. Note that the definition of two-boundary non-crossing perfect matching allows $0$ and $n+1$ to have multiple matchings; these boundary points may also be unmatched. We can give a diagrammatic representation of $\mathfrak{p}\in\mathcal{P}(\mathcal{S})$ by connecting $(i,0)$ and $(j,0)$ by an arc in the upper half plane for all $\{i,j\}\in\mathfrak{p}$. If $\mathfrak{p}\in\mathcal{M}$ then this can be done in such a way that the arcs do not intersect except possibly at the boundary endpoints $(0,0)$ and $(n+1,0)$. For $\mathfrak p \in \mathcal{M}$ and $1 \leq i \leq n$ we write $m_i(\mathfrak p)\in [n+1]\setminus \{i\}$ for the unique element such that $\{i,m_i(\mathfrak{p})\}\in\mathfrak{p}$. Next we set $$\alpha_i(\mathfrak p) = \begin{cases} - &\quad \hbox{ if } m_i(\mathfrak p)<i , \\ + &\quad \hbox{ if } m_i(\mathfrak p)>i. \end{cases}$$ The map $\nu: \mathcal{M} \to \{ +,- \}^n$ defined by $\nu(\mathfrak p):=(\alpha_1(\mathfrak p), \ldots, \alpha_n(\mathfrak p))$, is a bijection (cf., e.g., [@PoThesis]). Consequently $\#\mathcal{M}=2^n$. Let $\mathbb{C}[\mathcal{M}]$ be the formal vector space over $\mathbb{C}$ with basis the two-boundary non-crossing perfect matchings $\mathfrak{p}\in\mathcal{M}$. We define now an action of the two-boundary Temperley-Lieb algebra $\textup{TL}(\underline{\delta})$ on $\mathbb{C}[\mathcal{M}]$, depending on two parameters $\beta_0,\beta_1\in\mathbb{C}^*$, in which the generators $e_j$ act by so-called matchmakers. The idea is as follows. If $1\leq i<n$ and $\mathfrak{p}\in\mathcal{M}$ then the action of $e_i$ on $\mathfrak{p}$ replaces all pairs containing $i$ and/or $i+1$ and adds pairs $\{i,i+1\}$ and $\{m_{i}(\mathfrak{p}),m_{i+1}(\mathfrak{p})\}$. When this does not produce a two-boundary non-crossing perfect matching, either because $m_i(\mathfrak{p})$ and $m_{i+1}(\mathfrak{p})$ are both boundary points or because $m_i(\mathfrak{p})=i+1$, then only the pair $\{i,i+1\}$ is matched and the omission of the second pair is accounted for by a suitable multiplicative constant (in this case, the deleted pair is either an arc from $j$ to $j$ for some $0\leq j\leq n+1$ or an arc from $0$ to $n+1$). Thus the action of $e_i$ on a two-boundary non-crossing perfect matching $\mathfrak{p}$ always has the effect that it matches up $i$ and $i+1$. A similar matchmaker interpretation is given for the boundary operators $e_0$ and $e_n$, which match up $0$ with $1$ and $n$ with $n+1$, respectively. We now give the precise formulation of the resulting matchmaker representation of $\textup{TL}(\underline{\delta})$. For $\mathfrak{p}\in\mathcal{P}(\mathcal{S})$ and $1\leq i<n$ write $\mathfrak{p}_i$ to be the element in $\mathcal{P}(\mathcal{S})$ obtained from $\mathfrak{p}$ by removing any pairs containing $i$ and/or $i+1$. Similarly, for $j=0,n$ we write $\mathfrak{p}_j$ to be the element in $\mathcal{P}(\mathcal{S})$ obtained from $\mathfrak{p}$ by removing any pair containing $1,n$, respectively. For $i \in {\mathbb{Z}}$ write $\text{pty}(i) = 0$ if $i$ is even and $\text{pty}(i)=1$ if $i$ is odd. Fix $\beta_0,\beta_1\in\mathbb{C}^*$. There exists a unique algebra homomorphism $$\omega = \omega^{\underline \delta}_{\beta_0,\beta_1}: \textup{TL}(\underline{\delta})\rightarrow\textup{End}_{\mathbb{C}}\bigl(\mathbb{C}[\mathcal{M}]\bigr)$$ such that $e_j\mathfrak{p}:=\omega(e_j)\mathfrak{p}$ for $0\leq j\leq n$ and $\mathfrak{p}\in\mathcal{M}$ is given by $$e_i \mathfrak p =\begin{cases} \delta \mathfrak p & \text{if } m_i(\mathfrak p)=i+1, \\ \delta_0^{\text{pty}(i-1)} (\mathfrak p_i \cup \{ i,i+1\}) & \text{if } m_i(\mathfrak p)=0=m_{i+1}(\mathfrak p), \\ \delta_n^{\text{pty}(n+1-i)} (\mathfrak p_i \cup \{ i,i+1 \}) & \text{if } m_i(\mathfrak p)=n+1=m_{i+1}(\mathfrak p), \\ \beta_{\text{pty}(i)} (\mathfrak p_i \cup \{ i,i+1 \}) & \text{if } m_i(\mathfrak p)=0, \, m_{i+1}(\mathfrak p)=n+1, \\ \mathfrak p_i \cup \{ i,i+1 \}\cup \{ m_i(\mathfrak p),m_{i+1}(\mathfrak p) \}, & \text{otherwise}, \end{cases}$$ for $1 \leq i < n$, and $$\begin{aligned} e_0 \mathfrak p &= \begin{cases} \delta_0 \mathfrak p,&\quad \text{if } m_1(\mathfrak p)=0, \\ \beta_0 ( \mathfrak p_0 \cup \{ 0,1\}) &\quad \text{if } m_1(\mathfrak p)=n+1, \\ \mathfrak p_0 \cup \{ 0,1 \}\cup\{ 0, m_\mathfrak p(1) \}, &\quad \text{otherwise}, \end{cases}\\ e_n \mathfrak p &= \begin{cases} \delta_n \mathfrak p &\quad \text{if } m_n(\mathfrak p)=n+1, \\ \beta_{\text{pty}(n)} ( \mathfrak p_n \cup \{ n,n+1 \}) &\quad \text{if } m_n(\mathfrak p)=0, \\ \mathfrak p_n \cup \{ n,n+1 \}\cup\{ m_n(\mathfrak p),n+1 \}, &\quad \text{otherwise}. \end{cases}\end{aligned}$$ It is sufficient to verify that the defining relations - of $\textup{TL}(\underline \delta)$ are satisfied by the matchmakers $\omega(e_j)$. This can be done by a straightforward case-by-case analysis, relying in part on the fact that if $m_i(\mathfrak p) = j$ for $1 \leq i,j \leq n$ and $\mathfrak{p}\in\mathcal{M}$, then $\text{pty}(i) \ne \text{pty}(j)$. An instructive example is the case $e_0 e_n \mathfrak p = e_n e_0 \mathfrak p$ for $\mathfrak p \in \mathcal{M}$ such that $\{1,n\} \in \mathfrak p$. Then $$\begin{gathered} e_0 e_n \mathfrak p = e_0 (\mathfrak p_n \cup \{ 1,n+1 \}\cup\{n,n+1\}) = \beta_0 (\mathfrak p_{0,n} \cup \{0,1\}\cup\{n,n+1\}) \\ e_n e_0 \mathfrak p = e_n (\mathfrak p_0 \cup \{ 0,1 \}\cup\{ 0,n\}) = \beta_{\text{pty}(n)} (\mathfrak p_{0,n} \cup \{0,1\}\cup\{n,n+1\}),\end{gathered}$$ where $\mathfrak p_{i,j}:=(\mathfrak p_i)_j = (\mathfrak p_j)_i$ for $0 \leq i \leq n$. Since $\{1,n\}\in\mathfrak{p}$ the parities of $1$ and $n$ must be different, hence $\text{pty}(n)=0$. Hence $e_0 e_n \mathfrak p = e_n e_0 \mathfrak p$, as requested. See, e.g., [@dG; @dGN] for a discussion of the diagrammatic representation of the action of Temperley-Lieb algebras on matchings. It is very helpful for direct computations. An example of such a diagrammatic computation for our representation $\omega^{\underline{\delta}}_{\beta_0,\beta_1}$ of $\textup{TL}(\underline{\delta})$ on $\mathbb{C}[\mathcal{M}]$ ($n=3$) is $$\begin{array}{ccccc} \begin{minipage}[c]{20mm} \begin{tikzpicture}[scale = 1/2] \draw[gray] (0,-4) -- (0,2); \draw[gray] (4,-4) -- (4,2); \draw[gray] (0,0) -- (4,0); \draw[gray] (0,-2) -- (4,-2); \draw[gray] (0,-4) -- (4,-4); \fill (1,0) circle (3pt); \fill (2,0) circle (3pt); \fill (3,0) circle (3pt); \fill (1,-2) circle (3pt); \fill (2,-2) circle (3pt); \fill (3,-2) circle (3pt); \fill (1,-4) circle (3pt); \fill (2,-4) circle (3pt); \fill (3,-4) circle (3pt); \draw (0,-.75) .. controls (1,-.75) .. (1,0) .. controls (1,1) .. (4,1); \draw (2,0) .. controls (2,1) and (3,1) .. (3,0) -- (3,-2) .. controls (3,-3) and (2,-3) .. (2,-2) -- (2,0); \draw (0,-1.25) .. controls (1,-1.25) .. (1,-2) -- (1,-4); \draw (2,-4) .. controls (2,-3) and (3,-3) .. (3,-4); \end{tikzpicture} \end{minipage} \vspace{1mm} &=& \beta_0 \, \begin{minipage}[c]{20mm} \begin{tikzpicture}[scale = 1/2] \draw[gray] (0,-2) -- (0,2); \draw[gray] (4,-2) -- (4,2); \draw[gray] (0,-2) -- (4,-2); \fill (1,-2) circle (3pt); \fill (2,-2) circle (3pt); \fill (3,-2) circle (3pt); \draw (2,0) .. controls (2,1) and (3,1) .. (3,0) .. controls (3,-1) and (2,-1) .. (2,0); \draw (0,.75) .. controls (1,.75) .. (1,0) -- (1,-2); \draw (2,-2) .. controls (2,-1) and (3,-1) .. (3,-2); \end{tikzpicture} \end{minipage} &=& \beta_0 \delta \, \begin{minipage}[c]{20mm} \begin{tikzpicture}[scale = 1/2] \draw[gray] (0,0) -- (0,2); \draw[gray] (4,0) -- (4,2); \draw[gray] (0,0) -- (4,0); \fill (1,0) circle (3pt); \fill (2,0) circle (3pt); \fill (3,0) circle (3pt); \draw (0,.75) .. controls (1,.75) .. (1,0) ; \draw (2,0) .. controls (2,1) and (3,1) .. (3,0); \end{tikzpicture} \end{minipage} \\ e_2 e_0 \bigl( (+,+,-) \bigr)&=& \beta_0 e_2 \bigl( (-,+,-) \bigr) &=& \beta_0 \delta (-,+,-) \end{array}$$ where the left and right vertical line should be thought of as the boundary point $0$ and $n+1$, respectively, and where we have represented $\mathfrak{p}\in\mathcal{M}$ by its image under the bijection $\nu:\mathcal{M}\overset{\sim}{\longrightarrow} \{+,-\}^3$. Note that the spin representation $\hat\rho^{\underline \kappa}_{\psi_0,\psi_n}$ is isomorphic to $\hat\rho^{\underline\kappa}_{\psi_0^\prime,\psi_n^\prime}$ if $\psi_0\psi_n=\psi_0^\prime\psi_n^\prime$ in view of Proposition \[rhopiequivalence\]. We now show that the $2^n$-dimensional $\textup{TL}(\underline{\delta})$-representations $\hat\rho^{\underline{\kappa}}_{\psi_0,\psi_n}$ and $\omega^{\underline{\kappa}}_{\beta_0,\beta_1}$ are isomorphic for generic parameters if $$\label{parameterproduct} \beta_0 \beta_1 = \psi_0^{-1} \psi_n^{-1} \cdot \begin{cases} \frac{(1+\kappa_0 \kappa_n^{-1} \psi_0 \psi_n)(1+\kappa_0^{-1} \kappa_n \psi_0 \psi_n)}{(\kappa \kappa_0^{-1} + \kappa^{-1} \kappa_0) (\kappa \kappa_n^{-1} + \kappa^{-1} \kappa_n)} &\quad \hbox{ if } n \text{ odd} \\ \frac{(1-\kappa^{-1} \kappa_0 \kappa_n \psi_0 \psi_n)(1-\kappa \kappa_0^{-1} \kappa_n^{-1} \psi_0 \psi_n)}{(\kappa \kappa_0^{-1} + \kappa^{-1} \kappa_0) (\kappa \kappa_n^{-1} + \kappa^{-1} \kappa_n)} &\quad \hbox{ if } n \text{ even} \end{cases}$$ by writing down an explicit intertwiner (for a different approach, see [@dGN]). To define the intertwiner we write for $h \in \{0,1\}$ and $\mathfrak p \in \mathcal{M}$, $$\begin{gathered} L_{0,h}(\mathfrak p) := \# \left\{ \{ i,0\} \in \mathfrak p \, \mid \, 1 \leq i \leq n \text{ and } \text{pty}(i)=h \right\}, \\ L_{n,h}(\mathfrak p) := \# \left\{ \{ i,n+1\} \in \mathfrak p \, \mid \, 1 \leq i \leq n \text{ and } \text{pty}(i)=h \right\}. \end{gathered}$$ A straightforward induction argument on $n$ shows that $$\label{Lsum}\sum_{j \in \{0,n\}} \sum_{h \in \{0,1\}} (-1)^h L_{j,h}(\mathfrak p) = -\text{pty}(n)$$ for all $\mathfrak p \in \mathcal{M}$. Define $$M(\mathfrak p) := \prod_{j \in \{0,n\} } \prod_{ h \in \{0,1\}} M_{j,h}^{L_{j,h}(\mathfrak p)},$$ with $M_{j,h}\in\mathbb{C}^*$ satisfying $$\label{Mrelations} \begin{gathered} M_{j,0} M_{j,1} = \psi_j^{-1} (\kappa \kappa_j^{-1} + \kappa^{-1} \kappa_j)^{-1}, \qquad j\in\{0,n\}, \\ M_{0,0} M_{n,1} = \beta_0 \cdot \begin{cases} (1+\kappa_0 \kappa_n^{-1} \psi_0 \psi_n)^{-1} &\quad \hbox{ if } n \textup{ odd} \\ (1-\kappa^{-1} \kappa_0 \kappa_n \psi_0 \psi_n)^{-1} &\quad \hbox{ if } n \textup{ even}. \end{cases} \end{gathered}$$ By the conditions fix $M(\mathfrak p)$ up to a multiplicative constant. Next, define an *oriented two-boundary non-crossing perfect matching* as a two-boundary non-crossing perfect matching $\mathfrak{p}$ with a chosen ordering of each pair in $\mathfrak{p}$. We write ordered pairs as $(i,j)$ and say that the pair (or, with the diagrammatic realization in mind, the associated connecting arc) is oriented from $i$ to $j$. We write $\vec{\mathcal{M}}$ for the set of oriented two-boundary non-crossing perfect matchings. Let $\text{Forg}: \vec{\mathcal{M}}\twoheadrightarrow\mathcal{M}$ be the canonical surjective map which forgets the orientation. For $\vec{\mathfrak{p}}\in\vec{\mathcal{M}}$ we write $\mathfrak{p}:=\textup{Forg}(\vec{\mathfrak{p}})$. Given $\vec{\mathfrak{p}} \in \overrightarrow{\mathcal{M}}$ and $i=1,\ldots,n$ we set $r_i(\vec{\mathfrak{p}})$ equal to $+$ or $-$ if the oriented arc connecting $i$ and $m_i(\mathfrak{p})$ is oriented outwards or inwards at $i$, respectively. In other words, $r_i(\vec{\mathfrak{p}})=+$ if $(i,m_i(\mathfrak{p}))\in\vec{\mathfrak{p}}$ and $r_i(\mathfrak{p})=-$ if $(m_i(\mathfrak{p}),i) \in\vec{\mathfrak{p}}$. We define a mapping $v: \overrightarrow{\mathcal{M}} \hookrightarrow ({\mathbb{C}}^2)^{\otimes n}$ by $$v(\vec{\mathfrak p}) = v_{r_1(\vec{\mathfrak{p}})} \otimes \cdots \otimes v_{r_n(\vec{\mathfrak{p}})}.$$ As an example with $n=4$ we have $v(\{ (0,1), (3,2), (5,4) \}) = v_- \otimes v_- \otimes v_+ \otimes v_-$. For $h \in \{0,1\}$ and $\vec{\mathfrak p} \in \overrightarrow{\mathcal{M}}$ we set $$\begin{gathered} N_{0,h}(\vec{\mathfrak p}) := \# \left\{ (i,0) \in \vec{\mathfrak p} \, \mid \, 1 \leq i \leq n \text{ and } \text{pty}(i)=h \right\}, \\ N_{n,h}(\vec{\mathfrak p}) := \# \left\{ (n+1,i) \in \vec{\mathfrak p} \, \mid \, 1 \leq i \leq n \text{ and } \text{pty}(n+1-i)=h \right\}.\end{gathered}$$ Finally, for $\vec{\mathfrak p} \in \overrightarrow{\mathcal{M}}$ define $$\text{or}(\vec{\mathfrak p}) = \# \left\{ (j,i) \in \vec{\mathfrak p} \, \mid \, 1 \leq i<j \leq n \right\} + N_{0,0}(\vec{\mathfrak p}) + N_{n,0}(\vec{\mathfrak p}).$$ \[isomorphicrepresentations\] Suppose that the parameters $(\underline{\delta},\beta_0,\beta_1)$ are related to $(\underline{\kappa},\psi_0,\psi_n)$ by and . For generic parameters the linear map $\Psi:=\Psi^{\underline{\kappa}}_{\psi_0,\psi_n}: \mathbb{C}[\mathcal{M}]\rightarrow\bigl(\mathbb{C}^2\bigr)^{\otimes n}$, defined by $$\Psi(\mathfrak{p}):=M(\mathfrak{p})\sum_{\vec{\mathfrak p} \in \textup{Forg}^{-1}(\mathfrak p)} (-\kappa)^{-\text{or}(\vec{\mathfrak p})} \biggl( \prod_{j \in \{0,n\}} (-\kappa_j)^{N_{j,0}(\vec{\mathfrak p})-N_{j,1}(\vec{\mathfrak p})} \psi_j^{N_{j,0}(\vec{\mathfrak p})+N_{j,1}(\vec{\mathfrak p})} \biggr) v(\vec{\mathfrak p})$$ for $\mathfrak{p}\in\mathcal{M}$, defines an isomorphism of $\textup{TL}(\underline{\delta})$-modules with respect to the $\textup{TL}(\underline{\delta})$-actions $\omega^{\underline{\delta}}_{\beta_0,\beta_1}$ on $\mathbb{C}[\mathcal{M}]$ and $\hat{\rho}^{\underline{\kappa}}_{\psi_0,\psi_n}$ on $\bigl(\mathbb{C}^2\bigr)^{\otimes n}$. The intertwining property $\Psi(\omega^{\underline{\delta}}_{\beta_0,\beta_1}(e_j)\mathfrak{p})= \hat{\rho}^{\underline{\kappa}}_{\psi_0,\psi_n}(e_j)(\mathfrak{p})$ for $0\leq j\leq n$ and $\mathfrak{p}\in\mathcal{M}$ follows by a careful case-by-case check. To show that $\Psi$ is an isomorphism for generic parameters, it suffices to show the modified linear map $\overline{\Psi}: \mathbb{C}[\mathcal{M}]\rightarrow\bigl(\mathbb{C}^2\bigr)^{\otimes n}$, defined by $$\overline{\Psi}(\mathfrak{p}):=\sum_{\vec{\mathfrak p} \in \textup{Forg}^{-1}(\mathfrak p)} (-\kappa)^{-\text{or}(\vec{\mathfrak p})} \biggl( \prod_{j \in \{0,n\}} (-\kappa_j)^{N_{j,0}(\vec{\mathfrak p})-N_{j,1}(\vec{\mathfrak p})} \psi_j^{N_{j,0}(\vec{\mathfrak p})+N_{j,1}(\vec{\mathfrak p})} \biggr) v(\vec{\mathfrak p})$$ for $\mathfrak{p}\in\mathcal{M}$, is generically a linear isomorphism. Since $\overline{\Psi}$ depends polynomially on the parameters $\kappa^{-1},\psi_0,\psi_n$, it suffices to show that it is a linear isomorphism when $\psi_0=\psi_n=\kappa^{-1}=0$. Then $$\overline{\Psi}_{\psi_0=\psi_n=\kappa^{-1}=0}(\mathfrak{p})=v_{\alpha_1(\mathfrak{p})}\otimes v_{\alpha_2(\mathfrak{p})}\otimes\cdots\otimes v_{\alpha_n(\mathfrak{p})}$$ for $\mathfrak{p}\in\mathcal{M}$, which defines a linear isomorphism $\overline{\Psi}: \mathbb{C}[\mathcal{M}]\overset{\sim}{\longrightarrow} \bigl(\mathbb{C}^2\bigr)^{\otimes n}$ since the map $\nu(\mathfrak{p}):=(\alpha_1(\mathfrak{p}),\ldots,\alpha_n(\mathfrak{p}))$ is a bijection $\nu: \mathcal{M}\overset{\sim}{\longrightarrow}\{+,-\}^n$. Integrable models {#integrablesection} ================= Baxterization {#Baxt} ------------- In [@Jo §5] Jones analysed when braid group representations can be “Baxterized”. Baxterization means that the action of the $\sigma_i$ ($1\leq i<n$) extend to $R$-matrices with spectral parameter, the essential building blocks for integrable lattice models of vertex type. The result [@dGN Prop 2.18] solves the Baxterization problem for the spin representations of two-boundary Temperley-Lieb algebras. However its proof, which is by direct computations, does not give insight in the Baxterization procedure. Cherednik’s theory on double affine Hecke algebras gives a natural Baxterization procedure for arbitrary representations of affine Hecke algebras. We explain this here briefly for the affine Hecke algebra $H(\underline{\kappa})$ of type $\widetilde{C}_n$. It reproduces for the spin representations the earlier mentioned result [@dGN Prop. 2.18] of de Gier and Nichols. \[Baxterization\] Let $\pi: H(\underline{\kappa})\rightarrow{\textup{End}}_{\mathbb{C}}(V)$ be a representation of $H(\underline{\kappa})$ and set $$\begin{split} K_0^V(x;\underline{\kappa};\upsilon_0,\upsilon_n)&:=\frac{\pi(T_0^{-1})+(\upsilon_0^{-1}-\upsilon_0)x-x^2\pi(T_0)}{\kappa_0^{-1}(1-\kappa_0\upsilon_0x)(1+\kappa_0\upsilon_0^{-1}x)},\\ R_i^V(x;\underline{\kappa};\upsilon_0,\upsilon_n)&:=\frac{\pi(T_i^{-1})-x\pi(T_i)}{\kappa^{-1}(1-\kappa^2x)},\qquad 1\leq i<n,\\ K_n^V(x;\underline{\kappa};\upsilon_0,\upsilon_n)&:=\frac{\pi(T_n^{-1})+(\upsilon_n^{-1}-\upsilon_n)x-x^2\pi(T_n)}{\kappa_n^{-1}(1-\kappa_n\upsilon_nx) (1+\kappa_n\upsilon_n^{-1}x)} \end{split}$$ as elements in $\mathbb{C}(x)\otimes_{\mathbb{C}}{\textup{End}}_{\mathbb{C}}(V)$. Then we have, as endomorphisms of $V$ with rational dependence on the spectral parameters, $$\begin{split} K_0^V(x)R_1^V(xy)K_0^V(y)R_1^V(y/x)&=R_1^V(y/x)K_0^V(y)R_1^V(xy)K_0^V(x),\\ R_i^V(x)R_{i+1}^V(xy)R_i^V(y)&=R_{i+1}^V(y)R_i^V(xy)R_{i+1}^V(x),\qquad 1\leq i <n-1,\\ K_n^V(y)R_{n-1}^V(xy)K_n^V(x)R_{n-1}^V(x/y)&=R_{n-1}^V(x/y)K_n^V(x)R_{n-1}^V(xy)K_n^V(y), \end{split}$$ and $$\begin{split} K_0^V(x)K_0^V(x^{-1})&={\textup{Id}}_V=K_n^V(x)K_n^V(x^{-1}),\\ R_i^V(x)R_i^V(x^{-1})&={\textup{Id}}_V,\qquad 1\leq i<n, \end{split}$$ as well as $$\begin{split} \lbrack K_0^V(x),R_i^V(y)\rbrack&=0,\qquad 2\leq i<n,\\ \lbrack K_n^V(x),R_i^V(y)\rbrack&=0,\qquad 1\leq i<n-1,\\ \lbrack R_i^V(x),R_j^V(y)\rbrack&=0,\qquad |i-j|\geq 2,\\ \lbrack K_0^V(x),K_n^V(y)\rbrack&=0, \end{split}$$ where we have suppressed the dependence on $\underline{\kappa}$, $\upsilon_0$ and $\upsilon_n$. The original action of the affine Hecke algebra is recovered by specializing the spectral parameter to $0$, $$K_0^V(0)=\kappa_0\pi(T_0^{-1}),\qquad R_i^V(0)=\kappa\pi(T_i^{-1}),\qquad K_n^V(0)=\kappa_n\pi(T_n^{-1})$$ for $1\leq i<n$. When specializing the spectral parameter to $1$ we recover the identity: $$\begin{split} K_0^V(1) &= {\textup{Id}}_V = K_n^V(1), \\ R_i^V(1) &= {\textup{Id}}_V, \qquad 1 \leq i < n. \end{split}$$ We suppress the dependence on $V$ in the notations if $V$ is the spin representation $\bigl(\rho^{\underline{\kappa}}_{\psi_0,\psi_n},\bigl(\mathbb{C}^2\bigr)^{\otimes n}\bigr)$. Proposition \[Baxterization\] extends results of Cherednik (see, e.g., [@CBook §1.3.2] and references therein) to the “Koornwinder case” (also known as the $C^\vee C$-case). Before sketching the proof of Proposition \[Baxterization\], we first show that it produces Baxterizations of the solution $\Upsilon$ of the quantum Yang-Baxter equation and of the solutions $\overline{K}$ and $K$ of the associated reflection equations when it is applied to the spin representations of the affine Hecke algebra $H(\underline{\kappa})$ (see Subsection \[Spinrep\] for the definitions of $\Upsilon,\overline{K}$ and $K$). Recall that $P:\mathbb{C}^2\otimes\mathbb{C}^2 \rightarrow\mathbb{C}^2\otimes\mathbb{C}^2$ is the flip operator. \[rk\] For parameters $\underline{\kappa}$, $\upsilon_0,\upsilon_n, \psi_0$ and $\psi_n$ consider the matrices $$\begin{aligned} \bar k(x) &:= \frac{\kappa_0}{(1-\kappa_0 \upsilon_0 x)(1+ \kappa_0 \upsilon_0^{-1} x)} \left( \begin{smallmatrix} (\kappa_0^{-1}-\kappa_0) x^2 + (\upsilon_0^{-1} - \upsilon_0) x & \psi_0 (1-x^2) \\ \psi_0^{-1}(1-x^2) & \kappa_0^{-1} -\kappa_0 + (\upsilon_0^{-1} - \upsilon_0) x \end{smallmatrix} \right), \\ r(x) &:= \frac{1}{1-\kappa^2 x} \left( \begin{smallmatrix} 1 - \kappa^2 x & 0 & 0 & 0 \\ 0 & \kappa(1-x) & 1-\kappa^2 & 0 \\ 0 & (1-\kappa^2)x & \kappa (1-x) & 0 \\ 0 & 0 & 0 & 1 - \kappa^2 x \end{smallmatrix} \right), \\ k(x) &:= \frac{\kappa_n}{(1-\kappa_n \upsilon_n x)(1+ \kappa_n \upsilon_n^{-1} x)} \left( \begin{smallmatrix} \kappa_n^{-1}-\kappa_n + (\upsilon_n^{-1} - \upsilon_n) x & \psi_n^{-1} (1-x^2) \\ \psi_n(1-x^2) & (\kappa_n^{-1}-\kappa_n)x^2 + (\upsilon_n^{-1} - \upsilon_n) x \end{smallmatrix} \right). \end{aligned}$$ Then 1. $r(x)$ satisfies the quantum Yang-Baxter equation with spectral parameter $$\label{YBEnew} r_{12}(x) r_{13}(xy) r_{23}(y) = r_{23}(y) r_{13}(xy) r_{12}(x)$$ as endomorphisms of $\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2$. Furthermore, $r(0)=\kappa\Upsilon^{-1}$ (Baxterization), $r(x)r_{21}(x^{-1})=\textup{Id}_{\mathbb{C}^2\otimes\mathbb{C}^2}$ (unitarity) and $r(1)=P$ (regularity). 2. $\overline{k}(x)$ satisfies the reflection equation with spectral parameter $$\label{REleftnew} r_{12}(x/y) \bar k_2(x) r_{21}(xy) \bar k_1(y) = \bar k_1(y) r_{12}(xy) \bar k_2(x) r_{21}(x/y)$$ as endomorphisms of $\mathbb{C}^2\otimes\mathbb{C}^2$. Furthermore, $\overline{k}(0)=\kappa_0\overline{K}^{-1}$ (Baxterization), $\overline{k}(x)\overline{k}(x^{-1})=\textup{Id}_{\mathbb{C}^2}$ (unitarity) and $\overline{k}(1)=\textup{Id}_{\mathbb{C}^2}=\overline{k}(-1)$ (regularity). 3. $k(x)$ satisfies the reflection equation with spectral parameter $$\label{RErightnew} r_{12}(x/y) k_1(x) r_{21}(xy) k_2(y) = k_2(y) r_{12}(xy) k_1(x) r_{21}(x/y)$$ as endomorphisms of $\mathbb{C}^2\otimes\mathbb{C}^2$. Furthermore, $k(0)=\kappa_nK^{-1}$ (Baxterization), $k(x)k(x^{-1})=\textup{Id}_{\mathbb{C}^2}$ (unitarity) and $k(1)=\textup{Id}_{\mathbb{C}^2}=k(-1)$ (regularity). Apply Proposition \[Baxterization\] to the $H(\underline{\kappa})$-module $\bigl(\rho^{\underline{\kappa}}_{\psi_0,\psi_n},\bigl(\mathbb{C}^2\bigr)^{\otimes n}\bigr)$ and use that the associated linear operators $K_0(x), R_i(x) $ and $K_n(x)$ on $\bigl(\mathbb{C}^2\bigr)^{\otimes n}$ are given by $\overline{k}_1(x), (r(x)\circ P)_{i\,i+1}$ and $k_n(x)$, respectively ($1\leq i<n$). The solution $r(x)$ of the quantum Yang-Baxter equation is gauge-equivalent to the $R$-matrix of the XXZ spin-$\frac{1}{2}$ chain. Corollary \[rk\] thus gives rise to a three-parameter family of solutions of the associated reflection equation (with the free parameters being $\kappa_0,\upsilon_0,\psi_0$). This three-parameter family of solutions of the reflection equation was written down before in [@dVGR] (also see [@Ne]). The proof of Proposition \[Baxterization\] uses Noumi-Sahi’s [@N; @Sa] extension of Cherednik’s [@CBook] double affine Hecke algebra and its basic representation, notions that also play a central role in the theory on Koornwinder polynomials; see [@N; @Sa] and Subsection \[Kosection\]. We sketch now some of the key steps of the proof of Proposition \[Baxterization\]. Fix $q\in\mathbb{C}^*$ and a square root $q^{\frac{1}{2}}$ once and for all. A $q$-dependent action of the affine Weyl group $W=W_0\ltimes\mathbb{Z}^n$ on $T=\bigl(\mathbb{C}^*\bigr)^n$ is given by $$\begin{split} s_0\bm{t}&:=(qt_1^{-1},t_2,\ldots,t_n),\\ s_i\bm{t}&:=(t_1,\ldots,t_{i-1},t_{i+1},t_i,t_{i+2},\ldots,t_n),\qquad 1\leq i<n,\\ s_n\bm{t}&:=(t_1,\ldots,t_{n-1},t_n^{-1}). \end{split}$$ Note that $\tau( \mu)\bm{t}=q^{ \mu}\bm{t}:=(q^{\mu_1}t_1,\ldots,q^{\mu_n}t_n)$ for $ \mu = (\mu_1,\ldots,\mu_n) \in \mathbb{Z}^n$. By transposition, $W$ acts on the fields $\mathbb{C}(T)$ and $\mathcal{M}(T)$ of rational and meromorphic functions on $T$ by field automorphisms. The following lemma is equivalent to Proposition \[Baxterization\] because of the Coxeter presentation of $W$ with respect to the simple reflections $s_0,\ldots,s_n$. \[reform\] Let $\pi: H(\underline{\kappa})\rightarrow{\textup{End}}_{\mathbb{C}}(V)$ be a representation. There exist unique $C_w^V=C_w^V(\cdot;\underline{\kappa};\upsilon_0,\upsilon_n)\in\mathbb{C}(T)\otimes{\textup{End}}_{\mathbb{C}}(V)$ ($w\in W$) satisfying the cocycle conditions $$\begin{aligned} C_e^V(\bm{t})&={\textup{Id}}_V,\\ \label{cocycle} C_{ww^\prime}^V(\bm{t})&=C_w^V(\bm{t})C_{w^\prime}^V(w^{-1}\bm{t}),\qquad \forall\, w,w^\prime\in W\end{aligned}$$ (with $e\in W$ the unit element) and satisfying $$C_{s_0}^V(\bm{t})=K_0^V(q^{\frac{1}{2}}/t_1),\qquad C_{s_i}^V(\bm{t})=R_i^V(t_i/t_{i+1}),\qquad C_{s_n}^V(\bm{t})=K_n^V(t_n)$$ for $1\leq i<n$. Define the algebra $\mathbb{C}(T)\#W$ of $q$-difference reflection operators with rational coefficients as the vector space $\mathbb{C}(T)\otimes \mathbb{C}[W]$ with the multiplication rule $$(p\otimes w)(r\otimes w^\prime):=p(w\cdot r)\otimes ww^\prime,\qquad p,r\in\mathbb{C}(T),\,\,\, w,w^\prime\in W,$$ where $w\cdot r$ is the result of the $q$-dependent action of $w\in W$ on $r\in \mathbb{C}(T)$. We simply write $pw$ for $p\otimes w\in\mathbb{C}(T)\#W$. The key of the proof of Lemma \[reform\] is to write the simple reflections $s_i \in\mathbb{C}(T)\otimes\mathbb{C}[W]$ ($0\leq i\leq n$) in terms of Noumi’s [@N] two-parameter family of realizations of the affine Hecke algebra $H(\underline{\kappa})$ as a subalgebra of $\mathbb{C}(T)\#W$, which is given as follows. Define $c_i=c_i(\cdot;\underline{\kappa};\upsilon_0,\upsilon_n)\in\mathbb{C}(T)$ for $0\leq i\leq n$ by the explicit formulas $$\begin{split} c_0(\bm{t})&:=\kappa_0^{-1}\frac{(1-q^{\frac{1}{2}}\kappa_0\upsilon_0t_1^{-1})(1+q^{\frac{1}{2}}\kappa_0\upsilon_0^{-1}t_1^{-1})} {(1-qt_1^{-2})},\\ c_i(\bm{t})&:=\kappa^{-1}\frac{(1-\kappa^2t_i/t_{i+1})}{(1-t_i/t_{i+1})},\\ c_n(\bm{t})&:=\kappa_n^{-1}\frac{(1-\kappa_n\upsilon_nt_n)(1+\kappa_n\upsilon_n^{-1}t_n)}{(1-t_n^2)} \end{split}$$ for $1\leq i<n$. \[thmN\] Let $\upsilon_0,\upsilon_n\in\mathbb{C}^*$. There exists a unique unital algebra embedding $$\iota=\iota_{\upsilon_0,\upsilon_n}^{\underline{\kappa}}:H(\underline{\kappa})\hookrightarrow \mathbb{C}(T)\#W$$ satisfying $$\iota(T_i)=\kappa_i+c_i(s_i-e),\qquad 0\leq i\leq n.$$ The Noumi-Sahi extension of Cherednik’s double affine Hecke algebra is the subalgebra of $\mathbb{C}(T)\#W$ generated by $\iota_{\upsilon_0,\upsilon_n}^{\underline{\kappa}}(H(\underline{\kappa}))$ and by the algebra $\mathbb{C}[T]$ of regular functions on $T$. The following consequence should be compared to [@St1 Prop. 3.5]. \[corCrit\] Let $\pi: H(\underline{\kappa})\rightarrow {\textup{End}}_{\mathbb{C}}(V)$ be a representation. There exists a unique algebra homomorphism $$\nabla^V: \mathbb{C}(T)\#W\rightarrow \mathbb{C}(T)\#W\otimes{\textup{End}}_{\mathbb{C}}(V)$$ satisfying $\nabla^V(p)=p\otimes{\textup{Id}}_V$ for $p\in\mathbb{C}(T)$ and $$\nabla^V(\iota(T_i))=s_i\otimes \pi(T_i)+(c_i-\kappa_i)(s_i-e)\otimes {\textup{Id}}_V,\qquad 0\leq i\leq n.$$ Consider the induced $\mathbb{C}(T)\#W$-module $$\textup{Ind}_{H(\underline{\kappa})}^{\mathbb{C}(T)\#W}\bigl(V\bigr)=\mathbb{C}(T)\#W\otimes_{H(\underline{\kappa})}V,$$ where we identify $H(\underline{\kappa})$ with the subalgebra $\iota(H(\underline{\kappa}))$ of $\mathbb{C}(T)\#W$. Using the linear isomorphism $$\mathbb{C}(T)\otimes \iota(H(\underline{\kappa}))\overset{\sim}{\longrightarrow}\mathbb{C}(T)\#W$$ defined by the multiplication map, the representation map $\nabla$ of $\textup{Ind}_{H(\underline{\kappa})}^{\mathbb{C}(T)\#W}\bigl(V\bigr)$ becomes an algebra map $$\nabla: \mathbb{C}(T)\#W\rightarrow {\textup{End}}_{\mathbb{C}}(\mathbb{C}(T)\otimes_{\mathbb{C}}V).$$ View $\mathbb{C}(T)\#W\otimes_{\mathbb{C}}\pi(H(\underline{\kappa}))$ as a subalgebra of ${\textup{End}}_{\mathbb{C}}(\mathbb{C}(T)\otimes_{\mathbb{C}}V)$, with $\mathbb{C}(T)\#W$ acting on $\mathbb{C}(T)$ as $q$-difference reflection operators. It then suffices to show that $\nabla(p)=\nabla^V(p)$ and $\nabla(T_i)=\nabla^V(T_i)$ for $p\in\mathbb{C}(T)$ and $0\leq i\leq n$. This follows by a direct computation using the commutation relations $$T_ip=(s_i\cdot p)T_i+(\kappa_i-c_i)(p-s_i\cdot p),\qquad 0\leq i\leq n,\,\,\, p\in\mathbb{C}(T)$$ in $\mathbb{C}(T)\#W$. Observe that $$\begin{split} \nabla^V(s_i)&=\nabla^V(c_i^{-1}(T_i-\kappa_i+c_i))\\ &=c_i^{-1}s_i\otimes\pi(T_i)+c_i^{-1}(c_i-\kappa_i)(s_i-e)\otimes\textup{Id}_V+c_i^{-1}(c_i-\kappa_i)e\otimes\textup{Id}_V\\ &=(c_i^{-1}e\otimes\pi(T_i)+c_i^{-1}(c_i-\kappa_i)e\otimes\textup{Id}_V)(s_i\otimes\textup{Id}_V)\\ &=C_{s_i}^V(\cdot)(s_i\otimes{\textup{Id}}_V) \end{split}$$ for $i=0,\ldots,n$, where the last equality follows from the fact that $$\begin{split} \frac{\pi(T_0)+c_0(\bm{t})-\kappa_0}{c_0(\bm{t})}&=K_0^V(q^{\frac{1}{2}}t_1^{-1})=C_{s_0}^V(\bm{t}),\\ \frac{\pi(T_i)+c_i(\bm{t})-\kappa}{c_i(\bm{t})}&=R_i^V(t_i/t_{i+1})=C_{s_i}^V(\bm{t}),\qquad 1\leq i<n,\\ \frac{\pi(T_n)+c_n(\bm{t})-\kappa_n}{c_n(\bm{t})}&=K_n^V(t_n)=C_{s_n}^V(\bm{t}), \end{split}$$ which in turn can be checked by a direct computation. Combined with Corollary \[corCrit\] this proves Lemma \[reform\]. Finally note that $$\nabla^V(w)=C_w^V(\cdot)(w\otimes{\textup{Id}}_V),\qquad w\in W.$$ The reflection quantum KZ equations {#ReflqKZeqns} ----------------------------------- Let $\mathcal{M}(T)$ be the field of meromorphic functions on $T=\bigl(\mathbb{C}^*\bigr)^n$. Let $V$ be a $H(\underline{\kappa})$-module and $\upsilon_0,\upsilon_n\in\mathbb{C}^*$. We say that $f\in\mathcal{M}(T)\otimes V$ is a solution of the associated reflection quantum Knizhnik-Zamolodchikov (KZ) equations if $$\nabla^V(\tau(\lambda))f=f\qquad \forall\,\, \lambda\in\mathbb{Z}^n.$$ We write $\textup{Sol}_{KZ}(V)=\textup{Sol}_{KZ}(V;\underline{\kappa};\upsilon_0,\upsilon_n)\subseteq \mathcal{M}(T)\otimes V$ for the associated space of solutions of the reflection quantum KZ equations. Note that for $\lambda\in\mathbb{Z}^n$, $$\bigl(\nabla^V(\tau(\lambda))f\bigr)(\bm{t})=C_{\tau(\lambda)}(\bm{t})f(q^{-\lambda}\bm{t})$$ with transport operators $C_{\tau(\lambda)}\in \mathbb{C}(T)\otimes\textup{End}_{\mathbb{C}}(V)$. Consequently the reflection quantum KZ equations form a consistent system of first order, linear $q$-difference equations. Note also that $\textup{Sol}_{KZ}(V)$ is a left $W_0$-module with the action given by $$\bigl(w\cdot f\bigr)(\bm{t}):=\bigl(\nabla^V(w)f\bigr)(\bm{t})=C_w(\bm{t})f(w^{-1}\bm{t}),\qquad w\in W_0.$$ The reflection quantum KZ equations are equivalent to $$C_{\tau_i}^V(\bm{t})f(q^{-\epsilon_i}\bm{t})=f(\bm{t}) \qquad \forall\, i=1,\ldots,n,$$ where $\{\epsilon_i\}_{i=1}^n$ is the standard orthonormal basis of $\mathbb{R}^n$. Using the expression of $\tau_i\in W$ as product of simple reflections and the cocycle property , $C_{\tau_i}^V$ takes on the explicit form $$\label{explicitC} \begin{split} C_{\tau_i}^V&(\bm{t})=R_{i-1}^V(t_{i-1}/t_i)R_{i-2}^V(t_{i-2}/t_i)\cdots R_1^V(t_1/t_i)K_0^V(q^{\frac{1}{2}}/t_i)\\ &\times R_1^V(q/t_1t_i)R_2^V(q/t_2t_i)\cdots R_{i-1}^V(q/t_{i-1}t_i)R_i^V(q/t_it_{i+1})\cdots R_{n-1}^V(q/t_it_n)\\ &\times K_n^V(q/t_i)R_{n-1}^V(qt_n/t_i)\cdots R_i^V(qt_{i+1}/t_i). \end{split}$$ [**(i)**]{} In Section \[Solsection\] we construct for principal series representations solutions of the associated reflection quantum KZ equations in terms of nonsymmetric Koornwinder polynomials. As a special case this construction gives rise to solutions for the spin representations $\bigl(\rho^{\underline{\kappa}}_{\psi_0,\psi_n},\bigl(\mathbb{C}^2\bigr)^{\otimes n}\bigr)$ of $H(\underline{\kappa})$ in view of Proposition \[rhopiequivalence\]. Alternatively, for special classes of parameters solutions of the reflection quantum KZ equations for spin representations can be constructed using a vertex operator approach (see [@JKKKM; @JKKMW; @We] for such a treatment involving diagonal $K$-matrices, and [@BKojima] for a recent extension to triangular $K$-matrices) or by a generalized Bethe ansatz method [@RSV] (for diagonal $K$-matrices).\ [**(ii)**]{} Write $\textup{Sol}_{KZ}^{\underline{\kappa}}(V)^{W_0}$ for the subspace of $W_0$-invariant solutions of the reflection quantum KZ equations. Then $f\in\textup{Sol}_{KZ}^{\underline{\kappa}}(V)^{W_0}$ if and only if $$\begin{split} K_0^V(q^{\frac{1}{2}}/t_1)f(s_0\bm{t})&=f(\bm{t}),\\ R_i^V(t_i/t_{i+1})f(s_i\bm{t})&=f(\bm{t}),\qquad 1\leq i<n,\\ K_n^V(t_n)f(s_n\bm{t})&=f(\bm{t}). \end{split}$$ It is in this form that the reflection quantum KZ equations often appear in the literature on spin chains, see, e.g., [@dFZJ; @dGPS; @dGP2; @PoThesis; @ZJ2]. The XXZ spin chain {#spsection} ------------------ One of the main aims in the study of spin chain models is to find the spectrum of the quantum Hamiltonian, a distinguished operator acting on the state space in which the combined states of $n$ individual spins reside, and to find a complete set of eigenfunctions. The method of commuting transfer operators, pioneered by Baxter (see [@Ba] and references therein) and elaborated upon principally by the Faddeev school (e.g. [@Fa; @Sk1982; @KBI]), produces a generating function of quantum Hamiltonians from the basic data of the models ($R$- and $K$-matrices). In many cases the basic data arise from the representation theory of quantum groups or Hecke algebras. We consider now briefly the construction of the quantum integrable model associated to the basic data $r(x)$, $k(x)$ and $\overline{k}(x)$, see Corollary \[rk\] (in this case the basic data thus are obtained by a Baxterization procedure from the spin representation $\hat{\rho}^{\underline{\kappa}}_{\psi_0,\psi_n}$). The associated state space is the representation space $({\mathbb{C}}^2)^{\otimes n}$ of the spin representation. To write down the associated transfer operators we first consider a larger space, which for the pertinent case is $\underset{(0)}{{\mathbb{C}}^2} \otimes ({\mathbb{C}}^2)^{\otimes n} $, where the additional copy of ${\mathbb{C}}^2$ numbered 0 is referred to as the auxiliary space. Given a spectral parameter $x \in {\mathbb{C}}^*$ and an $n$-tuple of so-called inhomogeneities $\bm t = (t_1,\ldots,t_n) \in T=({\mathbb{C}}^*)^n$, one constructs from the matrices $\check r(x) = r(x) \circ P$ and $k(x)$ the *monodromy operator* $$\begin{aligned} U_0(x;\bm t) &= \check r_{01}(x t_1^{-1}) \check r_{12}(x t_2^{-1}) \cdots \check r_{n\!-\!1 \, n}(x t_n^{-1}) k_n(x) \check r_{n\!-\!1 \, n}(x t_n) \cdots \check r_{12}(x t_2) \check r_{01}(x t_1) \\ &= r_{01}(x t_1^{-1}) \cdots r_{0n}(x t_n^{-1}) k_0(x) r_{n0}(x t_n) \cdots r_{10}(x t_1) ,\end{aligned}$$ acting on $\underset{(0)}{{\mathbb{C}}^2}\otimes\bigl(\mathbb{C}^2\bigr)^n= \underset{(0)}{{\mathbb{C}}^2} \otimes \underset{(1)}{{\mathbb{C}}^2} \otimes \cdots \otimes \underset{(n)}{{\mathbb{C}}^2}$ (with the sublabels indicating the numbering of the tensor legs on which the operator is acting). Fix a square root $\kappa^{\frac{1}{2}}$ of $\kappa$ and set $$\theta:=\left(\begin{matrix} \kappa^{-\frac{1}{2}} & 0\\ 0 & \kappa^{\frac{1}{2}}\end{matrix}\right).$$ The *transfer matrix* is the endomorphism of $\bigl(\mathbb{C}^2\bigr)^n$ given by $$T(x;\bm t) = {\mathop{\textup{Tr}}}_0 \bigl(\theta_0 \, \bar k_0(\kappa^2 x) \theta_0 U_0(x;\bm t)\bigr),$$ where ${\mathop{\textup{Tr}}}_0$ stands for the partial trace over the auxiliary space. The $\kappa^2$-shift in the argument of $\bar k$ is absent from the standard approach in [@Sk1988; @MN; @Ne]. It appears here to allow us to relate the transfer operators to the transport operators $C_{\tau_i}(\bm t)$, see Proposition \[transferoperatorandcocycles\]. The following result is the well-known statement that transfer operators are generating functions of commuting quantum Hamiltonians. \[commutingtransferoperators\] We have $$\label{eqn:commutingtransferoperators} [T(x;\bm t),T(y;\bm t)]=0.$$ as endomorphisms of $\bigl(\mathbb{C}^2\bigr)^{\otimes n}$ depending rationally on $\mathbf{t}\in T$. The proof is a straightforward modification of Sklyanin’s [@Sk1988] proof for $P$- and $T$-symmetric $R$-matrices; note that the presence of the inhomogeneities does not affect the argument at all. We remark that $r$ satisfies PT-symmetry, $r_{21}(x) = r^T_{12}(x)$, and crossing unitarity: $$\theta^{-1}_1 \theta^{-1}_2 r^{T_1}_{12}(\kappa^{-4}x^{-1}) \theta_1 \theta_2 r^{T_2}_{12}(x) = \Phi(x) {\textup{Id}}_{{\mathbb{C}}^2\otimes {\mathbb{C}}^2}, \qquad \Phi(x) = \frac{(1-x)(1-\kappa^4 x)}{(1-\kappa^2 x)^2}$$ where $T_i$ denotes the partial transpose with respect to the tensor leg labelled $i$. Crucially, note that , the reflection equation for $\bar k$, can be re-written as $$\label{reformRE} \begin{gathered} r_{12}(y/x) \bar k^{T_1}_1(\kappa^2 x) r_{21}((\kappa^{4}xy)^{-1}) \bar k^{T_2}_2(\kappa^2 y) = \hspace{30mm} \\ \hspace{30mm} = \bar k^{T_2}_2(\kappa^2 y) r_{12}((\kappa^{4}xy)^{-1}) \bar k^{T_1}_1(\kappa^2 x) r_{21}(y/x), \end{gathered}$$ which is equivalent to [@MN Eqn. (10)]. To obtain this reformulation of , we have used that $r^T_{12}(x) = r_{12}(x^{-1})|_{\kappa \to \kappa^{-1}}$ and $\bar k(x)$ does not depend on $\kappa$, and that $\bar k^T(x) = \bar k(x)|_{\psi_0 \to \psi_0^{-1}}$ and $r(x)$ does not depend on $\psi_0$. Using the proof is essentially the one implicitly present in [@MN]. The transport operators $C_{\tau_i}(\bm t)$ of the reflection quantum KZ equation associated to the spin representation $\widehat{\rho}^{\underline{\kappa}}_{\psi_0,\psi_n}$ are linear operators on $\bigl(\mathbb{C}^2\bigr)^n$ depending rationally on $\bm t\in T$. They are explicitly expressed in terms of $r(x)$, $k(x)$ and $\overline{k}(x)$ by $$\label{explicitCrho} \begin{split} &C_{\tau_i}(\bm t)=\check{r}_{i-1\,i}(t_{i-1}/t_i)\check{r}_{i-2\,i-1}(t_{i-2}/t_{i-1})\cdots \check{r}_{12}(t_1/t_i)\\ &\,\,\times \overline{k}_1(q^{\frac{1}{2}}/t_i)\check{r}_{12}(q/t_1t_i)\cdots\check{r}_{i-1\,i}(q/t_{i-1}t_i)\\ &\,\,\times \check{r}_{i\,i+1}(q/t_it_{i+1})\cdots\check{r}_{n-1\,n}(q/t_it_n)k_n(q/t_i) \check{r}_{n-1\,n}(qt_n/t_i)\cdots\check{r}_{i\,i+1}(qt_{i+1}/t_i) \end{split}$$ in view of and the proof of Corollary \[rk\]. Note that the cocycle property of $C_w$ $(w\in W)$ implies that $$C_{\tau_i}(\bm t)C_{\tau_j}(q^{-\epsilon_i}\bm t)=C_{\tau_j}(\bm t)C_{\tau_i}(q^{-\epsilon_j}\bm t),\qquad 1\leq i,j\leq n,$$ where $\{\epsilon_i\}_i$ denotes the standard orthonormal basis of $\mathbb{R}^n$. To relate the transport operators to the transfer operator we need the *boundary crossing symmetry* of the $K$-matrix $\overline{k}(x)$, $$\label{reflectioncrossingsymmetry} {\mathop{\textup{Tr}}}_0 \bigl( \theta_0 \bar k_0(\kappa^2 x) \theta_0 \check r_{01}(x^2) \bigr) = \Phi_\mathrm{bdy}(x) \bar k_1(x)$$ as linear operators on $\underset{(1)}{\mathbb{C}^2}$, where $$\Phi_{\mathrm{bdy}}(x):=\kappa\frac{(1-\kappa_0\upsilon_0x)(1+\kappa_0\upsilon_0^{-1}x)(1-\kappa^4x^2)} {(1-\kappa^2\kappa_0\upsilon_0x)(1+\kappa^2\kappa_0\upsilon_0^{-1}x)(1-\kappa^2x^2)}.$$ See [@GZ] for a discussion of the notion of boundary crossing symmetries. \[transferoperatorandcocycles\] For $i=1,\ldots,n$ we have $$\label{interpolants} \begin{split} T(t_i^{-1};\bm t)&=\Phi_\mathrm{bdy}(t_i^{-1})C_{\tau_i}(\bm t)|_{q=1},\\ T(t_i;\bm t)&=\Phi_\mathrm{bdy}(t_i)C_{\tau_i}(\bm t)^{-1}|_{q=1}. \end{split}$$ The transfer operator satisfies $$\begin{split} T(x;\bm t) \check r_{i \, i+1}(t_i/t_{i+1}) &= \check r_{i \, i+1}(t_i/t_{i+1}) T(x;s_i \bm t), \qquad 1 \leq i < n, \\ T(x;\bm t) k_n(t_n) &= k_n(t_n) T(x;s_n \bm t). \end{split}$$ On the other hand, the transport operators satisfy $$\begin{split} C_{\tau_{i+1}}(\bm t)&=\check{r}_{i\,i+1}(t_i/t_{i+1})C_{\tau_i}(\bm t)\check{r}_{i\,i+1}(t_{i+1}/qt_i),\qquad 1\leq i<n,\\ C_{\tau_n}(\bm t)^{-1}&=k_n(t_n/q)C_{\tau_n}(s_n\tau_n^{-1}\bm t)k_n(t_n^{-1}). \end{split}$$ Hence it suffices to prove the first equality of for $i=1$. Using the regularity $\check{r}(1)={\textup{Id}}_{\mathbb{C}^2\otimes\mathbb{C}^2}$ of the $R$-matrix the desired equality $$T(t_1^{-1};\bm t)=\Phi_\mathrm{bdy}(t_1^{-1})C_{\tau_1}(\bm t)|_{q=1}$$ follows by a direct computation using the definition of $T(x;\bm t)$, the boundary crossing symmetry , and . A detailed study of the relation between transfer operators and transport operators in the context of reflection quantum KZ equations is part of ongoing joint work with N. Reshetikhin. The Hamiltonian {#Hsection} --------------- Write $\bm 1 = (1,\ldots,1) \in T$ and $'$ for the derivative with respect to the spectral parameter. We suppose in this subsection that the parameters are generic. The following definition corresponds to [@MN Eqn. (25)]. The Hamiltonian for the XXZ Heisenberg spin chain with general boundary conditions is defined as $$\begin{aligned} H^\text{XXZ}_\text{bdy} &= \frac{\kappa-\kappa^{-1}}{2} \frac{\mathrm{d}}{\mathrm{d}x} \log\Bigl( \frac{T(x;\bm 1)}{{\mathop{\textup{Tr}}}\bigl( \theta \bar k(\kappa^2 x) \theta \bigr)}\Bigr)\Bigr|_{x=1}- C_0 {\textup{Id}}_{({\mathbb{C}}^2)^{\otimes n}} \\ &= (\kappa-\kappa^{-1}) \biggl( \sum_{i=1}^{n-1} \check{r}'_{i \, i \! + \! 1}(1) + \frac{ k'_n(1)}{2} + \frac{{\mathop{\textup{Tr}}}_0\bigl(\theta_0 \bar k_0(\kappa^2) \theta_0 \check{r}'_{01}(1)\bigr)}{{\mathop{\textup{Tr}}}\bigl(\theta \bar k(\kappa^2) \theta\bigr)} \biggr) - C_0 {\textup{Id}}_{({\mathbb{C}}^2)^{\otimes n}}\end{aligned}$$ where $$C_0 = \frac{-1}{\kappa(1+\kappa^2)} \frac{(\kappa^2-\kappa_0 \upsilon_0)(\kappa^2 + \kappa_0 \upsilon_0^{-1})}{(1-\kappa_0 \upsilon_0)(1+\kappa_0 \upsilon_0^{-1})}.$$ To find the spectrum of the Hamiltonian, one can now use the fact that, by construction, it commutes with all $T(x;\bm 1)$. It is therefore sufficient to find a complete set of common eigenfunctions of the $T(x;\bm 1)$, for which one typically uses the algebraic Bethe ansatz and related methods, which for *diagonal* boundary conditions was first done by Sklyanin [@Sk1988]. For the non-diagonal case, it may be possible to use a variant of the algebraic Bethe ansatz involving so-called dynamical $R$-and $K$-matrices (see [@FK] for such a treatment in a special case). Recall the Pauli spin matrices $$\sigma^X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad \sigma^Y = \begin{pmatrix} 0 & -\sqrt{-1} \\ \sqrt{-1} & 0 \end{pmatrix}, \qquad \sigma^Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},$$ and the auxiliary matrices $$\sigma^+ = \frac{1}{2} \left( \sigma^X + \sqrt{-1} \sigma^Y \right) = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \quad \sigma^- = \frac{1}{2} \left( \sigma^X - \sqrt{-1} \sigma^Y \right) = \begin{pmatrix}0 & 0 \\ 1 & 0 \end{pmatrix}.$$ From straightforward calculations the following statement is easily obtained (cf., e.g., [@Ne] for a similar statement). We have $$\label{XXZHamiltonian} \begin{split} H^\text{XXZ}_\text{bdy} &= \frac{1}{2} \Biggl( \sum_{i=1}^{n-1} \bigl( \sigma^X_i \sigma^X_{i\!+\!1} + \sigma^Y_i \sigma^Y_{i\!+\!1} + \frac{\kappa+\kappa^{-1}}{2} \sigma^Z_i \sigma^Z_{i\!+\!1} \bigr) + \\ & \quad+ \frac{\kappa-\kappa^{-1}}{2} \biggl( \frac{(1+\kappa_0 \upsilon_0)(1-\kappa_0 \upsilon_0^{-1}) \sigma^Z_1 + 4\kappa_0\bigl(\psi_0 \sigma^+_1 + \psi_0^{-1} \sigma^-_1 \bigr)}{(1+\kappa_0 \upsilon_0^{-1})(1-\kappa_0 \upsilon_0)} + \\ & \qquad \qquad \quad - \frac{(1+\kappa_n \upsilon_n)(1-\kappa_n \upsilon_n^{-1}) \sigma^Z_n - 4 \kappa_n \bigl( \psi_n^{-1} \sigma^+_n + \psi_n \sigma^-_n \bigr) }{(1+\kappa_n \upsilon_n^{-1})(1-\kappa_n \upsilon_n)} \biggr) \Biggr) + \hspace{-3mm} \\ & \qquad +\tilde C \, {\textup{Id}}_{({\mathbb{C}}^2)^{\otimes n}}, \end{split}$$ where $$\tilde C = \frac{\kappa-\kappa^{-1}}{2} \sum_{i \in \{0,n\}} \frac{1+\kappa_i^2}{(1-\kappa_i \upsilon_i)(1+\kappa_i \upsilon_i^{-1})} - \frac{n-1}{4} (\kappa+\kappa^{-1}).$$ The first line of is the Hamiltonian of the Heisenberg XXZ spin-$\frac{1}{2}$ chain. The second and third lines describe explicit three-parameter integrable boundary conditions for the left and right boundary of the spin chain, respectively. \[Hamiltonian2BTL\] Write $d_i=1$ for $1 \leq i < n$ and for $i=0,n$ write $$d_i = \frac{-\kappa_i(\kappa \kappa_i^{-1} + \kappa^{-1} \kappa_i)}{(1-\kappa_i \upsilon_i)(1+\kappa_i \upsilon_i^{-1})}.$$ Then $$H^\text{XXZ}_\text{bdy} = \sum_{i=0}^n d_i \hat{\rho}^{\underline{\kappa}}_{\psi_0,\psi_n}(e_i).$$ This follows immediately from the observations that $$\begin{gathered} \check{r}'_{i \, i\!+\!1}(1) = \frac{1}{\kappa-\kappa^{-1}} \hat \rho(e_i) , \qquad \frac{k'_n(1)}{2} = d_n \hat \rho(e_n) , \displaybreak[2] \\ \frac{{\mathop{\textup{Tr}}}_0\bigl( \theta_0 \bar k_0(\kappa^2) \theta_0 \check r'_{01}(1)\bigr)}{{\mathop{\textup{Tr}}}\bigl(\theta \bar k(\kappa^2) \theta\bigr)} = d_0 \hat \rho(e_0) + \frac{C_0}{\kappa-\kappa^{-1}} {\textup{Id}}_{({\mathbb{C}}^2)^{\otimes n}}. \qedhere\end{gathered}$$ With the notations of Proposition \[Hamiltonian2BTL\], the linear operator $$H^{\text{loop}}_{\text{bdy}}:=\sum_{i=0}^nd_i\omega^{\underline{\delta}}_{\beta_0,\beta_1}(e_i)$$ on $\mathbb{C}[\mathcal{M}]$, with $\omega^{\underline{\delta}}_{\beta_0,\beta_1}$ the matchmaker representation and the parameters $\underline{\delta},\beta_0,\beta_1$ related to $\underline{\kappa},\psi_0,\psi_n$ by and , is the quantum Hamiltonian of the Temperley-Lieb loop model (also known as the dense loop model) with general three-parameter open boundary conditions on both the left and the right boundaries, cf., e.g., [@dG; @dGN] for special cases. Theorem \[isomorphicrepresentations\] and Proposition \[Hamiltonian2BTL\] establish the link between the XXZ spin-$\frac{1}{2}$ chain and the Temperley-Lieb loop model for general integrable boundary conditions, not only on the level of the quantum Hamiltonian but also on the level of the underlying quantum symmetry algebra (the two-boundary Temperley-Lieb algebra). See [@MNGB] for a detailed discussion on Temperley-Lieb loop models and their relation to XXZ spin chains. Solutions of reflection quantum KZ equations {#Solsection} ============================================ The nonsymmetric difference Cherednik-Matsuo correspondence {#dCMsection} ----------------------------------------------------------- In this subsection we extend some of the main results of [@St1] to the Koornwinder setup. We do not dive into the detailed proofs, which are rather straightforward adjustments of the proofs in [@St1]. We do give precise references to the corresponding statements in [@St1]. Theorem \[thmN\] gives rise to an algebra homomorphism $$\varpi_{\underline{\kappa};\upsilon_0,\upsilon_n}: H(\underline{\kappa})\rightarrow\textup{End}_{\mathbb{C}}\bigl(\mathcal{M}(T)\bigr)$$ defined by $$\bigl(\varpi_{\underline{\kappa};\upsilon_0,\upsilon_n}(T_j)f\bigr)(\bm{t}):=\kappa_jf(\bm{t})+c_j(\bm{t};\underline{\kappa};\upsilon_0,\upsilon_n)(f(s_j\bm{t})-f(\bm{t})).$$ Note that the subspace $\mathbb{C}[T]$ of regular functions on $T$ is an invariant subspace. The resulting $H(\underline{\kappa})$-module is the analog of Cherednik’s basic representation. For our purposes it is convenient to consider the basic representation for inverted parameters $\underline{\kappa}^{-1},\upsilon_0^{-1},\upsilon_n^{-1}$ (the deformation parameter $q$ will always remain unchanged). We therefore write $\varpi:=\varpi_{\underline{\kappa}^{-1};\upsilon_0^{-1},\upsilon_n^{-1}}$ for the associated representation map in the remainder of this section. The operators $\varpi(Y^\lambda)$ ($\lambda\in\mathbb{Z}^n$) on $\mathcal{M}(T)$ are pairwise commuting $q$-difference reflection operators, known as nonsymmetric Koornwinder operators. Their common eigenspaces are denoted by $$\textup{Sp}_K(\gamma;\underline{\kappa};\upsilon_0,\upsilon_n):=\{ f\in\mathcal{M}(T) \,\, | \,\, \varpi_{\underline{\kappa}^{-1}; \upsilon_0^{-1},\upsilon_n^{-1}}(Y_i)f=\gamma_if\quad \forall\, i=1,\ldots,n\}$$ for $\gamma\in T$. If $\gamma\in T_I^{\underline{\kappa}^{-1}}$ (see Definition \[basicdefprincipal\]) then we have the natural subspace $$\begin{split} \textup{Sp}_K^{I}(\gamma;\underline{\kappa};\upsilon_0,\upsilon_n):=&\{ f\in\mathcal{M}(T) \,\, | \,\, \varpi_{\underline{\kappa}^{-1};\upsilon_0^{-1},\upsilon_n^{-1}}(h)f= \chi_{I,\gamma}^{\underline{\kappa}^{-1}}(h)f\qquad \forall\, h\in H_I(\underline{\kappa}^{-1})\}\\ =&\{ f\in \textup{Sp}_K(\gamma;\underline{\kappa};\upsilon_0,\upsilon_n) \,\, | \,\, \varpi_{\underline{\kappa}^{-1};\upsilon_0^{-1},\upsilon_n^{-1}}(T_i)f=\kappa_i^{-1}f\quad \forall i\in I\} \end{split}$$ of the common eigenspace $\textup{Sp}_K^{\underline{\kappa}}(\gamma)$. In writing these common eigenspaces we suppress the dependence on the parameters if no confusion can arise. \[nsbhf\] For $|q|<1$, for generic $\underline{\kappa},\upsilon_0,\upsilon_n$ and for $\gamma\in T$ satisfying $$\label{conditionsspecial} \upsilon_0\upsilon_n^{-1}\gamma_i^{\pm 1}\not\in -q^{\frac{1}{2}+\mathbb{Z}_{\geq 0}}\qquad \forall\, i\in\{1,\ldots,n\},$$ we write $\mathcal{E}_\gamma$ to be the nonsymmetric basic hypergeometric function associated to the Koornwinder root datum with multiplicity function $(k_0,k_{\vartheta},k_\theta,k_{2a_0},k_{2\theta})$ being $(\underline{\kappa}^{-1},\upsilon_0^{-1},\upsilon_n^{-1})$ (see [@St2 Def. 2.14 & §5.2] for the definition and the notations). Then $\mathcal{E}_\gamma\in\textup{Sp}_K(\gamma;\underline{\kappa};\upsilon_0,\upsilon_n)$. Here the conditions ensure that the nonsymmetric basic hypergeometric function may be specialized to $\gamma$ in its spectral parameter (cf. [@St2 Thm. 2.13(ii)]). The transformation property [@St2 Thm. 2.13(ii)(2)] of the nonsymmetric basic hypergeometric function with respect to the action of the double affine Hecke algebra implies that $$\varpi_{\underline{\kappa}^{-1};\upsilon_0,\upsilon_n}(T_i)\mathcal{E}_\gamma=\kappa_i^{-1}\mathcal{E}_\gamma,\qquad i\in I$$ if $\gamma\in T_I^{\underline{\kappa}^{-1}}$ since the numerators of $c_i(\cdot; \upsilon_n,\kappa,\kappa_n;\upsilon_0,\kappa_0)$ vanish if $\gamma\in T_I^{\underline{\kappa}^{-1}}$ and $i\in I$. Hence $$\mathcal{E}_\gamma\in\textup{Sp}_K^I(\gamma;\underline{\kappa};\upsilon_0,\upsilon_n)$$ if $\gamma\in T_I^{\underline{\kappa}^{-1}}$. The nonsymmetric difference Cherednik-Matsuo correspondence in the present setup gives a bijective correspondence between $\textup{Sp}_K^{I}(\gamma;\underline{\kappa};\upsilon_0,\upsilon_n)$ and the $W_0$-invariant solutions of the reflection quantum KZ equations associated to the principal series module $M_I^{\underline{\kappa}}(\gamma)$. The nonsymmetric difference Cherednik-Matsuo correspondence was considered before in [@KT; @St1; @Ka] in different setups. We follow closely [@St1]. To formulate the nonsymmetric difference Cherednik-Matsuo correspondence, we need to introduce a bit more notations and some basic facts on Coxeter groups and Hecke algebras first. The affine Weyl group $W$, which is a Coxeter group with simple reflections $s_i$ ($i=0,\ldots,n$), has a length function $l: W\rightarrow \mathbb{Z}_{\geq 0}$ defined as follows. The length of the unit element $e\in W$ is zero. For $w\in W$, the length $l(w)$ is the minimal number of simple reflections needed to write $w$ as product of simple reflections. An expression $w=s_{i_1}\cdots s_{i_r}$ with $r=l(w)$ is called a reduced expression. The length function of the Weyl group $W_0$, viewed as Coxeter group with simple reflections $s_i$ ($1\leq i\leq n$), coincides with the length function $l$ restricted to $W_0$. Let $I\subseteq\{1,\ldots,n\}$. There is a distinguished set $W_0^I$ of coset representatives of $W_0/W_{0,I}$, called the minimal coset representatives of $W_0/W_{0,I}$. It is defined as $$W_0^I:=\{w\in W_0 \,\, | \,\, l(wv)=l(w)+l(v)\quad \forall\, v\in W_{0,I}\}.$$ There exist unique $w_0\in W_0$ and $w_{0,I}\in W_{0,I}$ of maximal length. In fact, $w_0$ simply acts on $\mathbb{R}^n$ by multiplication by $-1$. For $I\subseteq\{1,\ldots,n\}$ we write $w_0^I:=w_0w_{0,I}^{-1}$. For $i\in I$ let $i_I^*\in\{1,\ldots,n\}$ be the unique index such that $w_0^Is_i=s_{i_I^*}w_0^I$. Set $I^*:=\{i_I^*\,\, | \,\, i\in I\}$. The case particularly relevant for our applications corresponds to the subset $J=\{1,\ldots,n-1\}$, in which case $w_{0,J}$ is the symmetric group element characterized by $w_{0,J}(\epsilon_i)=\epsilon_{n+1-i}$ for $i=1,\ldots,n$. Thus $i_J^*=n-i$ ($1\leq i<n$) and $J^*=J$. If $w=s_{i_1}\cdots s_{i_r}$ is a reduced expression, then $$T_w:=T_{i_1}T_{i_2}\cdots T_{i_r}$$ is a well-defined element of the affine Hecke algebra $H(\underline{\kappa})$. The principal series $M_I^{\underline{\kappa}}(\gamma)=H(\underline{\kappa})\otimes_{H_I(\underline{\kappa})} \mathbb{C}_{I,\gamma}$ for $\gamma\in T_I^{\underline{\kappa}}$ has a natural basis $\{v_w^{I}(\gamma;\underline{\kappa})\}_{w\in W_0^I}$ given by $v_w^{I}(\gamma;\underline{\kappa}):=\pi_{I,\gamma}^{\underline{\kappa}}(T_w)(1\otimes_{H(\underline{\kappa})}1)$. The following theorem gives the nonsymmetric difference Cherednik-Matsuo correspondence. \[nsCMtheorem\] Let $I\subseteq\{1,\ldots,n\}$ and $\gamma\in T_I^{\underline{\kappa}}$. Then $w_0^I\gamma^{-1}\in T_{I^*}^{\underline{\kappa}^{-1}}$ and the linear map $\mathcal{M}(T)\rightarrow \mathcal{M}(T)\otimes M_I^{\underline{\kappa}}(\gamma)$, defined by $$\phi\mapsto \sum_{w\in W_0^I}\varpi_{\underline{\kappa}^{-1};\upsilon_0^{-1},\upsilon_n^{-1}}(T_{w(w_0^I)^{-1}})\phi\otimes v_w^I(\gamma;\underline{\kappa})$$ restricts to a linear isomorphism $$\alpha_{\upsilon_0,\upsilon_n}^{\underline{\kappa}}: \textup{Sp}_K^{I^*}(w_0^I\gamma^{-1};\underline{\kappa};\upsilon_0,\upsilon_n)\overset{\sim}{\longrightarrow} \textup{Sol}_{KZ}(M_I^{\underline{\kappa}}(\gamma);\underline{\kappa};\upsilon_0,\upsilon_n)^{W_0}.$$ The proof is similar to the proof of [@St1 Prop. 4.7]. We will sketch here the main steps. Write $\nabla=\nabla^{M_I^{\underline{\kappa}}(\gamma)}$. By a direct computation, analogous to the proof of [@St1 Cor. 4.4], one shows that a meromorphic $M_I^{\underline{\kappa}}(\gamma)$-valued function $$\psi=\sum_{w\in W_0^I}\phi_w\otimes v_w^I(\gamma;\underline{\kappa})$$ on $T$ is $\nabla(W_0)$-invariant if and only if $$\phi_w=\varpi_{\underline{\kappa}^{-1};\upsilon_0^{-1},\upsilon_n^{-1}}(T_{w(w_0^I)^{-1}})\phi\qquad \forall\,w\in W_0^I$$ with $\phi\in\mathcal{M}(T)$ satisfying the invariance property $$\label{phivar} \varpi_{\underline{\kappa}^{-1};\upsilon_0^{-1},\upsilon_n^{-1}}(T_i)\phi=\kappa_i^{-1}\phi\qquad \forall\, i\in I^*.$$ So we need to investigate what the necessary and sufficient additional conditions on $\phi\in\mathcal{M}(T)$ satisfying are to ensure that the associated $\nabla(W_0)$-invariant $M_I^{\underline{\kappa}}(\gamma)$-valued function $$\psi=\sum_{w\in W_0^I} \varpi_{\underline{\kappa}^{-1};\upsilon_0^{-1},\upsilon_n^{-1}}(T_{w(w_0^I)^{-1}})\phi\otimes v_w^I(\gamma;\underline{\kappa})$$ on $T$ becomes $\nabla(W)$-invariant. The additional equation that $\psi$ should satisfy is $\nabla(s_0)\psi=\psi$. An algebraic computation analogous to the proof of [@St1 Prop. 4.7] shows that this is equivalent to the additional requirement on $\phi$ that $\phi\in\textup{Sp}_K(w_0^I\gamma^{-1};\underline{\kappa};\upsilon_0,\upsilon_n)$. This completes the proof of the theorem. \[differenceremark\] [**(i)**]{} The difference Cherednik-Matsuo correspondence is a correspondence between the common eigenspace of the (higher order) Koornwinder $q$-difference operators [@Ko; @N] and the full space $\textup{Sol}_{KZ}(M_\emptyset^{\underline{\kappa}}(\gamma);\underline{\kappa};\upsilon_0,\upsilon_n)$ of solutions of the reflection quantum KZ equations. It can be obtained from a spinor version of the nonsymmetric difference Cherednik-Matsuo correspondence by a symmetrization procedure, cf. [@St1 §5]. A distinguished $W_0$-invariant solution of the spectral problem of the Koornwinder $q$-difference operators is the symmetric basic hypergeometric function $\mathcal{E}_\gamma^+$ of Koornwinder type, obtainable from $\mathcal{E}_\gamma$ (see Example \[nsbhf\]) by a symmetrization procedure [@St2 §2.6]. For $n=1$ the symmetric basic hypergeometric function $\mathcal{E}_\gamma^+$ is the Askey-Wilson function [@IR; @KS; @Stlink], which is a nonpolynomial eigenfunction of the Askey-Wilson [@AW] second-order $q$-difference operator, alternatively expressible as a very-well-poised ${}_8\phi_7$ series.\ [**(ii)**]{} For $\gamma\in T_I^{\underline{\kappa}}$ the canonical map $M_\emptyset^{\underline{\kappa}}(\gamma)\rightarrow M_I^{\underline{\kappa}}(\gamma)$ induces a linear map $$\label{map} \textup{Sol}_{KZ}(M_\emptyset^{\underline{\kappa}}(\gamma);\underline{\kappa};\upsilon_0,\upsilon_n)\rightarrow \textup{Sol}_{KZ}(M_I^{\underline{\kappa}}(\gamma);\underline{\kappa},\upsilon_0,\upsilon_n),$$ cf, [@St1 (4.15)]. Combined with [**(i)**]{}, it leads to the construction of solutions of the reflection quantum KZ equations from common eigenfunctions of the (higher order) Koornwinder $q$-difference operators. It is an interesting open problem to understand the behaviour of the asymptotic basis of solutions of the reflection quantum KZ equations and their connection coefficients (see [@St3]) under the map . It is expected to lead to elliptic solutions of dynamical Yang-Baxter equations and reflection equations, cf. [@St3 §1.4]. Nonsymmetric Koornwinder polynomials {#Kosection} ------------------------------------ Let $\lambda\in\mathbb{Z}^n$ and define $\gamma_\lambda=(\gamma_{\lambda,1},\ldots,\gamma_{\lambda,n})\in T$ by $$\gamma_{\lambda,i}=q^{\lambda_i}(\kappa_0\kappa_n)^{-\eta(\lambda_i)}\bigl(\prod_{j<i}\kappa^{\eta(\lambda_j-\lambda_i)}\bigr) \bigl(\prod_{j>i}\kappa^{-\eta(\lambda_i-\lambda_j)}\bigr)\bigl(\prod_{j\not=i}\kappa^{-\eta(\lambda_i+\lambda_j)}\bigr)$$ where $\eta(x)=1$ if $x>0$ and $\eta(x)=-1$ if $x\leq 0$. Note that if $\lambda_1\leq\lambda_2\leq\cdots\leq\lambda_n\leq 0$ then $$\gamma_\lambda=\bigl(\kappa_0\kappa_n\kappa^{2(n-1)}q^{\lambda_1},\ldots,\kappa_0\kappa_n\kappa^2q^{\lambda_{n-1}},\kappa_0\kappa_nq^{\lambda_n}).$$ Another special case is $\bm{m}=(m,\ldots,m)\in\mathbb{Z}^n$ with $m\in\mathbb{Z}$, $$\label{gammaspecial} \gamma_{\bm{m}}= \begin{cases} (\kappa_0^{-1}\kappa_n^{-1}q^m,\kappa_0^{-1}\kappa_n^{-1}\kappa^{-2}q^m,\ldots,\kappa_0^{-1}\kappa_n^{-1}\kappa^{2(1-n)}q^m)\quad &\hbox{if }\, m\in\mathbb{Z}_{>0},\\ (\kappa_0\kappa_n\kappa^{2(n-1)}q^m,\kappa_0\kappa_n\kappa^{2(n-3)}q^m,\ldots,\kappa_0\kappa_nq^m)\quad &\hbox{if }\, m\in\mathbb{Z}_{\leq 0}. \end{cases}$$ In the remainder of the paper we assume that the parameters $\kappa_0,\kappa,\kappa_n,q$ are sufficiently generic, meaning that $\gamma_\lambda\not=\gamma_\mu$ if $\lambda\not=\mu$. Then $$\label{Spintersection} \textup{Sp}_K^{\emptyset}(\gamma_\lambda^{-1};\underline{\kappa};\upsilon_0,\upsilon_n) \cap\mathbb{C}[T]=\textup{span}_{\mathbb{C}}\{P_\lambda\}$$ is one-dimensional for all $\lambda\in\mathbb{Z}^n$. We can choose $P_\lambda=P_\lambda(\cdot;\underline{\kappa};\upsilon_0,\upsilon_n)$ such that the coefficient of $\bm{t}^\lambda=t_1^{\lambda_1}t_2^{\lambda_2}\cdots t_n^{\lambda_n}$ in the expansion of $P_\lambda(\bm{t})$ in monomials $\bm{t}^\mu$ ($\mu\in\mathbb{Z}^n$) is one. $P_\lambda$ is called the monic nonsymmetric Koornwinder polynomial of degree $\lambda\in\mathbb{Z}^n$. Since $\mathbb{C}[T]=\bigoplus_{\lambda\in\mathbb{Z}^n}\mathbb{C}P_\lambda$ the discrete set $\{\gamma_\lambda^{-1}\}_{\lambda\in\mathbb{Z}^n}\subset T$ is called the polynomial spectrum of the $q$-difference reflection operators $\varpi_{\underline{\kappa}^{-1};\upsilon_0^{-1},\upsilon_n^{-1}}(Y_i)$ ($1\leq i\leq n$). \[polred\] By [@St0 Thm. 6.9], the proof of [@St2 Thm. 2.13(ii)] and [@St2 (2.5)], the nonsymmetric Koornwinder polynomial of degree $\lambda$ equals $\mathcal{E}_{\gamma_\lambda^{-1}}$ (up to a multiplicative constant), with $\mathcal{E}_\gamma$ the nonsymmetric basic hypergeometric function as discussed in Example \[nsbhf\]. The following lemma is convenient for later purposes. \[invariancelem\] Let $i\in\{1,\ldots,n\}$. The following two conditions are equivalent. 1. $s_i\lambda=\lambda$. 2. $\varpi(T_i)P_\lambda=\kappa_i^{-1}P_\lambda$. If $I\subseteq\{1,\ldots,n\}$ and $s_i\lambda=\lambda$ for all $i\in I$, then $P_\lambda\in \textup{Sp}_K^I(\gamma_\lambda^{-1};\underline{\kappa};\upsilon_0,\upsilon_n)$. The first part of the lemma is [@StBook Prop. 4.15]. For the second part, suppose that $I$ is a subset of $\{1,\ldots,n\}$ and suppose that $\lambda\in\mathbb{Z}^n$ satisfies $s_i\lambda=\lambda$ for all $i\in I$. Then [@StBook Prop. 3.5] shows that $\gamma_\lambda^{-1}\in T_I^{\underline{\kappa}^{-1}}$. The statement then follows from the first part of the lemma. For a more detailed discussion on nonsymmetric Macdonald-Koornwinder polynomials see, e.g., [@StBook]. A suitable symmetrized version of the monic nonsymmetric Koornwinder polynomials give the monic Koornwinder polynomials [@Ko]. They are $W_0$-invariant Laurent polynomials in the variables $t_1,\ldots,t_n$, and common eigenfunctions of the (higher order) Koornwinder $q$-difference operators [@Ko; @N]. For $n=1$ the Koornwinder $q$-difference operator is the Askey-Wilson second-order $q$-difference operator [@AW] and the Koornwinder polynomials are the celebrated monic Askey-Wilson [@AW] polynomials, see, e.g., [@StBook §3.8] for a detailed discussion. Laurent polynomial solutions of the reflection quantum KZ equations ------------------------------------------------------------------- Let $I\subseteq\{1,\ldots,n\}$ and $\gamma\in T_I^{\underline{\kappa}}$. Then $w_0^I\gamma^{-1}\in T_{I^*}^{\underline{\kappa}^{-1}}$, see Theorem \[nsCMtheorem\](i). Hence for generic $\upsilon_0,\upsilon_n\in\mathbb{C}^*$ and $|q|<1$, $$\textup{Sp}_K^{I^*}(w_0^{I}\gamma^{-1}; \underline{\kappa};\upsilon_0,\upsilon_n)\not=\{0\}$$ in view of Example \[nsbhf\] (the same is true for $|q|>1$, using the nonsymmetric basic hypergeometric function for $|q|>1$ as constructed in [@St0]). By Theorem \[nsCMtheorem\] we conclude that nontrivial $W_0$-invariant solutions of the reflection quantum KZ equations associated to $(M_I^{\underline{\kappa}}(\gamma),\underline{\kappa},\upsilon_0,\upsilon_n)$ generically exist. In this subsection we focus on the $W_0$-invariant [*Laurent polynomial*]{} solutions of the reflection quantum KZ equations. Let $V$ be a $H(\underline{\kappa})$-module and $\upsilon_0,\upsilon_n\in\mathbb{C}^*$. We say that $(V,\upsilon_0,\upsilon_n)$ admits $W_0$-invariant Laurent polynomial solutions of the reflection quantum KZ equations if $$\textup{Sol}_{KZ}(V;\underline{\kappa};\upsilon_0,\upsilon_n)^{W_0}\cap\bigl(\mathbb{C}[T]\otimes V\bigr)\not=\{0\}.$$ A nonzero $f$ from this space is called a nontrivial $W_0$-invariant Laurent polynomial solution of the reflection quantum KZ equations associated to $(V,\upsilon_0,\upsilon_n)$. Recall that we assume that the multiplicity parameters are generic, i.e. that $\gamma_\lambda\not=\gamma_\mu$ if $\lambda\not=\mu$. \[polprop\] Let $I\subseteq\{1,\ldots,n\}$ and $\gamma\in T_I^{\underline{\kappa}}$. Then $(M_I^{\underline{\kappa}}(\gamma), \upsilon_0,\upsilon_n)$ admits $W_0$-invariant Laurent polynomial solutions of the reflection quantum KZ equations if and only if $w_0^I\gamma=\gamma_\lambda$ with $\lambda\in\mathbb{Z}^n$ satisfying $s_i\lambda=\lambda$ for all $i\in I^*$. The associated nontrivial $W_0$-invariant Laurent polynomial solutions of the reflection quantum KZ equations are the nonzero multiples of $$\alpha_{\upsilon_0,\upsilon_n}^{\underline{\kappa}}(P_\lambda)=\sum_{w\in W_0^I}\varpi_{\underline{\kappa}^{-1};\upsilon_0^{-1},\upsilon_n^{-1}}(T_{w(w_0^I)^{-1}})P_\lambda\otimes v_w^I(\gamma;\underline{\kappa}).$$ This follows from Theorem \[nsCMtheorem\], Lemma \[invariancelem\], and the fact that $\mathbb{C}[T]=\bigoplus_{\lambda\in\mathbb{Z}^n}\mathbb{C}P_\lambda$. We now come to our main application of Proposition \[polprop\], by applying it to the spin representations $\rho_{\psi_0,\psi_n}^{\underline{\kappa}}\simeq \pi_{J,\zeta}^{\underline{\kappa}}$. Recall here that $J=\{1,\ldots,n-1\}$ and that $$\zeta=(\psi_0\psi_n\kappa^{n-1},\psi_0\psi_n\kappa^{n-3},\ldots,\psi_0\psi_n\kappa^{1-n}),$$ see Proposition \[rhopiequivalence\]. Let $\lambda\in\mathbb{Z}^n$. Then $s_i\lambda=\lambda$ for all $i\in J^*$ if and only if $\lambda=\bm{m}$ for some $m\in\mathbb{Z}$. On the other hand, note that $$w_0^J\zeta=(\psi_0^{-1}\psi_n^{-1}\kappa^{n-1},\psi_0^{-1}\psi_n^{-1}\kappa^{n-3},\ldots,\psi_0^{-1}\psi_n^{-1}\kappa^{1-n}).$$ \[mainthm\] For generic parameters, the spin representation $(M_J^{\underline{\kappa}}(\zeta),\upsilon_0,\upsilon_n)$ admits nontrivial $W_0$-invariant Laurent polynomial solutions if and only if $$\label{mcondition} \psi_0\psi_nq^m=\bigl(\kappa_0\kappa_n\kappa^{n-1}\bigr)^{\eta(m)}$$ for some $m\in\mathbb{Z}$ where, recall, $\eta(x)=1$ if $x>0$ and $\eta(x)=-1$ if $x\leq 0$. The associated nontrivial $W_0$-invariant Laurent polynomial solutions of the reflection quantum KZ equations are multiples of $$\alpha_{\upsilon_0,\upsilon_n}^{\underline{\kappa}}(P_{\bm{m}})= \sum_{w\in W_0^J}\varpi_{\underline{\kappa}^{-1};\upsilon_0^{-1},\upsilon_n^{-1}}(T_{w(w_0^J)^{-1}})P_{\bm{m}}\otimes v_w^I(\zeta;\underline{\kappa}).$$ In view of the previous proposition, it suffices to note that for a given $m\in\mathbb{Z}$ we have $w_0^J\zeta=\gamma_{\bm{m}}$ if and only if holds true. Different examples of $W_0$-invariant Laurent polynomial solutions of reflection quantum KZ equations are given in [@Ka; @ZJ1; @dFZJ; @dGPS; @PoThesis]. We finally stress the importance of polynomial solutions of the quantum KZ equations, cf. [@dFZJ; @dFZJ2; @ZJ2; @Pa], in particular in relation to the Razumov-Stroganov conjectures [@RS] (recently proved by direct combinatorial methods in [@CS]). Similarly, polynomial solutions of the reflection quantum KZ equations play a similar role for the refinements of the Razumov-Stroganov conjectures for open boundaries from [@dGR], cf., e.g., [@dGP2; @ZJ1; @KaPrep]. We remark though that in the context of the Razumov-Stroganov conjectures the double affine Hecke algebra parameters $q,\underline{\kappa},\upsilon_0,\upsilon_n$ are specialized to particular non-generic values, in contrast to the setup of Theorem \[mainthm\], where the parameters $q,\underline{\kappa},\upsilon_0,\upsilon_n$ are assumed to be generic. [99]{} R. Askey, J. Wilson, [*Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials*]{}, Mem. Amer. Math. Soc. [**54**]{} (1985), no. 319. D. Allcock, [*Braid pictures for Artin groups*]{}, Trans. Amer. Math. Soc. [**354**]{}, no. 9 (2002), 3455–3474. P. Baseilhac, K. Koizumi, [*Exact spectrum of the XXZ open spin chain from the $q$-Onsager representation theory*]{}, J. Stat. Mech. Theory Exp. [**2007**]{}, no. 9, P09006, 27 pp. P. Baseilhac, T. Kojima, [*Correlation functions of the half-infinite XXZ spin chain with a triangular boundary*]{}, Nucl. Phys. B, [**880**]{} (2014), 378–413. R.J. Baxter, [*Exactly Solved Models in Statistical Mechanics*]{}, Academic Press, New York (1982). H. Bethe, [*Zur Theorie der Metalle*]{}, Zeitschrift für Physik [**71**]{} (1931) no. 3-4, 205-226. L. Cantini, A. Sportiello, [*Proof of the Razumov-Stroganov conjecture*]{}, J. of Combinatorial Theory, Series A [**118**]{} (2011), no. 5, 1549-1574. I. Cherednik, [*Quantum Knizhnik-Zamolodchikov equations and affine root systems*]{}, Comm. Math. Phys. [**150**]{} (1992), 109–136. I. Cherednik, [*A unification of Knizhnik-Zamolodchikov and Dunkl operators via affine Hecke algebras*]{}, Invent. Math. [**106**]{} (1991), 411–431. I. Cherednik, [*Double affine Hecke algebras*]{}, Cambridge Univ. Press, London Math. Soc. Lecture Note Series [**319**]{} (2005). P. Di Francesco, P. Zinn-Justin, [*Quantum Knizhnik-Zamolodchikov equation, generalized Razumov-Stroganov sum rules and extended Joseph polynomials*]{}, J. Phys. A: Math. Gen. [**38**]{} (2006) L815. P. Di Francesco, P. Zinn-Justin, [*Quantum Knizhnik-Zamolodchikov equation: reflecting boundary conditions and combinatorics*]{}, J. Stat. Mech. Theory Exp. [**2007**]{}, no. 12, P12009, 30pp. J.-F. van Diejen, [*Commuting difference operators with polynomial eigenfunctions*]{}, Compositio Math. [**95**]{} (1995), no. 2, 183–233. L.D. Faddeev, [*Instructive history of the quantum inverse scattering method*]{}, Acta Appl. Math. [**39**]{} (1995), 69-84. G. Filali, [*Elliptic dynamical reflection algebra and partition function of SOS model with reflecting end*]{}, J. Geom. Phys. [**61**]{} (2011), no. 10, 1789–1796. G. Filali, N. Kitanine, [*Spin chains with non-diagonal boundaries and trigonometric SOS model with reflecting end*]{}, SIGMA [**7**]{} (2011), Paper 012, 22 pages. I. Frenkel, N. Reshetikhin, [*Quantum affine algebras and holonomic difference equations*]{}, Comm. Math. Phys. [**146**]{} (1992), 1–60. G. Gasper, M. Rahman, [*Basic Hypergeometric Series*]{}, Encycl. Math. and Appl. [**96**]{}, Cambridge University Press 2004 (2nd edition). S. Ghoshal, A.B. Zamolodchikov, [*Boundary $S$ matrix and boundary state in two-dimensional integrable quantum field theory*]{}, Int. J. Mod. Phys. A [**9**]{} (1994), no. 21, 3841-3885. J. de Gier, [*Loops, matchings and alternating-sign matrices*]{}, Discrete Math. [**298**]{} (2005), no. 1-3, 365–388. J. de Gier, A. Nichols, [*The two-boundary Temperley-Lieb algebra*]{}, J. Algebra [**321**]{} (2009), no. 4, 1132–1167. J. de Gier, A.K. Ponsaing, K. Shigechi, [*The exact size ground state of the $O(n=1)$ loop model with open boundaries*]{}, J. Stat. Mech. (2009), no. 4, P04010. J. de Gier, P. Pyatov, [*Bethe Ansatz for the Temperley-Lieb loop model with open boundaries*]{}, J. Stat. Mech. Theory Exp. [**2004**]{}, no. 3, 002, 27pp. J. de Gier, P. Pyatov, [*Factorized solutions of Temperley-Lieb qKZ equations on a segment*]{}, Adv. Theor. Math. Phys. [**14**]{} (2010), 795–877. J. de Gier, V. Rittenberg, [*Refined Razumov-Stroganov conjectures for open boundaries*]{}, J. Stat. Mech. Theory Exp. [**2004**]{}, no. 9, 009, 14pp. J.S. Geronimo, P. Iliev, [*Multivariable Askey-Wilson function and bispectrality*]{}, Ramanujan J. [**24**]{} (2011), no. 3, 273–287. W. Heisenberg, [*Zur Theorie der Ferromagnetismus*]{}, Zeitschrift für Physik [**49**]{} (1928), no. 9-10, 619–636. T. Inami, H. Konno, [*Integrable $XYZ$ spin chain with boundaries*]{}, J. Phys. A: Math. Gen. [**27**]{} (1994), no. 24, L913–L918. M. Ismail, M. Rahman, [*The associated Askey-Wilson polynomials*]{}, Trans. Amer. Math. Soc. [**328**]{} (1991), 201–237. M. Jimbo, R. Kedem, T. Kojima, H. Konno, T. Miwa, [*XXZ chain with a boundary*]{}, Nucl. Phys. B [**441**]{} (1995), no. 3, 437–470. M. Jimbo, R. Kedem, H. Konno, T. Miwa, R. Weston, [*Difference equations in spin chains with a boundary*]{}, Nucl. Phys. B [**448**]{} (1995), no. 3, 429–456. M. Jimbo, T. Miwa, [*Algebraic analysis of solvable lattice models*]{}, CBMS Regional Conference Series in Mathematics, [**85**]{}., Amer. Math. Soc., Providence, RI, 1995. V.F.R. Jones, [*On a certain value of the Kauffman polynomial*]{}, Comm. Math. Phys. [**125**]{} (1989), 459–467. M. Kasatani, [*Boundary quantum Knizhnik-Zamolodchikov equation*]{}, in: “New trends in quantum integrable systems”, 157–171, World Sci. Publ., Hackensack, NJ, 2011. M. Kasatani, [*The polynomial representation of the double affine Hecke algebra of type $(C_n^\vee,C_n)$ for specialized parameters*]{}, arXiv:0807.2714. M. Kasatani, Y. Takeyama, [*The quantum Knizhnik-Zamolodchikov equation and non-symmetric Macdonald polynomials*]{}, Funkcialaj Ekvacioj [**50**]{} (2007), 491–509. E. Koelink, J.V. Stokman, [*The Askey-Wilson function transform*]{}, Int. Math. Res. Not. IMRN [**2000**]{}, no. 19, 1005–1042. S. Kolb, [*Quantum symmetric Kac-Moody pairs*]{}, arXiv:1207.6036. T.H. Koornwinder, [*Askey-Wilson polynomials for root systems of type BC*]{}, in: “Hypergeometric functions on domains of positivity, Jack polynomials, and applications” (Tampa, FL, 1991), 189–204, Contemp. Math., [**138**]{}, Amer. Math. Soc., Providence, RI, 1992. V.E. Korepin, N.M. Bogoliubov and A.G. Izergin, [*Quantum inverse scattering method and correlation functions*]{}, Cambridge Monographs on Mathematical Physics, Cambridge Univ. Press (1993). G. Letzter, [*Quantum zonal spherical functions and Macdonald polynomials*]{}, Adv. Math. [**189**]{} (2004), no. 1, 88–147. G. Lusztig, [*Affine Hecke algebras and their graded version*]{}, J. Amer. Math. Soc. [**2**]{} (1989), 599–635. I.G. Macdonald, [*Affine Hecke algebras and orthogonal polynomials*]{}, Cambridge Univ. Press (2003). S. Mitra, B. Nienhuis, J. de Gier, M.T. Batchelor, [*Exact expressions for correlations in the ground state of the dense $O(1)$ loop model*]{}, J. Stat. Mech. Theory Exp. [**2004**]{}, no. 9, 010, 24 pp. L. Mezincescu, R.I. Nepomechie, [*Integrable open spin chains with non-symmetric $R$-matrices*]{}, J. Phys. A: Math. Gen. [**24**]{} (1991), L17–L23. A. Morin-Duchesne, Y. Saint-Aubin, [*A homomorphism between link and XXZ modules over the periodic Temperley-Lieb algebra*]{}, J. Phys. A [**46**]{} (2013), no. 28, 285207, 34 pp. R.I. Nepomechie, [*Bethe ansatz solution of the open XXZ chain with nondiagonal boundary terms*]{}, J. Phys. A [**37**]{} (2004), no. 2, 433–440. M. Noumi, [*Macdonald-Koornwinder polynomials and affine Hecke ring*]{}, in: “Various aspects of hypergeometric functions” (Kyoto, 1994), Surikaisekikenkyusho Kokyuroku [**919**]{} (1995), 44-55. M. Noumi, M.S. Dijkhuizen, T. Sugitani, [*Multivariable Askey-Wilson polynomials and quantum complex Grassmannians*]{}, in “Special functions, $q$-series and related topics (Toronto, ON, 1995), 167–177, Fields Instit. Commun., [**14**]{} , Amer. Math. Soc., Providence, RI, 1997. M. Noumi, J.V. Stokman, [*Askey-Wilson polynomials: an affine Hecke algebra approach*]{}, in ”Laredo Lectures on Orthogonal Polynomials and Special Functions“, 111–144, Adv. Theory Spec. Funct. Orthogonal Polynomials, Nova Sci. Publ., Hauppauge, NY, 2004. A.A. Oblomkov, J.V. Stokman, [*Vector valued spherical functions and Macdonald-Koornwinder polynomials*]{}, Compos. Math. [**141**]{} (2005), no. 5, 1310–1350. V. Pasquier, [*Quantum Incompressibility and Razumov-Stroganov type conjectures*]{}, Ann. Henri Poincar[é]{} [**7**]{} (2006) no. 3, 397-421. W. Pauli, [*Zur Quantenmechanik des magnetischen Elektrons*]{}, Zeitschrift für Physik, [**43**]{} (1927) no. 9-10, 601-623 A.K. Ponsaing, [*Finite size lattice results for the two-boundary Temperley-Lieb loop model*]{}, PhD thesis, Univ. of Melbourne (2010). M. Rahman, [*An integral representation of a ${}_{10}\phi_9$ and continuous bi-orthogonal ${}_{10}\phi_9$ rational functions*]{}, Canad. J. Math. [**38**]{} (1986), 605–618. E.M. Rains, [*Transformations of elliptic hypergeometric integrals*]{}, Ann. of Math. (2) [**171**]{} (2010), no. 1, 169–243. A.V. Razumov, Yu.G. Stroganov, [*Combinatorial nature of ground state vector of $O(1)$ loop model*]{}, Theor. Math. Phys. [**138**]{} (2004), 333–337 (translation). N. Reshetikhin, J.V. Stokman, B.H.M. Vlaar, [*Reflection quantum Knizhnik-Zamolodchikov equations and Bethe vectors*]{}, arXiv:1305.1113. S.N.M. Ruijsenaars, [*Complete integrability of relativistic Calogero-Moser systems and elliptic function identities*]{}, Comm. Math. Phys. [**175**]{} (1995), 75–121. S. Sahi, [*Nonsymmetric Koornwinder polynomials and duality*]{}, Ann. of Math. (2) [**150**]{} (1999), no. 1, 267–282. E.K. Sklyanin, [*Quantum variant of the method of the inverse scattering problem*]{}, J. Sov. Math. [**19**]{} (1982), no. 5, 1564-1596. E.K. Sklyanin, [*Boundary conditions for integrable quantum systems*]{}, J. Phys. A: Math. Gen. [**21**]{} (1988), 2375–2389. F. Smirnov, [*A general formula for soliton form factors in the quantum sine-Gordon model*]{}, J. Phys. A [**19**]{} (1986), no. 10, L575-L578. V.P. Spiridonov, [*Theta hypergeometric integrals*]{}, St. Petersburg Math. J. [**15**]{} (2003), 161–215. V.P. Spiridonov, [*Essays on the theory of elliptic hypergeometric functions*]{}, Russian Math. Surveys [**63**]{} (2008), no. 3, 405–472. V.P. Spiridonov, [*Elliptic beta integrals and solvable models of statistical mechanics*]{}, in ”Algebraic aspects of Darboux transformations, quantum integrable systems and supersymmetric quantum mechanics“, 181–211, Contemp. Math., [**563**]{}, Amer. Math. Soc., Providence, RI, 2012. J.V. Stokman, [*An expansion formula for the Askey-Wilson function*]{}, J. Approx. Theory [**114**]{} (2002), no. 2, 308–342. J.V. Stokman, [*Difference Fourier transforms for nonreduced root systems*]{}, Selecta Math. (N.S.) [**9**]{} (2003), no. 3, 409–494. J.V. Stokman, [*Quantum affine Knizhnik-Zamolodchikov equations and quantum spherical functions, I*]{}, Int. Math. Res. Not IMRN [**2011**]{}, no. 5, 1023–1090. J.V. Stokman, [*The $c$-function expansion of a basic hypergeometric function associated to root systems*]{}, Ann. of Math. (2) [**179**]{} (2014), no. 1, 253–299. J.V. Stokman, [*Connection coefficients for basic Harish-Chandra series*]{}, Adv. Math. [**250**]{} (2014), 351–386. J.V. Stokman, [*Macdonald-Koornwinder polynomials*]{}, arXiv:1111.6112. To be part of volume 5: ”Multivariable Special Functions" of the Askey-Bateman project. T. Takebe, [*A system of difference equations with elliptic coefficients and Bethe vectors*]{}, Comm. Math. Phys. [**183**]{} (1997), no. 1, 161–181. H.J. de Vega, A. Gonz[á]{}lez-Ruiz, [*Boundary $K$-matrices for the XYZ, XXZ and XXX spin chains*]{}, J. Phys. A: Math. Gen. [**27**]{} (1994), 6129–6137. R. Weston, [*Correlation functions and the boundary qKZ equation in a fractured XXZ chain*]{}, J. Stat. Mech. Theory Exp. [**2011**]{}, no. 12, P12002, 28pp. P. Zinn-Justin, [*Loop model with mixed boundary conditions, qKZ equation and alternating sign matrices*]{}, J. Stat. Mech. Theory Exp. [**2007**]{}, no. 1, P01007, 16pp. P. Zinn-Justin, [*Six-vertex, loop and tiling models: integrability and combinatorics*]{}, arXiv:0901.0665. P. Zinn-Justin, [*Sum rule for the eight-vertex model on its combinatorial line*]{}, arXiv:1202.4420.

--- abstract: 'In this work, we demonstrate a Chinese classical poetry generation system called Deep Poetry. Existing systems for Chinese classical poetry generation are mostly template-based and very few of them can accept multi-modal input. Unlike previous systems, Deep Poetry uses neural networks that are trained on over 200 thousand poems and 3 million ancient Chinese prose. Our system can accept plain text, images or artistic conceptions as inputs to generate Chinese classical poetry. More importantly, users are allowed to participate in the process of writing poetry by our system. For the user’s convenience, we deploy the system at the WeChat applet platform, users can use the system on the mobile device whenever and wherever possible. The demo video of this paper is available at <https://youtu.be/jD1R_u9TA3M>.' author: - | Yusen Liu,[^1] Dayiheng Liu,Jiancheng Lv[^2]\ College of Computer Science, Sichuan University\ 24 Yihuan Road\ Chengdu 610065, China\ {liuyusen96, losinuris}@gmail.com, lvjiancheng@scu.edu.cn bibliography: - '258-References.bib' title: 'Deep Poetry: A Chinese Classical Poetry Generation System' --- Introduction ============ Chinese classical poetry plays a special and significant role in Chinese history. Poetry is the treasure of Chinese culture. During ancient times, many poets and poetry hobbyists were devoted to poetry and created a lot of poems. However, only a few best scholars can write coherent and beautiful poetry nowadays, most people either pay less attention to poetry or suffer from the high threshold of composing a poem. Therefore, we present a Chinese classical poetry generation system to arouse people’s interest in poetry and make writing poetry easier. Most recently, the automatic poetry generation has received great attention, neural networks have made this task a great development, including recurrent neural network [@zhang2014chinese], encoder-decoder sequence to sequence model [@yi2017generating], and neural attention-based model [@wang2016chinese], etc. These studies are of great significance to entertainment and education. However, most of the released Chinese poetry generation systems are template-based. The biggest drawback of this method is the generated poems are mostly monotonous because the templates restrict the content of the results. Moreover, these systems restrict users from entering text only and are unable to generate poetry according to images and artistic conceptions, such as the *Daoxiangju* system[^3] and the *Microsoft Quatrain*.[^4] Moreover, without the assistance of the teacher, it is difficult for users to compose poems using the systems above. To solve these problems, this paper introduces a new system that can accept plain text, images or artistic conceptions as input and allow users to participate in the process of generating. Our system not only can entertain people but also serve as a tool for poetry enthusiasts to make their works perfect. In other words, Our system plays the role of the assistant helping users polish their poems. The *jiuge* system[^5] is a human-machine collaborative poetry generation system using neural networks [@guo2019jiuge]. Comparing with the *jiuge*, our model is different from them and our system uses a better way to help users write poems. Furthermore, rather than a web application, we deploy our system at the WeChat applet platform. WeChat is the largest social media app in China with over 1 billion monthly active users. This means that users can access our system easily and use it on the mobile device whenever and wherever possible. In summary, the contributions of our Chinese classical poetry generation system are as follows: 1) Multi-modal input such as plain text, images or artistic conceptions can be accepted by our system. 2) Our system allows users to participate in the process of generating. 3) Our system is deployed at the WeChat applet platform for easy access. {width="95.00000%"} Architecture ============ As shown in Fig. \[fig1\], the model which our system used consists of three components: a powerful method to process input, a self-attention neural network to generate poetry, and a screening mechanism for results. The model is mainly based on our previous work [@liu2018multi]. Our system allows multi-modal input then generating poetry according to the input. For each image, we map it into at least two themes which recognized by Clarifai. The Clarifai API offers image recognition as a service. Then we retrieve and expand all related phrases. The phrases are from *ShiXueHanYing*, which is a poetic phrase taxonomy. Each theme in *ShiXueHanYing* contains dozens of related phrases. If the inputs are artistic conceptions, we will retrieve and expand all related phrases directly. Moreover, when users cooperate with our system to write poems, they can input sentences of any length and the system will generate some candidates using the human-authored prefixes as input. The second component is a generator that can automatically generate poetry according to the processed input. A self-attention neural network is used as our poetry generator. The embedding of each phrase that retrieves from the previous step is fed into the neural network. Moreover, our system can control the genre, we provide several genres to users: quatrain with five or seven characters and acrostic poetry. For acrostic poetry, the generator will be attached to a condition that the first word of each line is fixed. Last but not least, the poetry will be generated word by word and a softmax logistic loss is employed to train the network. The last component is a screening mechanism for results that generate with beam search (beam size=N). We design a mechanism to check these N candidates and remove the bad results. If the generated poems break the rhyming rules or have repetitive words, they will be classified as bad results by the screening mechanism and be abandoned. Demonstration ============= The main functions of our system are generating poetry with multi-modal input, assisting users to write poetry and a word puzzle about poetry. For the first part, the system can generate poetry automatically using the plain text, images or artistic conceptions as input. Moreover, our system can generate acrostic poetry, which is a special genre in Chinese poetry. Users give somebody’s name or short plain text to our system, then our system generates a poem using each word of the input as the first word of each line. After the poetry completes, it can be transferred to an exquisite card, the users can download it and share it with others. As for the second part, our system allows users to write a satisfying poem with our system collaboratively. For each line, the users can write as long as they want, maybe a character, a word, or a sentence. The system will complete the whole line according to the prefix that the users have given. These generated candidates can serve as a reference for users and help them complete their poems. It is meant for people who are suffering from being unable to write a poem. After four lines, our system can generate a title according to the poem. For the word puzzle part, the system rearranges the words in popular poems and users should put them in the right order. In general, the game can improve user viscosity and arouse people’s interest in classical poetry. Conclusion and Future work ========================== We propose Deep Poetry, a multi-function Chinese classical poetry generation system in this demo. Our system not only accepts multi-modal input but also allow users to participate in the process of writing poems, it is of great significance to both ordinary and professional users. Moreover, we add a word puzzle game to arouse public interest in traditional culture. without the hassle of installing the application program, our system can be easily accessed at the WeChat applet platform. In future work, we intend to integrate more genres generation into our system. Acknowledgments =============== This work is supported by the National Key R&D Program of China under contract No. 2017YFB1002201, the National Natural Science Fund for Distinguished Young Scholar (Grant No. 61625204), and partially supported by the Key Program of National Science Foundation of China (Grant No. 61836006). [^1]: Equal contribution. [^2]: Correspondence to Jiancheng Lv. [^3]: http://www.poeming.com [^4]: http://duilian.msra.cn/jueju [^5]: https://jiuge.thunlp.cn

--- abstract: 'We compute time-periodic and relative-periodic solutions of the free-surface Euler equations that take the form of overtaking collisions of unidirectional solitary waves of different amplitude on a periodic domain. As a starting guess, we superpose two Stokes waves offset by half the spatial period. Using an overdetermined shooting method, the background radiation generated by collisions of the Stokes waves is tuned to be identical before and after each collision. In some cases, the radiation is effectively eliminated in this procedure, yielding smooth soliton-like solutions that interact elastically forever. We find examples in which the larger wave subsumes the smaller wave each time they collide, and others in which the trailing wave bumps into the leading wave, transferring energy without fully merging. Similarities notwithstanding, these solutions are found quantitatively to lie outside of the Korteweg-de Vries regime. We conclude that quasi-periodic elastic collisions are not unique to integrable model water wave equations when the domain is periodic.' address: 'Dept of Mathematics, University of California, Berkeley, CA 94720-3840' author: - Jon Wilkening date: 'April 21, 2014' title: ' Relative-Periodic Elastic Collisions of Water Waves' --- =1 [^1] Introduction ============ A striking feature of multiple-soliton solutions of integrable model equations such as the Korteweg-deVries equation, the Benjamin-Ono equation, and the nonlinear Schrödinger equation is that they interact elastically, leading to time-periodic, relative-periodic, or quasi-periodic dynamics. By contrast, the interaction of solitary waves for the free-surface Euler equations is inelastic. However, it has been observed many times in the literature [@chan:street:70; @cooker:97; @maxworthy:76; @su:mirie; @mirie:su; @zou:su; @craig:guyenne:06; @milewski:11] that the residual radiation after a collision of such solitary waves can be remarkably small. In the present paper we explore the possibility of finding nearby time-periodic and relative-periodic solutions of the Euler equations using a collision of unidirectional Stokes waves as a starting guess. Such solutions demonstrate that recurrent elastic collisions of solitary waves in the spatially periodic case do not necessarily indicate that the underlying system is integrable. A relative-periodic solution is one that returns to a spatial phase shift of its initial condition at a later time. This only makes sense on a periodic domain, where the waves collide repeatedly at regular intervals in both time and space, with the locations of the collisions drifting steadily in time. They are special cases (with $N=2$) of quasi-periodic solutions, which have the form $u(x,t)=U(\vec\kappa x+\vec \omega t + \vec\alpha)$ with $U$ an $N$-periodic continuous function, i.e. $U\in C\big(\mathbb{T}^N\big)$, and $\vec\kappa$, $\vec\omega$, $\vec\alpha\in\mathbb{R}^N$. Throughout the manuscript, we will use the phrase “solitary waves” in a broad sense to describe waves that, most of the time, remain well-separated from one another and propagate with nearly constant speed and shape. “Stokes waves” will refer to periodic progressive solutions of the free-surface Euler equations of permanent form, or waves that began at $t=0$ as a linear superposition of such traveling waves. They comprise a special class of solitary waves. “Solitons” will refer specifically to superpositions of ${\operatorname}{sech}^2$ solutions of the KdV equation on the whole line, while “cnoidal solutions” will refer to their spatially periodic, multi-phase counterparts; see §\[sec:kdv\] for elaboration. It was found in [@water2] that decreasing the fluid depth causes standing waves to transition from large-scale symmetric sloshing behavior in deep water to pairs of counter-propagating solitary waves that collide repeatedly in shallow water. In the present work, we consider unidirectional waves of different amplitude that collide due to taller waves moving faster than shorter ones. We present two examples of solutions of this type: one where the resulting dynamics is fully time-periodic; and one where it is relative-periodic, returning to a spatial phase shift of the initial condition at a later time. Both examples exhibit behavior typical of collisions of KdV solitons. In the first, one wave is significantly larger than the other, and completely subsumes it during the interaction. In the second, the waves have similar amplitude, with the trailing wave bumping into the leading wave and transferring energy without fully merging. Despite these similarities, the amplitude of the waves in our examples are too large for the assumptions in the derivation of the KdV equation to hold. In particular, the larger wave in the first example is more than half the fluid depth in height, and there is significant vertical motion of the fluid when the waves pass by. A detailed comparison of the Euler and KdV equations for waves with these properties is carried out in §\[sec:kdv\]. A review of the literature on water wave collisions and the accuracy of the KdV model of water waves is also given in that section. Rather than compute such solutions by increasing the amplitude from the linearized regime via numerical continuation, as was done for counter-propagating waves in [@water2], we use collisions of right-moving Stokes waves as starting guesses. The goal is to minimally “tune” the background radiation generated by the Stokes collisions so that the amount coming out of each collision is identical to what went into it. In the first example of §\[sec:num\], we find that the tuned background radiation takes the form of a train of traveling waves of smaller wavelength moving to the right more slowly than either solitary wave. By contrast, in the counter-propagating case studied in [@water2], it consists of an array of smaller-wavelength standing waves oscillating rapidly relative to the time between collisions of the primary waves. In the second example of §\[sec:num\], the background radiation is essentially absent, which is to say that the optimized solution is free from high-frequency, low-amplitude disturbances in the trough, and closely resembles a relative-periodic cnoidal solution of KdV. We call the collisions in this solution “elastic” as they repeat forever, unchanged up to spatial translation, and there are no features to distinguish radiation from the waves themselves. This process of tuning parameters to minimize or eliminate small-amplitude oscillations in the wave troughs is reminiscent of Vanden-Broeck’s work [@vandenBroeck91] in which oscillations at infinity could be eliminated from solitary capillary-gravity waves by choosing the amplitude appropriately. To search for relative periodic solutions, we use a variant of the overdetermined shooting method developed by the author and collaborators in previous work to study several related problems: time-periodic solutions of the Benjamin-Ono equation [@benj1; @benj2] and the vortex sheet with surface tension [@vtxs1; @vtxs2]; Hopf bifurcation and stability transitions in mode-locked lasers [@lasers]; cyclic steady-states in rolling treaded tires [@tires1]; self-similarity (or lack thereof) at the crests of large-amplitude standing water waves [@water1]; harmonic resonance and spontaneous nucleation of new branches of standing water waves at critical depths [@water2]; and three-dimensional standing water waves [@water3d]. The three approaches developed in these papers are the adjoint continuation method [@benj1; @lasers], a Newton-Krylov shooting method [@tires1], and a trust region shooting method [@water2] based on the Levenberg-Marquardt algorithm [@nocedal]. We adopt the latter method here to exploit an opportunity to consolidate the work in computing the Dirichlet-Neumann operator for many columns of the Jacobian simultaneously, in parallel. One computational novelty of this work is that we search directly for large-amplitude solutions of a nonlinear two-point boundary value problem, without using numerical continuation to get there. This is generally difficult. However, in the present case, numerical continuation is also difficult due to non-smooth bifurcation “curves” riddled with Cantor-like gaps [@plotnikov01], and the long simulation times that occur between collisions in the unidirectional case. Our shooting method has proven robust enough to succeed in finding time-periodic solutions, when they exist, with a poor starting guess. False positives are avoided by resolving the solutions spectrally to machine accuracy and overconstraining the minimization problem. Much of the challenge is in determining the form of the initial condition and the objective function to avoid wandering off in the wrong direction and falling into a nonzero local minimum before locking onto a nearby relative-periodic solution. Equations of motion {#sec:eqm} =================== The equations of motion of a free surface $\eta(x,t)$ evolving over an ideal fluid with velocity potential $\phi(x,y,t)$ may be written [@whitham74; @johnson97; @craik04; @craik05] $$\begin{aligned} \label{eq:ww} \eta_t &= \phi_y - \eta_x\phi_x, \\[-3pt] \notag \varphi_t &= P\left[\phi_y\eta_t - \frac{1}{2}\phi_x^2 - \frac{1}{2}\phi_y^2 - g\eta\right], $$ where subscripts denote partial derivatives, $\varphi(x,t) = \phi(x,\eta(x,t), t)$ is the restriction of $\phi$ to the free surface, $g=1$ is the acceleration of gravity, $\rho=1$ is the fluid density, and $P$ is the projection $$Pf = f - \frac{1}{2\pi}\int_0^{2\pi} f(x)\,dx,$$ where we assume a $2\pi$-periodic domain. The velocity components $u=\phi_x$ and $v=\phi_y$ at the free surface can be computed from $\varphi$ via $$\label{eq:uv:from:G} \begin{pmatrix} \phi_x \\ \phi_y \end{pmatrix} = \frac{1}{1+\eta'(x)^2}\begin{pmatrix} 1 & -\eta'(x) \\ \eta'(x) & 1 \end{pmatrix} \begin{pmatrix} \varphi'(x) \\ {\mathcal{G}}\varphi(x) \end{pmatrix},$$ where a prime denotes a derivative and ${\mathcal{G}}$ is the Dirichlet-Neumann operator [@craig:sulem:93] $$\label{eq:DNO:def} {\mathcal{G}}\varphi(x) = \sqrt{1+\eta'(x)^2}\,\, {\frac{\partial \phi}{\partial n}}(x+i\eta(x)) = \phi_y - \eta_x\phi_x$$ for the Laplace equation, with periodic boundary conditions in $x$, Dirichlet conditions ($\phi=\varphi$) on the upper boundary, and Neumann conditions ($\phi_y=0$) on the lower boundary, assumed flat. We have suppressed $t$ in the notation since time is frozen in the Laplace equation. We compute ${\mathcal{G}}\varphi$ using a boundary integral collocation method [@lh76; @baker:82; @krasny:86; @mercer:92; @baker10] and advance the solution in time using an 8th order Runge-Kutta scheme [@hairer:I] with 36th order filtering [@hou:li:07]. See [@water2] for details. Computation of relative-periodic solutions {#sec:method} ========================================== Traveling waves have the symmetry that $$\label{eq:init} \eta(x,0) \, \text{ is even}, \qquad \varphi(x,0) \, \text{ is odd.}$$ This remains true if $x$ is replaced by $x-\pi$. As a starting guess for a new class of time-periodic and relative-periodic solutions, we have in mind superposing two traveling waves, one centered at $x=0$ and the other at $x=\pi$. Doing so will preserve the property (\[eq:init\]), but the waves will now interact rather than remain pure traveling waves. A solution will be called *relative periodic* if there exists a time $T$ and phase shift $\theta$ such that $$\label{eq:ts:def} \eta(x,t+T) = \eta(x-\theta,t), \qquad\quad \varphi(x,t+T) = \varphi(x-\theta,t)$$ for all $t$ and $x$. Time-periodicity is obtained as a special case, with $\theta\in2\pi\mathbb{Z}$. We can save a factor of 2 in computational work by imposing the alternative condition $$\label{eq:even:odd} \eta(x+\theta/2,T/2) \, \text{ is even}, \qquad\quad \varphi(x+\theta/2,T/2) \, \text{ is odd.}$$ From this, it follows that $$\begin{aligned} \eta(x+\theta/2,T/2) &= \eta(-x+\theta/2,T/2) = \eta(x-\theta/2,-T/2), \\ \varphi(x+\theta/2,T/2) &= -\varphi(-x+\theta/2,T/2) = \varphi(x-\theta/2,-T/2).\end{aligned}$$ But then both sides of each equation in (\[eq:ts:def\]) agree at time $t=-T/2$. Thus, (\[eq:ts:def\]) holds for all time. In the context of traveling-standing waves in deep water [@trav:stand], it is natural to define $T$ as twice the value above, replacing all factors of $T/2$ by $T/4$. That way a pure standing wave returns to its original configuration in time $T$ instead of shifting in space by $\pi$ in time $T$. In the present work, we consider pairs of solitary waves moving to the right at different speeds, so it is more natural to define $T$ as the first (rather than the second) time there exists a $\theta$ such that (\[eq:ts:def\]) holds. Objective function {#sec:obj:fun} ------------------ We adapt the overdetermined shooting method of [@water1; @water2] to compute solutions of (\[eq:init\])–(\[eq:even:odd\]). This method employs the Levenberg-Marquardt method [@nocedal] with delayed Jacobian updates [@water2] to solve the nonlinear least squares problem described below. For (\[eq:init\]), we build the symmetry into the initial conditions over which the shooting method is allowed to search: we choose an integer $n$ and consider initial conditions of the form $$\label{eq:init:trav} \hat\eta_k(0) = c_{2|k|-1}, \qquad\quad \hat\varphi_k(0) = \pm ic_{2|k|},$$ where $k\in\{\pm1,\pm2,\dots,\pm \frac{n}{2}\}$ and $\hat\eta_k(t)$, $\hat\varphi_k(t)$ are the Fourier modes of $\eta(x,t)$, $\varphi(x,t)$. The numbers $c_1,\dots,c_n$ are assumed real and all other Fourier modes (except $\hat\eta_0$) are zero. We set $\hat\eta_0$ to the fluid depth so that $y=0$ is a symmetry line corresponding to the bottom wall. This is convenient for computing the Dirichlet-Neumann operator [@water2]. In the formula for $\hat\varphi_k$, the minus sign is taken if $k<0$ so that $\hat\varphi_{-k} =\overline{\hat\varphi_k}$. We also solve for the period, $$\label{eq:T:theta} T=c_{n+1}. $$ The phase shift $\theta$ is taken as a prescribed parameter here. Alternatively, in a study of traveling-standing waves [@trav:stand], the author defines a traveling parameter $\beta$ and varies $\theta=c_{n+2}$ as part of the algorithm to obtain the desired value of $\beta$. This parameter $\beta$ is less meaningful for solitary wave collisions in shallow water, so we use $\theta$ itself as the traveling parameter in the present study. We also need to specify the amplitude of the wave. This can be done in various ways, e.g. by specifying the value of the energy, $$E = \frac{1}{2\pi}\int_0^{2\pi} {\textstyle}\frac{1}{2}\varphi{\mathcal{G}}\varphi + \frac{1}{2}g\eta^2\,dx,$$ by constraining a Fourier mode such as $\hat\eta_1(0)$, or by specifying the initial height of the wave at $x=0$: $$\eta(0,0) = \hat\eta_0 + \sum_{k=1}^{n/2} 2c_{2k-1}.$$ Thus, to enforce (\[eq:even:odd\]), we can minimize the objective function $$\label{eq:f} f(c) = \frac{1}{2} r(c)^Tr(c),$$ where $$\begin{aligned} \label{eq:r:def} r_1 = \big(\;\text{choose one:} \quad & E-a \quad,\quad \hat\eta_1(0)-a \quad,\quad \eta(0,0)-a \;\big), \\ \notag r_{2j} = {\operatorname{Im}}\{e^{ij\theta/2}\hat\eta_j(T/2)\}, \qquad &r_{2j+1} = {\operatorname{Re}}\{e^{ij\theta/2}\hat\varphi_j(T/2)\}, \qquad (1 \le j\le M/2).\end{aligned}$$ Here $a$ is the desired value of the chosen amplitude parameter. Alternatively, we can impose (\[eq:ts:def\]) directly by minimizing $$\label{eq:f1} \tilde f = \frac{1}{2}r_1^2 + \frac{1}{4\pi} \int_0^{2\pi} \left(\big[\eta(x,T)-\eta(x-\theta,0)\big]^2 + \big[\varphi(x,T)-\varphi(x-\theta,0)\big]^2\right)dx,$$ which also takes the form $\frac{1}{2}r^Tr$ if we define $r_1$ as above and $$\label{eq:r1:def} \begin{aligned} r_{4j-2}+ir_{4j-1} &= \sqrt{2}\left[ \hat\eta_j(T) - e^{-ij\theta}\hat\eta_j(0) \right], \\ r_{4j}+ir_{4j+1} &= \sqrt{2}\left[ \hat\varphi_j(T) - e^{-ij\theta}\hat\varphi_j(0) \right], \end{aligned} \qquad\quad (1\le j\le M/2).$$ Note that $f$ measures deviation from evenness and oddness of $\eta(x+\theta/2,T/2)$ and $\varphi(x+\theta/2,T/2)$, respectively, while $\tilde f$ measures deviation of $\eta(x+\theta,T)$ and $\varphi(x+\theta,T)$ from their initial states. In the first example of §\[sec:num\], we minimize $\tilde f$ directly, while in the second we minimize $f$ and check that $\tilde f$ is also small, as a means of validation. The number of equations, $m=M+1$ for $f$ and $m=2M+1$ for $\tilde f$, is generally larger than the number of unknowns, $n+1$, due to zero-padding of the initial conditions. This adds robustness to the shooting method and causes all Fourier modes varied by the algorithm, namely those in (\[eq:init:trav\]), to be well-resolved on the mesh. Computation of the Jacobian --------------------------- To compute the $k$th column of the Jacobian $J=\nabla_c r$, which is needed by the Levenberg-Marquardt method, we solve the linearized equations along with the nonlinear ones: $$\label{eq:q:qdot} {\frac{\partial }{\partial t}} \begin{pmatrix} q \\ \dot q \end{pmatrix} = \begin{pmatrix} F(q) \\ DF(q)\dot q \end{pmatrix}, \quad \begin{aligned} q(0) &= q_0 = (\eta_0,\varphi_0), \\ \dot q(0) &= \dot q_0 = \partial q_0/\partial c_k. \end{aligned}$$ Here $q=(\eta,\varphi)$, $\dot q=(\dot\eta,\dot\varphi)$, $F(q)$ is given in (\[eq:ww\]), $DF$ is its derivative (see [@water2] for explicit formulas), and a dot represents a variational derivative with respect to perturbation of the initial conditions, not a time derivative. To compute $\partial r_i/\partial c_k$ for $i\ge2$ and $k\le n$, one simply puts a dot over each Fourier mode on the right-hand side of (\[eq:r:def\]) or (\[eq:r1:def\]), including $\hat\eta_j(0)$ and $\hat\varphi_j(0)$ in (\[eq:r1:def\]). If $k=n+1$, then $c_k=T$ and $${\frac{\partial r_{2j}}{\partial T}} = {\operatorname{Im}}\{e^{ij\theta/2}(1/2)\partial_t\hat\eta_j(T/2)\}, \qquad {\frac{\partial (r_{4j}+ir_{4j+1})}{\partial T}} = \sqrt{2}\big[\partial_t\hat\varphi_j(T)\big]$$ in (\[eq:r:def\]) and (\[eq:r1:def\]), respectively, with similar formulas for $\partial(r_{4j-2}+ir_{4j-1})/\partial T$ and $\partial r_{2j+1}/\partial T$. The three possibilities for $r_1$ are handled as follows: $$\begin{aligned} &\text{case 1:} \quad {\frac{\partial r_1}{\partial c_k}} = \dot E = \frac{1}{2\pi}\int_0^{2\pi} \left[\dot\varphi\eta_t - \dot\eta\varphi_t \right]_{t=0}dx, \quad (k\le n), \qquad {\frac{\partial r_1}{\partial c_{n+1}}} = 0, \\ &\text{case 2:} \quad {\frac{\partial r_1}{\partial c_k}} = {{\dot\eta}^{\scriptscriptstyle\bm\wedge}}_1(0) = \delta_{k,1}, \quad (k\le n+1),\\ &\text{case 3:} \quad {\frac{\partial r_1}{\partial c_k}} = \dot\eta(0,0) = 2\delta_{k,\text{odd}}, \quad (k\le n), \qquad {\frac{\partial r_1}{\partial c_{n+1}}} = 0,\end{aligned}$$ where $\delta_{k,j}$ and $\delta_{k,\text{odd}}$ equal 1 if $k=j$ or $k$ is odd, respectively, and equal zero otherwise. The vectors $\dot q$ in (\[eq:q:qdot\]) are computed in batches, each initialized with a different initial perturbation, to consolidate the work in computing the Dirichlet-Neumann operator during each timestep. See [@water2; @trav:stand] for details. Numerical results {#sec:num} ================= As mentioned in the introduction, our idea is to use collisions of unidirectional Stokes (i.e. traveling) waves as starting guesses to find time-periodic and relative periodic solutions of the Euler equations. We begin by computing traveling waves of varying wave height and record their periods. This is easily done in the framework of §\[sec:method\]. We set $\theta=\pi/64$ (or any other small number) and minimize $\tilde f$ in (\[eq:f1\]). The resulting “period” $T$ will give the wave speed via $c=\theta/T$. Below we report $T=2\pi c$, i.e. $T$ is rescaled as if $\theta$ were $2\pi$. We control the amplitude by specifying $\hat\eta_1(0)$, which is the second option listed in §\[sec:method\] for defining the first component $r_1$ of the residual. A more conventional approach for computing traveling waves is to substitute $\eta(x-ct)$, $\varphi(x-ct)$ into (\[eq:ww\]) and solve the resulting stationary problem (or an equivalent integral equation) by Newton’s method [@chen80a; @chandler:93; @milewski:11]. Note that the wave speed $c$ here is unrelated to the vector $c$ of unknowns in (\[eq:init:trav\]). ![\[fig:bif:stokes\] Plots of wave height and first Fourier mode versus period for Stokes waves with wavelength $2\pi$ and fluid depth $h=0.05$. The temporal periods are $6T_A=137.843\approx 137.738 = 5T_C$.](figs/bif_stokes){width="3.3in"} ![\[fig:align:stokes\] Collision of two right-moving Stokes waves that nearly return to their initial configuration after the interaction. (left) Solutions A and C were combined via (\[eq:AandC\]) and evolved through one collision to $t=137.738$. (right) Through trial and error, we adjusted the amplitude of the smaller Stokes wave and the simulation time to obtain a nearly time-periodic solution. ](figs/align_stokes){width="\linewidth"} With traveling waves in hand, out next goal is to collide two of them and search for a nearby time-periodic solution, with $\theta=0$. As shown in Figure \[fig:bif:stokes\], varying $\hat\eta_1(0)$ from 0 to $7.4\times 10^{-4}$ causes the period of a Stokes wave with wavelength $\lambda=2\pi$ and mean fluid depth $h=0.05$ to decrease from $T_O=28.1110$ to $T_A=22.9739$, and the wave height (vertical crest-to-trough distance) to increase from 0 to $0.02892$. Solution C is the closest among the Stokes waves we computed to satisfying $5T_C=6T_A$, where $p=5$ is the smallest integer satisfying $\frac{p+1}{p}T_A<T_O$. We then combine solution A with a spatial phase shift of solution C at $t=0$. The resulting initial conditions are $$\label{eq:AandC} \begin{aligned} \eta^{A+C}_0(x) &= h + \big[\eta^A_0(x)-h\big] + \big[\eta^C_0(x-\pi)-h\big], \\ \varphi^{A+C}_0(x) &= \varphi^A_0(x) + \varphi^C_0(x-\pi), \end{aligned}$$ where $h=0.05$ is the mean fluid depth. Plots of $\eta_0^A(x)$, $\eta_0^C(x-\pi)$, $\varphi_0^A(x)$ and $\varphi_0^C(x-\pi)$ are shown in Figure \[fig:AandC\]. If the waves did not interact, the combined solution would be time-periodic (to the extent that $5T_C=6T_A$, i.e. to about $0.076\%$). But the waves do interact. In addition to the complicated interaction that occurs when they collide, each slows the other down between collisions by introducing a negative gradient in the velocity potential between its own wave crests. Indeed, as shown in the right panel of Figure \[fig:AandC\], the velocity potential increases rapidly across a right-moving solitary wave and decreases elsewhere to achieve spatial periodicity. The decreasing velocity potential induces a background flow opposite to the direction of travel of the other wave. In the left panel of Figure \[fig:align:stokes\], we see that the net effect is that neither of the superposed waves has returned to its starting position at $t=5T_C$, and the smaller wave has experienced a greater net decrease in speed. However, as shown in the right panel, by adjusting the amplitude of the smaller wave (replacing solution C by B) and increasing $T$ slightly to $138.399$, we are able to hand-tune the Stokes waves to achieve $\tilde f\approx5.5\times10^{-8}$, where $\theta$ is set to zero in (\[eq:f1\]). Note that as $t$ varies from 0 to $T/10$ in the left panel of Figure \[fig:evol:kdv1\], the small wave advances by $\pi$ units to the right while the large wave advances by $1.2\pi$ units. The waves collide at $t=T/2$. This generates a small amount of radiation, which can be seen at $t=T$ in the right panel of Figure \[fig:align:stokes\]. Some radiation behind the large wave is present for all $t>0$, as shown in Figure \[fig:pov:kdv1\]. Before minimizing $\tilde f$, we advance the two Stokes waves to the time of the first collision, $t=T/2$. At this time, the larger solitary wave has traversed the domain 3 times and the smaller one 2.5 times, so their peaks lie on top of each other at $x=0$. The reason to do this is that when the waves merge, the combined wave is shorter, wider, and smoother than at any other time during the evolution. Quantitatively, the Fourier modes of $\hat\eta_k(t)$ and $\hat\varphi_k(t)$ decay below $10^{-15}$ for $k\ge600$ at $t=0$, and $k\ge200$ when $t=T/2$. Thus, the number of columns needed in the Jacobian is reduced by a factor of 3, and the problem becomes more overdetermined, hence more robust. For the calculation of a time-periodic solution, we let $t=0$ correspond to this merged state, which affects the time labels when comparing Figures \[fig:evol:kdv1\] and \[fig:evol:kdv2\]. As a final initialization step, we project onto the space of initial conditions satisfying (\[eq:init:trav\]) by zeroing out the imaginary parts of $\hat\eta_k(0)$ and the real parts of $\hat\varphi_k(0)$, which are already small. Surprisingly, this improves the time-periodicity of the initial guess in (\[eq:f1\]) to $\tilde f = 2.3\times 10^{-8}$. ![\[fig:evol:kdv1\] Evolution of two Stokes waves that collide repeatedly, at times $t\approx T/2+kT$, $k\ge0$. (left) Traveling solutions A and B in Figure \[fig:bif:stokes\] were initialized with wave crests at $x=0$ and $x=\pi$, respectively. The combined solution is approximately time-periodic, with period $T=138.399$. (right) The same solution, at later times, starting with the second collision ($t=3T/2$).](figs/evol_kdv1){width="\linewidth"} ![\[fig:pov:kdv1\] A different view of the solutions in Figure \[fig:evol:kdv1\] shows the generation of background waves. Shown here are the functions $\eta(x+8\pi t/T,t)$, which give the dynamics in a frame moving to the right fast enough to traverse the domain four times in time $T$. In a stationary frame, the smaller and larger solitary waves traverse the domain 5 and 6 times, respectively.](figs/pov_kdv1){height="2in"} We emphasize that our goal is to find *any* nearby time-periodic solution by adjusting the initial conditions to drive $\tilde f$ to zero. Energy will be conserved as the solution evolves from a given initial condition, but is only imposed as a constraint (in the form of a penalty) on the search for initial conditions when the first component of the residual in (\[eq:r:def\]) is set to $r_1=E-a$. In the present calculation, we use $r_1=\eta(0,0)-a$ instead. In the second example, presented below, we will constrain energy. In either case, projecting onto the space of initial conditions satisfying (\[eq:init:trav\]) can cause $r_1$ to increase, but it will decrease to zero in the course of minimizing $\tilde f$. This projection is essential for the symmetry arguments of §\[sec:obj:fun\] to work. ![\[fig:evol:kdv2\] Time-periodic solutions near the Stokes waves of Figure \[fig:evol:kdv1\]. (left) $h=0.05$, $\eta(0,0) = 0.0707148$, $T=138.387$, $\tilde f=4.26\times10^{-27}$. (right) $h=0.0503$, $\eta(0,0)=0.0707637$, $T=138.396$, $\tilde f=1.27\times 10^{-26}$. The background radiation was minimized by hand in the right panel by varying $h$ and $\eta(0,0)$.](figs/evol_kdv2){width="\linewidth"} ![\[fig:pov:kdv2\] Same as Figure \[fig:pov:kdv1\], but showing the time-periodic solutions of Figure \[fig:evol:kdv2\] instead of the Stokes waves of Figure \[fig:evol:kdv1\]. The Stokes waves generate new background radiation with each collision while the time-periodic solutions are synchronized with the background waves to avoid generating additional disturbances. ](figs/pov_kdv2){height="2in"} We minimize $\tilde f$ subject to the constraint $\eta(0,0)=0.0707148$, the third case described in §\[sec:method\] for specifying the amplitude. This causes $\tilde f$ to decrease from $2.3\times 10^{-8}$ to $4.26\times 10^{-27}$ using $M=1200$ grid points and $N=1200$ time-steps (to $t=T$). The results are shown in the left panel of Figures \[fig:evol:kdv2\] and \[fig:pov:kdv2\]. The main difference between the Stokes collision and this nearby time-periodic solution is that the Stokes waves generate additional background ripples each time they collide while the time-periodic solution contains an equilibrium background wave configuration that does not grow in amplitude after the collision. While the background waves in the counter-propagating case (studied in [@water2]) look like small-amplitude standing waves, these background waves travel to the right, but slower than either solitary wave. After computing the $h=0.05$ time-periodic solution, we computed 10 other solutions with nearby values of $h$ and $\eta(0,0)$ to try to decrease the amplitude of the background radiation. The best solution we found (in the sense of small background radiation) is shown in the right panel of Figures \[fig:evol:kdv2\] and \[fig:pov:kdv2\], with $h=0.0503$ and $\eta(0,0)=0.0707637$. The amplitude of the background waves of this solution are comparable to that of the Stokes waves after two collisions. Our second example is a relative periodic solution in which the initial Stokes waves (the starting guess) are B and C in Figure \[fig:bif:stokes\] instead of A and C. As before, solution C is shifted by $\pi$ when the waves are combined initially, just as in (\[eq:AandC\]). Because the amplitude of the larger wave has been reduced, the difference in wave speeds is smaller, and it takes much longer for the waves to collide. If the waves did not interact, we would have $$\label{eq:cBcC} c_{B,0} = 0.23246089, \quad c_{C,0} = 0.22808499, \quad T_0 = \frac{2\pi}{c_{B,0}-c_{A,0}} = 1435.86,$$ where wave B crosses the domain $53.1230$ times in time $T_0$ while wave C crosses the domain $52.1230$ times. The subscript 0 indicates that the waves are assumed not to interact. Since the waves do interact, we have to evolve the solution numerically to obtain useful estimates of $T$ and $\theta$. We arbitrarily rounded $T_0$ to 1436 and made plots of the solution at times $\Delta t = T_0/1200$. We found that $\eta$ is nearly even (up to a spatial phase shift) for the first time at $463\Delta t=554.057$. This was our initial guess for $T/2$. The phase shift required to make $\eta(x+\theta/2,T/2)$ approximately even and $\varphi(x+\theta/2,T/2)$ approximately odd was found by graphically solving $\varphi(x,T/2)=0$. This gives the initial guess $\theta/2=2.54258$. This choice of $T$ and $\theta$ (with $\eta^{B+C}$ and $\varphi^{B+C}$ as initial conditions) yields $f=2.0\times10^{-11}$ and $\tilde f=1.5\times10^{-10}$. We then minimize $f$ holding $E$ and $\theta$ constant, which gives $f=2.1\times10^{-29}$ and $\tilde f=3.0\times10^{-26}$. We note that $\tilde f$ is computed over $[0,T]$, twice the time over which the solution was optimized by minimizing $f$, and provides independent confirmation of the accuracy of the solution and the symmetry arguments of §\[sec:obj:fun\]. The results are plotted in Figure \[fig:evol:kdv3\]. We omit a plot of the initial guess (the collision of Stokes waves) as it is indistinguishable from the minimized solution. In fact, the relative change in the wave profile and velocity potential is about $0.35$ percent, $$\left(\frac{ \|\eta_\text{Stokes} - \eta_\text{periodic}\|^2 + \|\varphi_\text{Stokes} - \varphi_\text{periodic}\|^2}{ \|\eta_\text{Stokes} - h\|^2 + \|\varphi_\text{Stokes}\|^2} \right)^{1/2} \le 0.0035,$$ and $T/2$ changes even less, from 554.057 (Stokes) to 554.053 (periodic). By construction, $E$ and $\theta/2$ do not change at all. It was not necessary to evolve the Stokes waves to $T/2$, shift space by $\theta/2$, zero out Fourier modes that violate the symmetry condition (\[eq:init\]), and reset $t=0$ to correspond to this new initial state. Doing so increases the decay rate of the Fourier modes (slope of $\ln|\hat\eta_k|$ vs $k$) by a factor of 1.24 in this example, compared to 3.36 in the previous example, where it is definitely worthwhile. ![\[fig:evol:kdv3\] A relative-periodic solution found using a superposition of the Stokes waves labeled B and C in Figure \[fig:bif:stokes\] as a starting guess. Unlike the previous case, the waves do not fully merge at $t=T/2$. ](figs/evol_kdv3){width="\linewidth"} The large change from $T_0/2 = 717.93$ to $T/2=554.053$ is due to nonlinear interaction of the waves. There are two main factors contributing to this change in period. The first is that the waves do not fully combine when they collide. Instead, the trailing wave runs into the leading wave, passing on much of its amplitude and speed. The peaks remain separated by a distance of $d=0.52462$ at $t=T/2$, the transition point where the waves have the same amplitude. Thus, the peak separation changes by $\pi-d$ rather than $\pi$ in half a period. The second effect is that the larger wave slows down the smaller wave more than the smaller slows the larger. Recall from Fig. \[fig:AandC\] that each wave induces a negative potential gradient across the other wave that generates a background flow opposing its direction of travel. Quantitatively, when the waves are well separated, we find that the taller and smaller waves travel at speeds $c_B=0.231077=0.994049c_{B,0}$ and $c_C=0.226153=0.991531c_{C,0}$, respectively. The relative speed is then $(c_B-c_C) = 1.12526(c_{B,0}-c_{C,0})$. Thus, $$\label{eq:ineq} \frac{\pi-d}{c_B-c_C} < \frac{T}{2} < \frac{\pi-d}{c_{B,0}-c_{C,0}} < \frac{T_0}{2} = \frac{\pi}{c_{B,0}-c_{C,0}}, $$ with numerical values $531.5<554.1<598.0<717.9$. This means that both effects together have overestimated the correction needed to obtain $T$ from $T_0$. This is because the relative speed slows down as the waves approach each other, which is expected since the amplitude of the trailing wave decreases and the amplitude of the leading wave increases in this interaction regime. Indeed, the average speed of the waves is $$\label{eq:average:speed} \overline{c_B} = \frac{\theta/2 - d/2}{T/2} = 0.993388c_{B,0}, \qquad \overline{c_C} = \frac{\theta/2 + d/2 - \pi}{T/2} = 0.991737c_{C,0},$$ which are slightly smaller and larger, respectively, than their speeds when well separated. Note that $T/2$ in (\[eq:ineq\]) may be written $T/2=(\pi-d)/(\overline{c_B} - \overline{c_C})$. We used $\theta/2=2.54258+40\pi$ in (\[eq:average:speed\]) to account for the 20 times the waves cross the domain $(0,2\pi)$ in time $T/2$ in addition to the offset shown in Figure \[fig:evol:kdv3\]. Comparison with KdV {#sec:kdv} =================== In the previous section, we observed two types of overtaking collisions for the water wave: one in which the larger wave completely subsumes the smaller wave for a time, and one where the two waves remain distinct throughout the interaction. Similar behavior has of course been observed for the Korteweg-de Vries equation, which was part of our motivation for looking for such solutions. Lax [@lax:1968] classified overtaking collisions of two KdV solitons as bimodal, mixed, or unimodal. Unimodal and bimodal waves are analogous to the ones we computed above, while mixed mode collisions have the larger wave mostly subsume the smaller wave at the beginning and end of the interaction, but with a two-peaked structure re-emerging midway through the interaction. Lax showed that if $1<c_1/c_2<A=(3+\sqrt{5})/2$, the collision is bimodal; if $c_1/c_2>3$, the collision is unimodal; and if $A<c_1/c_2<3$, the collision is mixed. Here $c_1$ and $c_2$ are the wave speeds of the trailing and leading waves, respectively, at $t=-\infty$. Leveque [@leveque:87] has studied the asymptotic dynamics of the interaction of two solitons of nearly equal amplitude. Zou and Su [@zou:su] performed a computational study of overtaking water wave collisions, compared the results to KdV interactions, and found that the water wave collisions ceased to be elastic at third order. Craig *et. al.* [@craig:guyenne:06] also found that solitary water waves collide inelastically. This does not conflict with our results since we optimize the initial conditions to make the collision elastic. Head on collisions have been studied numerically by Su and Mirie [@su:mirie; @mirie:su], experimentally by Maxworthy [@maxworthy:76], and by a mixture of analysis and computation by Craig *et. al.* [@craig:guyenne:06]. Validation of KdV as a model of water waves has also been studied extensively. A formal derivation may be found in Ablowitz and Segur [@ablowitz:segur]. Rigorous justification has been given by Bona, Colin and Lannes [@bona:lannes], building on earlier work by Craig [@craig:kdv] as well as Schneider and Wayne [@schneider:wayne]. According to [@bona:lannes], some gaps still exist in the theory in the spatially periodic case. Experimental studies of the validity of KdV for describing surface waves have been performed by Zabusky and Galvin [@zabusky:galvin] as well as Hammack and Segur [@hammack:segur:74]. Recently, Ostrovsky and Stepanyants [@ostrovsky] have compared internal solitary waves in laboratory experiments to the predictions of various model equations, including KdV, and give a comprehensive overview of the literature on this subject [@ostrovsky]. Our objective in this section is to determine quantitatively whether the solutions of the water wave equations that we computed in §\[sec:num\] are well-approximated by the KdV equation. Following Ablowitz and Segur [@ablowitz:segur], we introduce a small parameter ${\varepsilon}$ and dimensionless variables $$\hat y = \frac{y}{h}, \qquad \hat x = \sqrt{{\varepsilon}}\frac{x}{h}, \qquad \hat t = \sqrt{\frac{{\varepsilon}g}{h}} t, \qquad \hat \eta = \frac{\eta}{{\varepsilon}h}, \qquad \hat \phi = \frac{\phi}{\sqrt{{\varepsilon}g h^3}},$$ where $h$ is the fluid depth. We assume the bottom boundary is at $y=-h$ rather than 0 in this derivation, so that $\hat y$ runs from $-1$ to ${\varepsilon}\hat\eta$. The Laplacian becomes $\Delta_{\varepsilon}= h^{-2}\big( {\varepsilon}\partial_{\hat{x}}^2 + \partial_{\hat{y}}^2 \big)$, which allows for $\hat\phi = \hat\phi_0 + {\varepsilon}\hat\phi_1 + {\varepsilon}^2\hat\phi_2 + \cdots$ to be computed order by order, with leading terms satisfying $$\hat\phi_{0,\hat y} = 0, \qquad \hat\phi_1 = -\frac{1}{2}(1+\hat y)^2\hat\phi_{0,\hat x\hat x}, \qquad \hat\phi_2 = \frac{1}{24}(1+\hat y)^4\hat\phi_{0,\hat x\hat x\hat x\hat x}.$$ Here we used $\Delta\phi=0$ and $\phi_y(x,-h)=0$. Note that $\hat\phi_0$ is independent of $\hat y$, and agrees with the velocity potential $\phi$ on the bottom boundary, up to rescaling: $$\hat\phi_0(\hat x,\hat t) = ({\varepsilon}gh^3)^{-1/2}\phi(x,-h,t).$$ From the equations of motion, $\eta_t = \phi_y - \eta_x\phi_x$ and $\phi_t + \frac{1}{2}\phi_x^2 + \frac{1}{2}\phi_y^2 + g\eta = 0$, one finds that $$\begin{aligned} \hat\eta_{\hat t} + \hat u_{\hat x} &= {\varepsilon}\big\{ {\textstyle}\frac{1}{6}\hat u_{\hat x\hat x\hat x} - (\hat\eta\hat u)_{\hat x} \big\} + O({\varepsilon}^2), \\ \hat u_{\hat t} + \hat\eta_{\hat x} &= {\varepsilon}\big\{ {\textstyle}\frac{1}{2} \hat u_{\hat x\hat x\hat t} - \frac{1}{2}\partial_{\hat x}(\hat u)^2 \big\} + O({\varepsilon}^2),\end{aligned}$$ where $\hat u(\hat x,\hat t) = \partial_{\hat x}\hat\phi_0(\hat x,\hat t)$. Expanding $\hat\eta=\hat\eta_0 + {\varepsilon}\hat\eta_1 +\cdots$, $\hat u=\hat u_0 + {\varepsilon}\hat u_1 +\cdots$, we find that $$\begin{aligned} \hat\eta_0 = f(\hat x - \hat t; \tau) + g(\hat x + \hat t; \tau), \\ \hat u_0 = f(\hat x - \hat t; \tau) - g(\hat x + \hat t; \tau), \end{aligned} \qquad \begin{aligned} 2f_\tau + 3ff_r + (1/3)f_{rrr} &= 0, \\ -2g_\tau + 3gg_l + (1/3)g_{lll} &= 0, \end{aligned}$$ where we have introduced characteristic coordinates $r=\hat x-\hat t$, $l = \hat x + \hat t$ as well as a slow time scale $\tau={\varepsilon}\hat t$ to eliminate secular growth in the solution with respect to $r$ and $l$ at first order in ${\varepsilon}$; see [@ablowitz:segur] for details. The notational conflict of $g(l,\tau)$ with the acceleration of gravity, $g$, is standard, and will not pose difficulty below. ![\[fig:kdv:cmp1\] Comparison of the solutions of the KdV and Euler equations, initialized identically with the superposition of Stokes waves labeled A and B in Figure \[fig:bif:stokes\]. The final time $T$ is set to $138.399$, as in Fig. \[fig:align:stokes\], when the Euler solution nearly returns to its initial configuration after a single overtaking collision. ](figs/kdv1){width="\linewidth"} In our case, the waves travel to the right, so we may set $g(l,\tau)=0$ in the formulas above. Returning to dimensional variables, we then have $$\eta(x,t) = h{\varepsilon}f\left(\sqrt{\varepsilon}\left(\frac{x}{h} - \sqrt{\frac{g}{h}} t \right),\sqrt{\frac{g}{h}}{\varepsilon}^{3/2}t\right),$$ which satisfies $$\label{eq:dim:kdv} \eta_t + \alpha \eta_x + \frac{3\sqrt{gh}}{2h}\eta\eta_x + \frac{1}{6}\sqrt{gh}\,h^2\eta_{xxx} = 0,$$ where $\alpha=\sqrt{gh}$. Note that ${\varepsilon}$ drops out. For comparison with the results of §\[sec:num\], we will add $h$ to $\eta$ and set $\alpha=-\frac{1}{2}\sqrt{gh}$ instead. In Figure \[fig:kdv:cmp1\], we compare the solution of (\[eq:dim:kdv\]), with initial condition $\eta(x,0) = \eta_0^{A+B}(x)$, defined similarly to $\eta_0^{A+C}(x)$ in (\[eq:AandC\]), to the solution of the free-surface Euler equations shown in Figs. \[fig:align:stokes\] and \[fig:evol:kdv1\]. Shortly after the waves are set in motion, the KdV solution develops high-frequency oscillations behind the larger peak that travel left and quickly fill up the computational domain with radiation. The solution of the Euler equations remains much smoother. The large peak of the KdV solution also travels $3.4\%$ faster, on average, than the corresponding peak of the Euler solution, so that at $t=138.399$, when the taller Euler wave has traversed the domain 6 times, the taller KdV wave has traversed it $6.206$ times. For our purposes, these discrepancies are much too large for KdV to be a useful model, and we conclude that the first example in §\[sec:num\] is well outside of the KdV regime. In this comparison, timestepping the KdV equation was done with the 8 stage, 5th order implicit/explicit Runge-Kutta method of Kennedy and Carpenter [@carpenter]. Spatial derivatives were computed spectrally using the 36th order filter of Hou and Li [@hou:li:07]. We found that 2048 spatial grid points and 96000 timesteps was sufficient to reduce the error at $t=138.399$ below $5\times 10^{-6}$ near the larger peak and below $6\times 10^{-7}$ elsewhere, based on comparing the solution to one with 3072 grid points and 192000 timesteps. Our solutions of the Euler equations are much more accurate since there are no second or third spatial derivative terms present to make the equations stiff. Thus, we can use 8th or 15th order explicit timestepping rather than 5th order implicit/explicit timestepping. Monitoring energy conservation and performing mesh refinement studies suggests that we obtain 13–14 digits of accuracy in the solutions of the Euler equations, at which point roundoff error prevents further improvement in double-precision arithmetic. ![\[fig:kdv:cmp2\] Comparison of the solutions of the KdV and Euler equations, both initialized with the superposition of Stokes waves labeled B and C in Figure \[fig:bif:stokes\]. $T=1108.11$ here. ](figs/kdv2){width="\linewidth"} In Figure \[fig:kdv:cmp2\], we repeat this computation using initial conditions corresponding to the superposition of Stokes waves $\eta_0^{B+C}(x)$, which was used as a starting guess for the second example of §\[sec:num\]. This time the KdV solution does not develop visible high-frequency radiation in the wave troughs, and the solutions of KdV and Euler remain close to each other for much longer. However, the interaction time for a collision also increases, from $T=138.399$ in the first example to $T=1108.11$ here. In Fig. \[fig:kdv:cmp2\], by $t=T/6$, the taller KdV and Euler waves have visibly separated from each other, and by $t=T/2$, when the Euler waves have reached their minimum approach distance, the KdV solution is well ahead of the Euler solution. Thus, while there is good qualitative agreement between the KdV and Euler solutions, they do not agree quantitatively over the time interval of interest. From this point of view, the second example of §\[sec:num\] also lies outside of the KdV regime. An alternative measure of the agreement between KdV and Euler is to compare the solutions from §\[sec:num\] with nearby relative-periodic solutions of KdV. In other words, we wish to quantify how much the initial conditions and period have to be perturbed to convert a relative-periodic solution of the Euler equations into one for the KdV equations. Since we used a superposition of Stokes waves for the initial guess to find time-periodic and relative-periodic solutions of the Euler equations, we will use a similar superposition (of cnoidal waves) for KdV. The vertical crest-to-trough heights of the three Stokes waves considered in §\[sec:num\] are $$\label{eq:H:ABC} H_A = 0.028918699, \qquad H_B = 0.004973240, \qquad H_C = 0.002683648.$$ Well-known [@kdv:1895; @dingemans] periodic traveling wave solutions of (\[eq:dim:kdv\]) are given by $$\begin{gathered} \eta(x,t) = h - H + \frac{H}{m}\left(1 - \frac{E(m)}{K(m)}\right) + H{\operatorname}{cn}^2\left( 2K(m)\frac{x-ct}{\lambda}\bigg\vert m\right), \\ \lambda = \sqrt{\frac{16mh^3}{3H}}\,K(m), \qquad c = \left[1 - \frac{H}{2h} + \frac{H}{mh}\left(1 - \frac{3E(m)}{2K(m)}\right)\right]\sqrt{gh},\end{gathered}$$ where we added $h$ to $\eta$ to match the change in $\alpha$ from $\sqrt{gh}$ to $-\frac{1}{2}\sqrt{gh}$ in (\[eq:dim:kdv\]). Here $K(m)$ and $E(m)$ are the complete elliptic integrals of the first and second kind, respectively, and ${\operatorname}{cn}(z|m)$ is one of the Jacobi elliptic functions [@dingemans; @gradshteyn]. In our case $\lambda=2\pi$, $g=1$ and $h=0.05$. For each $H$ in (\[eq:H:ABC\]), we solve the $\lambda$ equation for $m$ using Mathematica [@mma], and then evaluate $\eta(x,0)$ on a uniform grid that is fine enough that its Fourier coefficients decay below machine roundoff. The values of $m' = 1-m$ are $$m'_A = 1.81924\times10^{-35}, \qquad m'_B = 1.98689\times10^{-14}, \qquad m'_C = 1.79643\times10^{-10}.$$ This approach requires extended precision arithmetic to compute $m$ and evaluate $\eta$, but the running time takes only a few seconds on a typical laptop. A periodized version of the simpler ${\operatorname}{sech}^2$ formula could be used for the first two waves, but decays too slowly for wave $C$ to be spatially periodic to roundoff accuracy. Once these cnoidal waves have been computed, we superpose their initial conditions to form $\eta_0^{A+B}$ and $\eta_0^{B+C}$, just as in §\[sec:num\]. It is well-known that a superposition of $N$ cnoidal waves retain this form when evolved via KdV, with $N$ amplitude and $N$ phase parameters governed by an ODE describing pole dynamics in the complex plane [@kruskal:pole; @airault:kdv; @deconinck:segur]. In the $N=2$ case, the solutions are relative-periodic. ![\[fig:kdv:cmp5\] Comparison of time-periodic solution found in §\[sec:num\] to nearby relative-periodic two phase cnoidal solution of KdV. The periods are $T=138.387$ and $113.079$, respectively. ](figs/kdv5){width=".98\linewidth"} ![\[fig:kdv:cmp4\] Comparison of relative-periodic solution found in §\[sec:num\] to nearby relative-periodic two phase cnoidal solution of KdV. The periods are $T=1108.11$ and $1068.73$, respectively. ](figs/kdv4){width=".98\linewidth"} Figures \[fig:kdv:cmp5\] and \[fig:kdv:cmp4\] compare the time-periodic and relative-periodic solutions of the Euler equations, computed in §\[sec:num\], to these cnoidal solutions of KdV. Since the periods are different, only the initial conditions are compared. In the larger-amplitude example, shown in Fig. \[fig:kdv:cmp5\], the Euler solution is not as flat in the wave trough as the cnoidal solution due to an additional oscillatory component (the “tuned” radiation). From the difference plot in the right panel, we see that the crest-to-trough amplitude of these higher frequency oscillations is roughly $6\times10^{-4}$, or $2.1\%$ of the wave height $H_A$. The Euler solution is time-periodic with period $T_\text{Euler}=138.387$ while the cnoidal solution is relative-periodic, returning to a spatial phase shift of its initial condition at $T_\text{KdV}=113.079$, which differs from $T_\text{Euler}$ by $18\%$. In the smaller-amplitude example, shown in Fig. \[fig:kdv:cmp4\], both solutions have smooth, flat wave troughs, and it is difficult to distinguish one from the other in the left panel. The crest-to-trough amplitude of the difference in the right panel is roughly $5.5\times10^{-5}$, or $1.1\%$ of $H_B$. The relative change in period is $(T_\text{Euler}-T_\text{KdV})/ T_\text{Euler} = 3.6\%$. While the left panels of Figures \[fig:kdv:cmp5\] and \[fig:kdv:cmp4\] show close agreement between relative-periodic solutions of the Euler and KdV equations at $t=0$, it should be noted that the wave amplitudes of the cnoidal solutions were chosen to minimize the discrepancy in these figures. The change in period by $18\%$ and $3.6\%$, respectively, is perhaps a better measure of agreement. ![\[fig:kdv:cmp7\] Comparison of KdV and Euler solutions, both initialized with a 2-phase cnoidal wave with peaks matching the heights of the Stokes waves labeled A and B (left) or B and C (right) in Fig. \[fig:bif:stokes\]. Here $T=138.387$ (left) and $T=1068.73$ (right). ](figs/kdv7){width=".98\linewidth"} A final comparison of the two equations is made in Fig. \[fig:kdv:cmp7\], where we evolve the Euler equations with the KdV initial conditions. This requires an initial condition for $\varphi(x)=\phi(x,\eta(x))$, where we have suppressed $t$ in the notation for this discussion since it is held fixed at 0. Based on the derivation presented above, we first solve $\phi_x(x,0) = \sqrt{g/h}[\eta(x)-h]$ for $\phi$ on the bottom boundary. We then use the approximation $$\varphi(x) \approx \phi(x,0) - \frac{\eta(x)^2}{2}\phi_{xx}(x,0) + \frac{\eta(x)^4}{24}\phi_{xxx}(x,0)$$ to evaluate $\phi$ on the free surface. In the left panel of Figure \[fig:kdv:cmp7\], the larger wave grows and overturns before $t=T/400$ when evolved under the Euler equations, instead of traveling to the right when evolved via KdV. To handle wave breaking, we switched to an angle-arclength formulation of the free-surface Euler equations [@hls94; @vtxs1]. In the small-amplitude example in the right panel, the Euler solution develops visible radiation and falls slightly behind the KdV solution, although the phases are closer at $T/2$ than the result of evolving the Stokes waves under KdV in Figure \[fig:kdv:cmp2\]. We also tried evaluating $$\phi(x,y)=\sqrt{g/h} \sum_{k=1}^\infty 2k^{-1}\hat\eta_k \sin(kx)\cosh(ky)$$ at $y=\eta(x)$ to obtain the initial condition for $\varphi(x)$, where $\hat\eta_k$ are the Fourier modes of $\eta(x)$ at $t=0$, but the results were worse for the large-amplitude example — the wave breaks more rapidly — and were visually indistinguishable in the small-amplitude example from the results plotted in Fig. \[fig:kdv:cmp7\]. In summary, the large-amplitude time-periodic solution of the Euler equations found in §\[sec:num\] is well outside of the KdV regime by any measure, and the small-amplitude relative-periodic solution is closer, but not close enough to achieve quantitative agreement over the entire time interval of interest. Conclusion ========== We have demonstrated that the small amount of background radiation produced when two Stokes waves interact in shallow water can often be tuned to obtain time-periodic and relative-periodic solutions of the free-surface Euler equations. Just as for the Korteweg-de Vries equation, the waves can fully merge when they collide or remain well-separated. However, the comparison is only qualitative as the waves are too large to be well-approximated by KdV theory. In future work, we will study the stability of these solutions using Floquet theory. Preliminary results suggest that the first example considered above is unstable to harmonic perturbations while the second example is stable. In the stable case, an interesting open question is whether the Stokes waves used as a starting guess for the minimization algorithm, which have the same energy as the relative-periodic solution found, might remain close to it forever, executing almost-periodic oscillations around it. Presumably $\theta$ would need to be varied slightly for this to be true, since $\theta$ is a free parameter that we selected by hand to obtain a small value of $\tilde f$ for the initial guess. Another open question is whether there are analogues for the Euler equations of $N$-phase quasi-periodic solutions of the KdV equation with $N\ge3$. We are confident that the methods of this paper could be used to construct degenerate cases of $N\ge3$ solitary water waves colliding elastically in a time-periodic or relative-periodic fashion, along the lines of what was done for the Benjamin-Ono equation in [@benj2]. Computing more general quasi-periodic dynamics of the form $\eta(x,t)=H(\vec\kappa x + \vec\omega t + \vec\alpha)$, $\varphi(x,t)=\Phi(\vec\kappa x + \vec\omega t + \vec\alpha)$ with $H,\Phi\in C(\mathbb{T}^N)$ and $\vec\kappa$, $\vec\omega$, $\vec\alpha\in\mathbb{R}^N$ seems possible in principle using a more sophisticated shooting method to determine $H$, $\Phi$ and $\vec\omega$. Existence of such solutions for the Euler equations would show that non-integrable equations can also support recurrent elastic collisions even if they cannot be represented as $N$-phase superpositions of elliptic functions. [10]{} Mark J. Ablowitz and Harvey Segur, *Solitons and the inverse scattering transform*, SIAM, Philadelphia, 1981. H. Airault, H. P. McKean, and J. Moser, *Rational and elliptic solutions of the [Korteweg-de Vries]{} equation and a related many-body problem*, Comm. Pure Appl. Math. **30** (1977), 95–148. D. M. Ambrose and J. Wilkening, *Global paths of time-periodic solutions of the [Benjamin]{}–[Ono]{} equation connecting pairs of traveling waves*, Comm. App. Math. and Comp. Sci. **4** (2009), no. 1, 177–215. [to3em]{}, *Computation of symmetric, time-periodic solutions of the vortex sheet with surface tension*, Proc. Nat. Acad. Sci. **107** (2010), no. 8, 3361–3366. [to3em]{}, *Computation of time-periodic solutions of the [Benjamin]{}–[Ono]{} equation*, J. Nonlinear Sci. **20** (2010), no. 3, 277–308. [to3em]{}, *Dependence of time-periodic votex sheets with surface tension on mean vortex sheet strength*, Procedia IUTAM **11** (2014), 15–22. G. R. Baker, D. I. Meiron, and S. A. Orszag, *Generalized vortex methods for free-surface flow problems*, J. Fluid Mech. **123** (1982), 477–501. G. R. Baker and C. Xie, *Singularities in the complex physical plane for deep water waves*, J. Fluid Mech. **685** (2011), 83–116. J. L. Bona, T. Colin, and D. Lannes, *Long wave approximations for water waves*, Arch. Rational Mech. Anal. **178** (2005), 373–410. R. K.-C. Chan and R. Street, *A computer study of finite amplitude water waves*, J. Comput. Phys. **6** (1970), 68–94. G. A. Chandler and I. G. Graham, *The computation of water waves modelled by [Nekrasov’s]{} equation*, SIAM J. Numer. Anal. **30** (1993), no. 4, 1041–1065. B. Chen and P. G. Saffman, *Numerical evidence for the existence of new types of gravity waves of permanent form on deep water*, Stud. Appl. Math. **62** (1980), 1–21. M. J. Cooker, P. D. Weidman, and D. S. Bale, *Reflection of a high-amplitude solitary wave at a vertical wall*, J. Fluid Mech. **342** (1997), 141–158. W. Craig, *An existence theory for water waves and the [Boussinesq]{} and [Korteweg-de Vries]{} scaling limits*, Comm. Partial Diff. Equations **10** (1985), 787–1003. W. Craig, P. Guyenne, J. Hammack, D. Henderson, and C. Sulem, *Solitary water wave interactions*, Phys. Fluids **18** (2006), 057106. W. Craig and C. Sulem, *Numerical simulation of gravity waves*, J. Comput. Phys. **108** (1993), 73–83. A. D. D. Craik, *The origins of water wave theory*, Ann. Rev. Fluid Mech. **36** (2004), 1–28. [to3em]{}, *George gabriel stokes on water wave theory*, Ann. Rev. Fluid Mech. **37** (2005), 23–42. B. Deconinck and H. Segur, *Pole dynamics for elliptic solutions of the [KdV]{} equation*, Mathematical Physics, Analysis and Geometry **3** (2000), 49–74. Maarten W. Dingemans, *Water wave propagation over uneven bottoms, part 2. non-linear wave propagation*, World Scientific, Singapore, 1997. S. Govindjee, T. Potter, and J. Wilkening, *Cyclic steady states of treaded rolling bodies*, Int. J. Numer. Meth. Engng. (2014), (accepted). I. S. Gradshteyn and I. M. Ryzhik, *Table of integrals, series and products*, 7th ed., Academic Press, Amsterdam, 2007. E. Hairer, S. P. Norsett, and G. Wanner, *Solving ordinary differential equations [I]{}: Nonstiff problems, 2nd edition*, Springer, Berlin, 2000. J. L. Hammack and H. Segur, *The [Korteweg-de Vries]{} equation and water waves. part 2. comparison with experiments*, J. Fluid Mech. **65** (1974), 289–314. T. Y. Hou and R. Li, *Computing nearly singular solutions using pseudo-spectral methods*, J. Comput. Phys. **226** (2007), 379–397. T. Y. Hou, J. S. Lowengrub, and M. J. Shelley, *Removing the stiffness from interfacial flows with surface tension*, J. Comput. Phys. **114** (1994), 312–338. R. S. Johnson, *A modern introduction to the mathematical theory of water waves*, Cambridge University Press, Cambridge, UK, 1997. C. A. Kennedy and M. H. Carpenter, *Additive [Runge]{}-[Kutta]{} schemes for convection-diffusion-reaction equations*, Appl. Numer. Math. **44** (2003), no. 1–2, 139–181. D. J. Korteweg and G. de Vries, *On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves*, Philosophical Magazine **39** (1895), 422–443. R. Krasny, *Desingularization of periodic vortex sheet roll-up*, J. Comput. Phys. **65** (1986), 292–313. M. D. Kruskal, *The [Korteweg-de Vries]{} equation and related evolution equations*, Nonlinear wave motion, Lectures in Appl. Math., vol. 15, American Mathematical Society, Providence, 1974, pp. 61–83. P. D. Lax, *Integrals of nonlinear equations of evolution and solitary waves*, Comm. Pure Appl. Math. **21** (1968), 467–490. R. J. LeVeque, *On the interaction of nearly equal solitons in the kdv equation*, SIAM J. Appl. Math. **47** (1987), 254–262. M. S. Longuet-Higgins and E. D. Cokelet, *The deformation of steep surface waves on water. [I]{}. a numerical method of computation*, Proc. Royal Soc. A **350** (1976), 1–26. T. Maxworthy, *Experiments on collisions between solitary waves*, J. Fluid Mech. **76** (1976), 177–185. G N Mercer and A J Roberts, *[Standing waves in deep water: Their stability and extreme form]{}*, Phys. Fluids A **4** (1992), no. 2, 259–269. P. A. Milewski, J.-M. Vanden-Broeck, and Z. Wang, *Dynamics of steep two-dimensional gravity–capillary solitary waves*, J. Fluid Mech. **664** (2010), 466–477. R. M. Mirie and C. H. Su, *Collisions between two solitary waves, [Part II]{}*, J. Fluid Mech. **115** (1982), 475–492. J. Nocedal and S. J. Wright, *Numerical optimization*, Springer, New York, 1999. L. A. Ostrovsky and Y. A. Stepanyants, *Internal solitons in laboratory experiments: Comparison with theoretical models*, Chaos **15** (2005), 037111:1–28. P. Plotnikov and J. Toland, *Nash-moser theory for standing water waves*, Arch. Rat. Mech. Anal. **159** (2001), 1–83. Chris H. Rycroft and Jon Wilkening, *Computation of three-dimensional standing water waves*, J. Comput. Phys. **255** (2013), 612–638. G. Schneider and C. E. Wayne, *The long-wave limit for the water wave problem. i. the case of zero surface tension*, Comm. Pure Appl. Math. **53** (2000), 1475–1535. C. H. Su and R. M. Mirie, *On head-on collisions between two solitary waves*, J. Fluid Mech. **98** (1980), 509–525. J.-M. Vanden-Broeck, *Elevation solitary waves with surface tension*, Phys. Fluids A **3** (1991), 1989–1993. G. B. Whitham, *Linear and nonlinear waves*, Wiley, New York, 1974. J. Wilkening, *Breakdown of self-similarity at the crests of large amplitude standing water waves*, Phys. Rev. Lett **107** (2011), 184501. [to3em]{}, *Traveling-standing water waves*, (2014), (in preparation). J. Wilkening and J. Yu, *Overdetermined shooting methods for computing standing water waves with spectral accuracy*, Comput. Sci. Disc. **5** (2012), 014017:1–38. M. O. Williams, J. Wilkening, E. Shlizerman, and J. N. Kutz, *Continuation of periodic solutions in the waveguide array mode-locked laser*, Physica D **240** (2011), no. 22, 1791–1804. , *Mathematica, version 8.0*, Champaign, IL, 2010. N. J. Zabusky and C. J. Galvin, *Shallow-water waves, the [Korteweg-de Vries]{} equation and solitons*, J. Fluid Mech. **47** (1970), 811–824. Q. Zou and C. H. Su, *Overtaking collisions between two solitary waves*, Phys. Fluids **29** (1986), 2113–2123. [^1]: This research was supported in part by the Director, Office of Science, Computational and Technology Research, U.S. Department of Energy under Contract No. DE-AC02-05CH11231, and by the National Science Foundation through grant DMS-0955078.

--- abstract: 'We present a short and self-contained proof of the following result: a random time is an honest time avoiding all stopping times if and only if it coincides with the (first) time that a nonnegative local martingale with continuous supremum process and zero terminal value achieves its overall maximum, given that this time of maximum is almost surely finite.' address: 'Constantinos Kardaras, Statistics Department, London School of Economics and Political Science, 10 Houghton Street, London, WC2A 2AE, UK.' author: - Constantinos Kardaras bibliography: - 'hon\_times\_new.bib' title: On the characterisation of honest times avoiding all stopping times --- The Characterisation Result {#sec: hon times avoid stop times} =========================== Consider a filtered probability space ${(\Omega, \, {\mathbf{F}}, \, {\mathbb{P}})}$, where ${\mathbf{F}}= ({\mathcal{F}}_t)_{t \in {{{\mathbb R}_+}}}$ is a filtration satisfying the usual conditions of right-continuity and saturation by ${\mathbb{P}}$-null sets of ${\mathcal{F}}{\, := \,}\bigvee_{t \in {\mathbb R}_+} {\mathcal{F}}_t$. All (local) martingales and supermartingales on ${(\Omega, \, {\mathbf{F}}, \, {\mathbb{P}})}$ are assumed to have [càdlàg]{} paths. A *random time* is a ${[0,\infty]}$-valued, ${\mathcal{F}}$-measurable random variable. The random time $\rho$ is said to *avoid all stopping times* if ${\mathbb{P}}[\rho = \tau] = 0$ holds whenever $\tau$ is a (possibly, infinite-valued) stopping time. The random time $\rho$ is called an *honest time* if for all ${t \in {\mathbb R}_+}$ there exists an ${\mathcal{F}}_t$-measurable random variable $R_t$ such that $\rho = R_t$ holds on ${\left\{\rho \leq t\right\}}$. Honest times constitute the most important class of random times outside the realm of stopping times. They have been extensively studied in the literature, especially in relation to filtration enlargements. It is impossible to present here the vast literature on the subject of honest times; we indicatively mention the early papers [@MR509204] [@MR511775], [@MR519998] and [@MR519996], as well as the monographs [@MR604176] and [@MR884713]. Lately, there has been considerable revival to the study of honest times, due to questions arising from the field of Financial Mathematics—see, for example, [@MR1802597], [@NP], [@FJS] and the references therein. Use ${\mathcal{M}_0}$ to denote the set of all nonnegative local martingales $L$ with $L_0 = 1$, $L_\infty {\, := \,}\lim_{t \to \infty} L_t = 0$, and such that its supremum process $L^* {\, := \,}\sup_{t \in [0, \cdot]} L_t$ is continuous, all the previous holding in the ${\mathbb{P}}$-a.s. sense. (Note that the limit in the definition of $L_\infty$ exists in the ${\mathbb{P}}$-a.s. sense, in view of the nonnegative supermartingale convergence theorem.) For $L \in {\mathcal{M}_0}$, define $$\label{eq: rml} {\rho^L_{\min}}{\, := \,}\inf {\left\{{t \in {\mathbb R}_+}{\ | \ }L_t = L^*_\infty\right\}},$$ where by convention we set ${\rho^L_{\min}}= \infty$ if the latter set is empty. Since $L$ is [càdlàg]{}, it is straightforward that $L_{{\rho^L_{\min}}} = L^*_\infty$ holds on the event ${\left\{{\rho^L_{\min}}< \infty\right\}}$. For $L \in {\mathcal{L}_0}$ and ${t \in {\mathbb R}_+}$, define the ${\mathcal{F}}_t$-measurable random variable $R^L_t {\, := \,}\inf {\left\{s \in [0, t] {\ | \ }L_s = L^*_t\right\}} \wedge t$, and note that ${\rho^L_{\min}}= R^L_t$ holds on ${\left\{{\rho^L_{\min}}\leq t\right\}}$. Therefore, ${\rho^L_{\min}}$ is an honest time. Use ${\mathcal{L}_0}$ to denote the set of all $L \in {\mathcal{M}_0}$ such that additionally ${\mathbb{P}}{\left[{\rho^L_{\min}}< \infty\right]} = 1$ holds. When $L \in {\mathcal{L}_0}$, it is true that ${\rho^L_{\min}}$ avoids all stopping times, and that it is actually the canonical example of an honest time avoiding all stopping times, as the following result shows. \[thm: main\] For a random time $\rho$, the following two statements are equivalent: 1. $\rho$ is an honest time avoiding all stopping times. 2. $\rho = {\rho^L_{\min}}$ holds for some $L \in {\mathcal{L}_0}$. Under the additional assumption that all martingales on ${(\Omega, \, {\mathbf{F}}, \, {\mathbb{P}})}$ have ${\mathbb{P}}$-a.s. continuous paths, a version of Theorem \[thm: main\] was established in [@MR2247846]; the case where martingales on ${(\Omega, \, {\mathbf{F}}, \, {\mathbb{P}})}$ do not necessarily have ${\mathbb{P}}$-a.s. continuous paths is treated in [@NP], using previously-established results regarding so-called processes of class $(\Sigma)$. Note that in both of the previous papers, the fact that $\rho$ avoids all stopping times in an *assumption*, and statements (1) and (2) of Theorem \[thm: main\] are replaced by $(1')$ $\rho$ is the end of an optional set (which can be seen to be equivalent to $\rho$ being an honest time) and $(2')$ $\rho = {\rho^L_{\max}}$ holds for some $L \in {\mathcal{M}_0}$ (where ${\rho^L_{\max}}$ is given in below), respectively. Remark \[rem: M vs L\] that follows clarifies further this point. In Section \[sec: proof\], we provide a short and self-contained proof of Theorem \[thm: main\], which does not rely on any previous results. \[rem: M vs L\] Instead of ${\rho^L_{\min}}$ of , the *last* time of supremum of $L$ is used in [@MR2247846] and [@NP], via considering $$\label{eq: rmal} {\rho^L_{\max}}{\, := \,}\sup {\left\{{t \in {\mathbb R}_+}{\ | \ }L_t = L^*_\infty\right\}}.$$ Furthermore, the requirement that the corresponding random time is ${\mathbb{P}}$-a.s. finite is thereby absent. In the case where $L$ has ${\mathbb{P}}$-a.s. continuous paths, both conditions ${\mathbb{P}}{\left[{\rho^L_{\max}}< \infty\right]} = 1$ and that $L$ sampled at ${\rho^L_{\max}}$ actually achieves the maximum when $L \in {\mathcal{M}_0}$ are automatically satisfied. In contrast, in the case where jumps may be present the extra requirement ${\mathbb{P}}{\left[{\rho^L_{\min}}< \infty\right]} = 1$ for $L \in {\mathcal{M}_0}$ to belong to ${\mathcal{L}_0}$ is essential for the validity of Theorem \[thm: main\]. Indeed, consider a complete probability space $(\Omega, \, {\mathcal{F}}, \, {\mathbb{P}})$ that supports a ${\mathbb R}_+$-valued random variable $\tau$ such that ${\mathbb{P}}{\left[\tau > t\right]} = e^{-t}$ holds for all ${t \in {\mathbb R}_+}$. Define ${\mathbf{F}}$ as the (usual augmentation of the) smallest filtration which makes $\tau$ a stopping time. Then, define the process $L$ via $L_t = \exp(t) {\mathbb{I}}_{{\left\{\tau > t\right\}}}$ for all ${t \in {\mathbb R}_+}$. It is straightforward to check that $L \in {\mathcal{M}_0}$. However, since ${\mathbb{P}}{\left[L_\tau = 0\right]} = 1$, it is immediate that ${\mathbb{P}}{\left[{\rho^L_{\min}}= \infty\right]} = 1$, which shows in a dramatic fashion the failure of ${\rho^L_{\min}}$ to avoid all stopping times. Whenever $L \in {\mathcal{L}_0}$ and $\rho$ is *any* random time such that ${\mathbb{P}}{\left[L_\rho = L^*_\infty\right]} = 1$ holds, Lemma \[lem: unique\] in Section \[sec: proof\] implies that ${\mathbb{P}}{\left[\rho = {\rho^L_{\min}}\right]} = 1$. Therefore, a process $L \in {\mathcal{L}_0}$ has a ${\mathbb{P}}$-a.s. unique time of maximum. In particular, ${\mathbb{P}}{\left[{\rho^L_{\min}}= {\rho^L_{\max}}< \infty\right]} = 1$ holds whenever $L \in {\mathcal{L}_0}$. Proof of Theorem \[thm: main\] {#sec: proof} ============================== During the course of the proof of Theorem \[thm: main\], and in an effort to be as self-contained as possible, we shall provide details for every step. For a random time $\sigma$ and a process $X = (X_t)_{{t \in {\mathbb R}_+}}$, $X^\sigma = (X_{\sigma \wedge t})_{{t \in {\mathbb R}_+}}$ will denote throughout the process $X$ stopped at $\sigma$. For any unexplained, but fairly standard, notation and facts regarding stochastic analysis, we refer the reader to [@MR1780932]. A couple of auxiliary results ----------------------------- The two results presented below will be used in the proof of both implications later. The first auxiliary result is a slightly elaborate version of Doob’s maximal identity—see [@MR2247846]. It will be useful throughout, sometimes in its “conditional” version. The second result implies in particular that ${\rho^L_{\min}}$ avoids all stopping times whenever $L \in {\mathcal{L}_0}$. \[lem: Doob\_maximal\] Let $L$ be a nonnegative local martingale with $L_0 = 1$. Then, ${\mathbb{P}}{\left[ L^*_\infty > x\right]} \leq 1/x$ holds for all $x \in (1, \infty)$. Furthermore, ${\mathbb{P}}{\left[ L^*_\infty > x\right]} = 1/x$ holds for all $x \in (1, \infty)$ if and only if $L \in {\mathcal{M}_0}$. For $x \in (1, \infty)$, define the stopping time $\tau_x {\, := \,}\inf {\left\{t \in {\mathbb R}_+ {\ | \ }L_t > x\right\}}$, and note that ${\left\{L^*_\infty > x\right\}} = {\left\{\tau_x < \infty\right\}}$. Since ${\mathbb{E}}{\left[L^*_{\tau_x}\right]} \leq x + {\mathbb{E}}{\left[L_{\tau_x}\right]} \leq x+1$, $L^{\tau_x}$ is a uniformly integrable martingale for all $x \in (1, \infty)$. It follows that $x {\mathbb{P}}{\left[ L^*_\infty > x\right]} = x {\mathbb{P}}[\tau_x < \infty] = {\mathbb{E}}[x {\mathbb{I}}_{{\left\{\tau_x < \infty\right\}}}] \leq {\mathbb{E}}[L_{\tau_x}] = 1$ for $x \in (1, \infty)$, with equality holding if and only if ${\mathbb{P}}[L_{\tau_x} = x {\mathbb{I}}_{{\left\{\tau_x < \infty\right\}}}] = 1$. Whenever $L \in {\mathcal{M}_0}$, the equality ${\mathbb{P}}[L_{\tau_x} = x {\mathbb{I}}_{{\left\{\tau_x < \infty\right\}}}] = 1$ is immediate for all $x \in (1, \infty)$. Conversely, assume that ${\mathbb{P}}[L_{\tau_x} = x {\mathbb{I}}_{{\left\{\tau_x < \infty\right\}}}] = 1$ holds for all for $x \in (1, \infty)$. It is clear that $L^*$ must have ${\mathbb{P}}$-a.s. continuous paths; furthermore, since ${\mathbb{P}}\big[ \bigcup_{{n \in {\mathbb N}}} {\left\{\tau_n = \infty\right\}} \big] = 1$, ${\mathbb{P}}[ L_\infty = 0] = 1$ follows. Therefore, $L \in {\mathcal{M}_0}$. \[lem: unique\] Suppose that $L \in {\mathcal{M}_0}$, and let $\rho$ be any random time such that ${\mathbb{P}}\big[L_\rho = L^*_\infty \big] = 1$. Then, $L \in {\mathcal{L}_0}$, ${\mathbb{P}}[\rho = {\rho^L_{\min}}] = 1$, and $\rho$ avoids all stopping times. Since ${\mathbb{P}}{\left[\rho = \infty, L_\rho = L^*_\infty\right]} = 0$ holds for $L \in {\mathcal{M}_0}$, ${\mathbb{P}}{\left[\rho = \infty\right]} = 0$ follows. In view of the obvious equality ${\mathbb{P}}\big[ {\rho^L_{\min}}\leq \rho] = 1$, we obtain $L \in {\mathcal{L}_0}$. Note the following set-inclusions for each ${t \in {\mathbb R}_+}$, valid modulo ${\mathbb{P}}$: $${\left\{\sup_{v \in [t, \infty)} L_v > L^*_t\right\}} \subseteq {\left\{{\rho^L_{\min}}> t\right\}}, \quad {\left\{\rho > t\right\}} \subseteq {\left\{\sup_{v \in [t, \infty)} L_v \geq L^*_t\right\}}.$$ (The fact that ${\mathbb{P}}{\left[\rho < \infty\right]}$ is used in the second set-inequality above.) A use of a conditional version of Lemma \[lem: Doob\_maximal\] gives $${\mathbb{P}}{\left[ \sup_{v \in [t, \infty)} L_v \geq L^*_t {\ | \ }{\mathcal{F}}_t \right]} = \frac{L_t}{L^*_t} = {\mathbb{P}}{\left[ \sup_{v \in [t, \infty)} L_v > L^*_t {\ | \ }{\mathcal{F}}_t \right]}, \quad \text{for } {t \in {\mathbb R}_+}.$$ It follows that ${\mathbb{P}}{\left[\rho > t\right]} \leq {\mathbb{P}}{\left[{\rho^L_{\min}}> t\right]}$ holds for all ${t \in {\mathbb R}_+}$. Combined with ${\mathbb{P}}\big[ {\rho^L_{\min}}\leq \rho < \infty] = 1$, it follows that ${\mathbb{P}}\big[ \rho = {\rho^L_{\min}}\big] = 1$. Given ${\mathbb{P}}\big[ \rho = {\rho^L_{\min}}\big] = 1$, for $\rho$ to avoid all stopping times, it suffices that ${\rho^L_{\min}}$ does so. Pick some stopping time $\tau$; it will be shown in the sequel that ${\mathbb{P}}[{\rho^L_{\min}}= \tau {\ | \ }{\mathcal{F}}_\tau] = 0$ holds. Indeed, on ${\left\{\tau = \infty\right\}} \cup {\left\{\tau < \infty, \, L_\tau < L^*_{\tau}\right\}}$, ${\mathbb{P}}[{\rho^L_{\min}}= \tau {\ | \ }{\mathcal{F}}_\tau] = 0$ trivially holds. (Recall that ${\mathbb{P}}{\left[{\rho^L_{\min}}= \infty\right]} = 0$.) On ${\left\{\tau<\infty, \, L_\tau = L^*_{\tau}\right\}}$, where in particular $L_\tau > 0$, a conditional form of Lemma \[lem: Doob\_maximal\] gives that ${\mathbb{P}}\big[ \sup_{t \in [\tau, \infty)} L_t > L^*_\tau {\ | \ }{\mathcal{F}}_\tau \big] = L_\tau / L^*_\tau = 1$ holds; therefore, ${\mathbb{P}}[{\rho^L_{\min}}= \tau {\ | \ }{\mathcal{F}}_\tau] = 0$. Proof of implication $(2) \Rightarrow (1)$ ------------------------------------------ It has already been established that ${\rho^L_{\min}}$ is an honest time if $L \in {\mathcal{L}_0}$. Implication $(2) \Rightarrow (1)$ then follow immediately from Lemma \[lem: unique\], which in particular implies that ${\rho^L_{\min}}$ avoids all stopping times whenever $L \in {\mathcal{L}_0}$. Proof of implication $(1) \Rightarrow (2)$ ------------------------------------------ Throughout the proof of implication $(1) \Rightarrow (2)$, fix an honest time $\rho$ avoiding all stopping times. Let $Z$ be the the nonnegative [càdlàg]{} Azéma supermartingale (see [@MR604176] and the references therein) that satisfies $Z_t = {\mathbb{P}}[\rho > t {\ | \ }{\mathcal{F}}_t]$ for all $t \in {\mathbb R}_+$. The next result follows from [@MR604176 Lemma 4.3(i) and Proposition 5.1]—we provide its proof for completeness. \[lem: Z is one\] Suppose that $\rho$ is an honest time avoiding all stopping times. Then, ${\mathbb{P}}[Z_\rho = 1] = 1$. Let $(R^0_t)_{{t \in {\mathbb R}_+}}$ be an adapted process such that $\rho = R^0_t$ holds on ${\left\{\rho \leq t\right\}}$ for all ${t \in {\mathbb R}_+}$. Note that the adapted process $(R^0_t \wedge t)_{{t \in {\mathbb R}_+}}$ has the same property as well; therefore, we may assume that $R^0_t \leq t$ holds for all ${t \in {\mathbb R}_+}$. With ${\mathbb{D}}$ denoting a dense countable subset of ${\mathbb R}_+$, define the process $R {\, := \,}\lim_{{\mathbb{D}}\ni t \downarrow \cdot} \big( \sup_{s \in {\mathbb{D}}\cap (0, t)} R^0_s \big)$; then, $R$ is right-continuous, adapted and non-decreasing, and $R_t \leq t$ still holds for all ${t \in {\mathbb R}_+}$. Furthermore, since for ${s \in {\mathbb R}_+}$ and ${t \in {\mathbb R}_+}$ with $s \leq t$, $\rho = R^0_s = R^0_{t}$ holds on ${\left\{\rho \leq s\right\}} \subseteq {\left\{\rho \leq t\right\}}$, it follows that $\rho = R_t$ holds on ${\left\{\rho \leq t\right\}}$ for all ${t \in {\mathbb R}_+}$. Define a ${\left\{0, 1\right\}}$-valued optional process $I$ via $I_t = {\mathbb{I}}_{{\left\{R_t = t\right\}}}$ for $t \in {\mathbb R}_+$. The properties of $R$ can be seen to imply ${\left\{I =1\right\}} \subseteq {[\kern-0.15em[ 0 ,\rho ]\kern-0.15em]}$, as well as $I_\rho = 1$ on ${\left\{\rho < \infty\right\}}$; since ${\mathbb{P}}[\rho = \infty] = 0$ holds due to the fact that $\rho$ avoids all stopping times, we conclude that ${\mathbb{P}}[I_\rho = 1] = 1$. Fix a finite stopping time $\tau$. Using again the fact that $\rho$ avoids all stopping times, $Z_\tau = {\mathbb{P}}[\rho \geq \tau {\ | \ }{\mathcal{F}}_\tau]$ holds. Then, $I_\tau \in {\mathcal{F}}_\tau$ and ${\left\{I = 1\right\}} \subseteq {[\kern-0.15em[ 0, \rho ]\kern-0.15em]}$ imply that ${\mathbb{E}}{\left[I_\tau Z_\tau\right]} = {\mathbb{E}}{\left[I_\tau {\mathbb{I}}_{{\left\{\tau \leq \rho\right\}}}\right]} = {\mathbb{E}}{\left[I_\tau\right]}$. Since $I$ is ${\left\{0,1\right\}}$-valued and $Z$ is $[0,1]$-valued, ${\mathbb{E}}{\left[I_\tau Z_\tau\right]} = {\mathbb{E}}{\left[I_\tau\right]}$ implies that ${\left\{I_\tau = 1\right\}} \subseteq {\left\{Z_\tau = 1\right\}}$. Since the latter holds for all finite stopping times $\tau$ and both $I$ and $Z$ are optional, the optional section theorem implies that ${\left\{I = 1\right\}} \subseteq {\left\{Z = 1\right\}}$, modulo ${\mathbb{P}}$-evanescence. Then, ${\mathbb{P}}{\left[I_\rho = 1\right]} = 1$ implies ${\mathbb{P}}{\left[Z_\rho = 1\right]} = 1$. Continuing, let $A$ be the unique (up to ${\mathbb{P}}$-evanescence) adapted, [càdlàg]{}, nonnegative and nondecreasing process such that ${{\mathbb{E}}_{\mathbb{P}}}[V_\rho] = {{\mathbb{E}}_{\mathbb{P}}}{\left[ \int_0^\infty V_t {\mathrm d}A_t\right]}$ holds for all nonnegative optional processes $V$—in other words, $A$ is the dual optional projection of ${\mathbb{I}}_{{[\kern-0.15em[ \rho, \infty [\kern-0.15em[}}$. Note the equality ${\mathbb{E}}{\left[A_\tau - A_{\tau -}\right]} = {\mathbb{P}}[\rho = \tau] = 0$, holding for all finite stopping times $\tau$, which implies by the optional section theorem that $A_0 = 0$ and $A$ has ${\mathbb{P}}$-a.s. continuous paths. Then, $Z = M - A$ is the Doob-Meyer decomposition of $Z$, where $M$ is the nonnegative martingale such that $M_t = {{\mathbb{E}}_{\mathbb{P}}}{\left[A_\infty {\ | \ }{\mathcal{F}}_t\right]}$ holds for all $t \in {{{\mathbb R}_+}}$. For each ${n \in {\mathbb N}}$, define the stopping time $\zeta_n {\, := \,}\inf {\left\{{t \in {\mathbb R}_+}{\ | \ }Z_t < 1/n\right\}}$, and set $\zeta {\, := \,}\lim_{n \to \infty} \zeta_n = \inf {\left\{{t \in {\mathbb R}_+}{\ | \ }Z_{t-} = 0 \text{ or } Z_t = 0\right\}}$. Define the $[0,1]$-valued continuous nondecreasing adapted process $K = 1 - \exp \big( - \int_0^{\zeta \wedge \cdot} (1 / Z_t) {\mathrm d}A_t \big)$. Since $A$ has continuous paths, ${\mathbb{P}}{\left[K_{\zeta_n} < 1\right]} = 1$ holds for all ${n \in {\mathbb N}}$. Defining $L^n {\, := \,}Z^{\zeta_n} / (1 - K^{\zeta_n})$, the integration-by-parts formula gives $L^n = 1 + \int_0^{\zeta^n \wedge \cdot} (L^n_t / Z_t) {\mathrm d}M_t$, implying that $L^n$ is a nonnegative local martingale for all ${n \in {\mathbb N}}$. For $m \leq n$, it holds that $L^m = L^n$ on ${[\kern-0.15em[ 0, \zeta_m ]\kern-0.15em]}$; from this consistency property and the nonnegative martingale convergence theorem, it easily follows that there exists a local martingale $L$ such that $L = L^n$ holds on ${[\kern-0.15em[ 0, \zeta_n ]\kern-0.15em]}$ for all ${n \in {\mathbb N}}$ and $L_t = \lim_{n \to \infty} L^n_{\zeta_n}$ holds for all $t \geq \zeta$. Since $K = K^\zeta$, $L = L^{\zeta}$ and $Z = Z^{\zeta}$, we conclude that $Z = L (1 - K)$ holds. By the integration-by-parts formula, $Z = 1 + \int_0^\cdot (1 - K_t) {\mathrm d}L_t - \int_0^\cdot L_t {\mathrm d}K_t$ holds; comparing with the Doob-Meyer decomposition $Z = M - A$ of $Z$, and recalling that $A_0 = 0$, we obtain that $A = \int_0^\cdot L_t {\mathrm d}K_t$. \[lem: K uniform\] Suppose that $\rho$ is an honest time avoiding all stopping times. Then, with the above notation, $K_\rho$ has the standard uniform law. For $u \in {[0, 1)}$, define the stopping time $\eta_u {\, := \,}\inf {\left\{t \in {{{\mathbb R}_+}}{\ | \ }K_t \geq u\right\}}$, with the usual convention $\eta_u = \infty$ if the last set is empty. Since $K$ has ${\mathbb{P}}$-a.s. continuous paths, $K_{\eta_u} = u$ holds ${\mathbb{P}}$-a.s. on ${\left\{\eta_u < \infty\right\}}$ for all $u \in {[0, 1)}$. Recalling that $A = \int_0^\cdot L_t {\mathrm d}K_t$, a change of variables gives $$\label{eq: time change} \int_0^\infty f(K_t) {\mathrm d}A_t = \int_0^\infty f(K_t) L_t {\mathrm d}K_t = \int_0^1 L_{\eta_u} {\mathbb{I}}_{{\left\{\eta_u < \infty\right\}}} f(u) {\mathrm d}u, \quad \text{for any Borel } f: {[0, 1)}\mapsto {\mathbb R}_+.$$ The facts that $Z \leq 1$ and $(1 - K) \geq 1-u$ hold up to ${\mathbb{P}}$-evanescence on ${[\kern-0.15em[ 0, \eta_u ]\kern-0.15em]}$ imply that ${\mathbb{P}}{\left[L^*_{\eta_u} \leq 1/(1 - u)\right]} = 1$ holds for all $u \in {[0, 1)}$. Therefore, ${\mathbb{E}}[L_{\eta_u}] = 1$ holds for all $u \in {[0, 1)}$. Since ${\mathbb{P}}[\rho = \infty] = 0$, it follows that ${\mathbb{P}}[Z_\infty = 0] = 1$; then, ${\mathbb{P}}[Z_\infty = L_\infty (1 - K_\infty)] = 1$ implies ${\mathbb{P}}{\left[K_\infty < 1, \, L_\infty > 0\right]} = 0$. Therefore, for $u \in {[0, 1)}$, the set-inclusion ${\left\{\eta_u = \infty\right\}} \subseteq {\left\{K_\infty < 1\right\}}$ implies ${\mathbb{P}}{\left[L_{\eta_u} {\mathbb{I}}_{{\left\{\eta_u < \infty\right\}}}= L_{\eta_u}\right]} = 1$. Then, ${\mathbb{E}}[L_{\eta_u}] = 1$ gives ${\mathbb{E}}{\left[ L_{\eta_u} {\mathbb{I}}_{{\left\{\eta_u < \infty\right\}}}\right]} = 1$ for $u \in {[0, 1)}$. By Fubini’s Theorem and , we obtain ${\mathbb{E}}{\left[f(K_\rho)\right]} = {\mathbb{E}}{\left[\int_0^\infty f(K_t) {\mathrm d}A_t\right]} = \int_0^1 f(u) {\mathrm d}u$. Since the latter holds for any Borel $f: {[0, 1)}\mapsto {\mathbb R}_+$, it follows that $K_\rho$ has the standard uniform law. Suppose now that $\rho$ is an honest time avoiding all stopping times, and recall all notation introduced above. Since ${\mathbb{P}}[Z_\rho = L_\rho (1 - K_\rho)] = 1$, Lemma \[lem: Z is one\] gives ${\mathbb{P}}{\left[L_\rho = 1 / (1 - K_\rho)\right]} = 1$; then, ${\mathbb{P}}[L_\rho > x] = {\mathbb{P}}[K_\rho > 1 - 1/x] = 1/x$ for all $x \in (1, \infty)$ follows from Lemma \[lem: K uniform\]. As ${\mathbb{P}}{\left[L_\rho \leq L^*_\infty\right]} = 1$, Lemma \[lem: Doob\_maximal\] implies both that $L \in {\mathcal{L}_0}$ and that ${\mathbb{P}}{\left[L_\rho = L^*_\infty\right]} = 1$. Then, Lemma \[lem: unique\] gives $L \in {\mathcal{L}_0}$ and ${\mathbb{P}}{\left[\rho = {\rho^L_{\min}}\right]} = 1$. Finally, since ${\mathcal{F}}_0$ contains all ${\mathbb{P}}$-null sets of ${\mathcal{F}}$, one can alter $L$ on a set of zero ${\mathbb{P}}$-measure and have that $\rho = {\rho^L_{\min}}$ identically holds. This establishes implication $(1) \Rightarrow (2)$.

--- abstract: 'Following previous work [@Bzdak:2012tp], we propose to analyze the rapidity dependence of transverse momentum and transverse-momentum multiplicity correlations. We demonstrate that the orthogonal polynomial expansion of the latter has the potential to discriminate between models of particle production.' author: - Adam Bzdak title: 'Rapidity dependence of transverse-momentum multiplicity correlations' --- Introduction {#sec: introduction} ============ One of the central problems in high-energy hadronic collisions is to understand the longitudinal structure of systems created in proton-proton (p+p), proton-nucleus (p+A) and nucleus-nucleus (A+A) collisions. Not long ago, it was argued that an event-by-event long-range fluctuation of the fireball rapidity distribution results in rather peculiar two- and multi-particle rapidity correlations [@Bzdak:2012tp; @Bzdak:2015dja]. Recent measurement by the ATLAS Collaboration at the LHC [Aaboud:2016jnr]{} revealed new and rather unexpected scaling results on asymmetric rapidity fluctuations in p+p, p+A and A+A interactions. Recently, this problem has drawn a noticeable theoretical [Jia:2015jga,Bozek:2015tca,Monnai:2015sca,Broniowski:2015oif,Schenke:2016ksl,Bzd-Dus,Ke:2016jrd,He:2017laa]{} and experimental [@Aaboud:2016jnr; @star:a1; @alice:a1] interest, see also [Bozek:2010vz,Bialas:2011xk,Bialas:2011bz,Csernai:2012mh,Pang:2012he,Vovchenko:2013viu,Olszewski:2015xba,Casalderrey-Solana:2013sxa,Vechernin:2015upa,Pang:2015zrq]{} for recent related studies. To summarize the main idea, the single-particle rapidity distribution in each event, $N(y)$, can be written as $$\frac{N(y)}{\left\langle N(y)\right\rangle }=1+a_{0}+a_{1}y+..., \label{eq:N(y)}$$where $a_{0}$ describes the rapidity independent fluctuation of the fireball. $a_{1}$ represents the fluctuating long-range forward-backward rapidity asymmetry.[^1] This coefficient can be driven for example by the difference in the number of left- and right-going sources of particles, e.g., wounded nucleons [@Bialas:1976ed; @Bialas:2004su]. $\left\langle N(y)\right\rangle $ is the average rapidity distribution in a given centrality class. By definition $\left\langle a_{i}\right\rangle =0$. It is straightforward to calculate the two-particle rapidity correlation [@Bzdak:2012tp] $$\frac{C(y_{1},y_{2})}{\left\langle N(y_{1})\right\rangle \left\langle N(y_{2})\right\rangle }=\left\langle a_{0}^{2}\right\rangle +\left\langle a_{1}^{2}\right\rangle y_{1}y_{2}+... \label{eq:C-NN}$$ where[^2] $$C(y_{1},y_{2})=\left\langle N(y_{1})N(y_{2})\right\rangle -\left\langle N(y_{1})\right\rangle \left\langle N(y_{2})\right\rangle .$$ As seen in Eq. (\[eq:C-NN\]), the long-range fluctuation of the fireball rapidity distribution, parameterized by fluctuating $a_i$, results in rather nontrivial correlations. The first term corresponds to a well-known rapidity independent multiplicity fluctuation, and it can be driven by, e.g., the impact parameter or volume fluctuation. The second term, $\sim y_{1}y_{2}$, is related to the fluctuating forward-backward asymmetry in rapidity. In the wounded nucleon model [@Bialas:1976ed; @Bialas:2004su] $\langle a_{1}^{2}\rangle \sim \langle (w_{L}-w_{R})^{2}\rangle ,$ where $w_{L(R)}$ is the number of left(right)-going wounded nucleons [@Bzdak:2012tp]. Recently, the ATLAS Collaboration observed $\langle a_{1}^{2}\rangle y_{1}y_{2}$ in the two-particle rapidity correlation functions measured in p+p, p+Pb and Pb+Pb collisions [@Aaboud:2016jnr]. They found that at a given event multiplicity $N_{\mathrm{ch}}$, $\langle a_{1}^{2}\rangle $ approximately scales with $1/N_{\mathrm{ch}}$ and numerically is very similar for all colliding systems.[^3] This surprising result still calls for a quantitative explanation. Transverse-momentum multiplicity correlations ============================================= It is proposed here to analyze, in a similar way, the rapidity dependence of transverse momentum and especially transverse-momentum multiplicity correlations. Analogously to Eq. (\[eq:N(y)\]) we have $$\frac{P_{t}(y)}{\left\langle P_{t}(y)\right\rangle }=1+b_{0}+b_{1}y+...,$$where $P_{t}(y)$ is the average (in one event) transverse momentum of particles in a given rapidity bin $y$ $$P_{t}(y)=\frac{1}{N}\sum\nolimits_{i=1}^{N}p_{t}^{(i)},$$ where $N$ is the number of particles (in a given event) at $y$. Here $p_{t}^{(i)}$ is the transverse momentum magnitude of the $i$-th particle. $\left\langle P_{t}(y)\right\rangle $ is the average of $P_{t}(y)$ over many events in a given centrality class. We note that $\left\langle b_{i}\right\rangle =0$, in close analogy to the $a_{i}$ coefficients. The transverse momentum correlation function (studied extensively in the literature for rather different reasons, see, e.g., [Gavin:2006xd,Gavin:2016hmv,Braun:2003fn]{}) reads $$\frac{C_{[P,P]}(y_{1},y_{2})}{\left\langle P_{t}(y_{1})\right\rangle \left\langle P_{t}(y_{2})\right\rangle }=\left\langle b_{0}^{2}\right\rangle +\left\langle b_{1}^{2}\right\rangle y_{1}y_{2}+..., \label{eq:C-PP}$$ where $$C_{[P,P]}(y_{1},y_{2})\equiv \left\langle P_{t}(y_{1})P_{t}(y_{2})\right\rangle -\left\langle P_{t}(y_{1})\right\rangle \left\langle P_{t}(y_{2})\right\rangle.$$ The first term in Eq. (\[eq:C-PP\]) describes an event-by-event rapidity independent transverse momentum fluctuation. This could be driven for example by an event-by-event long-range multiplicity fluctuation (if event multiplicity is correlated with $P_{t}$). The second term describes the forward-backward rapidity asymmetric transverse momentum fluctuation. A possible source of this effect is the forward-backward fireball multiplicity fluctuation. It would be especially interesting to measure an event-by-event relation between $a_{i}$ and $b_{i}$ coefficients. In order to do this, one can construct a simple correlation function $$C_{[N,P]}(y_{1},y_{2})\equiv \left\langle N(y_{1})P_{t}(y_{2})\right\rangle -\left\langle N(y_{1})\right\rangle \left\langle P_{t}(y_{2})\right\rangle ,$$ witch correlates multiplicity and transverse momentum, see, e.g., [@Braun:2003fn]. This results in $$\frac{C_{[N,P]}(y_{1},y_{2})}{\left\langle N(y_{1})\right\rangle \left\langle P_{t}(y_{2})\right\rangle }=\left\langle a_{0}b_{0}\right\rangle +\left\langle a_{1}b_{1}\right\rangle y_{1}y_{2}+...$$ The meaning of mixed coefficients $\left\langle a_{i}b_{k}\right\rangle $ is easy to understand. The first term describes the relation between rapidity independent fluctuation of multiplicity and transverse momentum. The second term is particularly interesting and describes how rapidity asymmetry in multiplicity is related to rapidity asymmetry of transverse momentum. If the particle multiplicity and $P_{t}$ are not correlated then $\left\langle a_{i}b_{k}\right\rangle =\left\langle a_{i}\right\rangle \left\langle b_{k}\right\rangle =0$. In general, the above correlation functions can be expanded in terms of the orthogonal polynomials [@Bzdak:2012tp]. For example $$\frac{C_{[N,P]}(y_{1},y_{2})}{\left\langle N(y_{1})\right\rangle \left\langle P_{t}(y_{2})\right\rangle }=\sum\nolimits_{i,k}\left\langle a_{i}b_{i}\right\rangle T_{i}(y_{1})T_{k}(y_{2}),$$with $T_{i}$ being, e.g., the Chebyshev or the Legendre polynomials [Bzdak:2012tp,Jia:2015jga]{}, and analogously for Eqs. (\[eq:C-NN\]) and (\[eq:C-PP\]). Discussion and conclusions ========================== Several comments are in order. Consider a set of events with $a_{1}>0$, i.e., the fireball multiplicity is larger for positive $y$, $N(y) \sim a_{1}y$. The question is what is the rapidity dependence of the transverse momentum in this case. If $P_{t}$ is also larger for positive $y$ then $b_{1}>0$ and thus $\langle a_{1}b_{1}\rangle >0$. This scenario is expected in a typical hydrodynamical framework, see, e.g., [@Bozek:2013sda]. For example, in the color glass condensate (CGC) framework [Gelis:2010nm,Blaizot:2016qgz]{} one could expect a rather different conclusion. Consider a proton-proton event, where the two protons are characterized by different saturation scales, $Q_{1}$ and $Q_{2}$. The importance of such fluctuations was recently discussed in Refs. [Marquet:2006xm,McLerran:2015lta,McLerran:2015qxa,Mantysaari:2016jaz,Bzd-Dus]{}. Here $Q_{1}^{2}=Q_{0,1}^{2}e^{+\lambda y}$ and $Q_{2}^{2}=Q_{0,2}^{2}e^{-\lambda y}$ with $\lambda \sim 0.3$, see, e.g., [@Praszalowicz:2015dta]. We choose $Q_{0,1}>Q_{0,2}$ so that in a given rapidity bin, say $|y|<2$, $Q_{1}>Q_{2}$, resulting in rapidity asymmetric $N(y)$. In this case [@Dumitru:2001ux; @Bozek:2013sda] $$\begin{aligned} N(y) &\sim &S_{t}Q_{2}^{2}\left[ 2+\ln \left( Q_{1}^{2}/Q_{2}^{2}\right) \right] , \\[5pt] P_{t}(y) &\sim &\frac{2Q_{1}-\frac{2}{3}Q_{2}}{1+\ln \left( Q_{1}/Q_{2}\right) },\end{aligned}$$ that is, in CGC the multiplicity is driven by the smaller scale in contrast to the transverse momentum controlled by the larger one [@Dumitru:2001ux]. Since $Q_{1}^{2}\sim e^{+\lambda y}$ and $Q_{2}^{2}\sim e^{-\lambda y}$, the multiplicity and the transverse momentum rapidity asymmetries have different signs. If $N(y)$ is growing with rapidity, then $P_{t}(y)$ is decreasing with $y$. Consequently $a_{1} \gtrless 0$ means $b_{1} \lessgtr 0$ and $\left\langle a_{1}b_{1}\right\rangle <0$. Clearly, this observation should be treated with caution and more detailed calculations are warranted, see, e.g., [@Duraes:2015qoa; @Deja:2017dqh]. The sole purpose of this exercise was to demonstrate that the sign of $\left\langle a_{1}b_{1}\right\rangle $ is not at all obvious, and could potentially discriminate between different models of particle production. As discussed earlier, the ATLAS Collaboration reported a surprising scaling of $\langle a_{1}^{2}\rangle $ in p+p, p+Pb and Pb+Pb collisions [@Aaboud:2016jnr]. At a given event multiplicity $N_{\mathrm{ch}}$, $\langle a_{1}^{2}\rangle $ scales with $1/N_{\rm ch}$ and is quantitatively very similar for all three systems. It would be very interesting to see if $\langle b_{1}^{2}\rangle $ and $\left\langle a_{1}b_{1}\right\rangle $ satisfy similar scaling. Obviously, it would be also desired to study higher order correlation functions [@Bzdak:2015dja; @DiFrancesco:2016srj]. An alternative way to analyze the above correlation functions is the principal component analysis, discussed in Ref. [@Bhalerao:2014mua]. In conclusion, it is proposed to analyze the rapidity dependence of transverse momentum and in particular transverse-momentum multiplicity correlation functions using the orthogonal polynomial expansion. A careful study of the coefficients $\langle a_{i}^{2}\rangle ,$ $\langle b_{i}^{2}\rangle $ and $\langle a_{i}b_{k}\rangle $ could potentially discriminate between different models of particle production, and reveal detailed information on the longitudinal structure of systems created in p+p, p+A and A+A collisions. **Acknowledgments** We thank Piotr Bożek and Volker Koch for useful comments. This work is supported by the Ministry of Science and Higher Education (MNiSW) and by the National Science Centre, Grant No. DEC-2014/15/B/ST2/00175, and in part by DEC-2013/09/B/ST2/00497. [99]{} A. Bzdak and D. Teaney, Phys. Rev. C [**87**]{}, no. 2, 024906 (2013) \[arXiv:1210.1965 \[nucl-th\]\]. A. Bzdak and P. Bozek, Phys. Rev. C [**93**]{}, no. 2, 024903 (2016) \[arXiv:1509.02967 \[hep-ph\]\]. M. Aaboud [*et al.*]{} \[ATLAS Collaboration\], Phys. Rev. C [**95**]{}, no. 6, 064914 (2017) \[arXiv:1606.08170 \[hep-ex\]\]. J. Jia, S. Radhakrishnan and M. Zhou, Phys. Rev. C [**93**]{}, no. 4, 044905 (2016) \[arXiv:1506.03496 \[nucl-th\]\]. P. Bozek, W. Broniowski and A. Olszewski, Phys. Rev. C [**92**]{}, no. 5, 054913 (2015) \[arXiv:1509.04124 \[nucl-th\]\]. A. Monnai and B. Schenke, Phys. Lett. B [**752**]{}, 317 (2016) \[arXiv:1509.04103 \[nucl-th\]\]. W. Broniowski and P. Bozek, Phys. Rev. C [**93**]{}, no. 6, 064910 (2016) \[arXiv:1512.01945 \[nucl-th\]\]. B. Schenke and S. Schlichting, Phys. Rev. C [**94**]{}, no. 4, 044907 (2016) \[arXiv:1605.07158 \[hep-ph\]\]. A. Bzdak and K. Dusling, Phys. Rev. C [**93**]{}, no. 3, 031901 (2016) \[arXiv:1511.03620 \[hep-ph\]\]; Phys. Rev. C [**94**]{}, 044918 (2016) \[arXiv:1607.03219 \[hep-ph\]\]. W. Ke, J. S. Moreland, J. E. Bernhard and S. A. Bass, arXiv:1610.08490 \[nucl-th\]. R. He, J. Qian and L. Huo, arXiv:1702.03137 \[nucl-th\]. S. Jowzaee \[STAR Collaboration\], talk at the the XXVI international conference on ultrarelativistic heavy-ion collisions (Quark Matter 2017), Chicago, 2017. Saehanseul Oh \[ALICE Collaboration\], poster at the the XXVI international conference on ultrarelativistic heavy-ion collisions (Quark Matter 2017), Chicago, 2017. P. Bozek, W. Broniowski and J. Moreira, Phys. Rev. C [**83**]{}, 034911 (2011) \[arXiv:1011.3354 \[nucl-th\]\]. A. Bialas and K. Zalewski, Nucl. Phys. A [**860**]{}, 56 (2011) \[arXiv:1101.1907 \[hep-ph\]\]. A. Bialas, A. Bzdak and K. Zalewski, Phys. Lett. B [**710**]{}, 332 (2012) \[arXiv:1107.1215 \[hep-ph\]\]. L. P. Csernai, G. Eyyubova and V. K. Magas, Phys. Rev. C [**86**]{}, 024912 (2012) Erratum: \[Phys. Rev. C [**88**]{}, 019902 (2013)\] \[arXiv:1204.5885 \[hep-ph\]\]. L. Pang, Q. Wang and X. N. Wang, Phys. Rev. C [**86**]{}, 024911 (2012) \[arXiv:1205.5019 \[nucl-th\]\]. V. Vovchenko, D. Anchishkin and L. P. Csernai, Phys. Rev. C [**88**]{}, no. 1, 014901 (2013) \[arXiv:1306.5208 \[nucl-th\]\]. A. Olszewski and W. Broniowski, Phys. Rev. C [**92**]{}, no. 2, 024913 (2015) \[arXiv:1502.05215 \[nucl-th\]\]. J. Casalderrey-Solana, M. P. Heller, D. Mateos and W. van der Schee, Phys. Rev. Lett.  [**112**]{}, no. 22, 221602 (2014) \[arXiv:1312.2956 \[hep-th\]\]. V. Vechernin, Nucl. Phys. A [**939**]{}, 21 (2015). L. G. Pang, H. Petersen, G. Y. Qin, V. Roy and X. N. Wang, Eur. Phys. J. A [**52**]{}, no. 4, 97 (2016) \[arXiv:1511.04131 \[nucl-th\]\]. A. Bialas, M. Bleszynski and W. Czyz, Nucl. Phys. B [**111**]{}, 461 (1976). A. Bialas and W. Czyz, Acta Phys. Polon. B [**36**]{}, 905 (2005) \[hep-ph/0410265\]. S. Gavin and M. Abdel-Aziz, Phys. Rev. Lett.  [**97**]{}, 162302 (2006) \[nucl-th/0606061\]. S. Gavin, G. Moschelli and C. Zin, Phys. Rev. C [**94**]{}, no. 2, 024921 (2016) \[arXiv:1606.02692 \[nucl-th\]\]. M. A. Braun, R. S. Kolevatov, C. Pajares and V. V. Vechernin, Eur. Phys. J. C [**32**]{}, 535 (2004) \[hep-ph/0307056\]. P. Bozek, A. Bzdak and V. Skokov, Phys. Lett. B [**728**]{}, 662 (2014) \[arXiv:1309.7358 \[hep-ph\]\]. F. Gelis, E. Iancu, J. Jalilian-Marian and R. Venugopalan, Ann. Rev. Nucl. Part. Sci.  [**60**]{}, 463 (2010) \[arXiv:1002.0333 \[hep-ph\]\]. J. P. Blaizot, Rept. Prog. Phys.  [**80**]{}, no. 3, 032301 (2017) \[arXiv:1607.04448 \[hep-ph\]\]. C. Marquet, G. Soyez and B. W. Xiao, Phys. Lett. B [**639**]{}, 635 (2006) \[hep-ph/0606233\]. L. McLerran and M. Praszalowicz, Annals Phys.  [**372**]{}, 215 (2016) \[arXiv:1507.05976 \[hep-ph\]\]. L. McLerran and P. Tribedy, Nucl. Phys. A [**945**]{}, 216 (2016) \[arXiv:1508.03292 \[hep-ph\]\]. H. Mäntysaari and B. Schenke, Phys. Rev. D [**94**]{}, no. 3, 034042 (2016) \[arXiv:1607.01711 \[hep-ph\]\]. M. Praszalowicz and A. Francuz, Phys. Rev. D [**92**]{}, no. 7, 074036 (2015) \[arXiv:1507.08186 \[hep-ph\]\]. A. Dumitru and L. D. McLerran, Nucl. Phys. A [**700**]{}, 492 (2002) \[hep-ph/0105268\]. F. O. Duraes, A. V. Giannini, V. P. Goncalves and F. S. Navarra, Phys. Rev. C [**94**]{}, no. 2, 024917 (2016) \[arXiv:1510.04737 \[hep-ph\]\]. K. Deja and K. Kutak, Phys. Rev. D [**95**]{}, no. 11, 114027 (2017) \[arXiv:1703.07959 \[hep-ph\]\]. P. Di Francesco, M. Guilbaud, M. Luzum and J. Y. Ollitrault, Phys. Rev. C [**95**]{}, no. 4, 044911 (2017) \[arXiv:1612.05634 \[nucl-th\]\]. R. S. Bhalerao, J. Y. Ollitrault, S. Pal and D. Teaney, Phys. Rev. Lett.  [**114**]{}, no. 15, 152301 (2015) \[arXiv:1410.7739 \[nucl-th\]\]. [^1]: We are not interested in statistical fluctuations, which can also generate nonzero values of $a_i$. These are removed by measuring correlation functions. [^2]: For clarity we skip $\left\langle a_{0}a_{1}\right\rangle$, which vanishes in symmetric (e.g. p+p) collisions. [^3]: One would expect rather different results in, e.g., peripheral Pb+Pb and ultracentral p+p collisions ($N_{\mathrm{ch}} \sim 150$ [@Aaboud:2016jnr]). The ATLAS result suggests that the number of particle sources (at a given $N_{\mathrm{ch}}$) and their fluctuations are actually similar in all measured systems.

--- abstract: 'The relation between quasar variability and parameters such as luminosity and redshift has been a matter of hot debate over the last few years with many papers on the subject. Any correlations which can be established will have a profound effect on models of quasar structure and evolution. The sample of quasars with redshifts in ESO/SERC field 287 contains over 600 quasars in the range $0 < z < 3.5$ and is now large enough to bin in luminosity and redshift, and give definitive measures of the correlations. We find no significant correlation between amplitude and redshift, except perhaps at very low redshift, but an inverse correlation between amplitude and luminosity. This is examined in the context of various models for quasar variability.' author: - 'M.R.S. Hawkins' date: 'Received &lt;date&gt; / accepted &lt;date&gt;' title: 'Quasar variability: correlations with amplitude' --- Introduction ============ One of the most important constraints on the structure of quasars is variability. Short term variations set limits on the size of the emitting region, and differences in the nature of the variation in the Xray, optical and radio domains give clues to the underlying structure. Variations on longer timescales of a few years are not so easily accounted for by the current black hole paradigm, and microlensing has been put forward as an alternative explanation for the observations. Although the most extensive monitoring of quasar variability has been done in optical wavebands, there is still little consensus even over the broad picture. One of the first problems is to parametrise the variation in such a way that it can be compared with models, or even more general expectations. The timescale of variation and the amplitude are the two parameters which have been mostly studied, although some authors have succeeded in confusing the two. This typically involves claiming that in a short run of data, objects varying on a short timescale will achieve a larger amplitude than those varying on a longer timescale, and so amplitude can be taken as an (inverse) measure of timescale (Hook et al. [@hoo94]).\ In this paper we shall confine our attention to the correlation of amplitude with other parameters. Amplitude is an easy parameter to measure, and involves none of the difficulties inherent in estimating timescale of variation. There are modes of variability in which some uncertainty will be introduced by the length of the run of observations, but this is a problem which can arise in any time series analysis. Most attention has so far been given to possible correlations of amplitude with redshift or luminosity (eg Trevese et al. [@tre89], Giallongo et al. [@gia91], Hook et al. [@hoo94], Cristiani et al. [@cri96]), but even in such straightforward situations there has been little agreement.\ The reasons for the lack of consensus are not easy to understand, but it is clear that any effect which is present is not large compared with the cosmic scatter in the data, which in the case of amplitude appears to be about one magnitude, much larger than any photometric errors. Perhaps the most likely cause of disagreement is in the selection of samples for analysis. In order to obtain a meaningful measure of correlation it is essential that unbiassed samples are used, and that they cover a large range of redshift and luminosity. In this paper particular attention is paid to the sample selection process, and by extending the analysis to high redshifts the baseline for the measurement of correlations is greatly extended over earlier samples.\ Quasar samples ============== Observational material ---------------------- The parent sample of quasars for the analysis of variability amplitude comes from a large scale survey and monitoring programme being undertaken in the ESO/SERC field 287, at 21h 28m, -45$^{\circ}$. Quasars have been selected according to a number of criteria, including colour, variability, radio emission and objective prism spectra, or a combination of these techniques. Redshifts for over 600 quasars have so far been obtained, and several complete samples defined within specific limits of magnitude, redshift and position on the sky (Hawkins & Véron [@haw95]). A detailed description of the survey is given by Hawkins & Véron ([@haw95]) and Hawkins ([@haw96]). Briefly, a large set of UK 1.2 Schmidt plates spanning 20 years was scanned with the COSMOS and SuperCOSMOS measuring machines to provide a catologue of some 200,000 objects in the central 19 square degrees of the plate. These were calibrated with CCD frames to provide light curves in $B_{J}$ and $R$, and colours in $UBVRI$. Photometric errors have been discussed in detail in earlier papers (Hawkins 1996 and references therein) and are $\sim 0.08$ magnitudes for any individual machine measurement, and only weakly dependent on magnitude. There are approximately four plates in each year which reduces the error to $\sim 0.04$ magnitudes per epoch. This is small compared with the amplitudes of interest in this paper, and no attempt has been made to deconvolve it. Sample selection ---------------- For the analysis in this paper three samples will be defined. The first (UVX) is based only on position on the sky and ultraviolet excess. The area on the sky containing the sample is defined by a number of AAT AUTOFIB fields and a 2dF field (Folkes et al. [@fol99] and references therein), covering a total area of 7.0 square degrees. Within this area all objects with $U-B < -0.2$ and $B_{J} < 21.0$ were observed. This was extended to $B_{J} < 21.5$ in the 2dF field. The ultra-violet excess (UVX) cut although necessary to give a relatively clean sample of quasar candidates, has the well known limitation of only being effective for redshifts $z < 2.2$. Beyond this redshift quasars become red in $U-B$ as the Lyman forest enters the $U$ band. For samples at higher redshift, variability has been found to provide a very useful criterion for quasar selection (Hawkins [@haw96]). The second sample for consideration herei (VAR), was selected on this basis, with the requirement that the object should lie in the same area of sky as for the first sample with a magnitude limit $B_{J} < 21$, or anywhere in the measured area of the plate with a magnitude limit $B_{J} < 19.5$, and should have an amplitude $\delta m > 0.35$. The defining epoch for the magnitudes was the year 1977. This sample was selected without any reference to colour and so can be used to measure trends over a large range of redshift ($z < 3.5$). The third sample (AMP) was selected in a similar manner, but over the whole measured area of the plate (19 square degrees), and with an amplitude cut $\delta m > 1.1$. There is some evidence that these large amplitude objects form a distinct group, which is discussed below.\ The development of fibre fed spectrographs has meant that every object included within a given set of search criteria can be observed to give a high completeness level. In the case of fields observed with AUTOFIB, the existence of forbidden regions in the 40 arcminute field, and the very variable throughput for different fibres aligned at different positions on the spectrograph slit meant that up to 20% of spectra did not have sufficient signal to provide an unambiguous redshift. These objects where re-observed later with the faint object spectrograph EFOSC on the ESO 3.6m telescope at La Silla. The 2dF observations where of very uniform quality, and the redshift measures or other classification were almost 100% successful from the fibre-feed spectra.\ Details of all three samples, containing a total of 384 quasars, are given in the Appendix. $B$ magnitudes are in the $B_{J}$ system defined by the IIIa-J emulsion and the GG395 filter, and refer to the year 1977. The amplitude $\delta m$ is the difference between maximum and minmum magnitude achieved over the 20 year run of data. The samples which each quasar belongs to are indicated by 1, 2 and 3 corresponding to VAR, UVX and AMP respectively, as defined above. Data for many of the quasars have already been published by Hawkins & Véron ([@haw95]), but are given again here for completeness. Any small differences in the parameters are the result of further refinement of the calibration, and more extended monitoring of the light curves.\ UVX selected samples have been used many times in the past for quasar surveys, and the constraints are quite well known. Variability selection has been less often used, and some additional comments are appropriate. In particular, there is the important question of completeness. Hawkins & Véron ([@haw95]) use a small sample of quasars in field 287 from Morris et al. ([@mor91]) selected by objective prism to test completeness. In 1993, 79% of the objective prism quasars would have also been detected according to the variability criterion $\delta m > 0.35$; by 1997 this figure had risen to 93%, with two quasars remaining below the variability threshold. In fact both of these quasars are clearly variable, with amplitudes 0.32 and 0.29 magnitudes. Strangely, they lie at either extreme of the redshift and luminosity range of the sample, with redshifts 3.23, 0.29 and luminosities -27.55, -22.33 respectively. Fig. \[1\](a) shows the distribution of epoch at which quasars first satisfy the detection criterion given by Hawkins ([@haw96]). This is generally speaking equivalent to attaining an amplitude of 0.35 magnitudes. The distribution peaks between 2 and 4 years, and most quasars have satisfied the criterion after 7 years. Fig. \[1\](b) shows the cumulative distribution of detection epochs, illustrating the point that after 15 years nearly all quasars have varied sufficiently to put them in the variability selected sample.\ The distribution of amplitudes is best shown with the UVX sample, which was selected without any reference to variability. Fig.  \[2\](a) is a histogram of amplitudes over a 20 year baseline, and shows a median amplitude of around 0.7 magnitudes. As would be expected from Fig. \[1\], nearly all of the sample lies above the threshold amplitude of the variability selected sample of 0.35 mags. With this in mind it is worth investigating the possibility of using the variability selected sample to look for correlations at higher redshift. Fig.  \[2\](b) shows a histogram of amplitudes for the VAR sample, which from the way it was selected has a cut-off at an amplitude of 0.35 mags. More interestingly, the distribution as a whole peaks at an amplitude of about 0.7, similar to the UVX sample. This peak is well clear of the cut-off, and implies that only a small fraction of quasars are missed from the VAR sample. There may however be a problem detecting the most luminous quasars, ($M_{B} < -27$) for which there is evidence for relatively small variations (Cristiani et al. [@cri96]). Nonetheless, the sample can be used with caution to extend quasar correlations to high redshift.\ Amplitude correlations ====================== The top two panels in Fig.  \[3\] show the distribution of amplitudes as a function of redshift and luminosity for the UVX sample. The most striking thing about these figures is the marked drop in amplitude towards low redshift and luminosity. This is illustrated more clearly in the top two panels of Fig.  \[4\], where the data is binned in intervals of 0.5 in redshift and unit absolute magnitude, and the mean plotted with $\sqrt{N}$ error bars. The fall towards low redshift and luminosity is highly significanti (a 3-$\sigma$ effect), although it is not clear from these data alone whether it is primarily a luminosity or redshift effect. The correlation coeffecient for the top left panel of Fig.  \[3\] with $z < 0.5$ is 0.46, and for the top right panel with $M_{B} > -22$ is -0.67, confirming the reality of the correlation. There is also some evidence for a decrease in amplitude for the most luminous quasars ($M_{B} < -25$), and an even weaker decline for high redshift objects. Correlation coefficients for all the data in the top two panels are 0.12 and -0.10 for left and right panels respectively, emphasising the weakness of the effect. This is another manifestation of the well known degeneracy between redshift and luminosity. This point will be investigated further below by dividing the data into sub-samples.\ The bottom two panels in Figs  \[3\] and  \[4\] show similar plots for the VAR sample. The greater redshift range allowed by variability selection still shows little trend of amplitude with redshift, but the decrease in amplitude for luminous quasars is shown to continue to greater luminosities, although the effect is inevitably lessened by the absence of quasars with $\delta m < 0.35$.\ It has been mentioned above that there is a degeneracy between redshift and luminosity. This results from the fact that in a magnitude limited sample high redshift quasars tend to be high luminosity objects, and vice versa. Thus a trend with one parameter will be mimicked by a trend with the other, and the true relation will be hard to disentangle. This degeneracy can in principle be broken byi binning the data in redshift and luminosity, and the result of doing this is shown in Figs.  \[5\] and  \[6\]. The VAR sample was used as it covers a wider range of luminosity and redshift, making binning feasible. The left hand panel shows amplitude as a function of luminosity in two redshift bins, $z < 1.5$ and $z > 1.5$. Data for the two redshift ranges overlap nicely, and clearly show a decrease in the relation between amplitude and luminosity. The right hand panel shows amplitude as a function of redshift in two luminosity bins. In this case there is some evidence for an increase in the relation at low redshift, but it is essentially flat beyond $z = 0.5$ for both luminosity ranges. It is however worth noting that the less luminous quasars have larger amplitudes, as expected from the data in the left hand panel. It would thus appear that while for $M_{B} > -25$ amplitude does indeed decrease with luminosity, it does not change with redshift for $z > 0.5$.\ Examination of Fig.  \[3\], especially the bottom panels, suggests that there may be population of low luminosity and/or low redshift quasars distinct from the parent population. This is particularly evident in the bottom left hand panel of Fig.  \[3\], which is uniformly populated between amplitudes of 1.1 and 1.8 up to a redshift of 2 at which point there is a sharp cut-off with no amplitudes greater than 1.1 at higher redshift. To investigate this population with better statistics all variables with $\delta m > 1.1$ in the measured area of field 287 were observed on the 3.6m at La Silla to confirm their identification as quasars and measure redshifts. This became the AMP sample. Fig.  \[7\] shows amplitude as a function of both redshift and luminosity, and it will be seen that there is indeed a cut-off in redshift at $z \sim 2$ and $M_{B} \sim -25$. It is clear that this cut-off must in fact be related to luminosity. If it were a redshift cut-off there is no reason why such objects should not be seen with greater luminosity. On the other hand if it were a luminosity cut-off, then this combined with a magnitude limit will indeed produce an effective cut-off in redshift.\ Discussion ========== Attempts to measure correlations of amplitude (or some related variability parameter) with redshift and luminosity have a long history, which is summarised by Hawkins ([@haw96]). Among recent work, a useful place to start is with the paper by Hook et al. ([@hoo94]). They analyse a sample of $\sim 300$ quasars in the SGP area from 12 UK 1.2m Schmidt plates taken in 5 separate yearly epochs spanning 16 years. They find a convincing anti-correlation between their variability parameter $\sigma_{v}$ (a measure of variation about the mean) and luminosity, but a much weaker anti-correlation with redshift. They attribute this to the degeneracy between redshift and luminosity in their sample. The same sample is re-analysed by Cid Fernandes et al. ([@cid96]) who use a variability index related to variance, and also one related to the structure function. They concur with Hook et al. ([@hoo94]) that there is an anti-correlation between their variability indices and luminosity, but claim a positive correlation with redshift. This effect is not apparent to the eye, but is interpreted as a variability-wavelength dependence rather than an intrinsic variability-redshift dependence. The net result in an un-binned sample cancels out with the luminosity variability relationship. Cid Fernandes et al.’s claim for a positive correlation between amplitude and redshift appears to be motivated at least in part by expectations arising from a paper by Di Clemente et al. ([@dic96]). This interesting paper examines the relation between their variability parameter $S_{1}$ (an amplitude based on the structure function) and wavelength. Their sample is composed of PG quasars (Schmidt and Green [@sch83]), which are mostly low redshift and relatively low luminosity objects. With the help of archival IUE observations they find that $S_{1}$ decreases with wavelength. This effect can clearly be seen in the light curve from the intensive monitoring programme of the Seyfert galaxy NGC 5548 (Clavel et al. [@cla91]), which has a larger amplitude at shorter wavelength. Fig.  \[8\] shows the relation between amplitude and wavelength taken from the light curves, with a best-fit quadratic curve.\ The relation between rest wavelength and amplitude is essentially equivalent to the relation between redshift and amplitude, where one is seeing progressively shorter wavelengths at higher redshift. This is illustrated in the top panel of Fig.  \[9\] which shows the UVX sample with two curves superimposed. The solid line is converted from Fig.  \[8\] and does not appear to follow the trend of the data, but the large scatter makes it hard to construct a convincing test. Nonetheless it suggests that any relation which holds for Seyfert galaxies might have to be modified for quasars. The dotted line is from Di Clemente et al. ([@dic96]), and shows a very small effect, which does nonetheless follow the flat distribution of the data.\ The decrease of amplitude for the smallest redshift and luminosity seen in the top two panels of Fig.  \[4\] has a number of possible explanations. It could be the effect of the underlying galaxy dominating any change in nuclear brightness for low luminosity objects, it could be a consequence of the small optical depth to microlensing at low redshift, or it could be a consequence of the wavelength dependence of variability (Cristiani et al. [@cri97]). The present dataset is not adequate to settle the question, which is best done by looking at luminous quasars at very low redshift and Seyfert galaxies at high redshift.\ All the plots in Figs  \[3\] and  \[4\] show a trend of decreasing amplitude towards higher redshift or more luminous objects. Although the trend as a function of luminosity is more marked, the old problem of degeneracy makes it hard to say for certain that it is a luminosity effect which is being observed. However, if we look at Fig.  \[6\] where the data are binned in luminosity and redshift we see that while there is no significant trend of amplitude with redshift in either luminosity bin, there is a marked inverse correlation between amplitude and luminosity.\ The relation between amplitude and luminosity is in agreement with that found in earlier work (Hook et al. [@hoo94], Hawkins [@haw96], Cristiani et al. [@cri96]) and may well turn out to be a useful way of distinguishing between various schemes for quasar variability. The evidence for a constant amplitude with redshift is more debatable. It would appear to be consistent with the early claim of Hook et al. ([@hoo94]) for a weak anti-correlation with redshift which they ascribed to degeneracy with luminosity. It is also in agreement with the results of Cristiani et al. in the observer’s frame. When they correct their structure function to the quasar rest frame they inevitably imprint a positive correlation between their variability parameter and redshift.\ To investigate this dependence of amplitude on redshift for the present sample, Fig.  \[10\] shows the epoch at which quasars achieve their maximum amplitude as a function of redshift. Apart from very low redshifts ($z < 0.3$) this relation is flat, implying that at least over the 21 years of the present dataset, time dilation effects will not bias the measurement of amplitude. Since the conclusions of Cid Fernandes et al. ([@cid96]) are largely based on a sample for which a time dilation correction has been applied, it is not feasible to make a direct comparison with the present work. However, it appears that the main difference between their results and those of Hook et al. is in the definition of a variability parameter and the method of analysis (both papers are based on the SGP sample).\ There are perhaps three currently discussed schemes for quasar variability. The least well constrained is the accretion disk model, where instabilities are propagated across the disk leading to variation in light. The details of this approach have proved hard to work out, especially in the context of the constraints imposed by existing observations, but it does not seem to lead to an inverse correlation between amplitude and luminosity. An interesting recent attempt to model variation on the basis of accretion disk instabilities by Kawaguchi et al. ([@kaw98]) may provide a means for producing the observed variations. It does however appear to predict variations which are either too asymmetric or of too small an amplitude to be consistent with the current observations. The timescales which they predict are also rather short, around 200 days for reasonable input parameters, and much shorter than the observed timescale of a few years.\ An alternative approach, developed by Terlevich and his collaborators, accounts for the variation by postulating that the quasar is powered by a series of supernova explosions. Qualitatively, this model can account for the observed relation between luminosity and amplitude, and works quite well for Seyfert galaxies (Aretxaga & Terlevich [@are94]). However, for quasars (Aretxaga et al. [@are97]) large numbers of supernovae are required to achieve the luminosity, which results in smaller variations. For example, even a relatively modest quasar with absolute magnitude $M_{B} \sim -26$ would need some 300 type II supernovae per year to power it, which given typical decay times would lead to very little variation at all. This clearly conflicts with observations in this paper which show that most quasars vary by around 0.5 to 1 magnitude on a timescale of a few years.\ The third way of explaining quasar variability is to invoke microlensing. This approach has been explored in several recent papers (Hawkins [@haw93], [@haw96], Hawkins & Taylor [@haw97]), and seems to account well for a number of statistical properties of quasar light curves. It can also explain the inverse correlationi between luminosity and amplitude in a natural way. It is well known that when a point source is microlensed by a population of compact bodies of significant optical depth, the lenses combine non-linearly to form a caustic pattern which produces sharp spikes in the resulting light curves (Schneider & Weiss [@sch87]). As the source becomes comparable in size with, or larger than, the Einstein radius of the lenses the amplitude decreases (Refsdal & Stabell [@ref91]). Thus if one assumes a uniform temperature for quasar disks, and that the luminosity is determined by the disk area, the larger more luminous disks will be amplified less. Using a relation between source size (in terms of Einstein radius) and amplitude given by Refsdal & Stabell ([@ref91]) (eqn 1), one can thus derive a relation between amplitude $\delta m$ and absolute magnitude $M$ of the form: $$\delta m = 10^{0.2(M+c)}$$ where $c$ is a constant. The bottom panel of Fig.  \[9\] shows a plot of amplitude versus absolute magnitude for the UVX sample with this relation superimposed. The constant $c$ was adjusted to allow the curve to track the upper envelope of the points, which has the effect of defining the quasar disk size. This ranges from 1.1 Einstein radii for $M_{B} = -23$ to 11 for $M_{B} = -28$. The points scatter downwards from the upper envelope because the quasars do not necessarily attain their maximum possible amplitude. Although one cannot say that the curve provides a fit to the data, the trend is certainly well represented.\ The possibility of a distinct population of large amplitude, low luminosity quasars suggested by Fig.  \[7\] may also be used to test the models of quasar variability. Again, an accretion disk provides no obvious mechanism for such an effect. The Christmas Tree model certainly does imply large amplitude variations for low luminosity objects, where each event is of comparable brightness to the nucleus itself. Perhaps the most natural explanation comes from microlensing, where the nucleus of low luminosity quasars would plausibly become very small compared with the Einstein radii of the lenses, resulting in large amplifications from caustic crossing events.\ Conclusions =========== In this paper we have examined the dependance of amplitude on redshift and luminosity for large samples of quasars selected on the basis of ultra-violet excess and variability. The quasars span a redshift range $0 < z < 3.5$ and a luminosity range $-20 > M_{B} > -28$. There is evidence for a correlation of amplitude with luminosity and/or redshift for the sample as a whole, but when it is binned in redshift the correlation with luminosity becomes significant. This result could be strengthened by the possible existence of a population of large amplitude low luminosity objects in the sample. No convincing evidence is found for a correlation between amplitude and redshift, either for the sample as a whole or when it is binned in luminosity.\ Various models of quasar variability are examined with respect to the observed correlations. It is concluded that any straightforward interpretation of an accretion disk model is incompatible with the data. The Christmas Tree model has some merits, especially in the regime of low luminosity quasars and Seyfert galaxies, but for luminous quasars the rate of supernovae required is too large to be compatible with the observed variability. The microlensing model can be used to explain all the data, although it does require that the Einstein radius of the microlensing bodies is comparable in size to the quasar nucleus.\ I thank Andy Lawrence and Omar Almaini for making some excellent suggestions for improvements to the paper. References {#references .unnumbered} ========== Aretxaga I., Terlevich R., 1994, MNRAS 269, 462 Aretxaga I., Cid Fernandes R., Terlevich R., 1997, MNRAS 286, 271 Cid Fernandes R., Aretxaga I., Terlevich R., 1996, MNRAS 282, 1191 Clavel J. et al., 1991, ApJ 366, 64 Cristiani S., Trentini S., La Franca F., Andreani P., 1996, A&A 321, 123 Cristiani S., Trentini S., La Franca F., Aretxaga I., Andreani P., Vio R., Gemmo A., 1996, A&A 306, 395 Di Clemente A., Giallongo E., Natali G., Trèvese D., Vagnetti F., 1996, ApJ 463, 466 Folkes S. et al., 1999, MNRAS 308, 459 Giallongo E., Trevese D., Vagnetti F., 1991, ApJ 377, 345 Hawkins M.R.S., 1993, Nat 366, 242 Hawkins M.R.S., 1996, MNRAS 278, 787 Hawkins M.R.S., Taylor A.N., 1997, ApJ 482, L5 Hawkins M.R.S. & Véron P., 1995, MNRAS 275, 1102 Hook I.M., McMahon R.G., Boyle B.J., Irwin M.J., 1994, MNRAS 268, 305 Kawaguchi T., Mineshige S., Umemura M., Turner E.L., 1998, ApJ 504, 671 Morris S.L., Weymann R.J., Anderson S.F., Hewett P.C., Foltz C.B., Chaffee F.H., Francis P.J., MacAlpine G.M., 1991, AJ 102, 1627 Refsdal S., Stabell R., 1991, A&A 250, 62 Schmidt M., Green R.F., 1983, ApJ 269, 352 Schneider P., Weiss A., 1987, A&A 171, 49 Trevese D., Pitella G., Kron R.G., Bershady M., 1989, AJ 98, 108 Appendix {#appendix .unnumbered} ======== ---- ---- ------- ----- ---- ------ ------- ------- ------ ------- -------- --- --- --- 21 16 28.38 -44 0 57.3 19.34 -0.42 0.64 2.062 -26.04 1 0 0 21 16 22.50 -44 29 13.9 19.33 -0.77 1.19 1.270 -25.03 1 0 3 21 16 44.32 -44 9 28.0 20.29 -0.59 1.72 1.375 -24.24 0 0 3 21 16 13.41 -46 42 47.3 18.71 -0.40 0.87 1.498 -26.00 1 0 0 21 16 25.78 -46 10 42.1 18.92 -0.37 0.54 0.748 -24.32 1 0 0 21 16 55.05 -44 39 37.1 17.89 -0.09 0.80 1.480 -26.80 1 0 0 21 17 24.47 -42 41 4.1 19.51 -0.35 1.12 0.363 -22.17 0 0 3 21 16 46.14 -46 14 21.7 19.08 -0.23 0.59 0.297 -22.17 1 0 0 21 17 30.01 -44 2 40.4 18.30 -0.45 0.44 1.710 -26.69 1 0 0 21 17 25.61 -45 4 58.1 20.40 0.25 0.72 2.313 -25.22 1 0 0 21 17 26.97 -45 30 20.9 20.28 1.15 1.06 2.993 -25.86 1 0 0 21 17 21.81 -47 3 49.0 18.80 0.01 0.45 2.260 -26.77 1 0 0 21 17 53.12 -46 0 44.8 19.71 0.31 0.82 2.955 -26.41 1 0 0 21 17 55.61 -46 9 17.1 20.38 0.10 0.82 2.120 -25.06 1 0 0 21 17 48.23 -46 47 36.2 19.35 -0.50 0.48 1.038 -24.59 1 0 0 21 17 57.12 -46 20 47.4 20.50 0.46 0.84 2.884 -25.57 1 0 0 21 18 21.07 -44 59 51.9 19.19 -0.12 0.59 2.194 -26.32 1 0 0 21 18 34.84 -43 49 45.2 19.29 -0.43 0.90 1.121 -24.81 1 0 0 21 18 34.30 -44 0 44.9 20.44 1.01 0.70 3.140 -25.80 1 0 0 21 18 41.77 -43 24 38.6 20.78 -0.47 1.57 1.076 -23.23 0 0 3 ---- ---- ------- ----- ---- ------ ------- ------- ------ ------- -------- --- --- --- ---- ---- ------- ----- ---- ------ ------- ------- ------ ------- -------- --- --- --- 21 18 47.48 -43 1 9.5 18.81 -0.62 0.75 2.123 -26.63 1 0 0 21 18 11.92 -46 56 3.6 18.35 -0.37 0.41 1.720 -26.65 1 0 0 21 18 55.69 -43 3 36.1 18.66 -0.01 0.70 2.201 -26.85 1 0 0 21 19 0.54 -43 48 0.5 20.72 0.63 0.82 3.064 -25.47 1 0 0 21 18 29.22 -47 2 34.9 18.36 -0.42 1.05 1.332 -26.10 1 0 0 21 19 14.89 -43 7 27.0 19.81 0.43 0.84 2.929 -26.29 1 0 0 21 19 37.27 -43 23 9.2 20.31 0.07 1.04 2.730 -25.65 1 0 0 21 19 20.05 -45 30 52.3 20.38 0.00 0.51 2.410 -25.32 1 0 0 21 19 45.46 -43 32 19.3 20.65 -0.65 1.40 0.652 -22.29 0 0 3 21 19 47.61 -43 46 23.9 19.16 -0.40 1.02 1.000 -24.70 1 0 0 21 19 37.16 -45 48 22.3 19.98 0.03 0.81 2.210 -25.54 1 0 0 21 19 55.45 -44 17 54.7 20.73 0.59 0.79 3.080 -25.47 1 0 0 21 20 14.92 -42 55 30.5 20.89 0.15 0.53 0.619 -21.94 1 0 0 21 19 49.62 -45 49 8.4 20.65 -0.26 0.67 1.715 -24.35 1 2 0 21 20 18.19 -44 10 40.6 20.69 -0.10 1.10 0.580 -22.00 0 0 3 21 20 28.49 -43 38 36.6 19.23 -0.35 0.51 0.840 -24.26 1 0 0 21 20 4.87 -46 34 17.3 19.56 0.02 0.59 2.375 -26.11 1 0 0 21 20 21.60 -45 52 12.2 20.22 0.85 0.60 2.989 -25.92 1 0 0 21 20 50.81 -43 27 55.6 17.92 -0.49 0.63 1.240 -26.39 1 0 0 21 20 58.76 -44 13 48.6 20.94 0.02 0.81 1.741 -24.09 1 0 0 21 20 47.86 -45 32 44.2 20.03 1.10 0.61 2.941 -26.08 1 0 0 21 20 57.48 -46 20 36.2 20.09 -0.35 0.64 0.739 -23.12 1 2 0 21 21 26.81 -43 51 34.2 19.20 -0.49 0.79 1.946 -26.06 1 0 0 21 21 20.68 -44 45 7.7 20.29 0.98 0.46 2.963 -25.83 1 0 0 21 21 19.34 -45 5 0.8 21.18 -0.07 1.24 0.748 -22.06 0 0 3 21 21 12.02 -46 4 1.4 20.34 0.09 0.95 2.260 -25.23 1 0 0 21 21 22.69 -45 58 25.6 20.74 -0.32 0.60 1.650 -24.17 0 2 0 21 21 41.90 -44 6 57.6 17.80 -0.46 0.52 1.735 -27.22 1 0 0 21 21 33.83 -45 15 9.2 20.39 -0.11 1.83 0.758 -22.88 0 0 3 21 21 23.86 -46 38 7.0 19.92 -0.21 1.64 0.912 -23.74 0 0 3 21 21 31.37 -46 13 36.3 20.51 -0.48 0.59 1.047 -23.45 0 2 0 21 21 43.13 -44 56 36.3 20.40 0.38 0.63 2.950 -25.71 1 0 0 21 21 40.68 -45 51 23.9 18.89 -0.43 0.46 0.947 -24.85 1 2 0 21 21 42.39 -45 49 26.6 20.10 -0.28 1.10 0.520 -22.36 1 2 3 21 21 35.46 -46 42 27.4 19.11 -0.41 0.76 1.347 -25.38 1 0 0 21 21 41.39 -45 59 53.9 19.73 -0.58 0.90 0.887 -23.87 1 2 0 21 21 39.65 -46 41 17.2 20.35 -0.52 1.12 1.352 -24.15 0 0 3 21 21 56.68 -45 8 33.6 18.83 -0.31 0.65 1.353 -25.67 1 0 0 21 22 12.50 -43 9 36.5 20.83 0.46 0.65 3.060 -25.36 1 0 0 21 22 0.52 -45 57 24.8 17.75 -0.42 0.46 0.953 -26.01 0 2 0 21 22 25.85 -42 43 24.2 20.16 0.17 0.68 2.457 -25.58 1 0 0 21 22 30.46 -43 28 38.6 20.01 -0.10 1.14 0.401 -21.89 0 0 3 21 22 25.60 -44 22 18.5 18.87 0.22 0.49 2.465 -26.88 1 0 0 21 22 16.68 -46 31 54.8 20.27 -0.10 1.60 0.820 -23.17 0 0 3 21 22 39.06 -44 33 4.8 20.32 0.05 0.80 2.600 -25.54 1 0 0 21 22 41.46 -44 47 46.1 19.92 0.06 1.05 1.137 -24.21 1 0 0 21 22 52.08 -43 33 58.6 18.71 -0.07 0.45 0.552 -23.88 1 0 0 21 22 35.59 -46 22 5.7 19.06 -0.37 0.81 2.093 -26.35 1 0 0 21 22 48.18 -45 55 30.6 19.96 -0.53 0.51 1.425 -24.65 1 2 0 21 22 49.75 -45 58 24.4 19.41 -0.31 0.86 0.986 -24.42 1 2 0 21 22 56.31 -45 57 17.2 20.13 -0.28 0.78 0.858 -23.40 1 2 0 21 23 1.03 -46 3 38.1 19.42 -0.61 0.96 1.384 -25.13 1 2 0 21 23 3.18 -46 5 48.0 20.53 0.70 0.42 3.220 -25.76 1 0 0 21 23 21.19 -43 38 29.5 18.37 -0.18 0.66 0.480 -23.92 1 0 0 21 23 3.71 -46 54 52.7 18.63 -0.24 0.79 1.429 -25.98 1 0 0 ---- ---- ------- ----- ---- ------ ------- ------- ------ ------- -------- --- --- --- ---- ---- ------- ----- ---- ------ ------- ------- ------ ------- -------- --- --- --- 21 23 14.08 -45 48 53.9 19.12 -0.22 0.80 1.549 -25.66 1 0 0 21 23 31.47 -42 56 35.7 19.49 0.23 1.26 0.141 -20.15 1 0 3 21 23 13.92 -46 54 33.0 20.34 -0.10 2.31 0.422 -21.67 0 0 3 21 23 37.93 -43 46 58.0 18.91 -0.44 0.58 1.000 -24.95 1 0 0 21 23 43.22 -44 29 5.9 19.66 -0.66 0.84 1.353 -24.84 1 2 0 21 23 45.16 -44 29 9.8 20.06 -0.08 1.17 1.586 -24.77 0 0 3 21 23 48.33 -45 38 4.0 19.50 -0.28 0.39 1.642 -25.40 1 0 0 21 24 7.74 -44 12 53.9 19.39 -0.16 1.50 0.542 -23.16 1 0 3 21 24 5.49 -44 43 57.5 19.83 -0.32 0.26 1.589 -25.01 0 2 0 21 24 9.52 -44 12 58.3 20.26 -0.43 1.20 1.239 -24.05 0 0 3 21 24 3.30 -46 27 37.3 20.78 0.00 0.52 1.140 -23.36 1 0 0 21 24 7.41 -46 5 44.6 20.45 0.06 1.51 0.705 -22.66 1 0 3 21 24 11.89 -45 40 17.0 18.78 -0.52 0.52 1.395 -25.78 1 0 0 21 24 19.46 -45 15 1.7 20.03 -0.51 0.61 1.935 -25.22 1 2 0 21 24 30.39 -43 3 34.7 19.46 -0.39 0.54 1.286 -24.93 1 0 0 21 24 33.74 -44 12 3.5 20.01 -0.34 1.88 1.625 -24.87 0 0 3 21 24 21.85 -46 52 0.4 19.90 0.58 0.58 2.495 -25.87 1 0 0 21 24 49.75 -43 20 36.0 19.01 0.07 0.64 0.820 -24.43 1 0 0 21 24 46.25 -44 32 24.7 20.66 0.16 0.47 0.677 -22.37 1 0 0 21 24 46.53 -45 53 46.9 20.48 -0.62 1.10 1.961 -24.79 0 2 3 21 24 54.69 -45 4 54.4 20.48 -0.32 0.30 0.388 -21.35 0 2 0 21 24 56.51 -45 53 2.6 19.76 0.03 0.61 2.322 -25.86 1 0 0 21 25 5.28 -44 48 42.3 19.91 -0.40 0.82 1.733 -25.11 0 2 0 21 25 25.85 -43 35 13.5 20.32 0.01 0.59 2.260 -25.25 1 0 0 21 25 23.50 -44 24 24.2 20.01 -0.16 0.69 2.500 -25.77 1 0 0 21 25 23.30 -45 16 17.9 20.62 -0.28 0.66 0.242 -20.19 1 2 0 21 25 28.96 -45 9 50.0 19.41 -0.38 0.93 1.590 -25.43 1 2 0 21 25 28.12 -45 36 53.9 20.44 0.24 0.63 2.766 -25.54 1 0 0 21 25 28.84 -46 39 17.1 19.11 -0.41 0.45 1.601 -25.74 1 0 0 21 25 46.60 -44 32 58.3 20.39 -0.04 0.56 2.503 -25.39 1 0 0 21 25 53.43 -43 43 39.9 20.44 -0.29 0.65 0.451 -21.71 1 2 0 21 25 56.64 -44 8 32.1 18.48 -0.33 0.63 0.366 -23.22 1 0 0 21 25 49.78 -46 11 32.1 20.32 -0.78 1.12 1.305 -24.10 1 2 3 21 25 49.84 -46 47 52.4 19.07 -0.45 0.50 1.888 -26.13 1 0 0 21 26 10.02 -42 56 30.8 17.79 -0.28 0.67 0.405 -24.13 1 0 0 21 26 1.19 -47 8 26.6 19.28 -0.10 0.52 0.698 -23.81 1 0 0 21 26 7.35 -45 34 52.2 21.27 -0.22 0.51 0.719 -21.88 0 2 0 21 26 12.68 -47 8 30.4 20.04 0.02 0.68 2.200 -25.47 1 0 0 21 26 33.72 -45 58 45.3 18.17 -0.23 0.80 1.579 -26.65 1 2 0 21 26 40.88 -43 34 19.4 18.48 -0.27 0.86 0.584 -24.23 1 2 0 21 26 43.20 -43 36 13.8 20.71 -0.49 0.68 1.280 -23.67 1 2 0 21 26 39.70 -45 2 43.9 20.54 -0.49 0.77 2.156 -24.93 0 2 0 21 26 41.19 -45 31 12.6 19.93 -0.56 0.71 0.980 -23.89 1 2 0 21 26 45.68 -43 50 6.3 20.08 -0.37 1.14 1.120 -24.02 1 2 3 21 26 50.42 -43 46 50.6 20.29 -0.47 1.05 1.110 -23.79 1 2 0 21 26 49.67 -45 26 56.3 18.59 -0.47 0.44 1.382 -25.95 1 2 0 21 26 52.97 -46 19 0.7 18.90 -0.66 0.67 1.880 -26.29 1 0 0 21 27 0.30 -44 56 34.0 20.33 -0.16 1.94 0.522 -22.14 0 0 3 21 26 58.26 -47 2 58.9 20.69 0.08 0.90 2.592 -25.16 1 0 0 21 27 8.64 -44 29 24.6 20.83 -0.20 0.90 2.122 -24.61 0 2 0 21 27 13.56 -44 35 17.7 18.82 -0.57 0.69 2.015 -26.51 1 2 0 21 27 19.33 -42 42 44.3 17.71 -0.41 0.66 0.799 -25.67 1 0 0 21 27 14.13 -45 55 19.9 20.12 0.15 0.66 2.440 -25.61 1 0 0 21 27 19.62 -43 45 11.4 20.08 -0.46 0.72 1.722 -24.92 1 2 0 21 27 17.65 -46 1 39.1 19.98 -0.94 1.37 1.285 -24.41 1 2 3 ---- ---- ------- ----- ---- ------ ------- ------- ------ ------- -------- --- --- --- ---- ---- ------- ----- ---- ------ ------- ------- ------ ------- -------- --- --- --- 21 27 21.35 -44 56 24.6 19.83 0.43 0.63 2.880 -26.24 1 0 0 21 27 31.32 -44 45 6.4 18.72 -0.35 0.78 1.536 -26.04 1 0 0 21 27 39.34 -43 8 36.7 20.46 0.00 0.84 2.674 -25.45 1 0 0 21 27 37.22 -45 28 57.0 18.60 0.84 0.44 2.730 -27.36 1 0 0 21 27 52.99 -43 50 37.0 19.40 -0.15 0.64 0.493 -22.94 1 0 0 21 27 54.41 -45 52 55.5 19.94 -0.48 1.17 1.101 -24.12 0 0 3 21 27 55.93 -45 49 0.5 19.04 -0.15 1.07 0.871 -24.52 1 0 0 21 27 55.24 -47 6 50.1 18.42 -0.24 0.57 0.578 -24.27 1 0 0 21 27 59.09 -45 11 9.3 19.83 -0.89 0.83 2.001 -25.49 1 2 0 21 28 2.17 -45 25 10.3 19.01 -0.37 0.59 0.961 -24.76 0 2 0 21 28 4.28 -46 39 7.9 20.73 0.05 0.71 2.610 -25.13 1 0 0 21 28 5.82 -46 33 24.7 17.44 0.19 0.65 0.153 -22.37 1 0 0 21 28 14.96 -42 45 41.9 19.44 -0.38 0.64 1.420 -25.16 1 0 0 21 28 13.84 -44 43 32.1 20.49 -0.23 0.62 1.229 -23.81 1 2 0 21 28 17.05 -44 34 56.0 19.52 -0.53 0.68 1.434 -25.10 1 2 0 21 28 21.96 -46 10 53.2 17.65 -0.50 0.40 0.833 -25.82 1 2 0 21 28 24.69 -45 37 3.4 19.77 0.00 0.70 0.204 -20.67 1 0 0 21 28 40.98 -45 20 20.8 20.56 -0.25 1.75 0.544 -22.00 0 0 3 21 28 42.00 -43 30 59.2 18.91 -0.60 0.42 1.905 -26.30 1 0 0 21 28 44.77 -46 4 16.3 20.51 -0.20 1.12 0.831 -22.95 1 2 0 21 28 56.74 -45 48 1.9 20.45 -0.42 0.98 1.087 -23.59 0 2 0 21 29 1.26 -46 6 30.4 20.07 -0.40 0.81 1.025 -23.84 1 2 0 21 29 4.08 -42 55 55.0 20.65 -0.46 1.10 2.070 -24.74 0 0 3 21 29 4.72 -44 42 39.7 19.22 -0.38 0.99 1.025 -24.69 1 2 0 21 29 8.42 -45 15 39.0 19.80 -0.67 0.61 1.260 -24.55 1 2 0 21 29 10.18 -46 17 2.3 20.74 -0.30 1.31 0.998 -23.11 1 2 0 21 29 10.53 -44 50 18.5 19.07 -0.68 1.03 1.942 -26.18 1 2 0 21 29 24.73 -45 27 7.0 20.21 -0.66 0.84 1.375 -24.32 1 2 0 21 29 27.63 -45 1 15.8 20.89 -0.08 0.56 2.170 -24.59 1 0 0 21 29 32.92 -45 51 30.9 20.58 -0.35 0.82 1.286 -23.81 1 2 0 21 29 38.02 -45 12 11.6 19.31 0.11 0.90 2.180 -26.18 1 0 0 21 29 38.26 -44 13 47.1 19.85 -0.45 0.83 1.452 -24.80 1 2 0 21 29 39.50 -46 24 19.2 17.80 -0.30 0.67 0.435 -24.27 1 0 0 21 29 39.63 -46 29 40.3 19.71 0.02 0.65 2.465 -26.04 1 0 0 21 29 41.25 -45 22 49.0 20.21 9.99 0.48 3.580 -26.30 1 0 0 21 29 42.67 -45 16 27.2 20.62 -0.54 0.73 1.030 -23.30 1 2 0 21 29 45.45 -46 53 47.6 18.85 -0.02 0.61 2.230 -26.69 1 0 0 21 29 47.37 -46 2 7.8 18.97 -0.27 0.51 0.299 -22.29 1 2 0 21 29 50.05 -45 46 56.6 20.29 -0.25 0.62 2.034 -25.06 0 2 0 21 29 51.34 -44 25 9.8 20.71 -0.19 0.86 0.744 -22.52 1 0 0 21 29 57.55 -46 53 51.1 19.85 0.02 0.62 2.208 -25.67 1 0 0 21 29 56.13 -44 23 10.5 20.76 -0.67 0.78 2.011 -24.57 0 2 0 21 30 1.27 -46 3 4.9 19.61 -0.14 0.52 2.244 -25.94 1 0 0 21 30 1.99 -44 3 36.7 20.88 0.68 0.74 2.970 -25.25 1 0 0 21 30 2.98 -45 8 45.2 19.07 0.05 1.38 0.740 -24.15 1 0 3 21 30 8.22 -43 52 59.0 19.93 0.11 0.84 2.634 -25.95 1 0 0 21 30 16.27 -43 10 28.9 21.37 -0.16 1.41 0.914 -22.30 0 0 3 21 30 20.02 -44 40 42.8 20.25 -0.07 0.47 0.610 -22.55 1 0 0 21 30 20.78 -44 45 36.3 20.00 -0.48 0.59 1.460 -24.66 0 2 0 21 30 22.37 -44 50 58.9 19.65 -0.12 0.61 0.725 -23.52 1 0 0 21 30 24.58 -44 42 19.8 20.29 0.96 0.40 3.040 -25.89 1 0 0 21 30 27.93 -45 55 32.9 18.91 -0.31 0.40 1.556 -25.88 1 2 0 21 30 30.00 -45 45 38.2 20.65 0.23 0.59 2.645 -25.24 1 0 0 21 30 30.16 -44 45 24.4 19.71 -0.63 0.64 1.324 -24.74 0 2 0 21 30 30.63 -43 6 51.6 19.51 -0.23 0.69 1.645 -25.40 1 2 0 ---- ---- ------- ----- ---- ------ ------- ------- ------ ------- -------- --- --- --- ---- ---- ------- ----- ---- ------ ------- ------- ------ ------- -------- --- --- --- 21 30 38.92 -43 50 22.4 20.22 -0.69 1.31 1.229 -24.08 1 2 3 21 30 41.40 -43 48 1.0 19.56 -0.40 0.61 1.597 -25.29 1 2 0 21 30 45.86 -45 56 38.6 20.58 -0.26 0.61 1.498 -24.13 1 2 0 21 30 50.85 -44 32 7.2 18.85 -0.27 0.54 0.793 -24.51 1 2 0 21 31 1.26 -46 7 4.0 20.73 -0.39 0.41 2.010 -24.60 0 2 0 21 31 0.28 -44 36 23.0 19.99 -0.36 0.47 1.234 -24.31 1 2 0 21 31 7.60 -46 12 42.8 20.71 -0.40 0.83 1.978 -24.58 0 2 0 21 31 4.84 -44 14 6.4 20.23 -0.44 0.62 0.999 -23.63 0 2 0 21 31 12.57 -46 0 12.2 19.90 0.10 0.65 2.719 -26.05 1 0 0 21 31 14.89 -46 56 9.1 19.26 -0.30 0.37 0.980 -24.56 1 0 0 21 31 9.87 -43 39 29.7 20.76 -0.33 0.65 1.390 -23.79 1 2 0 21 31 15.15 -45 7 35.1 19.19 -0.73 0.52 1.966 -26.09 1 2 0 21 31 15.80 -45 1 19.0 19.67 -0.22 1.15 0.418 -22.32 0 0 3 21 31 21.94 -46 38 23.6 18.71 -0.45 0.54 1.720 -26.29 1 0 0 21 31 15.90 -42 43 18.9 20.20 0.08 0.99 0.365 -21.50 1 0 0 21 31 22.52 -43 43 53.5 20.55 -0.41 0.70 0.844 -22.95 1 2 0 21 31 26.19 -45 15 3.1 20.69 -0.72 1.12 1.920 -24.54 1 2 3 21 31 27.03 -45 4 49.3 19.85 0.00 1.12 0.715 -23.29 1 0 0 21 31 23.41 -42 59 60.0 20.13 -0.39 0.80 1.648 -24.78 1 2 0 21 31 37.75 -45 48 53.2 20.65 -0.20 1.37 0.781 -22.68 0 0 3 21 31 35.99 -44 10 45.4 21.12 -0.62 1.25 0.529 -21.38 0 2 3 21 31 33.39 -42 57 51.0 18.22 -0.67 0.74 2.096 -27.19 1 2 0 21 31 43.86 -46 57 13.0 17.97 0.03 0.54 0.684 -25.08 1 0 0 21 31 36.21 -44 3 24.2 20.35 -0.36 0.63 1.660 -24.58 0 2 0 21 31 36.77 -43 39 43.3 19.90 -0.35 0.74 0.843 -23.59 0 2 0 21 31 38.88 -43 54 22.2 20.40 -0.34 0.42 1.216 -23.87 0 2 0 21 31 38.93 -42 47 29.3 20.39 -0.36 1.23 0.982 -23.43 1 2 3 21 31 43.86 -43 39 11.8 20.52 -0.53 0.55 0.715 -22.62 0 2 0 21 31 50.98 -43 19 14.5 20.05 -0.38 0.51 1.656 -24.87 0 2 0 21 31 54.25 -44 29 23.0 19.14 -0.49 0.50 1.414 -25.45 1 0 0 21 32 2.95 -45 16 9.7 18.91 -0.55 0.79 1.890 -26.29 1 0 0 21 31 56.70 -42 55 39.3 21.08 -0.36 1.70 1.730 -23.93 0 2 3 21 32 10.46 -46 14 20.6 19.60 0.34 0.72 2.769 -26.39 1 0 0 21 32 4.70 -42 45 41.3 19.02 -0.61 0.70 1.992 -26.29 1 0 0 21 32 13.49 -45 16 50.0 17.68 -0.59 0.55 0.507 -24.72 1 2 0 21 32 13.00 -44 14 14.0 19.65 -0.22 0.63 1.645 -25.26 1 2 0 21 32 11.55 -43 24 45.8 19.87 -0.35 1.71 0.860 -23.67 1 2 3 21 32 18.25 -45 6 59.8 19.20 -0.45 0.61 0.520 -23.26 1 0 0 21 32 18.33 -44 18 48.4 19.87 -0.35 0.97 1.451 -24.77 1 2 0 21 32 21.72 -43 24 47.5 20.43 -0.11 0.52 2.240 -25.12 1 0 0 21 32 23.85 -43 48 10.8 20.19 0.23 0.86 2.790 -25.81 1 0 0 21 32 23.30 -43 9 42.2 20.82 -0.56 0.45 0.959 -22.95 0 2 0 21 32 27.48 -44 5 24.8 20.41 -0.28 0.98 0.478 -21.87 1 2 0 21 32 36.44 -46 34 0.5 19.46 -0.53 1.33 1.320 -24.99 1 0 3 21 32 35.27 -46 11 31.5 19.01 -0.46 0.66 1.600 -25.84 1 2 0 21 32 26.58 -43 18 4.3 19.66 -0.31 0.82 1.710 -25.33 1 2 0 21 32 31.07 -44 18 57.9 19.08 -0.77 1.25 1.258 -25.26 1 2 3 21 32 37.49 -45 8 31.6 20.12 -0.56 0.59 1.377 -24.41 1 2 0 21 32 37.63 -44 51 1.2 18.55 -0.41 0.91 0.920 -25.13 1 0 0 21 32 38.00 -44 20 41.8 20.61 -0.53 1.12 0.655 -22.34 1 2 0 21 32 54.14 -47 5 8.9 19.38 -0.44 1.59 0.244 -21.44 1 0 3 21 32 53.63 -45 58 49.2 20.29 -0.41 0.76 1.003 -23.57 1 2 0 21 32 51.16 -44 31 17.9 21.33 -0.09 1.76 0.430 -20.72 0 0 3 21 33 3.03 -46 20 24.7 20.87 0.27 0.78 2.760 -25.11 1 0 0 21 32 55.05 -43 21 44.1 17.93 0.27 0.48 2.420 -27.78 1 0 0 ---- ---- ------- ----- ---- ------ ------- ------- ------ ------- -------- --- --- --- ---- ---- ------- ----- ---- ------ ------- ------- ------ ------- -------- --- --- --- 21 33 2.60 -44 16 33.7 20.06 -0.29 0.28 0.327 -21.40 0 2 0 21 33 6.66 -44 28 26.5 20.80 -0.68 0.99 0.544 -21.76 1 2 0 21 33 7.60 -44 16 50.7 20.76 9.99 0.75 3.340 -25.61 1 0 0 21 33 13.11 -45 20 2.1 20.65 0.34 0.49 2.898 -25.43 1 0 0 21 33 7.84 -43 4 13.9 20.25 0.19 1.00 2.470 -25.50 1 0 0 21 33 12.34 -42 58 51.0 19.40 -0.22 0.43 1.190 -24.83 1 2 0 21 33 22.63 -44 20 23.7 19.96 -0.37 0.45 0.999 -23.90 1 2 0 21 33 31.09 -46 11 38.3 19.72 -0.37 0.65 0.888 -23.89 1 2 0 21 33 31.37 -46 13 34.8 19.58 -0.44 1.36 1.448 -25.06 1 2 3 21 33 21.32 -43 30 55.6 21.29 -0.55 0.92 1.720 -23.71 0 2 0 21 33 23.97 -43 36 17.3 20.42 -0.20 0.46 0.669 -22.58 1 2 0 21 33 29.69 -44 4 8.7 20.37 -0.22 0.90 1.756 -24.67 1 2 0 21 33 39.37 -46 3 3.6 20.75 0.03 1.45 0.423 -21.26 1 0 3 21 33 26.75 -42 55 1.3 20.45 -0.14 1.57 0.580 -22.24 0 0 3 21 33 38.98 -45 34 51.6 19.44 -0.24 1.54 0.868 -24.12 1 2 3 21 33 41.36 -45 53 0.8 18.34 -0.54 0.36 1.000 -25.52 1 2 0 21 33 51.84 -46 17 51.0 20.75 -0.49 1.21 1.571 -24.06 0 2 0 21 33 53.98 -46 17 24.2 19.65 -0.40 0.85 0.844 -23.85 1 2 0 21 33 58.56 -47 8 7.8 19.35 -0.62 1.10 1.097 -24.70 1 0 3 21 33 45.77 -43 58 44.9 18.27 -0.33 0.83 1.560 -26.53 1 2 0 21 33 45.15 -43 49 23.8 20.09 -0.59 0.77 1.850 -25.06 1 2 0 21 33 47.57 -44 17 35.9 20.35 0.44 0.45 2.838 -25.69 1 0 0 21 33 48.75 -44 28 48.7 20.21 -0.51 0.51 2.059 -25.17 0 2 0 21 34 1.71 -46 46 47.6 19.90 0.63 0.74 3.065 -26.29 1 0 0 21 33 53.80 -44 56 27.5 20.08 0.03 0.62 2.745 -25.89 1 0 0 21 33 46.40 -43 21 34.2 20.24 -0.69 0.83 2.022 -25.10 1 2 0 21 33 51.02 -43 6 10.0 20.81 -0.55 0.85 1.400 -23.76 1 2 0 21 34 8.30 -45 6 3.0 20.59 -0.31 0.53 0.880 -23.00 1 2 0 21 34 2.22 -43 10 38.6 21.12 -0.50 0.56 0.524 -21.36 0 2 0 21 34 8.13 -43 51 26.6 19.56 -0.24 1.03 1.663 -25.37 1 2 0 21 34 19.95 -46 2 17.9 20.41 -0.65 1.52 1.292 -23.99 1 2 3 21 34 8.81 -43 11 33.7 19.92 -0.24 0.74 1.650 -24.99 1 2 0 21 34 19.94 -45 16 58.6 20.44 -0.48 0.84 1.106 -23.63 1 2 0 21 34 17.90 -43 50 8.8 19.88 -0.35 0.82 1.558 -24.91 1 2 0 21 34 13.93 -42 57 43.8 19.30 0.11 0.60 2.608 -26.56 1 0 0 21 34 18.64 -43 52 39.7 20.71 -0.61 0.64 1.370 -23.81 1 2 0 21 34 37.07 -46 45 4.8 19.37 -0.23 0.82 0.704 -23.74 1 0 0 21 34 30.32 -44 35 55.7 20.27 -0.72 1.48 1.319 -24.17 0 0 3 21 34 29.90 -44 3 5.5 20.47 -0.12 1.11 1.576 -24.35 1 0 0 21 34 47.85 -46 2 39.1 20.15 -0.52 0.56 1.340 -24.33 0 2 0 21 34 50.90 -46 1 49.4 17.81 0.12 0.46 0.528 -24.68 1 0 0 21 34 37.41 -43 40 37.4 19.39 -0.25 0.48 0.837 -24.09 0 2 0 21 34 49.15 -44 43 0.2 19.53 0.15 0.64 2.526 -26.27 1 0 0 21 35 0.95 -46 11 36.5 18.87 -0.41 0.27 1.903 -26.34 0 2 0 21 34 43.15 -43 6 56.5 19.67 -0.36 0.81 1.423 -24.93 1 2 0 21 35 6.63 -46 32 27.6 18.66 -0.08 1.02 2.222 -26.87 1 0 0 21 35 2.90 -45 49 32.5 20.06 -0.59 0.72 1.832 -25.07 1 2 0 21 35 4.13 -45 47 3.5 20.36 -0.46 1.15 1.845 -24.79 0 2 0 21 35 0.77 -44 16 28.6 20.67 -0.27 0.26 0.285 -20.49 0 2 0 21 34 56.25 -43 13 47.4 20.08 -0.30 0.85 2.141 -25.38 1 2 0 21 35 19.47 -46 20 47.4 18.41 -0.21 0.53 0.505 -23.99 1 0 0 21 35 0.16 -43 4 14.3 19.60 -0.26 1.96 0.511 -22.82 1 2 3 21 35 4.58 -43 46 13.0 20.87 0.50 0.59 3.100 -25.35 1 0 0 21 35 14.93 -45 16 31.0 20.94 -0.20 1.21 0.655 -22.01 1 2 3 21 35 5.14 -43 3 48.9 19.62 -0.45 0.47 2.029 -25.72 1 2 0 ---- ---- ------- ----- ---- ------ ------- ------- ------ ------- -------- --- --- --- ---- ---- ------- ----- ---- ------ ------- ------- ------ ------- -------- --- --- --- 21 35 12.41 -44 2 15.0 19.30 -0.33 0.34 0.929 -24.40 0 2 0 21 35 28.41 -45 59 2.1 19.95 -0.56 0.72 0.568 -22.70 1 2 0 21 35 13.86 -43 24 17.3 19.18 -0.42 0.77 1.935 -26.07 1 2 0 21 35 28.59 -45 29 56.1 20.77 -0.27 0.64 2.148 -24.69 1 2 0 21 35 28.35 -45 23 57.7 18.69 -0.47 0.66 1.097 -25.36 1 2 0 21 35 22.69 -44 7 50.2 20.89 -0.36 0.75 1.168 -23.30 1 2 0 21 35 20.13 -42 53 29.7 17.89 -0.45 0.64 1.469 -26.78 1 2 0 21 35 23.26 -43 20 7.1 20.33 -0.33 1.00 1.535 -24.43 1 2 0 21 35 34.64 -43 49 30.6 21.12 -0.54 0.86 1.395 -23.44 0 2 0 21 35 41.40 -44 0 49.2 18.45 -0.32 0.87 0.461 -23.75 0 2 0 21 35 38.64 -43 26 25.7 20.65 0.06 0.46 2.483 -25.11 1 0 0 21 35 43.33 -44 8 11.1 20.59 -0.48 1.23 1.785 -24.49 0 2 0 21 35 47.12 -44 1 23.2 20.41 -0.18 0.60 1.708 -24.58 1 0 0 21 35 43.26 -43 20 54.9 20.38 -0.11 0.55 2.186 -25.12 1 0 0 21 36 6.35 -46 26 0.5 19.48 -0.37 1.31 0.920 -24.20 0 0 3 21 35 53.01 -44 21 44.3 18.69 -0.15 0.70 0.408 -23.25 1 0 0 21 35 54.17 -44 14 32.4 21.14 -0.51 0.27 0.412 -20.82 0 2 0 21 35 54.42 -43 45 24.2 20.47 -0.43 0.98 1.633 -24.42 1 2 0 21 36 19.27 -46 48 48.0 18.88 -0.11 0.54 1.124 -25.23 1 0 0 21 36 17.27 -44 30 11.6 19.80 -0.46 0.41 0.931 -23.91 0 2 0 21 36 28.18 -45 38 4.6 19.14 -0.22 0.59 1.695 -25.83 1 0 0 21 36 21.01 -43 17 51.2 20.29 -0.39 0.98 1.048 -23.67 0 2 0 21 36 31.16 -44 20 53.9 21.14 -0.35 0.29 0.433 -20.92 0 2 0 21 36 28.71 -43 47 29.6 18.96 -0.34 0.61 1.520 -25.78 1 2 0 21 36 30.45 -43 1 22.7 18.12 -0.57 0.62 1.343 -26.36 1 0 0 21 36 39.18 -44 11 35.1 19.63 -0.41 0.69 1.925 -25.61 1 2 0 21 36 52.42 -45 29 1.7 19.43 -0.09 0.75 1.174 -24.77 1 0 0 21 36 34.83 -42 48 0.0 20.10 0.08 0.40 0.168 -19.92 1 0 0 21 37 4.86 -46 20 11.6 19.22 -0.02 0.52 2.495 -26.55 1 0 0 21 36 42.71 -43 30 28.1 20.86 -0.33 1.41 1.028 -23.06 1 2 3 21 36 49.17 -44 20 47.4 20.23 -0.30 0.35 0.275 -20.85 0 2 0 21 36 42.52 -43 28 46.6 18.38 -0.43 0.54 0.250 -22.50 1 2 0 21 36 59.31 -45 17 26.5 20.67 -0.38 0.86 1.105 -23.40 1 2 0 21 36 44.29 -42 51 36.2 19.67 -0.41 0.78 1.510 -25.06 1 2 0 21 37 8.20 -45 23 36.4 20.45 -0.38 0.94 1.645 -24.46 1 2 0 21 37 2.10 -44 22 56.2 20.64 -0.52 0.75 2.050 -24.73 0 2 0 21 37 12.13 -45 20 33.6 20.70 -0.34 0.58 2.113 -24.73 0 2 0 21 37 6.81 -43 54 27.7 19.70 -0.77 0.75 1.915 -25.52 1 2 0 21 37 25.87 -45 29 27.9 20.33 -0.39 0.60 1.438 -24.30 1 2 0 21 37 14.45 -44 6 39.4 20.30 -0.05 0.62 2.250 -25.26 1 0 0 21 37 17.79 -44 21 10.6 19.72 -0.35 1.42 0.346 -21.86 1 2 3 21 37 17.76 -44 10 44.8 20.87 -0.42 0.65 0.798 -22.51 0 2 0 21 37 10.59 -43 5 26.6 20.92 -0.42 1.32 1.393 -23.64 1 2 3 21 37 29.30 -44 49 49.2 19.54 0.49 0.72 0.136 -20.02 1 0 0 21 37 15.41 -43 1 47.3 19.83 -0.22 1.11 1.520 -24.91 1 2 3 21 37 31.36 -44 22 18.3 19.42 0.32 0.54 1.320 -25.03 1 0 0 21 37 53.50 -46 45 21.6 20.42 0.24 1.11 2.800 -25.59 1 0 0 21 37 32.47 -43 36 56.0 19.53 -0.10 0.48 2.310 -26.08 1 0 0 21 37 38.02 -43 43 5.3 19.16 -0.74 0.50 1.964 -26.12 1 2 0 21 38 0.90 -46 11 10.3 19.48 0.04 0.66 2.287 -26.11 1 0 0 21 37 51.68 -44 35 48.8 18.34 -0.34 0.35 0.632 -24.54 1 0 0 21 37 45.95 -43 56 50.9 20.02 -0.63 1.67 2.039 -25.34 0 2 3 21 38 17.41 -46 35 7.3 21.02 -0.37 1.40 0.780 -22.31 0 0 3 21 38 15.98 -46 21 31.8 19.45 -0.16 0.61 0.618 -23.38 1 0 0 21 38 16.48 -45 12 28.1 19.34 -0.24 0.25 2.164 -26.14 0 2 0 ---- ---- ------- ----- ---- ------ ------- ------- ------ ------- -------- --- --- --- ---- ---- ------- ----- ---- ------ ------- ------- ------ ------- -------- --- --- --- 21 38 11.93 -44 20 28.7 19.66 -0.59 0.51 2.102 -25.76 1 2 0 21 38 5.54 -43 20 26.1 19.91 -0.17 0.54 0.574 -22.76 1 0 0 21 38 41.58 -46 50 33.2 18.26 -0.20 1.02 0.762 -25.02 1 0 0 21 38 14.85 -43 45 49.7 19.26 -0.48 0.93 1.320 -25.19 1 2 0 21 38 12.87 -43 9 10.4 20.22 -0.17 2.08 0.687 -22.84 1 0 3 21 38 25.45 -44 5 15.3 19.85 -0.31 0.73 0.862 -23.69 1 2 0 21 38 15.73 -42 55 44.8 20.13 -0.23 0.83 1.597 -24.72 1 2 0 21 38 33.14 -44 16 15.2 20.53 -0.33 1.73 0.623 -22.32 0 0 3 21 38 28.59 -43 7 4.9 20.50 -0.30 0.26 0.255 -20.42 0 2 0 21 38 43.40 -44 32 17.9 0.00 -0.51 1.44 1.660 -44.93 0 0 3 21 38 31.80 -43 13 51.2 20.06 -0.75 0.79 1.287 -24.33 1 2 0 21 38 56.08 -45 26 4.1 20.08 -0.51 0.36 1.197 -24.16 1 2 0 21 38 56.93 -45 16 15.5 20.50 -0.50 1.50 2.013 -24.83 0 2 3 21 39 0.08 -45 20 38.8 20.01 -0.20 0.64 2.210 -25.51 1 2 0 21 38 53.72 -44 38 39.4 19.50 -0.53 0.61 1.078 -24.52 1 0 0 21 39 12.24 -46 14 18.8 20.15 1.14 0.59 3.340 -26.22 1 0 0 21 38 57.01 -44 16 52.4 19.31 -0.37 0.62 1.735 -25.71 1 2 0 21 38 54.54 -43 41 29.9 20.74 -0.33 1.02 2.210 -24.78 1 2 0 21 38 59.14 -43 50 57.6 18.71 0.21 0.40 0.142 -20.94 1 0 0 21 38 52.05 -42 58 42.7 20.26 -0.46 0.56 0.937 -23.46 1 2 0 21 38 54.79 -42 55 28.0 19.87 -0.20 1.18 0.542 -22.68 1 2 3 21 39 25.50 -45 20 14.0 19.97 -0.69 0.79 1.394 -24.59 1 2 0 21 39 20.18 -44 43 60.0 20.96 0.09 0.77 2.380 -24.71 1 0 0 21 39 27.52 -45 21 30.5 20.73 -0.24 0.79 1.671 -24.21 1 2 0 21 39 9.07 -42 54 48.2 20.62 -0.32 0.73 1.905 -24.59 0 2 0 21 39 22.45 -43 32 4.6 18.40 0.11 0.61 2.190 -27.10 1 0 0 21 39 32.94 -43 55 45.4 19.47 -0.51 0.64 0.679 -23.56 1 2 0 21 39 44.37 -44 49 37.2 20.10 -0.14 1.46 1.640 -24.80 0 0 3 21 39 27.24 -43 8 28.0 20.86 -0.20 1.04 0.752 -22.39 1 2 0 21 40 3.43 -46 20 2.2 18.78 -0.22 0.65 0.548 -23.79 1 0 0 21 39 30.84 -42 59 26.0 19.81 -0.35 0.59 1.890 -25.39 1 2 0 21 39 42.04 -43 28 17.4 20.55 -0.29 0.85 0.708 -22.57 0 2 0 21 39 48.87 -43 36 18.4 20.86 -0.56 1.21 1.379 -23.68 0 0 3 21 40 16.52 -45 52 37.0 18.11 -0.34 0.71 1.688 -26.85 1 0 0 21 39 51.98 -43 14 14.2 21.21 -0.15 1.48 1.645 -23.70 0 0 3 ---- ---- ------- ----- ---- ------ ------- ------- ------ ------- -------- --- --- ---

--- abstract: 'We computed flat rotation curves from scalar-tensor theories in their weak field limit. Our model, by construction, fits a flat rotation profile for velocities of stars. As a result, the form of the scalar field potential and DM distribution in a galaxy are determined. By taking into account the constraints for the fundamental parameters of the theory $(\lambda, \, \alpha)$, it is possible to obtain analytical results for the density profiles. For positive and negative values of $\alpha$, the DM matter profile is as cuspy as NFW’s.' address: | $^1$Depto. de Física, Instituto Nacional de Investigaciones Nucleares, A.P. 18-1027, 11801 D.F., México.\ $^2$Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, A.P. 70-543, 04510 D.F., México. author: - 'Jorge L Cervantes-Cota$^1$, M A Rodríguez-Meza$^1$ and Dario Nuñez$^2$' title: 'Flat rotation curves using scalar-tensor theories' --- Introduction ============ Recently, we have worked on some of the effects that general scalar-tensor theories (STT) of gravity yield on astrophysical scales [@Ro01; @RoCe04; @RoCePeTlCa05]. By taking the weak field limit of STT, three parameters appear [@PiOb86]: the Newtonian constant at infinity $G_{\infty}$, a Compton length-scale ($\lambda=\hbar/m c $) coming from the effective mass of the Lagrangian potential, and the strength of the new scalar gravitational interaction ($\alpha$). For point-like masses, the new Newtonian potential is of a Yukawa type [@PiOb86; @FiTa99] and, in general, analytical expressions can be found for spherical [@RoCe04] and axisymmetric systems [@RoCePeTlCa05]. Using these solutions we build a galactic model that is consistent with a flat rotation velocity profile which resembles the observations [@SoRu01]. In the present short contribution, we outline a way to construct such a galactic model using STT. STT and constraints {#st-constraints} =================== A typical spiral galaxy consists of a disk, a bulge, and a dark matter (DM) halo. A real halo is roughly spherical in shape and contains most of the matter, up to 90%, of the system. Thus, being the halo the main component, our galactic model will then consist of a spherical system of unknown DM. Therefore, in order compare with observations we assume that test particles –stars and dust– follow DM particles. To construct our halo model, we take the weak field limit of general STT and consider a spherically symmetric fluctuation of the scalar field around some fixed background field. The STT is then characterized by a background value of the scalar field, $\langle \phi \rangle$, a Compton lengthscale, $\lambda$, and a strength $\alpha$ [@PiOb86]. For point-like sources, the new Newtonian potential is well known to be of a Yukawa type [@FiTa99]: $$\label{phi_New} \Phi_N = -\frac{1}{\langle \phi \rangle} \frac{M}{r} (1+ \alpha \, \rm{e}^{-r/\lambda}) \, .$$ If one fixes the background field to be the inverse of the Newtonian constant, $\langle \phi \rangle= G_{N}^{-1}$ , then for $r \gg \lambda$ the new Newtonian potential coincides with the standard Newtonian one. Given this, for $r \ll \lambda$ one finds deviations of the order of $(1+\alpha)$ to the Newtonian dynamics. This setting is, however, very constrained by local (solar system) deviations of the Newtonian force, in order for $\alpha$ to be less than $10^{-10}$ [@FiTa99]. Alternatively, one can choose the setting $\langle \phi \rangle= G_{N}^{-1}(1+\alpha)$ and the new potential coincides with the Newtonian for $r \ll \lambda$ and deviates by $1/(1+\alpha)$ for $r \gg \lambda$. If one thinks of a galactic system, physical scales are around the tens of kiloparsecs –and so is our typical $\lambda$–, then for distances bigger than this, one expects constrictions, e.g. from cluster dynamics or cosmology. Several authors [@constraints] have considered these deviations, giving a rough estimate within the range of $-1 < \alpha \lesssim 5$. For example, the value of $\alpha=-0.5$ yields an asymptotic growing factor of $2$ in $G_N$, whereas the value of $\alpha=3.0$ reduces $G_N$ asymptotically by one fourth. The galactic model {#gal-model} ================== For a general density distribution the equations governing the weak energy (Newtonian limit) of STT are [@He91; @RoCe04]: $$\begin{aligned} \nabla^2 \Phi_{N} &=& \frac{G{_N}}{1+\alpha} \, \left[ 4\pi \rho - \frac{1}{2} \nabla^2 \bar{\phi} \right] \; , \label{pares_eq_h00}\\ \nabla^2 \bar{\phi} - m^2 \bar{\phi} &=& - 8\pi \alpha\rho \; , \label{pares_eq_phibar}\end{aligned}$$ where $\bar{\phi} $ is the scalar field fluctuation and $\rho$ is the density distribution which contains baryons, DM particles, or other types of matter. For definiteness we can think of DM, but its profile is unknown yet. This will be determined by setting the dynamics of test particles, i.e., by demanding the new Newtonian potential to be the one that solves the rotation curves of test particles –stars and dust. That is, we require $$\label{ph-rot-prof} v_{c}^{2} = r \frac{d\Phi_N}{dr} = {\rm const.} \, ,$$ where this constant is chosen to fit rotation velocities in spirals [@SoRu01]; this is the simplest model. We then proceed to solve for $\Phi_N$, and its solution is: $$\label{ph-N} \Phi_N = v_{c}^{2} \, \ln(r) \, .$$ Substituting this result into the original system, (\[pares\_eq\_h00\], \[pares\_eq\_phibar\]), gives $$\label{rho_sol} %\fl \rho \equiv \rho_{DM} = \frac{v_{c}^{2}}{4 \pi G_N r^2} + \frac{m^2}{8 \pi (1+\alpha)} \bar{\phi} \, .$$ For our convenience, we define the following densities: $ \rho^{*} \equiv \frac{v_{c}^{2}}{4 \pi G_N r^2} $ and $ \rho_{\bar \phi} \equiv \frac{m^2}{8 \pi (1+\alpha)} \bar{\phi} $, where $\rho^{*}$ is the density that the system would have to achieve flat rotation curves if system would be treated solely with Newtonian physics, and $ \rho_{\bar \phi}$ is the contribution of the scalar field, which should be obtained by integrating the equation: $$\label{nablaphi} \nabla^2 \bar{\phi} - \frac{m^2}{1+\alpha} \bar{\phi} = - 8 \pi \alpha \rho^{*} \, .$$ By comparing (\[pares\_eq\_phibar\]) with (\[nablaphi\]), it seems natural to identify $m = m^{*} \sqrt{1+\alpha}$ to convert (\[nablaphi\]) to a type (\[pares\_eq\_phibar\]) equation, now for $m^{*}$ and $\rho^{*}$. The solution is therefore given by [@RoCe04]: $$\begin{aligned} \label{pares_eq_finalphi} %\fl \bar{\phi}(r) &=& 8 \pi \alpha a^{2} \left[ \frac{\rm{e}^{-r_{a}/ \lambda^{*}_{a}}}{r_a} \int_0^{r_{a}} dx \; x \; \sinh(x/ \lambda^{*}_{a}) \rho^{*}(x) \right. \nonumber \\ && \left. + \frac{\sinh(r/ \lambda^{*}_{a})}{r_a} \int_{r_{a}}^{R_{a}} dx \; x \; \rm{e}^{-x/ \lambda^{*}_{a}} \rho^{*}(x) \right] \, ,\end{aligned}$$ where $\lambda^{*} \equiv \lambda \sqrt{1+\alpha}$ and $a$ is a length scale of the spherical system, e.g. related to the distance at which stars possess flat rotation curves; the size of the halo is denoted by $R$. Once (\[pares\_eq\_finalphi\]) is solved, its solution is substituted into (\[rho\_sol\]) to find the DM distribution, $\rho_{DM}$. The procedure outlined here is straightforward. Thus, using our given $\rho^{*}$ the above expression can analytically be integrated to have $$\begin{aligned} \label{phi-bar-sol} %\fl \bar{\phi}(r) &=& \frac{2 \alpha \, a^{2} v_{c}^{2} }{G_{N} r_{a}} \left[ \rm{e}^{-\frac{r_a}{\lambda^{*}_a }} \, \rm{sinhIntegral}\left(\frac{r_a}{\lambda^{*}_a }\right) \nonumber \right. \\ && \left. -\rm{sinh}\left(\frac{r_a}{\lambda^{*}_a}\right) \, \rm{expIntegralEi}\left(-\frac{r_a}{\lambda^{*}_a } \right) \right] \, . \end{aligned}$$ Thus, the DM profile becomes $$\begin{aligned} \label{rho-dm-sol} %\fl \rho_{DM} &=& \frac{v_{c}^{2}}{4 \pi G_{N}} \left\{ \frac{1}{a^{2} r_a^2} \right. \nonumber \\ && + \frac{\alpha}{1+\alpha} \frac{1}{\lambda_{a}^{2}\, r_{a} } \left[ \rm{e}^{-\frac{r_a}{\lambda^{*}_a }} \, \rm{sinhIntegral}\left(\frac{r_a}{\lambda^{*}_a }\right) %\right. \right. \nonumber \\ &&\left. \left. %\qquad \qquad \qquad - \rm{sinh}\left(\frac{r_a}{\lambda^{*}_a}\right) \, \rm{expIntegralEi}\left(-\frac{r_a}{\lambda^{*}_a } \right) \right] \right\} \, . \end{aligned}$$ Following, we plot in figure \[all-densities\] the density profiles $\rho_{DM}$, $\rho^{*} $ and $ \rho_{\bar \phi} $ for $\lambda=1$, $\alpha=1$. The main contribution to $\rho_{DM}$ comes from $\rho^{*} $, which is an inverse squared function of the radius. The DM profile is therefore cuspy near the galactic centre, similar to the NFW’s obtained from simulations [@Na96-97]. We have computed the best fits to these curves for $r\ll a$, given by $\rho_{DM} \sim \frac{0.08}{r^{1.99}}$ and $ \rho_{\bar \phi} \sim \frac{0.06}{r^{0.21}}$. On the other hand, the best fits for $r\gg a$ are $\rho_{DM} \sim \frac{0.16}{r^{2.05}}$ and $ \rho_{\bar \phi} \sim \frac{0.08}{r^{2.12}}$. In figure \[various-lambda\] we plot the DM profile for various $\lambda$ values, resulting in small changes in the slope. For negative $\alpha$ values, (\[phi-bar-sol\]) becomes negative, but the DM profile does not become shallower. We have computed the case $\lambda=1.0$, $\alpha=-0.5$ for $r\ll a$ and found $\rho_{DM} \sim \frac{0.08}{r^{2.00}}$, while for $r\gg a$ we obtained $\rho_{DM} \sim \frac{0.02}{r^{1.83}}$. Conclusions =========== We have constructed a spherically galactic halo model in which we fit the new gravitational potential to match a flat rotation profile for test particles –stars and dust. Once this is done, one computes the scalar field fluctuation and DM profiles resulting from this potential. We have found analytical results for our model, Eqs.(\[phi-bar-sol\]) and (\[rho-dm-sol\]). These equations contain the information of the scalar field ($\lambda$, $\alpha$), which determine the DM profile. We have plotted our results for $\alpha=1.0$ and various $\lambda$s. These profiles have an inner region as cuspy as the NFW’s. Negative values of $\alpha$ do not yield a shallower DM profile neither. This result may however not surprise since we are not resolving the inner structure of the rotation velocity curves. We only considered its flat asymptotic behaviour; in a forthcoming paper we will study it. This work was supported by CONACYT grant numbers 44917 and U47209-F. [99]{} Rodríguez–Meza M A, Klapp J, Cervantes–Cota J L, Dehnen H, 2001, in Macias A, Cervantes–Cota J L, Lämmerzahl C, eds, Exact solutions and scalar fields in gravity: Recent developments. Kluwer Academic/Plenum Publishers, New York, p. 213 Rodríguez–Meza M. A., Cervantes–Cota J. L., 2004, [*MNRAS*]{} [**350**]{} 671 Rodríguez–Meza M A et al 2005, [*Gen. Rel. Grav.*]{} [**37**]{} 823 Pimentel L O and Obregón O 1986 [*Astrophys. Space Sci.*]{} [**126**]{} 231-4 Fischbach E, Talmadge C L, 1999, [*The search for Non–Newtonian gravity.*]{} Springer–Verlag, New York Sofue Y and Rubin V 2001 [*Ann Rev Astron Astrophys*]{} [**39**]{} 137-174 Nagata R, Chiba T, Sugiyama N 2002 Phys. Rev. D [**66**]{} 103510; Umezu K, Ichiki K, Yahiro M 2005 Phys. Rev. D [**72**]{} 044010; Shirata A, Shiromizu T, Yoshida N, Suto Y 2005 Phys. Rev. D [**71**]{} 064030 Helbig T 1991 [*ApJ*]{} [**382**]{} 223 Navarro J. F. et al 1996 [*ApJ*]{} [**462**]{} 563; 1997 [*ApJ*]{} [**490**]{} 493

--- author: - | Jin U Kang$^1$,$^2$ and Grigoris Panotopoulos$^2$\ $^1$Department of Physics, Kim Il Sung University,\ Pyongyang, Democratic People’s Republic of Korea\ $^2$ASC, Department of Physics LMU,\ Theresienstr. 37, 80333 Munich, Germany\ [E-mail: Jin.U.Kang@physik.uni-muenchen.de, Grigoris.Panotopoulos@physik.uni-muenchen.de]{} title: 'Dark matter in supersymmetric models with axino LSP in Randall-Sundrum II brane model' --- Introduction ============ There are good theoretical reasons for which particle physics proposes that new exotic particles must exist. In particular, the strong CP problem and the hierarchy problem motivate symmetries and particles beyond the standard model of particle physics. On one hand, supersymmetry (SUSY) is an ingredient that appears in many theories for physics beyond the standard model. SUSY solves the hierarchy problem and predicts that every particle we know should be escorted by its superpartner. In order for the supersymmetric solution of the hierarchy problem to work, it is necessary that the SUSY becomes manifest at relatively low energies, less than a few $TeV$, and therefore the required superpartners must have masses below this scale (for supersymmetry and supergravity see e.g. [@nilles]). On the other hand, the strong CP problem can be solved naturally by implementing the Peccei-Quinn (PQ) mechanism [@quinn]. An additional global $U(1)$ symmetry referred to as PQ symmetry broken spontaneously at the PQ scale can explain the smallness of the CP-violating $\Theta$-vacuum term in quantum chromodynamics (QCD). The pseudo Nambu-Goldstone boson associated with this spontaneous symmetry breaking is the axion [@wilczek], which has not yet been detected. Axinos, the superpartners of axions, are special because they have unique properties: They are very weekly interacting and their mass can span a wide range, from very small ($ \sim eV$)to large ($ \sim GeV$) values. What is worth stressing is that, in contrast to the neutralino and the gravitino, axino mass does not have to be of the order of the SUSY breaking scale in the visible sector, $M_{SUSY} \sim 100GeV-1TeV$. One of the theoretical problems in modern cosmology is to understand the nature of cold dark matter in the universe. There are good reasons, both observational and theoretical, to suspect that a fraction of $0.22$ of the energy density in the universe is in some unknown “dark” form. Many lines of reasoning suggest that the dark matter consists of some new, as yet undiscovered, massive particle which experiences neither electromagnetic nor color interactions. In SUSY models which are realized with R-parity conservation the lightest supersymmetric particle (LSP) is stable. A popular cold dark matter candidate is the LSP, provided that it is electrically and color neutral. Certainly the most theoretically developed LSP is the lightest neutralino [@neutralino]. However, there are other dark matter candidates as well, for example the gravitino [@gravitino; @steffen1] and the axino [@axino; @steffen2], the superpartner of axion [@wilczek] which solves the QCD problem via the Peccei-Quinn mechanism [@quinn]. In this article we work in the framework of Randall-Sundrum type II brane model (RSII), we assume that the axino is the LSP and address the question whether the axino can play the role of dark matter in the universe, and for which range for axino mass and five-dimensional Planck mass. Our work is organized as follows: The article consists of four section, of which this introduction is the first. In the second section we present the theoretical framework, while in section 3 we show the results of our analysis. Finally we conclude in the last section. The theoretical framework ========================= The brane model --------------- Over the last years the brane-world models have been attracting a lot of attention as a novel higher-dimensional theory. Brane models are inspired from M/string theory and although they are not yet derivable from the fundamental theory, at least they contain the basic ingredients, like extra dimensions, higher-dimensional objects (branes), higher-curvature corrections to gravity (Gauss-Bonnet) etc. Since string theory claims to give us a fundamental description of nature it is important to study what kind of cosmology it predicts. Furthermore, despite the fact that supersymmetric dark matter has been analyzed in standard four-dimensional cosmology, it is challenging to discuss it in alternative gravitational theories as well. Neutralino dark matter in brane cosmology has been studied in [@okada1], while axino dark matter in brane-world cosmology has been studied in [@panotop]. In brane-world models it is assumed that the standard model particles are confined on a 3-brane while gravity resides in the whole higher dimensional spacetime. The model first proposed by Randall and Sundrum (RSII) [@rs], is a simple and interesting one, and its cosmological evolutions have been intensively investigated. An incomplete list can be seen e.g. in [@langlois]. In the present work we would like to study axino dark matter in the framework of RSII model. According to that model, our 4-dimensional universe is realized on the 3-brane with a positive tension located at the UV boundary of 5-dimensional AdS spacetime. In the bulk there is just a cosmological constant $\Lambda_{5}$, whereas on the brane there is matter with energy-momentum tensor $\tau_{\mu \nu}$. Also, the five dimensional Planck mass is denoted by $M_{5}$ and the brane tension is denoted by $T$. If Einstein’s equations hold in the five dimensional bulk, then it has been shown in [@shiromizu] that the effective four-dimensional Einstein’s equations induced on the brane can be written as $$G_{\mu \nu}+\Lambda_{4} g_{\mu \nu}=\frac{8 \pi}{m_{pl}^2} \tau_{\mu \nu}+(\frac{1}{M_{5}^3})^2 \pi_{\mu \nu}-E_{\mu \nu}$$ where $g_{\mu \nu}$ is the induced metric on the brane, $\pi_{\mu \nu}=\frac{1}{12} \: \tau \: \tau_{\mu \nu}+\frac{1}{8} \: g_{\mu \nu} \: \tau_{\alpha \beta} \: \tau^{\alpha \beta}-\frac{1}{4} \: \tau_{\mu \alpha} \: \tau_{\nu}^{\alpha}-\frac{1}{24} \: \tau^2 \: g_{\mu \nu}$, $\Lambda_{4}$ is the effective four-dimensional cosmological constant, $m_{pl}$ is the usual four-dimensional Planck mass and $E_{\mu \nu} \equiv C_{\beta \rho \sigma} ^\alpha \: n_{\alpha} \: n^{\rho} \: g_{\mu} ^{\beta} \: g_{\nu} ^{\sigma}$ is a projection of the five-dimensional Weyl tensor $C_{\alpha \beta \rho \sigma}$, where $n^{\alpha}$ is the unit vector normal to the brane. The tensors $\pi_{\mu \nu}$ and $E_{\mu \nu}$ describe the influence of the bulk in brane dynamics. The five-dimensional quantities are related to the corresponding four-dimensional ones through the relations $$m_{pl}=4 \: \sqrt{\frac{3 \pi}{T}} \: M_{5}^3$$ and $$\Lambda_{4}=\frac{1}{2 M_{5}^3} \left( \Lambda_{5}+\frac{T^2}{6 M_{5}^3} \right )$$ In a cosmological model in which the induced metric on the brane $g_{\mu \nu}$ has the form of a spatially flat Friedmann-Robertson-Walker model, with scale factor $a(t)$, the Friedmann-like equation on the brane has the generalized form [@langlois] $$H^2=\frac{\Lambda_{4}}{3}+\frac{8 \pi}{3 m_{pl}^2} \rho+\frac{1}{36 M_{5}^6} \rho^2+\frac{C}{a^4}$$ where $C$ is an integration constant arising from $E_{\mu \nu}$. The cosmological constant term and the term linear in $\rho$ are familiar from the four-dimensional conventional cosmology. The extra terms, i.e the “dark radiation” term and the term quadratic in $\rho$, are there because of the presence of the extra dimension. Adopting the Randall-Sundrum fine-tuning $$\Lambda_{5}=-\frac{T^2}{6 M_{5}^3}$$ the four-dimensional cosmological constant vanishes. In addition, the dark radiation term is severely constrained by the success of the Big Bang Nucleosynthesis (BBN), since the term behaves like an additional radiation at the BBN era [@orito]. So, for simplicity, we neglect the term in the following analysis. The five-dimensional Planck mass is also constrained by the BBN, which is roughly estimated as $M_{5} \geq 10 \: TeV$ [@cline]. The generalized Friedmann equation takes the final form $$H^2=\frac{8 \pi G}{3} \rho \left (1+\frac{\rho}{\rho_0} \right )$$ where $$\rho_0=96 \pi G M_{5}^6$$ with $G$ the Newton’s constant. One can see that the evolution of the early universe can be divided into two eras. In the low-energy regime $\rho \ll \rho_0$ the first term dominates and we recover the usual Friedmann equation of the conventional four-dimensional cosmology. In the high-energy regime $\rho_0 \ll \rho$ the second term dominates and we get an unconventional expansion law for the universe. In between there is a transition temperature $T_t$ for which $\rho(T_t)=\rho_0$. Once $M_{5}$ is given, the transition temperature $T_{t}$ is determined as $$T_{t}=1.6 \times 10^{7} \: \left ( \frac{100}{g_{eff}} \right )^{1/4} \: \left ( \frac{M_{5}}{10^{11} \: GeV} \right )^{3/2} \: GeV$$ where $g_{eff}$ counts the total number of relativistic degrees of freedom. The particle physics model -------------------------- The extension of standard model (SM) of particle physics based on SUSY is the minimal supersymmetric standard model (MSSM) [@mssm]. It is a supersymmetric gauge theory based on the SM gauge group with the usual representations (singlets, doublets, triplets) and on $\mathcal{N}=1$ SUSY. Excluding gravity, the massless representations of the SUSY algebra are a chiral and a vector supermultiplet. The gauge bosons and the gauginos are members of the vector supermultiplet, while the matter fields (quarks, leptons, Higgs) and their superpartners are members of the chiral supermultiplet. The Higgs sector in the MSSM is enhanced compared to the SM case. There are now two Higgs doublets, $H_u, H_d$, for anomaly cancelation requirement and for giving masses to both up and down quarks. After electroweak symmetry breaking we are left with five physical Higgs bosons, two charged $H^{\pm}$ and three neutral $A,H,h$ ($h$ being the lightest). Since we have not seen any superpartners yet SUSY has to be broken. In MSSM, SUSY is softly broken by adding to the Lagrangian terms of the form - Mass terms for the gauginos $\tilde{g}_i$, $M_1, M_2, M_3$ $$M \tilde{g} \tilde{g}$$ - Mass terms for sfermions $\tilde{f}$ $$m_{\tilde{f}}^2 \tilde{f}^{\dag} \tilde{f}$$ - Masses and bilinear terms for the Higgs bosons $H_u, H_d$ $$m_{H_u}^2 H_u^{\dag} H_u+m_{H_d}^2 H_d^{\dag} H_d+B \mu (H_u H_d + h.c.)$$ - Trilinear couplings between sfermions and Higgs bosons $$A Y \tilde{f}_1 H \tilde{f}_2$$ In the unconstrained MSSM there is a huge number of unknown parameters [@parameters] and thus little predictive power. However, the Constrained MSSM (CMSSM) or mSUGRA [@msugra] is a framework with a small controllable number of parameters, and thus with much more predictive power. In the CMSSM there are four parameters, $m_0, m_{1/2}, A_0, tan \beta$, which are explained below, plus the sign of the $\mu$ parameter from the Higgs sector. The magnitude of $\mu$ is determined by the requirement for a proper electroweak symmetry breaking, its sign however remains undetermined. We now give the explanation for the other four parameters of the CMSSM - Universal gaugino masses $$M_1(M_{GUT})=M_2(M_{GUT})=M_3(M_{GUT})=m_{1/2}$$ - Universal scalar masses $$m_{\tilde{f}_i}(M_{GUT})=m_0$$ - Universal trilinear couplings $$A_{i j}^u(M_{GUT}) = A_{i j}^d(M_{GUT}) = A_{i j}^l(M_{GUT}) = A_0 \delta_{i j}$$ - $$tan \beta \equiv \frac{v_1}{v_2}$$ where $v_1, v_2$ are the vevs of the Higgs doublets and $M_{GUT} \sim 10^{16}~GeV$ is the Grand Unification scale. Analysis and results ==================== We consider eight benchmark models (shown in Table 1 and Table 2) for natural values of $m_0, m_{1/2}$, representative values of $tan \beta$ and fixed $A_0=0, \mu >0$. In these models the lightest neutralino (denoted by $\chi$) or the lightest stau (denoted by $\tilde{\tau}$) is the lightest of the usual superpartners and thus the NLSP. Furthermore the following experimental constraints (for the lightest Higgs mass and a rare decay) [@precision; @Yao:2006px] are satisfied $$\begin{aligned} m_h & > & 114.4~GeV \\ BR(b \rightarrow s \gamma) & = & (3.39_{-0.27}^{+0.30}) \times 10^{-4}\end{aligned}$$ [|c|c|c|c|c|c|]{} Model & $m_0 \: (GeV)$ & $m_{1/2} \: (GeV)$ & $tan \beta$ & $m_{\chi} \: (GeV)$ & $\Omega_{\chi} h^2$\ A & 200 & 500 & 15 & 205.42 & 0.64\ B & 400 & 800 & 25 & 337.95 & 1.82\ C & 1000 & 600 & 30 & 252.41 & 7.37\ D & 350 & 450 & 20 & 184.46 & 1.2\ [|c|c|c|c|c|c|]{} Model & $m_0 \: (GeV)$ & $m_{1/2} \: (GeV)$ & $tan \beta$ & $m_{\tilde{\tau}} \: (GeV)$ & $\Omega_{\tilde{\tau}} h^2$\ E & 50 & 500 & 10 & 187.78 & 0.0088\ F & 60 & 600 & 11 & 223.47 & 0.012\ G & 70 & 700 & 12 & 258.95 & 0.017\ H & 100 & 800 & 15 & 295.93 & 0.022\ At this point we remark that any viable model should also satisfy two more mass limits [@Yao:2006px] $$\begin{aligned} m_{\tilde{\tau}_1} & > & 81.9~GeV \\ m_{\tilde{\chi}_1^{\pm}} & > & 94~GeV\end{aligned}$$ However, in the models we consider here the NLSP mass is at least $m_{NLSP} \simeq 184~GeV$ and therefore further imposing limits of ${\cal O}(100~GeV)$ on other sparticles is meaningless. The SUSY spectrum (as well as the Higgs bosons masses) and the neutralino relic density have been computed using the web site [@kraml], and the top quark mass is fixed to $m_t=172.7~GeV$ [@cdf]. Furthermore, following [@okada2] for the stau relic density we have made use of the simple formula $$\Omega_{\tilde{\tau}}h^2=\left( \frac{m_{\tilde{\tau}}}{2~TeV} \right )^2$$ Before proceeding any further a couple of remarks are in order. First, we mention that in principle the saxion (a scalar field in the same supermultiplet with axion and axino) could have important cosmological consequences. Here, however, we shall assume that the saxion mass is such that its cosmological consequences are negligible. This kind of assumption was also made in [@steffen2]. Furthermore, in two previous works [@steffen2; @panotop] the axino dark matter in standard and brane cosmology was considered, in which the axino thermal production only was taken into account. There it was found that the reheating temperature (in standard cosmology) or the transition temperature (in brane cosmology) had to be bounded from above, $T_{R,t} \leq 10^6~GeV$. However, at this temperature the strong coupling constant is of the order one, $g_s \sim 1$, a fact which may render the whole discussion invalid[^1]. That is why in the present work we have chosen to only consider the non-thermal production from the NLSP decay. If we restrict ourselves to small $M_5$ or $T_t$ we can neglect the thermal production mechanism as being negligible compared to the non-thermal production mechanism. Finally, in principle one should also impose the BBN constraints (see e.g. [@kohri]) if the NLSP decays after BBN time. However in the axino dark matter case the BBN constraints are easily avoided because the NSLP has a relatively short lifetime and decays well before BBN [@axino]a. For the axino abundance we take into account the non-thermal production (NTP) and we impose the WMAP constraint for cold dark matter [@wmap] $$0.075 < \Omega_{cdm} h^2=\Omega_{\tilde{\alpha}} h^2 < 0.126$$ In the NTP case the contribution to the axino abundance comes from the decay of the NLSP $$\Omega_{\tilde{\alpha}} h^2 = \frac{m_{\tilde{\alpha}}}{m_{NLSP}} \: \Omega_{NLSP} h^2$$ with $m_{\tilde{\alpha}}$ the axino mass, $m_{NLSP}$ the mass of the NLSP and $\Omega_{NLSP} h^2$ the NLSP abundance had it did not decay into the axino. Now we need to take into account the effect of the novel law for expansion of the universe. The relic density of a particle of mass $m$ is modified as follows [@okada3] $$\frac{\Omega^{(b)}}{\Omega^{(s)}}=0.54 \: \frac{x_t}{x_{d}^{(s)}}$$ in the limit $x_t \gg x_d$ and in the S-wave approximation, where the index b stands for “brane”, the index s stands for “standard”, $x_t=m/T_t$ and $x_d=m/T_d$, with $T_t$ the transition temperature and $T_d$ the decoupling temperature of the particle of mass $m$. In standard cosmology $x_d^{(s)} \simeq 30$. In a given particle physics model the axino abundance in terms of $M_5$ and $m_{\tilde{\alpha}}$ is given by $$\Omega_{\tilde{\alpha}} h^2 = 0.9 \times 10^{7.5} \: \left ( \frac{m_{\tilde{\alpha}}}{GeV} \right ) \: \Omega_{NLSP}^{(s)} h^2 \: \left ( \frac{M_5}{GeV} \right )^{-3/2}$$ For each benchmark model we have obtained plots (for example we show the plots for models A and E, for the rest of the models there are similar plots) which show the axino abundance $\Omega_{\tilde{\alpha}} h^2$ as a function of the axino mass $m_{\tilde{\alpha}}$ for several values of the five-dimensional Planck mass $M_5$. Figure $1$ corresponds to the neutralino NLSP case (the values that we have used are $10^4~GeV$, $10^5~GeV$, and $10^6~GeV$ from top to bottom), while figure $2$ corresponds to the stau NLSP case (same values of $M_5$). We see that there is always one allowed range for the axino mass from the milli-$GeV$ range to a few $GeV$. If however $M_5$ is high enough, in the stau NLSP case the axino has to be very heavy. In this case, using the formula above for the axino abundance and imposing the condition that $m_{\tilde{\alpha}} \leq 10~GeV$, it is easy to show that for $\Omega_{NLSP}^{(s)} h^2=0.01$ the fundamental Planck mass is bounded from above, $M_5 \leq 7.4 \times 10^4~GeV$. This can be shown in figure 3. Conclusions =========== We have studied axino dark matter in the brane-world cosmology. The theoretical framework for our work is the CMSSM for particle physics and RS II for gravity, which predicts a generalized Friedmann-like equation for the evolution of the universe. We assume that axino is the LSP and the lightest neutralino or the lightest stau is the NLSP. For the axino abundance we have taken into account the non-thermal production and have imposed the cold dark matter constraint $0.075 < \Omega_{cdm} h^2 < 0.126$. The formula valid in standard four-dimensional cosmology is corrected taking into account the novel expansion law for the universe. We have considered eight benchmark models (four for the neutralino NLSP and four for the stau NLSP case) for natural values of $m_0$ and $m_{1/2}$ and representative values of $tan \beta$. In these models the neutralino or the stau is the lightest of the usual superpartners (and thus the NLSP, since we assume that the axino is the LSP) and experimental constraints are satisfied. For each benchmark model we have produced plots of the axino abundance as a function of the axino mass for several different values of the five-dimensional Planck mass. The obtained plots show that in general the axino can be the cold dark matter in the universe for axino masses from $0.001~GeV$ up to a few $GeV$. Furthermore, in the stau NLSP case an upper bound on the five-dimensional Planck mass is obtained, $M_5 \leq 7.4 \times 10^{4}~GeV$. Acknowledgments {#acknowledgments .unnumbered} =============== J.U K is supported by the German Academic Exchange Service (DAAD), and G. P. is supported by project “Particle Cosmology”. [999]{} H.P.Nilles, “Supersymmetry, Supergravity And Particle Physics,” Phys. Rept.  [**110**]{} (1984) 1;\ H. E. Haber and G. L. Kane, “The Search For Supersymmetry: Probing Physics Beyond The Standard Model,” Phys. Rept.  [**117**]{} (1985) 75. R.D.Peccei and H.R.Quinn, “CP Conservation In The Presence Of Instantons,” Phys. Rev. Lett.  [**38**]{} (1977) 1440;\ R.D.Peccei and H.R.Quinn, “Constraints Imposed By CP Conservation In The Presence Of Instantons,” Phys. Rev. D [**16**]{} (1977) 1791. S.Weinberg, “A New Light Boson?,” Phys. Rev. Lett.  [**40**]{} (1978) 223;\ F.Wilczek, “Problem Of Strong P And T Invariance In The Presence Of Instantons,” Phys. Rev. Lett.  [**40**]{} (1978) 279. J. Edsjo and P. Gondolo, “Neutralino relic density including coannihilations,” Phys. Rev. D [**56**]{} (1997) 1879 \[arXiv:hep-ph/9704361\];\ J. Edsjo, M. Schelke, P. Ullio and P. Gondolo, “Accurate relic densities with neutralino, chargino and sfermion coannihilations in mSUGRA,” JCAP [**0304**]{} (2003) 001 \[arXiv:hep-ph/0301106\];\ H. Baer, A. Mustafayev, S. Profumo, A. Belyaev and X. Tata, “Neutralino cold dark matter in a one parameter extension of the minimal supergravity model,” Phys. Rev. D [**71**]{} (2005) 095008 \[arXiv:hep-ph/0412059\];\ A. Djouadi, M. Drees and J. L. Kneur, “Neutralino dark matter in mSUGRA: Reopening the light Higgs pole window,” Phys. Lett. B [**624**]{} (2005) 60 \[arXiv:hep-ph/0504090\];\ J. F. Gunion, D. Hooper and B. McElrath, “Light neutralino dark matter in the NMSSM,” Phys. Rev. D [**73**]{} (2006) 015011 \[arXiv:hep-ph/0509024\];\ G. B. Gelmini and P. Gondolo, “Neutralino with the right cold dark matter abundance in (almost) any supersymmetric model,” Phys. Rev. D [**74**]{} (2006) 023510 \[arXiv:hep-ph/0602230\]. J. R. Ellis, K. A. Olive, Y. Santoso and V. C. Spanos, “Gravitino dark matter in the CMSSM,” Phys. Lett. B [**588**]{} (2004) 7 \[arXiv:hep-ph/0312262\];\ L. Roszkowski, R. Ruiz de Austri and K. Y. Choi, “Gravitino dark matter in the CMSSM and implications for leptogenesis and the LHC,” JHEP [**0508**]{} (2005) 080 \[arXiv:hep-ph/0408227\];\ D. G. Cerdeno, K. Y. Choi, K. Jedamzik, L. Roszkowski and R. Ruiz de Austri, “Gravitino dark matter in the CMSSM with improved constraints from BBN,” JCAP [**0606**]{} (2006) 005 \[arXiv:hep-ph/0509275\];\ F. D. Steffen, “Gravitino dark matter and cosmological constraints,” JCAP [**0609**]{} (2006) 001 \[arXiv:hep-ph/0605306\]. J. Pradler and F. D. Steffen, “Constraints on the reheating temperature in gravitino dark matter scenarios,” arXiv:hep-ph/0612291. L. Covi, J. E. Kim and L. Roszkowski, “Axinos as cold dark matter,” Phys. Rev. Lett.  [**82**]{} (1999) 4180 \[arXiv:hep-ph/9905212\];\ L. Covi, H. B. Kim, J. E. Kim and L. Roszkowski, “Axinos as dark matter,” JHEP [**0105**]{} (2001) 033 \[arXiv:hep-ph/0101009\];\ L. Covi, L. Roszkowski, R. Ruiz de Austri and M. Small, ‘Axino dark matter and the CMSSM,’’ JHEP [**0406**]{} (2004) 003 \[arXiv:hep-ph/0402240\]. A. Brandenburg and F. D. Steffen, “Axino dark matter from thermal production,” JCAP [**0408**]{} (2004) 008 \[arXiv:hep-ph/0405158\]. T. Nihei, N. Okada and O. Seto, “Neutralino dark matter in brane world cosmology,” Phys. Rev. D [**71**]{} (2005) 063535 \[arXiv:hep-ph/0409219\]. G. Panotopoulos, “Axino dark matter in brane world cosmology,” JCAP [**0508**]{} (2005) 005 \[arXiv:hep-ph/0506093\]. L.Randall and R.Sundrum, “An alternative to compactification,” Phys. Rev. Lett.  [**83**]{} (1999) 4690 \[arXiv:hep-th/9906064\]. P.Binetruy, C.Deffayet and D.Langlois, “Non-conventional cosmology from a brane-universe,” Nucl. Phys. B [**565**]{} (2000) 269 \[arXiv:hep-th/9905012\];\ P.Binetruy, C.Deffayet, U.Ellwanger and D.Langlois, “Brane cosmological evolution in a bulk with cosmological constant,” Phys. Lett. B [**477**]{} (2000) 285 \[arXiv:hep-th/9910219\];\ C. Deffayet, G. R. Dvali and G. Gabadadze, “Accelerated universe from gravity leaking to extra dimensions,” Phys. Rev. D [**65**]{} (2002) 044023 \[arXiv:astro-ph/0105068\];\ E.Kiritsis, N.Tetradis and T.N.Tomaras, “Induced gravity on RS branes,” JHEP [**0203**]{} (2002) 019 \[arXiv:hep-th/0202037\];\ E.Kiritsis, G.Kofinas, N.Tetradis, T.N.Tomaras and V.Zarikas, “Cosmological evolution with brane-bulk energy exchange,” JHEP [**0302**]{} (2003) 035 \[arXiv:hep-th/0207060\];\ R. A. Brown, R. Maartens, E. Papantonopoulos and V. Zamarias, “A late-accelerating universe with no dark energy - and no big bang,” JCAP [**0511**]{} (2005) 008 \[arXiv:gr-qc/0508116\];\ G. Kofinas, G. Panotopoulos and T. N. Tomaras, “Brane-bulk energy exchange: A model with the present universe as a global attractor,” JHEP [**0601**]{} (2006) 107 \[arXiv:hep-th/0510207\];\ G. Panotopoulos, “Sneutrino inflation in Gauss-Bonnet brane-world cosmology, the gravitino problem and leptogenesis,” Nucl. Phys. B [**745**]{} (2006) 49 \[arXiv:hep-ph/0511040\]. T.Shiromizu, K.i.Maeda and M.Sasaki, “The Einstein equations on the 3-brane world,” Phys. Rev. D [**62**]{} (2000) 024012 \[arXiv:gr-qc/9910076\];\ A. N. Aliev and A. E. Gumrukcuoglu, “Gravitational field equations on and off a 3-brane world,” Class. Quant. Grav.  [**21**]{} (2004) 5081 \[arXiv:hep-th/0407095\]. K.Ichiki, M.Yahiro, T.Kajino, M.Orito and G.J.Mathews, “Observational constraints on dark radiation in brane cosmology,” Phys. Rev. D [**66**]{} (2002) 043521 \[arXiv:astro-ph/0203272\]. J.M.Cline, C.Grojean and G.Servant, “Cosmological expansion in the presence of extra dimensions,” Phys. Rev. Lett.  [**83**]{} (1999) 4245 \[arXiv:hep-ph/9906523\]. P. Fayet, “Spontaneously Broken Supersymmetric Theories Of Weak, Electromagnetic And Strong Interactions,” Phys. Lett. B [**69**]{}, 489 (1977);\ S. Dimopoulos and H. Georgi, “Softly Broken Supersymmetry And SU(5),” Nucl. Phys. B [**193**]{} (1981) 150. S. Dimopoulos and D. W. Sutter, “The Supersymmetric Flavor Problem,” Nucl. Phys. B [**452**]{} (1995) 496 \[arXiv:hep-ph/9504415\]. A. H. Chamseddine, R. Arnowitt and P. Nath, “Locally Supersymmetric Grand Unification,” Phys. Rev. Lett.  [**49**]{} (1982) 970;\ R. Barbieri, S. Ferrara and C. A. Savoy, “Gauge Models With Spontaneously Broken Local Supersymmetry,” Phys. Lett. B [**119**]{} (1982) 343;\ L. J. Hall, J. D. Lykken and S. Weinberg, “Supergravity As The Messenger Of Supersymmetry Breaking,” Phys. Rev. D [**27**]{} (1983) 2359. S. Heinemeyer, “Electroweak precision data and gravitino dark matter,” arXiv:hep-ph/0611372. W. M. Yao [*et al.*]{} \[Particle Data Group\], “Review of particle physics,” J. Phys. G [**33**]{} (2006) 1. http://cern.ch/kraml/comparison;\ B. C. Allanach, S. Kraml and W. Porod, “Theoretical uncertainties in sparticle mass predictions from computational tools,” JHEP [**0303**]{} (2003) 016 \[arXiv:hep-ph/0302102\];\ G. Belanger, S. Kraml and A. Pukhov, “Comparison of SUSY spectrum calculations and impact on the relic density constraints from WMAP,” Phys. Rev.  D [**72**]{} (2005) 015003 \[arXiv:hep-ph/0502079\]. \[CDF Collaboration\], “Combination of CDF and D0 results on the top-quark mass,” arXiv:hep-ex/0507091. T. Asaka, K. Hamaguchi and K. Suzuki, “Cosmological gravitino problem in gauge mediated supersymmetry breaking models,” Phys. Lett.  B [**490**]{} (2000) 136 \[arXiv:hep-ph/0005136\];\ N. Okada and O. Seto, “Gravitino dark matter from increased thermal relic particles,” arXiv:0710.0449 \[hep-ph\]. R. H. Cyburt, J. R. Ellis, B. D. Fields and K. A. Olive, “Updated nucleosynthesis constraints on unstable relic particles,” Phys. Rev.  D [**67**]{} (2003) 103521 \[arXiv:astro-ph/0211258\];\ M. Kawasaki, K. Kohri and T. Moroi, “Hadronic decay of late-decaying particles and big-bang nucleosynthesis,” Phys. Lett.  B [**625**]{} (2005) 7 \[arXiv:astro-ph/0402490\];\ M. Kawasaki, K. Kohri and T. Moroi, “Big-bang nucleosynthesis and hadronic decay of long-lived massive particles,” Phys. Rev.  D [**71**]{} (2005) 083502 \[arXiv:astro-ph/0408426\]. D. N. Spergel [*et al.*]{}, “Wilkinson Microwave Anisotropy Probe (WMAP) three year results: Implications for cosmology,” arXiv:astro-ph/0603449. N. Okada and O. Seto, “Relic density of dark matter in brane world cosmology,” Phys. Rev. D [**70**]{} (2004) 083531 \[arXiv:hep-ph/0407092\]. [^1]: We would like to thank F. D. Steffen for pointing this out.

--- abstract: | Using the geometrical thermodynamic approach, we study phase transition of Brans–Dicke Born–Infeld black holes. We apply introduced methods and describe their shortcomings. We also use the recently proposed new method and compare its results with those of canonical ensemble. By considering the new method, we find that its Ricci scalar diverges in the places of phase transition and bound points. We also show that the bound point can be distinguished from the phase transition points through the sign of thermodynamical Ricci scalar around its divergencies. *Keywords: black hole solutions; phase transition; geometrical thermodynamics; Brans-Dicke theory; Born Infeld theory.* author: - 'S. H. Hendi$^{1,2}$[^1], M. S. Talezadeh$^{1}$ and Z. Armanfard$^{1}$' title: | Phase transition of black holes in Brans–Dicke Born–Infeld gravity\ through geometrical thermodynamics --- Introduction ============ General relativity is accepted as a standard theory of gravitation and is able to pass more observational tests [@GR]. Although, this theory is successful in various domains, it cannot describe some experimental evidences such as the accelerating expansion of the Universe [@Expansion1; @Expansion2; @Expansion3]. Moreover, the general relativity theory is not consistent with Mach’s principle nor Dirac’s large number hypothesis . In addition, one needs further accurate observations to fully confirm (or disprove) the validity of general relativity in the high curvature regime such as black hole systems and other massive objects. Therefore, in recent years, more attentions have been focused on alternative theories of gravity. The most considerable alternative theories of gravity is the scalar-tensor theories. One of the good examples of these theories is Brans–Dicke (BD) theory which was introduced in $1961$ to combine the Mach’s principle with the Einstein’s theory of gravity [@BD]. It is worthwhile to mention that BD theory is one of the modified theories of general relativity which can be used for several cosmological problems like inflation, cosmic acceleration and dark energy modeling [@BD-Example1; @BD-Example2; @BD-Example3]. Also, it has a customizable parameter ($\omega$) which indicates the strength of coupling between the matter and scalar fields. The action of $4-$dimensional BD theory can be written as $$S=\frac{1}{16\pi }\int d^{4}x\sqrt{-g}\left(\Phi R-\frac{\omega }{\Phi} (\nabla \Phi)^{2}\right),$$ where $R$ and $\Phi$ are, respectively, the Ricci scalar and self gravitating scalar field. It is interesting to note that $4-$dimensional stationary vacuum BD solution is just the Kerr solution with a trivial scalar field [@Gao4]. In addition, Cai and Myung proved that $4-$dimensional solution of BD-Maxwell theory reduces to the Reissner–Nordström solution with a constant scalar field [@sh5a; @sh5b; @sh55a; @sh55b]. However, the solutions of BD-Maxwell gravity in higher dimensions will be reduced to the Reissner–Nordström solutions with a non-trivial scalar field because of the fact that higher dimensional stress energy tensor of Maxwell field is not traceless (conformally invariant). One of the most prominent problems which makes BD theory non-straightforward is the fact that the field equations of this theory are highly nonlinear. To deal with this issue, one could apply conformal transformation on known solutions of other modified theories like dilaton gravity [@Dilaton]. For instance, nonlinearly charged dilatonic black hole solutions and their BD counterpart in an energy dependent spacetime have been obtained by applying a conformal transformation [BDvsDilaton]{}. The first attempt for modifying the Maxwell theory to a consistent theory for describing point charges was made in $1912$ by Gustav Mie [@Mie1; @Mie2]. After that, Born and Infeld introduced a gauge-invariant nonlinear electrodynamic model to find a classical theory of point-like charges with finite energy density [@BI]. Born-Infeld (BI) theory was more interesting since it was obtained by using loop correction analysis of Quantum Field theory. Recently, Tseytlin has shown that BI theory can be derived as an effective theory of some string theory models [@Low-energy1; @Low-energy2; @Low-energy3; @Low-energy4; @Low-energy5; @Low-energy6]. Nowadays, the effects of BI electrodynamics coupled to various gravity theories have been considered by many authors in the context of black holes [@Blackhole1; @Blackhole2; @Blackhole3; @Blackhole4; @Blackhole5; @Blackhole6; @Blackhole7; @Blackhole8; @Blackhole9; @Blackhole10; @Blackhole11; @Blackhole12; @Blackhole13; @Blackhole14; @Blackhole15; @Blackhole16; @Blackhole17; @Blackhole18; @Blackhole19; @Blackhole20; @Blackhole21; @Blackhole22; @Blackhole23; @Blackhole24; @Blackhole25], rotating black branes [@Rotating1; @Rotating2; @Rotating3; @Rotating4; @Rotating5; @Rotating6; @Rotating7], wormholes [@Wormhole1; @Wormhole2; @Wormhole3; @Wormhole4], superconductors [@Super1; @Super2; @Super3; @Super4; @Super5; @Super6] and other aspects of physics [@BIpmi1; @BIpmi2]. On the other side, black hole thermodynamics became an interesting topic after the works of Hawking and Beckenstein [@HP1; @HP2; @HP3; @HP4; @HP5; @HP6]. Besides, based on the AdS/CFT correspondence, black hole thermodynamics was considered as the first step for constructing quantum gravity. In recent years, phase transition and critical behavior of the black holes have attracted more attentions among physicists. Generally, at the critical point where phase transition occurs, one may find a discontinuity of state space variable such as heat capacity [@HC]. In addition to heat capacity, there are various approaches for studying phase transition. One of such interesting methods is based on geometrical technique. Geometrical thermodynamic method was started by Gibbs and Caratheodory [@Callen]. Regarding this method, one could build a phase space by employing thermodynamical potential and its corresponding extensive parameter. Meanwhile, divergence points of Ricci scalar of thermodynamical metric provide information about phase transition points of thermodynamical systems. First time, Weinhold introduced a new metric on the equilibrium thermodynamical phase space [@Weinhold1; @Weinhold2] and after that another thermodynamical metric was defined by Ruppeiner from a different point of view [@Ruppeiner1; @Ruppeiner2]. It is worthwhile to mention that, there is a conformally relation between Ruppeiner and Weinhold metrics with the inverse of temperature as a conformal factor [@Salamon]. None of Weinhold and Ruppeiner metrics were invariant under Legendre transformation. Recently, Quevedo [@Quevedo1; @Quevedo2] removed some problems of Weinhold and Ruppeiner methods by proposing a Legendre invariant thermodynamical metric. Although Quevedo could solve some problems which previous metrics were involved with, it has been confronted with another problems in some specific systems. To solve these problems, a new method was proposed in Ref. [@HPEM1; @HPEM2; @HPEM3] which is known as HPEM metric. It was shown that HPEM metric is completely consistent with the results of the heat capacity in canonical ensemble in different gravitational systems. In this paper, we are going to consider black hole solutions of BD-BI as well as Einstein-BI-dilaton gravity and study their phase transition based on geometrical thermodynamic methods. We compare our results with those of other methods such as extended phase space thermodynamics. FIELD EQUATION AND CONFORMAL TRANSFORMATIONS \[FE\] =================================================== The $(n+1)-$dimensional BD-BI theory action containing a scalar field $\Phi $ and a self-interacting potential $V(\Phi )$ is as follows$$I_{BD-BI}=-\frac{1}{16\pi }\int_{M}d^{n+1}x\sqrt{-g}\left( \Phi R-\frac{% \omega }{\Phi }\left( \nabla \Phi \right) ^{2}-V\left( \Phi \right) +% \mathcal{L}(\mathcal{F})\right), \label{action}$$ where $\omega $ is a coupling constant and $\mathcal{L}(\mathcal{F})$ is the BI theory Lagrangian $$\mathcal{L}(\mathcal{F})=4\beta ^{2}\left( 1-\sqrt{1+\frac{\mathcal{F}}{% 2\beta ^{2}}}\right), \label{BI-Lagrangian}$$ in which $\beta$ and $\mathcal{F}=F_{\mu \nu}F^{\mu \nu}$ are BI parameter and Maxwell invariant, respectively. It is worth mentioning that $\mathcal{L}% (\mathcal{F})$ will be reduced to the standard Maxwell form $\mathcal{L}(% \mathcal{F})=-\mathcal{F}$ as $\beta \rightarrow \infty $. The field equations of gravitational, scalar and electromagnetic fields can be obtained by varying the action (\[action\]) $$\begin{aligned} G_{\mu \nu } &=&\frac{\omega }{\Phi ^{2}}\left( \nabla _{\mu }\Phi \nabla _{\nu }\Phi -\frac{1}{2}g_{\mu \nu }(\nabla \Phi )^{2}\right) -\frac{V(\Phi )% }{2\Phi }g_{\mu \nu }+\frac{1}{\Phi }\left( \nabla _{\mu }\nabla _{\nu }\Phi -g_{\mu \nu }\nabla ^{2}\Phi \right) \nonumber \\ &&+\frac{2}{\Phi }\left( \frac{F_{\mu \lambda }F_{\nu }^{\text{ }\lambda }}{% \sqrt{1+\frac{\mathcal{F}}{2\beta ^{2}}}}+\frac{1}{4}g_{\mu \nu }\mathcal{L}(% \mathcal{F})\right), \label{FBD1} \\ \nabla ^{2}\Phi &=&\frac{1}{2\left[ \left( n-1\right) \omega +n\right] }% \left( (n-1)\Phi \frac{dV(\Phi )}{d\Phi }-\left( n+1\right) V(\Phi )+\left( n+1\right) \mathcal{L}(\mathcal{F})+\frac{4\mathcal{F}}{\sqrt{1+\frac{% \mathcal{F}}{2\beta ^{2}}}}\right), \label{FBD2}\end{aligned}$$ $$\nabla _{\mu }\left( \frac{F^{\mu \nu }}{\sqrt{1+\frac{\mathcal{F}}{2\beta ^{2}}}}\right) =0. \label{FBD3}$$ It is not easy to solve Eqs. (\[FBD1\])-(\[FBD3\]) because there exist second order of scalar field in the denominator of field equation (\[FBD1\]). In order to overcome such a problem, we can use a suitable conformal transformation and convert the BD-BI theory to the Einstein-BI-dilaton gravity. The suitable conformal transformation is as follows $$\bar{g}_{\mu \nu }=\Phi ^{2/(n-1)}g_{\mu \nu }, \label{CT}$$ $$\begin{aligned} \bar{\Phi} &=&\frac{n-3}{4\alpha }\ln \Phi , \label{Phibar} \\ \alpha &=&(n-3)/\sqrt{4(n-1)\omega +4n}. \label{alpha}\end{aligned}$$ The Einstein-BI-dilaton gravity action and its related field equations can be obtained from the BD-BI action and its related field equations by applying the mentioned conformal transformation [@BDvsDilaton] $$\overline{I}_{G}=-\frac{1}{16\pi }\int_{\mathcal{M}}d^{n+1}x\sqrt{-\overline{% g}}\left\{ \overline{\mathcal{R}}-\frac{4}{n-1}(\overline{\nabla }\overline{% \Phi })^{2}-\overline{V}(\overline{\Phi })+\overline{L}\left( \overline{% \mathcal{F}},\overline{\Phi }\right) \right\} , \label{con-ac}$$$$\begin{aligned} \overline{\mathcal{R}}_{\mu \nu } &=&\frac{4}{n-1}\left( \overline{\nabla }% _{\mu }\overline{\Phi }\overline{\nabla }_{\nu }\overline{\Phi }+\frac{1}{4}% \overline{V}(\overline{\Phi })\overline{g}_{\mu \nu }\right) -\frac{1}{n-1}% \overline{L}(\overline{\mathcal{F}},\overline{\Phi })\overline{g}_{\mu \nu }+% \frac{2e^{-\frac{4\alpha \overline{\Phi }}{n-1}}}{\sqrt{1+\overline{Y}}}% \left( \overline{F}_{\mu \eta }\overline{F}_{\nu }^{\eta }-\frac{\overline{% \mathcal{F}}}{n-1}\overline{g}_{\mu \nu }\right) , \label{FE1} \\ \overline{\nabla }^{2}\overline{\Phi } &=&\frac{n-1}{8}\frac{\partial \overline{V}(\overline{\Phi })}{\partial \overline{\Phi }}+\frac{\alpha }{% 2(n-3)}\left( (n+1)\overline{L}(\overline{\mathcal{F}},\overline{\Phi })+% \frac{4e^{-\frac{4\alpha \overline{\Phi }}{n-1}}\overline{\mathcal{F}}}{% \sqrt{1+\overline{Y}}}\right) , \label{FE2}\end{aligned}$$$$\overline{\nabla }_{\mu }\left( \frac{e^{-\frac{4\alpha \overline{\Phi }}{n-1% }}}{\sqrt{1+\overline{Y}}}\overline{F}^{\mu \nu }\right) =0, \label{FE3}$$where $\overline{\nabla }$ is the covariant differentiation with respect to the metric $\overline{g}_{\mu \nu }$and $\overline{\mathcal{R}}$ is its Ricci scalar. The potential $\overline{V}\left( \overline{\Phi }\right) $ and the Lagrangian $\overline{L}\left( \overline{F},\overline{\Phi }\right) $ will take the following forms [@BDvsDilaton] $$\overline{V}(\overline{\Phi })=\Phi ^{-(n+1)/(n-1)}V(\Phi ), \label{poten}$$$$\overline{L}\left( \overline{\mathcal{F}},\overline{\Phi }\right) =4\beta ^{2}e^{-4\alpha \left( n+1\right) \overline{\Phi }/\left[ \left( n-1\right) \left( n-3\right) \right] }\left( 1-\sqrt{1+\frac{e^{16\alpha \overline{\Phi }/\left[ \left( n-1\right) \left( n-3\right) \right] }\overline{\mathcal{F}}% }{2\beta ^{2}}}\right) . \label{LFP}$$ In the limits of $\beta \rightarrow \infty $ and $\beta \rightarrow 0$, the Lagrangian will be $\overline{L}\left( \overline{\mathcal{F}},\overline{\Phi }\right) =-e^{-4\alpha \overline{\Phi }/\left( n-1\right) }\overline{% \mathcal{F}}\ $and$\ \overline{L}\left( \overline{\mathcal{F}},\overline{% \Phi }\right) \rightarrow 0$, respectively, as expected. In previous equations, we have used the following notations $$\overline{L}\left( \overline{\mathcal{F}},\overline{\Phi }\right) =4\beta ^{2}e^{-4\alpha \left( n+1\right) \overline{\Phi }/\left[ \left( n-1\right) \left( n-3\right) \right] }\overline{L}\overline{(Y}), \label{LFP2}$$$$\begin{aligned} \overline{L}\left( \overline{Y}\right) &=&1-\sqrt{1+\overline{Y}}, \label{L(Y)} \\ \overline{Y} &=&\frac{e^{16\alpha \overline{\Phi }/\left[ \left( n-1\right) \left( n-3\right) \right] }\overline{\mathcal{F}}}{2\beta ^{2}}. \label{Y}\end{aligned}$$ By considering the conformal relation between these two theories it can be understood that if  $\left( \overline{g}_{\mu \nu },\overline{F}_{\mu \nu },% \overline{\Phi }\right) $  are the solutions to the field equations of Einstein-BI-dilaton gravity (\[FE1\])-(\[FE3\]), then the solutions of BD-BI theory could be obtained by the following form $$\left[ g_{\mu \nu },F_{\mu \nu },\Phi \right] =\left[ \exp \left( -\frac{% 8\alpha \overline{\Phi }}{\left( n-1\right) (n-3)}\right) \overline{g}_{\mu \nu },\overline{F}_{\mu \nu },\exp \left( \frac{4\alpha \overline{\Phi }}{n-3% }\right) \right] . \label{BDsol}$$ Black hole solutions in Einstein-BI-dilaton gravity and BD-BI theory \[Sol\] ---------------------------------------------------------------------------- ### **Einstein frame:** In this section, we briefly obtain the Einstein-BI-dilaton gravity solutions and then by using the conformal transformation, we calculate the solutions of BD-BI theory [@HMAT]. We assume the following metric with various horizon topology $$d\overline{s}^{2}=-Z(r)dt^{2}+\frac{dr^{2}}{Z(r)}+r^{2}R^{2}(r)d\Omega _{k}^{2}, \label{metric}$$ where $d\Omega_{k}^{2}$ is an $(n-1)$-dimensional hypersurface of Euclidean metric with constant curvature $(n-1)(n-2)k$ and volume $\varpi_{n-1}$ with the following explicit form $$d\Omega _{k}^{2}=\left\{ \begin{array}{cc} d\theta _{1}^{2}+\sum\limits_{i=2}^{n-1}\prod\limits_{j=1}^{i-1}\sin ^{2}\theta _{j}d\theta _{i}^{2} & k=1 \\ d\theta _{1}^{2}+\sinh ^{2}\theta _{1}d\theta _{2}^{2}+\sinh ^{2}\theta _{1}\sum\limits_{i=3}^{n-1}\prod\limits_{j=2}^{i-1}\sin ^{2}\theta _{j}d\theta _{i}^{2} & k=-1 \\ \sum\limits_{i=1}^{n-1}d\phi _{i}^{2} & k=0% \end{array} \right. . \label{k}$$ In order to obtain consistent solutions, we should consider a suitable functional form for the potential, $\mathbf{\overline{V}}(\overline{\Phi})$. It was shown that the proper potential is a Liouville-type one with both topological and BI correction terms, as [@BDvsDilaton] $$\mathbf{\overline{V}}(\overline{\Phi })=2\Lambda \exp \left( \frac{4\alpha \overline{\Phi }}{n-1}\right) +\frac{k(n-1)(n-2)\alpha ^{2}}{b^{2}\left( \alpha ^{2}-1\right) }\exp \left( \frac{4\overline{\Phi }}{(n-1)\alpha }% \right) +\frac{W(r)}{\beta ^{2}}. \label{liovilpoten}$$ It is notable to mention that in the limit of $\alpha \rightarrow 0$ (absence of dilaton field) and $\beta \rightarrow \infty $, $\mathbf{% \overline{V}}(\overline{\Phi })$ reduces to $2\Lambda$, as expected [pakravan]{}. Now, regarding the field equations (\[FE1\])-(\[FE3\]), metric (\[metric\]) and the potential $\mathbf{\overline{V}}(\overline{\Phi% })$, it is a matter of calculation to show that $$\begin{aligned} F_{tr} &=&E(r)=\frac{qe^{\left( \frac{4\alpha \overline{\Phi }(r)}{n-1}% \right) }}{[r R(r)]^{(n-1)}\sqrt{1+\frac{e^{(\frac{8\alpha \overline{\Phi }% (r)}{n-3})}q^{2}[r R(r)]^{-2(n-1)}}{\beta ^{2}}}}, \label{E} \\ \overline{\Phi } &=&\frac{(n-1)\alpha }{2(1+\alpha ^{2})}\ln \left( \frac{b}{% r}\right), \label{phi}\end{aligned}$$$$W(r)=\frac{4q(n-1)\beta ^{2}R(r)}{\left( 1+\alpha ^{2}\right) r^{\gamma }b^{n\gamma }}\int \frac{E(r)}{r^{n(1-\gamma )-\gamma }}dr+\frac{4\beta ^{4}% }{R(r)^{\frac{2(n+1)}{n-3}}}\left( 1-\frac{E(r)R(r)^{(n-3)}}{qr^{1-n}}% \right) -\frac{4q\beta ^{2}E(r)}{r^{n-1}}(\frac{r}{b})^{\gamma (n-1)}, \label{W}$$$$\begin{aligned} Z(r) &=&-\frac{k\left( n-2\right) \left( \alpha ^{2}+1\right) ^{2}\left( \frac{r}{b}\right)^{2\gamma }}{\left( \alpha ^{2}+n-2\right) \left( \alpha ^{2}-1\right) }+\left( \frac{(1+\alpha ^{2})^{2}r^{2}}{(n-1)}\right) \frac{% 2\Lambda \left( \frac{r}{b}\right) ^{-2\gamma }}{(\alpha ^{2}-n)}-\frac{m}{% r^{(n-1)(1-\gamma )-1}} \nonumber \\ &&-\frac{4(1+\alpha ^{2})^{2}q^{2}(\frac{r}{b})^{2\gamma (n-2)}}{(n-\alpha ^{2})r^{2(n-2)}}\left( \frac{1}{2(n-1)}\digamma _{1}(\eta )-\frac{1}{\alpha ^{2}+n-2}\digamma _{2}(\eta )\right), \label{f}\end{aligned}$$ $$R(r)= \exp \left( \frac{2\alpha \overline{\Phi }}{n-1}\right) =\left( \frac{b% }{r}\right)^{\gamma }, \label{R(r)}$$ where $m$ is an integration constant related to mass and $b$ is another constant related to scalar field, and $$\begin{aligned} \digamma _{1}(\eta ) &=&\text{ }_{2}F_{1}\left( \left[ \frac{1}{2},\frac{% (n-3)\Upsilon }{\alpha ^{2}+n-2}\right] ,\left[ 1+\frac{(n-3)\Upsilon }{% \alpha ^{2}+n-2}\right] ,-\eta \right) , \nonumber \\ \digamma _{2}(\eta ) &=&\text{ }_{2}F_{1}\left( \left[ \frac{1}{2},\frac{% (n-3)\Upsilon }{2(n-1)}\right] ,\left[ 1+\frac{(n-3)\Upsilon }{2(n-1)}\right] ,-\eta \right) , \nonumber \\ \Upsilon &=&\frac{\alpha ^{2}+n-2}{2\alpha ^{2}+n-3}, \nonumber \\ \eta &=&\frac{q^{2}(\frac{r}{b})^{2\gamma (n-1)(n-5)/(n-3)}}{\beta ^{2}r^{2(n-1)}}, \nonumber \\ \gamma&=& \frac{\alpha^2}{1+\alpha^2}, \nonumber\end{aligned}$$ It is worthwhile to mention that, the dilatonic Maxwell solutions [Sheikhi]{} can be achieved from the obtained solutions in the limit of $\beta \rightarrow \infty $. The divergencies of scalar curvatures at the origin guarantee the existence of singularity. We interpret such a singularity as black hole since it is covered by an event horizon [@BDvsDilaton]. ### **Jordan frame:** To obtain the black hole solutions of BD-BI theory, first, by using the conformal transformation (\[poten\]) the $\mathbf{V}(\Phi )$ would be $$\mathbf{V}(\Phi )=2\Lambda \Phi ^{2}+\frac{k(n-1)(n-2)\alpha ^{2}}{% b^{2}\left( \alpha ^{2}-1\right) }\Phi ^{\lbrack (n+1)(1+\alpha ^{2})-4]/[(n-1)\alpha ^{2}]}+\Phi ^{(n+1)/(n-1)}\frac{W(r)}{\beta ^{2}}. \label{V(phi)}$$ Also, by considering the following $(n+1)$-dimensional metric $$ds^{2}=-A(r)dt^{2}+\frac{dr^{2}}{B(r)}+r^{2}H^{2}(r)d\Omega _{k}^{2}, \label{metric1}$$ one can find the following solutions through conformal transformation $$\begin{aligned} A(r) &=&\left( \frac{r}{b}\right) ^{4\gamma /\left( n-3\right) }Z\left( r\right) , \label{A(r)} \\ B(r) &=&\left( \frac{r}{b}\right) ^{-4\gamma /\left( n-3\right) }Z\left( r\right) , \label{B(r)} \\ H(r) &=&\left( \frac{r}{b}\right) ^{-\gamma (\frac{n-5}{n-3})}, \label{H(r)} \\ \Phi \left( r\right) &=&\left( \frac{r}{b}\right) ^{-\frac{2\gamma \left( n-1\right) }{n-3}}. \label{Phi}\end{aligned}$$ It is notable that, like Einstein frame, these solutions can be interpreted as black holes which are covered by event horizon. Thermodynamic properties: Dilatonic-BI vs BD-BI black holes \[p-vb\] ==================================================================== Thermodynamic quantities: ------------------------- In the following, we give a brief review regarding thermodynamic quantities of the black hole solutions in both frames. The Hawking temperature of the black hole can be obtained by using the surface gravity interpretation ($% \kappa$) through the following relation $$T=\frac{\kappa }{2\pi }=\frac{1}{2\pi }\sqrt{-\frac{1}{2}\left( \nabla _{\mu }\chi _{\nu }\right) \left( \nabla ^{\mu }\chi ^{\nu }\right) }=\left\{ \begin{array}{cc} \frac{Z^{\prime }(r_{+})}{4\pi }, & \text{dilatonic BI} \\ \nonumber & \\ \frac{1}{4\pi }\sqrt{\frac{B(r)}{A(r)}}A^{\prime }(r_{+}), & \text{BD-BI}% \end{array}% \right. , \label{Td1}$$in which $\chi =\partial /\partial t$ is the time like null Killing vector. It is easy to show that the Hawking temperature in both frames is uniform as $$T=\frac{\left( \alpha ^{2}+1\right) }{2\pi \left( n-1\right) }\left[ \frac{% -k\left( n-2\right) (n-1)}{2\left( \alpha ^{2}-1\right) r_{+}}\left( \frac{b% }{r_{+}}\right) ^{-2\gamma }-\Lambda r_{+}\left( \frac{b}{r_{+}}\right) ^{2\gamma }+\Gamma_{+} \right] ,\text{ dilatonic BI \& BD-BI} \label{Td2}$$where $$\Gamma_{+} =-\frac{\left( \alpha ^{2}+1\right) ^{2}q^{2}}{2\pi (n-1)}\left( \frac{r_{+}}{b}\right) ^{2\gamma \left( n-2\right) }r_{+}^{3-2n}\digamma _{1}(\eta_{+} ). \label{GAMMA}$$ Following Refs. [@BDvsDilaton; @Cai], the finite mass and entropy of the black hole in both Einstein and Jordan frames are $$\begin{aligned} M &=&\frac{\varpi _{n-1}b^{(n-1)\gamma }}{16\pi }\left( \frac{n-1}{1+\alpha ^{2}}\right) m, \label{Md} \\ S &=&\frac{\varpi _{n-1}b^{(n-1)\gamma }}{4}r_{+}^{(n-1)\left( 1-\gamma \right)}. \label{Sd}\end{aligned}$$ In addition, the electric charge $Q$ of the black holes can be obtained via the Gauss’s law $$Q=\frac{q}{4\pi }. \label{Qd}$$ Also, one can obtain the electric potential as $$\begin{aligned} U &=&\left(\frac{r_{+}}{b}\right)^{4\gamma+1 }\frac{b\beta (\alpha ^{2}+1)}{ (5\alpha ^{2}+1)}\;{}_{2}F_{1}\left( \left[ \frac{1}{2},\frac{5\alpha ^{2}+1}{6(2\alpha ^{2}+1)}\right] ,\left[ \frac{17\alpha ^{2}+7}{6(2\alpha ^{2}+1)}\right] ,\frac{\beta ^{2}b^{6}}{q^{2}}\left(\frac{r_{+}}{b}\right)^{6\gamma+6 }\right) .\label{U}\end{aligned}$$ It is straightforward to show that the mentioned conserved and thermodynamical quantities satisfy the first law of thermodynamics as $$dM=TdS+UdQ. \label{1stLAW}$$ Heat capacity and thermal stability ----------------------------------- Here, we want to investigate thermal stability of the black holes. Due to the set of state functions and thermodynamic variables of a system, one may study the thermodynamic stability from different points of view through various ensembles. One of the common methods to study phase transition is regarding the canonical ensemble. In this ensemble, thermal stability of a system will be ensured by positivity of the heat capacity. One can obtain the heat capacity relation with fixed charge as $$C_{Q}=\frac{\left( \frac{\partial M}{\partial S}\right) _{Q}}{\left( \frac{% \partial ^{2}M}{\partial S^{2}}\right) _{Q}}=\frac{M_{S}}{M_{SS}}=T\left( \frac{\partial S}{\partial T}\right) _{Q}, \label{CQ}$$where $M_{S}=\frac{\partial M}{\partial S}$ and $M_{SS}=\frac{\partial ^{2}M% }{\partial S^{2}}$. From the nominator of heat capacity, it is evident that the temperature ($M_{S}$) has crucial role on the sign of $C_{Q}$. In addition, divergence points of heat capacity are indicating second order phase transition. Hence, these divergencies are utilized for calculating critical values and investigating the critical behavior of the black hole. Now, for studying phase transition, we introduce various geometrical thermodynamic methods and compare their results with those of arisen from the heat capacity. GEOMETRICAL STUDY OF THE PHASE TRANSITION ----------------------------------------- One of the basic motivations for considering the geometrical thermodynamics comes from the fact that this formalism helps us to describe in an invariant way the thermodynamic properties of a given thermodynamical system in terms of geometric structures. Also, this method is a strong machinery for describing phase transition of the black holes. Another motivation is to give an independent picture regarding thermodynamical aspects of a system. In addition to some useful information about bound points, phase transitions and thermal stability conditions, this method contains information regarding molecular interaction around phase transitions for thermodynamical systems. In other words, by studying the sign of thermodynamical Ricci scalar around phase transition points, one can extract information whether interaction is repulsive or attractive. Based on such motivations, it will be interesting to investigate black hole phase transition in the context of geometrical thermodynamics, as an independent approach. In order to study the phase transition, one can employ thermodynamical quantities to build geometrical spacetime. There are several metrics in the context of geometrical thermodynamics which one can use them to study phase transition and critical behavior. The well-known thermodynamical metrics are Weinhold, Ruppeiner, Quevedo and HPEM as the recently proposed method. As we mentioned, in some specific types of systems the Weinhold, Ruppeiner and Quevedo metrics are not applicable and they will face some problems. Here, we want to discuss these thermodynamical metrics and their possible mismatched problems. Thermodynamical metric was first introduced by Weinhold [@Weinhold1; @Weinhold2]. This thermodynamical metric is given by $$dS_{W}^{2}=g_{ab}^{W}dX^{a}dX^{b}, \label{Weinmetric}$$ where $g_{ab}^{W}=\partial ^{2}M\left( X^{c}\right) /\partial X^{a}\partial X^{b}$, $X^{a}\equiv X^{a}\left( S,N^{i}\right) $ and $N^{i}$ denotes other extensive variables of the system. By calculating $M$ as a function of extensive quantities (such as entropy and electric charge) and using Weinhold metric (\[Weinmetric\]), one can find the Ricci scalar. It is expected that the singular points of the Weinhold Ricci scalar match to the root or divergence points of the heat capacity, to indicate the bound point or the phase transition ones. We plot Fig. \[FigW\] to investigate the mentioned behavior. $% \begin{array}{cc} \epsfxsize=5.5cm \epsffile{Wein11.eps} & \epsfxsize=5.5cm \epsffile{Wein22.eps} % \epsfxsize=5.5cm \epsffile{Wein33.eps}% \end{array} $ After that, Ruppeiner [@Ruppeiner1; @Ruppeiner2] has defined another thermodynamical metric with the following form $$dS_{R}^{2}=g_{ab}^{R}dX^{a}dX^{b}, \label{Rupp}$$ where $g_{ab}^{R}=-\partial ^{2}S\left( X^{c}\right) /\partial X^{a}\partial X^{b}$ and $X^{a}\equiv X^{a}\left( M,N^{i}\right) $. In the Ruppeiner metric, thermodynamical potential is entropy. It is worthwhile mentioning that these two metrics are conformally related to each other [@Salamon]. We plot Fig. \[FigR\] to show that the Ruppeiner Ricci scalar divergencies are not matched with those of heat capacity. $% \begin{array}{cc} \epsfxsize=7cm \epsffile{Rup1.eps} & \epsfxsize=7cm \epsffile{Rup2.eps}% \end{array} $ As we have shown, calculating thermodynamical Ricci scalar of these two thermodynamical metrics indicates that the results were not completely consistent with the results of heat capacity in the canonical ensemble. In order to remove some failures of the Weinhold and Ruppeiner metrics, recently, another metric which is Legendre invariant has been introduced by Quevedo [@Quevedo1; @Quevedo2]. The Quevedo metric has the following form $$ds_{Q}^{2}=\Omega (-M_{SS}dS^{2}+M_{QQ}dQ^{2}) \label{quevedo}$$ where the conformal coefficient $\Omega$ is $$\Omega =\left( SM_{S}+QM_{Q}\right) . \label{Omega}$$ $% \begin{array}{cc} \epsfxsize=7cm \epsffile{Qued11.eps} & \epsfxsize=7cm \epsffile{Qued12.eps} \end{array} $ Considering Figs. \[FigW\]–\[FigQ\], we find that by using these three well-known metrics, there is at least a mismatch between heat capacity divergencies and thermodynamical Ricci scalar divergencies (of these three metrics). Therefore, these metrics are not appropriate tools for investigation of our black hole phase transitions and related critical behavior. In other words, the method of geometrical thermodynamics which has been reported in [@Niu] is not an applicable method in the scalar field theory. Very recently, a new metric was proposed by Hendi, et al (HPEM metric) to solve this problem. This method is applied for various gravitating systems and it is shown that the root and divergence points of the heat capacity coincide with the divergence points of the HPEM Ricci scalar (see Figs. \[FigHPEMbeta\]–\[FigHPEMn6\], for more details). The generalized HPEM metric with $n$ extensive variables ($n\geq 2$) has the following form [@HPEM1; @HPEM2; @HPEM3] $$ds_{HPEM}^{2}=\frac{SM_{S}}{\left( \Pi _{i=2}^{n}\frac{\partial ^{2}M}{% \partial \chi _{i}^{2}}\right) ^{3}}\left( -M_{SS}dS^{2}+\sum_{i=2}^{n}\left( \frac{\partial ^{2}M}{\partial \chi _{i}^{2}}\right) d\chi _{i}^{2}\right), \label{HPEM}$$ where $\chi_{i}$’s ($\chi_{i}\neq S$) are extensive parameters. It is notable that HPEM metric is the same as that presented by Quevedo (with the same $"-,+,+,..."$ signature), but with different conformal factor and therefore it is expected to enjoy Legendre invariancy. In what follows, we will investigate the stability and phase transition of the physical BD-BI black holes in the context of the heat capacity and geometrical thermodynamics by using HPEM metric. [c]{} $n$   $r_{0}$    $r_{d_{1}}$   $r_{d_{2}}$ ------- ------------ ---------------- ------------- $5$ $0.2052\ $ $0.3820$ $2.5814$ $6$   $0.2842\ $ $0.4772\ $ $3.5605$ $7$  $0.3422\ $ $0.5390\ $ $4.7619$ \ Table I: critical points of BD-BI theory for $q=0.1$, $\Lambda =-1$, $\omega =10$, $b=1$ and $\beta =1.5$. [c]{} ------------------------------------------------------- $\beta $ $r_{0}$    $r_{d_{1}}$ $r_{d_{2}}$ ----------- -------------- -------------- ------------- $0.1$  $\ 0.0056$ $0.0112$ $1.7512$ $1.0$  $\ $0.1251$ $1.7512$ 0.0604$ $1.5$   $\ 0.0902\ $ $0.1996$ $1.7512$ $5.0$   $\ 0.2056$ $0.3523 $ $1.7512$ $100.0$   $\ 0.2400$ $0.3600 $ $1.7512$ $200.0$   $\ 0.2400$ $0.3600 $ $1.7512$ ------------------------------------------------------- \ Table II: critical points of BD-BI theory for $q=0.1$, $\Lambda =-1$, $% \omega =10$, $b=1$ and $n=4$. [c]{} $\omega $  $r_{0}$    $r_{d_{1}}$   $r_{d_{2}}$ ------------ ------------ ---------------- ------------- $0.2$  $0.0482\ $ $0.1132$ $1.9022$ $2$   $0.0742\ $ $0.1658\ $ $1.8038$ $200$   $0.0974\ $ $0.2128$ $1.7318$ \ Table III: critical points of BD-BI theory for $q=0.1$, $\Lambda =-1$, $n=4$, $b=1$ and $\beta =1.5$. $% \begin{array}{cc} \epsfxsize=6cm \epsffile{Newbeta11.eps} & \epsfxsize=6cm % \epsffile{Newbeta12.eps} \\ \epsfxsize=6cm \epsffile{Newbeta51.eps} & \epsfxsize=6cm % \epsffile{Newbeta52.eps}% \end{array} $ $% \begin{array}{cc} \epsfxsize=6cm \epsffile{omega21.eps} & \epsfxsize=6cm % \epsffile{omega22.eps} \\ \epsfxsize=6cm \epsffile{omega2001.eps} & \epsfxsize=6cm % \epsffile{omega2002.eps}% \end{array} $ $% \begin{array}{cc} \epsfxsize=7cm \epsffile{New6d1.eps} & \epsfxsize=7cm \epsffile{New6d2.eps}% \end{array} $ $% \begin{array}{ccc} \epsfxsize=5.6cm \epsffile{diffbeta1.eps} & \epsfxsize=5.6cm \epsffile{diffbeta2.eps} & \epsfxsize=5.6cm \epsffile{diffbeta3.eps} % \end{array} $ As the first significant point which must be taken into deep consideration, one should regard the sign of the temperature. The positivity of the temperature denotes a physical black hole; Whereas the negativity of $T$ represents a non-physical system. The temperature behavior has been shown in figures, too. As we can see, there is a lower bound for the horizon radius ($r_{0}$), in which for $r_{+}< r_{0}$, we encounter with a non-physical black hole, owing to negative sign of temperature. In contrast, in the case of $r_{+}> r_{0}$, we confront a physical system due to the positivity of the temperature. In other words, the horizon radius of physical black holes are located in this region. Figure \[FigHPEMbeta\] shows that for the special values of the electric charge, nonlinearity parameter and BD-coupling coefficient, we can obtain three characteristic points. One of them refers to the root of heat capacity (or temperature) which is known as $r_{0}$ and others are related to the divergence points of heat capacity which are denoted as $r_{d_{1}}$and $r_{d_{2}}$ ($r_{d_{1}} < r_{d_{2}}$). We also find that all divergence points of the Ricci scalar (for HPEM metric) are coincide with these three points. Here, we use some tables to study the influences of different parameters (dimensions, nonlinearity parameter and BD-coupling coefficient) on the mentioned characteristic points. These tables provide information regarding the lower bound of horizon radius, two points of phase transition (for the case of BD-BI) and their dependencies to the variation of dimensions, nonlinearity parameter and coupling coefficient. Regarding the tables and Figs. \[FigHPEMbeta\]–\[FigHPEMomega\], it is evident that one root and two divergence points for the heat capacity are almost observed. It is worthwhile to mention that, the region of $r_{0}<r_{+}<r_{d_1}$, (positive sign of heat capacity) shows the stability of the system. In contrast, one can find that for the region of $r_{d_1}<r_{+}<r_{d_2}$, the heat capacity has negative sign which indicates instability. In addition, at region $r_{+}>r_{d_2}$, the system is in the stable state due to the positive sign of heat capacity (see Figs. 4-6 for more details). According to table I, one can conclude that the lower bound radius and two divergence points are increasing functions of the dimensions. Also, according to table II, the lower bound of horizon radius and the first divergence point ($r_{d_1}$) will increase by increasing $\beta $ (the nonlinearity parameter), whereas the second point of divergency remains steady over this change. Considering figures and table II, it is obvious that by increasing $\beta$, root and the first divergence point of heat capacity will increase up to a point and then any increment in this parameter would have negligible effect on these values. To put in other words, it can be interpreted that in large $\beta $, we will face the Brans-Dicke-Maxwell behavior [@BD-Max]. For large $\beta $, the obtained values for lower bound horizon radius and divergence point are the same as the obtained values for the Brans-Dicke-Maxwell case [@BD-Max]. It is notable that the unstable region (between two divergencies, where the heat capacity is negative) is larger in small $\beta $ than the large one (Brans-Dicke-Maxwell case) as it would be expected, which is due to the nature of nonlinearity that would cause the instability of system to increase. Meanwhile, $r_{0}$ and $r_{d_1}$ have ascending functions and $r_{d_2}$ will be declined by increasing $\omega $ (see table III). Generally, from what has been discussed above, dimensionality $n$ and BD-coupling coefficient $\omega $ are playing the main role in changes of the location of larger divergence point. Conclusion ========== In this paper, the main goal was studying thermodynamical behavior of the BD-BI and Einstein-BI-dilaton black hole solutions. Since both of these solutions had very similar thermodynamical behavior in the context of geometrical thermodynamics, we have just considered the BD-BI ones. We have investigated the stability and phase transition in the canonical ensemble through the use of heat capacity. We have found that for having a physical black hole (positive temperature), there should be a restriction on the value of the horizon radius, which lead to a physical limitation point. This point was a border between non-physical and physical black hole horizon radius. Moreover, investigating the phase transition of the black holes exhibited that there exist second order phase transition points. In other words, the heat capacity had one real positive root and two divergence points. It was shown that these points (the root and divergence points of heat capacity) were affected by variation of the BI-parameter, BD coupling constant and dimensions. From presented tables and figures, we have found that the effect of dimensions on the larger divergence point was more than other factors and in contrast, the BI-parameter had no sensible effect on this value. The effect of BD-coupling constant on these three points was so small in a way that by applying a dramatic change in this constant, we observed a small change in the value of such characteristic points. It was illustrated that in the context of thermal stability there exist four regions, specified by the root and two divergence points of the heat capacity. The root of heat capacity was referred as the lower bound of horizon radius that separated the non-physical black holes from the physical ones. Between the two divergencies, we encountered an unstable state and after the second divergence point black hole obtained a stable state. It is notable that for small $\beta$, because of the nonlinearity effect, the unstable region is larger than the Maxwell case (large $\beta$) [@BD-Max]. Eventually, we employed the geometrical thermodynamic method to study the phase transition. We have shown that Weinhold, Ruppeiner and Quevedo metrics failed to provide a consistent result with the heat capacity’s result. In other words, their thermodynamical Ricci scalar’s divergencies did not match with the root and divergencies of the heat capacity, exactly. In some of these methods we encountered extra divergency which did not coincide with any of the phase transition points. At last, using the HPEM metric, we achieved desirable results. It was shown that all the divergence points of the Ricci scalar of the mentioned metric covered the divergencies and root of the heat capacity. It is worth mentioning that the behavior of the curvature scalar was different near its divergence points. In other words, the divergence points of the Ricci scalar related to root of the heat capacity could be distinguished from the divergencies related to phase transition points based on the curvature scalar behavior. Regarding the used method of this paper, it is interesting to extend obtained results to an energy dependent spacetime and discuss the role of gravity’s rainbow [@raibow1; @rainbow2; @rainbow3; @rainbow4]. We leave this issue for future work. **Competing Interests** The authors declare that there is no conflict of interests regarding the publication of this paper. We would like to thank the anonymous referee for his valuable comments. We also acknowledge M. Momennia and S. Panahiyan for reading the manuscript. We wish to thank Shiraz University Research Council. This work has been supported financially by the Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), Iran. [99]{} M. Laue, Jahrb. Radioaktiv. Elektron. **14** (1917) 263. S. Perlmutter et al., Astrophys. J. **517** (1999) 565. S. Perlmutter, M. S. Turner and M. White, Phys. Rev. Lett. **83** (1999) 670. A. G. Riess et al., Astrophys. J. **607** (2004) 665. P. A. M. Dirac, Proc. R. Soc. Lond. A **165** (1938) 199. S. Weinberg, *Gravitation and Cosmology* (Wiley, New York 1972). C. Brans and R. H. Dicke, Phys. Rev. **124** (1961) 925. N. Banerjee and D. Pavon, Phys. Rev. D **63** (2001) 043504. M. Sharif and S. Waheed, Eur. Phys. J. C **72** (2012) 1876. M. K. Mak and T. Harko, Europhys. Lett. **60** (2002) 155. S. W. Hawking, Commun. Math. Phys. **25** (1972) 167. R. G. Cai and Y. S. Myung, Phys. Rev. D **56** (1997) 3466. M. H. Dehghani, J. Pakravan and S. H. Hendi, Phys. Rev. D **74** (2006) 104014. S. H. Hendi, J. Math. Phys. **49** (2008) 082501. S. H. Hendi and R. Katebi, Eur. Phys. J. C **72** (2012) 2235. M. H. Dehghani and M. Shojania, Can. J. Phys. **80** (2002) 951. S. H. Hendi, M.S. Talezadeh, Gen. Relativ. Gravit. **49** (2017) 12. G. Mie, Ann. Phys. (Berlin) **37** (1912) 511. G. Mie Ann. Phys. (Berlin) **39** (1912) 1. M. Born and L. Infeld, Proc. R. Soc. A **144** (1934) 425. E. Fradkin and A. Tseytlin, Phys. Lett. B **163** (1985) 123. R. Matsaev, M. Rahmanov and A. Tseytlin, Phys. Lett. B **193** (1987) 205. E. Bergshoeff, E. Sezgin, C. Pope and P. Townsend, Phys. Lett. B **188** (1987) 70. C. Callan, C. Lovelace, C. Nappi and S. Yost, Nucl. Phys. B **308** (1988) 221. O. Andreev and A. Tseytlin, Nucl. Phys. B **311** (1988) 221. R. Leigh, Mod. Phys. Lett. A **04** (1989) 2767. M. H. Dehghani, N. Alinejadi and S. H. Hendi, Phys. Rev. D **77** (2008) 104025. M. H. Dehghani and S. H. Hendi, Phys. Rev. D **73** (2006) 084021. M. Allahverdizadeh, S. H. Hendi, J. P. S. Lemos and A. Sheykhi, Int. J. Mod. Phys. D **23** (2014) 1450032. D. C. Zou, S. J. Zhang and B. Wang, Phys. Rev. D **89** (2014) 044002. R. Banerjee and D. Roychowdhury, Phys. Rev. D **85** (2012) 104043. A. Lala and D. Roychowdhury, Phys. Rev. D **86** (2012) 084027. R. Banerjee and D. Roychowdhury, Phys. Rev. D **85** (2012) 044040. P. Li, R. H. Yue and D. C. Zou, Commun. Theor. Phys. **56** (2011) 845. D. C. Zou, Z. Y. Yang, R. H. Yue and P. Li, Mod. Phys. Lett. A **26** (2011) 515. A. Ghodsi and D. M. Yekta, Phys. Rev. D **83** (2011) 104004. R. G. Cai and Y. W. Sun, JHEP **09** (2008) 115. S. H. Mazharimousavi, M. Halilsoy and Z. Amirabi, Phys. Rev. D **78** (2008) 064050. W. A. Chemissany, Mees de Roo and S. Panda, Class. Quantum Gravit. **25** (2008) 225009. Y. S. Myung, Y. W. Kim and Y. J. Park, Phys. Rev. D **78** (2008) 084002. Y. S. Myung, Y. W. Kim and Y. J. Park, Phys. Rev. D **78** (2008) 044020. O. Miskovic and R. Olea, Phys. Rev. D **77** (2008) 124048. I. Zh. Stefanov, S. S. Yazadjiev and M. D. Todorov, Phys. Rev. D **75** (2007) 084036. S. Fernando, Phys. Rev. D **74** (2006) 104032. R. G. Cai, D. W. Pang and A. Wang, Phys. Rev. D **70** (2004) 124034. M. Aiello, R. Ferraro and G. Giribet, Phys. Rev. D **70** (2004) 104014. T. K. Dey, Phys. Lett. B **595** (2004) 484. T. Tamaki, JCAP **05** (2004) 004. S. Fernando and D. Krug, Gen. Relativ. Gravit. **35** (2003) 129. M. Wirschins, A. Sood and J. Kunz, Phys. Rev. D **63** (2001) 084002. M. Cataldo and A. Garcia, Phys. Lett. B **456** (1999) 28. S. H. Hendi, J. Math. Phys. **49** (2008) 082501. M. H. Dehghani, S. H. Hendi, A. Sheykhi and H. Rastegar Sedehi, JCAP **02** (2007) 020. M. H. Dehghani and S. H. Hendi, Int. J. Mod. Phys. D **16** (2007) 1829. M. H. Dehghani and H. Rastegar Sedehi, Phys. Rev. D **74** (2006) 124018. S. H. Hendi, Phys. Rev. D **82** (2010) 064040. D. J. Cirilo-Lombardo, Gen. Relativ. Gravit. **37** (2005) 847. V. Ferrari, L. Gualtieri, J. A. Pons and A. Stavridis, Mon. Not. Roy. Astron. Soc. **350** (2004) 763. H. Q. Lu, L. M. Shen, P. Ji, G. F. Ji and N. J. Sun, Int. J. Theor. Phys. **42** (2003) 837. M. H. Dehghani and S. H. Hendi, Gen. Relativ. Gravit. **41** (2009) 1853. E. F. Eiroa and G. F. Aguirre, Eur. Phys. J. C **72** (2012) 2240. S. H. Hendi, Adv. High Energy Phys. **2014** (2014) 697863. W. Yao and J. Jing, JHEP **05** (2014) 058. S. Gangopadhyay, Mod. Phys. Lett. A **29** (2014) 1450088. S. Gangopadhyay and D. Roychowdhury, JHEP **05** (2012) 156. S. Gangopadhyay and D. Roychowdhury, JHEP **05** (2012) 002. J. Jing, L. Wang, Q. Pan and S. Chen, Phys. Rev. D **83** (2011) 066010. J. Jing and S. Chen, Phys. Lett. B **686** (2010) 68. S. H. Hendi, M. R. Hadizadeh and R. Katebi, Iran. J. Sci. Technol. Trans. Sci. DOI 10.1007/s40995-016-0060-5. B. Vaseghi,G. Rezaei, S. H. Hendi and M. Tabatabaei, Quantum Matter **2** (2013) 194. S. W. Hawking, Nature **248** (1974) 30. J. D. Beckenstein, Phys. Rev. D **7** (1973) 2333. S. Hawking and D. N. Page, Commun. Math. Phys. **87** (1983) 577. R. G. Cai, S. P. Kim and B. Wang, Phys. Rev. D **76** (2007) 024011. R. G. Cai, L. M. Cao and Y. W. Sun, JHEP **11** (2007) 039. M. Eune, W. Kim and S. H. Yi, JHEP **03** (2013) 020. P. C. W. Davies, Proc. Roy. Soc. Lond. A **353** (1977) 499. H. B. Callen, *Thermodynamics and an introduction to thermostatistics*, (Wiley, New York 1985). F. Weinhold, J. Chem. Phys. **63** (1975) 2479. F. Weinhold, J. Chem. Phys. **63** (1975) 2484. G. Ruppeiner, Phys. Rev. A **20** (1979) 1608. G. Ruppeiner, Rev. Mod. Phys. **67** (1995) 605. P. Salamon, J. Nulton and E. Ihrig, J. Chem. Phys. **80** (1984) 436. H. Quevedo, J. Math. Phys. **48** (2007) 013506. H. Quevedo and A. Sanchez, JHEP **09** (2008) 034. S. H. Hendi, S. Panahiyan, B. Eslam Panah and M. Momennia, Eur. Phys. J. C **75** (2015) 507. S. H. Hendi, S. Panahiyan and B. Eslam Panah, Adv. High Energy Phys. **2015** (2015) 743086. S. H. Hendi, A. Sheykhi, S. Panahiyan and B. Eslam Panah, Phys. Rev. D **92** (2015) 064028. S. H. Hendi, R. Moradi, Z. Armanfard and M. S. Talezadeh, Eur. Phys. J. C **76** (2016) 263. M. H. Dehghani, J. Pakravan and S. H. Hendi, Phys. Rev. D **74** (2006) 104014. A. Sheykhi, Phys. Rev. D **76** (2007) 124025. R. G. Cai and Y. S. Myung, Phys. Rev. D **56** (1997) 3466. C. Niu and Y. Tian and X. N. Wu, Phys. Rev. D **85** (2012) 024017. S. H. Hendi, S. Panahiyan, B. Eslam Panah and Z. Armanfard, Eur. Phys. J. C **76** (2016) 396. G. Amelino-Camelia, Int. J. Mod. Phys. D **11** (2002) 35. G. Amelino-Camelia, New J. Phys. **6** (2004) 188. J. Mogueijo and L. Smolin, Class. Quantum Gravit. **21** (2004) 1725. S. H. Hendi, M. Momennia, B. Eslam Panah and M. Faizal, Astrophys. J. **827** (2016) 153. [^1]: email address: hendi@shirazu.ac.ir

--- abstract: 'In this study we report a method for the preparation of freestanding magnetocaloric thin films. Non-stoichiometric Heusler alloys Ni-Mn-Sn, Ni-Co-Mn-Sn and Ni-Co-Mn-Al are prepared via sputter deposition. A sacrifial vanadium layer is added between the substrate and the Heusler film. By means of selective wet-chemicals etching the vanadium layer can be removed. Conditions for the crystallization of Vanadium layers and epitaxial growth of the Heusler films are indicated. Magnetic and structural properties of freestanding and as-prepared films are compared in detail. The main focus of this study is on the influence of substrate constraints on the Martensitic transistion.' author: - 'L. Helmich' - 'N. Teichert' - 'W. Hetaba' - 'A. Behler' - 'A. Waske' - 'S. Klimova' - 'A. Hütten' title: Vanadium sacrificial layers as a novel approach for the fabrication of freestanding Heusler Shape Memory Alloys --- Introduction ============ Magnetocaloric materials for room-temperature cooling applications have attracted strong interested mainly for two reasons. On the one hand magnetic cooling devices obviate the need for greenhouse gases as freezing agents. On the other hand there are promising material candidates with high cooling efficiency gains.[@Sandeman2012] Magnetocaloric effects (MCE) emerge in any magnetic material due to the interdependence of thermal and magnetic properties. MCE can be induced by application and removal of an external magnentic field.[@Planes2009]. One can distinguish between two classes of MCE: Materials which heat up upon magnetization show the direct MCE whereas materials that cool down upon magnetization show the inverse magnetocaloric effect.[@Gomez2013] This work is related to three different non-stoichiometric Heusler alloys namely Ni-Mn-Sn, Ni-Co-Mn-Sn and Ni-Co-Mn-Al. All of them are known to show the inverse magnetocaloric effect.[@Yuzuak2013; @Kainuma2008] Depending on the experimental conditions there are two measures for the MCE. These are the isothermal entropy change $\Delta S$ and the adiabatic temperature change $\Delta T_{\text{ad}}$.[@Sandeman2012] Recently there has been a lot a progress in the investigation of $\Delta T_{\text{ad}}$ in bulk materials. Although, to the best of the authors knowledge, this property not been reported yet for Heusler alloys in thin films. However, direct measurements of the adiabatic temperature change in thin films are a challenging task. Thin films with a thickness of hundreds of nanometers provide an amount of mass that is still significantly smaller than the necessary minimum sample sizes even differential scanning calorimeters with high sensitivity would need.[@Jeppesen2008] Furthermore thin films are commonly grown on single crystaline substrates with a thickness of hundreds of micrometers. Since these substrates provide a huge heat sink $\Delta T_{\text{ad}}$ cannot be determined directly on these samples. In this study we report on a method in order to elude this issue, i.e. adding a sacrificial Vanadium layer between the substrate and the magnetocaloric Heusler alloy. In a subsequent wet-chemical treatment the vanadium layer can be etched selectively thus resulting in a freestanding Heusler film. Experimental details ==================== Vanadium layers and all Heusler films are grown on MgO(001) single crystalline substrates using a ultrahigh vacuum sputtering system with a base pressure typically better than $1\times10^{-9}$ mbar. The 3 inch sputter sources are arranged in a confocal sputter-up geometry. The distance between target and sample is 21 cm. The Heusler alloy films are deposited from elemental Ni, Co, Mn, Sn, and Al targets. The substrate temperature of $500^\circ\text{C}$ is applied during the deposition process. To ensure a homogeneous stoichiometry the sample is rotated with 5 rpm. Argon flow is regulated to a sputtering pressure of $2.3\times 10^{-3}$ mbar. To prevent surface oxidation all samples are coated with a $2.5$nm MgO capping layer deposited by electron beam evaporation. The stoichiometry of the films is determined by X-Ray fluorescence measurements. Due to the relatively low fluorescence yield of Alumnium the stoichiometry of NiCoMnAl samples are determined by inductively coupled plasma optical emission spectrometry (ICP-OES). In order to obtain depth profiles TEM lamellas are fabricated perpendicular to the sample surface. Subsequently EDX line scans are measured on these lamellas. The crystalline structure is analyzed via temperature dependent X-Ray diffraction. The samples is cooled in a custom-build liquid nitrogen cryostat. Therefore a temperature range from $-150^{\circ}\text{C}$ to $200^\circ\text{C}$ is accessible. XRD is measured in Bragg-Brentano geometry with Cu $K_\alpha$ radiation. Film thickness and density are measured via X-Ray reflectometry. The temperature dependent magnetization is measured using a vibrating sample magnetometer under 10 mT external field applied in in-plane direction. The electrical resistivity is measured using a standard 4-probe setup within a helium-cooled cryostat. The wet-chemical etching procedure is performed with the commercially available acid “Chromium Etchant No. 1” by MicroChemicals GmbH. Depending on the preparation conditions of the Vanadium layer it takes five to ten minutes in undiluted acid to remove the Heusler layer completely from the substrate. Afterwards, the brittle layer is washed in deionized water and ethanol. In order to contact the freestanding sample for four-probe measurements a special sample carrier is prepared. This carrier consists of a $\text{SiO}_x$ substrate with four micro-fabricated gold conduction lines on top. The Heusler layer is dried onto this sample carrier. This thin films then sticks to the sample carrier due to wetting effects. This procedure is also convenient to fasten thin films with thicknesses down to $20\text{nm}$ to a TEM grid. This provides practical advantages for TEM imaging since time-consuming preparation steps such as the thinning of the samples thus become unnecessary. Results and discussion ====================== Vanadium is an appropriate candidate as a sacrificial layer due to the small lattice mismatch to both the substrate and Heusler alloys. Lattice constants in table \[tab:latticeconst\] were determined by XRD. It is noteworthy that the lattice constants slightly shift for Vanadium buffered layers, i.e. $5.92\AA$ for NiCoMnSn. Material lattice const. $\left[\AA\right]$ ---------- ----------------------------------- Vanadium $3.05$ MgO(100) $4.21$ NiMnSn 5.98 NiCoMnSn 5.97 : lattice constants[]{data-label="tab:latticeconst"} Vanadium is known to crystallize in a body centered cubic structure and MgO in a face centered cubic structure. Vanadium can be grown epitactically on MgO with a mismatch of $2.3\%$ if the two lattices are twisted by $45$ degree to each other. Also the Heusler alloys can be grown epitactically on Vanadium with a lattice mismatch of $2.0\%$. Vanadium deposition at room temperature leads to an amorphous layer. Even post-annealing does not lead to crystallization. A minimum sample temperature of $200^\circ$C is necessary to ensure epitactical growth. An even higher deposition temperature results in a lower surface roughness. In former studies of Vanadium as spacer material in a Heusler sandwich structures it is reported that Vanadium is likely to interdiffuse into Heusler materials. This interdiffusion starts above a certain critical temperature and may cause significant problems. [@Slomka1999]. In order to determine this critical temperature an ex-situ post annealing study on MgO(001)/V(35nm)/NiMnSn(200nm)/MgO(2.5nm) is carried out. The samples are annealed for one hour each at different temperatures. However, for substrate temperatures higher than $550^\circ$C the Heusler peaks shift to larger lattice constants which indicates undesirable structural changes. Consequently interdiffusion determines an upper temperature limit. A depth profile of a Heusler sample which was deposited at $500^\circ$C substrate temperature is investigated by means of EDX. Results are shown in figure \[fig:edx\]. Within the range of uncertainties of this method no interdiffusion of Vanadium into the Heusler layer can be observed. This result is in agreement with a depth profile from Sputter-Auger-Electron-Spectroscopy (AES). ![EDX linescan on a TEM lamella: Depth profile on NiCoMnAl on V-Buffer. Lines between data points are a guide to the eye.[]{data-label="fig:edx"}](edx_linescan){width="\columnwidth"} ![XRD pattern of NiMnSn: Temperature series for different substrate temperatures during the deposition of NiMnSn.[]{data-label="fig:xrd_t_series"}](v_t_series){width="\columnwidth"} However, it not possible to fabricate a crystalline Heusler thin film at room temperature. NiMnSn samples are deposited at different substrate temperatures ranging from $20^\circ$C to $550^\circ$C. XRD pattern are shown in figure \[fig:xrd\_t\_series\]. The XRD analysis of this temperature series shows that those XRD peaks which belong to the Heusler emerge at substrate temperatures higher than $300^\circ$C. Therefore the appropriate interval for substrate temperatures during the deposition process is identified between $300^\circ$C and $500^\circ$. Furthermore a depth profile of the samples is measured by means of sputter AES. Within the uncertainties of this method no interdiffusion of Vanadium is observed. ![XRD pattern of NiMnSn: as prepared (red), freestanding on glass at $20^\circ$C (black) and $80^\circ$C (blue).[]{data-label="fig:xrd1"}](Graph2){width="\columnwidth"} A comparion of the structural properties between an as-prepared sample and a freestanding sample is carried out on $\text{Ni}_{50.2}\text{Mn}_{34.4}\text{Sn}_{15.4}$. Figure \[fig:xrd1\] shows the corresponding XRD pattern. The red curve belonging to the as-prepared samples clearly shows a cubic structure, i.e. Austenite, via the $(002)_A$ and $(004)_A$ peaks. The shoulder on the right hand side of the $(004)$ indicates that there is some martensitic contribution. Due to the distortion of the martensitic lattice the $(400)_M$ and $(004)_M$ Martensite peaks can only be observed in a $2\theta-\omega$ scan with a non-zero $\omega$-offset. The black curve is measured on the same sample after removing the substrate and drying the Heusler layer onto a glass sample carrier. The cubic Austenite-Peaks have vanished. Instead four Martensite peaks, namely $(002)_M$, $(200)_M$, $(400)_M$ and $(004)_M$ have emerged. In this case there is no $\omega$-offset needed to measure these peaks since the Heusler film does not dry smoothly on the glass surface but wrinkles. Consequently the film appears similarly to a polycrystalline sample in XRD. Both the red and the black curve are measured at the same temperature, but the crystal structure is clearly different. This means that the Austenite transition temperature is shifted to higher temperatures by removing the substrate. The substrate causes a strain in the Heusler film which hinders the martensitic transition. Hence by removing the substrate it is favorable for the sample to be in the martensitic state. Heating this sample to $80^\circ$ C restores the austenitic structure (blue curve). The two copper peaks result from the copper heating block underneath the sample which was only installed for this measurement. A similar broadening and displacement of the hysteresis due to substrate contraints has already been studied on epitaxial NiMnGa film. [@Buschbeck2009] ![XRD pattern of NiCoMnSn: as prepared (red) and after rapid thermal annealing for 30 seconds (black).[]{data-label="fig:rta1"}](RTA1){width="\columnwidth"} As already mentioned, Vanadium will not crystallize if it is deposited at room temperature. Therefore it is investigated whether a rapid thermal annealing (RTA) process leads to a subsequent crystallization. Figure \[fig:rta1\] shows the XRD pattern of this investigation on NiCoMnSn. The as-prepared sample (red curve) clearly does not show any crystalline Heusler structure. This sample was exposed to RTA for 30 seconds at 960 Watt resulting in a maximum temperature of $740^\circ$ C. The XRD results of this post-annealing process are shown in black. The distribution of the crystallites is investigated by means of an $\omega$-scan on the cubic (004) peak. This rocking curve shows a FWHM of $1.2^\circ$. The peaks at $2\theta = 48.7^\circ$ are due to $\text{Ni}_3\text{Mn}$-phases. It is noteworthy that there is no Vanadium peak to be seen in the XRD pattern. Thus, RTA is a convenient method to achieve crystallization of NiCoMnSn on an amorphous Vanadium layer. ![VSM measurement on NiCoMnSn in a constant measurement field of 100 Oe.[]{data-label="fig:vsm"}](VSM){width="\columnwidth"} The Martensite Phase of NiCoMnSn is known to show a considerably smaller magnetization than its Austenite Phase. Therefore the Martensitic transition can be observed by measuring the magnetic moment of the sample as a function of the temperature. A comparision of the magnetic properties of as-prepard and freestanding NiCoMnSn is carried out by means of vibration sample magnetometry (VSM). Results are shown in figure \[fig:vsm\]. NiMnSn has a Curie temperature of approximately 300 K[@Auge2012], i.e. room temperature. Substitution some amount of Nickel by Cobald leads to an increasing Curie temperature. For applications at room temperature a higher Curie temperature is obviously favorable. Both the as-prepared and the freestanding sample show the same behaviour on the heating branch, i.e. Austenite start and Austenite finish temperatures are in good agreement. In contrast the Martensite start temperature is shifted by 10 K and the Martensite finish temperature is shifted by 100 K to lower temperatures resulting in a broadening of the hysteresis. ![Temperature dependent XRD of NiCoMnAl. Curves are plotted from bottom to top in the sequence of measurements. (left) The Hysteresis of the Martensitic transformation is extracted form these data. Arrows are only intended as guide to the eye. (right)[]{data-label="fig:NiCoMnAl"}](NiCoMnAl1){width="\textwidth"} ![Temperature dependent XRD of NiCoMnAl. Curves are plotted from bottom to top in the sequence of measurements. (left) The Hysteresis of the Martensitic transformation is extracted form these data. Arrows are only intended as guide to the eye. (right)[]{data-label="fig:NiCoMnAl"}](NiCoMnAl2){width="\textwidth"} Magnetic field induced transformation in NiCoMnAl has been studied in bulk materials some time ago [@Kainuma2008]. However, we report on the transformation behavior of a thin film $\text{Ni}_{41}\text{Co}_{10.4}\text{Mn}_{34.8}\text{Al}_{13.8}$ sample on a Vanadium buffer which is investigated via temperature dependent XRD. A more detailed study on freestanding NiCoMnAl samples can be found elsewhere[@Teichert2014]. Results are shown in figure \[fig:NiCoMnAl\]. The measurement sequence is started at room temperature. At first the sample is heated to 370 K until the net area of the $(004)_A$ peak reaches its maximum value. Subsequently the sample is cooled down to 210 K, i.e. the minimum value of the $(004)_A$ peak and again heated up to $370 K$. The structural hysteresis is obtained from these data by plotting the $(004)_A$ peak net area as a function of the temperature. From the above figure it can clearly be seen that the shape of the Vanadium (002) peaks also changes during the Martensitic transformation. Therefore $\omega$-scans are performed on $2\theta = 60.87^\circ$ for each temperature step. It turns out that the FWHM of these rocking curves also changes from $0.67^\circ$ at 370 K to $0.99^\circ$ at 210 K. This indicates in change in the distribution of crystallites which in turn can be understood a strain effect due to the lattice interaction with the Heusler film. Conclusion ========== Freestanding magnetocaloric thin film can be prepared by selective wet-chemical etching of sacrificial Vanadium layers. A multilayer system of Vanadium and a MCE Heusler alloy can be grown epitactically at substrate temperatures between $300\circ$ C and $500^\circ$ C without interdiffusion issues. Releasing the MCE film from the substrate leads to a lowering of the Martensite transition temperature due to the lack of substrate constraints which hinder the transition. Crystalline Heusler film can also be prepared by deposition onto and amorphous Vanadium seed layer and a subsequent rapid thermal annealing process. The differences in the transition behavior can be investigated via VSM. Both Martensite start und Martensite finish temperatures are lower for freestanding films. This broadening of the hysteresis is possibly due to surface oxidation effects. NiCoMnAl magnetocaloric films can also be prepared in thin films. These films excert a strain to the Vanadium buffer during the Martensitic transition. Acknowledgement =============== The authors thank M. Meinert and C. Sterwerf for helpful discussions and K. Fritz for preliminary work on RTA. This work is supported by Deutsche Forschungsgemeinschaft (DFG) in the scope of project A6 within SPP 1599. [99]{} K. G. Sandeman, Scr. Mater. **67**, 556-571 (2012) A. Planes, J. Phys.: Condens. Matter **67**, 233201 (2009) J. R. Gomez, R. F. Garcia, A. M. Catoira, M. R. Gomez, Renew. Sust. Energ. Rev. **17**, 74-82 (2013) E. Yüzüak, I. Diner, Y. Elerman, N. Teichert, A. Hütten, Appl. Phys. Lett. **103**, 222403 (2013) R. Kainuma, W. Ito, R. Y. Umetsu, K. Oikawa, K. Ishida, Appl. Phys. Lett. **93**, 091906 (2008) S. Jeppesen, S. Linderoth, N. Pryds, L. T. Kuhn and J. B. Jensen, Rev. Sci. Instrum. **79**, 083901 (2008) J.-P. Slomka, M. Tolan, W. Press, M.R. Fritzsimmons, R. Siebrecht, D.W. Schubert, P. Simon, J. Appl. Phys. **86**, 9 5146 (1999) J. Buschbeck, R. Niemann, O.Heczko, M. Thomas, L. Schultz, S. Fähler, Acta Mater. **57**, 2516 (2009) A. Auge, N. Teichert, M. Meinert, G. Reiss, A. Hütten, E. Yüzüak, I. Dincer, Y. Elerman, I. Ennen, and P. Schattschneider, Phys. Rev. B **85**, 214118 (2012). N. Teichert et al., arXiv:1410.8583 (2014)

--- abstract: 'We review our recent contributions on shot noise for entangled electrons and spin-polarized currents in novel mesoscopic geometries. We first discuss some of our recent proposals for electron entanglers involving a superconductor coupled to a double dot in the Coulomb blockade regime, a superconductor tunnel-coupled to Luttinger-liquid leads, and a triple-dot setup coupled to Fermi leads. We briefly survey some of the available possibilities for spin-polarized sources. We use the scattering approach to calculate current and shot noise for spin-polarized currents and entangled/unentangled electron pairs in a novel beam-splitter geometry with a *local* Rashba spin-orbit (s-o) interaction in the incoming leads. For single-moded incoming leads, we find *continuous* bunching and antibunching behaviors for the *entangled* pairs – triplet and singlet – as a function of the Rashba rotation angle. In addition, we find that unentangled triplets and the entangled one exhibit distinct shot noise; this should allow their identification via noise measurements. Shot noise for spin-polarized currents shows sizable oscillations as a function of the Rashba phase. This happens only for electrons injected perpendicular to the Rashba rotation axis; spin-polarized carriers along the Rashba axis are noiseless. The Rashba coupling constant $\alpha$ is directly related to the Fano factor and could be extracted via noise measurements. For incoming leads with s-o induced interband-coupled channels, we find an additional spin rotation for electrons with energies near the crossing of the bands where interband coupling is relevant. This gives rise to an additional modulation of the noise for both electron pairs and spin-polarized currents. Finally, we briefly discuss shot noise for a double dot near the Kondo regime.' author: - 'J. C. [^1], P. Recher, D. S. Saraga, V. N. Golovach, G. Burkard, E. V. Sukhorukov, and D. Loss' title: 'Shot noise for entangled and spin-polarized electrons' --- Introduction ============ Fluctuations of the current away from its average usually contain supplementary information, not provided by average-current measurements alone. This is particularly true in the non-linear response regime where these quantities are not related via the fluctuation-dissipation theorem. At zero temperature, non-equilibrium current noise is due to the discreteness of the electron charge and is termed shot noise. This dynamic noise was first investigated by Schottky in connection with thermionic emission [@schottky]. Quantum shot noise has reached its come of age in the past decade or so and constitutes now an indispensable tool to probe mesoscopic transport [@blanter-buttiker]; in particular, the role of fundamental correlations such as those imposed by quantum statistics. More recently, shot noise has been investigated in connection with transport of entangled [@de]–[@prl-cgd] and spin polarized electrons [@gde], [@prl-cgd]–[@sej] and has proved to be a useful probe for both entanglement and spin-polarized transport. Entanglement [@entanglement] is perhaps one of the most intriguing features of quantum mechanics since it involves the concept of non-locality. Two-particle entanglement is the simplest conceivable form of entanglement. Yet, these Einstein-Podolsky-Rosen (EPR) pairs play a fundamental role in potentially revolutionary implementations of quantum computation, communication, and information processing [@als-review]. In this context, such a pair represents two qubits in an entangled state. The generation and detection of EPR pairs of photons has already been accomplished. On the other hand, research involving two-particle entanglement of massive particles (e.g. electrons) in a solid-state matrix is still in its infancy, with a few proposals for its physical implementation; some of these involve quantum-dot setups as sources of mobile spin-entangled electrons [@gde], [@P3RSL], [@Sar02]. Spin-polarized transport [@ohno-molen], [@egues], [@patrik], on the other hand, is a crucial ingredient in semiconductor spintronics where the spin (and/or possibly the charge) of the carriers play the relevant role in a device operation. To date, robust spin injection has been achieved in Mn-based semiconductor layers (*pin* diode structures) [@ohno-molen]. High-efficiency spin injection in other semiconductor systems such as hybrid ferromagnetic/semiconductor junctions is still challenging. It is clear that the ability to create, transport, coherently manipulate, and detect entangled electrons and spin-polarized currents in mesoscopic systems is highly desirable. Here we review some of our recent works [@gde], [@jsc-cgd], [@prl-cgd], [@P3RSL], [@Sar02], [@P8RL], [@GolovachLoss] addressing some of these issues and others in connection with noise. Shot noise provides an additional probe in these novel transport settings. We first address the production of mobile entangled electron pairs (Sec. \[entanglers\]). We discuss three proposals involving a superconductor coupled to two dots [@P3RSL], a superconductor coupled to Luttinger-liquid leads [@P8RL], and a triple-dot arrangement [@Sar02]. Our detailed analysis of these “entanglers” does not reveal any intrinsic limitation to their experimental feasibility. We also mention some of the available sources of spin-polarized electrons (Sec. \[spinfilters\]). Ballistic spin filtering with spin-selective semimagnetic tunnel barriers [@egues] and quantum dots as spin filters [@patrik] are also briefly discussed. We investigate transport of entangled and spin polarized electrons in a beam-splitter (four-port) configuration [@bs], [@henny] with a local Rashba spin-orbit interaction in the incoming leads [@feve], Fig. (\[bs-fig1\]). A local Rashba term provides a convenient way to coherently spin-rotate electrons as they traverse quasi one-dimensional channels, as was first pointed out by Datta and Das [@datta-das]. Within the scattering formalism [@blanter-buttiker], we calculate shot noise for both entangled and spin-polarized electrons. For entangled electrons, shot noise is particularly relevant as a probe for fundamental two-particle interference. More specifically, shot noise (charge noise) directly probes the orbital symmetry of the EPR pair wave function. However, the symmetry of the orbital degree of freedom (“the charge”) is intrinsically tied to that of the spin part of the pair wave function via the Pauli principle. That is, the total electron-pair wave function is antisymmetric thus imposing a fundamental connection between the spin and orbital parts of the pair wave function. Hence charge noise measurements probe in fact the spin symmetry of the pair. Moreover, if one can alter the spin state of the pair (say, via some proper coherent spin rotation) this will definitely influence shot-noise measurements. This is precisely what we find here for singlet and triplet pairs. The coherent local Rashba spin rotation in one of the incoming leads of our setup, continuously alters the (spin) symmetry of the pair wave function thus giving rise to sizable shot noise oscillations as a function of the Rashba phase. Noise measurements in our novel beam-splitter should allow one to distinguish entangled triplets from singlets and entangled triplets from the unentangled ones, through their Rashba phase. Entangled pairs display continuous bunching/antibunching behavior. In addition, triplets (entangled or not) defined along different quantization axes (*x*, *y*, or *z*) exhibit distinctive noise, thus allowing the detection of their spin polarization via charge noise measurements. Shot noise for spin-polarized currents also probes effects imposed by the Pauli principle through the Fermi functions in the leads. These currents also exhibit Rashba-induced oscillations for spin polarizations perpendicular to the Rashba rotation axis. We find zero shot noise for spin-polarized carriers with polarizations along the Rashba axis and for unpolarized injection. Moreover, the Rashba-induced modulations of the Fano factor for both entangled and spin-polarized electrons offer a direct way to extract the s-o coupling constant via noise measurements. We also consider incoming leads with two transverse channels. In the presence of a weak s-o induced interchannel coupling, we find an additional spin rotation due to the coherent transfer of carriers between the coupled channels in lead 1. This extra rotation gives rise to further modulation of the shot noise characteristics for both entangled and spin-polarized currents; this happens only for carriers with energies near the band crossings in lead 1. Finally, we briefly discuss shot noise for transport through a double dot near the Kondo regime [@GolovachLoss]. Sources of mobile spin-entangled electrons {#entanglers} ========================================== A challenge in mesoscopic physics is the experimental realization of an electron “entangler” – a device creating mobile entangled electrons which are spatially separated. Indeed, these are essential for quantum communication schemes and experimental tests of quantum non-locality with massive particles. First, one should note that entanglement is rather the rule than the exception in nature, as it arises naturally from Fermi statistics. For instance, the ground state of a helium atom is the spin singlet $|\!\!\uparrow \downarrow \rangle - |\!\!\downarrow \uparrow \rangle$. Similarly, one finds a singlet in the ground state of a quantum dot with two electrons. These “artificial atoms” [@Kou97] are very attractive for manipulations at the single electron level, as they possess tunable parameters and allow coupling to mesoscopic leads – contrary to real atoms. However, such “local” entangled singlets are not readily useful for quantum computation and communication, as these require control over each individual electron as well as non-local correlations. An improvement in this direction is given by two coupled quantum dots with a single electron in each dot [@ld], where the spin-entangled electrons are already spatially separated by strong on-site Coulomb repulsion (like in a hydrogen molecule). In this setup, one could create mobile entangled electrons by simultaneously lowering the tunnel barriers coupling each dot to separate leads. Another natural source of spin entanglement can be found in superconductors, as these contain Cooper pairs with singlet spin wave functions. It was first shown in Ref. [@P1CBL] how a non-local entangled state is created in two uncoupled quantum dots when coupled to the same superconductor. In a non-equilibrium situation, the Cooper pairs can be extracted to normal leads by Andreev tunnelling, thus creating a flow of entangled pairs [@P3RSL],[@P8RL],[@P13BVBF]–[@Bou02]. A crucial requirement for an entangler is to create [*spatially separated*]{} entangled electrons; hence one must avoid whole entangled pairs entering the same lead. As will be shown below, energy conservation is an efficient mechanism for the suppression of undesired channels. For this, interactions can play a decisive role. For instance, one can use Coulomb repulsion in quantum dots [@P3RSL],[@Sar02] or in Luttinger liquids [@P8RL],[@P13BVBF]. Finally, we mention recent entangler proposals using leads with narrow bandwidth [@P20Oliver] and/or generic quantum interference effects [@Bos02]. In the following, we discuss our proposals towards the realization of an entangler that produces mobile non-local singlets [@entang-triplet]. We set $\hbar=1$ in this section. Superconductor-based Electron entanglers {#ssentanglers} ---------------------------------------- Here we envision a [*non-equilibrium*]{} situation in which the electrons of a Cooper pair tunnel coherently by means of an Andreev tunnelling event from a SC to two separate normal leads, one electron per lead. Due to an applied bias voltage, the electron pairs can move into the leads thus giving rise to mobile spin entanglement. Note that an (unentangled) single-particle current is strongly suppressed by energy conservation as long as both the temperature and the bias are much smaller than the superconducting gap. In the following we review two proposals where we exploit the repulsive Coulomb charging energy between the two spin-entangled electrons in order to separate them so that the residual current in the leads is carried by non-local singlets. We show that such entanglers meet all requirements for subsequent detection of spin-entangled electrons via noise measurements (charge measurement, see Secs. \[earlier-results\] and \[noise-rashba\]) or via spin-projection measurements (Bell-type measurement, see Sec. \[bell\]). ### Andreev entangler with quantum dots {#ssecAndreev} The proposed entangler setup (see Fig. \[spintabl\]) consists of a SC with chemical potential $\mu_{S}$ which is weakly coupled to two quantum dots (QDs) in the Coulomb blockade regime [@Kou97]. These QDs are in turn weakly coupled to outgoing Fermi liquid leads, held at the same chemical potential $\mu_{l}$. A bias voltage $\Delta\mu=\mu_{S}-\mu_{l}$ is applied between the SC and the leads. The tunnelling amplitudes between the SC and the dots, and dots and leads, are denoted by $T_{SD}$ and $T_{DL}$, respectively (see Fig. \[spintabl\]). The two intermediate QDs in the Coulomb blockade regime have chemical potentials $\epsilon_{1}$ and $\epsilon_{2}$, respectively. These can be tuned via external gate voltages, such that the tunnelling of two electrons via different dots into different leads is resonant for $\epsilon_{1}+\epsilon_{2}=2\mu_{S}$ [@energyconservation]. As it turns out [@P3RSL], this two-particle resonance is suppressed for the tunnelling of two electrons via the same dot into the same lead by the on-site repulsion $U$ of the dots and/or the superconducting gap $\Delta$. Next, we specify the parameter regime of interest here in which the initial spin-entanglement of a Cooper pair in the SC is successfully transported to the leads. Besides the fact that single-electron tunnelling and tunnelling of two electrons via the same dot should be excluded, we also have to suppress transport of electrons which are already on the QD´s. This could lead to effective spin-flips on the QD´s, which would destroy the spin entanglement of the two electrons tunnelling into the Fermi leads. A further source of unwanted spin-flips on the QD´s is provided by its coupling to the Fermi liquid leads via particle-hole excitations in the leads. The QDs can be treated each as one localized spin-degenerate level as long as the mean level spacing $\delta\epsilon$ of the dots exceeds both the bias voltage $\Delta\mu$ and the temperature $k_{B}T$. In addition, we require that each QD contains an even number of electrons with a spin-singlet ground state. A more detailed analysis of such a parameter regime is given in [@P3RSL] and is stated here $$\label{regime} \Delta,U, \delta\epsilon>\Delta\mu>\gamma_{l},k_{B}T,{\rm and}\,\,\gamma_{l}>\gamma_{S}.$$ In (\[regime\]) the rates for tunnelling of an electron from the SC to the QDs and from the QDs to the Fermi leads are given by $\gamma_{S}=2\pi\nu_{S}|T_{SD}|^{2}$ and $\gamma_{l}=2\pi\nu_{l}|T_{DL}|^{2}$, respectively, with $\nu_{S}$ and $\nu_{l}$ being the corresponding electron density of states per spin at the Fermi level. We consider asymmetric barriers $\gamma_{l}>\gamma_{s}$ in order to exclude correlations between subsequent Cooper pairs on the QDs. We work at the particular interesting resonance $\epsilon_{1},\epsilon_{2}\simeq \mu_{S}$, where the injection of the electrons into different leads takes place at the same orbital energy. This is a crucial requirement for the subsequent detection of entanglement via noise [@gde; @prl-cgd]. In this regime, we have calculated and compared the stationary charge current of two spin-entangled electrons for two competing transport channels in a T-matrix approach.\ The ratio of the desired current for two electrons tunnelling into [*different*]{} leads ($I_{1}$) to the unwanted current for two electrons into the [*same*]{} lead ($I_{2}$) is [@P3RSL] $$\label{final} \frac{I_{1}}{I_{2}}= \frac{4{\cal E}^2}{\gamma^2} \left[\frac{\sin(k_{F}\delta r)}{k_{F}\delta r}\right]^2\,e^{-2\delta r/\pi\xi}, \quad\quad\quad \frac{1}{{\cal E}}=\frac{1}{\pi\Delta}+\frac{1}{U},$$ where $\gamma = \gamma_1 + \gamma_2$. The current $I_1$ becomes exponentially suppressed with increasing distance $\delta r=|{\bf r}_1-{\bf r}_2|$ between the tunnelling points on the SC, on a scale given by the superconducting coherence length $\xi$ which is the size of a Cooper pair. This does not pose a severe restriction for conventional s-wave materials with $\xi$ typically being on the order of $\mu {\rm m}$. In the relevant case $\delta r<\xi$ the suppression is only polynomial $\propto 1/(k_{F}\delta r)^2$, with $k_{F}$ being the Fermi wave vector in the SC. On the other hand, we see that the effect of the QDs consists in the suppression factor $(\gamma/{\cal E})^2$ for tunnelling into the same lead [@cost]. Thus, in addition to Eq. (\[regime\]) we have to impose the condition $k_F\delta r < {\cal E}/\gamma$, which can be satisfied for small dots with ${\cal E}/\gamma\sim 100$ and $k_F^{-1}\sim 1\, {\rm \AA}$. As an experimental probe to test if the two spin-entangled electrons indeed separate and tunnel to different leads we suggest to join the two leads 1 and 2 to form an Aharonov-Bohm loop. In such a setup the different tunnelling paths of an Andreev process from the SC via the dots to the leads can interfere. As a result, the measured current as a function of the applied magnetic flux $\phi$ threading the loop contains a phase coherent part $I_{AB}$ which consists of oscillations with periods $h/e$ and $h/2e$ [@P3RSL] $$\label{AB-osc} I_{AB}\sim \sqrt{8I_{1}I_{2}}\cos(\phi/\phi_{0})+I_{2}\cos(2\phi/\phi_{0}),$$ with $\phi_{0}=h/e$ being the single-electron flux quantum. The ratio of the two contributions scales like $\sqrt{I_{1}/I_{2}}$ which suggest that by decreasing $I_{2}$ (e.g. by increasing $U$) the $h/2e$ oscillations should vanish faster than the $h/e$ ones. We note that the efficiency as well as the absolute rate for the desired injection of two electrons into different leads can even be enhanced by using lower dimensional SCs [@P8RL; @P9JS] . In two dimensions (2D) we find that $I_{1}\propto 1/k_{F}\delta r$ for large $k_{F}\delta r$, and in one dimension (1D) there is no suppression of the current and only an oscillatory behavior in $k_{F}\delta r $ is found. A 2D-SC can be realized by using a SC on top of a two-dimensional electron gas (2DEG) [@P4Klapwijk], where superconducting correlations are induced via the proximity effect in the 2DEG. In 1D, superconductivity was found in ropes of single-walled carbon nanotubes [@P5Bouchiat]. Finally, we note that the coherent injection of Cooper pairs by an Andreev process allows the detection of individual spin-entangled electron pairs in the leads. The delay time $\tau_{\rm delay}$ between the two electrons of a pair is given by $1/\Delta$, whereas the separation in time of subsequent pairs is given approximately by $\tau_{\rm pairs} \sim 2e/I_{1}\sim \gamma_{l}/\gamma_{S}^{2}$ (up to geometrical factors) [@P3RSL]. For $\gamma_{S}\sim\gamma_{l}/10\sim 1\mu {\rm eV}$ and $\Delta\sim 1{\rm meV}$ we obtain that the delay time $\tau_{\rm delay}\sim 1/\Delta\sim 1{\rm ps}$ is much smaller than the delivery time $\tau_{\rm pairs}$ per entangled pair $2e/I_{1}\sim 40{\rm ns}$. Such a time separation is indeed necessary in order to detect individual pairs of spin-entangled electrons. ### Andreev entangler with Luttinger-liquid leads {#sssluttinger} Next we discuss a setup with an s-wave SC weakly coupled to the center (bulk) of two separate one-dimensional leads (quantum wires) 1,2 (see Fig. \[LLfig\]) which exhibit Luttinger liquid (LL) behavior, such as carbon nanotubes [@P6Bockrath; @P10Egger; @P11Kane]. The leads are assumed to be infinitely extended and are described by conventional LL-theory [@P7rev]. Interacting electrons in one dimension lack the existence of quasi particles like they exist in a Fermi liquid and instead the low energy excitations are collective charge and spin modes. In the absence of backscattering interaction the velocities of the charge and spin excitations are given by $u_{\rho}=v_{F}/K_{\rho}$ for the charge and $u_{\sigma}=v_{F}$ for the spin, where $v_{F}$ is the Fermi velocity and $K_{\rho}<1$ for repulsive interaction between electrons ($K_{\rho}=1$ corresponds to a 1D-Fermi gas). As a consequence of this non-Fermi liquid behavior, tunnelling into a LL is strongly suppressed at low energies. Therefore one should expect additional interaction effects in a two-particle tunnelling event (Andreev process) of a Cooper pair from the SC to the leads. We find that strong LL-correlations result in an additional suppression for tunnelling of two coherent electrons into the [*same*]{} LL compared to single electron tunnelling into a LL if the applied bias voltage $\mu$ between the SC and the two leads is much smaller than the energy gap $\Delta$ of the SC. To quantify the effectiveness of such an entangler, we calculate the current for the two competing processes of tunnelling into different leads ($I_{1}$) and into the same lead ($I_{2}$) in lowest order via a tunnelling Hamiltonian approach. Again we account for a finite distance separation $\delta r$ between the two exit points on the SC when the two electrons of a Cooper pair tunnel to different leads. For the current $I_{1}$ of the desired pair-split process we obtain, in leading order in $\mu/\Delta$ and at zero temperature [@P8RL; @P9JS] $$\label{LL1} I_{1}=\frac{I_{1}^{0}}{\Gamma(2\gamma_{\rho}+2)}\frac{v_{F}}{u_{\rho}} \left[\frac{2\Lambda\mu}{u_{\rho}}\right]^{2\gamma_{\rho}}, \,\,I_{1}^{0}=\pi e\gamma^{2}\mu F_{d}[\delta r],$$ where $\Gamma (x)$ is the Gamma function and $\Lambda$ is a short distance cut-off on the order of the lattice spacing in the LL and $\gamma=4\pi\nu_{S}\nu_{l}|t_{0}|^{2}$ is the dimensionless tunnel conductance per spin with $t_{0}$ being the bare tunnelling amplitude for electrons to tunnel from the SC to the LL-leads (see Fig. \[LLfig\]). The electron density of states per spin at the Fermi level for the SC and the LL-leads are denoted by $\nu_{S}$ and $\nu_{l}$, respectively. The current $I_{1}$ has its characteristic non-linear form $I_{1}\propto \mu^{2\gamma_{\rho}+1}$ with $\gamma_{\rho}=(K_{\rho}+K_{\rho}^{-1})/4-1/2>0$ being the exponent for tunnelling into the bulk of a [*single*]{} LL. The factor $F_{d}[\delta r]$ in (\[LL1\]) depends on the geometry of the device and is given here again by $F_{d}[\delta r]=[\sin(k_{F}\delta r)/k_{F}\delta r]^{2}\exp(-2\delta r/\pi\xi)$ for the case of a 3D-SC. In complete analogy to subsection \[ssecAndreev\] the power law suppression in $k_{F}\delta r$ gets weaker in lower dimensions. This result should be compared with the unwanted transport channel where two electrons of a Cooper pair tunnel into the same lead 1 or 2 but with $\delta r=0$. We find that such processes are indeed suppressed by strong LL-correlations if $\mu<\Delta$. The result for the current ratio $I_{2}/I_{1}$ in leading order in $\mu/\Delta$ and for zero temperature is [@P8RL; @P9JS] $$\frac{I_{2}}{I_{1}}=F_{d}^{-1}[\delta r]\sum\limits_{b=\pm 1}\,A_{b}\,\left(\frac{2\mu}{\Delta}\right)^{2\gamma_{\rho b}},\,\,\gamma_{\rho +}=\gamma_{\rho},\,\,\gamma_{\rho -}=\gamma_{\rho}+(1-K_{\rho})/2, \label{currentI222}$$ where $A_{b}$ is an interaction dependent constant [@LLfootnote]. The result (\[currentI222\]) shows that the current $I_{2}$ for injection of two electrons into the same lead is suppressed compared to $I_{1}$ by a factor of $(2\mu/\Delta)^{2\gamma_{\rho +}}$, if both electrons are injected into the same branch (left or right movers), or by $(2\mu/\Delta)^{2\gamma_{\rho -}}$ if the two electrons travel in different directions [@P21electronbunching]. The suppression of the current $I_{2}$ by $1/\Delta$ reflects the two-particle correlation effect in the LL, when the electrons tunnel into the same lead. The larger $\Delta$, the shorter the delay time is between the arrivals of the two partner electrons of a Cooper pair, and, in turn, the more the second electron tunnelling into the same lead will feel the existence of the first one which is already present in the LL. This behavior is similar to the Coulomb blockade effect in QDs, see subsection \[ssecAndreev\]. Concrete realizations of LL-behavior is found in metallic carbon nanotubes with similar exponents as derived here [@P10Egger; @P11Kane]. In metallic single-walled carbon nanotubes $K_{\rho}\sim 0.2$ [@P6Bockrath] which corresponds to $2\gamma_{\rho}\sim 1.6$. This suggests the rough estimate $(2\mu/\Delta)<1/k_{F}\delta r$ for the entangler to be efficient. As a consequence, voltages in the range $k_{B}T<\mu<100 \mu {\rm eV}$ are required for $\delta r\sim$ nm and $\Delta\sim 1{\rm meV}$. In addition, nanotubes were reported to be very good spin conductors [@P12Balents] with estimated spin-flip scattering lengths of the order of $\mu {\rm m}$ [@P13BVBF]. We remark that in order to use the beam-splitter setup to detect spin-entanglement via noise the two LL-leads can be coupled further to Fermi liquid leads. In such a setup the LL-leads then would act as QDs [@spinexc]. Another way to prove spin-entanglement is to carry out spin-dependent current-current correlation measurements between the two LLs. Such spin dependent currents can be measured e.g. via spin filters (Sec. \[spinfilters\]). Triple-quantum dot entangler {#triple-dot-entangler} ---------------------------- In this proposal [@Sar02], the pair of spin-entangled electrons is provided by the ground state of a single quantum dot [$ D_C \ $]{}with an even number of electrons, which is the spin-singlet [@singlet]; see Fig. \[figset\]. In the Coulomb blockade regime [@Kou97], electron interactions in each dot create a large charging energy $U$ that provides the energy filtering necessary for the suppression of the non-entangled currents. These arise either from the escape of the pair to the same lead, or from the transport of a single electron. The idea is to create a resonance for the joint transport of the two electrons from [$ D_C \ $]{}to secondary quantum dots [$ D_L \ $]{}and $D_R$, similarly to the resonance described in Sec. \[ssecAndreev\] . For this, we need the condition $ {\epsilon_L}+{\epsilon_R}=2 {\epsilon_C}$, where $ {\epsilon_L}$ and $ {\epsilon_R}$ are the energy levels of the available state in [$ D_L \ $]{}and $D_R$, and $2 {\epsilon_C}$ is the total energy of the two electrons in $D_C$. On the other hand, the transport of a single electron from [$ D_C \ $]{}to [$ D_L \ $]{}or [$ D_R \ $]{}is suppressed by the energy mismatch $ {\epsilon_C}\pm U \neq {\epsilon_L}, {\epsilon_R}$, where $ {\epsilon_C}\pm U $ is the energy of the $ 2^{\rm nd} / 1^{\rm st} $ electron in [$ D_C \ $]{}[@charging]. We describe the incoherent sequential tunneling between the leads and the dots in terms of a master equation [@Blu96] for the density matrix $ \rho$ of the triple-dot system (valid for $k_{\rm B}T > \gamma$). The stationary solution of the master equation is found with MAPLE, and is used to define stationary currents. Besides the entangled current $I_E$ coming from the [*joint*]{} transport of the electrons from [$ D_C \ $]{}to [$ D_L \ $]{}and $D_R$, the solitary escape of one electron of the singlet can create a non-entangled current $ I_1 $, as it could allow a new electron coming from the source lead to form a new spin-singlet with the remaining electron. Another non-entangled current $ \tilde{I}_1 $ can be present if only one electron is transported across the triple-dot system; see Fig. \[figset\](b). The definition of entangler [*qualities*]{} $ Q=I_E/I_1 $ and $ \tilde{Q}=I_E/\tilde{I_1} $ enables us to check the suppression of these non-entangled currents. In Fig. \[figres\] we present results in the case where $ {\epsilon_R}={\epsilon_C}$. This gives a two-electron resonance at $ {\epsilon_L}={\epsilon_C}={\epsilon_R}$, and create mobile entangled electrons with the same orbital energy, as required in the beam-splitter setup to allow entanglement detection [@gde], [@prl-cgd]. The exact analytical expressions are extremely lengthy, but we can get precise conditions for an efficient entangler regime by performing a Taylor expansion in terms of $ \alpha ,\gamma ,T_{0} $ (defined in Fig. \[figset\]). Introducing the conditions $ Q,\widetilde{Q}>Q^{\mathrm{min}}_{\mathrm{I}} $ away from resonance ($ {\epsilon_L}\neq {\epsilon_C}$) and $ Q,\widetilde{Q}>Q^{\mathrm{min}}_{\mathrm{II}} $ at resonance ($ {\epsilon_L}= {\epsilon_C}$), we obtain the conditions [@Sar02] $$\begin{aligned} && | {\epsilon_L}-{\epsilon_C}|< 2T_{0}/\sqrt{Q^{\mathrm{min}}_{\mathrm{I}}} \ , \label{qual1}\\ &&\gamma \sqrt{Q^{\mathrm{min}}_{\mathrm{II}}/8}<T_{0}<U\sqrt{4\alpha / \gamma Q_{\mathrm{II}}^{\mathrm{min}}}. \label{qual2}\end{aligned}$$ We need a large $ U $ for the energy suppression of the one-electron transport, and $ \gamma \ll T_0 $ because the joint transport is a higher-order process in $T_0$. The current saturates to $ I_E \to e \alpha $ when $ T^{4}_{0}\gg U^{2}\gamma \alpha /32 $ \[see Fig. \[figres\](d)\] when the bottleneck process is the tunneling of the electrons from the source lead to the central dot. We see in (c) that equal currents in the left and right drain lead, $ I_{L} = I_R$, are characteristic of the resonance $ {\epsilon_L}= {\epsilon_C}$, which provides an experimental procedure to locate the efficient regime. Taking realistic parameters for quantum dots [@Kou97; @Exp] such as $ I_E = 20 $ pA, $ \alpha =0.1$ $\mu$eV and $ U=1 $ meV, we obtain a maximum entangler quality $ Q_{\mathrm{II}}^{\mathrm{min}}=100 $ at resonance, and a finite width $ | {\epsilon_L}-{\epsilon_C}|\simeq 6$ $\mu$eV where the quality is at least $ Q_{\mathrm{I}}^{\mathrm{min}}=10 $. Note that one must avoid resonances with excited levels which could favour the undesired non-entangled one-electron transport. For this, one can either tune the excited levels away by applying a magnetic field, or require a large energy levels spacing $ \Delta \epsilon _{i} \simeq 2 U$, which can be found in vertical quantum dots or carbon nanotubes [@Kou97]. We can estimate the relevant timescales by simple arguments. The entangled pairs are delivered every $ \tau_{\mathrm{pairs}}\simeq 2/\alpha \simeq 13\, \mathrm{ns} $. The average separation between two entangled electrons within one pair is given by the time-energy uncertainty relation: $ \tau_{\mathrm{delay}}\simeq 1/U\simeq 0.6\, \mathrm{ps} $, while their maximal separation is given by the variance of the exponential decay law of the escape into the leads: $ \tau_{\mathrm{max}}\simeq 1/\gamma \simeq 0.6\, \mathrm{ns} $. Note that $ \tau_{\mathrm{delay}} $ and $ \tau_{\mathrm{max}} $ are both well below reported spin decoherence times (in bulk) of $ 100\, \mathrm{ns} $ [@Kik97]. Spin-polarized electron sources {#spinfilters} =============================== Here we briefly mention some of the possibilities for spin-polarized electron sources possibly relevant as feeding Fermi-liquid reservoirs to our beam-splitter configuration. Even though we are concerned here with mesoscopic *coherent* transport, we emphasize that the electron sources themselves can be diffusive or ballistic. Currently, there is a great deal of interest in the problem of spin injection in hybrid mesoscopic structures. At the simplest level we can say that the “Holy Grail” here is essentially the ability to spin inject *and* detect spin-polarized charge flow across interfaces. The possibility of controlling and manipulating the degree of spin polarization of the flow is highly desirable. This would enable novel spintronic devices with flexible/controllable functionalities. Recently, many different experimental possibilities for spin injection/detection have been considered: (i) all-optical [@irina] and (ii) all-electrical [@johnsson], [@jedema] spin injection and detection in semiconductors and metal devices, respectively, and (iii) electric injection with optical detection in hybrid (Mn-based) ferromagnetic/non-magnetic and paramagnetic/non-magnetic semiconductor *pin* diodes [@ohno-molen]. For an account of the experimental efforts currently underway in the field of spin-polarized transport, we refer the reader to Ref. [@als-review]. Below we focus on our proposals for spin filtering with a semimagnetic tunnel barrier [@egues] and a quantum dot [@patrik]. These can, in principle, provide alternative schemes for spin injection into our beam splitter. Quantum spin filtering ---------------------- Ballistic Mn-based tunnel junctions [@egues] offer an interesting possibility for generating spin-polarized currents. Here the s-d interaction in the paramagnetic layer gives rise to a spin-dependent potential. An optimal design can yield high barriers for spin-up and vanishingly small barriers for spin-down electrons. Hence, a highly spin-selective tunnel barrier can be achieved in the presence of an external magnetic field. Note that here *ballistic spin filtering* – due to the blocking of one spin component of the electron flow – is the relevant mechanism for producing a spin-polarized current. Earlier calculations have shown that full spin polarizations are attainable in ZnSe/ZnMnSe spin filters [@egues]. Quantum dots as spin filters ---------------------------- Spin polarized currents can also be generated by a quantum dot [@patrik]. In the Coulomb blockade regime with Fermi-liquid leads, it can be operated as an efficient spin-filter [@patrik-sf-memory] at the single electron level. A magnetic field lifts the spin degeneracy in the dot while its effect is negligible [@patrik-sf-filter] in the leads. As a consequence, only one spin direction can pass through the quantum dot from the source to the drain. The transport of the opposite spin is suppressed by energy conservation and singlet-triplet splitting. This filtering effect can be enhanced by using materials with different g-factors for the dot and the lead. To increase the current signal, one could also use an array of quantum dots, e.g. self-assembled dots. spin filters for spin detection and Bell inequalities {#bell} ----------------------------------------------------- Besides being a source of spin-polarized currents, such spin filters (with or without spin-polarized sources [@patrik],[@hanres]) could be used to measure electron spin, as they convert spin information into charge: the transmitted charge current depends on the spin direction of the incoming electrons [@ld]. Such filters could probe the degree of polarization of the incoming leads. In addition, Bell inequalities measurements could be performed with such devices [@P14Kawabata; @P15Martin]. Scattering formalism: basics ============================ *Current.* In a multi-probe configuration with incoming and outgoing leads related via the scattering matrix $\mathbf{s}_{\gamma \beta } $, the current operator in lead $\gamma $ within the Landauer-Büttiker [@buttiker] approach is given by $$\begin{aligned} \hat{I}_{\gamma }(t) &=&\frac{e}{h}\sum_{\alpha \beta }\!\!\int \!\!d\varepsilon d\varepsilon ^{\prime}e^{i(\varepsilon -\varepsilon^{\prime})t/\hbar }\mathbf{a}_{\alpha }^{\dagger }(\varepsilon )\mathbf{A}_{\alpha ,\beta }(\gamma ;\varepsilon ,\varepsilon ^{\prime}) \mathbf{a}_{\beta }(\varepsilon ^{\prime}), \nonumber \\ &&\mathbf{A}_{\alpha \beta }(\gamma ;\varepsilon ,\varepsilon^{\prime})=\delta _{\gamma \alpha }\delta _{\gamma \beta }\mathbf{1}-\mathbf{s}_{\gamma \alpha }^{\dagger }(\varepsilon )\mathbf{s}_{\gamma \beta }(\varepsilon ^{\prime }), \label{ceq1}\end{aligned}$$ where we have defined the two-component object $ \mathbf{a}^\dagger_{\alpha } (\varepsilon)=(a^\dagger_{\alpha, \uparrow} (\varepsilon), a^\dagger_{\alpha, \downarrow} (\varepsilon))$ with $ a^\dagger_{\alpha, \sigma} (\varepsilon)$ denoting the usual fermionic creation operator for an electron with energy $ \varepsilon $ and spin component $\sigma=\uparrow,\downarrow $ in lead $ \alpha $. Here the spin components $\sigma $ are along a properly defined quantization axis (e.g., $x$, $y$ or $z$). *Noise.* Let $\delta \hat{I}_{\gamma }(t)= \hat{I}_{\gamma }(t) - \langle I \rangle$ denote the current-fluctuation operator at time $t$ in lead $\gamma$ ($\langle I \rangle$: average current). We define noise between leads $\gamma$ and $\mu$ in a multi-terminal system by the average power spectral density of the symmetrized current-fluctuation autocorrelation function [@factor-of-two] $$S_{\gamma \mu }(\omega )=\frac{1}{2}\int \langle \delta \hat{I}_{\gamma }(t)\delta \hat{I}_{\mu }(t^{\prime })+\delta \hat{I}_{\mu }(t^{\prime })\delta \hat{I}_{\gamma }(t)\rangle e^{i\omega t}dt. \label{ceq2}$$ The angle brackets in Eq. (\[ceq2\]) denote either an ensemble average or an expectation value between relevant pairwise electron states. We focus on noise at zero temperatures. In this regime, the current noise is solely due to the discreteness of the electron charge and is termed *shot noise*. Scattering matrix ----------------- *Electron beam splitter.* This device consists of four quasi one-dimensional leads (point contacts) electrostatically defined on top of a 2DEG [@bs], [@henny]. An extra “finger gate” in the central part of the device acts as a potential barrier for electrons traversing the system, i.e., a “beam splitter”. That is, an impinging electron from, say, lead 1 has probability amplitudes $r$ to be reflected into lead 3 and $t$ to be transmitted into lead 4. *Beam splitter $\mathbf{s}$ matrix.* The transmission processes at the beam splitter can be suitably described in the language of the scattering theory: $s_{13}=s_{31}=r$ and $s_{14}=s_{41}=t$; similarly, $s_{23}=s_{32}=t$ and $s_{24}=s_{42}=r$, see Fig. \[fig2c\]. We also neglect backscattering into the incoming leads, $s_{12}=s_{34}=s_{\alpha\alpha}=0$. Note that the beam splitter $\mathbf{s}$ matrix is spin independent; this no longer holds in the presence of a spin-orbit interaction. We also assume that the amplitudes $r$ and $t$ are energy independent. The unitarity of $\mathbf{s}$ implies $|r|^2+|t|^2=1$ and $\mathrm{Re}(r^\ast t)=0$. Below we use the above scattering matrix to evaluate noise. Noise of entangled electron pairs: earlier results {#earlier-results} ================================================== *Singlet and triplets.* Let us assume that an entangler is now “coupled” to the beam-splitter device so as to inject entangled (and unentangled) electron pairs into the incoming leads, Fig. \[fig2c\]. This will certainly require some challenging lithographic patterning and/or elaborate gating structures. Let us consider the following two-electron states $$|S\rangle =\frac{1}{\sqrt{2}}\left[ a_{1\uparrow }^{\dagger }(\varepsilon _{1})a_{2\downarrow }^{\dagger }(\varepsilon _{2})-a_{1\downarrow }^{\dagger }(\varepsilon _{1})a_{2\uparrow }^{\dagger }(\varepsilon _{2})\right] |0\rangle ,\label{ceq4}$$ $$|Te\rangle =\frac{1}{\sqrt{2}}\left[ a_{1\uparrow }^{\dagger }(\varepsilon _{1})a_{2\downarrow }^{\dagger }(\varepsilon _{2})+a_{1\downarrow }^{\dagger }(\varepsilon _{1})a_{2\uparrow }^{\dagger }(\varepsilon _{2})\right] |0\rangle , \label{ceq5}$$ and $$|Tu_{\sigma }\rangle =a_{1\sigma }^{\dagger }(\varepsilon _{1})a_{2\sigma }^{\dagger }(\varepsilon _{2})|0\rangle , \qquad \sigma =\uparrow , \downarrow . \label{ceq6}$$ The above states correspond to the singlet $|S\rangle$, the entangled triplet $|Te\rangle $, and the unentangled triplets $|Tu_{\sigma }\rangle$, respectively, injected electron pairs. Note that $|0\rangle$ denotes the “lead vacuum”, i.e., an empty lead or a Fermi sea. Here we follow Ref. [@gde] and assume that the injected pairs have *discrete* energies $\varepsilon_{1,2}$. To determine the average current and shot noise for electron pairs we have to calculate the expectation value of the noise two-particle states in Eqs. (\[ceq4\])-(\[ceq6\]). In the limit of zero bias, zero temperature, and zero frequency, we find [@gde] $$S_{33}^{S/Te,u_{\sigma }}=\frac{2e^{2}}{h\nu }T(1-T)(1\pm \delta _{\varepsilon _{1},\varepsilon _{2}}), \label{ceq7}$$ for the shot noise in lead 3 for singlet (upper sign) and triplets (lower sign) with $T\equiv |t|^2$ (transmission coefficient). The corresponding currents in lead 3 are $I_3^{S,Te,u_{\sigma }}=I=\frac{e}{h\nu }$. Note the density of states factor $\nu$ in Eqs. (\[ceq7\]) arising from the discrete spectrum used [@discrete]. *Bunching and antibunching.* For $\varepsilon _{1}=\varepsilon _{2}$ the Fano factors corresponding to the shot noise in Eq. (\[ceq7\]) are $F^{S}=S_{33}^S/eI=4T(1-T)$, for the singlet and $F^{Te,u_{\sigma }}=0$, for all three triplets. Interestingly, the Fano factor for a singlet pair is enhanced by a factor of two as compared to the Fano factor $2T(1-T)$ for a *single* uncorrelated electron beam [@fu] impinging on the beam splitter; the Fano factor for the triplets is suppressed with respect to this uncorrelated case. This enhancement of $F^S$ and suppression of $F^{Te,u_{\sigma }}$ is due to *bunching* and *antibunching*, respectively, of electrons in the outgoing leads. This result offers the possibility of distinguishing singlet from triplet states via noise measurements (triplets cannot be distinguished among themselves here; a further ingredient is needed for this, e.g., a local Rashba interaction in one of the incoming leads). Electron transport in the presence of a *local* Rashba s-o interaction ====================================================================== The central idea here is to use the gate-controlled Rashba coupling to rotate the electron spins [@datta-das] traversing the Rashba-active region (lead 1 of the beam splitter), thus altering in a controllable way the resulting transport properties of the system. Below we first discuss the effects of the Rashba s-o interaction in one-dimensional systems; the incoming leads are essentially quasi one-dimensional wires, i.e, “quantum point contacts”. A local Rashba interaction can in principle be realized with an additional gating structure (top and back gates [@gatecontrol]). We focus on wires with one and two transverse channels [@moroz]. This latter case allows us to study the effects of s-o induced interband coupling on both current and shot noise. Rashba wires with uncoupled transverse channels ----------------------------------------------- ### Hamiltonian, eigenenergies and eigenvectors The Rashba spin-orbit interaction is present in low-dimensional systems with *structural* inversion asymmetry. Roughly speaking, this interaction arises from the gradient of the confining potential (“triangular shape”) at the interface between two different materials [@electric-field]. For a non-interacting one-dimensional wire with *uncoupled* transverse channels, the electron Hamiltonian in the presence of the Rashba coupling $\alpha$ reads [@rashba] $$H_n=-\frac{\hbar^2}{2m^{\ast}}\partial^2_x+ \epsilon_n + i\alpha\sigma_y \partial_x. \label{ceq11}$$ In Eq. (\[ceq11\]) $\partial_x\equiv\partial/\partial x$, $\sigma_y$ is the Pauli matrix, $m^{\ast}$ is the electron effective mass, and $\epsilon_n$ is the bottom of the n$^{th}$-channel energy band in absence of s-o interaction. For an infinite-barrier transverse confinement of width $w$, $\epsilon_n=n^2\pi^2 \hbar^2/(2mw^2)$. The Hamiltonian in (\[ceq11\]) yields the usual set of Rashba bands [@molen-flesh] $$\varepsilon^n_s =\hbar^{2}(k - sk_{R})^{2}/2m^{\ast} + \epsilon_n -\epsilon_{R}, \qquad s=\pm \label{ceq12}$$ where $k_{R}=m^{\ast}\alpha/\hbar^2$ and $\epsilon _{R}=\hbar ^{2}k_{R}^{2}/2m^{\ast }=m^{\ast}\alpha^2/2\hbar^2$ (“Rashba energy”). The corresponding wave functions are eigenvectors of $\sigma_y$ with the orbital part being a plane wave times the transverse-channel wave function. Figure \[fig3c\] shows that the parabolic bands are shifted sideways due to the Rashba interaction. Note that these bands are still identified by a unique spin index $s=\pm$ which in our convention corresponds to the eigenspinors $|\mp \rangle \sim |\uparrow\rangle \mp | \downarrow \rangle$ of $\sigma_y$. ### Boundary conditions and spin injection {#one-band-injection} Here we assume a unity transmission across the interface [@unity] depicted in Fig. \[fig3c\]. For a spin-up electron with wave vector $k_F$ entering the Rashba region at $x=0$, we have the following boundary conditions for the wave function and its derivative [@molen-flesh; @bc] $$|\uparrow \rangle e^{ik_Fx}|_{x\rightarrow 0^-} = \frac{1}{\sqrt{2}}[|+ \rangle e^{ik_2x}+|- \rangle e^{ik_1x}]_{x\rightarrow 0^+}, \label{ceq13}$$ and $$|\uparrow \rangle v_{k_F} e^{ik_Fx}|_{x\rightarrow 0^-} = \frac{1}{\sqrt{2}}[|+ \rangle v^R_{k_2}e^{ik_2x}+|- \rangle v^R_{k_1}e^{ik_1x}]_{x\rightarrow 0^+}, \label{ceq14}$$ with the Fermi and Rashba group velocities defined by $v_{k_F}=\hbar k_F/m^{\ast}$, $v^R_{k_1}=\frac{\hbar}{m^{\ast}}(k_1+k_R)$, and $v^R_{k_2}=\frac{\hbar}{m^{\ast}}(k_2-k_R)$. The wave vectors $k_1$ and $k_2$ are defined by the “horizontal” intersections with the Rashba bands $\varepsilon_{-}(k_1)= \varepsilon_{+}(k_2)$, see Fig. \[fig3c\]. This results in the condition $k_2-k_1=2k_R$ which implies that the Rashba group velocities are the same at these points: $v^R_{k_1}=v^R_{k_2}$. Equation (\[ceq14\]) is satisfied provided that [@molen-flesh] $$v_{k_F}=\frac{1}{2}(v^R_{k_1}+v^R_{k_2})=\frac{\hbar}{m^{\ast}} \sqrt{\frac{2m^{\ast}}{\hbar^2}(\varepsilon_F+\epsilon_R)}, \label{ceq18}$$ where the last equality follows from conservation of energy, $\varepsilon_{-}(k_1)= \varepsilon_{+}(k_2)=\varepsilon_F$. Note that the group velocity of the incoming spin-up electron is completely “transferred” to the Rashba states at the interface. *Spin-rotated state at $x=L$.* For an incoming spin-up electron, we have at the exit of the Rashba region the spin-rotated state $$\psi_{\uparrow,L} =\frac{1}{\sqrt{2}}[|+ \rangle e^{ik_2L}+|- \rangle e^{ik_1L}], \label{ceq19}$$ which is consistent with the boundary conditions (\[ceq13\]) and (\[ceq14\]). After some straightforward manipulations (and using $k_2-k_1=2k_R$), we find $$\psi_{\uparrow,L} =\left( \begin{array}{c} \cos \theta _{R}/2 \\ \sin \theta _{R}/2 \end{array} \right) e^{i(k_1+k_R)L}, \label{ceq20}$$ with the usual Rashba angle $\theta_R=2m^{\ast}\alpha L/\hbar^2$ [@datta-das; @apl-cgd]. A similar expression holds for an incoming spin-down electron. Note that the boundary conditions at $x=L$ are trivially satisfied since we assume unity transmission. The overall phase of the spinor in Eq. (\[ceq20\]) is irrelevant for our purposes; we shall drop it from now on. ### Rashba spin rotator From the results of the previous section we can now define a unitary operator which describes the action of the Rashba-active region on any incoming spinor $$\mathbf{U_{R}}=\left( \begin{array}{cc} \cos \theta _{R}/2 & -\sin \theta _{R}/2 \\ \sin \theta _{R}/2 & \cos \theta _{R}/2\end{array} \right). \label{ceq21}$$ Note that all uncoupled transverse channels are described by the same unitary operator $\mathbf{U_{R}}$. The above unitary operator allows us to incorporate the s-o induced precession effect straightforwardly into the scattering formalism (Sec. \[beam-splitter-rashba\]). Rashba wire with two coupled transverse channels {#two-bands} ------------------------------------------------ The Rashba s-o interaction also induces a coupling between the bands described in the previous section. Here we extend our analysis to the case of two *weakly* coupled Rashba bands. ### Exact and approximate energy bands Projecting the two-dimensional Rashba Hamiltonian [@rashba] onto the basis of the two lowest uncoupled Rashba states, we obtain the quasi one-dimensional Hamiltonian [@apl-cgd] $$H=\left[ \begin{array}{cccc} \varepsilon _{+}^{a}(k) & 0 & 0 & -\alpha d \\ 0 & \varepsilon _{-}^{a}(k) & \alpha d & 0 \\ 0 & \alpha d & \varepsilon _{+}^{b}(k) & 0 \\ -\alpha d & 0 & 0 & \varepsilon _{-}^{b}(k)\end{array}\right] , \label{ceq22}$$where the interband coupling matrix element is $d\equiv \langle \phi _{a}(y)|\partial /\partial y|\phi _{b}(y)\rangle $ and $\phi_n(y)$ is the transverse channel wave function. Here we label the uncoupled Rashba states by $n=a,b$ [@index]. The Hamiltonian above gives rise to two sets of parabolic Rashba bands for zero interband coupling $d=0$. These bands are sketched in Fig. \[fig5c\] (thin lines). Note that the uncoupled Rashba bands cross. For positive $k$ vectors the crossing is at $k_{c}=(\epsilon _{b}-\epsilon _{a})/2\alpha$. For non-zero interband coupling $d\neq0$ the bands anti-cross near $k_c$ (see thick lines); this follows from a straightforward diagonalization of the 4x4 matrix in Eq. (\[ceq22\]). We are interested here in the *weak* interband coupling limit. In addition, we consider electron energies near the crossing; away from the crossing the bands are essentially uncoupled and the problem reduces to that of the previous section. In what follows, we adopt a perturbative description for the energy bands near $k_c$ which allows us to obtain analytical results. *“Nearly-free electron bands”.* In analogy to the usual nearly-free electron approach in solids [@am], we restrict the diagonalization of Eq. (\[ceq22\]) to the 2x2 central block which corresponds to the degenerate Rashba states crossing at $k_c$ $$\tilde{H}=\left[ \begin{array}{cc} \varepsilon _{-}^{a}(k) & \alpha d \\ \alpha d & \varepsilon _{+}^{b}(k)\end{array} \right]. \label{ceq23}$$ To lowest order we find $$\varepsilon _{\pm }^{\mathrm{approx}}(k)=\frac{\hbar ^{2}k^{2}}{2m}+\frac{1}{2}\epsilon _{b}+\frac{1}{2}\epsilon _{a}\pm \alpha d. \label{ceq24}$$ The corresponding eigenvectors are the usual linear combination of the *zeroth order* degenerate states at the crossing $$|\psi _{\pm }\rangle =\frac{1}{\sqrt{2}}\left[ |-\rangle _{a}\pm |+\rangle _{b}\right], \label{ceq25}$$where the ket sub-indices denote the respective (uncoupled) Rashba channel \[for simplicity, we omit the orbital part of the wave functions in (\[ceq25\])\]. ### Boundary conditions and spin injection near the crossing {#two-band-injection} Here we extend the analysis in Sec. \[one-band-injection\] to the case of two interband-coupled bands. We first determine the $k$ points corresponding to the “horizontal intersections” near the crossing at $k_c$, i.e., $k_{c1}$ and $k_{c2}$, see Fig. \[fig5c\]. We need these points since incoming spin-up electrons will be primarily injected into those states (and also into $k_2$, conservation of energy). By defining $k_{c1}=k_c - \Delta/2$ and $k_{c2}=k_c+ \Delta/2$ and then imposing $\varepsilon _{+}^{\mathrm{approx}}(k_{c1})=\varepsilon _{-}^{\mathrm{approx}}(k_{c2})$ (assumed $\sim \varepsilon _{F}$) we find, $$\Delta =\frac{2m\alpha d}{\hbar ^{2}k_{c}}=2\frac{k_{R}}{k_{c}}d\mathrm{.} \label{ceq26}$$ For a spin-up electron in the lowest wire state in the “no-Rashba” region (channel $a$), we can again write at $x=0$ [@unity] $$\begin{aligned} | &\uparrow & \rangle e^{ikx}|_{x\rightarrow 0^{-}}= \nonumber \\ &&\frac{1}{\sqrt{2}}\left\{ \frac{1}{\sqrt{2}}\left[ |\psi _{+}\rangle e^{ik_{c1}x}+|\psi _{-}\rangle e^{ik_{c2}x}\right] +|+\rangle _{a}e^{ik_{2}x}\right\} _{x\rightarrow 0^{+}}, \label{ceq26a}\end{aligned}$$ in analogy to Eq. (\[ceq13\]). Note that we only need to include three intersection points in the above “expansion” since the incoming spin-up electron is in channel $a$. Equation (\[ceq26a\]) satisfies the continuity of the wave function. The boundary condition for the derivative of the wave function is also satisfied provide that $\Delta/4\ll k_F$. This condition is readily fulfilled for realistic parameters (Sec. \[estimates\]). Hence, fully spin-polarized injection into the Rashba region is still possible in the presence of a weak interband coupling. Here we are considering a fully spin-polarized injector so that the intrinsic limitation due to the “conductivity mismatch” [@schmidt] is not a factor. *Generalized spin-rotated state at $x=L$.* Here again we can easily determine the form of the state at the exit of the Rashba region. For an incoming spin-up electron in the lowest band of the wire, we find $$\Psi _{\uparrow ,L}=\frac{1}{2}e^{i(k_{c}+k_{R})L}\left( \begin{array}{c} \cos (\theta _{d}/2)e^{-i\theta _{R}/2}+e^{i\theta _{R}/2} \\ -i\cos (\theta _{d}/2)e^{-i\theta _{R}/2}+ie^{i\theta _{R}/2} \\ -i\sin (\theta _{d}/2)e^{-i\theta _{R}/2} \\ \sin (\theta _{d}/2)e^{-i\theta _{R}/2}\end{array}\right). \label{ceq26d}$$ A similar state holds for a spin-down incoming electron. The state (\[ceq26d\]) satisfies the boundary conditions at $x=L$ (again, provided that $\Delta \ll 4k_F$. Equation (\[ceq26d\]) essentially tells us that a weak s-o interband coupling gives rise to an additional spin rotation (besides $\theta_R$) described by the mixing angle $\theta_d=\theta_R d/k_c$. This extra modulation enhances spin control in a Datta-Das spin-transistor geometry. In Ref. [@apl-cgd] we show that the spin-resolved current in this case is $$I_{\uparrow ,\downarrow }=\frac{e}{h}eV[1\pm \cos (\theta _{d}/2)\cos \theta _{R}], \label{new-dd}$$ where $V$ is the source-drain bias. Novel Beam-splitter geometry with a local Rashba interaction {#beam-splitter-rashba} ============================================================ Figure \[bs-fig1\] shows an schematic of our proposed beam-splitter geometry with a local Rashba-active region of length $L$ in lead 1. Below we discuss its scattering matrix in the absence of interband coupling. In this case, each set of Rashba bands can be treated independently. *Combined $\mathbf{s}$ matrices.* An electron entering the system through port 1, first undergoes a unitary Rashba rotation $\mathbf{U_{R}}$ in lead 1 then reaches the beam splitter which either reflects the electron into lead 3 or transmits it into lead 4. This happens for electrons injected into either the first or the second set of uncoupled Rashba bands. Since the Rashba spin rotation is unitary, we can combine the relevant matrix elements of the beam-splitter $\mathbf{s}$ matrix, connecting leads 1 and 3 ($s_{14}=s_{41}$) and 1 and 4 ($s_{14}=s_{41}$), with the Rashba rotation matrix $\mathbf{U_{R}}$ thus obtaining effective *spin-dependent* $2\times 2$ matrices of the form $\mathbf{s^R_{13}}=\mathbf{s^R_{31}}=s_{13}\mathbf{U_{R}}$ A similar definition holds for $\mathbf{s^R_{14}}=s_{41}\mathbf{U_{R}}=\mathbf{s^R_{41}}$. Note also that $\mathbf{s_{23}}=\mathbf{s_{32}}=t\mathbf{1}$ and $\mathbf{s_{24}}=\mathbf{s_{42}}=r\mathbf{1}$ since no Rashba coupling is present in lead 2. All the other matrix elements are zero. Hence the new effective beam-splitter $\mathbf{s}$ matrix which incorporates the effect of the Rashba interaction in lead 1 reads $$\mathbf{s}=\left( \begin{array}{cccc} \mathbf{0} & \mathbf{0} & \mathbf{s^R_{13}} & \mathbf{s^R_{14}} \\ \mathbf{0} & \mathbf{0} & \mathbf{s_{23}} & \mathbf{s_{24}} \\ \mathbf{s^R_{31}} & \mathbf{s_{32}} & \mathbf{0} & \mathbf{0} \\ \mathbf{s^R_{41}} & \mathbf{s_{42}} & \mathbf{0} & \mathbf{0} \end{array}\right). \label{ceq28}$$ Note that incorporating the s-o effects directly into the beam-splitter scattering matrix makes it spin dependent. The Rashba interaction does not introduce any noise in lead 1. This is so because the electron transmission coefficient through lead 1 is essentially unity [@unity]; a quantum point contact is noiseless for unity transmission. *Coupled Rashba bands.* The interband-coupled case can, in principle, be treated similarly. However, we follow a different simpler route to determine the shot noise in this case. We discuss this in more detail in Sec. \[alternate-scheme\]. Noise of entangled and spin-polarized electrons in the presence of a local Rashba spin-orbit interaction {#noise-rashba} ======================================================================================================== Starting from the noise definition in (Eq. \[ceq2\]), we briefly outline here the derivation of noise expressions for pairwise electron states (entangled and unentangled) and spin-polarized electrons (Secs. \[entanglers\] and \[spinfilters\]). For each of these two cases, we present results with and without s-o induced interband coupling. Shot noise for singlet and triplets ----------------------------------- ### Uncoupled Rashba bands: single modulation $\theta_R$ {#noise-rashba-uncoupled} To determine noise, we calculate the expectation value of the noise operator (Eq. \[ceq2\]) between pairwise electron states. We have derived shot noise expressions for both singlet and triplet states for a generic *spin-dependent* $\mathbf{s}$ matrix. Our results quite generally show that unentangled triplets and the entangled triplet display distinctive shot noise for spin-dependent scattering matrices. Below we present shot noise formulas for the specific case of interest here; namely, the beam-splitter scattering matrix in the presence of a local Rashba term \[Eq. (\[ceq28\])\]. In this case, for singlet and triplets defined along different quantization axes ($\hat{x}$ and $\hat{z}$ are equivalent directions perpendicular to the Rashba rotation axis $-\hat{y}$), we find $$S_{33}^{S}(\theta _{R})=\frac{2e^{2}}{h\nu }T(1-T)[1+\cos (\theta _{R})\delta _{\varepsilon _{1},\varepsilon _{2}}], \label{ceq31}$$ $$S_{33}^{Te_y}(\theta _{R})=\frac{2e^{2}}{h\nu }T(1-T)[1-\cos (\theta _{R})\delta _{\varepsilon _{1},\varepsilon _{2}}], \label{ceq32}$$ $$S_{33}^{Te_{z}}(\theta _{R})=S_{s-o}^{T_{u_y}}(\theta _{R})=\frac{2e^{2}}{h\nu }T(1-T)(1-\delta _{\varepsilon _{1},\varepsilon _{2}}), \label{ceq33}$$ and $$S_{33}^{Tu_{\uparrow }}(\theta _{R})=S_{33}^{Tu_{\downarrow }}(\theta _{R})=\frac{2e^{2}}{h\nu }T(1-T)[1-\cos ^{2}(\theta _{R}/2)\delta _{\varepsilon _{1},\varepsilon _{2}}]. \label{ceq34}$$ Equations (\[ceq32\])–(\[ceq34\]) clearly show that entangled and unentangled triplets present distinct noise as a functions of the Rashba phase. Note that for $\theta_R=0$, we regain the formulas in Sec. \[earlier-results\]. Figure \[fano-b\] shows the “reduced” Fano factor $f=F/2T(1-T)$, $F=S_{33}/eI$ (here $I=e/h\nu$), as a function of the Rashba angle $\theta_R$ for the noise expressions (\[ceq31\])–(\[ceq34\]). We clearly see that singlet and triplet pairs exhibit distinct shot noise in the presence of the s-o interaction. The singlet $S$ and entangled (along the Rashba rotation axis $\hat y$) triplet $Te_y$ pairs acquire an oscillating phase in lead 1 thus originating intermediate degrees of bunching/antibunching (solid and dotted lines, respectively). Triplet states (entangled and unentangled) display distinctive noise as a function of the Rashba phase, e.g., $T{e_y}$ is noisy and $T{u_y}$ is noiseless. Hence entangled and unentangled triplets can also be distinguished via noise measurements. Note that for $\theta_R=0$ all three triplets exhibit identically zero noise \[see Eq.(\[ceq7\])\]. ### Interband-coupled Rashba bands: additional modulation $\theta_d$ {#alternate-scheme} Here we determine noise for injected pairs with energies near the crossing $\varepsilon(k_c)$ using an alternate scheme. We calculate the relevant expectation values of the noise by using pairwise states defined from the generalized spin-rotated state in Eq. (\[ceq26d\]) and its spin-down counterpart. Since these states already incorporate all the relevant effects (Rashba rotation and interband mixing), we can calculate noise by using the “bare” beam splitter matrix elements, generalized to account for two channels. The beam-splitter does not mix transverse channels; hence this extension is trivial, i.e., block diagonal in the channel indices. This approach was first developed in Ref. [@jsc-cgd]. *Rashba-evolved pairwise electron states.* The portion of an electron-pair wave function “propagating” in lead 1 undergoes the effects of the Rashba interaction: ordinary precession $\theta_R$ and additional rotation $\theta_d$. Using Eq. (\[ceq26d\]) (and its spin-down counterpart) we find the following states $$\begin{aligned} |S/Te_z\rangle_L &=&\frac{1}{2}[\cos (\theta _{d}/2)e^{-i\theta _{R}/2}+e^{i\theta _{R}/2}]\frac{|\uparrow \downarrow \rangle _{aa}\mp |\downarrow \uparrow \rangle _{aa}}{\sqrt{2}}+ \nonumber \\ &&\frac{1}{2}[-i\cos (\theta _{d}/2)e^{-i\theta _{R}/2}+ie^{i\theta _{R}/2}]\frac{|\downarrow \downarrow \rangle _{aa}\pm |\uparrow \uparrow \rangle _{aa}}{\sqrt{2}}+ \nonumber \\ &&\frac{1}{2}[-i\sin (\theta _{d}/2)e^{-i\theta _{R}/2}]\frac{|\uparrow \downarrow \rangle _{ba}\pm |\downarrow \uparrow \rangle _{ba}}{\sqrt{2}}+ \nonumber \\ &&\frac{1}{2}[\sin (\theta _{d}/2)e^{-i\theta _{R}/2}]\frac{|\downarrow \downarrow \rangle _{ba}\mp |\uparrow \uparrow \rangle _{ba}}{\sqrt{2}} \label{ceq36}.\end{aligned}$$ The notation $|T{e_z}\rangle_L$ and $|S\rangle_L$ emphasizes the type of injected pairs (singlets or triplets at $x=0$) propagating through the length $L$ of the Rashba-active region in lead 1. Similar expressions hold for $|Tu_{\uparrow,\downarrow} \rangle _{L}$. In addition, we use the shorthand notation $|\downarrow \uparrow \rangle _{ba} \equiv |\downarrow_{1b} \uparrow_{2a} \rangle$, denoting a pair with one electron in channel *b* of lead 1 and another in channel *a* of lead 2. Here we consider incoming pairs with $\hat z$ polarizations only. Despite the seemingly complex structure of the above pairwise states, they follow quite straightforwardly from the general state $\Psi _{\uparrow ,L}$ in (\[ceq26d\]) (and its counterpart $\Psi _{\downarrow ,L}$). For instance, the unentangled triplet $|Tu_\uparrow \rangle _{L}$ is obtained from the tensor product between $\Psi _{\uparrow ,L}$ \[which describes as electron crossing lead 1 (initially spin up and in channel *a*)\] and a spin-up state in channel $a$ of lead 2: $|Tu_\uparrow \rangle _{L} = |\Psi _{\uparrow ,L} \rangle \bigotimes |\uparrow \rangle_{2a}$. *Noise.* We can now use the above states to determine shot noise at the zero frequency, zero temperature, and zero applied bias. Using the shot-noise results of Sec. \[earlier-results\] (trivially generalized for two channels), we find for the noise in lead 3 $$\begin{aligned} S_{33}^{Tu_{\uparrow }}(\theta _{R},\theta _{d}) &=&S_{33}^{Tu_{\downarrow }}(\theta _{R},\theta _{d})=\frac{2e^{2}}{h\nu } T(1-T)\times \nonumber \\ && \left[ 1-\frac{1}{2}\left( 1 +\cos \frac{\theta _{d}}{2} \cos \theta _{R}-\frac{1}{2}\sin ^{2}\frac{\theta _{d}}{2}\right) \delta _{\varepsilon _{1},\varepsilon _{2}}\right], \qquad \label{ceq37}\end{aligned}$$ $$S_{33}^{Te_{z}}(\theta _{R},\theta _{d})=\frac{2e^{2}}{h\nu }T(1-T)\left[ 1-\frac{1}{2}\left( \cos ^{2}\frac{\theta _{d}}{2} +1\right) \delta _{\varepsilon _{1},\varepsilon _{2}}\right] , \label{ceq38}$$ and $$S_{33}^{S}(\theta _{R},\theta _{d})=\frac{2e^{2}}{h\nu }T(1-T)\left[ 1+\left( \cos \frac{\theta _{d}}{2}\cos \theta _{R}\right) \delta _{\varepsilon _{1},\varepsilon _{2}}\right]. \label{ceq39}$$ Equations (\[ceq37\])-(\[ceq39\]) describe shot noise only for injected pairs with energies near the crossing, say, within $\alpha d$ of $\varepsilon(k_c)$. Away from the crossing or for $d=0$, the above expressions reduce to those of Sec. \[noise-rashba-uncoupled\]. We can also define “reduced” Fano factors as before; the interband mixing angle $\theta_d$ further modulates the Fano factors. For conciseness, we present the angular dependence of the Fano factors in the next section. Shot noise for spin-polarized electrons {#noise-spin-pol} --------------------------------------- We have derived a general shot noise formula for the case of spin-polarized sources by performing the ensemble average in Eq. (\[ceq2\]) over appropriate thermal reservoirs. The resulting expression corresponds to the standard Landauer-Büttiker formula for noise with spin-dependent $\mathbf{s}$ matrices. Below we present results for the specific beam-splitter $\mathbf{s}$ matrix in (\[ceq28\]). ### Uncoupled-band case: single modulation $\theta_R$ For incoming leads with a degree of spin polarization $p$ and for the scattering matrix (\[ceq28\]), we find at zero temperatures $$S_{33}^{p} (\theta_R) = 2e I T(1-T)p \sin^{2}\frac{\theta _{R}}{2}, \label{ceq43}$$ where $I=2e^2V/[h(1+p)]$ is the average current in lead 3. The “reduced” Fano factor corresponding to Eq. (\[ceq43\]) is $f_{p} = p \sin^{2}(\theta _{R}/2)$. Figure \[fano-a\] shows $f_{p}$ as a function of the Rashba angle $\theta_R$. For spin polarized injection along the Rashba rotation axis ($-\hat y$) no noise results in lead 3. This is a consequence of the Pauli exclusion principle in the leads. Spin-polarized currents with polarization perpendicular to the Rashba axis exhibit sizable oscillations as a function of $\theta_R$. Full shot noise is obtained for $\theta_R=\pi$ since the spin polarization of the incoming flow is completely reversed within lead 1. *Probing/detecting spin-polarized currents.* Since *unpolarized* incoming beams in lead 1 and 2 yield zero shot noise in lead 3, the results shown in Fig. \[fano-a\] provide us with an interesting way to *detect* spin-polarized currents via their noise. In addition, noise measurements should also allow one to probe the direction of the spin-polarization of the injected current. *Measuring the s-o coupling.* We can express the s-o coupling constant in terms of the reduced Fano factor. For a fully spin-polarized beam ($p=1$), we have $$\alpha =\frac{\hbar ^{2}}{m^{\ast }L}\arcsin \sqrt{f_{p}}. \label{ceq46}$$ Equation (\[ceq46\]) provides a direct means of extracting the Rashba s-o coupling $\alpha$ via shot noise measurements. We can also obtain a similar expression for $\alpha$ from the unentangled triplet noise formula (\[ceq34\]). ### Interband-coupled case: extra modulation $\theta_d$ The calculation in the previous section can be extended to the interband-coupled case for electrons impinging near the anti crossing of the bands \[$\sim \varepsilon(k_c)$\]. Here we present a simple “back-of-the-envelope” derivation of the the shot noise for the fully spin-polarized current case $(p=1)$ from that of the spin-up unentangled triplet Eq. (\[ceq37\]). Here we imagine that the spectrum of the triplet $Tu_{\uparrow }$ forms now a continuum and integrate its noise expression (after making $\varepsilon_1=\varepsilon_2$) over some energy range to obtain the noise of a spin-polarized current. Assuming $T$ constant in the range ($\varepsilon_F,\varepsilon_F+eV$), we find to linear order in $eV$ $$S_{33}^{\uparrow }(\theta _{R},\theta _{d}) = eI T(1-T) \left( 1 -\cos \frac{\theta _{d}}{2} \cos \theta _{R}+\frac{1}{2}\sin ^{2}\frac{\theta _{d}}{2}\right). \label{ceq49}$$ Figures \[fano3d\](a) and \[fano3d\](b) illustrate the angular dependencies of the reduced Fano factors for both the spin-polarized case Eq. (\[ceq49\]) and that of the singlet Eq. (\[ceq39\]). Note that the further modulation $\theta_d$ due to interband mixing can drastically change the noise for both spin-polarized and entangled electrons. For the singlet pairs, for instance, it can completely reverse the bunching/antibunching features. Hence further control is gained via $\theta_d$ which can, in principle, be tuned independently of $\theta_R$ (see Sec. \[estimates\]). Realistic parameters: estimates for $\theta_R$ and $\theta_d$. {#estimates} -------------------------------------------------------------- We conclude this section by presenting some estimates for the relevant spin-rotation angles $\theta_R$ and $\theta_d$ for realistic system parameters. Let us assume, for the sake of concreteness, an infinite confining potential of width $w$. In this case, the transverse wire modes in absence of the Rashba interaction are quantized with energies $\epsilon_n=\hbar^2\pi^2 n^2/(2m^\ast w^2)$. Let us now set $\epsilon_b-\epsilon_a=3\hbar^2\pi^2/(2m^\ast w^2)=16\epsilon_R$ which is a “reasonable guess”. Since $\epsilon_R=m^\ast\alpha^2/2\hbar^2$, we find $\alpha=(\sqrt{3}\pi/4)\hbar^2/m^\ast w^2=3.45\times 10^{-11}$ eVm [@gatecontrol] (which yields $\epsilon_R\sim0.39$ meV) for $m^\ast=0.05m_0$ and $w=60$ nm. For the above choice of parameters, the energy at the crossing is $\epsilon^a_-(k_c)=\epsilon^b_+(k_c)=\epsilon(k_c)=24\epsilon_R\sim 9.36$ meV. Electrons with energies around this value are affected by the s-o interband coupling, i.e., they undergo the additional spin rotation $\theta_d$. The relevant wave vector at the crossing is $k_c=8\epsilon_R/\alpha$. Assuming the $L=69$ nm for the length of the Rashba channel, we find $\theta_R=\pi$ and $\theta_d=\theta_R d/k_c\sim \pi/2$ since $d/k_c=2/(3k_Rw)$ and $k_Rw=\sqrt{3}\pi/4\sim4/3$ for $\epsilon_b-\epsilon_a=16\epsilon_R$ which implies $d/k_c\sim 0.5$. The preceding estimates are conservative. We should point out that both $\theta_R$ and $\theta_d$ can, in principle, be varied independently via side gates. It should also be possible to “over rotate” $\theta_R$ (say, by using a larger $L$) and hence increase $\theta_d$. As a final point we note that $\Delta/4k_F \sim 0.05\ll 1$ \[$k_F$ is obtained by making $\varepsilon_F=\hbar^2k^2_F/2m^\ast=\epsilon(k_c)$\] which assures the validity of the boundary condition for the velocity operator. Relevant issues and outlook {#rel-issues} =========================== *Relevant time scales.* Typical parameters for a finite-size electron beam splitter (tunnel coupled to reservoirs) defined on a GaAs 2DEG are: a device size $L_0\sim 1$ $\mu$m, a Fermi velocity in the range $v_F\sim 10^4 - 10^5$ m/s and an orbital coherence length of $\sim 1$ $\mu$m [@bs]. These values lead to traversal times $\tau_t = L_0/v_F$ in the range $\sim 10-100$ ps; these are lower bounds for the actual dwell time $\tau_{\rm dwell}\sim 1/\gamma_R$ of the electrons in the beam splitter, where $\gamma_R$ is the tunnelling rate from the leads of the beam splitter to the reservoirs. Hence the electrons keep their orbital coherence across the beam-splitter at low temperatures. Moreover, long spin dephasing times in semiconductors ($\sim 100$ ns for bulk GaAs [@Kik97]) should allow the propagation of entangled electrons without loss of spin coherence. For the noise calculation with entangled/unentangled pairs, we have assumed discrete energy levels in the incoming leads. A “particle-in-a-box” estimate of the level spacing $\delta\varepsilon$ due to longitudinal quantization of the beam splitter leads yields $\delta\varepsilon \sim \hbar v_F/L_0\sim 0.01-0.1$ meV. The relevant broadening of these levels is given by the coupling $\gamma_R \ll \delta \varepsilon $, which justifies the discrete level assumption. Here we take $\gamma_R \lesssim \gamma \sim 1$ $\mu$eV, where $\gamma$ is the tunnelling rate from the entangler to the beam splitter (Sec. \[entanglers\]). In addition, the stationary state description we use requires that the electrons have enough time to “fill in” the extended states in the beam splitter before they leave to the reservoirs: $\tau_{\rm dwell} \gtrsim \tau_{\rm inj} \sim 1/\gamma$. Here $\tau_{\rm inj}$ is the injection time from the entangler to the beam splitter. To have well separated pairs of entangled electrons, we also need $\tau_{\rm delay} < \tau_{\rm pairs} \sim$ ns (Sec. \[entanglers\]), where $\tau_{\rm delay}$ ($\sim $ ps) is the time delay between two entangled electrons of the same pair, and $\tau_{\rm pairs}$ ($\sim $ ns) is the time separation between two subsequent pairs. *Interactions in the beam splitter.* For entangled electrons it would be advantageous to reduce electron-electron interaction in the beam splitter, which is the main source of orbital decoherence at low temperatures. This could be achieved by depleting the electron sea in the beam splitter, e.g., by using the lowest channel in a quantum point contact. A further possibility is to use a superconductor for the beam splitter [@leo]. A superconductor would have the advantage that the entangled electrons could be injected into the empty quasiparticle states right at their chemical potential. Because of the large gap $\Delta$ between these states and the condensate, the injected electron cannot exchange energy (nor spin) with the underlying condensate of the superconductor. An alternative way to detect entangled pairs would be to use a superconductor as an analyzer: arriving entangled (spin-singlet) pairs can enter the superconductor whereas any triplet state is not allowed. Thus, the current of entangled pairs is larger than otherwise. Noise of a double QD near the Kondo regime ========================================== Spin-flip processes in a spin $1/2$ quantum dot attached to leads result in a renormalization of the single-particle transmission coefficient ${\rm T}$, giving rise to the Kondo effect [@GRNgL] below the Kondo temperature $T_K$. Theoretical studies on shot noise in this system are available [@MeirGolub]–[@SchillerHershfield], and show that the noise $S$ obeys qualitatively the same formula as for noninteracting electrons but with a renormalized ${\rm T}$. Here, we consider a system where the spin fluctuations (that are enhanced near the Kondo regime) strongly affect the noise, resulting in some cases in super-Poissonian noise – a result which cannot be obtained from the “non-interacting” formula. We consider two lateral quantum dots (DD), connected in series between two metallic leads via tunnel contacts, see inset of Fig. \[DDKondo2\][*a*]{}. The dots are tuned into the Coulomb blockade regime, each dot having a spin $1/2$ ground state. The low energy sector of the DD consists of a singlet $|S\rangle$ and a triplet $|T\rangle\equiv\left\{|T_+\rangle,|T_0\rangle,|T_-\rangle\right\}$, with the singlet-triplet splitting $K$. The Kondo effect in this system has been studied extensively [@GolovachLoss]–[@AnoEto]. Two peculiar features in the linear conductance $G$ have been found: a peak in $G$ [*vs*]{} the inter-dot tunnel coupling $t_H$ (see Fig. \[DDKondo2\][*a*]{}), revealing the non-Fermi-liquid critical point of the two-impurity Kondo model (2IKM) [@ALJ]; and a peak in $G$ [*vs*]{} an applied perpendicular magnetic field $B$ (see Fig. \[DDKondo2\][*b*]{}), as a result of the singlet-triplet Kondo effect at $K=0$ [@GolovachLoss]. The problem of shot noise in DDs with Kondo effect is rather involved. Here we propose a phenomenological approach. For bias $\Delta\mu\gg T_K,K$, the scattering problem can be formulated in terms of the following scattering matrix $$\begin{aligned} {\rm s}&=& \left( \begin{array}{cc} r_S & t_S\\ t_S & r_S \end{array} \right)\left|S\right\rangle\left\langle S\right|+ \left( \begin{array}{cc} r_T & t_T\\ t_T & r_T \end{array} \right)\left|T\right\rangle\left\langle T\right|\nonumber\\ &&+\left( \begin{array}{cc} r_{TS} & t_{TS}\\ t_{TS} & r_{TS} \end{array} \right)\left|T\right\rangle\left\langle S\right|+ \left( \begin{array}{cc} r_{ST} & t_{ST}\\ t_{ST} & r_{ST} \end{array} \right)\left|S\right\rangle\left\langle T\right|, \label{Smatrix}\end{aligned}$$ where $t_{i(j)}$ and $r_{i(j)}$ are the transmission and reflection amplitudes. The spin fluctuations in the DD cause fluctuations in the transmission through the DD. The dominant mechanism is qualitatively described by the following stochastic model $$f(t)=\left[f_1(t)\left(1-F(t)\right)+f_2(t)F(t)\right] \left(1-\left|\dot{F}(t)\right|\right)+f_3(t)\left|\dot{F}(t)\right|,$$ where $f_i(t)=0,1$ is a white noise ($i=1,2,3$) with $\langle f_i(t)\rangle=\bar{f}_i$ and $\langle f_i(t)f_i(0)\rangle-\bar{f}_i^2=\bar{f}_i(1-\bar{f}_i) \delta(t/\Delta t)$, and $F(t)=0,1$ is a telegraph noise with $\bar{F}=\beta/(1+\beta)$ and $\langle F(t)F(0)\rangle-\bar{F}^2=\beta\exp(-ct)/(1+\beta)^2$, for $t\geq 0$. In this model, the time $t$ is discretized in intervals of $\Delta t=h/2\Delta\mu$. The derivative $\dot{F}(t)$ takes values $0,\pm 1$. The function $f_{1(2)}(t)$ describes tunnelling through the DD, with the DD staying in the singlet (triplet) state, while $f_3(t)$ describes tunnelling accompanied by the DD transition between singlet and triplet. The relation to formula (\[Smatrix\]) is given by: $\bar{f}_1=|t_S|^2=1-|r_S|^2$, $\bar{f}_2=|t_T|^2=1-|r_T|^2$, and $f_3=|t_{ST}|^2/\left(|t_{ST}|^2+|r_{ST}|^2\right)= |t_{TS}|^2/\left(|t_{TS}|^2+|r_{TS}|^2\right)$. The telegraph noise is described by two parameters: $\beta=w_{12}/w_{21}$ and $c=w_{12}+w_{21}$, where $w_{ij}$ is the probability to go from $i$ to $j$. The quantity of interest is the Fano factor $F=S/e|I|$. For a single-channel non-interacting system, one has $F=1-{\rm T}$. In order to show the effect of interaction, we introduce the factor $P=F/(1-{{\rm T}})$. The noise power at zero frequency is then given by $S=2eI_{\rm imp}{{\rm T}}(1-{{\rm T}})P$, where $I_{\rm imp}=2e\Delta\mu/h$. For the average transmission probability we obtain $${{\rm T}}\equiv\langle f\rangle=\frac{\bar{f}_1+\beta\bar{f}_2}{1+\beta}+ \frac{\beta c\Delta t}{(1+\beta)^2}\left(2\bar{f}_3- \bar{f}_1-\bar{f}_2\right).$$ The noise can be calculated as $S=2eI_{\rm imp}S_f$, with $S_f={{\rm T}}(1-{{\rm T}})+\Delta S_f$, where $$\begin{aligned} \Delta S_f&=& \frac{2\beta}{(1-q)(1+\beta)^2} \left\{q(\bar{f}_1-\bar{f}_2)^2+ \frac{c\Delta t(\bar{f}_1-\bar{f}_2)}{(1+\beta)}\times \right.\nonumber\\ && \left[\bar{f}_3(\beta-1)(q+1)+\bar{f}_1(1-\beta q)+\bar{f}_2(q-\beta)\right]+ \nonumber\\ && \left.\frac{(c\Delta t)^2}{4} \left[\left(2\bar{f}_3-\bar{f}_1-\bar{f}_2\right)^2- \left(\bar{f}_1-\bar{f}_2\right)^2\right]\right\},\end{aligned}$$ with $q=\exp(-c\Delta t)$. The factor $P$ is then given by $P=1+\Delta S_f/({{\rm T}}-{{\rm T}}^2)$. Deviations of $P$ from $P=1$ show the effect of interactions in the DD. We plot the Fano factor and the factor $P$ for a DD on Fig. \[DDKondo2\]. The results show that the spin fluctuations affect the shot noise in the regions where $K\lesssim T_K$. A peculiar feature in $P$ is found both at the 2IKM critical point (Fig. \[DDKondo2\][*a*]{}) and at the point of the singlet-triplet Kondo effect (Fig. \[DDKondo2\][*b*]{}). For $\Delta\mu\ll T_K$ the DD spin is screened, and correlations between two electrons passing through the DD occur only via virtual excitations of the Kondo state. The shot noise is expected to qualitatively obey the non-interacting formula with the renormalized ${{\rm T}}$. Summary ======= We presented our recent works on shot noise for spin-entangled electrons and spin-polarized currents in novel beam splitter geometries. After a detailed description of various schemes (“entanglers”) to produce entangled spin states, we calculated shot noise within the scattering approach for a beam splitter with and without a local s-o interaction in the incoming leads. We find that the s-o interaction significantly alters the noise. Entangled/unentangled pairs and spin-polarized currents show sizable shot noise oscillations as a function of the Rashba phase. Interestingly, we find an additional phase modulation due to s-o induced interband coupling in leads with two channels. Shot noise measurements should allow the identification/characterization of both entangled and unentangled pairs as well as spin-polarized currents. Finally, we find that the s-o coupling constant $\alpha$ is directly related to the Fano factor; this offers an alternative means of extracting $\alpha$ via noise. This work was supported by NCCR Nanoscience, the Swiss NSF, DARPA, and ARO. W. Schottky, Ann. Phys. 57 (1918) 541. Ya. M. Blanter and M. Büttiker, Phys. Rep. **336**, 1 (2000). D. Loss and E.V. Sukhorukov, Phys. Rev. Lett. **84**, 1035 (2000), cond-mat/9907129. G. Burkard, D. Loss, and E.V. Sukhorukov, Phys. Rev. B **61**, R16303 (2000), cond-mat/9906071. For an early account see D. P. DiVincenzo and D. Loss, J. Magn. Magn. Mat. **200**, 202 (1999), cond-mat/9901137. W. D. Oliver *et al.*, in *Quantum Mesoscopic Phenomena and Mesoscopic Devices in Microelectronics*, vol. 559 of NATO ASI Series C: Mathematical and Physical Sciences, eds. I. O. Kulik and R. Ellialtioglu (Kluwer, Dordrecht, 2000), pp. 457-466. F. Taddei and R. Fazio, Phys. Rev. B **65**, 075317 (2002). J. C. Egues, G. Burkard, and D. Loss, to appear in the Journal of Superconductivity; cond-mat/0207392. J. C. Egues, G. Burkard, and D. Loss, Phys. Rev. Lett. **89**, 176401 (2002); cond-mat/0204639. B. R. Bulka *et al.* Phys. Rev. B **60**, 12246 (1999). F. G. Brito, J. F. Estanislau, and J. C. Egues, J. Magn. Magn. Mat. **226-230**, 457 (2001). F. M. Souza, J. C. Egues, and A. P. Jauho, cond-mat/0209263. J. J. Sakurai, *Modern Quantum Mechanics*, San Fu Tuan, Ed., (Addison-Wesley, New York,1994); (Ch. 3, p. 223). See also J. I. Cirac, Nature **413**, 375 (2001). , Eds. D. D. Awschalom, D. Loss, and N. Samarth (Springer, Berlin, 2002). P. Recher, E.V. Sukhorukov, and D. Loss, Phys. Rev. B [**63**]{}, 165314 (2001); cond-mat/0009452. D. S. Saraga and D. Loss, cond-mat/0205553. R. Fiederling *et al.*, Nature **402**, 787 (1999); Y. Ohno *et al.,* Nature **402**, 790 (1999). See J. C. Egues Phys. Rev. Lett. **80**, 4578 (1998) and J. C. Egues *et al.* Phys. Rev. B **64**, 195319 (2001) for *ballistic* spin filtering in semimagnetic heterostructures. P. Recher, E. V. Sukhorukov, and D. Loss, Phys. Rev. Lett. **85**, 1962 (2000), cond-mat/0003089. P. Recher and D. Loss, Phys. Rev. B [**65**]{}, 165327 (2002), cond-mat/0112298. V.N. Golovach and D. Loss, cond-mat/0109155. R. C. Liu *et al.*, Nature (London), **391**, 263 (1998). M. Henny *et al.*, Science **284**, 296 (1999); W. D. Oliver *et al.*, Science **284**, 299 (1999). See also M. Büttiker, Science **284**, 275 (1999). G. Feve *et al.* (cond-mat/0108021) also investigate transport in a beam splitter configuration. These authors assume a “global” s-o interaction and formulate the scattering approach using Rashba states in single-moded leads. S. Datta and B. Das, Appl. Phys. Lett. **56**, 665 (1990). L.P. Kouwenhoven, G. Schön, L.L. Sohn, Mesoscopic Electron Transport, NATO ASI Series E:  Applied Sciences-Vol.345, 1997, Kluwer Academic Publishers, Amsterdam. D. Loss and D. P. DiVincenzo, Phys. Rev. A **57**, 120 (1998), cond-mat/9701055. M.-S. Choi, C. Bruder, and D. Loss, Phys. Rev. B [**62**]{}, 13569 (2000); cond-mat/0001011. C. Bena, S. Vishveshwara, L. Balents, and M.P.A. Fisher, Phys. Rev.  Lett. [**89**]{}, 037901 (2002). G.B. Lesovik, T. Martin, and G. Blatter, Eur. Phys. J. B [**24**]{}, 287 (2001). R. Mélin, cond-mat/0105073. V. Bouchiat *et al.*, cond-mat/0206005. W.D. Oliver, F. Yamaguchi, and Y. Yamamoto, Phys. Rev. Lett. [**88**]{}, 037901 (2002). S. Bose and D. Home, Phys. Rev. Lett. **88**, 050401 (2002). In principle, an entangler producing entangled triplets $|\!\!\uparrow \downarrow \rangle + |\!\!\downarrow \uparrow \rangle$ or [*orbital*]{} entanglement would also be desirable. This condition reflects energy conservation in the Andreev tunnelling event from the SC to the two QDs. This reduction factor of the current $I_{2}$ compared to the resonant current $I_{1}$ reflects the energy cost in the virtual states when two electrons tunnel via the same QD into the same Fermi lead and are given by $U$ and/or $\Delta$. Since the lifetime broadenings $\gamma_{1}$ and $\gamma_{2}$ of the two QDs 1 and 2 are small compared to $U$ and $\Delta$ such processes are suppressed. P. Recher and D. Loss, Journal of Superconductivity: Incorporating Novel Magnetism 15 (1): 49-65, February 2002; cond-mat/0205484. A.F. Volkov, P.H.C. Magnée, B.J. van Wees, and T.M. Klapwijk, Physica C [**242**]{}, 261 (1995). M. Kociak, A.Yu. Kasumov, S. Guéron, B. Reulet, I.I. Khodos, Yu.B. Gorbatov, V.T. Volkov, L. Vaccarini, and H. Bouchiat, Phys. Rev. Lett. [**86**]{}, 2416 (2001). M. Bockrath [*et al.*]{}, [*Nature*]{} [**397**]{}, 598 (1999). R. Egger and A. Gogolin, Phys. Rev. Lett. [**79**]{}, 5082 (1997); R. Egger, Phys. Rev.  Lett. [**83**]{}, 5547 (1999). C. Kane, L. Balents, and M.P.A. Fisher, Phys. Rev. Lett. [**79**]{}, 5086 (1997). L. Balents and R. Egger, Phys. Rev. B, [**64**]{} 035310 (2001). For a review see e.g. H.J. Schulz, G. Cuniberti, and P. Pieri, cond-mat/9807366; or J. von Delft and H. Schoeller, Annalen der Physik, Vol. [**4**]{}, 225-305 (1998). The interaction dependent constants $A_{b}$ are of order one for not too strong interaction between electrons in the LL but are decreasing when interaction in the LL-leads is increased [@P8RL]. Therefore in the case of substantially strong interaction as it is present in metallic carbon nanotubes, the pre-factors $A_{b}$ can help in addition to suppress $I_{2}$. Since $\gamma_{\rho -}>\gamma_{\rho +}$, it is more probable that two electrons coming from the same Cooper pair travel in the same direction than into different directions when injected into the same LL-lead. In order to have exclusively singlet states as an input for the beamsplitter setup, it is important that the LL-leads return to their spin ground-state after the injected electrons have tunnelled out again into the Fermi leads. For an infinite LL, spin excitations are gapless and therefore an arbitrary small bias voltage $\mu$ between the SC and the Fermi liquids gives rise to spin excitations in the LL. However, for a realistic finite size LL (e.g. a nanotube), spin excitations are gapped on an energy scale $\sim\hbar v_{F}/L$, where $L$ is the length of the LL. Therefore, if $k_{B}T,\mu<\hbar v_{F}/L$ only singlets can leave the LL again to the Fermi leads, since the total spin of the system has to be conserved. For metallic carbon nanotubes, the Fermi velocity is $\sim 10^{6}{\rm m/s}$, which gives an excitation gap of the order of a few meV for $L\sim$ $\mu$m; this is large enough for our regime of interest. A singlet-triplet transition for the ground state of a quantum dot can be driven by a magnetic field; see S. Tarucha *et al.*, Phys. Rev. Lett. **84**, 2485 (2000). This symmetric setup of the charging energy $U$ is obtained when the gate voltages are tuned such that the total Coulomb charging energies in [$ D_C \ $]{}are equal with zero or two electrons. K. Blum, *Density Matrix Theory and Applications* (Plenum, New York, 1996). T.H. Oosterkamp *et al.*, Nature (London) **395**, 873 (1998); T. Fujisawa *et al.*, Science **282**, 932 (1998). J.M. Kikkawa and D.D. Awschalom, Phys. Rev. Lett. **80**, 4313 (1998). I. Malajovich, J. M. Kikkawa, D. D. Awschalom, J. J. Berry, and D. D. Awschalom, Phys. Rev. Lett. **84**, 1015 (2000); I. Malajovich, J. J. Berry, N. Samarth, and D. D. Awschalom, Nature **411**, 770 (2001). M. Johnsson and R. H. Silsbee, Phys. Rev. Lett. **55**, 1790 (1985); M. Johnsson and R. H. Silsbee, Phys. Rev. B **37**, 5326 (1988); M. Johnsson and R. H. Silsbee, Phys. Rev. B **37**, 5712 (1988). F. J. Jedema, A. T. Filip, and B. J. van Wees, Nature **410**, 345 (2001); F. J. Jedema, H. B. Heersche, J. J. A. Baselmans, and B. J. van Wees, Nature **416**, 713 (2002). In addition, for fully spin-polarized leads the device can act as a single spin memory with read-in and read-out capabilities if the dot is subjected to a ESR source. This is true as long as the Zeeman splitting in the leads is much smaller than their Fermi energies. H.-A. Engel and D. Loss, Phys. Rev. B **65**, 195321 (2002), cond-mat/0109470. S. Kawabata, J. Phy. Soc. Jpn. [**70**]{}, 1210 (2001). N.M. Chtchelkatchev, G. Blatter, G.B. Lesovik, and T. Martin, cond-mat/0112094. M. Büttiker, Phys. Rev. B **46**, 12485 (1992); Th. Martin and R. Landauer, Phys. Rev. B **45**, 1742 (1992). For a recent comprehensive review on shot noise, see Ref. [@blanter-buttiker]. Our noise definition here differs by a factor of two from that in the review article by Blanter and Büttiker (Ref. [@blanter-buttiker]); these authors define their power spectral density of the noise with a coefficient two in front (see definition following Eq. (49) and footnote 4 in Ref. [@blanter-buttiker]). We use a standard Fourier transform (no factor of two in front) to define the noise spectral density. For a discrete energy spectrum we need to insert a density-of-states factor $\nu$ in the current and noise definitions; see Ref. [@gde]. Note that the uncorrelated-beam case here refers to a beam splitter configuration with only one of the incoming leads “open”. This is an important point since a beam splitter is noiseless for (unpolarized) uncorrelated beams in both incoming leads. G. Engels *et al.* Phys. Rev. B **55**, R1958 (1997); J. Nitta *et al.*, Phys. Rev. Lett. **78**, 1335 (1997); D. Grundler Phys. Rev. Lett. **84**, 6074 (2000); Y. sato *et al.* J. Appl. Phys. **89**, 8017 (2001). A. V. Moroz and C. H. W. Barnes, Phys. Rev. B **60**, 14272 (1999); F. Mireles and G. Kirczenow, *ibid.* **64**, 024426 (2001); M. Governale and U. Zülicke, Phys. Rev. B **66** 073311 (2002). G. Lommer *et al.*, Phys. Rev. Lett. **60**, 728 (1988), G. L. Chen *et al.*, Phys. Rev. B **47**, 4084(R) (1993), E. A. de Andrada e Silva *et al.*, Phys. Rev. B **50**, 8523 (1994), and F. G. Pikus and G. E. Pikus Phys. Rev. B **51**, 16928 (1995). Yu. A. Bychkov and E. I. Rashba, JETP Lett. **39**, 78 (1984). L. W. Molenkamp *et al.*, Phys. Rev. B **64**, R121202 (2001); M. H. Larsen *et al*., *ibid.* **66**, 033304 (2002). The Rashba-active region in lead 1 is (supposed to be) *electrostatically* induced. This implies that there is no band-gap mismatch between the Rashba region and the adjacent regions in lead 1 due to materials differences. There is, however, a small mismatch arising from the Rashba energy $\epsilon_R$; this is the amount the Rashba bands are shifted down with respect to the bands in the absence of s-o orbit in the channel. Since typically $\epsilon_R\ll \varepsilon_F$, we find that the transmission is indeed very close to unity (see estimate in Ref. [@prl-cgd]). Note that the velocity operator is not diagonal in the presence of the Rashba interaction. J. C. Egues, G. Burkard, and D. Loss, cond-mat/0209692. In the absence of the s-o interaction, we assume the wire has two sets of spin-degenerate parabolic bands for each $k$ vector. In the presence of s-o interaction but neglecting s-o induced interband coupling, there is a one-to-one correspondence between the parabolic bands with no spin orbit and the Rashba bands; hence they can both be labelled by the same indices. N. W. Ashcroft and N. D. Mermin, *Solid State Physics*, Ch. 9. (Holt, Rinehart, and Winston, New York, 1976). G. Schmidt, D. Ferrand, L. W. Molenkamp, A. T. Filip, and B. J. van Wees, Phys. Rev. B **62**, R4790 (2000). L. P. Kouwenhoven, [*private communication*]{}. L. I. Glazman and M.E. Raikh, JETP Lett. [**47**]{}, 452 (1988); T. K. Ng and P. A. Lee, Phys. Rev. Lett. [**61**]{}, 1768 (1988). Y. Meir and A. Golub, Phys. Rev. Lett. [**88**]{}, 116802 (2002). F. Yamaguchi and K. Kawamura, Physica B [**227**]{}, 116 (1996). A. Schiller and S. Hershfield, Phys. Rev. B [**58**]{}, 14978 (1998). G. Burkard, D. Loss, and D.P. DiVincenzo, Phys. Rev. B [**59**]{}, 2070 (1999), cond-mat/9808026. V. N. Golovach and D. Loss, cond-mat/0109155. W. Izumida and O. Sakai, Phys. Rev. B [**62**]{}, 10260 (2000). A. Georges and Y. Meir, Phys. Rev. Lett. [**82**]{}, 3508 (1999). T. Aono and M. Eto, Phys. Rev. B [**63**]{}, 125327 (2001). I. Affleck, A. W. W. Ludwig, and B. A. Jones, Phys. Rev. B [**52**]{}, 9528 (1995). [^1]: Permanent address: Department of Physics and Informatics, University of São Paulo at São Carlos, 13560-970 São Carlos/SP, Brazil.

[**The Genesis of Cosmological Tracker Fields**]{} 0.4in Vinod B.Johri$\dagger$ 0.2in [*Theoretical Physics Institute,School of Physics and Astronomy,\ University of Minnesota, Minneapolis, MN 55455,USA*]{} 0.2in [**Abstract** ]{} [The role of the quintessence field as a probable candidate for the repulsive dark energy, the conditions for tracking and the requisites for tracker fields are examined. The concept of ‘integrated tracking’ is introduced and a new criterion for the existence of tracker potentials is derived assuming monotonic increase in the scalar energy density parameter $\Omega_\phi$ with the evolution of the universe as suggested by the astrophysical constraints. It provides a technique to investigate generic potentials of the tracker fields. The general properties of the tracker fields are discussed and their behaviour with respect to tracking parameter $\epsilon$ is analyzed. It is shown that the tracker fields around the limiting value $\epsilon \simeq \frac 23$ give the best fit with the observational constraints.]{} PACS numbers: 98.80.Cq, 98.65.Dx, 98.70.Vc 0.2 in There is strong evidence, based on recent luminosity-redshift observations of Type $I_a$ supernovae \[1\] and consistently low measurements of matter density \[2\], to suggest that the major fraction of the energy content of the observable universe consists of an ’exotic matter’ with negative pressure, often referred to as ’dark energy’\[3\]. In cold dark matter(CDM) cosmology, the most probable candidates for dark energy are the cosmological constant $\Lambda$ and the weakly coupled scalar fields with negative pressure which might mimic $\Lambda$ to produce enough repulsive force to counter gravitational attraction and cause acceleration in the expanding universe at the present epoch. Comprehensive review reports on the observational,theoretical,physical and anthropic significance of the cosmological constant $\Lambda$ have been published by Zeldovich \[4\], Weinberg \[5\], Sahni and Starobinsky \[6\]. The cosmological constant seems to be a natural choice for the source of cosmic repulsion but it is hard to reconcile its constant value $\Lambda \sim 10^{-47} {GeV^4}$ (to be comparable with the present energy density of the universe) with the particle physics scales $ \sim 10^{56}{GeV}^4$ subsequent to inflation. Since $\Lambda$ has stayed constant through cosmic evolution, it demands setting up a new energy scale to explain as to why it should take 15 billion years of time for $\Lambda$ to dominate in the universe today (known as coincidence problem). To address this problem, a comprehensive study of the observational consequences of a dynamical $\Lambda$ term (representing vacuum energy) decaying with time \[7\] and the cosmological consequences of rolling scalar fields was undertaken \[8\]; subsequently, Caldwell et al \[12\] discussed the possibility that a significant contribution to the energy density of the universe might be from the scalar fields with an evolving equation of state, unlike radiation, matter or $\Lambda$ fields and proposed the nomenclature ’quintessence’ for such scalar fields which, during the process of roll-down, acquire negative pressure and might act as $\Lambda_{eff}$. But for the scalar energy density $\rho_{\phi}$ or $\Lambda_{eff}$ to be comparable with the present energy density $\rho_n$ of the universe, the initial conditions for the quintessence fields must be set up carefully and fine tuned. To overcome the ’fine tuning’ or the ’initial value’ problem, the notion of tracker fields \[13,14\] was introduced. It permits the quintessence fields with a wide range of initial values of $\rho_{\phi}$ to roll down along a common evolutionary track with $\rho_n$ and end up in the observable universe with $\rho_{\phi}$ comparable to $\rho_n$ at the present epoch. Thus, the tracker fields can get around both the coincidence problem and the fine tuning problem without the need for defining a new energy scale for $\Lambda_{eff}$. Although tracking is a useful tool to promote quintessence as a likely source of the Ômissing energyÕ in the universe, the concept of tracking as given by Steinhardt et al \[13,14\] does not ensure the physical viability of quintessence in the observable universe. It simply provides for synchronized scaling of the scalar field with the matter/radiation field in the expanding universe in such a way that at some stage (undefined and unrelated to observations), the scalar field energy starts dominating over matter and may induce acceleration in the hubble expansion. Since there is no control over the slow roll-down and the growth of the scalar field energy during tracking, the transition to the scalar field dominated phase may take place much later than observed. Moreover, any additional contribution to the energy density of the universe, such as quintessence, is bound to affect the dynamics of expansion and structure formation in the universe. As such, any physically viable scalar field must comply with the cosmological observations related to helium abundance, cosmic microwave background and galaxy formation, which are the pillars of the success of the standard cosmological model. A realistic theory of tracking of scalar fields must, therefore, take into account the astrophysical constraints arising from the cosmological observations. With this perspective in mind, we have introduced the notion of ‘integrated tracking’ in this paper which implies tracking compatible with astrophysical constraints. It provides a firm and credible foundation to the quintessence theory. Most of the investigations \[10,11,12,13\] on the scalar fields so far have been confined to exploring scalar potentials which roll down with the desired tracking behaviour to end up with dominance of quintessence energy. The first theoretical derivation of the tracking condition was attempted by Steinhardt et al \[14\] who put forth different criteria for tracking under varying conditions and discussed the tracking properties of certain exponential and inverse power law potentials. In this Letter, we report our investigations towards a systematic theory of Ôintegrated trackingÕ in which the tracking behaviour of the scalar field is closely related to the growth of its cosmological density parameter through the tracking parameter $\epsilon$, subject to astrophysical constraints as discussed in the paper. This approach provides us a powerful technique to study the general behaviour of the tracker fields with respect to $\epsilon$ and also a window to investigate the generic potentials of the tracker fields instead of dealing with isolated potentials and their properties. In general, the energy densities $\rho_n$ and $\rho_{\phi}$ scale down at different rates in the expanding universe. For a scalar field with potential $V(\phi) = \frac{1}{2}(\rho_{\phi} - p_{\phi})$ and kinetic energy $\frac{1}{2}{\dot\phi^2}$ = $\frac{1}{2}(\rho_{\phi}+p_{\phi})$,the equation of motion of the scalar field $$\ddot\phi+3H\dot\phi+V^{\prime}(\phi) = 0$$ leads to $\rho_{\phi} \sim a^{-3(1+w_\phi)}$ where $1+w_{\phi} = 1+\frac{p_{\phi}}{\rho_{\phi}}=\frac{\dot\phi^2}{\rho_{\phi}} = 2\eta$, $\eta$ denotes the ratio of kinetic energy to $\rho_\phi$. For the background energy, $\rho_n\sim a^{-3(1+w_n)}\sim\frac{1}{a^n}$, $n=4$(radiation) and $n=3$(matter). Obviously, the scalar field has a wider range of scaling as $\rho_{\phi} \sim a^{-6\eta}$, $(0\leq\eta\leq 1)$ depending upon the choice of $\eta$. When the kinetic energy is dominant, $\rho_{\phi}$ can scale down as steeply as $\frac{1}{a^6}$. It rolls down slowly as $V(\phi)$ starts dominating and the rolling reduces to a crawl as $\eta$ tends to zero. Therefore, the kinetic energy plays an important role in scaling down the energy of the scalar field. In order to solve the ’dark energy’ problem, we want domination of the scalar field ($\rho_\phi\ga \rho_n$) today but at the same time it is imperative that $\rho_\phi < \rho_n$ during radiation and matter dominated era and grows slowly to the present state so as not to interfere with the formation of galactic structure and the success of nucleosynthesis during cosmic evolution. Therefore, tracking requires proper synchronization of the scaling of the two fields so that $\rho_\phi$ rolls down slower than $\rho_n$ (i.e $w_\phi< w_n$) along a common evolutionary track and eventually overtakes it, causing acceleration in the cosmic expansion. This implies that $w_\phi<\frac13$ during radiation era, $w_\phi<0$ during matter domination and $w_\phi$ tends to $-1$ during scalar energy dominated phase, constraining $\eta$ to lie in the range $(0\leq\eta<\frac23)$. Naturally a fixed value of $\eta$ does not lead to tracking. It must vary with the roll down of the scalar field but its variation over the wide span of the cosmic time is so small that it may be regarded as almost a constant and the time derivatives of $\eta$ may be neglected. This assumption simplifies the dynamics of evolution of the tracker fields. Using Eq. (1), the logarithmic differentiation of $V(\phi) = (1-\eta)\rho_\phi$ yields an important condition to be satisfied by the tracker fields. $$\pm\frac{V'(\phi)}{V(\phi)} = 6\eta\frac{H}{\dot\phi} =\frac{\sqrt{6\eta}}{M_p\sqrt{\Omega_\phi}}$$ where $\Omega_\phi \equiv \frac{\rho_{\phi}}{\rho_\phi+\rho_n}$, $H$ is the Hubble constant given by $H^2=\frac{\rho_\phi+\rho_n}{3M_p^2}$ and $M_p=2.4\times10^{18} GeV$ is the reduced Planck mass. According to our notation, the prime denotes derivative with respect to $\phi$, an overdot denotes time-derivative and $\pm$ sign applies to $V'>0$ and $V'<0$ respectively. It is remarkable that the maintenance of the tracker condition (2) constrains the order of magnitude of the various terms involved. For instance, the middle term $\sqrt{\frac{\rho_\phi+\rho_n}{\rho_\phi}}$ in (2) must remain nearly of $O(1)$ throughout tracking. It follows, therefore, that the scalar fields with $\rho_\phi\ll\rho_n$ will remain frozen until hubble expansion slows down to the level when $\rho_n\approx\rho_\phi$; thereafter, the tracker condition holds good and tracking takes place. Thus, the tracker condition ensures that a large class of quintessence fields with widely diverse initial conditions $(\rho_\phi\ll\rho_n)$ would scale down to the same present state with $\rho_\phi\simeq\rho_n$. However, the scalar fields with $\rho_\phi\gg\rho_n$ violate astrophysical constraints given below although relation (2) continues to hold good. Hence the tracker condition (2) is not a sufficient condition for tracking and it has to be supplemented with more stringent requirements based on the astrophysical constraints discussed below. Observationally, the cosmological density parameter $\Omega_\phi$ of the scalar field is an important quantity since it measures the relative magnitude of the energy densities $\frac{\rho_\phi}{\rho_n}$ during cosmic evolution as given by $$\Omega_\phi = (1+a^{-3\epsilon})^{-1}$$ where $\epsilon\equiv w_n-w_\phi$, $0<\epsilon\leq1$. It may be used to regulate the tracking behaviour according to the following astrophysical constraints:\ I. $\Omega_\phi < 0.13 - 0.2$ at the nucleosynthesis epoch \[10\] around redshift $z=10^{10}$\ II. $\Omega_\phi < 0.5$ during galaxy formation epoch \[3\] around $z=2$ to 4\ III. $\Omega_\phi = 0.5$ at the onset of acceleration in cosmic expansion at redshift $z\,(0\leq z <2)$\ IV. $\Omega_\phi \simeq 0.65\pm\,0.05$ with $w_\phi\leq-0.4$ at the present epoch ($z=0$) \[15\]\ As stated above, the tracker condition given by Eq.(2) ensures slow rolling of the scalar potential $V(\phi)$ and may be regarded as a necessary condition for tracking but not as a criterion for tracker fields. To lay down a physical criterion for tracker fields, we take a clue from the astrophysical constraints I - IV which require progressive growth of $\Omega_\phi$ during tracking and postulate that $\dot\Omega_\phi > 0$ for tracker fields. This can be monitored by a single parameter $\epsilon$ (known as tracking parameter) since it reveals a clear picture of scaling of $\rho_\phi$ vs. $\rho_n$ throughout the range of tracking. The limiting value of $\epsilon$, as derived from Eq.(8) ensures the transition from matter to the scalar dominated phase. By logarithmic differentiation of Eq. (2), $\frac{\dot\Omega_\phi}{\Omega_\phi}$ may be expressed in terms of $V(\phi)$ and its derivatives as $$\frac{\dot\Omega_\phi}{\Omega_\phi} = \mp\,12\eta H(\Gamma-1)$$ where $\Gamma \equiv\frac{V''V}{V'^2}$. Again the time derivative of Eq.(3) gives $$\dot\epsilon\,\ ln\,a = \frac13\,\frac{\dot\Omega_\phi}{\Omega_\phi\Omega_n} - \epsilon H\,.$$ The choice of $V(\phi)$ may be further restricted by setting a stronger condition for tracking i.e $\dot\epsilon\geq 0$ which requires $$\dot\Omega_\phi \,\geq 3\epsilon H\Omega_\phi\Omega_n > 0\,.$$ Eqs. (4) and (6) lead to the final criterion for tracker fields $$\mp(\Gamma - 1) \,\geq \frac{\epsilon\Omega_n}{4\eta}\,.$$ [*The above inequality implies that a given scalar potential $V(\phi)$ will give rise to a tracker field if $\Gamma\leq 1-\frac{\epsilon\Omega_n}{4\eta}$ in case of increasing potential ($VÕ>0$) and $\Gamma \geq 1+ \frac{\epsilon\Omega_n}{4\eta}$ in case of decreasing potential ($VÕ<0$) where $\frac{\epsilon\Omega_n}{4\eta} <1$ and $\epsilon$ conforms to the astrophysical constraints I - IV.*]{} The tracking criterion for the quintessence potentials derived above is based on the restrictive assumption that $\Omega_\phi$ increases monotonically through most of the cosmological history. In fact, this assumption is motivated by the astrophysical Constraints I - IV listed above which demand the progressive increase in the fractional magnitude of $\Omega_\phi$ from nucleosynthesis epoch around $z\simeq 10^{10}$ through galaxy formation era to the present day (z=0). This assumption, although restrictive, seems to be quite natural and consistent with the thermal history of the universe. Dodelson et al \[17\] have suggested an alternative scenario for Ôtracking with oscillating energyÕ under which the scalar potential with sinusoidal modulation oscillates about the ambient energy density. The oscillating tracker potentials may satisfy the astrophysical constraints I - IV provided the magnitude and frequency of the oscillations are fine-tuned to comply with the specific requirements listed under the constraints. Similar existence conditions for the tracker fields have been obtained by Steinhardt et al \[14\] but they hold under the restriction $\Omega_n = 1$. The above results not only lay down the criterion for the existence of tracker fields but they also emphasize the importance of the tracker parameter $\epsilon$ which may be used with advantage to derive generic potentials for tracking fields as discussed below. Again, the astrophysical constraints during the phase transition from matter to scalar dominated phase, when $\Omega_n$ is significantly less than 1, may be utilized to limit the range of $\epsilon$ as shown below. This is to ensure that the roll down of the scalar field is not too slow and the universe must enter the phase of accelerating expansion at the right epoch at red shift $z_0\, (0<z_0<2)$ after the galactic structure has formed. The general behaviour of tracker fields, regardless of the form of $V(\phi)$, may be outlined by deducing the value of $\epsilon$ at the onset of acceleration subject to astrophysical constraints III and IV. Using the Friedmann equation during matter dominated era $$\frac{2\ddot a}{a}\, = -\frac{\rho_\phi[1+3w_\phi+(\rho_n/\rho_\phi)]}{3M_p^2},$$ the condition for onset of acceleration i.e. $\ddot a\ga 0$ when $\frac{\rho_n}{\rho_\phi}\simeq 1$, leads to the limiting value $\epsilon_0\ga\frac23$ around the transition to the scalar dominated phase. The relation $\frac{\rho_\phi}{\rho_n} = 2(1+z)^{-3\epsilon}$ involving redshift $z$, enables us to find $\rho_\phi$ at various landmark epochs in cosmic evolution for different values of $\epsilon$ and examine which of these lead to desired tracking behaviour. By choosing $\epsilon$ = 0.2, 0.6 and 0.7 successively in the redshift relation, we have shown by the indexed curves in the figure as to how the relative scaling of $\rho_r$, $\rho_m$ and $\rho_\phi$ takes place during different phases of evolution in the expanding universe. In fact, $0<\epsilon<\epsilon_0$ during matter and radiation dominated era; as such $\rho_\phi$ would track down closer to $\rho_n$ than shown in the figure. It is found that the models with $\epsilon = 0.6$ and $0.7$ satisfy all the requisite astrophysical constraints whereas $\epsilon = 0.2$ violates II and III. Therefore,the best-fit quintessence models correspond to $\epsilon\approx 0.66$ as also indicated by the limiting value derived above. This is also consistent with the concordance analysis \[15\], based on a comprehensive study of observational constraints on spatially flat cosmological models containing a mixture of matter and quintessence. For the known functional values of $\epsilon(\phi)$, the generic potentials for the tracker fields can be found by putting the tracker criterion in the form $$\pm\frac{V''V - V'^2}{V'^2}\,\geq \frac{\epsilon(1-\Omega_\phi)}{2(1+w_n-\epsilon)}$$ Inserting the value of $\Omega_\phi$ from Eq.(2), we get on simplification $$\pm\frac{\zeta'}{\zeta^2 - k^2}\geq \,f(\phi)\equiv \frac{\epsilon}{2(1+w_n-\epsilon)}$$ where $\zeta\equiv V'/V$ and $k^2=\frac{6\eta}{M_p^2}$. The above equation yields generic potential $V(\phi)$ for suitable choice of $f(\phi)$. For example, $\epsilon = 0$ corresponds to the generic potentials of the form $V(\phi) \sim exp[\beta\phi]$ for which $\Omega_\phi$ remains constant throughout. Astrophysical constraints I and II may be satisfied by choosing $\Omega_\phi <$ 0.15 as discussed in \[14\] but the tracking remains incomplete since there is no onset of acceleration as required by the constraint III. If $\epsilon$ = constant throughout, there is limited tracking since the ratio of the kinetic energy to the potential energy of the scalar field remains stationary throughout rolling with a ceiling fixed by the constraint II; further $V(\phi)$ never attains a constant value to play the role of $\Lambda$ in the universe. The generic potentials are of the hyperbolic form; in the particular case of matter dominated universe $(\Omega_n\simeq1)$, the potentials are of the inverse power law form. Power law potentials have been discussed extensively by several authors \[8,10,11,13,14,16\]. The detailed investigations involving mathematical theory of tracker fields, the stability and astrophysical consequences of tracker solutions will be published under a separate communication elsewhere. The author gratefully acknowledges useful discussions with Keith Olive and Panagiota Kanti and their keen interest and help in the preparation of this manuscript. $\dagger$Permanent address: Department of Mathematics and Astronomy, Lucknow University, Lucknow 226007, India. Email: vinodjohri@hotmail.com [99]{} S. Perlmutter et al, Nature (London) [**391**]{}, 51(1998); astro-ph/9712212, astro-ph/9812133; A.G. Riess et al., Astron. J [**116**]{}, 1009 (1998). S.D.M. White, J.F. Navarro, A.E. Evrard and C.S. Frenk, Nature (London) [**366**]{}, 429 (1993). M.S. Turner, astro-ph/9811366 (1998). Ya.B. Zeldovich, Sov. Phys.- Uspekhi [**11**]{}, 381 (1968). S. Weinberg, Rev. Mod. Phys [**61**]{}, 1 (1989). V. Sahni and A. Starobinsky, astro-ph/9904398v2. K. Freese, F.C. Adams, J.A. Frieman and E. Mottola, Nucl. Phys. [**B287**]{}, 797 (1987). B. Ratra and P.J.E. Peebles, Phys. Rev. D [**37**]{}, 3406 (1988). D. Wands, E.J. Copeland and A.R. Liddle, Ann. (N.Y.) Acad. Sci. [**688**]{}, 647 (1993); Phys. Rev. D [**57**]{}, 4686 (1998). P.G. Ferreira and M. Joyce, Phys. Rev. Lett. [**79**]{},4740 (1997); Phys. Rev. D [**58**]{}, 023503 (1998). A.R. Liddle and R.J. Scherrer, Phys. Rev.D [**59**]{}, 023509 (1998). R.R. Caldwell, R. Dave and P. Steinhardt, Phys. Rev. Lett [**80**]{}, 1582 (1998). I. Zlatev, L. Wang and P. Steinhardt, Phys. Rev. Lett. [**82**]{}, 896 (1999). P. Steinhardt, L. Wang and I. Zlatev, Phys. Rev. D [**59**]{}, 123504 (1999). L. Wang, R.R. Caldwell, J.P. Ostriker and P. Steinhardt, Astrophys. J. [**530**]{}, 17 (2000). M.S. Turner, Phys. Rev. D [**28**]{}, 1243 (1983). S. Dodelson, M. Kaplinghat and E. Stewart, astro-ph/0002360.

--- abstract: 'The correspondence principle in physics between quantum mechanics and classical mechanics suggests deep relations between spectral and geometric entities of Riemannian manifolds. We survey—in a way intended to be accessible to a wide audience of mathematicians—a mathematically rigorous instance of such a relation that emerged in recent years, showing a dynamical interpretation of certain Laplace eigenfunctions of hyperbolic surfaces.' address: - 'AP: University of Bremen, Department 3 – Mathematics, Bibliothekstr. 5, 28359 Bremen, Germany' - 'DZ: Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany and International Centre for Theoretical Physics, Strada Costiera, Trieste, Italy' author: - Anke Pohl - Don Zagier bibliography: - 'PZbib.bib' title: 'Dynamics of geodesics, and Maass cusp forms' --- Introduction ============ Suppose we have a huge space, such as the earth or a billiard table, and a small marble sitting on this space. We give this marble an initial push and observe its trajectory as it travels over the space. As we experienced from a very young age on, the marble goes straight until it hits an obstacle, e.g., the boundary of the billiard table, of which it reflects with outgoing angle equal to incoming angle, and then continues its straight path until the next obstacle where the same game restarts. ![Trajectory on a stadium-shaped billiard table.[]{data-label="fig:billiard"}](billiard){height="2.9cm"} In Figure \[fig:billiard\] this situation is depicted for a flat stadium-shaped billiard table. In Figure \[fig:bump\] it is shown for a disk with a bump in the middle, indicating that ‘straight path’ here means ‘path of minimal resistance’ or ‘path of minimal effort’. ![Trajectory on a disk with a bump in the middle. Height level curves are indicated by dotted circles.[]{data-label="fig:bump"}](circlebump){height="4.5cm"} In terms of physics, the motion of the marble is predicted by the laws of classical mechanics. In such a description, moving objects are often modeled as point particles, that is, as objects without size or dimension, identifying the object with its center of mass. In reality, any real-world object has a non-zero size, and the idealization as a point is not always desirable or correct. If we consider a very small marble which is almost a point, say of the size of an electron, or if we zoom in into our previous marble and try to describe the trajectory of a single electron of it then we notice that the classical mechanics model is not accurate on this subatomic level. One of the obstacles is the impossibility to determine simultaneously with absolute precision the position and momentum of the considered particle, as expressed by Heisenberg’s famous uncertainty principle. Thus, the classical mechanical principles of determinism and time reversibility are not valid anymore. On such small scale, a more accurate model is provided by quantum mechanics, which describes the probability with which the particle attains a specific position-momentum combination. The correspondence principle in physics states that, in the limit of passing to large scale, the predictions of quantum mechanics reproduce those of classical mechanics. However, the precise relation between classical and quantum mechanics is not yet fully understood, and its investigation gives rise to many interesting mathematical questions. In terms of mathematics, the classical mechanical aspects of the motion of the marble considered above translate to properties of the geodesic flow on a Riemannian manifold $X$, whereas the quantum mechanical description relates to the Laplace operator on $X$ and its ($L^2$-)eigenvalues and eigenfunctions. The correspondence principle then suggests an intimate relation between geometric-dynamical aspects of $X$ on the one hand, and its spectral aspects on the other hand: ------------------------------ ------------------------ --------------------- ----------------------------------------------------------------------------------------------- \[1mm\] \[-2mm\] classical mechanics $\leftrightsquigarrow$ geometric entities: $\begin{cases} \text{periodic geodesics} \\ \text{lengths of periodic geodesics} \end{cases}$ \[5mm\] quantum mechanics $\leftrightsquigarrow$ spectral entities: $\begin{cases} \text{Laplace eigenfunctions} \\ \text{Laplace eigenvalues} \end{cases}$ ------------------------------ ------------------------ --------------------- ----------------------------------------------------------------------------------------------- During the last century, many results showing relations between geometric-dynamical and spectral properties of Riemannian manifolds have been obtained. In Section \[sec:appetizer\] below we discuss—as an appetizer—the flat $1$-torus where a clear relation between the lengths of periodic geodesics (‘classical mechanical objects’) and the Laplace eigenvalues (‘quantum mechanical objects’) appears. The main aim of this article is to present a much deeper relation between periodic geodesics and Laplace eigenfunctions that has emerged in recent years, but now for a class of hyperbolic surfaces. In a nutshell, this goes as follows. A well-chosen discretization of the flow along the periodic geodesics gives rise to a one-parameter family of *transfer operators*, which are evolution operators that are reminiscent of weighted graph Laplacians and that also may be thought of as discretizations of the hyperbolic Laplacian. As such, these operators are simultaneously objects of classical and quantum mechanical nature, and therefore can serve as mediators between the dynamical and spectral entities of the hyperbolic surface under consideration. In our case, highly regular, rapidly decaying eigenfunctions (called *period functions*) of eigenvalue $1$ of the transfer operator with parameter $s$ are in bijection with rapidly decaying Laplace eigenfunctions (called *Maass cusp forms*) with spectral parameter $s$. This provides a purely dynamical characterization of the Maass cusp forms (not just their eigenvalues), shows a close dependence between periodic geodesics and these Laplace eigenfunctions, and provides a deep-lying mathematical realization of an instance of the correspondence principle. The modular surface was the first hyperbolic surface for which such a result could be established, through combination of work by E. Artin [@Artin], Series [@Series], Mayer [@Mayer_thermo; @Mayer_thermoPSL], Lewis [@Lewis], Bruggeman [@Bruggeman_lewiseq], Chang–Mayer [@Chang_Mayer_transop], and Lewis–Zagier [@LZ_survey; @LZ01]. Taking advantage of the constructions involved, an extension to a class of finite covers of the modular surface was achieved in the combination of [@Chang_Mayer_transop; @Deitmar_Hilgert; @Fraczek_Mayer_Muehlenbruch]. An alternative proof for the modular surface was provided in [@Mayer_Muehlenbruch_Stroemberg; @BM09]. The recent development of a new type of discretizations for geodesic flows on hyperbolic surfaces [@Pohl_Symdyn2d] and of a cohomological interpretation of the Maass cusp forms [@BLZm] allowed to prove such a relation between periodic geodesics and Laplace eigenfunctions for a large class of hyperbolic surfaces far beyond the modular surface and in a very direct way [@Moeller_Pohl; @Pohl_mcf_Gamma0p; @Pohl_mcf_general]. In Sections \[sec:mod\_surface\]–\[sec:recap\] below we survey this new approach, although in an informal way and restricting for simplicity to the modular surface. We attempt to provide sufficiently precise definitions and enough details to keep the exposition as understandable as possible without introducing too much technical material. As a general principle we invite all readers to rely on their intuitive understanding of the geometry and dynamics of Riemannian manifolds, to use the many figures as a support, and to ignore the exact expressions of all formulas. *Acknowledgements.* AP wishes to thank the Max Planck Institute for Mathematics in Bonn for hospitality and excellent working conditions during the preparation of this manuscript. Further, she acknowledges support by the DFG grants PO 1483/2-1 and PO 1483/2-2. An appetizer {#sec:appetizer} ============ In this section we will treat the ‘baby case’ of the flat *$1$-torus* $$\T \ceqq \R/\Z \ceqq [0,1]/{\{0\!=\!1\}},$$ and show an intimate and very clear relation between geometric and spectral entities, and hence a mathematical rigorous instance of the correspondence principle. Of course, this specific one-dimensional Riemannian manifold is much too simple to be representative of the general situation. However, it allows us to provide—without too much technical effort—a first instance of the relation between the geometry and the spectrum as motivated by the considerations from physics. We will also use this ‘baby example’ to carefully introduce the relevant geometrical and spectral concepts, whose counterparts in the situation of hyperbolic surfaces will be treated with less details. The flat 1-torus. ----------------- For a pictorial, but rather sketchy construction of the flat $1$-torus $\T$ we may imagine the set $\R$ of real numbers as a number line, and glue together this line at any two points that are separated by an integer distance. The glueing process can be visualized as rolling up the line to a unit circle. (See Figure \[fig:torus\].) Alternatively, we may take the interval $[0,1]$ and glue together its two endpoints $0$ and $1$. ![Rolling up $\R$ to form $\T$.[]{data-label="fig:torus"}](rolling) Both these geometric constructions indicate that $\T$ carries more structure than just being a set. In particular, measuring distances on $\T$ is possible, and a notion of derivatives exists. In order to be able to formulate such additional structures in precise terms and to work with them, we use a formula-based definition of $\T$. For that, we identify any two points of $\R$ that differ by an integer only. Thus, for each $t\in\R$, all points in the set $$\label{def:coset} \{ t+m \mid m\in\Z\}$$ are unified to a single element, which we denote by $[t]$. The torus $\T$, as a set, consists of all these elements. The glueing process in the pictorial construction is a visualization of the *projection map* $$\label{eq:projmap} \pi_{\mathbb{T}}\colon \R\to \mathbb{T},\quad t\mapsto [t].$$ This map is *locally injective*, which means that for any $t\in\R$ we find a small $\eps>0$ such that the restriction of $\pi_\T$ to the interval $(t-\eps,t+\eps)$ is injective. In rough terms, small pieces of the torus $\T$ look exactly like small pieces of $\R$. It is precisely this property which allows us to push certain structures of $\R$ to $\T$. Geometric entities. ------------------- We define the *distance* between two points $x,y\in\T$ to be the minimal distance between any two of their representatives in $\R$, hence $$d_\T(x,y) \ccoloneqq \min\big\{ d_\R(t_x,t_y) \ \big\vert\ [t_x] = x,\ [t_y]=y\big\},$$ where $$d_\R(t_x,t_y) \ccoloneqq | t_x -t_y |$$ is the usual euclidean distance on $\R$. A *straight path* or *geodesic* in $\T$ is—roughly said—a path such that for any two nearby points on the path no shorter way between them exists than the path itself. More precisely, a *path* on $\T$ is a differentiable map $p\colon I \to\T$, where $I\subseteq\R$ is an interval. The set $I$ should be thought of as a time interval, and $p(t)$ as the position where we are at time $t$ if we travel along the path $p$. The *speed* of $p$ is given by its derivative $p'$. The path $p$ is said to be of *unit speed* if $|p'(t)|=1$ for all $t\in I$. A path $p\colon I\to\T$ of unit speed is *straight* if for any $t\in I $ there exists $\eps>0$ such that for all $t_1,t_2\in (t-\eps,t+\eps)\cap I$ we have $$d_\T\big(p(t_1),p(t_2)\big) \ceqq \left| \int_{t_1}^{t_2} |p'(t)|\, dt\right| \ceqq \big|t_1-t_2\big| \ceqq d_\R(t_1,t_2).$$ That is, the distance between $p(t_1)$ and $p(t_2)$ equals the length of the path between $p(t_1)$ and $p(t_2)$, which here also equals the euclidean distance between $t_1$ and $t_2$. From now on, ‘geodesic’ will always mean a *unit speed, complete geodesic*, i.e., a straight path of unit speed with time interval $I=\R$. In everyday language, the notion of path usually does not refer to the motion, i.e., to a map $p\colon I \to\T$, but rather to the static object, i.e., to the image $p(I)$ of $p$. The orientation, however, is important: ‘the path from $a$ to $b$’. We too will use the notion of geodesic more flexibly and apply it to refer to either 1. \[geod1\] a geodesic $p\colon\R\to\T$ defined as above as a path, or 2. \[geod2\] The motivation for the second usage is that we are typically not interested in the specific time parametrization of a geodesic. The context should always clarify which version is being used. In our one-dimensional ‘baby example’ there are only two geodesics in the sense of (G\[geod2\]), namely those represented by the two geodesics in the sense of (G\[geod1\]) given by $$p_\pm \colon \R\to\mathbb{T},\quad t\mapsto [\pm t].$$ (See Figure \[fig:geod\_torus\].) ![The two periodic geodesics on $\T$.[]{data-label="fig:geod_torus"}](geodesics){height="2.3cm"} Both these geodesics are *periodic*, that is, they ‘close up’, or in rigorous terms, there exists $t_0 > 0$ such that for all $t\in \R$, $$p_\pm(t) \ceqq p_\pm (t+t_0).$$ The minimal such $t_0$ is called the *(primitive) period* or *(primitive) length* $\ell(p_\pm)$ of the geodesic $p_\pm$, which here is $\ell(p_\pm)=1$ in both cases. Periodicity and lengths are invariants under the equivalence of geodesics, and hence an intrinsic notion for geodesics in the sense of (G\[geod2\]). The *geometric entity* of $\mathbb{T}$ or, from the standpoint of the Introduction, the *classical mechanical object* we are interested in, is the *(primitive) geodesic length spectrum* $L_\T$, defined as the multiset ($=$ set with multiplicities) of lengths of the periodic geodesics in the sense of (G\[geod2\]). In our case, this is $$L_{\mathbb{T}} \ceqq\{ \text{lengths of periodic geodesics} \} \ceqq \{1,1\}.$$ Spectral entities. ------------------ The *spectral entity* or the *quantum mechanical object* is the Laplace spectrum of $\mathbb{T}$, which we now explain. The local injectivity of the projection map $\pi_\T$ from allows us to transfer all local notions from $\T$ to $\R$. In particular, a function $f\colon \mathbb{T}\to \C$ is *differentiable* if $$F\ccoloneqq f\circ\pi_\T \colon \R \to \C$$ is differentiable. The *derivative* of $f$ at $[t]\in\T$ is then the derivative of $F$ at $t\in\R$. The *Laplace operator* on $\mathbb{T}$ is $$\Delta_{\mathbb{T}} \ccoloneqq -\frac{d^2}{d t^2},$$ and a basis for its $L^2$-eigenfunctions is given by the family $$f_k\colon \mathbb{T} \to \C,\quad f_k\big([t]\big) \coloneqq e^{2\pi i k t}\qquad (k\in\Z).$$ A straightforward calculation shows that $$\Delta_{\mathbb{T}} f_k \ceqq (2\pi k)^2 f_k.$$ Thus, the *Laplace spectrum* of $\mathbb{T}$ is the multiset $$\sigma(\mathbb{T})\ceqq \{\text{Laplace eigenvalues}\} \ceqq \{ (2\pi k)^2 \mid k\in\Z\}.$$ Relation between geometric and spectral entities. ------------------------------------------------- A rather astonishing observation is that the geodesic length spectrum $L_\T$ of $\mathbb{T}$ and the Laplace spectrum $\sigma(\T)$ almost determine each other. To see this we consider the *dynamical zeta function* $$\zeta_{\mathbb{T}}(s) \ccoloneqq \prod_{\ell\in L_{\mathbb{T}}}\left( 1 - e^{-s\ell}\right) \ceqq \left( 1 - e^{-s}\right)^2.$$ Then $$\zeta_{\mathbb{T}}(s) = 0 \qquad\Longleftrightarrow\qquad s=2\pi i k \quad\text{for some $k\in\Z$,}$$ and the order of each zero is $2$. In order words, $$\label{eq:relation} \zeta_{\mathbb{T}}(s) = 0 \qquad\Longleftrightarrow\qquad (is)^2 \in \sigma(\mathbb{T}),$$ and the order of $s$ as a zero corresponds to the order of $(is)^2$ as eigenvalue, except for $s=0$, where the order of the Laplace eigenvalue $(is)^2 = 0$ is $1$, whereas the order of the zero $s=0$ of $\zeta_\T$ is $2$. Thus knowing the geodesic length spectrum $L_\T$, and hence the dynamical zeta function $\zeta_\T$, we can deduce all Laplace eigenvalues, and even their multiplicities up to the difficulty at $s=0$. Conversely, if we are given the Laplace spectrum $\sigma(\T)$ (with multiplicities), and hence all zeros of $\zeta_\T$ with almost all multiplicities, then we can easily deduce the exact formula of $\zeta_\T$ and thus the geodesic length spectrum. This ends the $1$-dimensional ‘appetizer’. In the rest of the paper we will study a $2$-dimensional case, again describing first the geometric side, then the spectral side, and then the relation between them. Of course, this case is much more involved, but we have tried to introduce the concepts in the torus case in such a way that they generalize naturally. Geometric and spectral sides of the modular surface {#sec:mod_surface} =================================================== In the previous section we considered the torus $\T$, which is a quotient of the flat manifold $\R$ by a discrete group action. From now on, we will consider hyperbolic surfaces, which are orbit spaces of the *hyperbolic plane* by discrete groups of isometries. For concreteness we will discuss only the *modular surface* $X=\PSL_2(\Z)\backslash\h$, even though the results hold for a much larger class. We will provide precise definitions further below in this section. In the course of the following four sections we will survey—as already mentioned in the Introduction—a rather deep relation between the geodesic flow on $X$ and the Maass cusp forms for the modular group $\PSL_2(\Z)$, resulting in a dynamical characterization of Maass cusp forms, or from a physics point of view, a description of certain quantum mechanical wave functions using only tools and objects from classical mechanics. The proof of this relation is split into three major steps: (I) A cohomological interpretation of Maass cusp forms, which we will explain in Section \[sec:mcf\] below. Representing Maass cusp forms faithfully as cocycle classes in suitable cohomology spaces provides an interpretation of these forms in a rather algebraic way which here simplifies to relate them to further objects. (II) A well-chosen discretization of the geodesic flow on $X$, which we will construct in Section \[sec:discretization\] below. This discretization extracts those geometric and dynamical properties from the geodesic flow on $X$ that are crucial for the relation to Maass cusp forms, and it discards all the other additional properties. This condensed, discrete version of the geodesic flow is also of a rather algebraic nature. (III) A connection between the discretization of the geodesic flow and the cohomology spaces, as discussed in Section \[sec:TO\] below. The central object mediating between these objects is the evolution operator (with specific weights, adapted to the spectral parameter of Maass cusp forms; a *transfer operator*) of the action map in the discrete version of the geodesic flow. We will see that the highly regular eigenfunctions of the evolution operator with parameter $s$ are building blocks for the cocycle classes of the Maass cusp forms with spectral parameter $s$, and will establish an explicit bijection between these eigenfunctions and the Maass cusp forms. The first two steps are independent of each other, and also the corresponding sections can be read independently. The third step necessarily takes advantage of the results from Sections \[sec:mcf\] and \[sec:discretization\], however only the final results are needed, not the information how to achieve these. In Section \[sec:recap\] below we will provide a brief overview of these steps. In the remainder of this section we introduce the geometric and spectral objects that we will need further on. The hyperbolic plane. --------------------- The *hyperbolic plane* is a certain two-dimensional manifold with Riemannian metric in which Euclid’s parallel axiom fails: on the hyperbolic plane, for every straight line $L$ (infinitely extended in both directions) and any point $p$ not on $L$ there are infinitely many lines $\tilde L$ passing through $p$ that do not intersect $L$. Abstractly, the hyperbolic plane is the unique two-dimensional connected, simply connected, complete Riemannian manifold with constant sectional curvature $-1$. There are many models for the hyperbolic plane. We use its *upper half plane model*[^1] $$\h \ccoloneqq \{z\in\C \mid \Ima z > 0\},$$ where the *line element of the Riemannian metric* is given by $$\label{eq:lineelement} ds^2_{x+iy} \ccoloneqq \frac{dx^2 + dy^2}{y^2}.$$ Informally, the Riemannian metric allows us to measure distances and angles. Angles in hyperbolic geometry are identical to the euclidean angles in $\h$. Distances between points however are changed in hyperbolic geometry when compared to euclidean geometry. From a euclidean point of view, hyperbolic distances between two points increase when these move nearer to the real axis $\R$. In the upper half plane model of the hyperbolic plane, the (G\[geod2\])-version of geodesics, i.e., infinite paths that are straight with respect to this metric, are the (oriented) semi-circles with center on $\R$ or the vertical rays based on the real axis. (See Figure \[fig:hypplane\].) ![Geodesics on $\h$.[]{data-label="fig:hypplane"}](hypplane){height="2.8cm"} A *Riemannian isometry* is a bijective map on $\h$ which preserves the distance between any two points. In particular, any Riemannian isometry maps geodesics to geodesics. The group of *orientation-preserving Riemannian isometries* on the hyperbolic plane is isomorphic to the (projective) matrix group $$G\ccoloneqq\PSL_2(\R) \ccoloneqq \SL_2(\R)/\{\pm \id\}.$$ The element $g\in G$ represented by the matrix $\textmat{a}{b}{c}{d}\in \SL_2(\R)$ is denoted by $g =\textbmat{a}{b}{c}{d}$, with square brackets. It then has one other representative in $\SL_2(\R)$, namely $\textmat{-a}{-b}{-c}{-d}$. The *action* of $G$ on $\h$ is given by $$\label{eq:action} \bmat{a}{b}{c}{d}\act z \ccoloneqq \frac{az+b}{cz+d}.$$ Occasionally, we will omit the dot $\cdot$ in the notation. The modular surface. -------------------- A subgroup of $G$ of particular importance is the *modular group* $$\Gamma \ccoloneqq \PSL_2(\Z).$$ It acts on $\h$ preserving the tesselation by triangles as indicated in Figure \[fig:tess\]. ![Tesselation of $\h$ by triangles.[]{data-label="fig:tess"}](tesselation) The *modular surface* is the orbit space $$X \ccoloneqq \Gamma\backslash\h,$$ that is, the space we obtain if we identify any two points of $\h$ that are mapped to each other by some element of $\Gamma$. A model is given by the (closed) *fundamental domain* $$\mc F_0\ccoloneqq \left\{ z \in \h \ \left\vert\ |z|\geq 1,\ |\Rea z |\leq \tfrac12\right.\right\}$$ (see Figure \[fig:funddom1\]). ![Fundamental domain $\mc F_0$ for $\Gamma$.[]{data-label="fig:funddom1"}](funddom1){width="0.7\linewidth"} It contains at least one point of any $\Gamma$-orbit. Only points in the boundary of $\mc F_0$ can be identified under the action of $\Gamma$, namely the left vertical boundary is mapped to the right one by the element $$\label{def:T} T \ccoloneqq \bmat{1}{1}{0}{1},$$ and the left bottom boundary (from $\overline{\varrho}$ to $i$) is mapped to the right bottom boundary (from $\varrho$ to $i$) by $$\label{def:S} S \ccoloneqq \bmat{0}{1}{-1}{0}.$$ If we glue $\mc F_0$ together according to these boundary identifications then we obtain the modular surface $X$, as illustrated in Figure \[fig:modsurface\]. This is just like what we did when we represented $\T = \R/\Z$ as $[0,1]/{\{0=1\}}$. ![The modular surface $X=\Gamma\backslash\h$.[]{data-label="fig:modsurface"}](orbifold){height="6cm"} Clearly, there is more than one fundamental domain for the modular surface. Another fundamental domain is, e.g., $$\mc F \ccoloneqq \left\{ z\in\h \ \left\vert\ |z-1|\geq 1,\ 0\leq \Rea z\leq \tfrac12 \right.\right\}$$ (see Figure \[fig:funddom2\]). ![Fundamental domain $\mc F$ for $\Gamma$.[]{data-label="fig:funddom2"}](funddom2){height="4.5cm"} It arises from $\mc F_0$ by cutting off the left half from $\mc F_0$ and gluing $S\act\mc F_L$ to the right half of $\mc F_0$. Thus, $$\mc F \ceqq S\act\mc F_L\;\cup\;(\mc F_0\smallsetminus\mc F_L).$$ For our constructions in Section \[sec:discretization\] below, the fundamental set $\mc F$ is more convenient than $\mc F_0$. The modular surface has an infinite ‘end’ of finite volume, called the *cusp*. In the fundamental domain $\mc F_0$ it is represented by the strip going to $\infty$. In terms of $\Gamma$, the presence of the element $T$ in $\Gamma$ caused the presence of this cusp. As we will see, this cusp and hence the element $T$ play a special role throughout. For completeness we remark that the modular surface is not a hyperbolic surface in the strict sense because it is not a Riemannian manifold but rather an orbifold. It has the two *conical singularities* at $i$ and $\varrho$ (see Figure \[fig:funddom1\] or \[fig:funddom2\]). At these points the structure of the quotient space $X=\Gamma\backslash\h$ is not smooth. The non-smoothness, however, does not influence any step in our argumentations. Geometric entity: geodesics. ---------------------------- Just as in the case of the torus, the ‘geometric entities’ for the modular surface are the periodic geodesics and their lengths. A geodesic on $X$ is the image under the projection map $$\label{def:projmap} \pi\colon \h \to X=\Gamma\backslash\h$$ of a geodesic on $\h$, as illustrated in Figure \[fig:modgeod\]. ![A geodesic on the modular surface.[]{data-label="fig:modgeod"}](orbigeodesic){height="5.5cm"} Geodesics on $\h$ are infinitely long, but geodesics on $X$ can be either infinitely long or else periodic and of finite length. The (primitive) geodesic length spectrum $L_X$ of $X$ is by definition the multiset of the lengths of periodic geodesics. The periodic geodesics on $X$ are closely related to those elements $g\in\Gamma$ with $|\tr(g)|>2$, the *hyperbolic elements*: For every periodic geodesic $\wh\gamma$ on $X$ and any representing geodesic $\gamma$ of $\wh\gamma$ on $\h$ (i.e., $\pi(\gamma) = \wh\gamma$) there exists a hyperbolic element $g\in\Gamma$ such that $g.\gamma$ is a time-shifted version of $\gamma$, i.e., there exists $t_g\in\R$ such that $$\label{eq:timeshift} g.\gamma(t) = \gamma(t+t_g)\qquad\text{for all $t\in\R$.}$$ (Note that $t_g\not=0$.) If in  the value $t_g$ is positive and minimal unter all positive choices for $g\in\Gamma$, then $g$ is *primitive hyperbolic*. An equivalent characterization is that $g$ is hyperbolic and not of the form $h^n$ with $h\in\Gamma$ and $n>1$. Conversely, whenever $\gamma$ is a geodesic on $\h$ and there exists $g\in\Gamma$ and $t_g\in\R$, $t_g\not=0$ such that holds, then $g$ is hyperbolic and $\pi(\gamma)$ is a periodic geodesic on $X$. Furthermore, every hyperbolic element in $\Gamma$ time-shifts a unique geodesic on $\h$. Under this assigment of primitive hyperbolic elements in $\Gamma$ to periodic geodesics on $X$, the set of periodic geodesics on $X$ is bijective to the set of conjugacy classes of the primitive hyperbolic elements in $\Gamma$, and the (primitive) geodesic length spectrum of $X$ is $$L_X \ceqq \left\{ 2\arcosh\Big(\tfrac{|\tr(g)|}{2}\Big) \ \left\vert\ g\in\HP \vphantom{\Big(\tfrac{|\tr(g)|}{2}\Big)} \right.\right\},$$ where $\HP$ is any set of representatives for the conjugacy classes of primitive hyperbolic elements in $\Gamma$. The smallest element in $L_X$ is $$2\arcosh\bigl(\tfrac{3}{2}\bigr) \ceqq 2\log\bigl(\tfrac{3+\sqrt{5}}2\bigr),$$ and by investigating the set of possible traces of the elements in $\Gamma$ one can find all elements in $L_X$ (with multiplicities) up to any given bound. The set $L_X$ is also closely related to the class numbers of indefinite binary quadratic forms. We refer the interested reader to [@Terras Exercises 18-20 in Section 3.7, and the paragraph below them] and omit any discussion of this relation here. Spectral entity: Laplace eigenfunctions. ---------------------------------------- We now introduce the spectral objects we are interested in: the Maass wave forms for $\Gamma$, and the more restrictive Maass cusp forms. The Laplacian on $\h$, the *hyperbolic Laplacian*, is $$\Delta \ccoloneqq -y^2 \big( \partial^2_x + \partial^2_y\big)\qquad (z=x+iy).$$ It is the differential operator on $\h$ that commutes with all elements of the group $G=\PSL_2(\R)$ of orientation-preserving Riemannian isometries; the factor $y^2$ corresponds to the factor $y^{-2}$ in the formula of the line element of the Riemannian metric in . Now let $u\colon\h\to\C$ be a $\Gamma$-invariant eigenfunction of $\Delta$, that is, a function satisfying $u(g\act z) = u(z)$ for all $g\in\Gamma$ and all $z\in\h$, and $$\label{eq:eigenfunction} \Delta u \ceqq s(1-s)u$$ for some $s\in\C$. Further below we will see that it is more convenient to work with the *spectral parameter* $s$ rather than with the eigenvalue $s(1-s)$ itself. We do not need to specify *a priori* the regularity of $u$: since the Laplace operator is elliptic with real-analytic coefficients, the function $u$ is automatically real-analytic. The invariance of $u$ under the element $T\in\Gamma$ from shows that $u$ is $1$-periodic, and hence has a Fourier expansion of the form $$u(x+iy) \ceqq \sum_{n\in\Z} a_n(y)\, e^{2\pi i n x}.$$ By separation of variables in we see that each function $a_n$ is a solution of a second-order differential equation (depending on $s$), a modified Bessel equation. This equation has two independent solutions, one exponentially big and one exponentially small as $y\to\infty$, except if $n=0$, where the two solutions are $y^{s}$ and $y^{1-s}$ for $s\not=\tfrac12$, and $y^{1/2}$ and $y^{1/2}\log y$ for $s=\tfrac12$. Therefore, if we assume in addition that $u$ has polynomial growth at infinity, in which case $u$ is called a *Maass wave form* for $\Gamma$, then the Fourier expansion becomes $$u(x+iy) \ceqq c_1 y^s + c_2 y^{1-s} + y^{\frac12} \sum_{\substack{n\in\Z\\ n\not=0}} A_n\, K_{s-\frac12}(2\pi |n| y)\, e^{2\pi i n x},$$ where the first two terms must be replaced by $c_1 y^{1/2} + c_2 y^{1/2}\log y$ if $s=\tfrac12$. Here $K_\nu$ denotes the modified Bessel function of the second kind with index $\nu\in\C$, whose precise definition plays no role here and is therefore omitted, and the $A_n$ are complex numbers that automatically have polynomial growth. If we further assume that $u$ has *rapid decay* at infinity then $c_1 = c_2 = 0$, and $$u(x+iy) \ceqq y^{\frac12} \sum_{\substack{n\in\Z\\ n\not=0}} A_n\, K_{s-\frac12}(2\pi |n| y)\,e^{2\pi i n x}.$$ In this case, $u$ is called a *Maass cusp form* with spectral parameter $s$. It is known that the real part of $s$ then always lies between $0$ and $1$. Since any Maass wave form $u$ is $\Gamma$-invariant, we can also consider $u$ as a true function on $X=\Gamma\backslash\h$, and characterize Maass cusp forms as eigenfunctions of $\Delta$ on $X$ having rapid decay as their argument tends to the cusp. The $\Gamma$-invariant eigenfunctions of $\Delta$ on $\h$ are the constant functions (with eigenvalue $0$) and the Maass cusp forms, whose eigenvalues are positive and tend to infinity, giving an $L^2$-Laplace spectrum $$\sigma(X) \ceqq \bigl\{ 0,\ 91.141\cdots,\ 148.432\cdots,\ 190.131\cdots,\ \ldots \bigr\}$$ whose elements are known numerically to high precision, but not in closed form. Dynamical zeta function. ------------------------ The analogue of the dynamical zeta function $\zeta_\T$ of the torus is the Selberg zeta function $Z_X$, which has an Euler product given by lengths of periodic geodesics and a Hadamard product in terms of Laplace eigenvalues. More precisely, $Z_X(s)$ is defined for $\Rea s >1$ by $$Z_X(s) \ceqq \prod_{\ell\in L_X} \prod_{k=0}^\infty \bigl( 1- e^{-(s+k)\ell}\bigl),$$ and the analogue of  is Selberg’s theorem that this function extends meromorphically to $\C$ and vanishes if $s$ is a spectral parameter. The cohomological interpretation of Maass cusp forms {#sec:mcf} ==================================================== We now turn to the first step in the passage from geodesics on the modular surface $X$ to Maass cusp forms for $\Gamma$: The interpretation of Maass cusp forms in terms of *parabolic $1$-cohomology* as provided in [@BLZm]. The essential part of this cohomological interpretation, of which we take advantage here, is that every Maass cusp form $u$ with spectral parameter $s$ is characterized by a vector $(c_g^u)_{g\in\Gamma}$ of functions given by integrals of the form $$c_g^u(t) \ceqq \int_{g^{-1}\infty}^\infty \omega_s(u,t) \qquad (t\in\R),$$ where $\omega_s(u,\cdot)$ is a certain closed form on $\h$ defined below and where the integration is along any path in $\h\cup\R\cup\{\infty\}$ from $g^{-1}\infty$ to $\infty$, with at most finitely many points in $$\mP^1(\R) \ccoloneqq \R\cup \{\infty\}.$$ The functions $(c_g^u)_{g\in\Gamma}$ satisfy certain relations among each other, so-called cocycle relations, showing that a suitable cohomology theory is the natural home of this setup. For completeness of exposition and for the convenience of the reader we provide a rather detailed definition of this cohomology (specialized to the modular group $\Gamma$), even though these details will not be needed further on. Readers who want to proceed faster to the final result are invited to skip the remaining part of this section after having read Theorem \[thm:BLZ\] below. They should interpret the space $H^1_\parab(\Gamma;\mc V_s^{\omega^*,\infty})$ as a vector space whose elements are equivalence classes of maps from $\Gamma$ to the space of highly regular functions on $\R$ (or rather on $\mP^1(\R)$), where the notion of ‘highly regular’ depends on the parameter $s$. Theorem \[thm:BLZ\] then states that the assignment of Maass cusp forms $u$ with spectral parameter $s$ to the equivalence class of the vector $(c_g^u)$ is bijective and linear. For the detailed description we start with a few preparations. The parabolic cohomology will then be seen a refinement of the standard group cohomology in order to account for the cusp of the modular surface and the rapid decay of the Maass cusp forms towards this cusp. The name *parabolic* alludes to the fact that elements in $G$ that stabilize a single point in $\mP^1(\R)$, such as $T$, are called parabolic. The upper half plane $\h$ has a dynamically defined boundary consisting of all ‘infinite endpoints’ of its geodesics. Considering Figure \[fig:hypplane\], this boundary is given by $\mP^1(\R)$. The action of $G$ on $\h$, as defined in , extends continuously to an action on $\h\cup \mP^1(\R)$ in the obvious way, replacing the right-hand side of  by $a/c$ if $z=\infty$ (and $c\not=0$) and by $\infty$ if $z=-d/c$. For any $s\in\C$, we define an action of $G$ on locally-defined functions on $\mP^1(\R)$ by setting $$\label{def:taus} \tau_s(g^{-1})f(t) \ccoloneqq \big(g'(t)\big)^s f(g\act t)$$ (sometimes also denoted $f\vert_{2s}g$) wherever it is defined. Let $\mc V_s^{\omega^*,\infty}$ (called the space of *smooth, semi-analytic vectors of the principal series representation with spectral parameter $s$ in the line model*) denote the space of smooth ($C^\infty$) functions $\varphi\colon\mP^1(\R)\to\C$ that are real-analytic on $\R$ up to a finite set that may depend on $\varphi$, with the action . Smoothness at the point $\infty$ here means that the map $$\tau_s(S)\varphi\colon t\mapsto |t|^{-2s} \varphi\big(-\tfrac1t\big)$$ extends smoothly to the point $0$ (recall the element $S$ from ). The vector space $Z^1_\parab(\Gamma;\mc V_s^{\omega^*,\infty})$ of *parabolic $1$-cocycles* is then the space of maps $c\colon\Gamma\to \mc V_s^{\omega^*,\infty}$ such that - for all $g,h\in\Gamma$, we have $$\label{eq:coc_relation} c_{gh} \ceqq \tau_s(h^{-1})c_g + c_h,$$ where $c_g$ denotes the function $c(g)$, and - there exists $\varphi\in \mc V_s^{\omega^*,\infty}$ such that $$c_T \ceqq \tau_s(T^{-1})\varphi - \varphi.$$ (For general discrete subgroups we would need a similar condition for representatives of each conjugacy class of parabolic elements.) The subspace $B^1_\parab(\Gamma;\mc V_s^{\omega^*,\infty})$ of *coboundaries* consists of the maps $c\colon\Gamma\to \mc V_s^{\omega^*,\infty}$ for which there exists $\varphi\in\mc V_s^{\omega^*,\infty}$ such that $$c_g \ceqq \tau_s(g^{-1})\varphi -\varphi$$ for every $g\in\Gamma$. The quotient $$H^1_\parab(\Gamma;\mc V_s^{\omega^*,\infty}) \ccoloneqq Z^1_\parab(\Gamma;\mc V_s^{\omega^*,\infty})/B^1_\parab(\Gamma;\mc V_s^{\omega^*,\infty})$$ is called the *space of parabolic $1$-cohomology classes* with values in $\mc V_s^{\omega^*,\infty}$. For any two real-analytic functions $u,v$ on $\h$ we define the *Green’s form* to be the real-analytic form $$[u,v] \ccoloneqq \frac{\partial u}{\partial z} \cdot v\cdot dz \cplus u\cdot\frac{\partial v}{\partial {\overline z}}\cdot d\overline z,$$ which is easily seen to be closed if $u$ and $v$ are eigenfunctions of $\Delta$ with the same eigenvalue. For any $s\in\C$ and any $t\in\R$ the function $R(t;\cdot)^s\colon \h\to\C$, where $$R(t;z)\ccoloneqq \Ima\frac{1}{t-z}\,,$$ is a eigenfunction with eigenvalue $s(1-s)$. Therefore, if $u$ is a Maass cusp form with spectral parameter $s$, then for any $t\in\R$ the $1$-form $$\omega_s(u,t) \ccoloneqq \big[u, R(t;\cdot)^s\big]$$ is closed. From this it follows that, for any $g\in\Gamma$, the integral $$\label{eq:paramintegral} c_g^u(t) \ccoloneqq \int_{g^{-1}\infty}^\infty \omega_s(u,t)$$ is independent of the chosen path from $g^{-1}\infty$ to $\infty$. The integral is convergent due to the rapid decay of $u$ at the cusp. The regularities of $u$ and $R(\cdot\,;\cdot)^s$ yield . Furthermore, the invariance of $u$ implies the transformation formula $$\label{eq:transform} \tau_s(g)\int_a^b \omega_s(u,t) \ceqq \int_{g\cdot a}^{g\cdot b} \omega_s(u,t) \qquad (g\in\Gamma,\ a,b\in\mP^1(\R))$$ and from this one easily deduces that the map $c^u$ satisfies the cocycle relation  and hence is a parabolic cocycle. Then we have: \[thm:BLZ\] For $s\in\C$, $\Rea s\in (0,1)$, the map $u\mapsto [c^u]$ defines a bijection $$\{\text{Maass cusp forms with spectral parameter $s$}\}\, \overset{\thicksim}{\longrightarrow}\, H^1_\parab(\Gamma;\mc V_s^{\omega^*,\infty}).$$ Discretization of geodesics {#sec:discretization} =========================== In this section we will discuss the second step in the passage from geodesics on the modular surface $X$ to Maass cusp forms for $\Gamma$: The construction of a discretization of the motion along the geodesics on $X$. We will show that the *discrete dynamical system* $$F\colon (0,\infty)\smallsetminus\Q \to (0,\infty)\smallsetminus\Q$$ given by the two branches $$\label{def:F} \begin{cases} (0,1)\smallsetminus\Q \stackrel{\sim}{\longrightarrow} (0,\infty)\smallsetminus\Q, & x\mapsto T_1^{-1} x = \frac{x}{1-x} \\[2mm] (1,\infty)\smallsetminus\Q \stackrel{\sim}{\longrightarrow} (0,\infty)\smallsetminus\Q, & x\mapsto T_2^{-1} x = x-1 \end{cases}$$ can be thought of as a discrete version of the geodesic flow on $X$: The map $F$ and its iterates capture the essential geometric and dynamical properties of the geodesic flow that will be needed for establishing the relation between the geodesics on $X$ and the Maass cusp forms for $\Gamma$. In particular, the orbits of the map $F$ describe the future behavior of (almost all) geodesics on $X$, and periodic geodesics on $X$ correspond to points $x\in(0,\infty)\smallsetminus\Q$ with periodic (i.e., finite) orbits under $F$.[^2] The construction of $F$ from the geodesic flow on $X$ proceeds in several steps: We first choose a ‘good’ cross section (in the sense of Poincaré) for the geodesic flow on $X$, i.e., a subset $\wh C$ of the unit tangent bundle of $X$ that is intersected by all periodic geodesics at least once, and each intersection between any geodesic on $X$ and $\wh C$ is discrete. We refer to the discussion below for precise definitions. The choice of $\wh C$ yields a first return map, which is the map that assigns to each element $\wh v\in\wh C$ the next intersection between $\wh C$ and the geodesic on $X$ starting at time $0$ in the direction $\wh v$. The first return map provides a first discretization of the geodesic flow on $X$. Then we choose a ‘good’ set of representatives for $\wh C$, i.e., a subset $C^*$ of the unit tangent bundle of $\h$ that is bijective to $\wh C$ with respect to the canonical quotient map. The specific properties of $C^*$ will allow us to semi-conjugate the first return map to a map on $(0,\infty)\smallsetminus\Q$, which is precisely the map $F$. The construction we will present below is a special case of the algorithm in [@Pohl_Symdyn2d] for finding good discretizations for geodesic flows on much much general hyperbolic surfaces. We refer to [@Pohl_Symdyn2d] for further details and all omitted proofs. As in Section \[sec:discretization\], readers who want to proceed faster to the final result are invited to skip the remaining part of this section. In Section \[sec:TO\] only the map $F$ will be needed, not the details of its construction. Geodesics. ---------- While in Section \[sec:mod\_surface\] we used the notion of geodesics in the sense of (G\[geod2\]) (adapted to the hyperbolic plane and the modular surface in place of the real line and the torus), we now also need geodesics in the sense of (G\[geod1\]). A geodesic $\gamma$ on $\h$ in the sense of (G\[geod1\]) is completely determined by requiring that it passes through a given point $z\in\h$ at time $t=0$ in a given direction. Recall that we consider only geodesics of unit speed, so that the speed in the given direction does not form another parameter. Therefore we may identify geodesics in the sense of (G\[geod1\]) with the set of all unit length direction vectors at all points of $\h$, thus, with the *unit tangent bundle* $S\h$ of $\h$. For $v\in S\h$ we let $\gamma_v\colon\R\to\h$ be the geodesic on $\h$ such that $$\label{def:geodH} \gamma_v'(0)=v.$$ Both the tangent vector $\gamma_v'(0)$ to $\gamma_v$ at time $t=0$ and the element $v\in S\h$ are combinations of position and direction, the position $\gamma_v(0)$ being the *base point* $\base(v)\in\h$. The *geodesic flow* on $\h$ (the motion along geodesics on $\h$) is the map $$\label{def:geodflowH} \R\times S\h \to S\h,\quad (t,v)\mapsto \gamma_v'(t).$$ The action of $G$ on $\h$ by Riemannian isometries induces an action of $G$ on $S\h$ by $$g\act v \ccoloneqq (g\act \gamma_v)'(0) \qquad (g\in G,\ v\in S\h).$$ The *unit tangent bundle* of $X$ is then just the quotient $$SX \ceqq \Gamma\backslash S\h.$$ We denote the projection map $$\label{def:projmaptangent} \pi\colon S\h\to SX$$ with the same symbol as the projection map $\h\to X$ of . The context always clarifies which one is meant. We typically denote a geodesic on $\h$ by $\gamma$ and a unit tangent vector in $S\h$ by $v$, and use $\wh\gamma$ and $\wh v$ for the corresponding geodesic $\pi(\gamma)$ on $X$ and unit tangent vector $\pi(v)\in SX$. In analogy with , for any $\wh v\in SX$ we let $\wh\gamma_v$ denote the geodesic on $X$ determined by $$\wh\gamma_v'(0) = \wh v.$$ Also the *geodesic flow* on $X$ is inherited from the geodesic flow on $\h$ as defined in , and hence is the map $$\R\times SX\to SX,\quad (t,\wh v)\mapsto \wh\gamma_v'(t).$$ Cross section. -------------- By a *cross section* we mean (slightly deviating from the standard definition) a subset $\wh C$ of $SX$ such that 1. every periodic geodesic on $X$ intersects $\wh C$. In other words, for any periodic geodesic $\wh\gamma$ there exists $t\in\R$ such that $\wh\gamma'(t) \in \wh C$. 2. each intersection of any geodesic on $X$ with $\wh C$ is discrete. In other words, for any geodesic $\wh\gamma$ and $t\in\R$ with $\wh\gamma'(t)\in\wh C$ there exists $\eps>0$ such that $$\wh\gamma'\big((t-\eps, t+\eps)\big) \cap \wh C = \big\{\wh\gamma'(t)\big\}.$$ We define a *set of representatives* $C^*$ for a cross section $\wh C$ to be a subset of $S\h$ that is bijective to $\wh C$ under the projection map $\pi$ from . (We write $C^*$ rather than $C$ because the latter traditionally denotes the full preimage of $\wh C$ in $S\h$.) Of course, to characterize a cross section $\wh C$ it suffices to provide a set of representatives, but choosing a cross section and a set of representatives that serves our purposes is an art. For the modular surface we will take $$C^* \ccoloneqq \{ v\in S\h \mid \base(v)\in i\R^+,\ \gamma_v(\infty) \in (0,\infty)\smallsetminus\Q \}$$ as set of representatives, where $$\gamma_v(\infty) \ccoloneqq \lim_{t\to\infty} \gamma_v(t).$$ ![The set of representatives $C^*$ and the cross section $\wh C$. The gray shadows indicate the directions of the elements of $\wh C$ and $C^*$.[]{data-label="fig:crosssection"}](orbicross){width="10.34cm"} The associated cross section $$\wh C \ccoloneqq \pi(C^*)$$ is the set of unit tangent vectors $\wh v\in SX$ sitting on the geodesic from $\pi(i)$ to $\pi(\infty)$ such that the geodesic emanating from $\wh v$ does not converge to the cusp $\pi(\infty)$ in future or past time. A pictorial representation of $C^*$ and $\wh C$ is given in Figure \[fig:crosssection\]. Discretization. --------------- We will now show how to relate the geodesic flow on $X$ to a discrete dynamical system on (a subset of) $\R_{>0}$. In the case of the modular surface, this construction is closely related to continued fractions, more precisely to Farey fractions. The reader interested in this connection may find the articles [@Artin; @Richards; @Series; @Katok_Ugarcovici] useful. Let $\wh v\in \wh C$ be an element of the cross section and consider the associated geodesic $\wh\gamma_v$ on $X$. By the choice of $\wh C$, the geodesic $\wh\gamma_v$ intersects $\wh C$ again in future time. Let $t_0>0$, the *first return time*, be the minimal positive number such that $$\wh w\ccoloneqq\wh\gamma_v'(t_0) \in \wh C.$$ (See Figure \[fig:cross2\].) ![The geodesic determined by $\wh v$ and its first return to $\wh C$.[]{data-label="fig:cross2"}](orbigeodesic2){height="6.5cm"} Let $v,w\in C^*$ be the elements in the set of representatives corresponding to $\wh v, \wh w$, and $\gamma_v, \gamma_w$ the associated geodesics on $\h$. (See Figure \[fig:next\].) ![Associated geodesics on $\h$.[]{data-label="fig:next"}](nextinter){height="5.8cm"} Since the unit tangent vector $\gamma_v'(t_0) \in S\h$ projects to $\wh w$ under $\pi$, that is, $$\pi\big( \gamma_v'(t_0) \big) \ceqq \wh w,$$ there exists a unique element $g\in\Gamma$ such that $$\gamma_v'(t_0) \ceqq g\act w\,.$$ This element is characterized by $$\label{eq:nextelem} \gamma_v'(t_0) \in g\act C^*,$$ i.e., by the first intersection of $\gamma_v$ with some $\Gamma$-translate of $C^*$ after passing through $v$. To find the element $g$ we consider the neighboring translates of the fundamental domain $\mc F$ and the relevant translates of $C^*$. ![Relevant $\Gamma$-translates of $\mc F$ and $C^*$.[]{data-label="fig:forward"}](forward){height="6.6cm"} We observe that the unit tangent vector $\gamma_v'(t_0)$ can be only in $T_1\act C^*$ or $T_2\act C^*$, where $$T_1 \ceqq \bmat{1}{0}{1}{1}\qquad\text{and}\qquad T_2\ceqq\bmat{1}{1}{0}{1},$$ as shown in Figure \[fig:forward\]. Explicitly, this unit tangent vector is in $T_1\act C^*$ if and only if $\gamma_v(\infty) \in (0,1)$, and it is in $T_2\act C^*$ if and only if $\gamma_v(\infty) \in (1,\infty)$. ![Next intersection.[]{data-label="fig:nextex"}](nextexample){height="5.5cm"} In Figure \[fig:nextex\] we have $g=T_1$, so that here $$w \ceqq T_1^{-1}\gamma_v'(t_0), \qquad \gamma_w(\infty) \ceqq T_1^{-1}\gamma_v(\infty).$$ We further observe that for every point $x\in (0,\infty)\smallsetminus\Q$, no matter which with $\gamma_v(\infty)=x$ we consider, we find the same value for the element $g\in\Gamma$ in . In other words, $g$ only depends on $x$, not on the specific element $v\in C^*$ with $\gamma_v(\infty)=x$. Therefore the procedure just described induces a *discrete dynamical system* $$\label{def:F2} F\colon (0,\infty)\smallsetminus\Q \to (0,\infty)\smallsetminus\Q,$$ where for each $x\in (0,\infty)\smallsetminus\Q$, we pick $v\in C^*$ such that $\gamma_v(\infty)=x$, let $g$ be the element in $\Gamma$ such that $\gamma_v'(t_0)\in g\act C^*$ and set $$F(x) \ccoloneqq g^{-1}\act x.$$ The set $\wh C$ is a cross section for the geodesic flow on $X$, and $C^*$ is a set of representatives for $\wh C$. The induced discrete dynamical system (as in ) is the map $F$ as given in . Transfer operators and Maass cusp forms {#sec:TO} ======================================= In this section we carry out the third and final step in the passage from geodesics on the modular surface $X$ to Maass cusp forms for $\Gamma$: Tie together the discrete dynamical system $F$ from Section \[sec:discretization\] and the parabolic interpretation of Maass cusp forms from Section \[sec:mcf\]. The mediating object between both sides is the *transfer operator family* $(\TO_s)_{s\in\C}$ associated to $F$. The *transfer operator* $\TO_s$ with parameter $s$ is the operator $$\label{def:TO} \TO_sf(t) \ccoloneqq \sum_{w\in F^{-1}(t)} |F'(w)|^{-s} f(w),$$ acting on functions $f\colon (0,\infty)\to\C$. This operator has its origin in the thermodynamic formalism of statistical mechanics. It is a generalization of the transfer matrix for systems, which is used to find equilibrium distributions. The weight, in particular its dependence, is motivated within this framework, where $s$ serves as an inverse Boltzmann constant and temperature. From a purely mathematical point of view, this operator can be seen as an evolution operator or as a graph Laplacian on a somewhat generalized graph, in both cases with appropriate weights. The explicit expression for $F$ allows us to evaluate  in our special case to $$\TO_sf(t) \ceqq f(t+1) \cplus (t+1)^{-2s} f\Bigl(\frac{t}{t+1}\Bigr),\qquad t>0,$$ or, using , to $$\TO_s \ceqq \tau_s(T_1^{-1}) \cplus \tau_s(T_2^{-1}).$$ (This simple formula is for the modular group only. For other groups one can have a vector of more complicated finite sums.) The correspondence that we have been aiming at is a bijection between the eigenfunctions of $\TO_s$ with eigenvalue $1$ and the Maass cusp forms with spectral parameter $s$. More precisely, we have the following theorem. Let $s\in\C$, $1>\Rea s>0$. Then the Maass cusp forms with spectral parameter $s$ are bijective to the real-analytic eigenfunctions $f$ of $\TO_s$ for which the map $$\label{eq:mapdecay} \begin{cases} f & \text{on $(0,\infty)$} \\ -\tau_s(S)f & \text{on $(-\infty,0)$} \end{cases}$$ extends smoothly to $0$. If $u$ is a Maass cusp form with spectral parameter $s$ then the associated eigenfunction of $\TO_s$ is $$\label{eq:f_path} f(t) \ccoloneqq \int_0^\infty \omega_s(u,t).$$ We will now explain the main steps of the proof with an emphasis on intuition and heuristics. Some steps will be omitted, most prominently some discussions of convergence and regularities. We hope to convince the reader that a major part of the proof is encoded in Figure \[fig:backwards\] and that the choice of the integral path in and the function in is natural. ![Relevant $\Gamma$-translates for proof of Theorem.[]{data-label="fig:backwards"}](backwards){width="6.5cm"} The exposition will show that the bijection claimed in the Theorem is not just proven by showing that the dimensions of both spaces are equal; we will provide an explicit map. **Proof (key elements).** We present the main part of the proof, split into four steps. **Step 1: Relation between $\TO_s$ and $C^*$.** We first reconsider the transfer operator $\TO_s$ and its domain. Let $f\colon (0,\infty)\to\C$ be a function in the domain of $\TO_s$. We may think of $f$ as being a mass distribution or density on $(0,\infty)$ of which the transfer operator evaluates its weighted evolution under one application of $F$. Recalling that $F$ is a discrete version of the geodesic flow on $X$, that $\TO_s$ is a weighted evolution operator of $F$, and that the essential ingredient of this discretization is the set $C^*$, we may intuitively think of $f$ as being a ‘shadow’ of some function $f^*$ on $C^*$ that is constant on any set of the form $$E_t\ccoloneqq \{ v\in C^* \mid \gamma_v(\infty) = t\}\qquad (t\in (0,\infty)).$$ Thus, $$f(t) \ceqq f^*(v) \qquad\text{for any $v\in E_t$.}$$ When developing the formula for $F$ we asked where the geodesics determined by the elements in $C^*$ go to. In the expression for $\TO_s$, the preimage of $F$ is used. Hence, when building $\TO_s$, we may alternatively ask where these geodesics come from. For the modular group $\Gamma$, the relevant sets are $T_1^{-1}C^*$ and $T_2^{-1}C^*$. (See Figure \[fig:backwards\].) **Step 2: Relation between Maass cusp forms and $C^*$.** Let $u$ be a Maass cusp form with spectral parameter $s$. We use the characterization of $u$ via a cocycle class in the space $H^1_\parab(\Gamma;\mc V_s^{\omega^*,\infty})$ from the Theorem in Section \[sec:mcf\], and then use the family of functions $(c_g^u)_{g\in\Gamma}$ from as a representative for this cocycle class. We think of each $c_g$ as being the integral along the geodesic from $g^{-1}\infty$ to $\infty$, or even better, as an integral over the set of unit tangent vectors to this geodesics. In particular, for $g=S$ we have $S^{-1}\infty=0$, so that $$\label{eq:cS} c_S^u(t) \ceqq \int_0^\infty \omega_s(u,t)\qquad (t\in\R)$$ is the integral along the geodesic from $0$ to $\infty$. Thus, in an intuitive way, we may think of $c_S^u$ as an integral over $C^*\cup S\act C^*$. **Step 3: From Maass cusp forms to eigenfunctions of $\TO_s$.** Let $u$ be a Maass cusp form with spectral parameter $s$ with associated function vector $(c_g^u)_{g\in\Gamma}$. We want to associate to $u$ in a natural way an eigenfunction $f$ of $\TO_s$ with eigenvalue $1$. The intuitive way of thinking of $c^u_S$ and any function $f$ as objects on $C^*$ suggests using $C^*$ as linking pin. Staying in this intuition, we should restrict $c_S^u$ to an integral over $C^*$ and use $f^* = c_S^u\vert_{C^*}$. In terms of the actual objects (and their rigorous definitions) we are led to set $$\label{eq:f_heur} f \ccoloneqq c_S^u\vert_{(0,\infty)},$$ which is precisely . We now show that indeed defines an eigenfunction of $\TO_s$ with eigenvalue $1$. So far we have used in , and hence in , the geodesic from $0$ to $\infty$ as path of integration. Since the form $\omega_s(u,t)$ is closed, we may change the path to be the geodesic from $0$ to $-1$ followed by the geodesic from $-1$ to $\infty$: $$\int_0^\infty \omega_s(u,t) \ceqq \int_0^{-1} \omega_s(u,t) \cplus \int_{-1}^\infty \omega_s(u,t).$$ Using the transformation formula  we now find $$\begin{aligned} f(t) & \ceqq \int_0^\infty \omega_s(u,t) \\ & \ceqq \int_{T_1^{-1}0}^{T_1^{-1}\infty} \omega_s(u,t) \cplus \int_{T_2^{-1}0}^{T_2^{-1}\infty}\omega_s(u,t) \\ & \ceqq \tau_s(T_1^{-1}) \int_0^\infty \omega_s(u,t) \cplus \tau_s(T_2^{-1}) \int_0^\infty \omega_s(u,t) \\ & \ceqq \tau_s(T_1^{-1})f(t) \cplus \tau_s(T_2^{-1})f(t).\end{aligned}$$ Therefore $f=\TO_sf$. **Step 4: From eigenfunctions of $\TO_s$ to Maass cusp forms.** Conversely, let $f$ be an eigenfunction $\TO_s$ with eigenvalue $1$. We want to associate to $f$ a Maass cusp form $u$ in a way which inverts the mapping from above and which is also natural. Instead of trying to do this directly, we will define a parabolic $1$-cocycle $c=c^f$ in $Z^1_\parab(\Gamma;\mc V_s^{\omega^*,\infty})$. The Theorem in Section \[sec:mcf\] then implies that the cocycle $c$ is indeed of the form $c=c^u$ for a unique Maass cusp form $u$. In order to define $c$ we prescribe it on the group elements $T$ and $S$ by setting $$c_T \ccoloneqq 0,$$ which is motivated by , and $$\label{eq:cs_from_f} c_S \ccoloneqq \begin{cases} f & \text{on $(0,\infty)$} \\ -\tau_s(S)f & \text{on $(-\infty,0)$,} \end{cases}$$ according to the heuristic above. The minus sign in the second row is motivated by the fact that $S$ ‘changes the direction’ of the geodesic from $0$ to $\infty$. In order to define $c_S$ also at $0$ (and $\infty$), we need to require that the right hand side of , equivalently , extends smoothly to $0$. Since $T$ and $S$ generate all of $\Gamma$, the cocycle relation  dictates the value of $c$ on all other elements. It remains to show that $c$ is well-defined, which here means that if a combination of $T$ and $S$ equals the identity then the corresponding combination of $c_T$ and $c_S$ vanish. To that end we use the presentation $$\Gamma \ceqq \left\langle\ S,\;T \ \left\vert\ S^2 = \big(T^{-1}S\big)^3 = \id\ \right.\right\rangle$$ and show that $$\tau_s(S)c_S\cplus c_S \quad\text{and}\quad \big(\tau_s\big((ST)^2\big) + \tau_s(ST) + 1\big)\big(\tau_s(S)c_{T^{-1}} + c_{S}\big)$$ vanish identically. For the first expression, this follows immediately from . For the second expression we use $c_T=0$ and find $$\begin{aligned} \big(\tau_s\big((ST)^2\big) & + \tau_s(ST) + 1\big)\big(\tau_s(S)c_{T^{-1}} + c_{S}\big) \\ & \ceqq \tau_s\big((ST)^2\big)c_S \cplus \tau_s(ST)c_S \cplus c_S \\ & \ceqq \begin{cases} -\tau_s\big(T_2^{-1}\big)f - \tau_s\big(T_1^{-1}\big)f + f & \text{on $(0,\infty)$} \\ \tau_s\big(T_1^{-1}S\big) \left[ -\tau_s\big(T_1^{-1}\big)f + f - \tau_s\big(T_2^{-1}\big)f\right] & \text{on $(-1,0)$} \\ \tau_s\big(T^{-1}S\big) \left[ f - \tau_s\big(T_2^{-1}\big)f - \tau_s\big(T_1^{-1}\big)f\right] & \text{on $(-\infty,-1)$,} \end{cases}\end{aligned}$$ which vanishes since $f=\TO_sf$. This calculation ![Relevant $\Gamma$-translates for proof of Theorem.[]{data-label="fig:readoff"}](cocyclesgeom){height="5cm"} can also be read off from Figure \[fig:readoff\], as the reader can verify. Recapitulation and closing comments {#sec:recap} =================================== We have surveyed an intriguing relation between the periodic geodesics on the modular surface $X=\Gamma\backslash\h$ (‘classical mechanical objects’) and the Maass cusp forms for $\Gamma$ (‘quantum mechanical objects’). For this, we started simultaneously on both ends: On the geometric side, we developed a discrete version of the (periodic part of the) geodesic flow on the modular surface by means of a cross section in the sense of Poincaré. We realized this discretization as a discrete dynamical system on $(0,\infty)$ by using a well-chosen representation of the cross section on the upper half plane. This step turns the geodesic flow into a discrete and somehow finite object while preserving its essential dynamical features. On the spectral side, we characterized the Maass cusp forms as cocycle classes in a certain cohomology space. The isomorphism from Maass cusp forms to cocycle classes is given by an integral transform, where a certain form is integrated along certain geodesics. Even though the cocycle classes remain objects of quantum mechanical nature, this characterization of Maass cusp forms constitutes a first and very important step towards the geometry and dynamics of the modular surface. Connecting these two sides is the family of transfer operators, which from their definition are purely classical mechanical objects but which clearly exhibit a quantum mechanical nature. These transfer operators depend heavily on the choice of the discretization. The proof of the isomorphism between eigenfunctions of the transfer operators and the parabolic cocycles clearly shows that the shape of the set of representatives is crucial. Here, it is the set of (almost) all unit tangent vectors that are based on the geodesic from $0$ to $\infty$ and that point ‘to the right’. This set of representatives and its $\Gamma$-translates can be seen as a geometric realization of the cohomology. The transfer operator then encodes the cocycle relation. An eigenfunction with eigenvalue $1$ of the transfer operator obeys a geometric variant of the cocycle relation, and hence can be related to an actual cocycle, which in turn characterizes a Maass cusp form. [^1]: Another widely known model for the hyperbolic plane is the Poincaré disk model, which prominently features in several of M. C. Escher’s pictures. [^2]: We remark that the formula for $F$ is identical to the map $\Phi$ given in [@Choie_Zagier Section 1.1, Lemma] in connection with the so-called rational period functions.

--- abstract: 'We return to the description of the damped harmonic oscillator by means of a closed quantum theory with a general assessment of previous works, in particular the Bateman-Caldirola-Kanai model and a new model recently proposed by one of the authors. We show the local equivalence between the two models and argue that latter has better high energy behavior and is naturally connected to existing open-quantum-systems approaches.' author: - 'M.C. Baldiotti[^1], R. Fresneda[^2], and D.M. Gitman[^3]' bibliography: - '/home/fresneda/work/mendeley.bib/library.bib' title: Quantization of the Damped Harmonic Oscillator Revisited --- Instituto de Física, Universidade de São Paulo,\ Caixa Postal 66318-CEP, 05315-970 São Paulo, S.P., Brasil Introduction ============ The problem of constructing a quantum theory for the damped harmonic oscillator as well as for similar dissipative systems, e.g. radiating point-like charge, has attracted attention for already more than 50 years. In spite of the success of interaction-with-reservoir approaches, we feel there is still room for some formal developments in the direction of a closed theory approach. In this article we analyze a novel Lagrangian model for the damped harmonic oscillator, which was recently proposed in [@Gitman2007kg] as a particular case of a general procedure for finding action functionals for non-Lagrangian equations of motion. Even though we find it is locally equivalent to the renowned Bateman-Caldirola-Kanai (BCK) model [@Bateman1931; @Caldirola1941; @Kanai1950] (see Section \[sec:Bateman-Caldirola-Kanai-Theory\] for a revision), a complete global equivalence is absent, and we believe it has some formal advantages over its predecessor, regarding the high energy behavior of solutions to the Schrödinger equation. Notwithstanding these formal discrepancies between the two models, we show that they share a very close physical interpretation with regard to their asymptotic behavior in time and to their physical functions. Ever since the proposition of the BCK model, there have been divided opinions as to whether it really describes the damped harmonic oscillator or dissipation. There are those who dispute it as a possible dissipative model [@Senitzky1960; @Ray1979; @Greenberger1979], and those, in addition to the original authors, who maintain it accounts for some form of dissipation [@Dodonov1978; @Dodonov1979]. As we explain further on, the BCK theory is not defined globally in time, and many of the pathologies usually appointed to the quantum theory can be seen as an artifact of its infinite time indefinition. Nonetheless, regardless of these disputes, there has been fruitful applications of the BCK theory as a model of dissipation, at least in the case of the canonical description of the Fabry-Pérot cavity [@Colegrave1981]. The underlying understanding of dissipative systems is that they are physically part of a larger system, and dissipation is a result of a non-elastic interaction between the reservoir and the subsystem. Thus, quantization can conceivably follow two different approaches: the first one takes the classical equations of motion of the system and applies to them formal quantization methods, trying to overcome the difficulties related to the “non-Lagrangian” nature of the system. Without attempting to exhaust all the literature on this matter, we cite, for instance, the use of canonical quantization of classical actions [@Hasse1975; @Okubo1981], Fermi quantization [@Latimer2005], path-integral quantization [@Kochan2010], doubling of the degrees of freedom [@BLASONE2004; @CELEGHINI1992], group theory methods [@Villarroel1984], complex classical coordinates [@Dekker1977], propagator methods [@Tikochinsky1978], non-linear Schrödinger equation [@kostin1972], and finally a constrained dynamics approach [@Gitman2007b]. The second approach aims at constructing a quantum theory of the subsystem by averaging over the reservoir, see e.g. [@Senitzky1960; @*Feynman1963; @*Haken1975; @*Caldeira:1982iu; @*Pedrosa1984; @*Walls1985; @*Grabert1988; @*Yu1994; @*Joichi1997]. It is immediately clear that two different outcomes must follow from these approaches. Indeed, in the first case, we consider the system as being closed, and thus naturally only pure states can result, and they represent physical states without any further restrictions. In the second case, the subsystem’s states will be necessarily described by a statistical operator, such that only mixed states are physically sensible. In spite of the general understanding that the second approach seems to be more physical and probably should produce a more adequate quantum theory for dissipative systems, in this article we wish once again to analyze the first, formal, approach. Its comparison to the second one will be given in a future work. Our analysis will be particular to the damped harmonic oscillator as an example of a dissipative system, and it will be devoted to the comparison of two different canonical quantizations. In Section 2 we revise the BCK theory, emphasizing problems which were not considered before. In Section 3 we present the first-order theory, define some useful physical quantities, and we also construct the coherent and squeezed states of the first-order theory in order to obtain the classical limit. In Section 4, we prove the local equivalence of the theories presented in Sections 2 and 3, and discuss possible divergences related to their global behavior. In Section 5 we present some final remarks and discuss our results as well as open problems. BCK Theory\[sec:Bateman-Caldirola-Kanai-Theory\] ================================================ One of the peculiarities of dissipative systems, which hindered early quantization attempts, was the non-Lagrangian nature of the classical equations of motion. In the particular case of the damped harmonic oscillator of constant frequency $\omega$ and friction coefficient $\alpha>0$, the second-order equation of motion$$\ddot{q}+2\alpha\dot{q}+\omega^{2}q=0\label{eq:eqm-dho}$$ cannot be directly obtained as the Euler-Lagrange (EL) equation of any Lagrangian, since it fails to satisfy the Helmholtz conditions [@Helmholtz1887]. Nevertheless, there is an equivalent second-order equation for which a variational principle can be found, namely,$$e^{2\alpha t}\left(\ddot{q}+2\alpha\dot{q}+\omega^{2}q\right)=0\,.\label{eq:equiv-dho-eqm}$$ The exponential factor is known as the integrating multiplier, and it is enough to make the above equation satisfy the Helmholtz conditions [@Starlet1982]. The fact that a Lagrangian can always be found for the one-dimensional problem such that its EL equation is equivalent to a given second-order equation was established by Darboux [@Darboux1894]. As was already mentioned, the equation (\[eq:eqm-dho\]) is traditionally considered to be non-Lagrangian, albeit the existence of a questionable action functional [@Ray1979; @Greenberger1979] which reproduces the equivalent equations of motion (\[eq:equiv-dho-eqm\]). In this respect, we have to mention that an action principle for the equation of motion (\[eq:equiv-dho-eqm\]) was first proposed by Bateman [@Bateman1931] in terms of the Lagrangian $$L_{B}=\frac{1}{2}\left(\dot{q}^{2}-\omega^{2}q^{2}\right)e^{2\alpha t}\,.\label{eq:caldirola-kanai}$$ If Bateman had constructed the corresponding Hamiltonian formulation, he would have discovered that the corresponding Hamiltonian theory is canonical without constraints and with Hamiltonian$$H_{BCK}\left(q,p\right)=\frac{1}{2}\left[e^{-2\alpha t}p^{2}+\omega^{2}e^{2\alpha t}q^{2}\right]\,.\label{eq:caldirola-kanai-hamiltonian}$$ This Hamiltonian was proposed independently by Caldirola and Kanai [@Caldirola1941; @Kanai1950] to describe the damped harmonic oscillator in the framework of quantum mechanics. Consequently, we write the subscript BCK (Bateman-Caldirola-Kanai) to label the Hamiltonian. Formal canonical quantization of the Lagrangian action (\[eq:caldirola-kanai\]) is straightforward,$$\left[\hat{q},\hat{p}\right]=i\,,\,\,\left[\hat{q},\hat{q}\right]=\left[\hat{p},\hat{p}\right]=0,\ \hat{H}_{BCK}=H_{BCK}\left(\hat{q},\hat{p}\right)\,,\label{eq:BCK-quantum-theory}$$ and coincides with Caldirola and Kanai’s quantum theory. Solutions to the Schrödinger equation with Hamiltonian $\hat{H}_{BCK}$ have been found in the form $$\psi_{n}^{BCK}\left(q,t\right)=\left(2^{n}n!\right)^{-1/2}\left(\frac{\tilde{\omega}}{\pi}\right)^{1/4}\exp\left(-iE_{n}t+\frac{\alpha t}{2}-\left(\tilde{\omega}+i\alpha\right)\frac{q^{2}}{2}e^{2\alpha t}\right)H_{n}\left(\sqrt{\tilde{\omega}}qe^{\alpha t}\right)\,,\label{eq:pseudostationary-state}$$ where $E_{n}=\tilde{\omega}\left(n+1/2\right)$, $\tilde{\omega}=\sqrt{\omega^{2}-\alpha^{2}}$ and $H_{n}$ are Hermite polynomials. These are the familiar pseudostationary states [@Kerner1958; @Bopp.F.1962; @Hasse1975] or loss-energy states [@Dodonov1978] which are also eigenstates of the Hamiltonian $\hat{H}_{BCK}+\frac{\alpha}{2}\left(\hat{q}\hat{p}+\hat{p}\hat{q}\right)$ with eigenvalues $E_{n}$. Even though $\left\vert \psi_{n}^{BCK}\right\vert ^{2}$ depends on time, the total probability $\int dq\left\vert \psi_{n}^{BCK}\right\vert ^{2}=1$ is time-independent, as can be seen by the transformation of variables $q\mapsto q^{\prime}=qe^{\alpha t}$. Moreover, the mean value of the BCK Hamiltonian in the pseudostationary states is constant, $$\left\langle \psi_{n}^{BCK}\right\vert \hat{H}_{BCK}\left\vert \psi_{n}^{CK}\right\rangle =\left\langle \psi_{n}^{BCK}\right\vert E_{n}-\frac{\alpha}{2}\left(\hat{q}\hat{p}+\hat{p}\hat{q}\right)\left\vert \psi_{n}^{BCK}\right\rangle =\frac{\omega^{2}}{\tilde{\omega}}\left(n+\frac{1}{2}\right)\,,\label{eq:average-ck-hamiltonian}$$ which is a reflection of the fact that in the classical theory defined by (\[eq:caldirola-kanai\]) the average of the Hamiltonian over the period of one oscillation is constant. On the other hand, mean values of the mechanical energy $E=\frac{1}{2}\left(\dot{q}^{2}+\omega^{2}q^{2}\right)$ decay exponentially with time, $\left\langle E\right\rangle _{n}=e^{-2\alpha t}\left\langle H_{BCK}\right\rangle _{n}$ [@Kerner1958; @Hasse1975]. Coherent states for the BCK theory are given in [@Dodonov1979], for which the uncertainty relations are$$\Delta q\Delta p=\frac{\omega}{2\tilde{\omega}}\geq\frac{1}{2}\,.\label{eq:manko-uncertainty}$$ Now we draw attention to the high-energy behavior of pseudostationary states. In the appendix we consider asymptotic (in $n$) pseudostationary functions, and for these one has the limiting eigenvalue equation$$\hat{H}_{BCK}\psi_{n}^{BCK}=\left[E_{n}-\frac{\alpha}{2}\left(\hat{q}\hat{p}+\hat{p}\hat{q}\right)\right]\psi_{n}^{BCK}=\left(E_{n}+\frac{i\alpha}{2}\right)\psi_{n}^{BCK}+O\left(n^{-1/4}\right)\,.\label{eq:limit-eigenequation}$$ The appearance of imaginary eigenvalues is actually explained by taking into account the domain of the operator $\hat{q}\hat{p}+\hat{p}\hat{q}$, which, as is shown in the appendix, does not include the asymptotic part of $\psi_{n}^{BCK}$. In this connection, it should be noted that the above is in contrast to [@Dodonov1978; @Dodonov1979a], where it is claimed that the pseudostationary states are eigenstates of $\hat{H}_{BCK}$ with eigenvalues $\left(\tilde{\omega}+i\alpha\right)\left(n+1/2\right)$. One overlooked aspect of the BCK theory is that EL equation (\[eq:equiv-dho-eqm\]) obtained from the Lagrangian (\[eq:caldirola-kanai\]) is only equivalent to the equation of motion of the damped harmonic oscillator (\[eq:eqm-dho\]) for finite times. The equation (\[eq:equiv-dho-eqm\]) is indicative that the theory described by (\[eq:caldirola-kanai\]) is not globally defined, i.e., it is not defined for infinite times. In effect, the manifold difficulties which appear in connection to the $t\rightarrow\infty$ limit, such as the violation of the Heisenberg uncertainty principle and the vanishing of the ground state energy for infinite times, can be seen as consequence of inadvertently assuming equations (\[eq:eqm-dho\]) and (\[eq:equiv-dho-eqm\]) are equivalent for all values of the time parameter. First-order action\[sec:First-order-action\] ============================================ Action, hamiltonization, and quantization ----------------------------------------- Next we consider the canonical quantization of the damped harmonic oscillator based on the alternative action proposed in [@Gitman2007kg]. The idea was to reduce the second-order equations (\[eq:eqm-dho\]) to the first-order system$$\dot{x}=y,\ \dot{y}=-\omega^{2}x-2\alpha y\,,\label{eq:1st-order-eqm-dho}$$ for which, according to the general theory [@Gitman2007a; @Gitman2006], the action functional has the form $$S=\frac{1}{2}\int dt\left[y\dot{x}-x\dot{y}-\left(y^{2}+2\alpha xy+\omega^{2}x^{2}\right)\right]e^{2\alpha t}\,.\label{eq:gitman-kupriyanov}$$ The EL equations of motion derived from (\[eq:gitman-kupriyanov\]) are locally equivalent to (\[eq:1st-order-eqm-dho\]),$$\frac{\delta S}{\delta x}=\left(\dot{y}+2\alpha y+\omega^{2}x\right)e^{2\alpha t}\,,\,\,\frac{\delta S}{\delta y}=\left(\dot{x}-y\right)e^{2\alpha t}\,.$$ Note that, as in the case of the BCK theory, the theory fails to describe the damped harmonic oscillator as the time approaches infinity, and thus one should expect problems in the quantum theory such as violation of the uncertainty principle for infinite times, for instance. This action describes a singular system with second-class constraints, and furthermore, these constraints are time-dependent (we follow the terminology of the book [@Gitman:1990qh]). Even though the constraints are explicitly time-dependent, it is still possible to write the Hamiltonian formalism with the help of Dirac brackets and perform the canonical quantization, as is explained in [@Gitman:1990qh]. In order to do this, one must extend the initial phase space of canonical variables $\eta=\left(x,y,p_{x},p_{y}\right)$ by the inclusion of the time $t$ and its associated momentum $\varepsilon$. As a result, the Poisson brackets between functions defined on the extended phase space is$$\left\{ F,G\right\} =\left(\frac{\partial F}{\partial x}\frac{\partial G}{\partial p_{x}}+\frac{\partial F}{\partial y}\frac{\partial G}{\partial p_{y}}+\frac{\partial F}{\partial t}\frac{\partial G}{\partial\varepsilon}\right)-F\leftrightarrow G\,.$$ With the above definition, the first-order equations of motion (\[eq:1st-order-eqm-dho\]) are equivalently written in terms of Dirac brackets, $$\dot{\eta}=\left\{ \eta,H\left(x,y,t\right)+\epsilon\right\} _{D\left(\phi\right)}\,,\,\,\phi_{x}=\phi_{y}=0\,,\label{eq:constrained-eqm}$$ where the Hamiltonian $H\left(x,y,t\right)$ and the constraints $\phi$ are $$\begin{aligned} & H\left(x,y,t\right)=\frac{1}{2}\left(y^{2}+2\alpha xy+\omega^{2}x^{2}\right)e^{2\alpha t}\,,\label{eq:kg-hamiltonian}\\ & \phi_{x}=p_{x}-\frac{1}{2}ye^{2\alpha t}\,,\,\,\phi_{y}=p_{y}+\frac{1}{2}xe^{2\alpha t}\,.\nonumber \end{aligned}$$ The nonzero commutation relations between the independent variables are$$\left\{ x,y\right\} _{D\left(\phi\right)}=e^{-2\alpha t}\,.\label{eq:classical-commutators}$$ Quantization of this system follows the general method described in [@Gitman:1990qh] for theories with time-dependent constraints. One introduces time-dependent operators $\hat{\eta}\left(t\right)$ which satisfy the differential equations $d\hat{\eta}/dt=\left.i\left\{ \eta,\epsilon\right\} _{D\left(\phi\right)}\right\vert _{\eta=\hat{\eta}}$ with initial conditions subject to an analog of the Dirac quantization, $$\begin{aligned} & \left[\hat{\eta}\left(0\right),\hat{\eta}\left(0\right)\right]=\left.i\left\{ \eta,\eta\right\} _{D\left(\phi_{0}\right)}\right\vert _{\eta=\hat{\eta}\left(0\right)}\,,\\ & \hat{p}_{x}\left(0\right)-\frac{1}{2}\hat{y}\left(0\right)=\hat{p}_{y}\left(0\right)+\frac{1}{2}\hat{x}\left(0\right)=0\,.\end{aligned}$$ The above operatorial constraints allows us to work only in terms of the independent operators $\hat{x}$ and $\hat{y}$. Let us define $\hat{x}\left(0\right)\equiv\hat{q}$ and $\hat{y}\left(0\right)\equiv\hat{p}$, so that the above quantum brackets have the familiar form $$\left[\hat{q},\hat{p}\right]=i\,,\,\,\left[\hat{q},\hat{q}\right]=\left[\hat{p},\hat{p}\right]=0,\ \ \hat{x}\left(0\right)\equiv\hat{q},\ \ \hat{y}\left(0\right)\equiv\hat{p}\,.\label{eq:qp-operators}$$ The differential equations for $\hat{x}$ and $\hat{y}$ can be easily integrated to$$\begin{aligned} & \hat{x}\left(t\right)=e^{-\alpha t}\hat{q}\,,\,\,\hat{x}\left(0\right)\equiv\hat{q}\,,\,\,\hat{y}\left(t\right)=e^{-\alpha t}\hat{p}\,,\,\,\hat{y}\left(0\right)\equiv\hat{p}\,.\label{eq:operator-solutions}\end{aligned}$$ The quantum Hamiltonian is obtained from the classical Hamiltonian (\[eq:kg-hamiltonian\]) as a function of the operators $\hat{x}\left(t\right)$ and $\hat{y}\left(t\right)$ (\[eq:operator-solutions\]): it does not depend on time at all, $$\hat{H}=H\left(\hat{x}\left(t\right),\hat{y}\left(t\right),t\right)=\frac{1}{2}\left[\hat{p}^{2}+\alpha\left(\hat{q}\hat{p}+\hat{p}\hat{q}\right)+\omega^{2}\hat{q}^{2}\right]\,,\label{eq:kg-quantum-hamiltonian}$$ where we have used Weyl (symmetric) ordering for the mixed product $2\alpha xy$. The Hamiltonian $\hat{H}$ governs the time-evolution of the state vector in the Schrödinger picture, and it has appeared in a number of different contexts: in [@Stevens1958; @Colegrave1981] with regard to the electromagnetic field in a resonant cavity; in the quantization of the complex symplectic theory [@Dekker1981]; and in [@Isar1999] in connection to the Lindblad theory of open quantum systems for the damped harmonic oscillator. Since the Hamiltonian is time-independent, the evolution operator is given simply by $U\left(t\right)=e^{-i\hat{H}t}$, with $\hat{H}$ given by (\[eq:kg-quantum-hamiltonian\]). The Heisenberg operators $\check{x}$ and $\check{y}$ corresponding to the classical variables $x$ and $y$ are$$\begin{aligned} & \check{x}=U^{-1}\hat{x}\left(t\right)U=e^{-\alpha t}\left(\cos\tilde{\omega}t+\frac{\alpha}{\tilde{\omega}}\sin\tilde{\omega}t\right)\hat{q}+\frac{1}{\tilde{\omega}}e^{-\alpha t}\sin\left(\tilde{\omega}t\right)\hat{p}\,,\\ & \check{y}=U^{-1}\hat{y}\left(t\right)U=e^{-\alpha t}\left(\cos\tilde{\omega}t-\frac{\alpha}{\tilde{\omega}}\sin\tilde{\omega}t\right)\hat{p}-\frac{\omega^{2}}{\tilde{\omega}}e^{-\alpha t}\sin\left(\tilde{\omega}t\right)\hat{q}\,,\\ & \check{x}\left(0\right)\equiv\hat{q}\,,\,\,\check{y}\left(0\right)\equiv\hat{p}\,,\,\,\left[\hat{q},\hat{p}\right]=i\,,\,\,\left[\hat{q},\hat{q}\right]=\left[\hat{p},\hat{p}\right]=0\,.\end{aligned}$$ From the above expressions, one also finds$$\frac{d\check{x}}{dt}=\check{y}\,,\,\frac{d\check{y}}{dt}=-\omega^{2}\check{x}-2\alpha\check{y}\,,$$ which coincide in form with the classical equations,$$\frac{d\check{x}}{dt}=\left.\left\{ x,H+\epsilon\right\} _{D\left(\phi\right)}\right\vert _{\eta=\check{\eta}}\,,\,\,\frac{d\check{y}}{dt}=\left.\left\{ y,H+\epsilon\right\} _{D\left(\phi\right)}\right\vert _{\eta=\check{\eta}}\,.\label{eq:heisenberg-equations}$$ Thus, Heisenberg equations (\[eq:heisenberg-equations\]) reproduce the classical equations of motion, and therefore mean values of $x$ and $y$ follow classical trajectories. Moreover, the only nonzero commutator becomes$$\left[\check{x},\check{y}\right]=ie^{-2\alpha t}\,,$$ which matches the classical Dirac bracket (\[eq:classical-commutators\]). Thus, the resulting quantum theory at least obeys the correspondence principle. One can easily find solutions of the Schrödinger equation by making a (time-independent) unitary transformation $\hat{H}_{\tilde{\omega}}=\hat{S}^{-1}\hat{H}\hat{S},$ $$\hat{S}=\exp\left(-\frac{i\alpha}{2}\hat{q}^{2}\right)\,,\,\,\hat{S}^{-1}\hat{p}\hat{S}=\hat{p}-\alpha\hat{q}\,,\label{eq:s-transform}$$ one obtains $$\hat{H}_{\tilde{\omega}}=\frac{1}{2}\left(\hat{p}^{2}+\tilde{\omega}^{2}\hat{q}^{2}\right),\ \ \tilde{\omega}=\sqrt{\omega^{2}-\alpha^{2}}\,.\label{eq:undamped-hamiltonian}$$ The Hamiltonian (\[eq:undamped-hamiltonian\]) has the familiar stationary states $$\begin{aligned} & \psi_{n}^{\tilde{\omega}}\left(t,q\right)=\exp\left(-itE_{n}\right)\psi_{n}^{\tilde{\omega}}\left(q\right)\,,\\ & \psi_{n}^{\tilde{\omega}}\left(q\right)=\left(2^{n}n!\right)^{-1/2}\left(\frac{\tilde{\omega}}{\pi}\right)^{1/4}\exp\left(-\frac{\tilde{\omega}}{2}q^{2}\right)H_{n}\left(\sqrt{\tilde{\omega}}q\right)\,,\\ & \hat{H}_{\tilde{\omega}}\psi_{n}^{\tilde{\omega}}=E_{n}\psi_{n}^{\tilde{\omega}}\,,\,\, E_{n}=\tilde{\omega}\left(n+\frac{1}{2}\right)\,.\end{aligned}$$ Therefore, the wave functions$$\psi_{n}\left(q\right)=\hat{S}\psi_{n}^{\tilde{\omega}}\left(q\right)=\left(2^{n}n!\right)^{-1/2}\left(\frac{\tilde{\omega}}{\pi}\right)^{1/4}\exp\left(-\left(\tilde{\omega}+i\alpha\right)\frac{q^{2}}{2}\right)H_{n}\left(\sqrt{\tilde{\omega}}q\right)\label{eq:GK-eigenfunctions}$$ are eigenfunctions of the Hamiltonian $\hat{H}$, and the solutions to the corresponding Schrödinger equation are$$\begin{aligned} & \psi_{n}\left(t,q\right)=\hat{S}\psi_{n}^{\tilde{\omega}}\left(t,q\right)=\exp\left(-itE_{n}\right)\psi_{n}\left(q\right)\,,\\ & \hat{H}\psi_{n}\left(q\right)=E_{n}\psi\left(q\right)\,.\end{aligned}$$ One can also check directly that (\[eq:GK-eigenfunctions\]) are eigenfunctions of $\hat{H}$ with eigenvalue $E_{n}$ by using properties of Hermite functions. We now define some useful quantities to be used in Subsection \[sub:Physical-Equivalence\] for the purpose of establishing the physical equivalence between the approaches presented here. Let us write the classical Lagrangian energy as $$\mathcal{E}_{L}\equiv\frac{\partial L}{\partial\dot{x}}\dot{x}+\frac{\partial L}{\partial\dot{y}}\dot{y}-L=\frac{1}{2}\left(y^{2}+2\alpha xy+\omega^{2}x^{2}\right)e^{2\alpha t}\,.$$ The corresponding Weyl-ordered Schrödinger operator for the Lagrangian energy is$$\hat{\mathcal{E}}_{L}=\left.\mathcal{E}_{L}\left(\eta\right)\right\vert _{\eta=\hat{\eta}}=\frac{1}{2}\left[\hat{p}^{2}+\alpha\left(\hat{q}\hat{p}+\hat{p}\hat{q}\right)+\omega^{2}\hat{q}^{2}\right]=\hat{H}\,,\label{eq:kg-conserved-energy}$$ which coincides with the Hamiltonian and thus with its Heisenberg representation, and is therefore conserved. Likewise, we define the following conserved “energy” in the BCK approach, $$\begin{aligned} & \mathcal{E}=\frac{1}{2}\left(\dot{q}^{2}+2\alpha q\dot{q}+\omega^{2}q^{2}\right)e^{2\alpha t}=H_{BCK}+\alpha qp\,,\\ & \frac{d}{dt}\mathcal{E}=\left(\dot{q}+\alpha q\right)e^{2\alpha t}\frac{\delta S}{\delta q}\,,\end{aligned}$$ which is constant on-shell. Its image as a Weyl-ordered Schrödinger operator is$$\hat{\mathcal{E}}=\hat{H}_{BCK}+\frac{\alpha}{2}\left(\hat{q}\hat{p}+\hat{p}\hat{q}\right)\,,\,\,\hat{\mathcal{E}}\psi_{n}^{BCK}=E_{n}\psi_{n}^{BCK}\,.\label{eq:BCK-conserved-energy}$$ Now consider the mechanical energy in the theory with Lagrangian (\[eq:gitman-kupriyanov\]) at $\alpha=0$: $$E_{M}=\frac{1}{2}\left(y^{2}+\omega^{2}x^{2}\right)\,.$$ The corresponding operator is equal to$$\hat{E}_{M}=\frac{1}{2}\left(\hat{p}^{2}+\omega^{2}\hat{q}^{2}\right)e^{-2\alpha t}=e^{-2\alpha t}\hat{H}-e^{-2\alpha t}\frac{\alpha}{2}\left(\hat{q}\hat{p}+\hat{p}\hat{q}\right)\,.\label{eq:1st-order-mech-energy}$$ The mean value of $\hat{E}_{M}$ in the energy eigenstates is$$\left\langle \psi_{n}\right\vert \hat{E}_{M}\left\vert \psi_{n}\right\rangle =e^{-2\alpha t}\frac{\omega^{2}}{\tilde{\omega}}\left(n+\frac{1}{2}\right)\,.$$ Finally, we consider the observable defined by the mechanical energy in the BCK description, $$E=\frac{1}{2}\left(\dot{q}^{2}+\omega^{2}q^{2}\right)=\frac{1}{2}\left(e^{-2\alpha t}p^{2}+\omega^{2}e^{2\alpha t}q^{2}\right)e^{-2\alpha t}=H_{BCK}e^{-2\alpha t}\,.\label{eq:bck-mech-energy}$$ Thus, by (\[eq:average-ck-hamiltonian\]), $$\left\langle \psi_{n}^{BCK}\right\vert \hat{E}\left\vert \psi_{n}^{BCK}\right\rangle =e^{-2\alpha t}\frac{\omega^{2}}{\tilde{\omega}}\left(n+\frac{1}{2}\right)=\left\langle \psi_{n}\right\vert \hat{E}_{M}\left\vert \psi_{n}\right\rangle \,.$$ Semiclassical description ------------------------- ### Coherent states Finally, we obtain semiclassical states for the damped harmonic oscillator from the coherent states of the simple harmonic oscillator using the unitary transformation $\hat{S}$ (\[eq:s-transform\]). To this end, we introduce first creation and annihilation operators $\hat{a}^{+}$ and $\hat{a}$ and the corresponding coherent states $\left\vert z\right\rangle $,$$\begin{aligned} & \hat{a}=\frac{1}{\sqrt{2\tilde{\omega}}}\left(\tilde{\omega}\hat{q}+i\hat{p}\right)\,,\,\,\hat{a}^{+}=\frac{1}{\sqrt{2\tilde{\omega}}}\left(\tilde{\omega}\hat{q}-i\hat{p}\right)\,,\,\,\left[\hat{a},\hat{a}^{+}\right]=1\,,\nonumber \\ & \left\vert z\right\rangle =D\left(z\right)\left\vert 0\right\rangle \,,\,\, D\left(z\right)=\exp\left(z\hat{a}^{+}-\bar{z}\hat{a}\right)\,,\,\, a\left\vert z\right\rangle =z\left\vert z\right\rangle \,.\label{eq:coherent-states}\end{aligned}$$ In terms of these creation and annihilation operators, the Hamiltonian (\[eq:undamped-hamiltonian\]) is $$\hat{H}_{\tilde{\omega}}=\tilde{\omega}\left(\hat{a}^{+}\hat{a}+\frac{1}{2}\right)\,.$$ Thus, the coherent states for the Hamiltonian $\hat{H}$ are $\hat{S}\left\vert z\right\rangle $ and the mean values of $\check{x}$ and $\check{y}$ in these coherent states are$$\begin{aligned} & \left\langle x\right\rangle \equiv\left\langle z\right\vert \hat{S}^{-1}\check{x}\hat{S}\left\vert z\right\rangle =\frac{1}{\sqrt{2\tilde{\omega}}}e^{-\alpha t}\left(ze^{-i\tilde{\omega}t}+\bar{z}e^{i\tilde{\omega}t}\right)\,,\\ & \left\langle y\right\rangle =i\sqrt{\frac{\tilde{\omega}}{2}}e^{-\alpha t}\left(\bar{z}e^{i\tilde{\omega}t}-ze^{-i\tilde{\omega}t}\right)-\alpha\left\langle x\right\rangle \,.\end{aligned}$$ One can now easily verify that the mean values of the coordinates $x$ and $y$ follow the classical trajectories,$$\frac{d}{dt}\left\langle x\right\rangle =\left\langle y\right\rangle \,,\,\,\frac{d}{dt}\left\langle y\right\rangle =-\omega^{2}\left\langle x\right\rangle -2\alpha\left\langle y\right\rangle \,.$$ The pathological behavior of the first-order theory with regard to the limit $t\rightarrow\infty$ can be seen here in the computation of the uncertainty relation $$\Delta x\Delta y=\frac{1}{2}e^{-2\alpha t}\frac{\omega}{\tilde{\omega}}\,.\label{eq:time-dep-uncertainty}$$ The unphysical result of the above violation of the uncertainty principle is an indication that the first-order theory is not defined for all values of the time parameter, as was pointed out in connection to the classical equations of motion and to the commutation relation between $\hat{x}$ and $\hat{y}$. Another indication of the failure of the theory at infinite times is in the observation that the radius of the trajectory of the mean values vanishes:$$\rho\left(z=re^{i\theta}\right)=\sqrt{\left\langle x\right\rangle ^{2}+\left\langle y\right\rangle ^{2}}=\sqrt{\frac{2}{\tilde{\omega}}}e^{-\alpha t}r\sqrt{1+\alpha^{2}\cos\left(\tilde{\omega}t-\theta\right)+\alpha\sin2\left(\tilde{\omega}t-\theta\right)}\,.$$ ### Squeezed-state One can also consider the family of conserved creation and annihilation operators$$\begin{aligned} & \hat{b}\left(t\right)=\cosh\xi e^{i\tilde{\omega}t}\hat{a}+\sinh\xi e^{-i\tilde{\omega}t}\hat{a}^{\dagger}\,,\,\,\hat{b}^{\dagger}\left(t\right)=\cosh\xi e^{-i\tilde{\omega}t}\hat{a}^{\dagger}+\sinh\xi e^{i\tilde{\omega}t}\hat{a}\,,\\ & \left[\hat{b}\left(t\right),\hat{b}^{\dagger}\left(t\right)\right]=1\,,\,\,\frac{d}{dt}\hat{b}=\frac{d}{dt}\hat{b}^{\dagger}=0\,,\end{aligned}$$ and construct squeezed coherent states $\left\vert z,\xi\right\rangle =\exp\left(z\hat{b}^{\dagger}-\bar{z}\hat{b}\right)\left\vert 0\right\rangle $ [@Dodonov]. For $\xi=0$ one arrives at the previous coherent states, $$\left\langle z,0\right\vert \hat{x}\left\vert z,0\right\rangle =\left\langle z\right\vert \hat{S}^{-1}\check{x}\hat{S}\left\vert z\right\rangle \,,\,\,\left\langle z,0\right\vert \hat{y}\left\vert z,0\right\rangle =\left\langle z\right\vert \hat{S}^{-1}\check{y}\hat{S}\left\vert z\right\rangle \,.$$ For arbitrary values of $\xi$ and with $\hat{x}$ and $\hat{y}$ given by (\[eq:operator-solutions\]), one has for the mean values of coordinates$$\begin{aligned} \left\langle z,\xi\right\vert \hat{x}\left\vert z,\xi\right\rangle & =\frac{e^{-\alpha t}}{\sqrt{2\tilde{\omega}}}\left[\left(e^{-i\tilde{\omega}t}z+e^{i\tilde{\omega}t}\bar{z}\right)\cosh\xi-\left(e^{i\tilde{\omega}t}z+e^{-i\tilde{\omega}t}\bar{z}\right)\sinh\xi\right]\,,\\ \left\langle z,\xi\right\vert \hat{y}\left\vert z,\xi\right\rangle & =i\sqrt{\frac{\tilde{\omega}}{2}}e^{-\alpha t}\left[\left(e^{i\tilde{\omega}t}\bar{z}-e^{-i\tilde{\omega}t}z\right)\cosh\xi-\left(e^{i\tilde{\omega}t}z-e^{-i\tilde{\omega}t}\bar{z}\right)\sinh\xi\right]-\alpha\left\langle z,\xi\right\vert \hat{x}\left\vert z,\xi\right\rangle \,.\end{aligned}$$ The main interest in squeezed states is that they allow one to change the uncertainty in either direction $x$ or $y$ by adjusting the parameter $\xi$. For example, the uncertainty in $x$ can be written for arbitrary values of $\xi$ as $$\left(\Delta x\right)^{2}=\frac{1}{2\tilde{\omega}}e^{-2\alpha t}\left[\left(\cosh\xi+\sinh\xi\right)^{2}-4\cosh\xi\sinh\xi\cos^{2}\tilde{\omega}t\right]\geqslant0\,,$$ and it reduces to the preceding coherent states calculations for $\xi=0$. Comparison of the BCK system and the first-order system ======================================================= BCK model as a transformation of the first-order system ------------------------------------------------------- Here we show how one can obtain the classical and quantum description of the BCK damped harmonic oscillator as a canonical transformation of the first-order approach (\[sec:First-order-action\]). At the classical level, both systems are transformed one into the other by means of the following time-dependent canonical transformation[^4]$$\begin{aligned} & q=\frac{1}{2}\left(x-2e^{-2\alpha t}p_{y}\right)\,,\,\, p=p_{x}+\frac{1}{2}e^{2\alpha t}y\,,\\ & \Omega^{1}=\frac{1}{2}y-p_{x}e^{-2\alpha t}\,,\,\,\Omega_{2}=p_{y}+\frac{1}{2}xe^{2\alpha t}\,,\end{aligned}$$ where $\left(q,p\right)$ and $\left(\Omega^{1},\Omega_{2}\right)$ are new pairs of canonical variables. In these new variables, the equation of motion become Hamiltonian:$$\begin{aligned} & \dot{q}=\left\{ q,H_{BCK}\right\} \,,\,\,\dot{p}=\left\{ p,H_{BCK}\right\} \,,\,\,\Omega=0\,,\\ & H_{BCK}=\frac{1}{2}\left[e^{-2\alpha t}p^{2}+\omega^{2}e^{2\alpha t}q^{2}\right]\,,\end{aligned}$$ where the Hamiltonian $H_{BCK}\left(q,p,t\right)$ is the canonically transformed Hamiltonian $H\left(x,y,t\right)$ (\[eq:kg-hamiltonian\]) on the equivalent constraint surface $\Omega=0$. It is useful to write the following relation between old coordinates $x$ and $y$ and the new variables $q$ and $p$:$$\begin{aligned} x & = & q+O\left(\Omega\right)\,,\nonumber \\ y & = & e^{-2\alpha t}p+O\left(\Omega\right)\,.\label{eq:physical-relations}\end{aligned}$$ This relation shows that $x$ is physically equivalent to $q$, while $y$ is physically equivalent to $e^{-2\alpha t}p$, since they coincide on the constraint surface $\Omega=0$. The quantum theory can be readily obtained by a quantum time-dependent canonical transformation[^5] $$\hat{D}=\exp\left[\frac{i\alpha t}{2}\left(\hat{q}\hat{p}+\hat{p}\hat{q}\right)\right]\,,\label{eq:time-dep-unitary-transf}$$ which is suggested from the classical generating function and the relationship between old and new variables. The effect of $\hat{D}$ on the canonical variables is to make the dilation $$\hat{D}^{-1}\hat{q}\hat{D}=e^{-\alpha t}\hat{q}\,,\,\,\hat{D}^{-1}\hat{p}\hat{D}=e^{\alpha t}\hat{p}\,.$$ The dilation operator $\hat{D}$ has been discussed in a more general setting in [@Onofri1978] in connection with the description of open systems by time-dependent Hamiltonians. There, as is the case here, the dilation operator simplifies the calculation of the evolution operator. The dilaton operator transforms the BCK Hamiltonian into the first-order Hamiltonian $\hat{H}$,$$\hat{H}=\hat{D}^{-1}\hat{H}_{BCK}\hat{D}-i\hat{D}^{-1}\frac{\partial\hat{D}}{\partial t}=\frac{1}{2}\left[\hat{p}^{2}+\alpha\left(\hat{q}\hat{p}+\hat{p}\hat{q}\right)+\omega^{2}\hat{q}^{2}\right]\,.$$ Since $\psi_{n}\left(q,t\right)$ satisfy the Schrödinger equation with $\hat{H}$, it follows that $\hat{D}\psi_{n}\left(q,t\right)$ satisfy the Schrödinger equation with $\hat{H}_{BCK}$. The wave functions $\hat{D}\psi_{n}\left(q,t\right)$ are indeed the pseudostationary states, as can be seen by direct application of $\hat{D}$. Thus, we write $\psi_{n}^{BCK}\left(q,t\right)\equiv\hat{D}\psi_{n}\left(q,t\right)$. Similarly, it also follows that $U=\hat{D}\exp\left(-it\hat{H}\right)$ satisfies the Schrödinger equation with Hamiltonian $\hat{H}_{BCK}$, $$U\left(t\right)=\hat{D}\exp\left(-i\hat{H}t\right)\,,\,\, i\frac{\partial U}{\partial t}=\hat{H}_{BCK}U\,,\,\, U\left(0\right)=I\,.$$ One should not be to eager to jump to the conclusion that the two theories here presented, the BCK theory and the first-order theory, are physically equivalent solely on the grounds of the time-dependent canonical transformation $\hat{D}$. The existence of this transformation *per se* is insufficient to prove physical equivalence, since in principle, and at least locally, one can always construct a time-dependent canonical transformation between any two given theories, classical or quantum. For instance, given two quantum theories $T_{1}$ and $T_{2}$, and corresponding evolution operators $U_{1}$ and $U_{2}$, one can always write a general solution $\psi_{2}\left(t\right)$ of the Schrödinger equation of $T_{2}$ in terms of the general solution $\psi_{1}\left(t\right)$ of $T_{1}$ by means of the transformation $\psi_{2}\left(t\right)=U_{2}U_{1}^{-1}\psi_{1}\left(t\right)$. Besides the necessary ingredient of the unitary transformation relating the two theories, it is imperative to show that the physical observables pertaining both theories are also unitarily equivalent in order to prove physical equivalence. A proof which we postpone to the next Section. Physical Equivalence\[sub:Physical-Equivalence\] ------------------------------------------------ In this Section we show that the two approaches presented in this article are physically equivalent. In order to sum up the previous results in a coherent whole, let us present both quantum theories anew. Starting with the BCK theory, it is defined by the Hamiltonian $\hat{H}_{BCK}$ (\[eq:BCK-quantum-theory\]) written in terms of the canonically conjugated operators $\hat{q}$ and $\hat{p}$ (\[eq:BCK-quantum-theory\]) with the usual realization in terms of multiplication and derivation operators in the Hilbert space $\mathcal{H}_{BCK}$ of square-integrable states $\psi_{BCK}\left(q\right)$ with measure$$\left\langle \psi_{BCK}\right.\left|\psi_{BCK}\right\rangle _{\mathcal{H}_{BCK}}=\int_{-\infty}^{+\infty}dq\bar{\psi}_{BCK}\left(q\right)\psi_{BCK}\left(q\right)\,.$$ The first-order theory is defined by the Hamiltonian $\hat{H}$ (\[eq:kg-quantum-hamiltonian\]) and canonically conjugated operators $\hat{q}$ and $\hat{p}$ (\[eq:qp-operators\]) realized as the usual multiplication and derivation operators in the Hilbert space $\mathcal{H}$ of square-integrable states $\psi\left(q\right)$ with measure$$\left\langle \psi\right.\left|\psi\right\rangle _{\mathcal{H}}=\int_{-\infty}^{+\infty}dq\bar{\psi}\left(q\right)\psi\left(q\right)\,.$$ We have seen that the time-dependent unitary transformation $\hat{D}$ (\[eq:time-dep-unitary-transf\]) maps the two Hilbert spaces, $$\hat{D}:\mathcal{H}\rightarrow\mathcal{H}_{BCK}\,,\,\,\psi\mapsto\psi_{BCK}=\hat{D}\psi\,,$$ and is a canonical transformation, $$\hat{H}_{BCK}=\hat{D}\hat{H}\hat{D}^{-1}+i\frac{\partial\hat{D}}{\partial t}\hat{D}^{-1}\,.$$ Therefore, for every $\psi\left(t\right)$ solution of the Schrödinger equation of of the first-order theory, $\hat{D}\psi\left(t\right)$ is a solution of the BCK Schrödinger equation. Furthermore, one has $\left\langle \psi_{BCK}\right.\left|\psi_{BCK}\right\rangle _{\mathcal{H}_{BCK}}=\left\langle \psi\right.\left|\psi\right\rangle _{\mathcal{H}}$, as can be seen by noting that $\hat{D}\psi\left(q\right)=e^{\alpha t/2}\psi\left(qe^{\alpha t}\right)$ and$$\left\langle \psi_{BCK}\right.\left|\psi_{BCK}\right\rangle _{\mathcal{H}_{BCK}}=\int_{-\infty}^{+\infty}dqe^{\alpha t}\bar{\psi}\left(qe^{\alpha t}\right)\psi\left(qe^{\alpha t}\right)=\int_{-\infty}^{+\infty}dq\bar{\psi}\left(q\right)\psi\left(q\right)=\left\langle \psi\right.\left|\psi\right\rangle _{\mathcal{H}}\,.$$ Finally, to complete the proof of the physical equivalence, it remains to show that any two physical observables $\hat{\mathcal{O}}_{BCK}$ and $\hat{\mathcal{O}}$ are $D$-equivalent, that is, $\hat{\mathcal{O}}_{BCK}=\hat{D}\hat{\mathcal{O}}\hat{D}^{-1}$. One can check that this is indeed the case for the physical observables previously considered, such as the mechanical energies (\[eq:1st-order-mech-energy\],\[eq:bck-mech-energy\]) $$\hat{E}=\hat{D}\hat{E}_{M}\hat{D}^{-1}\,,$$ and the conserved energies (\[eq:kg-conserved-energy\],\[eq:BCK-conserved-energy\]),$$\hat{\mathcal{E}}=\hat{D}\hat{\mathcal{E}}_{L}\hat{D}^{-1}\,.$$ As a result of the equivalence, one can easily obtain the BCK coherent states presented in [@Dodonov1979] by merely transforming the coherent states (\[eq:coherent-states\]) given in the context of the first-order theory. Taking into account the relation $y=e^{-2\alpha t}p+\left\{ \Omega\right\} $ between the physical variables of the two theories (\[eq:physical-relations\]), one can see why the uncertainty relations we obtain decay exponentially with time (\[eq:time-dep-uncertainty\]), while those (\[eq:manko-uncertainty\]) calculated in [@Dodonov1979] are constant. This has a simple explanation in terms of the physical equivalence of $y$ and $e^{-2\alpha t}p$: $y$ is physically equivalent to the physical momentum of the BCK oscillator, and it is in terms of the physical momentum that the Heisenberg uncertainty relations are violated (see Dekker [@Dekker1981] for a comprehensive review). Conclusion ========== In this article we have proved that the classical and quantum description of the damped harmonic oscillator by the BCK time-dependent Hamiltonian [@Bateman1931; @Caldirola1941; @Kanai1950] is locally equivalent to the first order approach given in terms of a constrained system [@Gitman2007b]. This equivalence allowed us to easily obtain the evolution operator for the BCK oscillator and many other results in a simpler manner, due to the time-independence of the quantum first-order Hamiltonian. As has been pointed out (see Dekker [@Dekker1981] for a comprehensive review), the BCK oscillator has a pathological behavior for infinite times, since the (mechanical) energy mean values and the Heisenberg uncertainty relation - between the coordinate and the physical momentum - go to zero as time approaches infinity, so that even the ground state’s energy eventually vanishes. Despite these shortcomings, the quantum theory has a well-defined classical limit, and for time values less than $\frac{1}{2\alpha}\ln\frac{\omega}{\tilde{\omega}}$ the Heisenberg uncertainty principle is not violated. It is our understanding that this unphysical behavior is a result of extending the proposed theories beyond their validity. The fact that both theories are not globally defined in time has its roots already at the classical level, as a consequence of the non-Lagrangian nature of the equations of motion of the damped harmonic oscillator. Finally, we recall the intriguing behavior of the asymptotic pseudostationary states, which so far has escaped notice from all works dedicated to the BCK oscillator. At first glance eq. (\[eq:limit-eigenequation\]) implies that the BCK Hamiltonian loses self-adjointness as $n$ approaches infinity. Fortunately, these asymptotic states are not in the domain of the BCK Hamiltonian and thus pose no threat whatsoever. On the other hand, there is no such constraint on the domain of the first-order Hamiltonian $\hat{H}$, where there is no upper bound to the energy eigenstates. This inconsistency can spoil the physical equivalence of the two theories at high energies, but it is not altogether unexpected. As the energy grows, that is to say, as $n$ grows, the wave functions spread farther out in space and become highly non-local. Any canonical transformation, on the contrary, is only locally valid, and thus our time-dependent dilation transformation $\hat{D}$ (\[eq:time-dep-unitary-transf\]) cannot guarantee there are no global issues in establishing the physical equivalence of the two theories (or any two theories, for that matter). This is not unexpected, since in general the non-local properties of the wave functions can spoil any equivalence that the two theories might have on the level of the Heisenberg equations. Darboux’s method for solving the inverse problem of the calculus of variations for second-order equations of motion of one-dimensional systems produces theories which, despite being classically equivalent, do not possess the expected quantum properties. Thus, whereas every classical one-dimensional system can be “Lagrangianized”, in quantum mechanics one still encounters one-dimensional systems which can still be called “non-Lagrangian”. For such systems, quantum anomalies resulting from limiting processes (in our case, the limit $t\rightarrow\infty$) can be identified already at the classical level by an asymptotic inequivalence between the original second-order equation and the EL equation resulting from the Lagrangianization process. We note that the quantum Hamiltonian (\[eq:kg-quantum-hamiltonian\]) naturally appears in the master equation for open quantum systems in the description of the damped harmonic oscillator in the Lindblad theory [@Isar1999] and Dekker’s complex symplectic approach [@DEKKER1979]. This relation has yet to be clarified, and it should prove useful in the comparison of the BCK theory and the first-order theory along the lines of a subsystem plus reservoir approach. #### Acknowledgements: M.C. Baldiotti and D.M. Gitman thank FAPESP for financial support. {#acknowledgements-m.c.-baldiotti-and-d.m.-gitman-thank-fapesp-for-financial-support. .unnumbered} Appendix ======== #### Large $n$ limit of $\hat{A}\psi_{n}^{BCK}$ {#large-n-limit-of-hatapsi_nbck .unnumbered} Let us compute the action of $\hat{A}=\hat{q}\hat{p}+\hat{p}\hat{q}$ on the functions $\psi_{n}^{BCK}$ (\[eq:pseudostationary-state\]),$$\begin{aligned} \left(\hat{q}\hat{p}+\hat{p}\hat{q}\right)\psi_{n}^{BCK} & = & -i\left(2q\frac{\partial}{\partial q}+1\right)\psi_{n}^{BCK}\nonumber \\ & = & i\sqrt{\left(n+2\right)\left(n+1\right)}\psi_{n+2}^{CK}-i\sqrt{n\left(n-1\right)}\psi_{n-2}^{BCK}\nonumber \\ & & -\frac{\alpha}{\tilde{\omega}}\left[\sqrt{\left(n+2\right)\left(n+1\right)}\psi_{n+2}^{CK}+\left(2n+1\right)\psi_{n}^{CK}+\sqrt{n\left(n-1\right)}\psi_{n-2}^{BCK}\right]\,.\label{eq:acalculus}\end{aligned}$$ The asymptotic form of the pseudostationary states for large values of $n$ is [@smirnov]$$\begin{aligned} \psi_{2n}^{BCK}\left(q,t\right) & = & \left(\frac{\tilde{\omega}}{\pi}\right)^{1/4}\exp\left(-i\alpha\frac{q^{2}e^{2\alpha t}}{2}\right)\left(-1\right)^{n}\sqrt{\frac{\left(2n-1\right)!!}{2n!!}}\cos\left(\sqrt{\left(4n+1\right)\tilde{\omega}}e^{\alpha t}q\right)+O\left(n^{-1/4}\right)\nonumber \\ & = & \psi_{2n}^{asym}+O\left(n^{-1/4}\right)\,,\nonumber \\ \psi_{2n+1}^{BCK}\left(q,t\right) & = & \left(\frac{\tilde{\omega}}{\pi}\right)^{1/4}\exp\left(-i\alpha\frac{q^{2}e^{2\alpha t}}{2}\right)\left(-1\right)^{n}\sqrt{\frac{\left(2n-1\right)!!}{2n!!}}\sin\left(\sqrt{\left(4n+3\right)\tilde{\omega}}e^{\alpha t}q\right)+O\left(n^{-1/4}\right)\nonumber \\ & = & \psi_{2n+1}^{asym}+O\left(n^{-1/4}\right)\,.\label{eq:asymptotic-waves}\end{aligned}$$ Taking into account that $\cos\sqrt{4n+\alpha}x-\cos\sqrt{4n+1}x=O\left(n^{-1/4}\right)$ for any real $\alpha$, and substituting the above in the right-hand side of (\[eq:acalculus\]), one has$$\left(\hat{q}\hat{p}+\hat{p}\hat{q}\right)\psi_{n}^{BCK}=-i\psi_{n}^{asym}+O\left(n^{-1/4}\right)\,.$$ On the other hand $\left(\hat{q}\hat{p}+\hat{p}\hat{q}\right)\psi_{n}^{BCK}=\left(\hat{q}\hat{p}+\hat{p}\hat{q}\right)\psi_{n}^{asym}+O\left(n^{-1/4}\right)$, so$$\left(\hat{q}\hat{p}+\hat{p}\hat{q}\right)\psi_{n}^{asym}=-i\psi_{n}^{asym}+O\left(n^{-1/4}\right)\,,$$ or to the same approximation,$$\left(\hat{q}\hat{p}+\hat{p}\hat{q}\right)\psi_{n}^{BCK}=-i\psi_{n}^{BCK}+O\left(n^{-1/4}\right)\,.$$ #### Proposition: The asymptotic pseudostationary functions $\psi_{n}^{asym}$ are not in the domain of $\hat{A}=\hat{q}\hat{p}+\hat{p}\hat{q}$. {#proposition-the-asymptotic-pseudostationary-functions-psi_nasym-are-not-in-the-domain-of-hatahatqhatphatphatq. .unnumbered} We first find the domain of $\hat{A}$ for which it is symmetric. Consider, for $\phi,\psi\in D\left(\hat{A}\right)$:$$\begin{aligned} \left\langle \phi,\hat{A}\psi\right\rangle -\left\langle \hat{A}\phi,\psi\right\rangle & = & -2i\int_{-\infty}^{\infty}dq\frac{d}{dq}\left(q\bar{\phi}\left(q\right)\psi\left(q\right)\right)\\ & = & -2i\lim_{q\rightarrow\infty}q\left(\bar{\phi}\left(q\right)\psi\left(q\right)+\bar{\phi}\left(-q\right)\psi\left(-q\right)\right)\,.\end{aligned}$$ Therefore, functions such that $\lim_{x\rightarrow\pm\infty}q\psi\left(q\right)\neq0$ are not in the domains of $\hat{A}$. On the other hand, we know that for large values of $n$ the pseudostationary functions have the asymptotic form (\[eq:asymptotic-waves\]). Thus, clearly $\lim_{q\rightarrow\pm\infty}q\psi_{n}^{asym}\left(q\right)\ne0$ and $\psi_{n}^{asym}\neq D\left(\hat{A}\right)$. Note that the closure of $\hat{A}$ is not affected by the exclusion of the asymptotic functions, since they can’t be the limit of any sequence. #### Proposition: $\hat{A}$ is self-adjoint. {#proposition-hata-is-self-adjoint. .unnumbered} It suffices to show that the equation $\hat{A}^{*}\psi=\pm i\psi$ does not have solutions in $D\left(\hat{A}^{*}\right)$ [@reedsimonv1]. The solutions are of the form$$\psi_{\pm}=x^{\lambda_{\pm}}\,,\,\,\lambda_{\pm}=\mp\frac{1}{2}\,,$$ which are not square-integrable in the interval $\left(-\infty,\infty\right)$. Therefore, the corresponding deficiency indices are $\left(0,0\right)$ and $\hat{A}$ is essentially self-adjoint. [^1]: baldiott@fma.if.usp.br [^2]: fresneda@gmail.com [^3]: gitman@dfn.if.usp.br [^4]: The generating function for these canonical transformations depending on the new and old momenta is$$F\left(p_{x},p_{y},p,\Omega_{2},t\right)=-2e^{-2\alpha t}\left(\Omega_{2}-p_{y}\right)p_{x}-2e^{-2\alpha t}pp_{y}+e^{-2\alpha t}p\Omega_{2}\,.$$ [^5]: A proof that $\hat{x}\hat{p}+\hat{p}\hat{x}$ is self-adjoint is provided in the appendix.

--- abstract: 'Non-bonded potentials are included in most force fields and therefore widely used in classical molecular dynamics (MD) simulations of materials and interfacial phenomena. It is commonplace to truncate these potentials for computational efficiency based on the assumption that errors are negligible for reasonable cutoffs or compensated for by adjusting other interaction parameters. Arising from a metadynamics study of the wetting transition of water on a solid substrate we find that the influence of the cutoff is unexpectedly strong and can change the character of the wetting transition from continuous to first order by creating artificial metastable wetting states. Common cutoff corrections such as the use of a force switching function, a shifted potential or a shifted force do not avoid this. Such a qualitative difference urges caution and suggests that using truncated non-bonded potentials can induce unphysical behavior that cannot be fully accounted for by adjusting other interaction parameters.' author: - Martin Fitzner - Laurent Joly - Ming Ma - 'Gabriele C. Sosso' - Andrea Zen - Angelos Michaelides bibliography: - 'main.bib' title: 'Communication: Truncated Non-Bonded Potentials Can Yield Unphysical Behavior in Molecular Dynamics Simulations of Interfaces' --- Short- to medium-range potentials such as the Lennard-Jones [@jones1924determination] or the Buckingham [@buckingham1938classical] potential are the backbone of classical MD simulations. They represent Pauli repulsion as well as non-directional dispersion attraction and there exist multiple flavors implemented in most MD codes under the term of non-bonded interactions. In practice there is a need to truncate these potentials since the number of neighbors that have to be considered for each entity grows enormously, drastically increasing the computational cost for the force calculation. Truncating between $r_\mathrm{c}$ = 2.5$\sigma$ and 3.5$\sigma$, where $\sigma$ is the characteristic interaction range, is a very common practice in MD studies [@frenkel2001understanding] and has become the minimum standard, assuming that errors arising from this are small enough. Several studies have reported that with these settings significant problems can arise. For instance the truncation can alter the phase diagram of the Lennard-Jones system [@smit1992phase; @wang2008homogeneous] or yield different values for interfacial free energies [@ismail2007surface; @valeriani2007comparison; @ghoufi2016computer; @ghoufi2017importance; @marcello2017LongRange]. These effects are quantitative in nature, meaning that they can in certain circumstances be analytically corrected for [@sun1998compass; @sinha2003surface; @werth2015long] or compensated for by other interaction parameters such as interaction strength or interaction range. The latter is important for the development of force fields where non-bonded potentials are often included and the cutoff can be seen as another fitting parameter. Naturally, a parametrization with a small cutoff would be preferred to another one if they deliver equal accuracy. This however is only true in the assumption that the underlying physical characteristics that are created by truncated and longer ranging potentials are the same. In this work we investigated the influence of the cutoff for the interfacial phenomenon of water-wetting on a solid substrate. We found that the effect of the cutoff of the water-substrate interaction was not only unexpectedly strong, but also changed the fundamental physics of the wetting transition in an unprecedented way by creating metastable wetting states that have also never been seen in experiments. We show that proposed cutoff corrections such as the use of a force switching function, a shifted potential or a shifted force did not fix this and could even worsen the effect. This finding shows that atomistic simulations of interfaces need to be treated with great care since unphysical behavior could occur and easily remain undetected. This is particularly relevant since a large number of MD studies using truncated potentials are reported each year. Our results suggest the use of much larger-than-common cutoffs or long-range versions of non-bonded potentials in MD studies of wetting and interfacial phenomena. ![a) Side view of the two wetting states for the small droplet. Water is blue and surface atoms are gray. b) Temperature of the wetting transition $T_\mathrm{w}$ (points) versus cutoff radius $r_\mathrm{c}$ and fit (red line). The $T_\mathrm{w}$ were obtained from the free energy profiles (see text) and we estimate errors to be $\pm 3$ K. $T_0$ is the converged wetting temperature.[]{data-label="FIG_1_TRANSITION_TEMPERATURE"}](./fig_1.pdf){width="6.6cm"} {width="16.2cm"} We investigated two droplets comprised of 3000 and 18000 water molecules which were represented by the coarse-grained mW model [@molinero_water_2009], on top of a rigid, pristine fcc(100) surface (lattice parameter 4.15 ). Whilst this substrate does not aim at representing any particular material, similar systems have been used to study ice nucleation [@reinhardt2012free; @cox2015molecular1; @cox2015molecular2; @fitzner_many_2015] or water-metal interfaces [@heinz2008accurate; @xu2015nanoscale]. The simulation cell had dimensions $17\times 17 \times 11$ nm which is enough to avoid interaction of the water molecules with their periodic images for all wetting states. Even though the liquid is rather non-volatile even at the highest temperature considered, we employed a reflective wall at the top of the cell to avoid evaporation and mimic experimental conditions. Our simulations were performed with the LAMMPS code [@plimpton1995fast], integrating the equations of motion with a timestep of 10 fs. This rather large timestep is commonly used in combination with the mW model and is acceptable for our system since during NVE simulations the total energy drift was found to be only about $2\times10^{-9}$ eV per water molecule per ps. In addition, we verified that we obtain the same results using standard protocols for updating the neighbor lists compared with unconditionally updating them every timestep. All production simulations were performed in the NVT ensemble with constant temperature maintained by a ten-fold Nosé-Hoover chain [@martyna1992nose] with a relaxation time of 1 ps. The substrate-water interaction was given by a distance ($r$) dependent Lennard-Jones potential $$U_\mathrm{LJ}(r)=4\epsilon\left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^{6} \right]$$ with $\epsilon = $ 29.5 meV, $\sigma$ = 2.5  truncated at a cutoff $r_\mathrm{c}$. This resulted in a maximum interaction energy of 154 meV for an adsorbed water monomer (weakly depending on the cutoff). Additionally we performed well-tempered metadynamics simulations [@laio2002escaping; @barducci2008well] for the smaller droplet with the PLUMED2 code [@tribello2014plumed]. In these simulations the Gaussian height, width, bias-factor and deposition stride were $2.16$ meV, $0.15$ , 20 and $20$ ps respectively. Metadynamics is usually applied to drive rare events such as nucleation [@sosso2016crystal; @sosso2016ice; @tribello2017analyzing; @cheng2017bridging] or protein folding [@bussi2006free; @laio2008metadynamics]. In our systems, this method helped to uncover the underlying free energy profile of wetting. We studied the wetting behavior of the larger droplet by performing standard MD runs at different temperatures first. As starting configurations we chose either a flat water film in direct contact or a spherical droplet placed above the substrate. Within at most 5 ns the simulation was equilibrated and a seemingly stable configuration was reached, where the water is either wetting (contact angle $\theta = 0^{\circ}$) or partially wetting ($0^\circ < \theta < 180^\circ$). An illustration of the two wetting states can be found in figure \[FIG\_1\_TRANSITION\_TEMPERATURE\]a. Initially we employed a radial cutoff at $r_\mathrm{c} = 3.0\sigma$ for the water-substrate interaction. With this setting we found that interestingly a wetting transition happened at finite angle $\theta_0 \approx 23^\circ$, i.e. a smaller non-zero contact angle was not possible. This behavior cannot be explained by the standard Young’s equation. However, upon increasing the cutoff we found that the wetting behavior drastically changed. First, the wetting temperature $T_\mathrm{w}$ at which the wetting transition took place increased as we increased the cutoff (figure \[FIG\_1\_TRANSITION\_TEMPERATURE\]b). Whilst $T_\mathrm{w}$ shows a clear convergence behavior with $r_\mathrm{c}$, it is unexpectedly slow. A reasonably converged wetting temperature $T_0$ is only reached for $r_\mathrm{c} > 7\sigma$. Second, we noticed that for an increasing cutoff the minimum possible contact angle $\theta_0$ got smaller and eventually vanished. Most importantly, we also found that for temperatures around $T_\mathrm{w}$ the stable configuration that was reached after the 5 ns could depend on the starting configuration for smaller cutoffs, while for larger $r_\mathrm{c}$ it always reached the same state. This suggests that for small $r_\mathrm{c}$ we actually found metastable wetting states that are absent for large $r_\mathrm{c}$. This also means that $T_\mathrm{w}$ cannot naively be defined through visual analysis of trajectories at different temperatures but needs to be defined by the free energy of wetting. For a first order phase transition we define $T_\mathrm{w}$ to be the temperature where the two basins (corresponding to wetting and partial wetting) have the same free energy. For a continuous phase transition $T_\mathrm{w}$ is the temperature where the single basin represents a contact angle of $\theta = 0^\circ$ for $T < T_\mathrm{w}$ and $\theta > 0^\circ$ for $T > T_\mathrm{w}$. Understanding the character of these wetting states with standard MD can prove difficult as the dependence on the starting configuration always leaves doubt on the outcome of the equilibrated configuration obtained from it. To clarify, we show the results from the metadynamics simulations in figure \[FIG\_2\_FREEENERGY\]. As a collective variable we chose the z-component of the center of mass of the water droplet ($\mathrm{COM}_\mathrm{z}$), where z is the surface normal direction. While this choice is not equivalent to the contact angle (as they are related in a non-linear manner) it is clear that significantly different values for $\mathrm{COM}_\mathrm{z}$ correspond to different contact angles and can therefore distinguish the different wetting states. For the smallest cutoff at $T_\mathrm{w}$ and around we found that two basins coexist, one being the flat film (COM$_\mathrm{z} \approx$ 4 ) and the other being a droplet with certain contact angle (COM$_\mathrm{z} \gtrsim 5$ ). These two states are separated by a significant barrier larger than 20 $k_\mathrm{B}T$, which explains why we observed metastable states in the unbiased simulations for small $r_\mathrm{c}$. This corresponds to a first-order phase transition between the wetting states. The occurrence of a minimum possible contact angle $\theta_0$ is explained by the existence of the second basin, which does not approach the wetting basin, but rather becomes less stable as temperature changes. However, this character faded as we increased $r_\mathrm{c}$. The barrier became smaller and the distance between the basins got smaller. For the largest cutoff investigated ($8\sigma$) we clearly see that only a single basin exists that changes its position with temperature. As a result no metastable wetting states exist and the phase transition is continuous. We note that in this case the estimate of $T_\mathrm{w}$ is more difficult than for the first order transitions, however in this work we aim at presenting qualitative results and from figure \[FIG\_2\_FREEENERGY\] it is clear that $T_\mathrm{w}$ is higher than for the smaller cutoffs. Only the results for the largest cutoff are in agreement with the fact that water wetting transitions are generally continuous when probed in experiments [@bonn2001wetting; @friedman2013wetting] and finite-angle wetting transitions have, to the best of our knowledge, never been observed experimentally. Therefore, the correct qualitative wetting behavior in our system is not achieved with standard cutoffs and if undetected could potentially lead to false conclusions. Differences between short and long-ranged interactions have been highlighted for other interfacial phenomena, such as drying [@evans2016critical] or grain boundary melting [@caupin2008absence]. We further study the effect of the most commonly used correction schemes to cutoffs: 1. A shifted potential (sp) which ensures that the value of the potential energy $U$ does not jump at the cutoff distance, given by: $$\begin{aligned} U_\mathrm{sp}(r) &= U_\mathrm{LJ}(r) - U_\mathrm{LJ}(r_\mathrm{c})\end{aligned}$$ The corresponding force $F$ remains unaltered: $$\begin{aligned} F_\mathrm{sp}(r) &= F_\mathrm{LJ}(r) \end{aligned}$$ 2. A switching function (switch) which brings the force to zero between an inner $r_\mathrm{c,1}$ and an outer cutoff $r_\mathrm{c,2}$ (we chose $3\sigma$ and $4\sigma$): $$\begin{aligned} F_\mathrm{switch}(r) &= F_\mathrm{LJ}(r) &r \le r_\mathrm{c,1}&\\ F_\mathrm{switch}(r) &= \sum_{k=0}^3 C_k(r-r_\mathrm{c,1})^{k} &r_\mathrm{c,1} < r \le r_\mathrm{c,2}& \nonumber\end{aligned}$$ where $C_k$ are constants determined to ensure a smooth behavior [@plimpton1995fast]. 3. A shifted-force potential (sf), which ensures that force and potential do not jump: $$\begin{aligned} U_\mathrm{sf}(r) &= U_\mathrm{LJ}(r) - U_\mathrm{LJ}(r_\mathrm{c}) - (r - r_\mathrm{c})F_\mathrm{LJ}(r_\mathrm{c}) \\ F_\mathrm{sf}(r) &= F_\mathrm{LJ}(r) - F_\mathrm{LJ}(r_\mathrm{c}) \nonumber\end{aligned}$$ The latter approach was found to give good results for a homogeneous system and even allowed for a reduction of the cutoff [@toxvaerd2011communication]. Our results for these three corrections can be found in figure \[FIG\_3\_SHIFTED\]. By definition and thus unsurprisingly, the shifted potential does not yield any significant difference (where the remaining minor deviations are due to the metadynamics sampling) over the plain cutoff since forces remain unaltered. The smooth cutoff via switching function seems to improve the situation, however the fact that the transition temperature lies between the ones we found for a plain cutoff at $3\sigma$ and $4\sigma$ suggests that the improvement stems from the effectively increased interaction range rather than the fact that the force vanishes smoothly. Interestingly, the shifted force with the same cutoff performs worst out of all candidates as the barrier increases by a factor of two, which increases the likelihood that simulations are performed in the metastable state without realizing it. The fact that none of the considered correction schemes significantly improved the character of the wetting free energy profile leads us to conclude that it is not the way in which the cutting is done that matters most, but rather the effective cutoff distance as well as the overall interaction strength at that distance. ![Free energy profiles of wetting approximately at the transition temperature with uncorrected setup (cut) and for different correction schemes \[shifted potential (sp), force switch (switch) and shifted force (sf)\] applied with a cutoff at $3\sigma$. None of the schemes show the correct behavior, which is shown in figure \[FIG\_2\_FREEENERGY\] to be a single basin.[]{data-label="FIG_3_SHIFTED"}](./fig_3.pdf){height="6.8cm"} As an initial attempt to understand the results obtained we looked at the potential energies of the various systems with the different cutoffs considered. This, however, did not reveal any obvious explanation. One possible interpretation for the creation of metastable states in our systems with shorter cutoff can be obtained by considering the droplet state (not assuming anything about the stability relative to the film state). For a transition towards the film state, there needs to be thermal fluctuations of water molecules that are above the contact layer in the downwards direction (the fact that COM$_\mathrm{z}$ has proven a good reaction coordinate supports this statement). With an infinite interaction range all molecules that are loosing height contribute to these fluctuations since they have an interaction with the substrate. Therefore we expect the interaction energy to change monotonically and the free energy to follow monotonically either up or down depending on the balance of the interfacial free energies (see figure \[FIG\_2\_FREEENERGY\], $r_\mathrm{c} = 8\sigma$). But if the interaction range is finite, not all molecules contribute to an increased interaction with the substrate even if they decrease their height (and subsequently weaken the water-water interaction of the system by leading to deviations from a perfect spherical droplet). In other words, there is a minimum distance from the substrate that has to be surpassed by a molecule for it to contribute to a fluctuation increasing the interaction energy, otherwise it will (on average) actually decrease the total interaction energy. This minimum fluctuation for a single molecule translates into the macroscopic states (droplet and film) being connected by a barrier shaped free energy profile rather than a monotonic one (see figure \[FIG\_2\_FREEENERGY\], $r_\mathrm{c} = 3\sigma$). The entropic contributions to the free energy are unlikely to change this, since they are essentially dominated by the environment a molecule is in (quasi-static contact layer or quasi-liquid water on top). The entropic change between these two states will be monotonic for a single water molecule and therefore also for the whole droplet. Finding a general recipe for how to avoid such unphysical wetting states is difficult. Other aspects like e.g. the substrate density or the liquid-liquid interaction strength will have an influence on how strongly the fluctuations in the droplet state are affected by $r_\mathrm{c}$. Generally, cutoffs that are deemed acceptable from the inter-molecular perspective do not necessarily mean that the interaction between macroscopic states such as a film/droplet and a substrate is sufficiently captured. This is especially important in an interfacial simulation setting such as a slab, where a cutoff-caused change in interaction from the substrate side is not compensated by an equal change from the vacuum side. Consequently, only employing much larger cutoffs or techniques to calculate the long-range part of the dispersion force [@in2007application; @isele2012development; @isele2013reconsidering] can ensure that unphysical effects are avoided. A minimal sanity check for future wetting studies could be to start simulations from both a wetting film and a spherical liquid snapshot. If both of them end up in the same configuration the existence of an unphysical metastable wetting state is unlikely. In light of the vast amount of work that is done in the MD community using similar interactions, our findings urge extreme caution when dealing with truncated non-bonded potentials in simulations of interfacial phenomena. We have seen both quantitative and qualitative differences for the wetting transition. The former could be accounted for by changing other interaction parameters to reproduce the transition at the right temperature $T_0$. This assumption is fundamental to fitting force fields with truncated potentials to obtain quantitative agreement with e.g. experimental values. But it does not hold for the character of the transition because it arises purely from the value of the cutoff itself. If the resulting metastability of states remains undetected, the use of truncated interaction potentials could lead to wrong inferences about physical properties being made. While this conclusion has resulted from a simulation of wetting, similar implications could hold for other interfacial phenomena such as capillary flow [@joly2011capillary; @gravelle2016anomalous], evaporation/condensation [@hens2014nanoscale; @nagayama2015molecular], mixtures [@iyer2013computer; @tran2014molecular; @radu2017enhanced] or heterogeneous nucleation [@reinhardt2014effects; @cabriolu2015ice; @bi2016heterogeneous; @qiu2017strength; @bourque2017heterogeneous] where it is commonplace to use truncated interactions. This work was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement number 616121 (HeteroIce project). A.M. is supported by the Royal Society through a Royal Society Wolfson Research Merit Award. We are grateful for computational resources provided by the London Centre for Nanotechnology and the Materials Chemistry Consortium through the EPSRC grant number EP/L000202. L.J. is supported by the French Ministry of Defense through the project DGA ERE number 2013.60.0013 and by the LABEX iMUST (ANR-10-LABX-0064) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). M.M. is supported by the Thousand Young Talent Program from the Organization Department of the CPC Central Committee.

--- author: - | , Thomas J. Maccarone\ School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, UK\ E-mail: - | Kieran O’Brien\ Department of Physics, University of California Santa Barbara, CA 93106, USA title: 'Fast multiwavelength variability from jets in X-ray binaries' --- Introduction ============ To date, jets have been discovered in a variety of astronomical objects over a wide range of gravitational regimes, from planetary nebulae to accreting white dwarfs, from stellar-mass black holes to supermassive AGN. Clearly, to reach a good understanding of such a general phenomenon would be interesting in its own right, but it would also be important given the important role that such jets have on the evolution of the launching systems (because of the power carried away from the accreting system), and given the influence they have on the surrounding media (e.g., the energetic feedback of relativistic jets in supermassive black holes has cosmological consequences). The wealth of multi-wavelength observations of X-ray binaries (XBs) over the past decade have made clear the ubiquity of jets in these systems [@fender06]. The study of XB jets can be an important tool to understand the physics of jets and their link with the accretion flow, as the luminosity (thus plausibly the accretion rate) in these systems shows variability over several order of magnitudes and on timescales ranging from milliseconds to decades. The presence of jets in XBs had been first proposed to explain the radio emission often observed in these objects and in analogy to the well known jets in AGN, and then confirmed in several cases though imaging. The overall spectrum of XBs is rather complex (see a simplified schematic view in Fig. \[sed\]): the soft X-ray flux is generally believed to come predominantly from an accretion disc around the compact object, whose emission can extend down to optical or infrared wavelengths, while the hard X-ray flux is thought to arise from a hot Comptonizing corona and/or from the jet. Recently it has been shown that also the infrared emission includes a substantial contribution from the relativistic jet, in the hard states of XBs [@corbelfender02; @russelletal07]. Despite the rapid increase in our phenomenological understanding of jets from XBs, we still lack a fundamental understanding of how jets are powered and collimated, or what the bulk and internal properties of the jets are. High-speed simultaneous optical/X-ray photometry of three accreting black holes (BHs) opened a new promising window. Complex correlated variability in the optical and X-ray emission [@spruitkanbach02] was seen from XTE J1118+480, while fast optical photometry of SWIFT J1753.5-0127 [@durantetal08] and GX 339–4 [@gandhietal08] revealed further complexity. [@malzacetal04] explained the behaviour observed in XTE J1118+480 through coupling of an optically emitting jet and an X-ray emitting corona in a common energy reservoir. An alternative explanation comes from the magnetically driven disc corona model [@merlonietal00]: magnetic flares happen in an accretion disc corona where thermal cyclo-synchrotron emission contributes significantly to the optical emission, while the X-rays are produced by Comptonization of the soft photons produced by dissipation in the underlying disc and by the synchrotron process itself. The two explanations differ substantially in the predictions at infrared wavelengths, where a jet appears as the most probable origin for the emission [@russelletal06 see also Fig. \[sed\]]. In order to solve these ambiguities and securely identify the jet variable component, we have started a large multi-wavelengh program, aimed at performing fast-timing simultaneous observations at different wavelengths of several XBs hosting both black holes and neutron stars. Here we report on the result of the first obtained dataset. ![Schematic representation of a typical broad-band energy spectrum of a black-hole X-ray binary in its hard state. The main spectral components are indicated and highly simplified. Several spectral components contribute to the optical emission, with different proportions depending on the source, while going toward longer wavelengths the jet becomes more and more dominant.[]{data-label="sed"}](BH_SED.pdf){width=".8\textwidth"} The first dataset: ISAAC + RXTE observations of GX 339–4 ======================================================== The BH candidate GX 339–4 is a recurrent X-ray transient [@markert73]. It has been detected as a highly variable source from radio through hard X-rays [@makishimaetal86; @corbeletal00; @coriatetal09 and references therein]. Optical spectroscopy indicates a mass function of 5.8 $\pm$ 0.5 M$\odot$ and a minimum distance of 6 kpc [@hynesetal03; @hynesetal04]. It is the first BH XB for which fast optical/X-ray correlated variability was observed [@motchetal82]. Multiwavelength campaigns clearly reveal a non-thermal contribution to the infrared emission in the hard state, most probably arising from a compact jet [@corbelfender02]. Thus, we observed this source with high time resolution simultaneously in infrared and X-rays, aiming at identifying a possible variable jet component. Data reduction -------------- We observed GX339-4 from ESO’s Paranal Observatory on 2008 August 18th. We obtained fast Ks-band photometry with ISAAC (Moorwood et al. 1998) mounted on the 8.2-m UT1/ANTU telescope. The 23” x 23” window used encompassed the target, a bright ’reference’ star located 13.6 arcsec south of our target and a fainter ’comparison’ star 8.9-arcsec north-east of GX339-4 (respectively 2MASS17024972-4847361; Ks=9.5 and 2MASS17024995-4847161; Ks=12.8). We used the “FastJitter mode” with a time resolution of 62.5 ms. This generated cubes of data with 2500 images in each cube and a small deadtime between cubes. The ULTRACAM pipeline[^1] was used for the data reduction. We performed aperture photometry of the three sources (target, reference and comparison stars) and used the bright reference star for relative photometry of the target and comparison stars. The positions of the aperture regions around the target and the comparison star were linked to the position of the bright reference star to allow for image motion and were updated at each time step. The atmospheric conditions were good and the resulting light curve for the comparison star was consistent with a constant, as expected. By combining all 250000 images, we estimate an average magnitude of Ks=12.4$\pm$0.2 for GX 339–4, which corresponds to an average flux of F$\sim 10^{-11} erg s^{-1} cm^{-2}$ . The time-stamp was generated from the DATE-OBS fits keyword, which represents the start time of the first image and the exposure time (DIT) of each subsequent image. A sample of the highly variable light curve for GX 339–4 is shown in the left-bottom panel of Fig. \[lcccf\]. ![[*Left top panel:*]{} A sample of the X-ray light curve of GX 339–4, obtained with the PCA onboard RXTE (2nd out of 3 satellite orbits). The data are background subtracted, in the 2-15 keV energy range, at 1-second time resolution. [*Left bottom panel:*]{} The simultaneous infrared light curve, obtained with ISAAC. We show the ratio between the source (average $4.4\times10^5$ counts/s) and the reference-star ($6\times10^6$ counts/s) count rates in the K$_S$ filter, at 1-second time resolution. The right ordinates show the de-reddened flux. We show the typical error bars in the top-left corner of each panel. [*Right:*]{} Cross-correlations of the X-ray and infrared light curves of GX 339–4 (positive lags mean infrared lags the X-rays). A strong, nearly symmetric correlation is evident in all the three time intervals, corresponding to different RXTE orbits. In the inset we show a zoom of the peaks, showing the infrared delay of $\sim$100 ms with respect to the X-rays. The inset also shows a slight asymmetry toward positive delays. []{data-label="lcccf"}](2lc_new_error.pdf "fig:"){width=".505\textwidth"} ![[*Left top panel:*]{} A sample of the X-ray light curve of GX 339–4, obtained with the PCA onboard RXTE (2nd out of 3 satellite orbits). The data are background subtracted, in the 2-15 keV energy range, at 1-second time resolution. [*Left bottom panel:*]{} The simultaneous infrared light curve, obtained with ISAAC. We show the ratio between the source (average $4.4\times10^5$ counts/s) and the reference-star ($6\times10^6$ counts/s) count rates in the K$_S$ filter, at 1-second time resolution. The right ordinates show the de-reddened flux. We show the typical error bars in the top-left corner of each panel. [*Right:*]{} Cross-correlations of the X-ray and infrared light curves of GX 339–4 (positive lags mean infrared lags the X-rays). A strong, nearly symmetric correlation is evident in all the three time intervals, corresponding to different RXTE orbits. In the inset we show a zoom of the peaks, showing the infrared delay of $\sim$100 ms with respect to the X-rays. The inset also shows a slight asymmetry toward positive delays. []{data-label="lcccf"}](3ccf_draft.pdf "fig:"){width=".5\textwidth"} Simultaneously with the infrared observations, GX 339–4 was observed with the Proportional Counter Array (PCA) onboard the [*Rossi X-ray Timing Explorer (RXTE)*]{}. Two proportional counter units (PCUs) were active during the whole observation. The X-ray data span three consecutive satellite orbits, for a total exposure of 4.6 ksec. The [Binned Mode]{} (8 ms time resolution) was used for this analysis, using the 2-15 keV energy range (channels 0-35). The barycenter correction for Earth and satellite motion was applied. Standard HEADAS 6.5.1 tools were used for data reduction. In the left-upper panel of Fig. \[lcccf\] we show a sample of the light curve, corresponding to the second [*RXTE*]{} orbit. Spectral fitting with a power-law with photon index 1.6 results in a 2–10 keV unabsorbed flux of $F_{\rm X}\sim1.4\times 10^{-10}$ erg s$^{-1}$ cm$^{-2}$ (corresponding to a luminosity of $L_{\rm X}\sim6\times 10^{35} ({d \over 6~kpc})^2$ erg s$^{-1}$). ![[*Left:*]{} Auto-correlations of the X-ray and infrared light curves of GX 339–4 for the 2nd RXTE orbit. The two auto-correlation functions are somewhat overlapping on timescales of $\sim$40 seconds or longer, but the optical one becomes clearly narrower on short timescales. [*Right:*]{} X-ray (2-15 keV) power spectrum of the second RXTE orbit (upper curve), together with the power spectrum of the simultaneous infrared light curve (lower curve). The Poissonian noise has been subtracted from both spectra. The peak at $\sim$6 Hz in the infrared spectrum is instrumental. The high-frequency portion of the infrared spectrum has yet un-modeled systematics, which however do not affect the results presented here.[]{data-label="acfpds"}](2acf_2.pdf "fig:"){width=".485\textwidth"} ![[*Left:*]{} Auto-correlations of the X-ray and infrared light curves of GX 339–4 for the 2nd RXTE orbit. The two auto-correlation functions are somewhat overlapping on timescales of $\sim$40 seconds or longer, but the optical one becomes clearly narrower on short timescales. [*Right:*]{} X-ray (2-15 keV) power spectrum of the second RXTE orbit (upper curve), together with the power spectrum of the simultaneous infrared light curve (lower curve). The Poissonian noise has been subtracted from both spectra. The peak at $\sim$6 Hz in the infrared spectrum is instrumental. The high-frequency portion of the infrared spectrum has yet un-modeled systematics, which however do not affect the results presented here.[]{data-label="acfpds"}](PDSdraft.pdf "fig:"){width=".5\textwidth"} Timing analysis --------------- Both datasets have an absolute time accuracy better than the time resolution used here: ISAAC data have a timing accuracy of about 10 ms (the readout time), while RXTE data have a timing accuracy of 2.5 $\mu$s [@jahodaetal06]. From the light curves shown in the left panels of Fig. \[lcccf\] a strong correlation between X-ray and infrared flux is evident. Both long, smooth variability and short, sharper flares appear with similar relative amplitude in the two energy bands. In order to measure any time delay, we calculated a cross-correlation function (CCF) for each of the three [*RXTE*]{} orbits, without applying any de-trending procedure. The results are shown in the right panel of Fig. \[lcccf\]. The strong correlation is confirmed. The CCF appears highly symmetric and relatively stable over the three time intervals, with the change in amplitude simply reflecting the different variability amplitude in the light curves themselves. In the inset, we show a zoom on the peak of the CCF, which shows how the infrared emission lags the X-rays by 0.1 seconds, to which we associate an uncertainty of 30$\%$ (which includes systematics). For each RXTE orbit, we also calculated the auto-correlation functions and the Fourier power spectra of both the infrared and the X-ray light curves (in Fig. \[acfpds\] we plot those corresponding to the 2nd RXTE orbit). The Fourier power spectra were calculated after filling the gaps in the infrared light curve with simulated Poissonian noise. Different filling methods do not change the resulting power spectra significantly, especially at high frequencies. Strong evidences for a flickering jet ------------------------------------- The main result of our work is the discovery of a strong correlation between the infrared and the X-ray variability in GX 339–4. The fact that the CCF is nearly symmetric and peaks at 100 ms, together with the optical ACFs being narrower than the X-ray ones (at least on timescales shorter than a few tens of seconds, see left panel of Fig.\[acfpds\]) rules out a reprocessing origin for the infrared variability. If the infrared radiation arose from reprocessing of X-rays by the outer disk, the short time delay would imply a highly inclined disk. This would produce an highly asymmetric CCF, with a tail at long lags [@obrienetal02]. Additionally, power spectral analysis shows significant infrared variability (at least 5% fractional rms, see right panel of Fig. \[acfpds\]) on timescales of $\sim$200 ms or shorter, which sets an upper limit of $\sim 6 \times$ 10$^9$ cm to the radius of the infrared-emitting region. From (5% of) the observed infrared average flux of $F\sim 1.5\times10^{-11}$ erg s$^{-1}$ cm$^{-2}$, we derive a minimum brightness temperature of $\sim2.5\,\times 10^6$ K. Optically thick thermal emission of the derived size and temperature would result in a 2–10 keV flux in excess of $10^{-5}$ erg s$^{-1}$ cm$^{-2}$, which is not observed in the data. These values represent very conservative estimates: a smaller region emitting the infrared radiation would result in a higher brightness temperature, which would in turn result in a higher expected X-ray luminosity. With similar arguments we exclude thermal Bremsstrahlung emission. The existence of an infrared lag is also inconsistent with the magnetic corona model [@merlonietal00], in which the same population of electrons produces the infrared synchrotron emission and the X-ray Compton emission. We conclude that the most plausible origin for the observed infrared variability is synchrotron emission from the inner jet. This is confirmed by nearly-simultaneous optical and infrared observations, obtained while the source was in the same low-luminosity state. Those data (Lewis et al., in prep.) show a flat or inverted spectrum (inconsistent with thermal emission from a disc or companion star), and long-timescale ($\sim$minutes) variability stronger in infrared than in optical. This result is a new, independent strong indication that jet synchrotron emission contributes significantly to the infrared radiation in this source. This is the first time that hard-state, compact jet emission has been securely identified to vary on sub-second timescales in an XB, although variability on similar dynamical timescales t$_{Dyn}$ (i.e., scaled to mass) had been already observed in Active Galactic Nuclei [@schodeletal07]. These data thus represent a further step forward towards a full unification of the accretion/ejection process over a broad range of black-hole masses. Jet speed and magnetic field ---------------------------- Our data strongly suggest that the variable infrared emission comes from the jet, although we cannot conclude whether it is optically–thick or –thin synchrotron. The X-ray emission is usually interpreted as Comptonized radiation from energetic plasma in the very inner regions of the accretion flow, although the actual emitting region is still an open issue (either a corona or the base of the jet itself, for a discussion see [@markoffetal05; @maccarone05]. Depending on the assumptions we make, we can obtain estimate of different parameters of the jet. Namely, if we assume that the infrared emission is thick-synchrotron radiation from the jet, the observed time delay between the infrared and the X-ray variability can give us an upper limit (given the unknown time for the ejection to take place) to the travel time of the variability – thus presumably the matter – along the jet. Thus, given a measure of the jet elongation we could estimate the jet speed. Unfortunately such a measure is not available for GX 339-4; however, a jet elongation measurement has been reported from 8.4 GHz observations of another BH XB, Cyg X-1 [@stirlingetal01]. Thus, within the standard model for compact jets [@BK79] and assuming that the main physical properties of the jet do not change, we can rescale the jet elongation measured at radio wavelengths in the BH Cyg X-1 down to the infrared wavelengths, obtaining a measurement of the distance of the infrared-emitting region in the jet from the black hole in GX 339-4. With the caveats of the many key needed assumptions (see [@casellaetal10] for a full discussion of the method and its underlying assumptions), we obtain a 3.3-$\sigma$ lower limit on the jet speed of $\Gamma > 2$. We conclude that, if the infrared synchrotron emission is optically-thick, these data suggest that the jets from accreting stellar-mass BHs are at least mildly relativistic, also in their common low/hard state. If, as is widely suggested, the jet speed corresponds to the escape speed at the launch point, this might imply that the jet is launched from a region very close to the black hole itself. If on the other hand both the infrared and the X-ray emission are from the very base of the jet, arising from thin-synchrotron radiation, then the above calculation does not hold anymore. Within such an assumption (we refer the reader again to [@casellaetal10] for a full discussion of the caveats), the observed time delay between the infrared and the X-ray variability would represent the cooling time of the emitting electron population, from which we obtain an estimate for the magnetic field intensity in the jet of B$\sim 10^4$ Gauss. The future ========== We have detected for the first time fast (sub-second) infrared variability from a jet in an X-ray binary, discovering a clear correlation with the known X-ray variability. We have shown that, within a (large) number of assumptions, this type of data allow us to put quantitative constraints to the jet speed and internal magnetic field. Clearly the obtained estimates have several caveats, or at least large uncertainties. Nevertheless, the potential of the method is revealed, as it offers for the first time the possibility to track the accreting matter from the inflow out in and along the outflow. Future monitoring observations with this technique will allow to refine these measurements, studying the relative dependency of these quantities with the varying accretion rate or total luminosity. In particular, observations performed simultaneously in (mid-)infrared, X-rays and optical at high time resolution will allow to securely disentangle the different varying components, and possibly separate emission from different regions along the jets. Observations in the hard state at different luminosities will provide information on the evolution of the jet characteristics (many of the uncertainties on the estimates reported above will not affect relative measurements), as for example possible jet acceleration. Similarly, observations in different states, possibly tracking the full spectral evolution during an outburst, will provide information on how and if the jet switches off (or on) during spectral transitions to (or out of) the soft state. Finally, similar observations of neutron-star X-ray binaries will allow to study how the jet properties depend on the type of accreting compact object, possibly unveiling the role of the event horizon and/or ergosphere in the jet-launching mechanism. The scheduling of such observations is at present severely complicated from the scarcity of permanently mounted fast photometers (needed given the transient nature of the studied objects), and from the lack of [*multi-wavelength*]{} ones. Nevertheless, some observations have been successfully performed (thanks to the efforts of planners and observers of several facilities, including RXTE, VLT, ULTRACAM and SPITZER), and the analysis is in progress, while others have been approved and will be performed in the following months. The increased statistics that will be available with the upcoming Extremely Large Telescopes will allow to apply these methods to a large number of sources, otherwise too faint to be observed with the telescopes operating today, eventually offering the possibility to perform population-statistics studies. [99]{} R. D. Blandford & A. Königl, 1979, ApJ, 232, 34 J. Cardelli, G. Clayton, J. Mathis, 1989, ApJ, 345, 245 P. Casella & A. Pe’er , 2009, ApJL, 703, L63 P. Casella, T. J. Maccarone, K. O’Brien, R. P. Fender, D. M. Russell, M. van der Klis, A. Pe’Er, D. Maitra, D. Altamirano, T. Belloni, G. Kanbach, M. Klein-Wolt, E. Mason, P. Soleri, A. Stefanescu, K. Wiersema, R. Wijnands, 2010, MNRAS Letters, 404, 21 S. Corbel, R. P. Fender, A. K. Tzioumis, M. Nowak, V. McIntyre, P. Durouchoux, R. Sood, 2000, A&A, 359, 251 S. Corbel, R. P. Fender, 2002, ApJ, 573, 35 M. Coriat, S. Corbel, M. M. Buxton, C. D. Bailyn, J. A. Tomsick, E. Körding, E. Kalemci, 2009, MNRAS, tmp1311 Cowley A. P., Schmidtke P. C., Hutchings J. B., Crampton D., 2002, AJ, 123, 1741 Dolan J. F., 1992, ApJ, 384, 249 Durant M., Gandhi P., Shahbaz T., Fabian A. P., Miller J., Dhillon V. S., Marsh T. R., 2008, ApJ, 682, 45 Eikenberry S. S., Patel S. G., Rothstein D. M., Remillard R., Pooley G. G., Morgan E. H., 2008, ApJ, 678, 369 Fender R.P., Gallo E., Jonker P.G., 2003, MNRAS, 343, 99 Fender R. P., Belloni T., Gallo E., 2004, MNRAS, 355, 1105 Fender R. P., 2006, Jets from X-ray binaries, in Compact Stellar X-Ray Sources, ed. W. H. G. Lewin & M. van der Klis (Cambridge: Cambridge University Press), 381-419 Gallo E., Fender R., Pooley G., 2003, MNRAS, 344, 60 Gallo E., 2007, AIP Conference Proceedings, 914, 715 Gandhi P., Makishima K., Durant M., Fabian A. C., Dhillon V. S., Marsh T. R., Miller J. M., Shahbaz T., Spruit H. C., 2008, MNRAS, 390, L29 Gandhi P., Dhillon V. S., Durant M., Fabian A. C., Kubota A., Makishima K, Malzac J., Marsh T. R., Miller J. M., Shahbaz T., Spruit H. C., Casella P., 2010, MNRAS, in press, arXiv:1005.4685 Gierlinski M., Zdziarski A. A., Poutanen J., Coppi P. S., Ebisawa K., Johnson W. N., 1999, MNRAS, 309, 496 Heinz S., Merloni A., 2004, MNRAS, 355, L1 Hynes R. I., Steeghs D., Casares J., Charles P. A., O’Brien K., 2003, ApJL, 583, L95 Hynes R. I., Steeghs D., Casares J., Charles P. A., O’Brien K., 2004, ApJ, 609, 317 Jahoda K., Markwardt C. B., Radeva Y., Rots A. H., Stark M. J., Swank J. H., Strohmayer T. E., Zhang W., 2006, ApJS, 163, 401 Jamil O., Fender R., Kaiser C., 2009, MNRAS, tmp1543 Kaiser C. R., 2006, MNRAS, 367, 1083 Maccarone T. J., 2005, MNRAS, 360, 68 Malzac J., Merloni A., Fabian A., 2004, MNRAS, 351, 253 Makishima K., Maejima Y., Mitsuda K., Bradt H. V., Remillard R. A., Tuohy I. R., Hoshi R., Nakagawa M., 1986, ApJ, 308, 635 Markert T. H., Canizares C. R., Clark G. W., Lewin W. H. G., Schnopper H. W., Sprott G. F., 1973, ApJ, 184, L67 Markoff S., Nowak M. A., Wilms J., 2005, ApJ, 635, 1203 Merloni A., Di Matteo T., Fabian A. C., 2000, MNRAS, 318, 15 Miller J. M., Reynolds C. S., Fabian A. C., Cackett E. M., Miniutti G., Raymond J., Steeghs D., Reis R., Homan J., 2008, ApJ, 679, 113 Moorwood A., et al., 1998, Messenger, 94, 7 Motch C., Ilovaisky S. A., Chevalier C., 1982, A&A, 109, 1 O’ Brien K., Horne K., Hynes R. I., Chen W., Haswell C. A., Still M. D., 2002, MNRAS, 334, 426 Pe’er A., Casella P., 2009, ApJ, 699, 1919 Russell D., Fender R., Hynes R., Brocksopp C., Homan J., Jonker P., Buxton M., 2006, MNRAS, 371, 1334 Russell D., Fender R., Jonker P., 2007, MNRAS, 379, 1108 Russell D., Maccarone T., Körding E., Homan J., 2007, MNRAS, 379, 1401 Schödel R., Krips M., Markoff S., Neri R., Eckart A., 2007, A&A, 463, 551 Spruit H. C., Kanbach G., 2002, A&A, 391, 225 Stirling A. M., Spencer R. E., de la Force C. J., Garrett M. A., Fender R. P., Ogley R. N., 2001, MNRAS, 327, 1273 Takahashi T., Tashiro M., Madejski G., Kubo H., Kamae T., Kataoka J., Kii T., Makino F., Makishima K., Yamasaki N., 1996, ApJL, 470, L89 Vadawale S., Rao A., Naik S., Yadav J., Ishwara-Chandra C., Pramesh Rao A., Pooley G., 2003, ApJ, 597, 1023 Ziolkowski J., 2005, MNRAS, 358, 851 [^1]: We thank Tom Marsh for the use of the software (http://deneb.astro.warwick.ac.uk/phsaap/software/).

--- abstract: 'Thermal models are very useful in the understanding of particle production in general and especially in the case of strangeness. We summarize the assumptions which go into a thermal model calculation and which differ in the application of various groups. We compare the different results to each other. Using our own calculation we discuss the validity of the thermal model and the amount of strangeness equilibration at CERN-SPS energies. Finally the implications of the thermal analysis on the reaction dynamics are discussed.' address: 'Department of Physics, University of Bielefeld, Universitätsstr., D-33615 Bielefeld, GERMANY' author: - Josef Sollfrank title: Chemical equilibration of strangeness --- BI-TP/97-21\ nucl-th/9707020\ Invited talk given at the [*Int. Symposium on Strangeness in Quark Matter 1997*]{}, Santorini (Greece), April 14 – 18, 1997 Introduction ============ Strange particles and other heavy flavors play always a special role in the analysis of hadronic collisions since they carry a new quantum number not present in the incoming nucleons or nuclei. Therefore they are considered as one of the most promising tools for learning more about the dynamics of heavy ion collisions. Especially, their use as a signature for Quark-Gluon Plasma (QGP) formation was proposed long ago [@Rafelski81; @Rafelski82]. The argument is based on the different time scale for equilibration of strangeness due to the reduced kinematic threshold in the QGP [@Koch86]. There is a clear enhancement of strange particle production in heavy ion collisions compared to nucleon-nucleon collisions [@Kinson95; @Gazdzicki95a]. Especially, the multistrange baryon yields increase drastically from $p+p$ to $A$+$A$ [@diBari95]. This interpretation of the measured abundances in favor of a QGP formation has basically two strategies. First, dynamical arguments show that the lifetime of a QGP is enough to reproduce the (still not complete) strangeness equilibration [@Letessier94a; @Rafelski96]. Second, the strange particle ratios are compatible with a sudden disintegrating QGP [@Letessier94b; @Letessier95]. However, the interpretation is still controversial [@Sollfrank95] and alternative explanations without QGP formation are also successful [@Sorge94; @ToporPop95; @Capella96]. The various microscopic production models for strange particles in heavy ion collisions go from perturbative calculations on parton level [@Rafelski96] to string-string interactions (color ropes) [@Sorge94] to hadronic rescattering mechanisms [@Capella96]. We leave out the complicated question of the dynamics of strange quark production and address the simpler question of whether strangeness production already saturates in heavy ion collisions, i.e. it reaches chemical equilibrium in the final state. This investigation concentrates only on the final freeze-out and it implicitly assumes that the particle production may be described in a statistical or even thermal model. Therefore one has first to address the more general question whether the overall particle production is given by a thermal production mechanism. The answers will in general depend on the particle species, the center of mass energy and the volume. The recent success in the statistical interpretation of particle production in elementary reactions like $e^+ + e^-$ [@Becattini96a] and $p + p$ ($\bar{p} + p$) [@Becattini97a] as well as in heavy ion reactions at various energies [@BraunMunzinger95; @BraunMunzinger96] triggered a revival of the thermal model applications. Therefore we will concentrate to review the status of the thermal model for particle production in general and in the special case of strangeness. In section 2 we will present a hitch hikers guide through thermal models addressing various points which differ in the application of the model by various groups. Section 3 is devoted to the discussion of our own calculation done for CERN-SPS energies. In section 4 we discuss the implications from the thermal analysis of particle production and give a general conclusion. The thermal model ================= A lot of publications have addressed the question of particle production within the thermal model. This can be seen in table \[summary\] where only recent ones have been included. The list is certainly not complete due to ignorance or due to series of publications of one group where we took only the most recent one. Even if the thermal formalism is used for the same collision system the results vary (see table \[summary\]) because the model is applied in various approximations and extensions which we would like to discuss first. This will help to understand the differences in the calculations and the conclusions about the physics of these reactions. [@llllll]{} & $\sqrt{s}$ & & & &\ collision & (MeV) & $T$ (MeV) &$\mu_B$ (MeV) & $\gamma_s$ & ref.\ $e^+$ + $e^{-}$ & 29.0 & $196\pm7 $ & - & 1 & [@Hoang88]\ $e^+$ + $e^{-}$ & 91.5 & $261\pm9 $ & - & 1 & [@Hoang94]\ $e^+$ + $e^{-}$ & 29.0 & $163.6\pm3.6 $ & - & $0.724\pm0.045$ & [@Becattini96b]\ $e^+$ + $e^{-}$ & 91.5 & $160.6\pm1.7 $ & - & $0.675\pm0.020$ & [@Becattini96b]\ p+p & 19.5 & $161\pm31 $ & $200\pm37$ & $0.22\0\pm0.05 $ & [@Sollfrank94]\ p+p & 19.5 & $190.8\pm27.4$ & - & $0.463\pm0.037$ & [@Becattini97a]\ p+p & 27.5 & $169.0\pm2.1 $ & - & $0.510\pm0.011$ & [@Becattini97a]\ Si+Au & 5.3 & $100.2$ & $559.5$ & 1 &[@Davidson91a]\ Si+Au & 5.3 & $127\pm8$ & $485\pm70$ & $0.5\pm0.2^{\rm a}$ &[@Letessier94c]\ Si+Au & 5.3 & $140\pm5$ & $555\pm33$ & $ 1 $ &[@Panagiotou96]\ Si+Au(Pb) & 5.3 & $130\pm10$ & $540$ & $ 1 $ &[@BraunMunzinger95]\ Si+Au & 5.3 & $110\pm5$ & $540\pm20$ & $ 1 $ &[@Cleymans97a]\ Au+Au & 4.7 & $100\pm4$ & - & $ 1 $ &[@Cleymans97b]\ S+S & 19.5 & $170$ & $257$ & $1$ &[@Davidson91b]\ S+S & 19.5 & $197\pm29$ & $267\pm21$ & $1.00\0\pm0.21$ &[@Sollfrank94]\ S+S & 19.5 & $185 $ & $301$ & $1$ &[@Tiwari96]\ S+S & 19.5 & $192\pm15$ & $222\pm10$ & $1$ &[@Panagiotou96]\ S+S & 19.5 & $182\pm9$ & $226\pm13$ & $0.73\0\pm0.04$ &[@Becattini97b]\ S+S & 19.5 & $202\pm13$ & $259\pm15$ & $0.84\0\pm0.07$ &[@Sollfrank97b]\ S+Ag & 19.5 & $191\pm17$ & $279\pm33$ & $1$ &[@Panagiotou96]\ S+Ag & 19.5 & $180.0\pm3.2$ & $238\pm12$ & $0.83\0\pm0.07$ &[@Becattini97b]\ S+Ag & 19.5 & $185\pm8$ & $244\pm14$ & $0.82\0\pm0.07$ &[@Sollfrank97b]\ S+Pb & 19.5 & $172\pm16$ & $292\pm42$ & $1$ &[@Andersen94]\ S+W & 19.5 & $190\pm10$ & $240\pm40$ & $0.7$ &[@Redlich94]\ S+W & 19.5 & $190$ & $223\pm19$ & $0.68\0\pm0.06$ &[@Letessier95]\ S+W & 19.5 & $196\pm9$ & $231\pm18$ & $1$ &[@Panagiotou96]\ S+Au(W,Pb) & 19.5 & $165\pm5$ & $175\pm5$ & $1$ &[@BraunMunzinger96]\ S+Au(W,Pb) & 19.5 & $160$ & $171$ & $1$ &[@Spieles97]\ S+Au(W,Pb) & 19.5 & $160.2\pm3.5$ & $158\pm4$ & $0.66\pm0.04$ &[@Sollfrank97b]\ \ $^{\rm a}$ only guessed. Basic particle yields --------------------- In the thermal model the particle yields are given by a temperature $T$ and a volume $V$ common for all particles. If one considers only particle ratios then in most of the applications the ratio is independent of $V$ as it was shown in [@Cleymans97c]. In addition the abundance of a particle depends on its conserved quantum numbers. This is either regulated by chemical potentials in the grand canonical description or by restricting the partition function only to states which have the same quantum number as the fireball, i.e. canonical treatment. The basic expression for the abundance $N_j$ of a particle of species $j$ is given by $$N_j = \lambda_j \frac{\partial \ln Z(T,V,...)}{\partial \lambda_j}.$$ All models fulfill this basic requirement with the exception of the work of Hoang for $e^+$ + $e^-$ collisions [@Hoang88; @Hoang94]. Some empirical formula for the particle yields is used, which is badly justified. Therefore all results in [@Hoang88; @Hoang94] should be taken with care. Statistic --------- It is obvious that one should use quantum statistics for calculating the partition function, i.e. to use Bose and Fermi statistics. However, for practical reasons one usually switches to the Boltzmann approximation. The error for pions is at $T = 150$ MeV ($T = 200$ MeV) 9.4% (11.3%), respectively and for kaons at the same temperature 0.5% (1.1%), respectively. This estimate suggest to use Bose statistic for pions while for all other heavier particles the Boltzmann approximation is valid. Note, that for entropy and pressure the differences between Boltzmann statistic and Bose/Fermi statistic are larger. Going to very low temperatures and/or high densities the use of the right statistic is unavoidable [@Lee93]. Canonical vs grand canonical ---------------------------- The strong interaction conserves exactly the quantum numbers charge $Q$, baryon number $B$ and strangeness S. This has to be taken care in a statistical approach and therefore the question arises which statistical ensemble concerning these quantum numbers one has to use [@canonical; @Redlich80; @Cleymans97b]. As a first estimate one usually considers the fluctuations $\Delta O $ of a conserved quantity $O$ in the grand canonical treatment [@Pathria72] $$\label{fluctuation} \Delta O = \sqrt{\langle O^2 \rangle - \langle O \rangle^2} \propto \sqrt{N},$$ where $N$ is the number of all particles carrying a non zero quantum number $O$. In order to be better than 10% this suggest to use canonical treatment when the number of corresponding particles is below 100. However, in practice – depending on the observable calculated – already a much smaller total amount $N$ of particles is enough to have accuracy better then 10% using the grand canonical ansatz. This was shown in [@canonical] and also recently in [@Cleymans97b], where particle ratios already seem to saturate for number of participating baryons of around 30–40. In the applications for heavy ion collisions the grand canonical treatment is justified when single particle yields are addressed [@Cleymans91]. We tested the difference in the resulting thermal parameters using canonical and grand canonical treatment for strangeness. We saw only minor differences between both calculations for ultra-relativistic heavy ion collisions. However, for strange particle correlations the use of the canonical ansatz is recommended [@Cleymans91]. If the thermal model is applied to inelastic two body collisions at high energies the analysis should be done using the canonical ensemble. Among the calculations in table \[summary\] this was only performed by Becattini [@Becattini96b; @Becattini97a]. All other calculations concerning $p$ + $p$ and $e^+$ + $e^-$ have to be taken with care. For example, we made a least mean square fit ($\chi^2$-fit) using the grand canonical ensemble for charge, baryon number and strangeness for particle production in p+p collisions at $\sqrt{s} = 27.5$ GeV. The experimental input data are the same as in the work of Becattini [@Becattini97a] and the resonances included are very much the same. The fit to the data gets worse but the fit temperature stays very much the same and turned out to be $T = 160$ MeV but the strangeness suppression $\gamma_s$ (see Section \[offequilibrium\]) is reduced to $\gamma_s = 0.35$ compared to $\gamma_s = 0.51$ in [@Becattini97a]. The canonical treatment suppresses the strangeness production due to associate pair production leading naturally to lower strangeness yields as compared to elementary reactions. The chemical potentials in the grand canonical approach are first of all Lagrangian multipliers for the conserved charge. However, the strange quark chemical potential $\mu_{\rm s}$ plays a special role in the interpretation of strange particle abundances [@Rafelski91]. It is zero in a strangeness neutral QGP but has usually non-zero values in an equilibrated hadron gas. The $\mu_{\rm s} = 0$ line in the $T$-$\mu_{\rm B}$ plane calculated for an equilibrated hadron gas is close to the expected QGP phase transition line [@Asprouli95; @Panagiotou96]. This similarity leads to speculations about the meaning of the $\mu_{\rm s} = 0$ line which were extensively discussed in [@Asprouli95; @Panagiotou96]. Isospin ------- Isospin breaking effects become important when target and projectile nuclei are large. In most of the thermal models isospin is neglected. We may estimate the effect of isospin breaking to be of order $1 - 2Z/A$. This leads in S+Au collisions to a correction of order 10 % and for Pb+Pb of order 20%. Therefore one should take charge and baryon number separately into account especially for isospin sensitive quantities like $\pi^+/\pi^-$ or $K^+/K^-$. All above listed models in table \[summary\] neglect isospin in the case of a heavy target. As an example one may discuss the $\pi^+/\pi^-$ ratio which is different from one in the low $p_T$-region of Au + Au [@Ahle95] and Pb + Pb [@Boggild96] collisions by chemical equilibration arguments. This ratio is sensitive to isospin violating weak decays as discussed in [@Arbex97]. But in addition the isospin asymmetry in the colliding nuclei plays an important role and this was neglected in [@Arbex97]. The isospin SU(2) may be treated exactly like in [@Redlich80] or to a very good approximation as an U(1) $\times$ U(1). Then one usually conserves baryon number and charge or on valence quark level the net number of u-quarks and net number of d-quarks. In a grand canonical treatment it is equivalent to introduce chemical potential for baryon number and charge or for up and down quarks. Finite volume correction ------------------------ Calculating the partition function one switches from the summation over the states to the integral representation $$\sum\limits_{k}^{\rm states} \longrightarrow \int \d^3 \vec{k} \; g(\vec{k}) = 4\pi \int \d k \; k^2 g(k),$$ where $g(k)$ is the density of state. If the volume is small $g(k)$ is very different from the infinite volume limit $V/(2\pi)^3$ and often approximated by [@Hill53; @Pathria72] $$\label{gk} g(k) = \frac{V}{(2\pi)^3} - \frac{S}{32 \pi^2 k} + \frac{L}{32 \pi^2 k^2},$$ where $V$ is the volume $S$ the surface and $L$ the circumference of the box. This is only an approximate formula derived for Dirichlet boundary conditions and the error is of the order of the last term. This kind of correction was applied in [@BraunMunzinger95; @BraunMunzinger96; @Davidson91a; @Davidson91b]. Going to $p$ + $p$ collisions one might think that this kind of correction might be very important. However, we like to argue that in this case and also for ultra-relativistic heavy ion collisions it is reasonable to use the continuous density of state for the following reason. In equation (\[gk\]) it is assumed that the states have to fulfill Dirichlet boundary conditions on the edge of the interaction volume. This is only true for an infinite high potential well. In reality there is no such large binding force for the particles squeezing its wave function to the reaction volume. The particle energies are much higher than a nuclear binding potential. Therefore we think that the use of the continuous density of state is appropriate as long as the thermal energies are above a mean field potential which serves as a confining box. Resonances ---------- It is now commonly agreed that the particle production is dominated by resonance production mechanisms. Therefore the resonance states are included in the partition function of all calculations in table \[summary\] except for [@Hoang88; @Hoang94]. On the other hand their decay is sometimes neglected [@Panagiotou96; @Tiwari96] when calculations are compared to experimental data. Some of the publications restrict their discussion to strange baryon ratios. Then the resonance contribution may be restricted to the inclusion of the $\Sigma^0$ decay which feeds into the $\Lambda$ yield. This gives a reasonable estimate for the fugacities. If, however, kaons are included in the chemical analysis one has to consider the sizeable feeding from resonances and its omission is hardly comprehensible. In the treatment of resonances the following items are important. ### Number of resonance states The modeling of a steady state hadron gas needs the input of all resonance states leading to the bootstrap description of thermodynamics of strong interacting matter [@Hagedorn71]. As a result one gets an exponential increasing density of resonance states $$\label{bootstrap} \rho(m) \propto m^{a}\exp(m/T_{\rm H})$$ with the consequence of a limiting temperature $T_{\rm H}$, called Hagedorn temperature. In the applications for hadronic collisions it is assumed that only a finite range in the resonance mass spectrum is equilibrated. The finite volume of the fireball leads to a finite total energy giving an upper limit in the mass spectrum to be considered. This estimate corresponds practically to an infinite mass spectrum. It arises the question whether already a much lower cut in the resonance spectrum can be justified. If chemical equilibrium is build up via secondary interactions than the limited life time of the fireball leads to a limited amount of equilibrated resonance states. If, however, the hadronization in elementary p + p collisions follows the chemical equilibrium abundances – and this seemed to be the case [@Becattini97a] – then it is hard to argue for the omission of the higher mass states, especially if the temperature is around 200 MeV. Nevertheless the known resonance states fade at masses around 2 GeV and therefore the calculations have to be restricted to a finite number of states. The applied cut in the resonance states of the various groups is arbitrary and given by practical considerations. It has been shown that there is a small influence on the extracted thermal parameters on the cut in the resonance spectrum [@Sollfrank94; @Becattini96a; @Becattini97a]. The temperature fitted to measured particle ratios shows a maximum at a resonance cut-off at $\approx 1.5$ GeV. It is interesting to note that fits to the experimental known density of states $\rho(m)$ by the bootstrap formula in equation (\[bootstrap\]) start to deviate at the same mass of 1.5 GeV [@Tounsi94]. Therefore one should be aware that thermal fits at temperatures higher than $\approx 170$ MeV are biased from the resonance spectrum which is taken into account. ### Branching of resonances Not only the poor knowledge of resonance states around 2 GeV has some influence on the analysis for high temperatures, but also the branching ratios. Already at 1.5 GeV they are starting of getting basically unknown and the various groups use some “educated guess” [@Cleymans97a]. ### Resonance width Usually the width of a resonance is neglected. This means that in the Boltzmann factor $\exp(-\sqrt{m^2 + p^2}/T)$ the mean mass $\bar{m}$ of a resonances is taken. This assumption was made in nearly all calculations of table \[summary\]. For broad resonances this is a bad approximation. One suggestion to improve this is to distribute the mass states of a resonance according to a Breit-Wigner form [@Becattini96a; @Becattini97a; @Becattini97b; @Sollfrank97b]. This means that the mass shell constraint in the Lorentz invariant momentum integration for the partition function is replaced by [@Sollfrank91] $$\begin{aligned} \label{width} \int \d^4 p \; \delta\left( p_\mu p^\mu - m_0^2\right) \theta \left(\sqrt{p_\mu p^\mu}\right) && \nonumber \\ \longrightarrow \int \d^4 p \; \frac{m_0\Gamma}{\left(p_\mu p^\mu - m_0^2\right)^2 - m_0^2\Gamma^2} \; \theta\left(\sqrt{p_\mu p^\mu} - m_{\rm thres}\right), &&\end{aligned}$$ where $\Gamma$ is the width of the resonance and $m_{\rm thres}$ a threshold for the resonance production. The inclusion of the width is important for the yield of the $\rho$-meson. It turned out to improve the fits in p + p collisions [@Becattini97a]. Repulsive interaction --------------------- The experimental particle yields and ratios are determined at (chemical) freeze-out. One usually assumes that the system is already such diluted that the interactions have effectively cease to exist. Then the problem of residual interactions don’t occur. However, in the analysis of SPS energies the chemical freeze-out temperature seems to be around 160–200 MeV (see table \[summary\]). At this temperature the particle density is still very high and one has to ask how density corrections influence the particle yields. In the bootstrap model of Hagedorn the dominant part of the attractive strong interaction is effectively taken into account by including the higher resonance states [@Hagedorn71]. However, at high densities the repulsive part have to be accounted for, too. It is popular to use an excluded volume correction [@Hagedorn80; @Cleymans86; @Rischke91; @Uddin94] where the hadrons are treated as finite size hard core particles. In the early suggestions [@Hagedorn80; @Cleymans86] the real physical particle density $n^{\rm phy}$ is related to the ideal or point particle density $n^{\rm point}$ by $$\label{excludedv} n^{\rm phy} = \alpha^{-1} n^{\rm point},$$ where $\alpha$ is either given by the total point particle energy density $\varepsilon_0$ and the bag constant $B$, i.e. $\alpha = 1 + \varepsilon_0/(4B)$ [@Hagedorn80], or by $\alpha = 1 + \sum_j V^0_j n_j^0$ where $V^0_j$ is a hard core volume and $n_j^0$ the point particle density of species $j$ [@Cleymans86]. The important point is that such a treatment don’t influence particle ratios since the factor is common for all particle species and therefore don’t change the thermal analysis. The volume $V$ which appears as a common factor has to be regarded as the point particle volume and the physical volume is then given by $\alpha V$. The above described correction is thermodynamically not consistent [@Hagedorn80; @Cleymans86] and improvements have been suggested [@Rischke91; @Uddin94; @Kapusta83] (for an extended discussion see [@Venugopalan92]). One may divide them basically into mean field approaches like [@Kapusta83] or thermodynamically consistent excluded volume corrections like [@Rischke91; @Uddin94] or both [@Rischke91]. These models contain additional parameters which characterize either the hard core size $V^0_j$ or the mean field coupling $K_j$ of a particle $j$. If $V^0_j$ or $K_j$ are different for various $j$ then they influence the particle ratios. The only publications in table \[summary\] which have such an influence on particle ratios due to repulsions are [@Davidson91a; @Davidson91b; @Tiwari96]. In [@Davidson91a; @Davidson91b] the effect is minimal since the mean field coupling $K_B = 680$ MeV fm$^3$ of baryons and anti-baryons is similar to the one of mesons $K_M = 600$ MeV fm$^3$. The studies in [@Tiwari96] show the repulsion effect for the heavy baryons. The extracted $\mu_B$ in [@Tiwari96] differs from the other results of table \[summary\]. Tiwari take for the hard core size of a particle the MIT bag model result of $V^0_j = m_j/(4B)$, i.e. the hard core volume scales with its mass. Therefore the massive baryons are additionally suppressed by their size. This has to be compensated by a higher $\mu_B$. Off-equilibrium phenomenology\[offequilibrium\] ----------------------------------------------- Since the life time of a fireball created in heavy ion collisions is very short and the dynamics is very rapid it cannot be expected that the production of particles of all kind follow the equilibrium statistics. In order to study the deviations from equilibrium quantitatively one introduces over-saturation/suppression factors $\gamma$ which measure the deviations from full equilibrium. For strangeness they were first introduced by Rafelski [@Rafelski91] in a phenomenological way. They were defined by $$\gamma = \frac{\rm actual\;density}{\rm equilibrium\;density}.$$ It has been shown [@Slotta95] that the thermodynamically correct way of defining such a parameter is to define them as fugacity $$\label{gammas} \gamma = \exp(\mu/T),$$ as it is done in a grand canonical approach. This is in accordance with the model of relative chemical equilibrium. This means that only a subset of particles is in chemical equilibrium among each other and the deviation to another set of particles is parametrized by $\gamma$. Examples for such applications are the strangeness suppression $\gamma_s$ [@Rafelski91], the pion chemical potential [@Kataja90] or general meson and baryon suppression factors [@Letessier94d]. With the help of suppression factors one is able to study the approach to chemical equilibrium. Some of the calculations in table \[summary\] allow for such a suppression and some don’t. Since we know from $p$ + $p$ collisions [@Becattini97a] that strangeness is suppressed by roughly a factor of 2 one should always allow the suppression possibility in heavy ion reactions. In table \[summary\] all calculation which don’t allow strangeness suppression have 1 in the column for $\gamma_s$. Restricted acceptance --------------------- If thermal equilibrium is established it is either globally or locally. In most of the cases as for example in S+S collisions at CERN-SPS the rapidity distributions of baryons and pions are very different in shape [@Bachler94]. This suggest that the freeze-out parameters vary locally. One way to demonstrate this was assuming a rapidity dependence of the baryon chemical potential as done in [@Slotta95]. However, the proper treatment of locally varying thermal parameters is via hydrodynamics [@Sollfrank97a]. On the other hand, if the differences are small, the use of one global fireball in the analysis of $4\pi$ data is appropriate and the result should be understood as a global average. The most reliable way to analyze particle yields and ratios is using $4\pi$ integrated data. This avoids problems due to kinematic cuts. Addressing particle ratios in restricted kinematic regions needs a model for the particle spectra, too. Particle spectra are much more sensitive to the dynamics and one needs more assumptions and knowledge about the space-time evolution in longitudinal and transverse direction. Therefore we recommend the use of $4\pi$ data for the study of chemical equilibration. Most of the analysis in table \[summary\] is applied to particle ratios in a restricted kinematic region. The reason is that most of the particle ratios are only measured in a kinematic window. This requires that the calculation is cut to the experimental acceptance and the knowledge of the particle spectra is unavoidable. We like to demonstrate how much results in principle may depend on assumptions about the longitudinal dynamics at freeze-out. The two extreme scenarios are a static fireball and the Bjorken boost-invariant scenario [@Bjorken83]. In a static fireball the rapidity distribution using Boltzmann approximation is given by $$\begin{aligned} \label{static} \frac{\d N^{\rm static}}{\d y} (y) &=& \frac{V g m^2 T}{2\pi^2}\exp\left[-(m\cosh(y) -\mu)/T\right] \nonumber \\ &&\times \left[ \frac{1}{2} + \frac{T}{m\cosh(y)} + \left(\frac{T}{m\cosh(y)}\right)^2 \right],\end{aligned}$$ while in the Bjorken case it is $$\label{bjo} \frac{\d N^{\rm Bjo}}{\d y} (y) \propto \frac{g m^2 T}{2\pi^2} \; \exp(\mu/T) \; {\rm K}_2(m/T).$$ It is $V$ the volume $g$ the spin degeneracy and $m$ the mass. As an example we show in figure \[rapidity\] the pion and proton rapidity distributions for a static fireball given by equation (\[static\]). The pion and proton distribution are normalized to one and therefore its ratio at midrapidity is one. In the Bjorken scenario the ratio of pions to protons is given by the integration over rapidity in equation (\[static\]) resulting in the expression of equation (\[bjo\]). In our example it would be $(dN^{p}/dy)/(dN^{\pi}/dy) = 0.54$. Since the mass of pions and protons are very different the effect is most pronounced in the example. We show in Section \[results\] that in practice the differences between both scenarios are minor using the example of S+Au collisions. The experimental rapidity distributions are broader than given by a static source (\[static\]) but not infinite broad like in the case of Bjorken scaling. Since the experimental width of various rapidity spectra is very similar a thermal analysis in a restricted rapidity range usually assumes Bjorken scaling and uses equation (\[bjo\]) for particle yields. Results from thermal model analysis \[results\] =============================================== The strength of the thermal model is that most of the particle ratios can be explained by only a few parameters. However, from table \[summary\] one cannot see how well the thermal model works and where the deviations start. Therefore we show the results of one calculation in more detail. We perform a thermal model calculation which has the following characteristic [@Sollfrank97b]: - All hadronic states up to 1.7 GeV in mass are included. - Pions follow the Bose statistic while for all other hadrons the Boltzmann approximation is used. - The resonances are populated including their width according to equation (\[width\]). The Breit-Wigner distribution in mass is restricted to a range of two times the width [@Becattini96a]. - When comparing to experimental results we include the feeding of resonances and in the case of S+Au we also include the $p_T$-cut of the experimental ratios. - For S+S and S+Ag strangeness is treated in the canonical formalism while for S+Au we use the grand canonical ensemble. Baryon number is regulated via a chemical potential. Isospin symmetry is assumed. - No finite size correction and no repulsive interaction is included. We analyze experimental particle yields in two ways. On the one hand we perform a $\chi^2$-fit to $4\pi$ data of S+S and S+Ag collisions and to central particle ratios in S+Au collisions. The second possibility is to display the experimental particle ratios in the $T$-$\mu_B$ plane and look for overlap regions of the various bands. [@lllllll]{} & S+S & & & S+Ag & &\ & calculation & data & ref. & calculation & data & ref.\ $h^-$ & 83.7 & $94\pm5$ & [@Bachler94] & 151 & $160\pm8$ & [@Rohrich94]\ $K^+$ & 12.7 & $12.5\pm0.4$ & [@Bachler93] & 23.0 & &\ $K^-$ & 7.13 & $6.9\pm0.4$ & [@Bachler93] & 13.3 & &\ $K_s^0$ & 9.70 & $10.5\pm1.7$ & [@Alber94] & 17.8 & $15.5\pm1.5$ & [@Alber94]\ $\Lambda$ & 8.69 & $9.4\pm1.0$ & [@Alber94] & 14.4 & $15.2\pm1.2$ & [@Alber94]\ $\bar{\Lambda}$& 1.84 & $2.2\pm0.4$ & [@Alber94] & 2.54 & $2.6\pm0.3$ & [@Alber94]\ $p - \bar{p}$ & 22.6 & $20.2\pm2.0$ & [@Bachler94] & 38.1 & $34\pm4$ & [@Rohrich94]\ $\bar{p}^{\rm a}$& 1.93 & $1.15\pm0.4$ & [@Alber96] & 2.99 & $2.0\pm0.8$ & [@Alber96]\ T (MeV) & $202\pm13$ & & & $185\pm8$ & &\ V (fm$^3$) & $81.5\pm39.4$ & & & $275\pm84 $ & &\ $\gamma_s$ & $0.84\pm0.07$ & & & $0.82\pm0.07$ & &\ $\lambda_q$ & $1.532\pm0.038 $ & & & $1.552\pm0.041$ & &\ $\chi^2$/dof & 11.6/4 & & & 6.72/2 & &\ \ $^{\rm a}$ The experimental value is extrapolated to $4\pi$ assuming the same rapidity shape as the $\bar{\Lambda}$. In table \[ss\] we show the result of a $\chi^2$-fit to $4\pi$ data from the NA35 collaboration (see references in table \[ss\]). In S+Ag collisions the assumed isospin symmetry is slightly violated. In order to estimate the size of the effect we have also done a fit using separate chemical potentials for up and down quarks and including the total net charge. In the case of S+Ag collisions the result is a slightly larger $\lambda_d = 1.572\pm0.053$ as compared to $\lambda_u = 1.521\pm0.034$. This is expected by the larger amount of incoming u-quarks. The calculations are very similar to the one of Becattini [@Becattini97b] but note that in the case of S+S collisions a different input set of experimental data is used. Therefore we get a much higher temperature for S+S while for S+Ag both calculations basically agree. The differences between them are small deviations in the input resonances states, their branching and the treatment of the $\eta$-$\eta^\prime$ mixing. Looking at the result in more detail one realizes that the thermal fit is not perfect, especially for S+S collisions ($\chi^2/{\rm dof} = 11.6/4$). The largest deviations are in the anti-baryon yields. The high absorption cross section of this particles may explain the deviations. A different way of displaying the quality of the thermal model approach is shown in figure \[ssfig\] where in the $T$-$\mu_B$-plane various bands indicate experimental particle ratios. We changed to particle ratios because they are nearly independent of the volume. Since we use the canonical ensemble for strangeness we have a small influence of the particle ratios on the volume. We calculated the used experimental particle ratios from table \[ss\]. The errors of the ratios are determined by adding the individual errors quadratically. The bands correspond to the upper and lower bound on the experimental ratio. Note that a fixed volume was used. In figure \[ssfig\] we see no real overlap region of all particle ratios. Especially, the ratios containing the $h^-$ fail to cover the $\chi^2$-fit point which is given by the filled circle. We like to point out the possible sign of an enhanced entropy production seen in the $h^-$ as discussed in [@Letessier95; @Gazdzicki95b]. Our present reevaluation of the experimental data confirms this possibility. We expect a stronger effect on an enhanced pion production in Pb+Pb collisions. In S+Ag collisions the quality of particle production using a thermal model is similar to the one in S+S as may be seen from the $\chi^2$ in table \[ss\] or in figure \[sagfig\] where we plotted again various ratios in the same way as in figure \[ssfig\]. All particle ratios, excluding $K_s^0/h^-$, have one common overlap at $T = 170\pm10 $ MeV and $\mu_B = 220\pm 10$ MeV. The inclusion of the $K_s^0/h^-$ ratio in the $\chi^2$-fit moves that result out of the otherwise common overlap. Excluding $K_s^0$ one gets for S+Ag collisions similar freeze-out parameters as for S+Au collisions below. [@lllllll]{} S+Au & cal. & data& target& rapidity& $p_T$cut& ref.\ $D_q^{\rm a}$ & 0.0782 & $ 0.088\pm0.007$ & Pb & 2.3–3.0 & 0 & EMU05[@emu05]\ $p/\pi^+$ & 0.188 & $0.19\pm0.03$ & Pb & 2.6–2.8 & 0 & NA44[@Murray94]\ $\bar{p}/\pi^-$ & 0.0262 & $0.024\pm0.009$ & Pb & 2.6–2.8 & 0 & NA44[@Murray94]\ $\bar{p}/p$ & 0.139 & $0.12\pm0.02$ & Pb & 2.65–2.95& 0 & NA44[@Jacak94]\ $\eta/\pi^0$ & 0.0816 & $0.15\pm0.02$ & Au & 2.1–2.9 & 0 & WA80[@Albrecht95]\ $R_K^{\rm b}$ & 2.03 & $2.14\pm0.06$ & W & 2.5–3.0 &1.0& WA85[@diBari95]\ $K^+/K^-$ & 1.57 & $1.67\pm0.15$ & W & 2.3–3.0 & 0.9& WA85[@diBari95]\ $K_s^0/\Lambda$ & 1.21 & $1.4\pm0.1$ & W & 2.5–3.0& 1.0 & WA85[@Abatzis96]\ $\bar{\Lambda}/\Lambda$ & 0.203 & $0.196\pm0.011$ & W & 2.3–3.0& 1.2 & WA85[@Evans96]\ $\Xi^-/\Lambda$ & 0.0967 & $0.097\pm0.006$ & W & 2.3–3.0& 1.2 & WA85[@Evans96]\ $\Xi^+/\Xi^-$ & 0.283 &$0.47\pm0.06$ & W & 2.3–3.0& 1.2 & WA85[@Evans96]\ $R_\Omega^{\rm c}$ & 0.145 &$0.8\pm0.4$ & W &2.5–3.0& 1.6& WA85[@diBari95]\ $\bar{\Omega}/\Omega$ & 0.430 &$0.57\pm0.41$ & W & 2.5–3.0 & 0 & WA85[@Abatzis93]\ \ $^{\rm a}$ $D_q = (h^+ - h^-)/(h^+ + h^-)$ $^{\rm b}$ $R_K = (K^+ + K^-)/K_s^0$ $^{\rm c}$ $R_\Omega= (\bar{\Omega}+\Omega)/(\Xi^+ + \Xi^-)$ There are not enough $4\pi$ data on S+Au collisions and therefore we switch to particle ratios. The analysis is inspired by the work of Braun-Munzinger [@BraunMunzinger96]. However we take in our analysis only a subset of particle ratios from their list, excluding all ratios which don’t cover midrapidity $y_{\rm cm} = 2.65$. For the particle yields we use the scaling assumption, i.e. equation (\[bjo\]) as it was done in [@BraunMunzinger96]. In addition we change to the grand canonical ensemble for strangeness. The result of the $\chi^2$-fit is given in table \[sau\]. We reproduce the temperature $T = 160$ MeV as it was assumed in [@BraunMunzinger96] but we got a slightly lower $\mu_B$ of 158 MeV. The main difference of our calculation to the one in [@BraunMunzinger96] is that we allow strangeness suppression. The ratios sensitive to $\gamma_s$ are $\Xi^-/ \Lambda$ and $R_\Omega= (\bar{\Omega}+\Omega)/(\Xi^+ + \Xi^-)$. The resulting low value of $\gamma_s = 0.65$ is not in agreement with the assumption of full chemical equilibrium for strangeness as it was assumed in many calculations. The result of table \[sau\] shows that a thermal hadron gas model even with no complete strangeness equilibration is not able to reproduce [*all*]{} experimental data. Some serious deviations are not in the table like $\bar{\Xi}/\bar{\Lambda} = 0.23 \pm 0.02$ [@Evans96] which in the thermal model is given by $\bar{\Xi}/\bar{\Lambda} = 0.135$. The disagreement in the multi-strange baryons might indicate the onset of non-equilibrium physics with the origin in a QGP formation [@Rafelski81]. This proposal was recently discussed in depth in [@Rafelski96]. We tested our result against the assumption of Bjorken scaling in rapidity and did the same fit assuming one static fireball, i.e. using equation (\[static\]) at $y=0$. We got a rather similar result of $T= 158\pm3$ MeV, $\lambda_q = 1.408\pm0.012$ and $\gamma_s = 0.74 \pm 0.05$. We see in the $\chi^2$-fit no large dependence on the assumption about the rapidity distribution. We show in figure \[saufig\] some selected ratios from table \[sau\] together with the point from the $\chi^2$-fit. Again we see no perfect agreement but a rather broad region where various bands concentrate. The statistical error in the $\chi^2$-fit is determined by the region where $\chi^2$ increases by one unit from its minimum. However, from the figures \[ssfig\], \[sagfig\] and \[saufig\] one sees that the systematic error of the model applied is much larger. From the figures we conclude that the freeze-out temperature in heavy-ion collisions is still very uncertain and one has in fact to take a range of $T^{\rm chem} = 150$–200 MeV. The results on sulphur induced collisions at CERN-SPS indicate no full strangeness equilibration but one is very close to it. The slightly higher $\gamma_s$ in the smaller collision systems may be a result of having no multi-strange particles in the $\chi^2$-fit. Conclusions from thermal models =============================== Since the application of a statistical model to multi-particle production by Fermi [@Fermi50] its validity is under debate. Therefore we would first like to make some general remarks. Thermodynamics is first of all a formalism which may be derived from statistical (quantum) mechanics in the infinite volume limit. The basic assumption going in is that all states which are allowed by conservation laws, including energy conservation, are equal probable. We get the microcanonical formalism. Going to the canonical formalism only states sharing the same total energy have the same probability which is proportional to $\propto \exp(-E/T)$. The equal probability is usually violated in elementary reactions where the final state probability is given by the corresponding matrix element. Therefore the general opinion is that in elementary reaction the thermal model has no justification. However, a large number of particles in the final state suggests to use a statistical approach. Therefore the thermal model was applied for elementary reactions and one gets reasonable agreement [@Becattini96a; @Becattini97a]. This observation suggest that if only enough energy is available the particle production is dominated by the statistical component and dynamical aspects are of minor importance. In addition Becattini [@Becattini96a; @Becattini97a] got the important result that there is an universal hadronization temperature in different elementary collision systems and it is independent on $\sqrt{s}$. The interpretation is that in the rest frame of the leading particle/parton the probability of producing a particle with energy $E$ is proportional to $\propto \exp(-E/T_h)$. The hadronization temperature $T_h$ may be identified with the Hagedorn temperature $T_H$. In the interpretation of Hagedorn [@Hagedorn71] it is not possible to create a system of higher temperature than $T_H$ unless there is a phase transition. The limiting temperature is seen in elementary reactions even at very high energies like in $p$+$\bar{p}$ at CERN [@Becattini97a]. The above described observation was also dubbed as [*statistical filling of phase space*]{}. We want to point out that there is a difference in the basic hadron production probability compared to the string models for hadronic reactions [@Becattini97b]. There the production probability goes basically like $\propto \exp(-m^2*k)$ [@Andersson83] with $k$ being a universal constant. The mechanism for chemical equilibration in nucleus-nucleus collisions is expected to be different from the one in elementary reactions. In nuclear reactions we assume that chemical equilibration is established by secondary interactions among the produced hadrons. Therefore we distinguish two mechanisms which bring the system to maximum entropy or chemical equilibrium. - The production of particles, i.e. the hadronization or fragmentation, follows a statistical law and already at their production they are distributed according to maximum entropy. The ensemble average is done by averaging over many events in the experimental analysis. - The maximum of entropy is build up in the classical sense by interactions among the particles until detailed balance is reached. The ensemble average is reached in each collision by the average over the lifetime of the system (engodic theorem). The thermodynamical formalism cannot distinguish between both scenarios and one has to use dynamical arguments to justify one or the other mechanism. Since at SPS-energies the chemical freeze-out temperatures and the chemical potentials are very similar in p+p collisions and in S+$A$ collisions (see table \[summary\]) one cannot use them to justify secondary interactions for chemical equilibrium. A superposition of p+p collisions explains most of the features in the nuclear collisions [@Jeon97] with the exception of strangeness. Therefore we emphasize the importance of the measurement of strangeness because there the difference in p+p to $A$+$A$ is most clearly seen. The freeze-out temperature is expected to decrease with increasing $A$, because freeze-out occurs when the mean free path is of the order of the size of the system. Such an effect is not observed in an unambiguous way, yet. So far only indications are seen as for example the decrease of chemical freeze-out in our analysis from S+S to S+Ag to S+Au. A clear sign for chemical equilibration due to secondary interactions would be a difference in the chemical freeze-out of p+p compared to a real heavy nucleus like Pb+Pb or Au+Au. Such an analysis hasn’t been performed yet but with the now analyzed data of Pb+Pb at CERN-SPS (see for example the various contributions to this proceedings) it will be possible soon. At the AGS we expect different freeze-out temperatures for Si+Au and Au+Au. The first results on Au+Au [@Cleymans97b] go in this direction but we have to wait for the completion of the experimental data analysis. Coming finally back to strangeness we have here a clear signal of the difference between elementary reactions and nuclear collisions. The strangeness suppression factor $\gamma_s$ is the quantitative measure for the strangeness enhancement in the thermal model. In p+p it is $\gamma_s \approx 0.5$ [@Becattini97a] but in nuclear collisions at CERN-SPS it is around 0.7-1 (see table \[summary\]). A strangeness enhancement is seen at the AGS, too [@Abbott90], but the situation is not so clear in terms of the thermal ansatz. A consistent study of the $\gamma_s$ dependence has not been made. First there is no thermal analysis of the p+p interaction at the corresponding $\sqrt{s}$ and second the analysis at AGS assumes mostly full strangeness equilibration as it is seen in table \[summary\]. However, we have already remarked [@Sollfrank95] that a $\gamma_s \approx 0.7$ is better for describing the data at AGS. We have shown that the thermal model fits are not perfect including all measured particle species. The deviations are a source of debate. The various interpretations are that the thermal description is not valid at all, the hadronization of a QGP leaves non-equilibrium tracks in special hadron ratios [@Rafelski96], anti-baryons exhibit a large absorption [@Spieles95] or the deviations are not serious [@BraunMunzinger96]. The final answer will be given in the expected high statistic data of Pb+Pb in the future. We summarize that there are strong signs of chemical equilibration in heavy ion collisions. Since there is already an equally good chemical equilibrium (excluding strangeness) in the basic p+p collisions chemical equilibrium cannot be used for justifying secondary hadronic collisions. (There are better signals for abundant secondary interactions like the collective flow studies [@Bearden97].) However, the strangeness production is very different between both collision systems and it may be used as the chemometer for chemical equilibrium. This work was supported by the Bundesministerium für Bildung und Forschung (BMBF) under grand no. 06 BI 556 (6). We gratefully acknowledge helpful discussions with H Satz, U Heinz, F Becattini, M Gaździcki and J Rafelski. [99]{} Rafelski J 1981 [*Workshop on future relativistic heavy ion experiments*]{} ed R Bock and R Stock (Darmstadt: GSI-report 81-6) p 282 Rafelski J and Müller B 1982 1066 Koch P, Müller B and Rafelski J 1986 [*Phys. Rep.*]{} [**142**]{} 167 Kinson J B 1995 A [**590**]{} 317c Gaździcki M 1995 A [**590**]{} 197c diBari N 1995 A [**590**]{} 307c Letessier J, Rafelski J and Tounsi A 1994 484; Letessier J, Rafelski J and Tounsi A 1997 363 Rafelski J, Letessier J and Tounsi A 1996 [*Acta Phys. Pol.*]{} B [**27**]{} 1035 Letessier J, Rafelski J and Tounsi A 1994 393 Letessier J, Tounsi A, Heinz U, Sollfrank J and Rafelski J 1995 D [**51**]{} 3408 Sollfrank J and Heinz U 1995 [*Quark-gluon plasma*]{} vol 2 ed R C Hwa (Singapore: World Scientific) p 555 Sorge H 1994 A [**566**]{} 633c Topor Pop V, Gyulassy M, Wang X N, Andrighetto A, Morando M, Pellegrini F, Ricci R A and Segato G 1995 C [**52**]{} 1618 Capella A 1996 400 Becattini F 1996 C [**69**]{} 485 Becattini F and Heinz U 1997 hep-ph/9702274 Braun-Munzinger P, Stachel J, Wessels J P and Xu N 1995 43 Braun-Munzinger P, Stachel J, Wessels J P and Xu N 1996 1 Hoang T F and Cork B 1988 C [**38**]{} 603 Hoang T F 1994 C [**62**]{} 343 Becattini F 1996 [*Proc. of the XXXIII Eloisatron workshop on universality features in multihadron production and the leading effect (Erice)*]{} to be published (hep-ph/9701275) Sollfrank J, Gaździcki M, Heinz U and Rafelski J 1994 C [**61**]{} 659 Davidson N J, Miller H G and von Oertzen D W 1991 554 Letessier J, Rafelski J and Tounsi A 1994 499 Asprouli M N and Panagiotou A D 1995 C [**51**]{} 1445 Panagiotou A D, Mavromanolakis G and Tzoulis J 1996 C [**53**]{} 1353 Cleymans J, Elliott D, Satz H and Thews R L 1997 C [**74**]{} 319 Cleymans J and Muronga A 1997 5; Cleymans J, Marais M and Suhonen E 1997 preprint nucl-th/9705014 Davidson N J, Miller H G, Quick R M and Cleymans J 1991 105; Davidson N J, Miller H G, von Oertzen D W and Redlich K 1992 C [**56**]{} 319 Tiwari V K, Singh S K, Uddin S and Singh C P 1996 C [**53**]{} 2388 Becattini F 1997 this proceedings Sollfrank J 1997 in preparation Andersen E 1994 433 Redlich K, Cleymans J, Satz H and Suhonen E 1994 A [**566**]{} 391c Spieles C, Stöcker H and Greiner C 1997 nucl-th/9704008 Cleymans J 1997 [*Proc. of Third Int. Conf. on Physics and Astrophysics of Quark-gluon plasma (Jaipur)*]{} to appear, nucl-th 9704046 Lee K S and Heinz U 1993 D [**47**]{} 2068 Rafelski J and Danos M 1980 279; Hagedorn R and Redlich K 1985 C [**27**]{} 541; Cleymans J, Suhonen E and Weber G M 1992 C [**53**]{} 485 Redlich K and Turko L 1980 C [**5**]{} 201 Pathria R K 1972 [*Statistical Mechanics*]{} (Oxford: Pergamon Press) p 487 Cleymans J and Koch P 1991 C [**52**]{} 137 Rafelski J 1991 333 Ahle L 1995 A [**590**]{} 249c Bøggild H 1996 343 Arbex N, Ornik U, Plümer M, Schlei B R and Weiner R M 1997 465 Hill D L and Wheeler J A 1953 1102 Hagedorn R 1971 CERN-report 71-12; Hagedorn R 1995 [*Hot Hadronic Matter: Theory and Experiment*]{}, ed J Letessier (New York: Plenum) p 13 Tounsi A, Letessier J and Rafelski J 1994 [*Hot Hadronic Matter: Theory and Experiment*]{}, ed J Letessier (New York: Plenum) p 105 Sollfrank J, Koch P and Heinz U 1991 C [**52**]{} 593 Hagedorn R and Rafelski J 1980 136 Cleymans J, Redlich K, Satz H and Suhonen E 1986 C [**33**]{} 151; Heinz U, Subramanian P R, Stöcker H and Greiner W 1986 1237 Rischke D H, Gorenstein M I, Stöcker H and Greiner W, 1991 C [**51**]{} 485 Uddin S and Singh C P 1994 C [**63**]{} 147 Kapusta J I and Olive K A 1983 A [**408**]{} 478 Venugopalan R and Prakash M 1992 A [**546**]{} 718 Slotta C, Sollfrank J and Heinz U 1995 [*Proc. of Strangeness in Hadronic Matter (Tucson)*]{} ed J Rafelski (Woodbury: AIP Press) p 462 Kataja M and Ruuskanen P V 1990 181 Letessier J, Rafelski J and Tounsi A 1994 394 Bächler J 1994 1419 Sollfrank J, Huovinen P, Kataja M, Ruuskanen P V, Prakash M and Venugopalan R 1997 C [**55**]{} 392 Bjorken J D 1983 D [**27**]{} 140 Röhrich D 1994 A [**566**]{} 35c Bächler J 1994 C [**58**]{} 367 Alber T 1994 C [**64**]{} 195 Alber T 1994 56 Gaździcki M 1995 C [**66**]{} 659; Gaździcki M 1997 this proceedings Takahashi Y private communication Murray M 1994 A [**566**]{} 515c Jacak B 1994 [*Hot and Dense Nuclear Matter*]{}, ed W Greiner (New York: Plenum) Albrecht R 1995 14 Abatzis S 1996 251 Evans D 1996 [*Heavy Ion Phys.*]{} [**4**]{} 79 Abatzis S 1993 615 Fermi E 1950 [*Progr. Theor. Phys.*]{} [**5**]{} 570 Andersson B, Gustafson G, Ingelman G and Sjöstrad T 1983 31 Jeon S and Kapusta J 1997 University of Minnesota preprint NUC-MINN-97/1-T, nucl-th/9703033 Abbott T 1990 847 Spieles C 1995 A [**590**]{} 271c Bearden I G 1997 2080 **Additional Figure**

--- abstract: 'Neural machine translation systems tend to fail on less decent inputs despite its great efficacy, which may greatly harm the credibility of these systems. Fathoming how and when neural-based systems fail on such cases is critical for industrial maintenance. Instead of collecting and analyzing bad cases using limited handcrafted error features, here we investigate this issue by generating adversarial samples via a new paradigm based on reinforcement learning. Our paradigm could expose pitfalls for a given performance metric, e.g. BLEU, and could target any given neural machine translation architecture. We conduct experiments of adversarial attacks on two mainstream neural machine translation architectures, RNN-search and Transformer. The results show that our method efficiently produces stable attacks with meaning-preserving adversarial samples. We also present a qualitative and quantitative analysis for the preference pattern of the attack, showing its capability of pitfall exposure.' author: - | Wei Zou$^{1}$Shujian Huang$^{1}$Jun Xie$^{2}$Xinyu Dai$^{1}$Jiajun Chen$^{1}$\ $^{1}$National Key Laboratory for Novel Software Technology, Nanjing University, China\ $^{2}$Tencent Technology Co, China\ `zouw@smail.nju.edu.cn`, `{huangsj,daixy, chenjj}@nju.edu.cn` \ `stiffxie@tencent.com`\ bibliography: - 'reference.bib' title: A Reinforced Generation of Adversarial Samples for Neural Machine translation --- Conclusion ========== We propose a new paradigm to generate adversarial samples for neural machine translation, which is capable of exposing translation pitfalls without handcrafted error features. Experiments show that our method achieves stable degradation with meaning preserving adversarial samples over different victim models. Please notice that our method can generate adversarial samples efficiently from monolingual data. As a result, mass production of adversarial samples for victim model’s analysis and further improvement of robustness become convenient, which we leave as the future work.

--- abstract: 'Numerical simulations have consistently shown that the reconnection rate in certain collisionless regimes can be fast, on the order of $0.1 {v_A}{B_u}$, where $v_A$ and ${B_u}$ are the Alfv[é]{}n speed and the reconnecting magnetic field upstream of the ion diffusion region. This particular value has been reported in myriad numerical simulations under disparate conditions. However, despite decades of research, the reasons underpinning this specific value remain mysterious. Here, we present an overview of this problem and discuss the conditions under which the “0.1 value” is attained. Furthermore, we explain why this problem should be interpreted in terms of the ion diffusion region length.' title: On the Value of the Reconnection Rate --- Introduction with Overview of the Problem ========================================= Magnetic reconnection is a fundamental plasma process that occurs in a wide variety of laboratory, space, and astrophysical plasmas. Its definition is meaningful in plasmas that are almost ideal, i.e., in those cases where magnetic field lines “move” with the plasma in the vast majority of the domain, while the breaking of the magnetic field line connectivity occurs only in very localized diffusion regions. This reconnection process can enable a rapid conversion of magnetic energy into thermal, supra-thermal, and bulk kinetic energy. As such, magnetic reconnection is believed to play a key role in many of the most striking and energetic phenomena such as sawtooth crashes, magnetospheric substorms, coronal mass ejections, stellar and gamma-ray flares [@Tajima1997; @Kulsrud2005; @Yamada2010]. In order to explain the magnetic energy conversion rates associated with these phenomena, it is essential to know the rate at which magnetic reconnection occurs. The reconnection rate quantifies the temporal rate of change of magnetic flux that undergoes the reconnection process. When the system under consideration is translationally invariant in one direction, the reconnection rate can be expressed as $$\label{} \frac{{d\Phi}}{{dt}} = \frac{d}{{dt}}\int_S {{\vec B} \cdot d{\vec S}} = \oint_{\partial S} {\vec E \cdot d\vec l} = \int_{X{\rm{-line}}} {{E_z}dl} \, .$$ Here, $\Phi$ is the magnetic flux through the surface $S$ bounded by the contour $\partial S$ encompassing the $X$-line. An $X$-line is the projection of an hyperbolic point for the magnetic field along the ignorable direction. Therefore, the reconnection rate is a measure of the rate at which magnetic flux is transported across the $X$-line. In a more general three-dimensional case, the evaluation of the reconnection rate is more subtle. A general approach [@Hesse2005] would be to quantify the reconnection rate as $$\label{Rec_Hesse} \frac{{d\Phi}}{{dt}} = \max \left( {\int {{E_\parallel }ds} } \right) \, ,$$ where $s$ represents the parametrization of the magnetic field lines, and the integral has to be performed over all field lines passing through the non-ideal region (where $E_\parallel = \vec E \cdot \vec B/ | {\vec B} | \neq 0$). The measure (\[Rec\_Hesse\]) is an attractive choice for quantifying the reconnection rate, but there are some caveats associated with it. Indeed, there could be some ambiguity related to the field line integration of $E_\parallel$, as in regions where magnetic field lines are stochastic [@Borgogno2005], or it may be not possible to distinguish between reconnection and simple diffusion [@Huang2014]. In addition, this measure can be applied only in the presence of a non-vanishing magnetic field. If this is not the case, the reconnection rate may be calculated by combining the line integrals of $E_\parallel$ along magnetic separators [@Greene1988; @LauFinn1990; @Wilmot2011], which are magnetic field lines connecting two null points (i.e., points at which $|{\vec B}|=0$). As this brief discussion may suggest, a completely general and practical measure of the reconnection rate is still lacking, and indeed it constitutes an important ongoing area of research (see, for example, the discussion given by @Dorelli2008 in the context of the Earth’s magnetosphere.) The problem of determining the reconnection rate of a magnetic reconnection process dates back to the 1950’s. At that time, the astrophysical community was trying to understand if magnetic reconnection could have served as the mechanism underlying solar flares, which are bursts of high-energy radiation from the Sun’s atmosphere that strongly affect the space weather surrounding the Earth. A simple resistive magnetohydrodynamic (MHD) model of magnetic field line merging was proposed by @Sweet, and then, with the contribution of @Parker, the reconnection rate was evaluated. They considered a quasi-stationary reconnection process occurring within a two-dimensional current sheet. Then, assuming an incompressible flow, the normalized reconnection rate (per unit length) can be shown to be $$\label{eq1} \frac{1}{{{v_A}B_{u}}} \frac{{d{\Phi}}}{{dt}} \sim S^{-1/2} (1 + P_m)^{1/4} \, .$$ In this formula, $S := v_A L/\eta$ and $P_m := \nu/\eta$ are the Lundquist number and the magnetic Prandtl number, respectively. As usual, $\eta$ indicates the magnetic diffusivity and $\nu$ the kinematic viscosity. The Lundquist number is evaluated using the current sheet half-length $L$ and the Alfv[é]{}n speed $v_A = B_u {\left( {{\mu _0}\rho } \right)^{-1/2}}$, where $B_u$ is the reversing magnetic field upstream of the current sheet. In reality, Eq. (\[eq1\]) is not exactly the Sweet-Parker formula for the reconnection rate, but represents its generalization to account for plasma viscosity [@Park1984]. The Sweet-Parker model of reconnection is faster than simple diffusion, but for very large $S$ systems, such as those found in most space and astrophysical environments, it is far too slow to explain the observed fast energy release rates. In order to bypass this limitation, @Petschek1964 proposed a different model in which a relatively short reconnection layer acts as a source for two pairs of slow mode shocks, allowing for much faster reconnection rates. This model was subsequently generalized by @Priest1986, who put forward a wider family of “almost-uniform models” that include Petschek’s model as a special case. However, these models have not been supported by numerical simulations [@Biskamp1986], which have shown that Petschek-like configurations cannot be sustained in the context of MHD with constant resistivity. Petschek’s mechanism can occur within the resistive-MHD framework if the plasma resistivity increases sharply in the reconnection layer [@Kulsrud2001; @Kulsrud2011], but the difficulties in firmly establishing the nature and details of such anomalous resistivity have led the scientific community to look for other alternatives. An important advance occurred when @BHYR_2009, and later @Cassak2009, showed that the predictions of the Sweet-Parker model break down for large $S$ values because of the occurrence of the plasmoid instability [@Biskamp1986; @Tajima1997; @Loureiro2007; @Comisso2016; @Comisso2016B]. In the high-Lundquist number regime, the reconnection process in the nonlinear regime becomes strongly time-dependent due to the continuous formation, merging, and ejection of plasmoids. An estimation of the time-averaged reconnection rate in this regime was proposed by @Huang2010, as well as by @Uzdensky2010, and it has been generalized to account for plasma viscosity as [@Comisso2015; @Comisso2016] $$\label{eq2} \frac{1}{{{v_A}{B_u}}}\left\langle {\frac{{d{\Phi}}}{{dt}}} \right\rangle \sim 10^{-2} {(1 + P_m)^{-1/2}} \, ,$$ where $\left\langle \ldots \right\rangle$ denotes time-average. This formula shows that, for high-Lundquist numbers, the (time-averaged) reconnection rate becomes independent of the Lundquist number (but not the magnetic Prandtl number) and much higher than the Sweet-Parker rate for very large $S$-values. Other MHD models of reconnection have also been investigated. In particular, since the pioneering work by @Matthaeus1986, turbulence effects have been shown to produce a distribution of reconnection sites that is capable of increasing the global reconnection rate [@Servidio2009]. The impact of turbulence and the plasmoid instability on the reconnection rate has caused a rethinking of magnetic reconnection in MHD plasmas. However, in many situations the current layers that form reach scales at which two fluid/kinetic effects become important. In all these cases, an MHD description fails to reproduce accurately the physics of the reconnection process, and two fluid and kinetic effects must be considered. For the aforementioned reasons, a complementary path in investigating fast magnetic reconnection has been pursued at least since the 1990’s by means of numerical simulations of Hall MHD, two-fluid and kinetic models. Several research groups have shown that collisionless effects were able to strongly speed up the reconnection process [@Aydemir1992; @OttPor1993; @Wang1993; @Mandt1994; @Biskamp1995; @Kleva1995; @MaBhatta1996; @Shay1999; @Grasso1999; @Birn_2001; @Porcelli2002]. In particular, numerical simulations consistently demonstrated that the reconnection rate in certain collisionless regimes becomes $$\label{eq3} \frac{1}{{{v_A}B_{u}}} \frac{{d{\Phi}}}{{dt}} \sim 0.1 \, ,$$ a value that is compatible with many observations and experiments [@Yamada2010], meaning that collisionless effects may be crucial to explain many magnetic reconnection phenomena. Note that even here (and in the following) we have considered the reconnection rate per unit length in the out-of-plane direction, whereas $v_A$ and $B_u$ are evaluated upstream of the ion diffusion region, which can be seen as the region where ${E_z} + {({{\vec v}_i} \times \vec B)_z} \ne 0$. Although the relation (\[eq3\]) was found to be valid only in the steady-state limit, or in the vicinity of the peak reconnection rate, it was nevertheless surprising to discover that ${({v_A}{B_u})^{ - 1}}d\Phi /dt$ seemed to be unaffected by the microphysics and macrophysics of specific models. This intriguing result led @Shay1999 to speculate that the aforementioned value could be universal. Such a conjecture stimulated a long debate in the plasma physics and astrophysics communities - one that continues to this day. *Is the reconnection rate value of $0.1$ truly universal?* *What are the physical reasons of this particular value?* In order to explain the fast reconnection rates observed in numerical simulations, @Shay1999 brought forward an argument by @Mandt1994, who proposed that fast magnetic reconnection is enabled by the presence of fast dispersive waves. These waves would speed up the reconnection process by giving rise to the development of a Petschek-type outflow configuration. In contrast, the absence of dispersive waves would lead to an extended Sweet-Parker-type layer, forcing collisionless reconnection to be slow in large systems [@Rogers2001]. This argument, however, was found not to be true. Indeed, numerical simulations have shown that fast magnetic reconnection also occurs in electron-positron plasmas, which do not support fast dispersive waves [@Bessho2005; @DaugKar2007; @Chacon2008; @Zenitani2008]. More recently, @Liu_2014, as well as @Stanier2015, have reconsidered this argument and have shown that, in an electron-ion plasma, fast reconnection is also manifested in the strongly magnetized limit (where fast dispersive waves are suppressed) defined by $\beta := 2{\mu_0}{n_0}{k_B}(T_e + T_i)/B^2 \ll {m_e}/{m_i}$ and $B_{u}^2 \ll ({m_e}/{m_i}){B^2}$. While several works have shown that fast dispersive waves are not required for fast magnetic reconnection, they have also confirmed that the maximum/steady-state reconnection rate satisfies Eq. (\[eq1\]) [e.g., @DaugKar2007; @Liu_2014; @Stanier2015]. There are also some works that have argued against the $\sim 0.1$ value of the maximum/steady-state reconnection rate [e.g., @Porcelli2002; @Fitzpatrick2004; @Bhatta2005; @Andres2016]. In light of the subtlety of the problem, we shall elucidate the conditions under which one should expect a maximum/steady-state reconnection rate $\sim 0.1$. We will also present some thoughts on this apparent commonality of the reconnection rate, which still remain a mystery, and constitutes an important unsolved problem in magnetic reconnection theory. We refer to this problem as the “$0.1$ problem”. Thoughts on the interpretation of this problem {#sec:solve_problem} ============================================== Hitherto, we have focused on summarizing some important discoveries and ideas that underlie the reconnection rate and the $0.1$ problem. In this section, we will present some thoughts as to how to interpret this problem. The first important point that cannot be overlooked is that [*not*]{} all of the collisionless reconnection processes give rise to a peak/steady-state reconnection rate $\sim 0.1$. Indeed, this value is attained only if the system under consideration is [*strongly unstable*]{} (e.g., the tearing stability parameter $\Delta '$ is greater than a certain threshold) and/or [*forced*]{} (e.g., the externally imposed flow or magnetic perturbation exceeds a certain threshold). In the following, we will assume that this is the case. Otherwise, the reconnection rate can be arbitrarily low (e.g., Rutherford-like evolution). In principle, there are no fundamental reasons to believe that different physical models - e.g., Hall-MHD, extended-MHD, multi-fluid, gyrofluid, hybrid, gyrokinetic, kinetic - which are characterized by different physics at the $X$-line, should yield the same peak/steady-state reconnection rate. Indeed, one may think that the commonality of the $\sim 0.1$ value is just a coincidence. However, if the same motif is repeated many times, that cannot be coincidence [@Christie1936], especially when one considers all the differences inherent in the many numerical simulations that have reported this reconnection rate. In particular, it appears that: 1. The steady-state reconnection rate is not linked to the microphysics of the electron diffusion region [e.g., @Shay1998; @Stanier2015]. 2. The $\sim 0.1$ value of the reconnection rate is independent of the system size [e.g., @Shay2004; @Comisso2013]. 3. The $\sim 0.1$ value occurs even when the field structures (e.g., current density) are very different [e.g., @Liu_2014; @Stanier2015]. 4. 3D simulations, despite exhibiting great differences in the structure of the reconnection layer, give reconnection rates similar to those of 2D simulations [e.g., @Daughton2014; @Guo2015]. 5. Simulations in turbulent scenarios lead to current sheets characterized by the same reconnection rate as in the standard laminar picture [e.g., @Wendel2013; @Daughton2014]. To correctly interpret the aforementioned results, we argue that is necessary to shift the focus from the reconnection rate itself, and we conjecture that the reconnection rate is actually [*not*]{} the real “universal quantity”, but it is derived from a more fundamental one, the [*aspect ratio of the ion diffusion region*]{} $\Delta /L$. It is not the former that is $\sim 0.1$, but it is the latter which takes on this value. Then, from mass conservation in steady state, $\vec \nabla \cdot (n {\vec v}) = 0$, one obtains $$\label{eq4} \frac{{{v_{in}}}}{{{v_{out}}}} \frac{{{n_{in}}}}{{{n_{out}}}} \sim \frac{\Delta}{L} \sim 0.1 \, .$$ It is only when the flow is incompressible, $\vec \nabla \cdot {\vec v} = 0$, and the outflow velocity is $v_{out} \sim v_A$, that the reconnection rate turns out to be $\sim 0.1$. The discrepancy between the aspect ratio and the reconnection rate can become particularly evident when considering magnetic reconnection in the relativistic regime. Indeed, the inflow velocity may increase due to Lorentz contraction as per the relation ${v_{in}}/{v_{out}} \sim ({\gamma _{out}}{n_{out}}/{\gamma_{in}}{n_{in}})\Delta /L$, where $n$ is the proper particle number density and $\gamma := {(1 - {v^2}/{c^2})^{ - 1/2}}$. This reconnection rate enhancement has been clearly found in electron-positron plasmas, which are characterized by $\Delta \sim L [ S^{-1}(1 + {P_m})^{1/2} + H^{-1}]^{1/2}$, where $H := f^{-1} (2L/\lambda_e)^2$ is the thermal-inertial number [@Comisso2014] and $f=K_3(\zeta)/K_2(\zeta)$ is the relativistic thermal factor ($K_n$ indicates the modified Bessel function of the second kind of order $n$ and $\zeta$ defines the ratio of rest-mass energy to thermal energy). Indeed, in the strictly collisionless regime, @Liu2015 have performed kinetic simulations that have shown an enhancement of ${v_{in}}/{v_{out}}$ consistent with the increase of ${\gamma_{in}}/{\gamma_{out}}$. There is another important point that needs to be considered. This involves an essential difference between collisional and collisionless reconnection. In collisional MHD models, the thickness of the diffusion region depends on its length $L$, precisely $$\label{} \frac{\Delta}{L^{1/2}} \sim {\left( {\frac{\eta}{v_A}} \right)^{1/2}} {(1 + {P_m})^{1/4}} \, .$$ On the other hand, in two-fluid/kinetic collisionless models, the width of the ion diffusion region depends only on the details of the microphysics of the reconnection process: $$\label{} \Delta \sim {\rho _{se}} = \frac{{{c_{se}}}}{{{\omega_{ci}}}}, \, \, {\rho_{s}} = \frac{{{c_{s}}}}{{{\omega _{ci}}}}, \, \, {\lambda_i} = \frac{c}{\omega_{pi}}, \, \, {\lambda_e} = \frac{c}{\omega_{pe}}, \, \, ....$$ This width is associated to very different length scales, such as the cold ion sound Larmor radius ${\rho _{se}}$, the ion sound Larmor radius ${\rho_{s}}$, the ion skin depth $\lambda_i$, the positron/electron skin depth $\lambda_e$, etc. Therefore, if $\Delta/L \sim 0.1$ holds true for all the different two-fluid/kinetic models, it means that $L$ self-adjust itself in such a way to match the observed aspect ratio. Given the above refocusing observations, the key question of the $0.1$ problem shifts to: *why does the ion diffusion region self-adjust in such a way that the length obeys $L \sim 10 \Delta$?* Here, without intending to furnish a solution to this problem, we provide qualitative arguments to illustrate why $L$ cannot be significantly different from $10 \Delta$. This can be shown heuristically in the following manner. Let us start by assuming $L = \Delta$ and examine its validity. This limit has been studied extensively in the past [e.g., @Priest1985; @Priest2000] and is relevant to the solution of the $0.1$ problem (e.g., P.A. Cassak and M.A. Shay, private communication, 2016). In this instance, it is possible to demonstrate that the reconnection rate vanishes in a plasma. To this purpose, one can exploit the symmetries of the problem, which dictate that $$\label{} {v_x}( \pm x, \mp y) = \pm {v_x}(x,y) \, , \quad {v_y}( \pm x, \mp y) = \mp {v_y}(x,y) \, ,$$ $$\label{} {B_x}( \pm x, \mp y) = \mp {B_x}(x,y) \, , \quad {B_y}( \pm x, \mp y) = \pm {B_y}(x,y) \, ,$$ if the reconnection occurs at a symmetric $X$-point configuration, with the $X$-point situated at the origin. Combining the above properties with the fact that the reconnection rate for $L = \Delta$ must be invariant under a point reflection of the velocity field $$\label{} ({v_x}, {v_y}, {B_x}, {B_y}) \mapsto (-{v_x}, -{v_y}, {B_x}, {B_y}) \, ,$$ it follows that the only possible solution is $v_{in} = v_{out} = 0$. This implies that the reconnection process chokes itself off if $L = \Delta$. It is straightforward to check that a steady reconnection process is also not possible for $L < \Delta$. Indeed, in this case, the current density at the $X$-point would act to decrease the inflow velocity. Next, we consider the case $L > \xi \Delta$, where $\xi \gg 1$ represents a coefficient that will be discussed soon. If a current sheet remains stable over time for arbitrarily large $\xi$, the possibility of obtaining a fast reconnection rate $\sim 0.1$ would be precluded. However, extended current sheets are subject to a tearing-like (plasmoid) instability [e.g., @Biskamp1986]. This implies that for $\xi > \xi_c$, with $\xi_c$ indicating a critical threshold value, the global current sheet breaks up and is replaced by a chain of plasmoids/flux ropes of different sizes separated by smaller current sheets [@Shibata2001]. In a reconnection layer dominated by the presence of plasmoids, the complexity of the dynamics gives rise to a strongly time-dependent process [e.g., @Daughton2009]. Nevertheless, if this process can reach a statistical steady state, we may expect that the current sheet located at the main $X$-point, which is the one that determines the global reconnection rate, should not be longer than the marginally stable sheet [@Huang2010; @Uzdensky2010; @Comisso2015; @Comisso2016]. Indeed, the fractal cascade arising from the plasmoid instability terminates when the length of the innermost local current layer is shorter than the critical length $L_c = \xi_c \Delta$. The local current sheet situated at the primary $X$-point could be subjected to continual stretching by plasmoids moving in the outflow direction, but the plasmoid instability occurs if its length exceeds $L_c$. Thus, it is reasonable to assume that the length of the local current sheet at the main $X$-point should not exceed $L_c$. As a consequence, the length of the main diffusion region remains bounded from above, but a clear-cut value of $\xi_c$ remains unknown. Although at present there are no analytical estimates of the aspect ratio of the main diffusion region, numerical simulations have found $\xi_c \sim 50$ in the collisionless regime [@Daughton2006; @Ji2011]. According to the above arguments, it is clear that the length of the ion diffusion region that determines the reconnection rate is bounded from above and below as $\Delta < L \lesssim 50 \Delta$ in a quasi-steady (or statistical steady state) strongly driven/unstable collisionless reconnection process. Final remarks ============= We have seen that the maximum/steady-state reconnection rate is regulated by the length of the ion diffusion region. However, so far we have not stressed the importance of the boundary conditions on the diffusion region length. Boundary condition may indeed have a strong impact on the length of the current sheets if the computational domain is not sufficiently large. Therefore, the choice of the boundary conditions require extra caution. For example, periodic boundary conditions may force the length of a current sheet to remain small, limiting the duration in which the results are physically meaningful [@Daughton2006]. The knowledge of the maximum/steady-state reconnection rate is crucial when trying to understand whether magnetic reconnection can be fast enough to account for the energy release time-scales observed in a specific system. This is because most of the magnetic flux reconnection takes place during this stage of the process. It is therefore not surprising that much of the magnetic reconnection research done to date has focused on this issue. However, we wish to end our discussion by noting that there are other important questions that lie beyond the paradigm of the maximum/steady-state reconnection rate. While this observable could be insensitive to many features of the specific model, the reconnection rate evolution is not. Indeed, it can be extremely different in diverse systems, since the initial evolution of any reconnection process depends on the details of the microphysics as well as the large-scale ideal-MHD conditions. This initial (typically linear) stage could be completely negligible in terms of magnetic flux reconnection, but it is crucial for determining whether a particular system has enough time to accumulate the magnetic energy that is mostly liberated during the faster stage of the reconnection process. This issue, which is commonly referred to as the onset problem, is also an important and active area of research. Acknowledgments {#acknowledgments .unnumbered} =============== It is a pleasure to acknowledge the fruitful and lively discussions held during the symposium “Solved and Unsolved Problems in Plasma Physics”, held in celebration of the career of Professor Nathaniel Fisch, who has made an art form of asking simple but deep questions that have led to remarkable discoveries. We are indebted to Russell Kulsrud for many enlightening discussions and his suggestion to exploit the symmetries of the problem to construct an elegant proof that steady-state reconnection cannot occur for $L = \Delta$. We are particularly grateful to Manasvi Lingam and Felipe Asenjo, who read the manuscript and provided important suggestions. Finally, we would like to acknowledge stimulating discussions with Paul Cassak, William Daughton, Will Fox, Daniela Grasso, Yi-Min Huang, Hantao Ji, Yi-Hsin Liu, Jonathan Ng, Adam Stanier and Masaaki Yamada. This research is supported by the NSF Grant Nos. AGS-1338944, AGS-1552142, and the DOE Grant No. DE-AC02-09CH-11466. [14]{} natexlab\#1[\#1]{} 2016 [*Phys. Plasmas*]{} **23**, 022903. 1992 [*Phys. Fluids B*]{} **4**, 3469. 2005 [*Phys. Rev. Lett.*]{} **95**, 245001. 2005 [*Phys. Plasmas*]{} **12**, 042305. 2009 [*Phys. Plasmas*]{} **16**, 112102. 2001 [*J. Geophys. Res.*]{} **106**, 3715. 1986 [*Phys. Fluids*]{} **29**, 1520. 1995 [*Phys. Rev. Lett.*]{} **75**, 3850. 2005 [*Phys. Plasmas*]{} **12**, 032309. 2009 [*Phys. Plasmas*]{} **16**, 120702. 2008 [*Phys. Rev. Lett.*]{} **101**, 025003. 1936 [*The A.B.C. Murders*]{}. Collins Crime Club. 2013 [*Phys. Plasmas*]{} **20**, 092118. 2014 [*Phys. Rev. Lett.*]{} **113**, 045001. 2015 [*Phys. Plasmas*]{} **22**, 042109. 2016 [*Phys. Plasmas*]{} **23**, 032111. 2016 [*General Theory of the Plasmoid Instability*]{} arXiv:1608.04692 2006 [*Phys. Plasmas*]{} **13**, 072101. 2007 [*Phys. Plasmas*]{} **14**, 072303. 2009 [*Phys. Rev. Lett.*]{} **103**, 065004. 2014 [*Phys. Plasmas*]{} **21**, 052307. 2008 [*Phys. Plasmas*]{} **15**, 056504. 2004 [*Plasmas*]{} **11**, 937. 1999 [*Plasma Phys. Controlled Fusion*]{} **41**, 1497. 1998 [*J. Geophys. Res.*]{} **93**, 8583. 2015 [*Astrophys. J.*]{} **806**, 167. 2005 [*Astrophys. J.*]{} **631**, 1227. 2010 [*Phys. Plasmas*]{} **17**, 062104. 2014 [*Astrophys. J.*]{} **793**, 106. 1995 [*Phys. Plasmas*]{} **2**, 23. 1996 [*Geophys. Res. Lett.*]{} **23**, 1673. 2011 [*Phys. Plasmas*]{} **18**, 111207. 2001 [*Earth Planets Space*]{} **53**, 417. 2005 [*Plasma Physics for Astrophysics*]{}. Princeton University Press. 2011 [*Phys. Plasmas*]{} **18**, 111201. 1990 [*Astrophys. J.*]{} **350**, 672. 2014 [*Phys. Plasmas*]{} **21**, 022113. 2015 [*Phys. Rev. Lett.*]{} **114**, 095002. 2007 [*Phys. Plasmas*]{} **14**, 100703. 1994 [*Geophys. Res. Lett.*]{} **21**, 73. 1986 [*Phys. Fluids*]{} **29**, 2513. 1993 [*Phys. Rev. Lett.*]{} **21**, 082114. 1984 [*Phys. Fluids*]{} **27**, 137. 1957 [*J. Geophys. Res.*]{} **62**, 509. 1964 in [*AAS/NASA Symposium on the Physics of Solar Flares*]{}, edited by W.N. Hess (NASA, Washington, DC), p.425. 2002 [*Plasma Phys. Control. Fusion*]{} **44**, 389. 1985 in [*Unstable Current Systems and Plasma Instabilities in Astrophysics, IAU 107*]{} edited by M.R. Kundu and G.D. Holman (Reidel, Dordrecht), p.233 1986 [*J. Geophys. Res.*]{} **9**, 5579. 2000 [*Magnetic Reconnection: MHD Theory and Applications*]{}. Cambridge University Press. 2001 [*Phys. Rev. Lett.*]{} **87**, 195004. 2009 [*Phys. Rev. Lett.*]{} **102**, 115003. 1998 [*Geophys. Res. Lett.*]{} **25**, 3759. 1999 [*Geophys. Res. Lett.*]{} **26**, 2163. 2004 [*Phys. Plasmas*]{} **11**, 2199. 2001 [*Earth, Planets Space*]{} **53**, 473. 2015 [*Phys. Plasmas*]{} **22**, 101203. 1958 in [*Electromagnetic Phenomena in Cosmical Physics*]{}, edited by B. Lehnert (Cambridge University Press, New York), p. 123. 1997 [*Plasma Astrophysics*]{}. Frontiers in Physics (Addison-Wesley). 2010 [*Phys. Rev. Lett.*]{} **105**, 235002. 1993 [*Phys. Rev. Lett.*]{} **70**, 1627. 2013 [*Phys. Plasmas*]{} **20**, 122105. 2011 [*Astrophys. J.*]{} **740**, 89. 2010 [*Rev. Mod. Phys.*]{} **82**, 603. 2008 [*Astrophys. J.*]{} **684**, 1477.

--- abstract: 'Touch sensing can help robots understand their surrounding environment, and in particular the objects they interact with. To this end, roboticists have, in the last few decades, developed several tactile sensing solutions, extensively reported in the literature. Research into interpreting the conveyed tactile information has also started to attract increasing attention in recent years. However, a comprehensive study on this topic is yet to be reported. In an effort to collect and summarize the major scientific achievements in the area, this survey extensively reviews current trends in robot tactile perception of object properties. Available tactile sensing technologies are briefly presented before an extensive review on tactile recognition of object properties. The object properties that are targeted by this review are shape, surface material and object pose. The role of touch sensing in combination with other sensing sources is also discussed. In this review, open issues are identified and future directions for applying tactile sensing in different tasks are suggested.' author: - 'Shan Luo^\*^, Joao Bimbo, Ravinder Dahiya and Hongbin Liu[^1] [^2] [^3] [^4]' bibliography: - 'main.bib' title: 'Robotic Tactile Perception of Object Properties: A Review' --- [Shell : Bare Demo of IEEEtran.cls for IEEE Journals]{} Tactile sensing, robot tactile systems, object recognition, sensor fusion, survey. Introduction {#sec:intro} ============ sense of touch is an irreplaceable source of information for humans while exploring the environment in their close vicinity. It conveys diverse sensory information, such as pressure, vibration, pain and temperature, to the central nervous system, assisting humans in perceiving their surroundings and avoiding potential injuries [@dahiya2012robotic]. Research has shown that, compared to vision and audition, the human sense of touch is superior at processing material characteristics and detailed shapes of objects [@lederman2009haptic; @dahiya2012robotic]. As for humans, it is essential that robots are also equipped with advanced touch sensing in order to be aware of their surroundings, keep away from potentially destructive effects and provide information for subsequent tasks such as in-hand manipulation. A general block diagram of a tactile sensing system [@dahiya2013directions; @dahiya2010tactile] is illustrated in Fig. \[fig:tactilesensingsystem\]. A tactile sensing system here refers to a system where a robot uses tactile sensors to sense the ambient stimuli through touch, acquiring information on the properties of objects, such as shape and material, providing action related information, such as object localization and slippage detection. On the left side of Fig. \[fig:tactilesensingsystem\], the tactile sensing process is divided into functional blocks that depict sensing, perception and action at different levels. We follow existing literature [@lederman2009haptic], using the term “perception” to refer to the process of observing object properties through sensing. The right side of Fig. \[fig:tactilesensingsystem\] shows the structural blocks of hardware that correspond to those functional blocks. The sensing process transduces the external stimuli (e.g., pressure, vibration and thermal stimulus), into changes on the sensing elements of the tactile sensors [@dahiya2013directions; @dahiya2010tactile]. This data is acquired, conditioned and processed using an embedded data processing unit, and then transferred to the higher perception level to construct a world model, perceive the properties of interacted objects, (e.g., shape and material properties). While perceiving, the sense of touch may possibly be fused with other sensing modalities such as vision and auditory perception. Control commands are ultimately to be exerted in order to obtain the desired actions by the controller. ![Hierarchical functional (left) and structural (right) block diagrams of robotic tactile sensing system [@dahiya2013directions; @dahiya2010tactile]. The perception blocks have been highlighted that the paper is contributing.[]{data-label="fig:tactilesensingsystem"}](tactilesensingsystem){width=".6\textwidth"} The last few decades have witnessed a tremendous progress in the development of tactile sensors, including diverse materials and methods explored to develop them, as extensively reported in [@dahiya2013directions; @dahiya2010tactile; @schmitz2011methods; @kaltenbrunner2013ultra; @yogeswaran2015new; @khan2015technologies]. Research into the interpretation of tactile data to extract the information it conveys, particularly in the task of object recognition and localization, is nowadays also beginning to attract increasing attention. A comprehensive discussion on tactile sensory data interpretation is needed since many commonly used techniques, for example those adapted from the field of computer vision, may not always be suitable for tactile data. This is because of the fundamentally different operating mechanisms of these two important sensory modalities. To assist the advancement of the research in information extraction from tactile sensory data, this survey reviews the state-of-the-art in tactile perception of object properties, such as material identification, object recognition and pose estimation. In addition, work on perception that combines touch sensing with other sensing modalities, e.g., movement sensors and vision, are also studied. The remainder of this paper is organized as follows: Available tactile sensors are first briefly introduced and compared in Section \[sec:tactileprecption\]. Research on material recognition via touch is then presented in Section \[materialrecognition\]. Works on tactile shape recognition and object pose estimation are reviewed on both local and global scales in Section \[shaperecognition\] and \[localization\] respectively. How vision and touch have been combined for object perception is discussed in Section \[sensingintegration\]. The last section concludes the paper and points to future directions in interpreting tactile data. Tactile perception as representation and interpretation of haptic sensing signals {#sec:tactileprecption} ================================================================================= Tactile sensing modalities {#sec:tactsens} -------------------------- As introduced in Section \[sec:intro\], tactile perception is the process of interpreting and representing touch sensing information to observe object properties. In the hierarchy presented in Fig. \[fig:tactilesensingsystem\], it is placed a level above sensing, and provides useful, task-oriented information for planning and control [@siciliano2016springer]. How tactile sensing information is interpreted and represented is closely linked with the type of hardware used and with the task to be fulfilled by the robot. Tactile sensing has been extensively reviewed in [@dahiya2013directions; @dahiya2010tactile; @argall2010survey; @kappassov2015tactile] and the tactile sensors can be categorized in multiple manners, such as according to their sensing principles [@dahiya2011towards], fabrication methods [@dahiya2013directions] and transduction principles [@dahiya2010tactile]. In this paper, we follow existing literature [@koiva2013highly; @buscher2015flexible] and choose to categorize tactile sensors according to the body parts they are analogous to. As shown in Fig. \[fig:biovsrobot\], tactile sensors in literature can be categorized into three types with respect to their spatial resolution, and an analogy can be made to corresponding parts of the human body. We provide a short description of each category in order to frame this review as follows. ![Tactile sensors of different types with the corresponding biological body parts in anatomy.[]{data-label="fig:biovsrobot"}](biovsrobot){width=".4\textwidth"} #### Single-point contact sensors, analogous to single tactile cells {#single-point-contact-sensors-analogous-to-single-tactile-cells .unnumbered} This kind of sensor is used to confirm the object-sensor contact and detect force or vibrations at the contact point. Depending on the sensing modalities, single-point contact sensors can be categorized into: 1) force sensors for measuring contact forces, where a typical example is the ATI Nano 17 force-torque sensor; 2) biomimetic whiskers, also known as dynamic tactile sensors, for measuring vibrations during contact [@kroemer2011learning; @fox2012tactile; @kuchenbecker2006improving; @mitchinson2014biomimetic; @huet2017tactile]. #### High spatial resolution tactile arrays, analogous to human fingertips {#high-spatial-resolution-tactile-arrays-analogous-to-human-fingertips .unnumbered} Most research in tactile sensing is carried out using this type of tactile sensors [@kappassov2015tactile; @dahiya2013directions], and example prototypes are tactile arrays of 3$\times$4 tactile sensing elements based on fiber optics [@xie2013fiber], tactile array sensors based on MEMS barometers [@tenzer2014feel] and fingertip sensors based on embedded cameras [@chorley2009development; @sato2010finger; @johnson2011microgeometry; @yamaguchi2016combining]. There are also a multitude of commercial sensors available, such as RoboTouch and DigiTacts from Pressure Profile Systems (PPS)[^5], tactile sensors from Weiss Robotics[^6], Tekscan tactile system[^7], BioTac multimodal tactile sensors from SynTouch[^8]. Among them, the most common are planar array sensors. #### Large-area tactile sensors, analogous to skin of human arms, back and other body parts {#large-area-tactile-sensors-analogous-to-skin-of-human-arms-back-and-other-body-parts .unnumbered} Unlike in fingertip tactile sensors, high spatial resolution is not essential in this type of tactile sensing. More importantly, they should be flexible enough to be attached to curved body parts of robots. The attention to developing this type of sensors has emerged in recent decades [@kaltenbrunner2013ultra; @polat2015synthesis; @hoffmann2017robotic]. Some researchers have developed this kind of tactile sensors for hands [@muscari2013real], arms and legs [@mittendorfer2011humanoid; @bartolozzi2016robots], front and back parts [@kaboli2015humanoids] of humanoid robots. For a comprehensive review of large area and flexible tactile skins, the reader is referred to [@dahiya2013directions; @khan2015technologies; @dang2017printable; @heidari2017bending]. Tactile Perception ------------------ Representations of tactile data are commonly either inspired by machine vision feature descriptors, where each tactile element is treated as an image pixel [@schneider2009object; @pezzementi2011tactile], biologically inspired[@Cannata2010], or resort to dimensionality reduction[@heidemann2004dynamic; @liu2012computationally; @goger2009tactile]. Tactile information can be interpreted according to the desired function of the robot. Relevant information that can be extracted from sensing data include shape, material properties, and object pose. In this review, we focus on existing methods to extract these parameters, which are fundamental in the field of robot grasping and manipulation [@yousef2011tactile], and have been used for the purpose of grasp control [@tegin2009demonstration], slippage detection and prevention [@song2014efficient], grasp stability assessment [@bekiroglu2011assessing], among others. Other applications where tactile perception has been successfully applied range from haptic cues for Minimally Invasive Surgery (MIS) [@li2014multi], creating interfaces for interactive games [@benali2004tactile] or medical training simulators [@coles2011role], and assisting underwater robotic operations [@aggarwal2014object]. Compared to the rapid development of tactile sensors, the interpretation of tactile sensors readings has not yet been fully taken into consideration. This is reflected by the small number of survey articles reported in the literature that focus on reviewing the computational intelligence methods applied in tactile sensing. The development of tactile sensors of increasing spatial resolution and fast temporal response provides an opportunity to apply state-of-the-art techniques of machine intelligence from multiple fields, such as machine learning, signal processing, computer vision and sensor fusion, in the field of tactile sensing. Accordingly, this paper explores recent advances in object shape and material recognition and pose estimation via tactile sensing and gives guidance towards possible future directions. In addition, it is also investigated how vision and touch sensing modalities can be combined for object recognition and pose estimation. Material recognition by tactile sensing {#materialrecognition} ======================================= The material properties of an object’s surface are one of the most important cues that a robot requires for the sake of effectively interacting with its surroundings. Vision has been a popular approach to recognize object material [@liu2010exploring; @sharan2013recognizing; @sun2016recognising]. However, vision alone can only recognize a previously known surface material, and cannot, on its own, estimate its physical parameters. In this respect, it is essential to utilize the sense of touch to identify the material properties. Surface texture (friction coefficients and roughness) and compliance are amongst the most crucial parameters for manipulating objects. Humans are extremely skilled in recognizing object material properties based on these cues [@lederman1990haptic] and a comprehensive review on human perception of material properties can be found in [@tiest2010tactual]. In robotics, researchers have endeavoured to enable a robot to identify the material properties at a level comparable to humans. These properties can be categorized into two different methods, i.e., surface texture based and object stiffness based. \[tab:materialrecognition\] ----------------------------------------------------------- ------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------- ------------------------------------------------------------------------------ ---------------- Classifier input Sensors (resolution) Classifier $\#$ of classes Ref. Year Acoustic signals Microphone Correlation 6 Roy et al. [@roy1996surface] 1996 PVDF output PVDF sensors; strain gauges Wavelet analysis and power spectrum analysis 30 Tanaka et al. [@tanaka2003haptic] 2003 Accelerometer sensor data Accelerometer Lookup table 5 Windau et al. [@windau2010inertia] 2010 Sensor array values MEMS based tactile array sensor (4$ \times $4) Maximum-likelihood estimation 5 Kim et al. [@kim2005texture] 2005 FFT, Numerical features, PCA data Dynamic tactile sensor *k*NN 22 Edwards et al. [@edwards2008extracting] 2008 FFT of PVDF output Finger with embedded strain gauges and PVDFs Naive Bayes classifier [@jamali2011majority; @jamali2009texture; @jamali2010material], decision trees 8 [@jamali2011majority] 7 [@jamali2010material] 10 [@jamali2009texture] Jamali et al. [@jamali2011majority; @jamali2009texture; @jamali2010material] 2009 2010 2011 Acoustic data; variable resistor output Microphone and variable resistor SOMs 8 Johnsson et al. [@johnsson2011sense] 2011 Force amplitude; tactile sensor output amplitude MEA-based tactile sensor (32 elements) [@dahiya2009development] RLS, SVMs, RELM 4 Decherchi et al. [@decherchi2011tactile] 2011 Accelerometer data Accelerometer sensor *k*NN, SVM 20 Sinapov et al. [@sinapov2011vibrotactile] 2011 Acceleration signals Tactile probe [@giguere2011simple] Neural Network 10 Giguere et al. [@giguere2011simple] 2011 Fourier transform Dynamic tactile sensor $\mu$MCA, WMCA 26 Kroemer et al. [@kroemer2011learning] 2011 Moments of tactile images Flexible array tactile sensor [@drimus2011classification] DTW, *k*NN 10 Drimus et al. [@drimus2014design; @drimus2011classification] 2011 2014 Acceleration signals Tactile probe [@dallaire2011artificial; @dallaire2014autonomous] Non-parametric approach, Dirichlet processes, SVM 28 Dallaire et al. [@dallaire2011artificial; @dallaire2014autonomous] 2011 2014 Cross-wavelet transform MEMS based tactile array sensor (2x2) *k*NN 5 Oddo et al. [@oddo2011roughness] 2011 Interaction forces Nano 17 Finite-element analysis 4 Sangpradit et al. [@sangpradit2011finite] 2011 Normal force and velocity BioTac (19 cells) Bayesian exploration 117 Fishel et al. [@fishel2012bayesian] 2012 Friction force Nano 17 *k*NN, Neural network, Naive Bayes classifier 12 Liu et al. [@liu2012surface] 2012 Sensor outputs Fabric sensor [@araki2012experimental] Naive Bayes classifier, Neural Network, DTW 3 Ho et al. [@araki2012experimental] 2012 Tactile images GelSight Hellinger similarity metric 24 Li et al. [@li2013sensing] 2013 Multimodal data Multimodal BioTac sensor Bayesian inference 10 Xu et al. [@xu2013tactile] 2013 PVDF output; pressure signals PVDF films, BioTac sensor (19 cells) FFT 8 Heyneman et al. [@heyneman2013slip] 2013 Accelerometer signals Accelerometer Neural network 7 Chathuranga et al. [@chathuranga2013investigation] 2013 Vibration, normal force, scanning speed Accelerometer, force sensor Majority voting, SVM 15 Romano et al. [@romano2014methods] 2014 Data of force sensors, proximity sensor and accelerometer Accelerometer, force sensor, proximity and temperature sensors SVM, EM 10 Kaboli et al. [@kaboli2014humanoids] 2014 Tactile images Tactile array sensor (38$ \times $41) Parzen windows estimation (PWE) classifier 4 Shill et al. [@shill2015tactile] 2015 Electrode data Multimodal BioTac sensor SVMs and Random Forests 49 Hoelscher et al. [@hoelscher2015evaluation] 2015 ----------------------------------------------------------- ------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------- ------------------------------------------------------------------------------ ---------------- Surface texture based tactile material recognition -------------------------------------------------- Surface texture information can be extracted through the investigation of friction coefficients, roughness and micro-structure patterns of objects. The former two can be obtained using a force or tactile sensor sliding on the object surface, whereas the latter can be attained through tactile images. The friction that arises at the contact while the sensor is sliding on an object’s surface can be used to recognize the surface materials. Among the methods of this kind, the use of acoustic signals resulted from the friction to recognize the object materials is low-cost and requires limited computational power [@shan2017knock]. In [@roy1996surface], a microphone is mounted on a robot leg which taps on the ground as the mobile robot moves, similar to the manner a blind person might tap his cane. The acoustic signature from tapping is then used to classify different floor materials. In [@edwards2008extracting], an artificial finger equipped with a microphone (i.e., dynamic tactile sensor) is used to collect frictional sound data that are mapped to frequency domain through Fast Fourier Transform (FFT) to detect different textures. To recognize different texture surfaces, in [@kroemer2011learning] mean Maximum Covariance Analysis ($\mu$MCA) and weakly paired MCA (WMCA) are used to analyze the acoustic data after Fourier transform, which were collected by a dynamic tactile sensor. In [@johnsson2011sense], a microphone-based texture sensor is employed and the textures can be classified using Self-Organizing Maps (SOMs). The use of acoustic signals has merits of low-cost and limited computational expense, however, ambient and motor noise may deteriorate the recognition performance. Strain gauges and force sensors are also used to detect vibrations during object-sensor interaction, in order to discriminate materials. By transferring raw data into the frequency domain by FFT, in [@jamali2010material], different materials are classified based on surface textures by analyzing induced vibration intensities. In these works, an artificial finger is used with strain gauges and polyvinylidene fluoride films (PVDFs) embedded in silicone. In [@liu2012surface], a dynamic friction model is applied to determine the surface physical properties while a robotic finger slides along the object surface with varying sliding velocity. The yarn tension data from a fabric sensor are also collected while the fingertip slides over objects in [@ho2012experimental], with multiple computational intelligence methods for recognition being compared. In [@dallaire2014autonomous], a tactile probe is proposed to measure the vibration signals while sliding to classify disks with different textures. This tactile probe is also used in [@giguere2011simple] to identify textures of different terrains. In [@sinapov2011vibrotactile], artificial fingernails with three-axis accelerometers are used to scratch on surfaces and a frequency-domain analysis of the vibrations is done by using machine learning algorithms, i.e., *k*NN and SVM. Similarly, vibrations are also used in [@romano2014methods], in line with contact forces, to classify different surfaces. In addition to vibrations and force data, the output of proximity sensor is used to distinguish different surface textures in [@kaboli2014humanoids]. Roughness is another important cue to discriminate between different object materials. In [@tanaka2003haptic], object roughness is computed based on the variances of strain gauge signals using wavelet analysis. A microelectromechanical systems (MEMS) based tactile sensor is used to discriminate the roughness of object surfaces in [@oddo2011roughness]. Using a BioTac sensor in [@fishel2012bayesian], Bayesian exploration is proposed to discriminate textures, which selects the optimal movements adaptively based on previous experience; good recognition performance is achieved for a large dataset of 117 textures. The micro-structure patterns of objects can also be utilized to recognize object materials, usually with tactile array sensors. By using the camera-based GelSight sensor [@johnson2011microgeometry], height maps of the pressed surfaces are treated as images to classify different surface textures using visual texture analysis [@li2013sensing]. Similarly, another camera-based tactile sensor TacTip is also used to analyze the object textures in [@winstone2013tactip]. In [@kim2005texture] a MEMS based tactile array sensor is employed to distinguish simple textures by using Maximum Likelihood (ML) estimation. The probability density function (PDF) of each texture type is created based on the mean and variance of the obtained tactile arrays and the textures are estimated by maximizing the PDFs. In [@shill2015tactile], tactile images are also utilized to classify terrains, i.e., wood, carpet, clay and grass. Object stiffness based tactile material recognition --------------------------------------------------- Object stiffness is also one of the critical material properties [@nanayakkara2016stable]. By using a BioTac sensor, the object compliance (the reciprocal of stiffness) can be estimated either using the contact angle of the fingertip [@xu2013tactile] or investigating BioTac electrode data [@su2012use; @hoelscher2015evaluation; @chu2013using]. In [@windau2010inertia] a robot leg equipped with an accelerometer is employed to actively knock on object surfaces and by analysing the sensor data, the hardness, elasticity and stiffness of the object can be revealed. In recent works [@yuan2016estimating; @yuan2017shape], the hardness of objects can also be estimated by processing the tactile image sequences from a GelSight sensor. In [@decherchi2011tactile], multiple computational algorithms are applied to classify various materials based on mechanical impedances using tactile data and it is found that SVM performs best. In [@drimus2014design], by using image moments of tactile readings as features, dynamic time warping is used to compare the similarity between time series of signals to classify objects into rigid and deformable. A force sensor is used to test the mechanical impedances of materials in terms of shear modulus, locking stretch and density in [@sangpradit2011finite]. A multi-indenting sensing device is proposed in [@faragasso2015multi] to measure the stiffness of the examined object, i.e., phantom soft tissue used in this work. In summary, object materials can be recognized by using different touch sensing cues based on surface textures, vibrations and mechanical impedances, summarized in Table \[tab:materialrecognitionsummary\] with a comparison of their pros and cons. \[tab:materialrecognitionsummary\] [m[1.5cm]{}| m[1.5cm]{}|m[2cm]{}| m[3cm]{}| m[3cm]{}| m[3cm]{}| m[0.5cm]{}]{} Methods & Motions & Data types & Advantages & Disadvantages & Applicable sensors & Ref.\ & Sliding, tapping, scratching & Frictions, acoustic data, acceleration, vibrations & Low cost and limited computational expenses & Interaction actions (sliding, tapping, scratching) may damage objects & Dynamic tactile sensors, force sensors, strain gauges and PVDF sensors & [@johnsson2011sense; @jamali2011majority; @sinapov2011vibrotactile]\ & Imprint & Tactile images & Micro-structure patterns of object textures can be captured in one tactile image & Low resolution of tactile sensors may cause difficulties for processing & Planar array sensors; potentially applied to curved tactile sensors & [@li2013sensing; @kim2005texture]\ Mechanical impedances based & Squeezing, knocking, pressing & Force variances, tactile images, acceleration & Limited computational costs; complementary to other information & Interaction actions (knocking, squeezing) may damage objects & Accelerometer-based sensors, force sensors and tactile array sensors & [@windau2010inertia; @drimus2011classification; @kaboli2014humanoids]\ Tactile object shape perception {#shaperecognition} =============================== Object shape perception is the ability to identify or reconstruct the shape of objects. The goals of shape perception vary in different tasks, from capturing the exact shape, like getting the point cloud of the object, to classification of shape elements or overall profiles. This capability is crucial for robots to perform multiple tasks such as grasping and in-hand manipulation. The more complete information is obtained about the object’s shape, the more capable the robot will be to plan and execute grasping trajectories and manipulation strategies. Research into shape recognition has been dominated by vision based methods [@lowe1999object; @felzenszwalb2005pictorial]. However, visual shape features cannot be observed when vision is occluded by hand or in poor illumination conditions. In contrast, tactile object shape perception is not affected by such factors and can observe detailed shapes by sensor-object interactions. The surge of high-performance tactile sensors gives rise to the emergence and rapid spread of algorithms to recognize object shapes via touch. The perception of object shapes can be done on two scales, i.e., local and global, as illustrated in Fig. \[fig:ballshape\]. The former can be revealed by a single touch through tactile image analysis. It is analogous to the human cutaneous sense of touch, which is localized in the skin. The latter reflects the contribution of both cutaneous and kinaesthetic feedback, e.g., contours that extend beyond the fingertip scale. In this case, intrinsic sensors, i.e., proprioceptors in joints, are also utilized to acquire the position and movement of the fingers/end-effectors that are integrated with local features to recognize the objects. Here the kinaesthetic cues are similar to human proprioception that refers to the awareness of the positions of the body parts. [0.24]{} ![Two ball shapes with different local shapes: one is a golf ball and the other is a tennis ball. At a global scale, both of them are balls. At a local scale, (a) has small pits whereas (b) has curvilinear shapes. []{data-label="fig:ballshape"}](golfball "fig:"){height="1.61cm" width="1.61cm"} [0.24]{} ![Two ball shapes with different local shapes: one is a golf ball and the other is a tennis ball. At a global scale, both of them are balls. At a local scale, (a) has small pits whereas (b) has curvilinear shapes. []{data-label="fig:ballshape"}](tennisball "fig:"){height="1.61cm" width="1.61cm"} Local shape recognition {#localshapeclassification} ----------------------- In terms of identifying local shapes, recent years saw a trend to treat pressure patterns as images, thereby extracting features based on the pressure distribution within the images [@pezzementi2011tactile]. The increasing spatial resolution and spatio-temporal response enable tactile sensors to demonstrate the ability to serve as an “imaging" device. A large number of researchers have applied feature descriptors from vision such as image moments [@pezzementi2011tactile; @corradi2015bayesian] to tactile data to represent local shapes. However, there are differences between vision and tactile imaging as listed in Table \[tab:visionvstactile\]. For vision, the field of view (FoV) is large and global since multiple objects can exist in a single camera image. A great amount of features can be obtained from one single image and it is relatively easy to collect data by cameras. On the other hand, it requires high computational resources to process the visual data; there are several fluctuation factors that affect the performance of extracted features, e.g., scaling, rotation, translation and illumination. Here scaling is caused by the distance from cameras to observed objects. In contrast, for tactile sensing, the FoV is small and local as direct sensor-object interactions need to be made. And the information available in one reading is limited due to the low sensor resolution (for instance, Weiss tactile sensor of 14$\times$6 sensing elements compared to a typical webcam of 1280$\times$1024 pixels), especially in terms of revealing the appearance and geometry of objects. Compared to vision, it is relatively expensive to collect tactile data, in terms of sensors and robot components but there is less effort involved, as it requires less computations to process the tactile data. In addition to these properties, features can be extracted from the tactile data that include surface texture [@jamali2011majority], mechanical impedance [@xu2013tactile] and local detailed shapes. In tactile imaging, the influence of scaling is removed as the real dimension and shape of the interacted object can be mapped to the tactile sensor directly, whereas the impact of rotation, translation and “illumination" remains. Here, “illumination" refers to different impressions of object shapes caused by forces of various magnitudes and directions, similar to the variety of light conditions in vision. [l|m[0.7cm]{}|m[0.6cm]{}|m[0.7cm]{}|m[0.7cm]{}|m[2.7cm]{} ]{} Modality & FoV & Info. & Compl. & Compu. & Invariance\ Visual & Global & Rich & Low & High & Scaling, rotation, translation, illumination\ Tactile & Local & Sparse & High & Low & Rotation, translation, “illumination"\ Note: FoV: Field of View; Info.: Information; Compl.: Complexity to collect data; Compu.: Computation. “illumination": different impressions caused by forces of various magnitudes and directions. ### Shape descriptors for tactile object recognition As raw tactile readings are to some extent redundant, features are utilized to represent the collected tactile data. The paradigm is to extract features based on the pressure distribution in tactile arrays and then feed the extracted features into classifiers to build a training model that can be used to recognize test object shapes. According to the descriptors used, methods for local shape recognition can be categorized as follows: #### Raw tactile readings as features {#raw-tactile-readings-as-features .unnumbered} This method avoids the feature extraction process and is easy to implement [@jimenez1997featureless; @liu2012tactile; @dang2011blind]. However, on the other hand it is sensitive to pattern variations in positions, orientations and sizes. In [@schneider2009object], the columns of each tactile matrix are concatenated to form a vector that is directly treated as a descriptor, i.e., a “do-nothing" descriptor. The feature is sensitive to the pose variances of objects, as a result, a single object is assigned with multiple identities if it is placed in different orientations to the robotic gripper. In [@pezzementi2011tactile], the method using a “do-nothing" descriptor is taken as a baseline and shows worse performance compared to the other methods. In [@martinez2013active], the tactile readings of iCub fingertip with 12 taxels are taken to classify regions that the fingertip taps into edges, plane and air. #### Statistical features {#statistical-features .unnumbered} It is effortless to obtain statistical features but the extracted features cannot be guaranteed to be useful. In [@schopfer2009using], statistical features are utilized, e.g., maximum, minimum and mean pressure values of each tactile reading, and positions of the center of gravity. As a result, the statistical features of 52 different types form a 155 dimensional feature vector. As the information that resides in some features is redundant, an entropy based method is applied to investigate the usefulness of the features but as a result low recognition rate of only around 60% is achieved. The statistical features have also been applied in other applications. In [@tawil2012interpretation; @tawil2011touch], touch attributes, e.g., pressure intensity and area of contact, are taken as features to classify the touch modalities. In [@chitta2011tactile], the internal states of bottles and cans are estimated using the designed statical features of the tactile data. \[tab:tactilefeatures\] [m[3.5cm]{}| m[4.8cm]{}| m[3.5cm]{} | m[2.5cm]{}| l]{} Descriptors & Sensors (resolution) & Classifier & Ref. & Year\ Raw tactile readings & K.U. Leuven (16$ \times $16) & Neural Network & Jimenez et al. [@jimenez1997featureless] & 1997\ Image moments & Bonneville (16$ \times $16) & Neural Network & Russell et al. [@russell2000object] & 2000\ PCA-based features & Weiss DSA100 (16$ \times $16) & Neural Network & Heidemann et al. [@heidemann2004dynamic] & 2004\ Statistical features & Weiss DSA100 (16$ \times $16) & Decision trees C4.5 & Sch[ö]{}pfer et al. [@schopfer2009using] & 2009\ Eigenvectors & Weiss DSA9335 (4$ \times $7) & *k*NN & G[ö]{}ger et al. [@goger2009tactile] & 2009\ Raw tactile readings & Weiss DSA9205 (14$ \times $6) & BoW, Bayes & Scheider et al. [@schneider2009object] & 2009\ Image moments, eigenvalues & Weiss DSA9330/9335(4$ \times $6, 4$ \times $7) & SOM, Bayes & Gorges et al. [@gorges2010haptic] & 2010\ Tactile vector, SIFT, MR-8, Hu’s moments, Polar Fourier & PPS DigiTacts (6$ \times $6), Tekscan (26$ \times $26) & Maximum likelihood estimator & Pezzementi et al. [@pezzementi2011tactile] & 2011\ Image moments & Weiss DSA9205 (14$ \times $6, 13$ \times $6) & AdaBoost, SVM, HMM & Bekiroglu et al. [@bekiroglu2011assessing] & 2011\ Eigenvectors, image moments & Weiss sensors (4$ \times $6, 4$ \times $7, 7$ \times $14), Schunk dexterous (13$ \times $6, 14$ \times $6) & SOM, BoW, ANN & Navarro et al. [@navarro2012haptic] & 2012\ Self-organizing features & iCub fingertip sensor (12 taxels) & STORK-GP & Soh et al. [@soh2012online] & 2012\ 512-feature vector & Tekscan (349 cells) & Neural Network & Liu et al. [@liu2012tactile] & 2012\ Eigenvectors, convexity & Tekscan (9$ \times $5) & Naive Bayes & Liu et al. [@liu2012computationally] & 2012\ Image moments & Flexible sensor [@drimus2014design] (8$ \times $8), Weiss (14$ \times $6) & *k*NN, DTW & Drimus et al. [@drimus2014design] & 2014\ Regional descriptors & PPS (32$ \times $32) & LSVM & Khasnobish et al. [@khasnobish2014object] & 2014\ PCA-based geometric features, Fourier descriptors & Flexible film [@liu2014low] (10$ \times $10) & MKL-SVM & Liu et al. [@liu2014low] & 2014\ Self-organizing features & Schunk Dextrous Model (12$ \times $6), Schunk dexterous (13$ \times $6, 14$ \times $6), Schunk Parallel (8$ \times $8), iCub (12 taxels) & Spatio-Temporal Hierarchical Matching Pursuit algorithm & Madry et al. [@madry2014st] & 2014\ Self-organizing features & TWENDY-ONE hand (241 cells) [@schmitz2014tactile] & Deep learning and dropout & Schmitz et al. [@schmitz2014tactile] & 2014\ Self-organizing features & iCub fingertip sensor (12 taxels) & OIESGP & Soh et al. [@soh2014incrementally] & 2014\ Image moments & Camera based sensor (640$ \times $480) [@corradi2015bayesian] & Naive Bayes & Corradi et al. [@corradi2015bayesian] & 2015\ Self-organizing features & Schunk Dexterous (13$ \times $6, 14$ \times $6), Schunk Parallel (8$ \times $8) & 3T-RTCN & Cao et al. [@cao2016efficient] & 2016\ Self-organizing features & BarrettHand with 3 fingers (8$ \times $3) & Joint sparse coding & Liu et al. [@liu2016object]  & 2016\ #### Descriptors adapted from computer vision {#descriptors-adapted-from-computer-vision .unnumbered} In these methods, tactile arrays are treated as images and thus vision descriptors can be applied and adapted. Several researchers took image moments as feature descriptors [@pezzementi2011tactile; @corradi2015bayesian; @russell2000object; @bekiroglu2011assessing; @drimus2014design]. For a tactile reading $f(x,y)$, the image moment $ m_{pq} $ of order $p+q$ can be calculated as follows: $$m_{pq}=\sum_{x}\sum_{y}x^{p}y^{q}f(x,y),$$ where $p$ and $q$ stand for the order of the moment, $x$ and $y$ stand for the horizontal and vertical positions of the cell in the tactile image, respectively. In most cases, the moments of order up to 2, i.e., $ (p+q)\in\{0,1,2\} $, are computed and other properties are based on them, like Hu’s moments used in [@pezzementi2011tactile] and Zernike moments used in [@corradi2015bayesian]. For example, in [@schmid2008opening], a tactile reading is approximated by an ellipse whose principal axes are represented by the second order central moments. Some other vision descriptors are also applied in tactile sensing like regional descriptors used in [@khasnobish2014object]. The Scale Invariant Feature Transform (SIFT) descriptor is created based on image gradients [@lowe2004distinctive] that has been proved robust to cope with object pose variations in vision applications. This is also particularly useful for tactile object recognition, since the touches could introduce unexpected object rotation and translation. SIFT has been explored in [@pezzementi2011tactile; @luo2014rotation; @luonovel; @luo2015tactile] for tactile recognition and a good performance can be achieved. As both visual images and tactile readings are present in numerical matrices, many other vision descriptors [@mikolajczyk2005performance; @gauglitz2011evaluation] also have the potential to be applied in tactile sensing, e.g., Shape Context [@belongie2002shape], SURF [@bay2006surf] and 3D descriptor SHOT [@tombari2010unique]. Vision descriptors have also been applied in other tactile applications [@nagatani2012can]. Inspired by the similarities between tactile patterns and grey-scale images, in [@ji2011histogram] tactile data are transformed into histograms as structured features to discriminate human-robot touch patterns. In [@schurmann2011modular], the pose of objects placed on the sensor, i.e., the orientation of a cup handle in this work, is estimated by applying Hough transform to the impressed tactile profiles. In [@wong2014spatial], the edge orientation is estimated with a BioTac sensor by using a support vector regression (SVR) model. Image moments have also been widely used in these applications. In [@bekiroglu2011assessing; @bekiroglu2011learning], image moments are utilized to represent the acquired tactile data, applied in grasp stability analysis. In a more recent work [@bekiroglu2013probabilistic], in addition to the image moments mentioned above, more haptic features including the 3D version of image moments and average normal vector are taken into consideration for the same task, together with features of other sensory streams, i.e., vision and proprioception. In [@licontrol], both Hough transform and image moments are employed and compared to predict the orientation of an object edge so as to perform tactile servoing. The GelSight tactile sensor has an extra-high resolution of 320$ \times $240 and provides the ability of using high level vision descriptors [@li2014localization]. It consists of a camera at the bottom and a piece of clear elastomer on the top. The elastomer is coated with a reflective membrane, which deforms to take the surface geometry of the objects on it. The deformation is then recorded by the camera under illumination from LEDs that project from various directions. With the use of GelSight sensor, a multi-scale Local Binary Pattern (LBP) descriptor is developed and used to extract both micro- and macro-structures of the objects from obtained tactile readings/images for texture classification [@li2013sensing]. #### PCA-based features {#pca-based-features .unnumbered} Principal Component Analysis (PCA) can be applied to tactile readings and the acquired principal components (PCs) are taken as features. It can reduce the redundancy of tactile data and is easy to implement but it lacks physical meaning [@gorges2010haptic]. In [@heidemann2004dynamic], a $ 16 \times 16 $ tactile reading is projected onto a feature space of a lower dimension and an iterative procedure is taken to compute PCs with the largest eigenvalues. In [@liu2012computationally], the pressure distribution is defined as $ M=[x \ y \ p]^T $, where $x$, $y$ are location coordinates in the sensor plane and $p$ is the pressure in this location. By covariance analysis of $M$, the resultant eigenvector lengths, principal axis direction and shape convexity are taken as features to recognize local shapes. The kernel PCA-based feature fusion is used in [@liu2014low] to fuse geometric features and Fourier descriptors (based on Fourier coefficients) to better discriminate objects. In [@li2014learning], PCA is applied to reduce the dimensionality of tactile readings. The obtained tactile features are then used for grasp stability assessment and grasp adaptation. In [@scho2007acquisition], PCA is also employed to extract features from tactile readings that is applied for object pose estimation[@bimbo2016hand]. #### Self-organizing features {#self-organizing-features .unnumbered} The aforementioned features are predefined and hand-crafted and they are fed into shallow classifiers, such as single-layer neural networks, *k*NN, and SVM. Methods of this type can be easily implemented but they can restrict the representation capability to serve different applications and may only capture insignificant characteristics for a task when using hand-crafted features [@madry2014st]. In contrast, there is no need to define the feature representation format a-priori when using multilayer/deep architecture methods to learn self-organizing features from raw sensor data. Soh et al. [@soh2012online] developed an on-line generative model that integrates a recursive kernel into a sparse Gaussian Process. The algorithm iteratively learns from temporal tactile data and produces a probability distribution over object classes, without constructing hand-crafted features. They also contribute discriminative and generative tactile learners [@soh2014incrementally] based on incremental and unsupervised learning. In [@madry2014st] and [@molchanov2016contact], unsupervised hierarchical feature learning using sparse coding is applied to extract features from sequences of raw tactile readings, for grasping and object recognition tasks. In [@schmitz2014tactile] denoising autoencoders with dropout are applied in tactile object recognition and a dramatic performance improvement of around 20% is observed in classifying 20 objects compared to using shallow neural networks and supervised learning. In [@cao2016efficient], the randomized tiling Convolutional Network (RTCN) is applied for the feature representation in tactile recognition and can achieve an extremely good recognition rate of 100% for most of the datasets tested in the paper. In [@liu2016object], a joint kernel sparse coding model is proposed for the classification of tactile sequences acquired from multiple fingers. As new techniques in deep learning and unsupervised learning emerge and grow rapidly in recent years [@bengio2013representation], it is promising to apply more such algorithms to acquire self-organizing features in tactile sensing. On the other hand, though deep learning shows tremendous promise for the object recognition tasks, online application of such a computationally intensive process is difficult and it is hard to tune the parameters of deep architectures; also, the complexity of these deep architectures translate into difficulties in the introspection and physical interpretation of the resulting model. \[tab:tactilefeaturesother\] [m[2.7cm]{}| m[5cm]{}| m[4.2cm]{}| m[3.2cm]{}| m[0.5cm]{}]{} Descriptors & Sensors (resolution) & Applications & Ref. & Year\ PCA & Tactile sensor (16$ \times $16) & Object pose estimation & Sch[ö]{}pfer et al. [@scho2007acquisition] & 2007\ Concatenated vector & PPS RoboTouch System (162 in total, 7 sensor pads, 22 or 24 cells each) & Grasp stability assessment & Dang et al. [@dang2011blind] & 2011\ Histograms & Triangular patches with 12 elements each & Identification of touch modalities & Ji et al. [@ji2011histogram] & 2011\ Hough Transform & Modular tactile sensor (16$ \times $16) & Object pose estimation & Schurmann et al. [@schurmann2011modular] & 2011\ Image moments & Weiss DSA9205 (14$ \times $6, 13$ \times $6) & Grasp stability assessment & Bekiroglu et al. [@bekiroglu2011assessing] & 2011\ Image moments & Schunk Dexterous Hand & Grasp stability assessment & Bekiroglu et al. [@bekiroglu2011learning] & 2011\ Statical features & Tactile sensors of PR2 gripper (22 elements) & Internal state estimation & Chitta et al. [@chitta2011tactile] & 2011\ Pyramid Lucas-Kanade & Nitta I-SCAN50 (44$ \times $44) & Slip perception & Ho et al. [@nagatani2012can] & 2012\ Statistical features & EIT-based sensitive skin (19 sensing cells) & Identification of touch modalities & Tawil et al. [@tawil2012interpretation; @silvera2014interpretation] & 2012; 2014\ 2D/3D image moments, average normal vector & Weiss DSA9205 (14$ \times $6, 13$ \times $6) & Grasp stability assessment & Bekiroglu et al. [@bekiroglu2013probabilistic] & 2013\ PCA & BioTac (19 cells) & Learning haptic adjectives & Chu et al. [@chu2013using] & 2013\ Raw tactile readings & iCub fingertip sensor (12 taxels) & Contour following & Martinez et al. [@martinez2013active; @martinez2013active1] & 2013\ Hough Transform, image moments & Modular tactile sensor (16$ \times $16) [@schurmann2011modular] & Tactile servoing & Li et al. [@licontrol] & 2013\ MLBP & GelSight (320$ \times $240) & Surface texture recognition & Li et al. [@li2013sensing] & 2013\ PCA & BioTac (19 cells) & Grasp stability assessment and grasp adaptation & Li et al. [@li2014learning] & 2014\ Self-organizing features & Schunk Dextrous Model (12$ \times $6), Schunk dexterous (13$ \times $6,14$ \times $6), Schunk Parallel (8$ \times $8), iCub (12 taxels) & Grasp stability assessment & Madry et al. [@madry2014st] & 2014\ BRISK & GelSight (320$ \times $240) & Localization and manipulation & Li et al. [@li2014localization] & 2014\ Statistical features & Cellular Skin [@mittendorfer2011humanoid] & Identification of touch modalities & Kaboli et al. [@kaboli2015humanoids] & 2015\ Image moments & Robotiq three-finger gripper (2$ \times $4, 3$ \times $3) & Grasp stability assessment & Hyttinen et al. [@hyttinen2015learning] & 2015\ ### Discussions of local shape descriptors A summary of the discussed tactile features is given in Table \[tab:tactilefeaturesummary\], with discussions and comparison of pros and cons, and guidance on the selection of tactile sensors. In the state-of-the-art literature, vision based descriptors that take tactile readings as images are widely employed and will also be the mainstream in extracting features for tactile object recognition and other applications. There is another trend to employ unsupervised learning and deep architectures to learn self-organizing features from raw tactile readings as an increasing number of such algorithms are being developed [@bengio2013representation]. In addition to shape recognition, the various descriptors discussed here have also been used to identify different contact patterns in several other tasks that can be grouped as grasp stability assessment [@madry2014st; @bekiroglu2013probabilistic; @bekiroglu2011assessing; @dang2011blind; @bekiroglu2011learning; @hyttinen2015learning], identification of touch modalities [@kaboli2015humanoids; @ji2011histogram; @tawil2012interpretation], object pose estimation [@schurmann2011modular], slip detection [@nagatani2012can], learning system states during in-hand manipulation [@stork2015learning], contour following [@martinez2013active], tactile servoing [@licontrol], surface texture recognition [@li2013sensing], localization and manipulation in assembly tasks [@li2014localization], and grasp adaptation [@li2014learning]. They can also be applied in other applications in the future research. \[tab:tactilefeaturesummary\] [m[1.5cm]{}| m[5cm]{}| m[5cm]{}| m[2.5cm]{}| m[1.5cm]{}]{} Feature type & Advantages & Disadvantages & Applicable sensors & Ref.\ Raw tactile readings & A “do-nothing" descriptor; easy to implement; applied to any type of tactile data & Lack of physical meaning; redundant; sensitive to variations of contact forces and poses & Planar array sensors; curved tactile sensors & [@schneider2009object; @dang2011blind; @martinez2013active]\ Statistical features & Easy to be computed; based on statistics; can be applied to any type of tactile data & Lack of physical meaning; redundant; sensitive to force and pose variance; hand-crafted & Planar array sensors; curved tactile sensors & [@schopfer2009using; @tawil2012interpretation; @tawil2011touch; @kaboli2014humanoids]\ Descriptors adapted from vision & Extract distinct features; invariant to force and pose variance; can be used to share information with vision & Some of the features are of high dimension compared to original reading; hard to design; predefined and hand-crafted & Planar array sensors; low curvature sensors & [@pezzementi2011tactile; @corradi2015bayesian; @russell2000object; @drimus2014design; @bekiroglu2013probabilistic; @bekiroglu2011learning; @schurmann2011modular]\ PCA-based features & Low dimensionality; statistics based; easy to implement & Lack of physical meaning; sensitive to force and pose variance & Planar array sensors; curved tactile sensors & [@li2014learning; @liu2012computationally; @bimbo2016hand]\ Self-organizing & No need to define the feature representation format a-priori; learn features from raw data & Lack of physical meaning; high computational complexity; hard to tune parameters & Planar array sensors; curved tactile sensors & [@soh2012online; @madry2014st; @schmitz2014tactile; @soh2014incrementally; @cao2016efficient]\ Global shape perception {#globalshaperecognition} ----------------------- The methods to recognize or reconstruct the global shape of objects with tactile sensing can be grouped into three categories with respect to the sensing inputs: 1) methods using the distributions of contact points obtained from single-point contact force sensors or tactile sensors; 2) methods based on analysing the pressure distributions in tactile arrays; 3) methods of combining both tactile patterns and contact locations. Here the global shape refers to the overall shape of objects, especially contours that extend beyond the fingertip scale. #### Points based recognition {#points-based-recognition .unnumbered} The methods of this type often employ techniques from computer graphics to fit the obtained cloud of contact points to a geometric model and outline the object contour. This method was widely used by early researchers due to the low resolution of tactile sensors and prevalence of single-point contact force sensors [@grimson1984model; @allen1989haptic; @charlebois1999shape; @okamura2001feature]. In [@allen1989haptic] resultant points from tactile readings are fit to super-quadric surfaces to reconstruct unknown shapes. In a similar manner, relying on the locations of contact points and hand pose configurations, a polyhedral model is derived to recover object shapes in [@casselli1995robustness]. These approaches are limited as objects are usually required to be fixed and stationary. Different from the point cloud based approaches, a non-linear model-based inversion is proposed in [@fearing1991using] to recover surface curvatures by using a cylindrical tactile sensor. In more recent works [@ibrayev2005semidifferential; @jia2006surface; @jia2010surface], the curvatures at curve intersection points are analyzed and thus a patch is described through polynomial fitting; in [@abraham2017ergodic], estimation of nonparametric shapes is demonstrated using binary sensing (collision and no collision) and ergodic exploration. In some other works tactile sensors are utilized to classify objects by taking advantage of the spatial distribution of the object in space. In [@pezzementi2011object] an object representation is constructed based on mosaics of tactile measurements, in which the objects are a set of raised letter shapes. In this work, the object recognition is regarded as a problem of estimating a consistent location within a set of object maps and thus histogram and particle filtering are used to estimate possible states (locations). A descriptor based on the histogram of triangles generated from three contact points was used in [@zhang2016triangle] to classify 10 classes of objects. This descriptor is invariant to object movements between touches but requires a large number of samples (grasps) to accurately classify the touched object. Kalman filters are applied in [@meier2011probabilistic] to generate 3D representations of objects from contact point clouds collected by tactile sensors and the objects are then classified by the Iterative Closest Point (ICP) algorithm. A similar method is employed in [@aggarwal2015haptic], for haptic object recognition in underwater environments. Through utilizing these methods, arbitrary contact shapes can be retrieved, however, it can be time consuming when investigating a large object surface as excessive contacts are required for recognizing the global object shape. #### Tactile patterns based recognition {#tactile-patterns-based-recognition .unnumbered} Another approach is to recognize the contact shapes using pressure distribution within tactile arrays. As a result of the increasing performance of tactile sensors, this approach has become increasingly popular in recent years. Various methods to recognize the local contact shapes have been reviewed and discussed in Section \[localshapeclassification\]. In terms of recognising the global object shape by analysing pressure distributions in tactile images collected at different contact locations, however, a limited number of approaches are available. One popular method is to generate a codebook of tactile features and use it to classify objects and a particular paradigm is the Bag-of-Features (BoF) model [@madry2014st; @schneider2009object; @pezzementi2011tactile]. The BoF model originates from the Bag-of-Words (BoW) model in natural language processing for text classification and has been widely utilized in the field of computer vision [@nowak2006sampling], thanks to the simplicity and power of the model. Inspired by the similar essence of vision and tactile sensing, Schneider et. al. [@schneider2009object] first applied the BoF model in tactile object recognition. In this framework, local contact features extracted from tactile readings in the training phase are clustered to form a dictionary and cluster centroids are taken as “codewords". Based on the dictionary, each tactile feature is assigned to its nearest codeword and a fixed length feature occurrence vector is generated to represent the object. It is easy to implement and can achieve an appropriate performance [@schneider2009object; @pezzementi2011tactile], however, only local contact patterns are taken and the distribution of the features in three-dimensional space is not incorporated. #### Object recognition based on both sensing modalities {#object-recognition-based-on-both-sensing-modalities .unnumbered} For humans, the sense of touch consists of both kinaesthetic and cutaneous sensing and these two sensing modalities are correlated [@lederman2009haptic]. Therefore, the fusion of the spatial information and tactile features could be beneficial for the object recognition tasks. This combination has already been proved to improve recognition capabilities in teleoperation experiments with human subjects both in identifying curvatures [@pattichizzo2013towards] and estimating stiffness [@pacchierotti2013improving]. In [@mcmath1991tactile; @petriu1992active], a series of local “tactile probe images" is assembled and concatenated together to obtain a “global tactile image" using 2D correlation techniques with the assistance of kinaesthetic data. However, in this work tactile features are not extracted whereas raw tactile readings are utilized instead, which would bring high computational cost when investigating large object surfaces. In [@johnsson2007neural], three different models are proposed based on proprioceptive and tactile data, using Self-Organising Maps (SOMs) and Neural Networks. In [@gorges2010haptic], the tactile and kinaesthetic data are integrated by decision fusion and description fusion methods. In the former, classification is done with two sensing modalities independently and recognition results are combined into one decision afterwards. In the latter, the descriptors of kinaesthetic data (finger configurations/positions) and tactile features for a single palpation are concatenated into one vector for classification. In other words, the information of the positions where specific tactile features are collected is lost. In both methods, the tactile and kinaesthetic information is not fundamentally linked. In a similar manner, in [@navarro2012haptic] the tactile and kinaesthetic modalities are fused in a decision fusion fashion. Both tactile features and joint configurations are clustered by SOMs and classified by ANNs separately and the classification results are merged to achieve a final decision. In a more recent work [@spiers2016single], the actuator positions of robot fingers and force values of embedded TakkTile sensors form the feature space to classify object classes using random forests but there are no exploratory motions involved, with data acquired during a single and unplanned grasp. In [@luo2016iterative], an algorithm named Iterative Closest Labeled Point (iCLAP) is proposed to recognize objects using both tactile and kinaesthetic information that has been shown to outperform those using either of the separate sensing modalities. In general, it is still an open question how to link both sensor locations and tactile images. \[tab:globalrecognition\] [m[1.2cm]{}|m[4cm]{}|m[3.5cm]{}|m[3.5cm]{}|m[2.3cm]{}]{} Modalities & Recognition methods & Advantages & Disadvantages & Applicable sensors\ Contact points & Graphical models (point clouds) [@casselli1995robustness]; polynomial fitting (surface curvatures) [@jia2010surface]; filtering (spatial distribution) [@pezzementi2011object; @meier2011probabilistic] & Arbitrary shapes can be retrieved; object graphical models can be built; spatial distribution is revealed & Time consuming when investigating large surfaces; excessive contacts required; local features are not revealed & Single-point contact sensors; planar/curved tactile sensors\ Tactile patterns & Bag-of-Features [@luonovel; @madry2014st; @schneider2009object; @pezzementi2011tactile] & Easy to implement; local shape features are employed & The distribution of the features in 3D space is not incorporated & Planar sensors or low curvature\ Both sensing modalities & Image stitching [@mcmath1991tactile; @petriu1992active]; Decision fusion [@gorges2010haptic; @navarro2012haptic]; Description fusion [@gorges2010haptic] & Both kinaesthetic and cutaneous cues are included & Hard to associate local patterns and kinaesthetic data; bring additional computational cost & Planar sensors or low curvature\ In summary, different strategies have been taken to recover the global shape of objects in the view of proprioception and tactile sensing, as summarized in Table \[tab:globalrecognition\] with a comparison of pros and cons of different methods. Some researchers take advantage of the distributions of contact points in space, e.g., points based methods, whereas some others utilize local patterns only, e.g., BoF framework; and it is also expected to achieve a better perception by combining both distributions of contact points in space and local contact patterns. Pose estimation via touch sensing {#localization} ================================= Effective object manipulation requires accurate and timely estimation of the object pose. This pose is represented by the object’s position and orientation with respect to the robot end-effector or to a global coordinate frame. Even small errors the estimate of the object’s location can lead to incorrect placement of the robot fingers on the object, generate wrong assumptions on grasp stability and compromise the success of a manipulation task. In fact, in-hand manipulation is, by definition, the task of changing an object’s pose from an initial to a final configuration [@hertkorn2013planning]. Thus, robust, accurate and fast perception of an object’s pose must be a crucial part of any sophisticated grasping and manipulation system. The most common means in robotics to estimate an object’s pose is using computer vision. However, when a robot approaches the object to be manipulated, it creates occlusions and vision cannot be relied upon. To cope with this problem, tactile sensing has been used to assist a robot in determining the pose of a touched object, either on its own or in combination with a vision system. In this review we classify existing techniques according to the sensing inputs: single-point contact sensor and tactile sensing arrays, on their own or used together with a vision system (contact-visual and tactile-visual), as shown in Table \[tab:slam\]. #### Single-point contact based {#single-point-contact-based .unnumbered} Due to the poor performance of tactile sensors, most early works tend to use single-point contact sensors, i.e., force sensors and dynamic tactile sensors, for localizing the objects or features. Early work on finding an object’s pose used only angle and joint-torque sensing and used an interpretation tree that contained possible correspondences between object vertices to fingers[@siegel1991finding]. In [@gadeyne2005bayesian], a force-controlled robot is used to localize objects using Markov Localization that is applied in a task of inserting a cube (manipulated object) into a corner (environment object) by a manipulator. This compliant motion problem is also compared with the data association [@montemerlo2003simultaneous], i.e., assigning measurements to landmarks, and global localization with different models in Simultaneous Localization And Mapping (SLAM) for mobile robotics. In both cases, Bayesian based approaches can provide a systematic solution to estimate both models and states/parameters simultaneously. In [@schaeffer2003methods], a set of algorithms are implemented for SLAM during haptic exploration but only simulation results are presented. Particle filtering is popular in (optical/acoustic-based) robot localization problems and may merit further investigation in tactile sensing, where objects could be modelled as clouds of particles distributed according to iteratively updated probability distributions. In [@corcoran2010measurement], particle filtering is applied to estimate the object pose and track the hand-object configurations, especially in the cases where objects are possibly moving in the robot hand. Also using particle filtering, small objects, e.g., buttons and snaps, are localized and manipulated in flexible materials such as fabrics that are prone to move during robot manipulations in [@platt2011using]. Improvements of the particle filter algorithm have been presented to address the problem of localizing an object via touch. In [@petrovskaya2006bayesian; @petrovskaya2007touch; @petrovskaya2011global], this novel particle filter, named Scaling Series, has each particle representing not a single hypothesis, but a region in the search space which is sequentially refined. Besides, annealing is used to improve sampling from the measurement model. This method was tested using a robot manipulator equipped with a 6D force/torque sensor. It is applied in two scenarios: 1) to localize (estimate positions and orientations), grasp and pick up a rectangular box; 2) to grasp a door handle. A method that includes memory from past measurements and a “scaled unscented transformation” is performed on the prediction step was presented in [@vezzani2016memory]. A “haptic map" is created from the proprioceptive and contact measurements during the training that is then used to localize the small objects embedded in flexible materials during robot manipulation. By using dynamic tactile sensors inspired by rat whiskers, the grid based SLAM is introduced in [@fox2012tactile] to navigate a robot with touch sensors by deriving timing information from contacts and a given map about edges in a small arena, in which particle filters are also used. In [@yu2015shape], SLAM is applied to recover the shape and pose of a movable object from a series of observed contact locations and contact normals with a pusher. Analogous to SLAM in mobile robotics, the object is taken as a rigid but moving environment and the pusher is taken as a sensor to get noisy observations of the locations of landmarks. However, compared to the exploration using visual feedback, tactile exploration is challenging in the sense that touch sensing is intrusive in nature, that is, the object (environment) is moved by the action of sensing. [m[1.7cm]{}| m[2.7cm]{}| m[1.5cm]{}| m[0.8cm]{}]{} Types & Sensors involved & Information & Example\ Visual & Cameras, lasers & Global & [@davison2007monoslam]\ Contact & Single-point contact sensors & Local & [@petrovskaya2011global]\ Tactile & Tactile sensors & Local & [@li2014localization]\ Contact-Visual & Cameras, single-point contact sensors & Global+local & [@Bhattacharjee2015combining]\ Tactile-Visual & Cameras, tactile sensors & Global+local & [@luo2015localizing]\ Note: Laser scanners and cameras (widely used in mobile robots) collect global information about the environment, whereas the contact sensors (single-point contact sensors and tactile sensors) provide local information. #### Tactile-based {#tactile-based .unnumbered} The increasing performance of tactile sensors provides the feasibility to localize objects or object features in robot hand using the information derived from tactile arrays. With the high-resolution GelSight sensor, collected tactile images can be localized [@li2014localization] within a height map via image registration to help localize objects in hand. A height map is first built based on the collected tactile readings. The keypoints are then localized from both the map and incoming tactile measurements. After that, feature descriptors are extracted from both and matched. In this manner, the pose of an assembly part in the robot hand can be estimated. Similarly to the way that local geometric features can be extracted using PCA, these features were used to determine an object’s pose. In [@bimbo2015global] a Monte Carlo method was used to find an object pose where the local geometry of the object at the contact location matched the PCA features obtained from the tactile sensor. Another approach relied on the fact that, when the robot is in contact with the object, the possible locations of the object must lie inside a contact manifold, a novel particle filter was developed that was both faster and more accurate than a standard particle filter [@koval2015pose]. #### Contact-Visual based {#contact-visual-based .unnumbered} Vision and tactile sensing share information of how objects are present in 3D space in the form of data points. For vision, a mesh can be generated based on point cloud from 3D cameras or laser scanners that can also be obtained by contact sensors (in this case, mostly single-point contact sensors). In [@hebert2011fusion], localizing object within hand is treated as a hybrid estimation problem, by fusing data from stereo vision, force-torque and joint sensors. Though reasonable object localization can be achieved, tactile information on the fingertips is not utilized, being however promising to obtain better performance of object localization by incorporating additional information. Based on the assumption that visually similar surfaces are likely to have similar haptic properties, vision is used to create dense haptic maps efficiently across visible surfaces with sparse haptic labels in [@Bhattacharjee2015combining]. Vision can also provide an approximate initial estimate of the object pose that is then refined by tactile sensing using local[@honda1998realtime; @bimbo2013combining] or global optimization[@bimbo2015global]. #### Tactile-Visual based {#tactile-visual-based .unnumbered} Vision and tactile sensing of humans have also been found to share the representations of objects [@lacey2007vision]. To put it in another way, there are some correspondences, e.g., prominent features, between vision and tactile sensing while observing objects. In [@luo2015localizing], it is proposed to localize a tactile sensor in a visual object map by sharing similar sets of features between visual map and tactile readings. It is treated as a probabilistic estimation problem and solved in a framework of recursive Bayesian filtering, where tactile patterns are for local information and vision is to provide a global map to localize sensor contact on the object. In summary, the localization of objects during manipulation using tactile sensing remains an open problem. While the most popular approaches rely on techniques similar to the ones used in SLAM applications, high accuracy and real-time performance is yet to be achieved. Furthermore, these approaches require a large number of contacts, which may be impractical for effective object manipulation. Nevertheless, it is foreseen that tactile sensors of different types will need to be used along with vision for solving the localization problem. Due to the reduced number of contacts, high resolution tactile sensors that can capture the object’s local shape can be useful to determine the pose using few contacts. Tactile sensing in sensor fusion {#sensingintegration} ================================ Robots must be equipped with different sensing modalities to be able to operate in unstructured environments. The fusion of these different sources of data into more meaningful, higher-level representations of the state of the world is also part of the process of perception (See Fig. \[fig:tactilesensingsystem\]). The information that is provided by the sensors may be redundant, reducing the uncertainty with which features are perceived by the system, and increasing the reliability of the system in case of failure [@felip2014multisensor]. Combining different sensing modalities which provide complementary information may also have a synergistic effect, where features in the environment can be perceived in situations where it would be impossible using the information from each sensor separately [@luo2002multisensor]. Furthermore, multiple sensors can provide more timely and less costly information, given the different operating speeds of the sensors and the fact that their information can be processed in parallel.\ In tasks that require interaction with the environment, tactile sensing can be combined with other sensing modalities to increase precision and robustness [@prats2009vision]. Most typical sensing arrangements in robot manipulation and other tasks that include physical interaction, are combinations of tactile sensing with vision, kinaesthetic cues, force-torque and range sensing [@luo1989multisensor]. We have discussed the works on integrating tactile sensing and kinaesthetic cues for global shape recognition in Section \[globalshaperecognition\]. In this section, we mainly focus on tactile sensing in sensor fusion with vision. In robotics, attempts to fuse vision and touch to recognize and represent objects can be dated back to the 1980’s [@allen1984surface]. In most applications, tactile sensing was utilized to support vision for improving performance in object recognition [@bjorkman2013enhancing], object reconstruction [@ilonen2013fusing] and grasping [@bekiroglu2013probabilistic]. With the increasing performance in the last few decades, tactile sensors have shown the potential to play a more significant role in tasks using information integrated from different modalities [@sinapov2011interactive; @hebert2011fusion; @araki2012online]. Thanks to the development of tactile sensor technologies, the role of tactile sensing in sensor fusion for multiple applications has evolved to a mature stage in a number of applications, which can be summarized as follows: #### Verifying contacts {#verifying-contacts .unnumbered} Most early researchers took tactile sensors as devices to verify contacts due to their low resolution. In this type of methods, rough object models are first built by vision and the description is then refined and detailed by tactile sensing. For instance, in [@allen1984surface] vision is first used to obtain object contours and edges as it can capture rich information rapidly, by taking bi-cubic surface patches as primitives. The tactile trace information is then added into the boundary curves acquired in the first step to achieve a more detailed description of the surfaces. In this work, tactile (haptic) sensing is utilized to trace the object surface to get 3D coordinates, surface normal and tangential information of contact points. A similar framework is also employed in [@allen1988integrating], where the role of tactile sensing is greater. The strategy consisted of exploring regions that are uncertain for vision, determining the features such as surfaces, holes or cavities, and outlining the boundary curves of these features. In [@allen1999integration] vision is used to estimate the contact positions along a finger and applied forces for grasping tasks, whereas tactile sensors and internal strain gauges are utilized to assist vision, especially in two cases: when vision is occluded or when it is needed to further confirm contact positions determined by vision. In [@dragiev2011gaussian], implicit surface potentials described by Gaussian Processes are taken as object models for internal shape representations. In the Gaussian Process of shape estimation, uncertain sensor channels, i.e., tactile, visual and laser, are equally integrated, to support the control of reach-and-grasp movements. In [@ilonen2013fusing], an optimal estimation method is proposed to learn object models during grasping via visual and tactile data by using iterative extended Kalman filter (EKF). A similar work is carried out in [@bjorkman2013enhancing] but Gaussian process regression is utilized instead of EKF: visual features are extracted first to form an initial hypothesis of object shapes; tactile measurements are then added to refine the object model. Though tactile array sensors are employed in these works, still only the locations of tactile elements that are in contact with the object are used, not including the information of pressure distribution in tactile arrays. A framework to detect and localize contacts with the environment using different sources of sensing information is presented in [@felip2014multisensor]. Each sensor generates contact hypotheses which are fused together using a probabilistic approach to obtain a map that contains the likelihood of contact at each location in the environment. This approach was tested in different platforms with various sensing modalities, such as tactile, force-torque, vision, and range sensing. #### Extracting features to assist vision {#extracting-features-to-assist-vision .unnumbered} In [@bekiroglu2013probabilistic], multiple sensory streams, i.e., vision, proprioception and tactile sensing, are integrated to facilitate grasping in a goal-oriented manner. In this work, image moments of tactile data (both 2D and 3D), are utilized; together with vision and action features, a feature set is formed for grasping tasks. In [@hebert2011fusion], the in-hand object location is estimated by fusing data from stereo vision, force-torque and joint sensors. The sensor fusion is achieved by simply concatenating features or data from single modalities into a joint vector. In [@guler2014s], vision and tactile sensing are proved to be complementary in the task of identifying the content in a container by grasping: as the container is squeezed by a robot hand, deformation of the container is observed by vision and the pressure distributions around the contact region are captured by tactile sensors. PCA is employed to extract PCs from vision, tactile and vision-tactile data, which are then fed into classifiers. It is concluded that combining vision and tactile sensing leads to the improvement of classification accuracy where one of them is weaker. In addition, some researchers have attempted to combine vision, tactile and force sensing to estimate the object pose in [@bimbo2015global]. #### Providing local and detailed information {#providing-local-and-detailed-information .unnumbered} In [@schmid2008opening], a multi-sensor control framework is created for a task of opening a door. The vision is used to detect the door handle prior to the gripper-handle contact and tactile sensors mounted on the robot grippers are used to measure the gripper orientation with respect to the door handle by applying moment analysis to tactile readings during the gripper-handle contact. Based on the information, the pose of grippers can be adjusted accordingly. In [@prats2010reliable], vision, force/torque sensors and tactile sensors are also combined in a control scheme for opening a door. A sensor hierarchy is established in which the tactile sensing is preferred over vision: the tactile sensing can obtain robust and detailed information about the object position, whereas vision provides global but less accurate data. In both works, tactile sensors play a greater role than just confirming contacts but the vision and tactile sensing are employed in different phases instead of achieving a synthesized perception. In recent work [@izatt2017tracking], a GelSight sensor is taken as a source of dense local geometric information that is incorporated in object point cloud from an RGB-D camera. #### Transferring knowledge with vision {#transferring-knowledge-with-vision .unnumbered} The information of object representations has been found to be shared by vision and tactile sensing of humans [@lacey2007vision] and the visual imagery is discovered to be involved in the tactile discrimination of orientation in normally sighted humans [@zangaladze1999involvement]. It has also been shown that the human brain employs shared models of objects across multiple sensory modalities such as vision and tactile sensing so that knowledge can be transferred from one to another [@amedi2002convergence]. This mechanism can also be applied in robotic applications. In [@kroemer2011learning], vision and tactile samples are paired to classify materials using a dynamic tactile sensor. In the training phase, material representations are learned by both tactile (haptic) and visual observations of object surfaces. A mapping matrix for transferring knowledge from vision to tactile domain is then learned by dimensionality reduction and at the test stage, materials can be classified with only tactile (haptic) information available based on the obtained shared models. In [@sanchez2013sensorimotor], vision and tactile feedback are associated by mapping visual and tactile features to object curvature classes. By taking advantage of the fact that both visual and tactile images are present in numerical matrices, a tactile sensor can be localized in a visual object map by sharing the same sets of features between the visual map and tactile readings [@luo2015localizing]. In recent works [@gao2016deep] and [@yuan2017connecting], deep networks, typically Convolutional Neural Networks (CNNs), are applied to both vision and hatpic data and the learned features from both modalities are then associated by a fusion layer for haptic classification. To conclude, as the performance of tactile sensors improves, tactile sensing plays an increasingly important role in the sensor fusion for multiple applications. In addition to providing information of contact locations like single-point contact sensors, more useful cues can be extracted from tactile readings, such as detailed local information and features to improve models built by vision. More importantly, knowledge can be transferred between vision and tactile sensing, which can improve the perception of the environment interacting with robots. Discussion and conclusion {#discussionandconclusion} ========================= The rapid development of tactile devices and skins over the last couple of years has provided new opportunities for applying tactile sensing in various robotic applications. Not only the types of tactile sensors are to be considered for an application, the suitable computational method to decipher the encoded information is also of the utmost importance. At the same time it has brought to the fore numerous challenges towards effective use of tactile data. This paper focuses on tactile object recognition of shape and surface material and the estimation of its pose. Knowledge of these properties is crucial for the successful application of robots in grasping and manipulation tasks. In this section the progress in the literature is summarised, open issues are discussed and future directions are highlighted. #### Selection of tactile sensors to meet the requirements of tactile object recognition and localization {#selection-of-tactile-sensors-to-meet-the-requirements-of-tactile-object-recognition-and-localization .unnumbered} Apart from the algorithms used to interpret sensor data, the recognition performance also depends on what tactile sensors are used. As high spatial and magnitude resolution provides more detailed information on the object, tactile sensors of higher resolution are preferable. But on the other hand, higher resolution will likely bring about higher costs, both in development and fabrication of the sensor. Most of the available tactile sensors used in robotic fingertips are of lower or similar resolution over an equivalent area compared to the spatial resolution of the human finger, i.e., the density of Merkel receptors in the fingertip (14$\times$14) [@pezzementi2011tactile]. For instance, the widely used tactile sensor modules in the literature Weiss DSA 9205 (14$\times$6) with a size of 24 mm by 51 mm; the Weiss DSA 9335 sensor module provides a similar spatial resolution of 3.8 mm with 28 sensor cells organized as 4$\times$7 matrices; the TakkTile array sensor senses independent forces over an 8$\times$5 grid with approximately 6 mm resolution [@tenzer2014feel]; the DigiTacts sensor from Pressure Profile Systems has 6$\times$6 sensor elements with 6$\times$6 mm resolution; a Tekscan sensor (model: 4256E) has a sensing pad of 24 mm by 35 mm with 9$\times$5 elements. To achieve good spatial resolution, these sensors are commonly costly. An exception is the GelSight tactile sensor that has high resolution of 320$\times$240 and is economical, as it utilizes a low-cost webcam to capture the object properties. But the inclusion of a webcam and illumination LEDs makes the whole sensor bulky and difficult to miniaturize. Therefore, researchers need to balance multiple factors when selecting tactile sensors, i.e., spatial and magnitude resolution, size and cost of the sensors. In real-time tactile localization applications, it is required that tactile sensors have quick responses and short latency. Human tactile receptors can sense the transient properties of a stimulus at up to 400 Hz [@jamali2011majority]. Since the output of single-point contact sensors, i.e., force sensors and whisker based sensors, is of low dimension, these sensors have typically high update rate and low latency. For instance, the data-acquisition of strain gauges and PVDFs embedded in a robotic finger can achieve a temporal resolution of 2.5 kHz [@jamali2011majority]. In contrast, tactile array sensors output data of higher resolution and therefore need to reduce the latency and improve response times. The maximum scanning rate for the Weiss DSA 9205 sensor is around 240 frames per second (fps) [@schneider2009object], while using a serial bus (RS-232) shared by multiple sensors for data transmission will limit the available frame rate for each sensor to approximate 30 Hz when six sensor pads are used [@meier2011probabilistic]; the update rate of the Tekscan pressure sensing is set as 200 Hz in [@liu2012tactile]; by using a GelSight sensor, the in-hand localization can be achieved at 10 fps [@li2014localization]. An alternative method to sense the transient stimulus is to use neuromorphic tactile systems that provides transient event information as the sensor interface with objects [@lee2015kilohertz; @bartolozzi2016neuromorphic]. The sensing element triggers an event when it detects relevant transient stimulus and this provides important local information in manipulation [@son1994tactile]. Many of the tactile sensors used in the literature are of rigid sensing pad and plated with an elastic cover such as Weiss modules and TakkTile sensors. As the data are read from sensing elements arranged in a plane, it reduces the complexity of data interpretation but these sensors are hard to be mounted on curved robot body parts. There are also flexible tactile sensors like DigiTacts from PPS and Tekscan tactile system. They can be attached to curved surfaces but the data interpretation becomes more complex. #### Tactile object recognition {#tactile-object-recognition .unnumbered} As for the local shape recognition by touch sensing, multiple feature descriptors have been proposed in the literature to represent features within tactile readings. The “do-nothing" descriptors, statistical features, and PCA-based features are simple and easy to implement but normally sensitive to pattern variances. Vision-based descriptors have been observed to be the mainstream by taking tactile readings as images. It can be foreseen that more vision descriptors will be applied in touch sensing to represent the information embedded in the tactile data which may provide cues of how to combine vision and touch sensing. In addition, unsupervised learning and deep architectures show promise in recent years by obtaining self-organizing features from raw tactile readings without pre-defining the feature structure. To seek shape representation appropriate for tactile sensing, these methods originated from vision could be further investigated by examining the differences between vision and tactile sensing mechanisms. To reveal the global object shapes, one method is to utilize the distribution of contact points that can retrieve arbitrary contact shapes but will be time consuming when investigating a large object surface, while also being sensitive to object movements. The other method is to use local tactile patterns and a paradigm of this method is Bag-of-Features that is becoming popular recently. It can reveal local features but the three-dimensional distribution information of the object is not incorporated. One future direction is to perceive global shapes by integrating tactile patterns and kinaesthetic cues to achieve a better recognition performance as cutaneous sensing and proprioception are correlated in object recognition. #### Object localization by touch sensing {#object-localization-by-touch-sensing .unnumbered} The object localization via touch could be a significant application that can facilitate object manipulation and can be complementary to visual localization. Currently, most of the attention regarding this topic has been paid to haptic-based localization with single-point contact sensors. But single-point contact sensors, like force sensors and whiskers, provide only limited information, i.e., contact locations of single points. Single-point contact sensors also require multiple contacts, which may cause the unpredicted motion of the object during the contacts. Thanks to the advancement of tactile devices, the works into other types of localization have emerged, especially using tactile array sensors to identify contact patterns. There exist multiple open issues expected to be investigated: 1) how to map the contact points on the sensor pad to the 3D space; 2) how to combine the data points acquired from vision and tactile sensing; 3) how to define the priority hierarchy when the information from different sensors conflicts. #### Tactile sensing in sensor fusion {#tactile-sensing-in-sensor-fusion .unnumbered} The use of tactile sensing has also been applied to material recognition and some other emerging applications such as slip detection, grasp stability assessment, identification of touch modalities and tactile servoing. The role of tactile sensing plays in sensor fusion has evolved from just verifying object-sensor contacts to extracting features to assist vision and provide detailed local information, and to transferring knowledge with vision. It is envisioned that tactile sensing will play a more important role in multiple applications. The development of a “sensor hierarchy" could be sought to establish the “preference" of sensor data. #### Deep learning in robotic tactile perception {#deep-learning-in-robotic-tactile-perception .unnumbered} After dominating the computer vision field, deep learning, especially CNNs, has also been applied in robotic tactile perception in recent years, from object shape recognition [@schmitz2014tactile; @cao2016efficient], hardness estimation [@yuan2017shape], to sharing features with vision [@gao2016deep; @yuan2017connecting]. When using deep learning, there are several issues need to be carefully taken care of. 1) ***Why deep learning?*** Deep learning can learn self-organising features instead of defining hand-crafted features a-priori and has shown considerable success in tasks such as object recognition. However, online application of such computationally intensive algorithm is difficult, especially for tasks need timely response such as grasping and manipulation, and it is hard to tune the parameters of deep architectures. On the other hand, it would be an overkill to put deep learning on small datasets and simple tasks; such algorithms can also potentially lose physical interpretation. 2) ***Data collection.*** Compared to datasets in vision, it is relatively expensive to collect tactile data, not only because of the required robot components but also the efforts that go into designing the data collection process. On the other hand, the same algorithm may achieve distinct performances on data collected from different tactile sensors. Due to such reasons, there are still no commonly used datasets for evaluating the processing algorithms in tactile sensing but a substantially large haptic dataset would benefit the field. 3) ***Labeling and ground truth.*** It is hard to obtain ground truth for tactile perception tasks using crowd-sourcing tools like Amazon Mechanical Turk in vision. In the current literature [@gao2016deep; @yuan2017connecting], tactile properties such as haptic adjectives (e.g., compressible or smooth) are labeled by limited number of human annotators which may bring bias to the ground truths. To this end, domain adaptation can be applied from vision to tactile sensing and unsupervised learning will be of benefit in the future research of tactile sensing. Acknowledgment {#acknowledgment .unnumbered} ============== The work presented in this paper was partially supported by the Engineering and Physical Sciences Council (EPSRC) Grant (Ref: EP/N020421/1), the King’s-China Scholarship Council PhD scholarship, the European Commission under grant agreements PITN–GA–2012–317488-CONTEST, and EPSRC Engineering Fellowship for Growth – Printable Tactile Skin (EP/M002527/1). [Shan Luo]{} is currently a Postdoctoral Research Fellow at Brigham and Women’s Hospital, Harvard Medical School. Previous to this position, he was a Research Fellow at University of Leeds and a Visiting Scientist at Computer Science and Artificial Intelligence Laboratory (CSAIL), Massachusetts Institute of Technology (MIT). He received the B.Eng. degree in Automatic Control from China University of Petroleum, Qingdao, China, in 2012. He was awarded the Ph.D. degree in Robotics from King’s College London, UK, in 2016. His research interests include tactile sensing, object recognition and computer vision. [Joao Bimbo]{} is currently a Postdoctoral Researcher at the Istituto Italiano di Tecnologia. He received his MSc. degree in Electrical Engineering from the University of Coimbra, Portugal, in 2011. He was awarded the Ph.D. degree in Robotics from King’s College London, UK, in 2016. His research interests include tactile sensing and teleoperation. [Ravinder Dahiya]{} is Professor and EPSRC Fellow in the School of Engineering at University of Glasgow, UK. He is Director of Electronics Systems Design Centre in University of Glasgow and leader of Bendable Electronics and Sensing Technologies (BEST) group. He completed Ph.D. at Italian Institute of Technology, Genoa (Italy). In past, he worked at Delhi University (India), Italian Institute of Technology, Genoa (Italy), Fondazione Bruno Kessler, Trento (Italy), and University of Cambridge (UK). His multidisciplinary research interests include Flexible and Printable Electronics, Electronic Skin, Tactile Sensing, robotics, and wearable electronics. He has published more than 160 research articles, 4 books (including 3 in various stage of publication) and 9 patents (including submitted). He has worked on and led many international projects. He is Distinguished Lecturer of IEEE Sensors Council and senior member of IEEE. Currently he is serving on the Editorial Boards of Scientific Reports (Nature Pub. Group), IEEE Transactions on Robotics and IEEE Sensors Journal and has been guest editor of 4 Special Journal Issues. He is member of the AdCom of IEEE Sensors Council and is the founding chair of the IEEE UKRI sensors council chapter. He was General Chair of IEEE PRIME 2015 and is the Technical Program Chair (TPC) of IEEE Sensors 2017. Prof. Dahiya holds prestigious EPSRC Fellowship and also received Marie Curie Fellowship and Japanese Monbusho Fellowship. He was awarded with the University Gold Medal and received best paper awards 2 times and another 2 second best paper awards (as co-author) in the IEEE international conferences. He has received numerous awards including the 2016 IEEE Sensors Council Technical Achievement award and 2016 Microelectronics Young Investigator Award. Dr. Dahiya was among Scottish 40under40 in 2016. [Hongbin Liu]{} is a Senior Lecturer (Associate Professor) in the Department of Informatics, King’s College London (KCL) where he is directing the Haptic Mechatronics and Medical Robotics Laboratory (HaMMeR) within the Centre for Robotics Research (CoRe). Dr. Liu obtained his BEng in 2005 from Northwestern Polytechnical University, China, MSc and PhD in 2006 and 2010 respectively, both from KCL. He is a Technical Committee Member of IEEE EMBS BioRobotics. He has published over 100 peer-reviewed publications at top international robotic journals and conferences. His research lies in creating the artificial haptic perception for robots with soft and compliant structures, and making use of haptic sensation to enable the robot to effectively physically interact with complex and changing environment. His research has been funded by EPSRC, Innovate UK, NHS Trust and EU Commissions. [^1]: ^\*^ Corresponding author. [^2]: S. Luo and H. Liu are with the Centre for Robotics Research, Department of Informatics, King’s College London, London WC2R 2LS, U.K. (e-mail: shan.luo, hongbin.liu@kcl.ac.uk). S. Luo is also with School of Civil Engineering and School of Computing, University of Leeds, Leeds LS2 9JT, U.K., and Brigham and Women’s Hospital, Harvard Medical School, 75 Francis St, Boston, MA 02115. [^3]: J. Bimbo is with the Department of Advanced Robotics, Istituto Italiano di Tecnologia, 16163 Genova, Italy (e-mail: joao.bimbo@iit.it). [^4]: R. S. Dahiya is with the Electronics and Nanoscale Engineering Research Division, University of Glasgow, Glasgow G12 8QQ, U.K. (e-mail: ravinder.dahiya@glasgow.ac.uk). [^5]: [^6]: [^7]: [^8]:

--- abstract: | Generalized Davenport-Schinzel sequences are sequences that avoid a forbidden subsequence and have a sparsity requirement on their letters. Upper bounds on the lengths of generalized Davenport-Schinzel sequences have been applied to a number of problems in discrete geometry and extremal combinatorics. Sharp bounds on the maximum lengths of generalized Davenport-Schinzel sequences are known for some families of forbidden subsequences, but in general there are only rough bounds on the maximum lengths of most generalized Davenport-Schinzel sequences. One method that was developed for finding upper bounds on the lengths of generalized Davenport-Schinzel sequences uses a family of sequences called formations. An $(r, s)$-formation is a concatenation of $s$ permutations of $r$ distinct letters. The formation width function ${\operatorname{fw}}(u)$ is defined as the minimum $s$ for which there exists $r$ such that every $(r, s)$-formation contains $u$. The function ${\operatorname{fw}}(u)$ has been used with upper bounds on extremal functions of $(r, s)$-formations to find tight bounds on the maximum possible lengths of many families of generalized Davenport-Schinzel sequences. Algorithms have been found for computing ${\operatorname{fw}}(u)$ for sequences $u$ of length $n$, but they have worst-case run time exponential in $n$, even for sequences $u$ with only three distinct letters. We present an algorithm for computing ${\operatorname{fw}}(u)$ with run time $O(n^{\alpha_r})$, where $r$ is the number of distinct letters in $u$ and $\alpha_r$ is a constant that only depends on $r$. We implement the new algorithm in Python and compare its run time to the next fastest algorithm for computing formation width. We also apply the new algorithm to find sharp upper bounds on the lengths of several families of generalized Davenport-Schinzel sequences with $3$-letter forbidden patterns. author: - | [**[Jesse Geneson]{}**]{}\ Iowa State University, Ames, IA 50011, USA\ [*geneson@gmail.com*]{} title: An algorithm for bounding extremal functions of forbidden sequences --- Introduction ============ Upper bounds on the lengths of generalized Davenport-Schinzel sequences have been used to bound the complexity of faces in arrangements of arcs with a limited number of pairwise crossings [@agsh], the complexity of unions of fat triangles [@petties], the complexity of lower envelopes of sets of polynomials of bounded degree [@DS], and the number of edges in $k$-quasiplanar graphs with no pair of edges intersecting in more than a bounded number of points [@fox]. The original Davenport-Schinzel sequences of order $s$, denoted $DS(n, s)$-sequences, are sequences with $n$ distinct letters and no adjacent same letters that avoid alternations of length $s+2$. Although generalized Davenport-Schinzel sequences are not well-understood in general, sharp bounds have been found on the maximum lengths of $DS(n,s)$ sequences for both fixed $s$ and $s = \Omega(n)$ [@niv; @pettie3; @welpet; @gends]. In [@gpt], we developed an algorithm for obtaining upper bounds on the lengths of generalized Davenport-Schinzel sequences, giving sharp bounds for many new families of forbidden patterns. We say that a sequence $u$ *contains* a sequence $v$ if some subsequence of $u$ is isomorphic to $v$. Otherwise $u$ *avoids* $v$. A sequence is called *$r$-sparse* if every contiguous subsequence of length $r$ has no repeated letters. If $u$ has $r$ distinct letters, then $\mathit{Ex}(u, n)$ is the maximum possible length of an $r$-sparse sequence that avoids $u$ with $n$ distinct letters. For example, the maximum possible length of a $DS(n, s)$-sequence is $\mathit{Ex}(u, n)$ when $u$ is an alternation of length $s+2$. Other than a few families of forbidden sequences $u$ with sharp bounds, only rough bounds are known for $\mathit{Ex}(u, n)$ in general. An *$(r, s)$-formation* is a concatenation of $s$ permutations of $r$ distinct letters. Formations were introduced in [@klazar1] as a tool for obtaining general upper bounds on $\mathit{Ex}(u, n)$. The extremal function ${\operatorname{F}}_{r, s}(n)$ is defined as the maximum length of an $r$-sparse sequence with $n$ distinct letters that avoids all $(r, s)$-formations. Klazar proved that if $u$ has length $s$ and $r$ distinct letters, then every $(r, s-1)$-formation contains $u$, so ${\operatorname{Ex}}(u, n) \leq {\operatorname{F}}_{r, s-1}(n)$. Nivasch improved the bound by proving that every $(r, s-r+1)$-formation contains $u$, so ${\operatorname{Ex}}(u, n) \leq {\operatorname{F}}_{r, s-r+1}(n)$ [@niv]. Nivasch also proved that ${\operatorname{F}}_{r, 2t-1}(n) = n2^{\frac{1}{(t-2)!}\alpha(n)^{t-2} \pm O(\alpha(n)^{t-3})}$ and ${\operatorname{Ex}}((a b)^t, n) = n2^{\frac{1}{(t-2)!}\alpha(n)^{t-2} \pm O(\alpha(n)^{t-3})}$ for all $r \geq 2$ and $t\geq 3$. The results of [@agshsh; @klazar1; @pettie; @pettie3] give that ${\operatorname{F}}_{r, 4}(n) = \Theta(n \alpha(n))$, ${\operatorname{Ex}}(ababa, n) = \Theta(n \alpha(n))$, ${\operatorname{Ex}}(abababa, n) = \Theta(n \alpha(n) 2^{\alpha(n)})$, and ${\operatorname{Ex}}((ab)^t a, n) = n2^{\frac{1}{(t-2)!}\alpha(n)^{t-2} \pm O(\alpha(n)^{t-3})}$ for all $t\geq 4$. Pettie [@pettie3] proved that ${\operatorname{F}}_{2, 6}(n) = \Theta(n \alpha(n) 2^{\alpha(n)})$ and ${\operatorname{F}}_{2, 2t}(n) = n2^{\frac{1}{(t-2)!}\alpha(n)^{t-2} \pm O(\alpha(n)^{t-3})}$ for all $t\geq 4$, but ${\operatorname{F}}_{r, 2t}(n) = n2^{\frac{1}{(t-2)!}\alpha(n)^{t-2}(\log{\alpha(n)}\pm O(1))}$ for all $r, t\geq 3$. In [@gpt], we defined a function called the [*formation width*]{} of $u$, denoted by ${\operatorname{fw}}(u)$, to be the minimum value of $s$ such that there exists an $r$ for which every $(r, s)$-formation contains $u$. We also defined the *formation length* of $u$, denoted by ${\operatorname{fl}}(u)$, to be the minimum value of $r$ for which every $(r, {\operatorname{fw}}(u))$-formation contains $u$. We found that ${\operatorname{Ex}}(u, n) = O({\operatorname{F}}_{{\operatorname{fl}}(u),{\operatorname{fw}}(u)}(n))$ for all sequences $u$ with $r$ distinct letters [@gpt], so we developed an algorithm to compute $\mathit{fw}(u)$ and we used $\mathit{fw}(u)$ to prove sharp bounds on $\mathit{Ex}(u, n)$ for several families of forbidden sequences $u$. We found that $\mathit{fw}((1 2 \ldots l)^{t})=2t-1$ for all $l\geq 2$ and $t\geq 1$, and thus $\mathit{Ex}((1 2 \ldots l)^{t}, n)=n2^{\frac{1}{(t-2)!}\alpha(n)^{t-2}\pm O(\alpha(n)^{t-3})}$ for all $l \geq 2$ and $t\geq 3$. This improved the upper bound from [@fox] on the number of edges in $k$-quasiplanar graphs with no pair of edges intersecting in more than $O(1)$ points. In [@gtseq], we developed a faster algorithm for computing formation width and used it to identify every sequence $u$ for which $u$ contains $ababa$ and $\mathit{fw}(u) = 4$. As a corollary, we showed that $\mathit{Ex}(u, n)=\Theta(n\alpha(n))$ for every such sequence $u$, which is the same as $\mathit{Ex}(a b a b a, n)$ up to a constant factor. Define a sequence $u$ to be *minimally non-linear* if ${\operatorname{Ex}}(u, n) = \omega(n)$ but ${\operatorname{Ex}}(u', n) = O(n)$ for all sequences $u'$ properly contained in $u$. Pettie [@petmnl] posed the problem of determining every minimally non-linear sequence $u$. Every known minimally non-linear sequence $u$ has formation width $4$, so identifying sequences with formation width $4$ provides candidates for other minimally non-linear sequences. In [@GT1], we used $\mathit{fw}(u)$ to find tight bounds on the lengths of generalized Davenport-Schinzel sequences that avoid $a b c (a c b)^t a b c$, answering a problem from [@gpt], as well as generalized Davenport-Schinzel sequences that avoid $a b c a c b (a b c)^t a c b$. In this paper, we present an algorithm for computing ${\operatorname{fw}}(u)$ for sequences $u$ with at most $r$ distinct letters that has run time bounded by a polynomial in the length of $u$, where the degree of the polynomial depends on $r$. We also include an implementation of the algorithm in Python, and we compare its run time to the previous fastest algorithm for computing formation width. Furthermore, we apply the algorithm to obtain sharp bounds on ${\operatorname{Ex}}(u, n)$ for sequences $u$ with $3$ distinct letters that contain alternations of length $6$, $7$, $8$, or $9$. Given an alternation $u = (a b)^t$ of even length, let $u'$ be any sequence obtained by replacing each $b$ in $u$ with $bc$ or $cb$. So $(a b c)^t$ and $(a b c)^{t-1} a c b$ are two of the $2^t$ possibilities for $u'$. One might expect that ${\operatorname{Ex}}(u', n) = \omega({\operatorname{Ex}}(u, n))$ in general, but we found in [@gpt] that ${\operatorname{fw}}((a b c)^t) = {\operatorname{fw}}((a b c)^{t-1} a c b) = {\operatorname{fw}}((a b)^t) = 2t-1$, which implies that ${\operatorname{Ex}}((a b c)^t, n) = n2^{\frac{1}{(t-2)!}\alpha(n)^{t-2}\pm O(\alpha(n)^{t-3})}$ and ${\operatorname{Ex}}((a b c)^{t-1} a c b, n) = n2^{\frac{1}{(t-2)!}\alpha(n)^{t-2}\pm O(\alpha(n)^{t-3})}$ for $t \geq 3$. A natural problem is to identify the sequences $u'$ for which ${\operatorname{Ex}}(u', n) = \Theta({\operatorname{Ex}}(u, n))$, as well as the more open-ended problem of identifying the sequences $u'$ for which ${\operatorname{Ex}}(u', n) \approx {\operatorname{Ex}}(u, n)$. The latter problem makes more sense than the former for $t \geq 4$, since the bounds on ${\operatorname{Ex}}((ab)^t, n)$ for $t \geq 4$ are not tight up to a constant factor. We approach this problem by using the new formation width algorithm to determine the sequences $u'$ for which ${\operatorname{fw}}(u') = {\operatorname{fw}}(u)$ for $t \leq 12$, and proving the next theorem for all $t \geq 3$. \[mainth\] If $u$ is one of the sequences $(a b c)^{t}$, $a b c (a c b)^{t-1}$, $a b c (a c b)^{t-2} a b c$, $a b c a c b (a b c)^{t-3} a c b$, $a b c a c b (a b c)^{t-2}$, $a b c a b c (a c b)^{t-2}$, $(a b c)^{t-2} a c b a c b$, $(a b c)^{t-2} a c b a b c$, or $(a b c)^{t-1} a c b$, then ${\operatorname{fw}}(u) = {\operatorname{fw}}((a b)^t)$ and ${\operatorname{Ex}}(u, n) = n2^{\frac{1}{(t-2)!}\alpha(n)^{t-2}\pm O(\alpha(n)^{t-3})}$ for all $t\geq 3$. Algorithms for ${\operatorname{fw}}(u)$ ======================================= There are three algorithms for formation width including the new one, which we call *BinaryFormation*, *FormationTree*, and *PermutationVector*. As mentioned in the introduction, ${\operatorname{fw}}(u)$ is defined as the minimum value of $s$ for which there exists an $r$ such that every $(r, s)$-formation contains $u$. From the definition, it might seem like we need to check all integers $r > 0$ to know that there is no $(r, s)$-formation that contains $u$ for a given $s$. However, we showed in the original paper on ${\operatorname{fw}}(u)$ that only finitely many values of $r$ need to be checked for each $s$ [@gpt]. This gives the first algorithm for computing formation width. BinaryFormation --------------- To state the next result, we define an $(r, s)$-formation $f$ to be *binary* if there exists a permutation $p$ on $r$ letters such that every permutation in $f$ is either equal to $p$ or the reverse of $p$. [@gpt] For any sequence $u$ with $r$ distinct letters, ${\operatorname{fw}}(u)$ is the minimum value of $s$ for which every binary $(r, s)$-formation contains $u$. This theorem gives an obvious algorithm for computing ${\operatorname{fw}}(u)$: 1. Let $r$ be the number of distinct letters in $u$ and let $s = 0$. 2. Check if every binary $(r, s)$-formation contains $u$. If so, return $s$. If not, increment $s$ and repeat this step. In the worst case, BinaryFormation has exponential run time in the length of $u$, so it is not practical for computing formation width of long sequences. In [@gpt] we showed that ${\operatorname{fw}}(u) = t$ for any sequence $u$ of length $t+1$ with two distinct letters. For sequences $u$ with three distinct letters, no polynomial time algorithm was known for computing ${\operatorname{fw}}(u)$. FormationTree ------------- In [@gtseq], we found a faster algorithm to compute ${\operatorname{fw}}(u)$, which allowed us to find all sequences $u$ for which ${\operatorname{fw}}(u) = 4$ and $u$ contains $a b a b a$. This answered a question that we asked in [@gpt]. The faster algorithm uses the binary tree structure of binary $(r, s)$-formations, where the root of the tree is the binary $(r, 1)$-formation with first permutation $1 \dots r$, and the binary $(r, s)$-formation $f$ is the parent of $f 1 \dots r$ and $f r \dots 1$. The main new idea of this algorithm was that we do not have to check the descendants of a binary $(r, s)$-formation $f$ once we know that $f$ contains $u$. 1. Let $r$ be the number of distinct letters in $u$ and let $s = 1$. If $u$ has length $0$, return $0$. 2. Let Fset be the set that contains only the $(r, 1)$-formation with first permutation $1 \dots r$. 3. Check if every binary $(r, s)$-formation in Fset contains $u$. If so, return $s$. If not, construct two new binary $(r, s+1)$-formations $f 1 \dots r$ and $f r \dots 1$ for each $f$ in Fset that avoids $u$, let Fset be the set of newly constructed formations, increment $s$, and repeat this step. Although FormationTree is much faster than BinaryFormation in practice, in the worst case it still has exponential run time in the length of $u$. For example, it is easy to see immediately that ${\operatorname{fw}}((a b c)^t) = 2t-1$ using the pigeonhole principle for the upper bound and an alternating binary formation for the lower bound, but this is very slow to compute with the FormationTree algorithm since we do not find any binary $(3, s)$-formations that contain $(a b c)^t$ until $s = t$, at which point we are checking for containment of $(a b c)^t$ in $2^{t-1}$ binary formations. PermutationVector ----------------- Given a sequence $u$ with $r$ distinct letters $1, \dots, r$ and length $n$, PermutationVector maintains a dynamic set of vectors, where each vector has $r!$ entries corresponding to permutations of the distinct letters of $u$ and each entry has a value between $1$ and $n$. Each round, we modify the set of vectors, and the output of ${\operatorname{fw}}(u)$ is the number of rounds it takes for the set of vectors to become empty. In this algorithm, we no longer keep track of any formations as we increment $s$, since there are exponentially many binary formations in terms of $s$. Instead, for each binary formation $f$ on the same letters as $u$, we define a vector $p_f(u)$ with an entry for each permutation $\pi$ of the distinct letters of $u$. If $v$ is the sequence obtained from applying $\pi$ to the letters of $u$, then the value of the $\pi$ entry of $p_f(u)$ is the maximum length of an initial segment of $v$ that is a subsequence of $f$. There are at most $n^{r!}$ possible vectors in any round of the algorithm. We can tell whether $f$ contains $u$ by checking if any of the entries of $p_f(u)$ is equal to the length of $u$. Given a $p_f(u)$ with no entries equal to the length of $u$, we can compute $p_{f 1 \dots r}(u)$: if $v$ is the sequence obtained from applying the permutation $\pi$ to the letters of $u$, we use the $\pi$ entry of $p_f(u)$ to determine the longest initial segment $v^{*}$ of $v$ that is a subsequence of $f$ and we let $v'$ be obtained from $v$ by removing $v^{*}$, and then we find the longest initial segment of $v'$ that is a subsequence of $1 \dots r$ and we add its length to the length of $v^{*}$ to get the $\pi$ entry of $p_{f 1 \dots r}(u)$. We also compute $p_{f r \dots 1}(u)$ similarly: we find the longest initial segment of $v'$ that is a subsequence of $r \dots 1$, and we add its length to the length of $v^{*}$ to obtain the $\pi$ entry of $p_{f r \dots 1}(u)$. 1. Let $r$ be the number of distinct letters in $u$ and let $s = 1$. If $u$ has length $0$, return $0$. 2. Let Vset be the set that contains only $p_{1 \dots r}(u)$. 3. Check if every vector in Vset has an entry equal to the length of $u$. If so, return $s$. If not, delete any vectors with entries equal to the length of $u$. For any vector $p_f(u)$ with no entries equal to the length of $u$, construct the two new vectors $p_{f 1 \dots r}(u)$ and $p_{f r \dots 1}(u)$ as described, let Vset be the set of newly constructed permutation vectors, increment $s$, and repeat this step. At each value of $s$ from $1$ to ${\operatorname{fw}}(u)$, we only check and replace at most $n^{r!}$ vectors, each with $r!$ entries. Computing each entry of $p_{f 1 \dots r}(u)$ and $p_{f r \dots 1}(u)$ from the corresponding entry in $p_f(u)$ takes $O(n)$ run time per entry, since finding the longest initial segment of $v'$ that is a subsequence of $1 \dots r$ or $r \dots 1$ takes $O(r)$ run time, and adding its length to the length of $v^{*}$ takes $O(\log{n})$ run time. Thus the run time is $O(n^{r!+2})$ when $r = O(1)$. Comparison of FormationTree and PermutationVector ------------------------------------------------- We compared the run times of FormationTree and PermutationVector on sequences $u$ with three and four distinct letters. We implemented PermutationVector in Python [@pvpy], and we ran this against the Python implementation of FormationTree from [@gtseq]. The computations were performed on an ASUS TUF Gaming FX504 with a 2.30GHz Intel i5-8300H CPU and 8GB of RAM, running Python 3.7.1 on Windows 10. We ran the two algorithms on sequences of the form $(a b c)^t$ and $(a b c d)^t$ for $1 \leq t \leq 10$. For each sequence, we performed twenty trials of each of the algorithms and computed the mean run time. FormationTree was faster for four of the sequences: $a b c$, $a b c a b c$, $a b c d$, and $a b c d a b c d$. For all other sequences, PermutationVector was faster. For computing ${\operatorname{fw}}((a b c)^{10})$, PermutationVector was almost $300$ times faster than FormationTree. \[uniform1000\] Sequence Mean (FT) Mean (PV) ------------------ ----------- ----------- -- -- $(a b c)$ 0.000004 0.000014 $(a b c)^2$ 0.000056 0.000067 $(a b c)^3$ 0.00035 0.000229 $(a b c)^4$ 0.001525 0.00061 $(a b c)^5$ 0.006406 0.001413 $(a b c)^6$ 0.026046 0.002916 $(a b c)^7$ 0.104506 0.005198 $(a b c)^8$ 0.403432 0.008571 $(a b c)^9$ 1.5426 0.013164 $(a b c)^{10}$ 6.197161 0.021925 $(a b c d)$ 0.000005 0.000065 $(a b c d)^2$ 0.000211 0.000316 $(a b c d)^3$ 0.001483 0.001117 $(a b c d)^4$ 0.007846 0.003879 $(a b c d)^5$ 0.038453 0.012018 $(a b c d)^6$ 0.171947 0.032365 $(a b c d)^7$ 0.752102 0.080562 $(a b c d)^8$ 3.189359 0.186831 $(a b c d)^9$ 13.48486 0.476468 $(a b c d)^{10}$ 60.345885 0.778753 : FormationTree (FT) versus PermuationVector (PV): comparison of run time in seconds for sequences of the form $(a b c)^t$ and $(a b c d)^t$ Applications ============ We computed all sequences with $3$ distinct letters that have formation width $x$ and alternation length $x+1$ for each $x = 5, 6, 7, 8$ using PermutationVector. These sequences are in the appendices \[x = 5\], \[x = 6\], \[x = 7\], and \[x = 8\], all on the alphabet $0, 1, 2$ with the letters making first appearances in that order. Combining ${\operatorname{Ex}}(u, n) = O({\operatorname{F}}_{{\operatorname{fl}}(u),{\operatorname{fw}}(u)}(n))$ with the bounds on ${\operatorname{F}}_{r, s}(n)$, we obtain the following theorem. 1. For all of the sequences $u$ in Appendix \[x = 5\], ${\operatorname{Ex}}(u, n) = \Theta(n 2^{\alpha(n)})$. 2. For all of the sequences $u$ in Appendix \[x = 6\], ${\operatorname{Ex}}(u, n) = \Theta(n \alpha(n) 2^{\alpha(n)})$. 3. For all of the sequences $u$ in Appendix \[x = 7\], ${\operatorname{Ex}}(u, n) = n 2^{\frac{\alpha(n)^2}{2}\pm O(\alpha(n))}$. 4. For all of the sequences $u$ in Appendix \[x = 8\], ${\operatorname{Ex}}(u, n) = O(n2^{\frac{\alpha(n)^{2}}{2}(\log{\alpha(n)}+O(1))})$ and ${\operatorname{Ex}}(u, n) = \Omega(n2^{\frac{\alpha(n)^2}{2} - O(\alpha(n))})$. In addition, we computed ${\operatorname{fw}}(u)$ for the more restricted family of sequences $u$ obtained by concatenating copies of $a b c$ and $a c b$, where the first permutation is $a b c$. We used PermutationVector to find all such sequences $u$ that have formation width $x$ and alternation length $x+1$ for $x = 5, 7, 9, \dots, 23$. They are listed in Appendix \[abcform\]. For each $x = 2t+5 \geq 11$, there are nine sequences $u$ that have formation width $x$ and alternation length $x+1$ such that $u$ is obtained by concatenating copies of $a b c$ and $a c b$, where the first permutation is $a b c$. These sequences are $(a b c)^{t+3}$, $a b c (a c b)^{t+2}$, $a b c (a c b)^{t+1} a b c$, $a b c a c b (a b c)^{t} a c b$, $a b c a c b (a b c)^{t+1}$, $a b c a b c (a c b)^{t+1}$, $(a b c)^{t+1} a c b a c b$, $(a b c)^{t+1} a c b a b c$, and $(a b c)^{t+2} a c b$. For $x = 5$ and $x = 7$, there are four and eight sequences respectively (all possible sequences obtained by concatenating copies of $a b c$ and $a c b$, where the first permutation is $a b c$). Interestingly, for $x = 9$ there are ten such sequences, even though there are nine for $x > 9$. Nine of the sequences for $x = 9$ have the same form as the nine sequences for each $x > 9$; the extra sequence for $x = 9$ is $a b c a b c a c b a c b a b c$. In [@gpt], we showed that ${\operatorname{fw}}((a b c)^t) = 2t-1$ and ${\operatorname{fw}}(a b c (a c b)^{t}) = 2t+1$ for $t \geq 0$. Furthermore in [@GT1] we proved that ${\operatorname{fw}}(a b c (a c b)^{t} a b c) = 2t+3$ and ${\operatorname{fw}}(a b c a c b (a b c)^t a c b) = 2t+5$ for $t \geq 0$. We use a computational method to handle the five other cases that appear for each $x > 9$ in Appendix \[abcform\], automating much of the proof with a Python script [@acbpy]. Also we note that the next result implies Theorem \[mainth\] as a corollary. \[halfmhalfc\] If $t \geq 1$ and $u$ is one of the sequences $a b c a c b (a b c)^{t}$, $a b c a b c (a c b)^{t}$, $(a b c)^{t} a c b a c b$, $(a b c)^{t} a c b a b c$, or $(a b c)^{t+1} a c b$, then ${\operatorname{fw}}(u) = 2t+3$. The lower bound ${\operatorname{fw}}(u) \geq 2t+3$ is immediate, so we prove that ${\operatorname{fw}}(u) \leq 2t+3$. By symmetry, it suffices to prove for $t \geq 1$ that ${\operatorname{fw}}(u) \leq 2t+3$ for each sequence $u = (a b c)^{t}b a c a b c$, $(a b c)^{t} b a c b a c$, $(a b c)^{t} a c b a c b$, $(a b c)^{t} a c b a b c$, or $(a b c)^{t+1} a c b$. Let $f$ be any binary $(3, 2t+3)$-formation with permutations $x y z$ and $z y x$. Then the first $2t-1$ permutations of $f$ have the subsequence $(x y z)^t$ or $(z y x)^t$. Without loss of generality, suppose the first $2t-1$ permutations of $f$ have the subsequence $(x y z)^t$. We consider two cases. For the first case, suppose that the first $2t-1$ permutations of $f$ have the subsequence $(x y z)^{t+1}$. This immmediately implies that $f$ has the subsequences $(x y z)^t x z y x z y$, $(x y z)^t x z y x y z$, and $(x y z)^{t+1} x z y$. Moreover, $f$ has the subsequence $(x y z)^t y x z x y z$ unless its last $12$ letters are $z y x x y z z y x z y x$, in which case it has the subsequence $(y z x)^t z y x y z x$. Furthermore, $f$ has the subsequence $(x y z)^t y x z y x z$ unless its last $12$ letters are $z y x x y z x y z z y x$, in which case it has the subsequence $(y z x)^t z y x z y x$. Thus we have shown that any binary $(3, 2t+3)$-formation that has the subsequence $(x y z)^{t+1}$ in its first $2t-1$ permutations contains $u$ for each $u = (a b c)^{t}b a c a b c$, $(a b c)^{t} b a c b a c$, $(a b c)^{t} a c b a c b$, $(a b c)^{t} a c b a b c$, or $(a b c)^{t+1} a c b$. For the second case, suppose that the first $2t-1$ permutations of $f$ have the subsequence $(x y z)^t$ and the subsequence $(z y x)^{t-1}$. We handle each sequence with a simple computation [@acbpy] in Python described below. 1. For $u = (a b c)^{t}b a c a b c$, we use [@acbpy] to verify that every binary $(3, 4)$-formation on the permutations $xyz$ and $zyx$ except for $xyzxyzzyxzyx$ has at least one of the following subsequences: $y x z x y z$, $x z y x y z x$, $x y x z y z x y$, $z y x y z x z y x$, $z y x z x y z y x z$, $z y x z y z x y x z y$. So we may assume that the last $12$ letters of $f$ are $xyzxyzzyxzyx$. If permutation $2t-1$ is $zyx$, then $f$ has the subsequence $(x y z)^t y x z x y z$. If permutation $2t-1$ is $x y z$, then $f$ has the subsequence $(z y x)^t y z x z y x$. 2. For $u = (a b c)^t b a c b a c$, we use [@acbpy] to verify that every binary $(3, 4)$-formation on the permutations $xyz$ and $zyx$ except for $xyz zyx xyz xyz$ has at least one of the following subsequences: $y x z y x z$, $x z y x z y x$, $x y x z y x z y$, $z y x y z x y z x$, $z y x z x y z x y z$, $z y x z y z x y z x y$. So we may assume that the last $12$ letters of $f$ are $xyz zyx xyz xyz$. If permutation $1$ is $zyx$, then $f$ has the subsequence $(z x y)^t x z y x z y$. If permutation $2t-1$ is $z y x$, then $f$ has the subsequence $(x y z)^t y x z y x z$. If permutations $1$ and $2t-1$ are both $x y z$, then $f$ has the subsequence $(x z y)^t z x y z x y$. 3. For $u = (a b c)^t a c b a c b$, we use [@acbpy] to verify that every binary $(3, 4)$-formation on the permutations $xyz$ and $zyx$ except for $zyx xyz zyx xyz$ and $zyx xyz xyz zyx$ has at least one of the following subsequences: $x z y x z y$, $x y x z y x z$, $x y z y x z y x$, $z y x z x y z x y$, $z y x z y z x y z x$, $z y x z y x y z x y z$. So we may assume that the last $12$ letters of $f$ are $zyx xyz zyx xyz$ or $zyx xyz xyz zyx$. If permutation $1$ is $zyx$, then in both cases $f$ has the subsequence $(z x y)^t z y x z y x$. If permutation $1$ is $xyz$, then in both cases $f$ has the subsequence $(x z y)^t x y z x y z$. 4. For $u = (a b c)^t a c b a b c$, we use [@acbpy] to verify that every binary $(3, 4)$-formation on the permutations $xyz$ and $zyx$ except for $zyx xyz zyx zyx$ has at least one of the following subsequences: $x z y x y z$, $x y x z y z x$, $x y z y x z x y$, $z y x z x y z y x$, $z y x z y z x y x z$, $z y x z y x y z x z y$. So we may assume that the last $12$ letters of $f$ are $zyx xyz zyx zyx$. If permutation $1$ is $zyx$, then $f$ has the subsequence $(z x y)^t z y x z x y$. If permutation $1$ is $x y z$, then $f$ has the subsequence $(x z y)^t x y z x z y$. 5. For $u = (a b c)^{t+1} a c b$, we use [@acbpy] to verify that every binary $(3, 4)$-formation on the permutations $xyz$ and $zyx$ except for $zyx zyx xyz zyx$ and $zyx xyz zyx xyz$ has at least one of the following subsequences: $x y z x z y$, $x y z x y x z$, $x y z x y z y x$, $z y x z y x z x y$, $z y x z y x z y z x$, $z y x z y x z y x y z$. So we may assume that the last $12$ letters of $f$ are $zyx zyx xyz zyx$ or $zyx xyz zyx xyz$. If permutation $1$ is $zyx$, then in both cases $f$ has the subsequence $(z x y)^{t+1} z y x$. If permutation $1$ is $x y z$, then in both cases $f$ has the subsequence $(x z y)^{t+1} x y z$. We finish with a conjecture based on Proposition \[halfmhalfc\] and the evidence in Appendix \[abcform\]. \[acbconj\] Suppose that $t \geq 6$. Among $(3, t)$-formations $u$ with first permutation $a b c$ and other permutations equal to $a b c$ or $a c b$, ${\operatorname{fw}}(u) = 2t-1$ if and only if $u$ is $(a b c)^{t}$, $a b c (a c b)^{t-1}$, $a b c (a c b)^{t-2} a b c$, $a b c a c b (a b c)^{t-3} a c b$, $a b c a c b (a b c)^{t-2}$, $a b c a b c (a c b)^{t-2}$, $(a b c)^{t-2} a c b a c b$, $(a b c)^{t-2} a c b a b c$, or $(a b c)^{t-1} a c b$. For any fixed $r$, PermutationVector has worst-case run time polynomial in $n$ for all sequences of length $n$ with at most $r$ letters, where the degree of the polynomial depends on $r$. However, PermutationVector can be slower than FormationTree when $n$ is not sufficiently large relative to $r$, as we saw with the sequences $abc$, $abcd$, $abcabc$, and $abcdabcd$. Clearly PermutationVector does not have run time polynomial in $n$ when $r = \Theta(n)$, since the vectors in the algorithm each have $r!$ entries. Is there an algorithm for computing ${\operatorname{fw}}(u)$ that has worst-case run time polynomial in $n$ for all sequences $u$ of length $n$, regardless of the number of distinct letters in $u$? P. Agarwal, M. Sharir, and P. Shor. Sharp upper and lower bounds on the length of general Davenport-Schinzel sequences. J. Combin. Theory Ser. A, 52:228-274, 1989. J. Cibulka and J. Kynčl. Tight bounds on the maximum size of a set of permutations with bounded vc-dimension. Journal of Combinatorial Theory Series A, 119: 1461-1478, 2012. H. Davenport and A. Schinzel. A combinatorial problem connected with differential equations. American J. Mathematics, 87:684-694, 1965. J. Fox, J. Pach, and A. Suk. The number of edges in k-quasiplanar graphs. SIAM Journal of Discrete Mathematics, 27:550-561, 2013. J. Geneson, A Relationship Between Generalized Davenport-Schinzel Sequences and Interval Chains. Electr. J. Comb. 22(3): P3.19, 2015. J. Geneson, Constructing sparse Davenport-Schinzel sequences. CoRR abs/1810.07175. J. Geneson, Forbidden formations in multidimensional 0-1 matrices. Eur. J. Comb. 78: 147-154, 2019. J. Geneson, Python code (fwpv.py) for the FormationTree and PermutationVector algorithms. Available at https://github.com/jgeneson/permutation-vector. J. Geneson, Python code (binf.py) for proof of Proposition \[halfmhalfc\] in ’An algorithm for bounding extremal functions of forbidden sequences’. Available at https://github.com/jgeneson/permutation-vector. J. Geneson, R. Prasad, and J. Tidor, Bounding Sequence Extremal Functions with Formations. Electr. J. Comb. 21(3): P3.24, 2014. J. Geneson and P. Tian, Formations and generalized Davenport-Schinzel sequences. CoRR abs/1909.10330, J. Geneson and P. Tian, Sequences of formation width 4 and alternation length 5. CoRR abs/1502.04095. M. Klazar. Generalized Davenport-Schinzel sequences: results, problems, and applications. Integers, 2:A11, 2002. M. Klazar. A general upper bound in the extremal theory of sequences. Commentationes Mathematicae Universitatis Carolinae, 33:737-746, 1992. G. Nivasch. Improved bounds and new techniques for Davenport-Schinzel sequences and their generalizations. J. ACM, 57(3), 2010. S. Pettie, Generalized Davenport-Schinzel sequences and their 0-1 matrix counterparts, J. Combin. Theory Ser. A 118 (2011) 1863-1895 S. Pettie. On the Structure and Composition of Forbidden Sequences, with Geometric Applications. SoCG 2011. S. Pettie. Sharp bounds on Davenport-Schinzel sequences of every order. J. ACM, 62(5):36, 2015. S. Pettie. Three generalizations of Davenport-Schinzel sequences. SIAM J. Discrete Mathematics, 29(4):2189-2238, 2015. M. Sharir and P. Agarwal. Davenport-Schinzel Sequences and their Geometric Applications. Cambridge University Press, 1995. J. Wellman and S. Pettie, Lower bounds on Davenport-Schinzel sequences via rectangular Zarankiewicz matrices. Discrete Mathematics 341(7): 1987-1993, 2018. $3$-letter sequences with formation width $5$ and alternation length $6$ {#x = 5} ======================================================================== 0101012, 0101021, 0101201, 0102101, 0120101, 0120202, 0121212, 01012012, 01012021, 01012201, 01021012, 01021021, 01021201, 01022101, 01201012, 01201021, 01201201, 01201202, 01201212, 01202012, 01202021, 01202101, 01202102, 01210202, 01210212, 01212012, 01212102, 01220101, 012012012, 012012021, 012012102, 012021012, 012021021, 012021102, 012102012, 012102102, 012201021 $3$-letter sequences with formation width $6$ and alternation length $7$ {#x = 6} ======================================================================== 01010102, 01010120, 01010210, 01012010, 01021010, 01201010, 01202020, 01212121, 010102102, 010102120, 010102210, 010120102, 010120120, 010120210, 010122010, 010210102, 010210120, 010210210, 010212010, 010221010, 012010102, 012010120, 012010210, 012012010, 012012020, 012012121, 012020120, 012020210, 012021010, 012021020, 012102020, 012102121, 012120121, 012121021, 012121210, 012201010, 0102102102, 0102102120, 0102120102, 0102120120, 0102210120, 0120102102, 0120102120, 0120102210, 0120120102, 0120120120, 0120120121, 0120120210, 0120121020, 0120121021, 0120121210, 0120210120, 0120210210, 0120211020, 0121020120, 0121020121, 0121021020, 0121021021, 0121210021, 0121210210, 0122010102, 0122010210 $3$-letter sequences with formation width $7$ and alternation length $8$ {#x = 7} ======================================================================== 010101012, 010101021, 010101201, 010102101, 010120101, 010210101, 012010101, 012020202, 012121212, 0101012012, 0101012021, 0101012201, 0101021012, 0101021021, 0101021201, 0101022101, 0101201012, 0101201021, 0101201201, 0101202101, 0101220101, 0102101012, 0102101021, 0102101201, 0102102101, 0102120101, 0102210101, 0120101012, 0120101021, 0120101201, 0120102101, 0120120101, 0120120202, 0120121212, 0120201202, 0120202012, 0120202021, 0120202102, 0120210101, 0120210202, 0121020202, 0121021212, 0121201212, 0121210212, 0121212012, 0121212102, 0122010101, 01012012012, 01012012021, 01012021012, 01012021021, 01012201021, 01021012012, 01021012021, 01021012201, 01021021012, 01021021021, 01021021201, 01021201021, 01021201201, 01022101012, 01022101201, 01201012012, 01201012021, 01201012201, 01201021012, 01201021021, 01201021201, 01201022101, 01201201012, 01201201021, 01201201201, 01201201202, 01201201212, 01201202012, 01201202021, 01201202101, 01201202102, 01201210202, 01201210212, 01201212012, 01201212102, 01202012012, 01202012021, 01202012102, 01202021012, 01202021021, 01202021102, 01202101012, 01202101021, 01202101201, 01202101202, 01202102012, 01202102021, 01202102101, 01202102102, 01202110202, 01210201202, 01210201212, 01210202012, 01210202102, 01210210202, 01210210212, 01210212012, 01210212102, 01212012012, 01212012102, 01212100212, 01212102012, 01212102102, 01220101021, 01220101201, 01220102101, 012012012012, 012012012021, 012012012102, 012012021012, 012012021021, 012012021102, 012012102012, 012012102102, 012021012012, 012021012021, 012021021012, 012021021021, 012102012012, 012102012102, 012102102012, 012102102102, 012201021021 $3$-letter sequences with formation width $8$ and alternation length $9$ {#x = 8} ======================================================================== 010101010, 0101010102, 0101010120, 0101010210, 0101012010, 0101021010, 0101201010, 0102101010, 0120101010, 0120202020, 0121212121, 01010102102, 01010102120, 01010102210, 01010120102, 01010120120, 01010120210, 01010122010, 01010210102, 01010210120, 01010210210, 01010212010, 01010221010, 01012010102, 01012010120, 01012010210, 01012012010, 01012021010, 01012201010, 01021010102, 01021010120, 01021010210, 01021012010, 01021021010, 01021201010, 01022101010, 01201010102, 01201010120, 01201010210, 01201012010, 01201021010, 01201201010, 01201202020, 01201212121, 01202012020, 01202020120, 01202020210, 01202021020, 01202101010, 01202102020, 01210202020, 01210212121, 01212012121, 01212102121, 01212120121, 01212121021, 01212121210, 01220101010, 010102102102, 010102102120, 010102120102, 010102120120, 010102210120, 010120102102, 010120102120, 010120102210, 010120120102, 010120120120, 010120120210, 010120210120, 010120210210, 010122010102, 010122010210, 010210102102, 010210102120, 010210102210, 010210120102, 010210120120, 010210120210, 010210122010, 010210210102, 010210210120, 010210210210, 010210212010, 010212010102, 010212010120, 010212010210, 010212012010, 010221010120, 010221010210, 010221012010, 012010102102, 012010102120, 012010102210, 012010120102, 012010120120, 012010120210, 012010122010, 012010210102, 012010210120, 012010210210, 012010212010, 012010221010, 012012010102, 012012010120, 012012010210, 012012012010, 012012012020, 012012012121, 012012020120, 012012020210, 012012021010, 012012021020, 012012102020, 012012102121, 012012120121, 012012121021, 012012121210, 012020120120, 012020120210, 012020121020, 012020210120, 012020210210, 012020211020, 012021010120, 012021010210, 012021012010, 012021012020, 012021020120, 012021020210, 012021021010, 012021021020, 012021102020, 012102012020, 012102012121, 012102020120, 012102021020, 012102102020, 012102102121, 012102120121, 012102121021, 012120120121, 012120121021, 012120121210, 012121002121, 012121020121, 012121021021, 012121021210, 012121210021, 012121210210, 012201010102, 012201010210, 012201012010, 012201021010, 0102102102102, 0102102102120, 0102102120102, 0102102120120, 0102120102102, 0102120102120, 0102120120102, 0102120120120, 0102210120120, 0120102102102, 0120102102120, 0120102120102, 0120102120120, 0120102210120, 0120120102102, 0120120102120, 0120120102210, 0120120120102, 0120120120120, 0120120120121, 0120120120210, 0120120121020, 0120120121021, 0120120121210, 0120120210120, 0120120210210, 0120120211020, 0120121020120, 0120121020121, 0120121021020, 0120121021021, 0120121210210, 0120210120120, 0120210120210, 0120210210120, 0120210210210, 0120211020120, 0121020120120, 0121020120121, 0121020121020, 0121020121021, 0121021020120, 0121021020121, 0121021021020, 0121021021021, 0121210210210, 0122010210210 Evidence for Conjecture \[acbconj\] {#abcform} =================================== In this section, we list all sequences obtained by concatenating copies of $a b c$ and $a c b$, where the first permutation is $a b c$, that have formation width $x$ and alternation length $x+1$ for $x = 5, 7, 9, \dots, 23$. On each line, we list a binary string $s$, a sequence $u$, and ${\operatorname{fw}}(u)$. The sequence $u$ is obtained from $s$ by replacing each $0$ with $021$ and each $1$ with $012$.\ 100 012021021 5\ 101 012021012 5\ 110 012012021 5\ 111 012012012 5\ 1000 012021021021 7\ 1001 012021021012 7\ 1010 012021012021 7\ 1011 012021012012 7\ 1100 012012021021 7\ 1101 012012021012 7\ 1110 012012012021 7\ 1111 012012012012 7\ 10000 012021021021021 9\ 10001 012021021021012 9\ 10110 012021012012021 9\ 10111 012021012012012 9\ 11000 012012021021021 9\ 11001 012012021021012 9\ 11100 012012012021021 9\ 11101 012012012021012 9\ 11110 012012012012021 9\ 11111 012012012012012 9\ 100000 012021021021021021 11\ 100001 012021021021021012 11\ 101110 012021012012012021 11\ 101111 012021012012012012 11\ 110000 012012021021021021 11\ 111100 012012012012021021 11\ 111101 012012012012021012 11\ 111110 012012012012012021 11\ 111111 012012012012012012 11\ 1000000 012021021021021021021 13\ 1000001 012021021021021021012 13\ 1011110 012021012012012012021 13\ 1011111 012021012012012012012 13\ 1100000 012012021021021021021 13\ 1111100 012012012012012021021 13\ 1111101 012012012012012021012 13\ 1111110 012012012012012012021 13\ 1111111 012012012012012012012 13\ 10000000 012021021021021021021021 15\ 10000001 012021021021021021021012 15\ 10111110 012021012012012012012021 15\ 10111111 012021012012012012012012 15\ 11000000 012012021021021021021021 15\ 11111100 012012012012012012021021 15\ 11111101 012012012012012012021012 15\ 11111110 012012012012012012012021 15\ 11111111 012012012012012012012012 15\ 100000000 012021021021021021021021021 17\ 100000001 012021021021021021021021012 17\ 101111110 012021012012012012012012021 17\ 101111111 012021012012012012012012012 17\ 110000000 012012021021021021021021021 17\ 111111100 012012012012012012012021021 17\ 111111101 012012012012012012012021012 17\ 111111110 012012012012012012012012021 17\ 111111111 012012012012012012012012012 17\ 1000000000 012021021021021021021021021021 19\ 1000000001 012021021021021021021021021012 19\ 1011111110 012021012012012012012012012021 19\ 1011111111 012021012012012012012012012012 19\ 1100000000 012012021021021021021021021021 19\ 1111111100 012012012012012012012012021021 19\ 1111111101 012012012012012012012012021012 19\ 1111111110 012012012012012012012012012021 19\ 1111111111 012012012012012012012012012012 19\ 10000000000 012021021021021021021021021021021 21\ 10000000001 012021021021021021021021021021012 21\ 10111111110 012021012012012012012012012012021 21\ 10111111111 012021012012012012012012012012012 21\ 11000000000 012012021021021021021021021021021 21\ 11111111100 012012012012012012012012012021021 21\ 11111111101 012012012012012012012012012021012 21\ 11111111110 012012012012012012012012012012021 21\ 11111111111 012012012012012012012012012012012 21\ 100000000000 012021021021021021021021021021021021 23\ 100000000001 012021021021021021021021021021021012 23\ 101111111110 012021012012012012012012012012012021 23\ 101111111111 012021012012012012012012012012012012 23\ 110000000000 012012021021021021021021021021021021 23\ 111111111100 012012012012012012012012012012021021 23\ 111111111101 012012012012012012012012012012021012 23\ 111111111110 012012012012012012012012012012012021 23\ 111111111111 012012012012012012012012012012012012 23\

--- abstract: 'Controlled heat transfer and thermal rectification in a system of two coupled cavities connected to thermal reservoirs are discussed. Embedding a dispersively interacting two-level atom in one of the cavities allows switching from a thermally conducting to resisting behavior. By properly tuning the atomic state and system-reservoir parameters, direction of current can be reversed. It is shown that a large thermal rectification is achievable in this system by tuning the cavity-reservoir and cavity-atom couplings. Partial recovery of diffusive heat transport in an array of $N$ cavities containing one dispersively coupled atom is discussed.' author: - Nilakantha Meher$^a$ - 'S. Sivakumar$^b$' title: Atomic switch for control of heat transfer in coupled cavities --- Introduction ============ Coherent and controlled transfer of photons is of fundamental interest in quantum information processing and communication [@Northup]. Recent developments in fabrication of suitably coupled cavities have made it possible to study their use in transferring information using photons as the carrier [@Meher; @Zhong; @Yang]. Highly tunable cavity couplings and resonance frequencies of coupled cavities make them suitable for photon transfer, quantum state transfer, entanglement generation, etc [@Majumdar; @Almeida; @Biella; @YangLiu; @Liao; @Liew]. Transport of photons in an array can be modified by embedding atoms or Kerr-medium in the cavities which modify the cavity resonance frequencies [@Imamoglu; @Felicetti; @Qin; @Zhou; @Brune]. This helps to realize phenomenon such as photon blockade [@Imamoglu], quantum state switching [@Meher], generation of cat states [@Brune2], localization and delocalization [@Schmidt; @Meher2], etc.\ In the ideal case of a cavity being completely isolated from its surroundings, its dynamics is unitary. However, complete isolation of a system is not feasible. In this case, the evolution is not unitary. Simplest of this situation corresponds to coupling the system to a reservoir at zero absolute temperature. On incorporating such a reservoir, many of the photon transport phenomena indicated previously can be explained. However, the dynamics differs if the cavities are coupled to heat reservoirs at non-zero temperatures, where cavities exchange energy with reservoirs. In this context, coupled cavities can be used to transport energy between the thermal reservoirs [@Manzano2; @Andr]. For a conventional bulk material, steady state heat transport is governed by $$\begin{aligned} \label{Fourier} \textbf{J}=-\kappa\nabla T,\end{aligned}$$ which is the Fourier’s law of heat conduction. Here $\textbf{J}$ is the thermal current and $\nabla T$ is the temperature gradient. The proportionality constant $\kappa$ is the thermal conductivity, which is positive for all known materials. This law is valid if the system is close to its equilibrium, in which case linear response theory is applicable [@Garrido; @Dhar; @Saito2; @Michel; @Dubi]. Transport of heat by magnetic excitations in spin chains [@Saito; @Landi; @Schachenmayer; @Werlang2; @Manzano], phonons in atomic lattices [@Thingna; @He], photons in cavity arrays [@Manzano2; @Asad; @Purkayastha], etc. have been investigated. Similar to electronic devices, several thermal devices such as thermal diodes [@Li], thermal transistors [@Li3], thermal ratchet [@NLi], thermal logic gates [@Li4], thermal memory [@Wang], etc. based on non-equilibrium dynamics have been proposed.\ A system away from equilibrium may violate the Fourier’s empirical law. There is no universal theory of heat transfer applicable to all nonequilibrium systems. A chain of coupled oscillators is known to violate the Fourier’s law of heat conduction in the sense that the thermal current is independent of system size and the heat transport is ballistic [@Reider; @Dhar; @Asad; @Zotos]. Diffusive transport can be recovered by including anharmonicity or dephasing [@Hu; @Kamanashish; @Asad; @Tejal]. The dynamics of nonequilibrium systems is conceptually rich with many unsolved problems.\ Another interesting phenomenon is thermal rectification, which is essential for realizing thermal diodes and transistors [@Starr; @Li]. A system shows thermal rectification if it possesses structural asymmetry allowing higher thermal current in one direction. Thermal rectification is known in the case of nanotubes [@Chang], quantum spin chains [@Zhang; @Yan; @Werlang], nonlinear oscillators [@Terr], two-level systems [@Segal], etc.\ In the present work, heat transfer in a system of two coupled cavities containing a single atom is discussed. The system-reservoir interaction is assumed to be of Lindblad type [@Lindblad]. Magnitude as well as direction of current can be controlled by suitably choosing the atomic state and the system-reservoir parameters. The system exhibits large thermal rectification for proper choices of the cavity-reservoir and cavity-atom couplings.\ The present paper is organized as follows. In Sec. \[Physical\], details of the system and its theoretical model are discussed to arrive at an expression for heat current. Also, various special cases of importance are indicated. Based on the dependence of the current on the reservoir temperatures and coupling parameters, violation of Fourier’s law is estabslished Sec. \[NTC\]. Thermal rectification behavior of the system is explored in Sec. \[RectificationSection\]. Generalization to $N$ cavities is discussed in Sec. \[NcavitySection\]. Results are summarized in Sec. \[Summary\]. Current in coupled cavities {#Physical} =========================== A system of two linearly coupled cavities is described by the Hamiltonian[@AgarwalBook], $$\begin{aligned} H_{1}=&\omega_L a^\dagger_L a_L+\omega_R a^\dagger_R a_R+J(a_L^\dagger a_R+a_L a_R^\dagger),\end{aligned}$$ where $\omega_L$ and $\omega_R$ are the resonance frequencies. The coupling strength between the cavities is $J$. In addition, a two-level atom is dispersively coupled to the right cavity and the corresponding atom-cavity interaction is governed by the Hamiltonian [@Gerry], $$\begin{aligned} H_{2}=&\frac{\omega_0}{2}\sigma_z+\chi(\sigma_+\sigma_-+a^\dagger_R a_R\sigma_z),\end{aligned}$$ where $\chi=g^2/(\omega_0-\omega_R)$ is assumed to be positive. The states $\ket{e}$ and $\ket{g}$ are respectively the excited and ground states of the two-level atom. The operators $\sigma_+=\ket{e}\bra{g}$ and $\sigma_-=\ket{g}\bra{e}$ are the raising and lowering operators for the atom respectively. The energy operator for the atom is $\sigma_z=\ket{e}\bra{e}-\ket{g}\bra{g}$. The coupling strength between the atom and the cavity field is $g$ and the atomic transition frequency is $\omega_0$. This is an effective interaction obtained from Jaynes-Cummings model, if the atom and the cavity are highly detuned so that $\Delta=(\omega_0-\omega_R) >>g$ and the mean number of photons $n$ is smaller than $\Delta^2/g^2$ [@Gerry; @Holland]. Dispersive coupling between atom and cavity has been used to realize the cat states of the cavity field [@Brune; @Brune2].\ The system considered in this work is a pair of linearly coupled cavities and a dispersively interacting atom in one of the cavities. Based on the discussion given above, the total Hamiltonian is $$\begin{aligned} \label{Hamiltonian} H=H_1+H_2=&\frac{\omega_0}{2}\sigma_z+\omega_L a^\dagger_L a_L+\omega_R a^\dagger_R a_R +\chi(\sigma_+\sigma_-+a^\dagger_R a_R\sigma_z)+J(a_L^\dagger a_R+a_L a_R^\dagger).\end{aligned}$$ This Hamiltonian conserves the respective total excitation numbers for the cavity fields and the atom in the absence of dissipation, *i.e.*, $[a_L^\dagger a_L+a_R^\dagger a_R,H]=0$ and $[\sigma_z,H]=0$. As a consequence, the atom and the field cannot exchange energy in the dispersive limit [@Brune].\ The system is coupled to two reservoirs, each modelled as a collection of independent oscillators [@Carmichael]. The reservoir Hamiltonian is taken to be $$\begin{aligned} H_{x}=\sum_{j}\omega_{xj} b_{xj}^\dagger b_{xj}, \end{aligned}$$ where $x=L$, $R$ is the index referring to the left reservoir and the right reservoir respectively. The creation and annihilation operators of the reservoirs obey the bosonic commutation relation $[b_{xj}, b^\dagger_{xk}]=\delta_{jk}$. The arrangement of the cavities and reservoirs is shown in Fig. \[CoupledCavity\]. The interaction Hamiltonian for the cavity-reservoir component is $$\begin{aligned} H_{I}=\left(\sum_{j}g_{Lj}\right.&(a_L^\dagger+a_L)(b_{Lj}+b_{Lj}^\dagger)\\ &\left.+\sum_{j}g_{Rj}(a_R^\dagger+a_R)(b_{Rj}+b_{Rj}^\dagger)\right),\end{aligned}$$ where $g_{Lj}(g_{Rj})$ is the coupling strength of left (right) cavity to $j$th mode of left (right) reservoir.\ ![Schematic representation of system of coupled cavities with a two-level atom embedded in the right cavity. Both the cavities are also coupled with their respective reservoirs.[]{data-label="CoupledCavity"}](Fig1.eps){width="8cm" height="4.3cm"} Under the Born-Markov and rotating wave approximations [@GardinerZoller; @Zoller], the reduced joint density matrix for the two cavities (traced over the reservoirs) obeys [@Carmichael] $$\begin{aligned} \label{Master} \frac{\partial \rho}{\partial t}=-i[H,\rho]+\mathcal{D}_L(\rho)+\mathcal{D}_R(\rho),\end{aligned}$$ where the Lindblad operators $$\begin{aligned} \label{Lindblad} \mathcal{D}_x(\rho)=&\frac{\Gamma_x(\bar n_x+1)}{2}(2a_x\rho a_x^\dagger-a_x^\dagger a_x \rho-\rho a_x^\dagger a_x)\nonumber\\ &+\frac{\Gamma_x \bar n_x}{2}(2a_x^\dagger\rho a_x-a_x a_x^\dagger \rho-\rho a_x a_x^\dagger),\end{aligned}$$ for $x=L,R$. The parameters $\Gamma_L$ and $\Gamma_R$ are related to the coupling strengths as [@Biehs] $$\begin{aligned} \Gamma_x=2\pi\sum_{j}g_{xj}^2\delta(\omega_{xj}-\omega_x).\end{aligned}$$ The two terms in Eqn. \[Lindblad\] correspond to energy flow from the system to the reservoir and vice-versa respectively. The dynamics generated by the master equation approach satisfies the detailed balance condition and gives the correct steady state if the different components of the system are weakly coupled [@Rivas; @Purkayastha2; @Manzano; @Santos].\ The reservoirs $R_L$ and $R_R$ are assumed to be in thermal equilibrium at temperatures $T_L$ and $T_R$ respectively. The density operators which characterize the states of the reservoirs are $$\begin{aligned} \varrho_x=\frac{e^{-H_x/k_BT_x}}{\text{Tr}\left(e^{-H_x/k_BT_x}\right)},\end{aligned}$$ with mean photon numbers $$\begin{aligned} \bar n_{x}=\frac{1}{\exp{(\omega_{x}/k_B T_x)}-1},\end{aligned}$$ where $x=L,R$.\ The dynamics of the system can be understood from the temporal evolution of expectation values of various operators. The expectation values satisfy \[EquationOfMotion\] $$\begin{aligned} &\frac{d}{dt}\langle a_L^\dagger a_L\rangle=iJ(\langle a_L a_R^\dagger\rangle-\langle a_L^\dagger a_R\rangle)-\Gamma_L \langle a_L^\dagger a_L\rangle+\Gamma_L \bar{n}_L,\\ &\frac{d}{dt}\langle a_R^\dagger a_R\rangle=-iJ(\langle a_L a_R^\dagger\rangle-\langle a_L^\dagger a_R\rangle)-\Gamma_R \langle a_R^\dagger a_R\rangle+\Gamma_R \bar{n}_R,\\ &\frac{d}{dt}\langle a_L^\dagger a_R\rangle=i\Delta_c\langle a_L^\dagger a_R\rangle-iJ(\langle a_L^\dagger a_L\rangle-\langle a_R^\dagger a_R\rangle)-i\chi\langle a_L^\dagger a_R\sigma_z\rangle-\gamma\langle a_L^\dagger a_R\rangle,\\ &\frac{d}{dt}\langle a_L a_R^\dagger\rangle=-i\Delta_c\langle a_L a_R^\dagger\rangle+iJ(\langle a_L^\dagger a_L\rangle-\langle a_R^\dagger a_R\rangle)+i\chi\langle a_L a_R^\dagger\sigma_z\rangle-\gamma\langle a_L a_R^\dagger \rangle,\\ &\frac{d}{dt}\langle a_L^\dagger a_L\sigma_z\rangle=iJ(\langle a_L a_R^\dagger \sigma_z\rangle-\langle a_L^\dagger a_R \sigma_z\rangle)-\Gamma_L \langle a_L^\dagger a_L \sigma_z\rangle+\Gamma_L \bar{n}_L\langle \sigma_z\rangle,\\ &\frac{d}{dt}\langle a_R^\dagger a_R\sigma_z\rangle=-iJ(\langle a_L a_R^\dagger\sigma_z\rangle-\langle a_L^\dagger a_R\sigma_z\rangle)-\Gamma_R \langle a_R^\dagger a_R\sigma_z\rangle+\Gamma_R \bar{n}_R\langle \sigma_z\rangle,\\ &\frac{d}{dt}\langle a_L^\dagger a_R\sigma_z\rangle=i\Delta_c\langle a_L^\dagger a_R\sigma_z\rangle-iJ(\langle a_L^\dagger a_L\sigma_z\rangle-\langle a_R^\dagger a_R\sigma_z\rangle)-i\chi\langle a_L^\dagger a_R\rangle-\gamma\langle a_L^\dagger a_R\sigma_z\rangle,\\ &\frac{d}{dt}\langle a_L a_R^\dagger\sigma_z\rangle=-i\Delta_c\langle a_L a_R^\dagger\sigma_z\rangle+iJ(\langle a_L^\dagger a_L\sigma_z\rangle-\langle a_R^\dagger a_R\sigma_z\rangle)+i\chi\langle a_L a_R^\dagger\rangle-\gamma\langle a_L a_R^\dagger\sigma_z \rangle,\end{aligned}$$ where $\gamma=(\Gamma_L+\Gamma_R)/2$ and $\Delta_c=\omega_L-\omega_R$. Here $\langle A \rangle=\text{Tr}[\rho A]$, where $\rho$ satisfies the master equation given in Eqn. \[Master\].\ As $[\sigma_z, H]=[\sigma_z,\mathcal{D}_L(\rho) ]=[\sigma_z, \mathcal{D}_R(\rho)]=0$, the evolution equation for $\langle\sigma_z\rangle$ is $d \langle \sigma_z\rangle/d t=0$. This indicates that the value of $\langle \sigma_z \rangle $ remains constant during time evolution as a consequence of the fact that the atom is dispersively coupled with the cavity field.\ Steady state current is defined *via* the continuity equation $$\begin{aligned} \label{Cont} \frac{d}{dt}\langle H\rangle=0,\end{aligned}$$ which expresses the conservation of the total energy in the system.\ With $\langle H \rangle=$Tr$[\rho H]$ and using Eqn. \[Master\] for evolving $\rho$, the continuity equation given in Eqn. \[Cont\] yields $$\begin{aligned} 0=\text{Tr}[H\mathcal{D}_L(\rho)+H\mathcal{D}_R(\rho)]=:I_L+I_R.\end{aligned}$$ Here $I_x=\text{Tr}[H\mathcal{D}_x(\rho)], x=L,R$. Further, $I_L$ refers to the thermal current from the left reservoir $R_L$ to the system and $I_R$ indicates the current from the right reservoir $R_R$ to the system. Using Eqn. \[Master\], the steady state heat current from the left reservoir to the right reservoir through the system is $$\begin{aligned} \label{CurrentExpression} I_L=\text{Tr}[H\mathcal{D}_L(\rho)]=\Gamma_L(I_{nd}-I_{coh}).\end{aligned}$$ Here $I_{nd}=(\bar{n}_L-\langle a_L^\dagger a_L \rangle_{ss})\omega_L$ is the current due to mean excitation number difference between the left reservoir and the left cavity, and $I_{coh}=\frac{1}{2}J(\langle a_L^\dagger a_R\rangle_{ss}+\langle a_L a_R^\dagger \rangle_{ss})$ is the current due to the total coherence in the system. Here $\langle \cdot \rangle_{ss}$ represents the steady state mean value. A similar expression for the steady state heat current from the right reservoir to the left reservoir is $$\begin{aligned} \label{RightCurrent} I_R&=\text{Tr}[H\mathcal{D}_R(\rho)],\nonumber\\ &=\Gamma_R(\bar{n}_R-\langle a_R^\dagger a_R \rangle_{ss})(\omega_R+\langle\sigma_z\rangle\chi)-\Gamma_R I_{coh}.\end{aligned}$$\ Steady state solutions are obtained by equating the time derivatives of the expectation values of the relevant operators given in Eqns. \[EquationOfMotion\]$a$-\[EquationOfMotion\]$h$ to zero. The steady state values are \[SteadyMean\] $$\begin{aligned} &\langle a_L^\dagger a_L \rangle_{ss}=\frac{C(\Gamma_L \bar{n}_L+\Gamma_R \bar{n}_R)+\Gamma_L\Gamma_R \bar{n}_L}{C(\Gamma_L+\Gamma_R)+\Gamma_L\Gamma_R},\\\nonumber\\ &\langle a_R^\dagger a_R \rangle_{ss}=\frac{C(\Gamma_L \bar{n}_L+\Gamma_R \bar{n}_R)+\Gamma_L\Gamma_R \bar{n}_R}{C(\Gamma_L+\Gamma_R)+\Gamma_L\Gamma_R},\\\nonumber\\ &\delta N= \langle a_L^\dagger a_L \rangle_{ss}-\langle a_R^\dagger a_R \rangle_{ss}=\frac{\Gamma_L\Gamma_R(\bar{n}_L-\bar{n}_R)}{C(\Gamma_L+\Gamma_R)+\Gamma_L\Gamma_R},\\\nonumber\\ &\text{and}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\ &\langle a_L^\dagger a_R \rangle_{ss}=-J\frac{\chi \langle\sigma_z\rangle+\Delta_c+i\gamma}{\chi^2-\Delta_c^2+\gamma^2-2i\gamma\Delta_c}\delta N,\end{aligned}$$ with $$\begin{aligned} C=2J^2\gamma\frac{\Delta_c^2+\chi^2+2\Delta_c\chi\langle \sigma_z\rangle+\gamma^2}{(\chi^2-\Delta_c^2+\gamma^2)^2+4\gamma^2\Delta_c^2}.\end{aligned}$$\ Using these steady state solutions, Eqn. \[CurrentExpression\] yields $$\begin{aligned} \label{nonresonantCurrent} I_L=J^2\delta N\frac{\Gamma_L\chi\langle\sigma_z\rangle(\chi^2-\Delta_c^2+\gamma^2)+(\omega_L\Gamma_R+\omega_R\Gamma_L)(\Delta_c^2+\gamma^2)+\chi^2(2\omega_L\gamma+\Delta_c\Gamma_L)+4\Delta_c\chi\langle\sigma_z\rangle \omega_L\gamma}{(\chi^2-\Delta_c^2+\gamma^2)^2+4\gamma^2\Delta_c^2}.\end{aligned}$$ In the absence of inter-cavity coupling $(J=0)$, the cavities equilibrate with their respective reservoirs with mean photon numbers $\bar{n}_L$ and $\bar{n}_R$. The currents $I_L$ and $I_R$ vanish since energy cannot flow from one cavity to another as $J=0$. If the coupling is non-zero and the reservoirs are at different temperatures, energy flows from one reservoir to other through the cavities.\ Interestingly, expression in Eqn. \[nonresonantCurrent\] shows that the current through the system explicitly depends on $\langle\sigma_z\rangle$, which, in turn, depends on the state of the atom. This dependency arises as the atom modifies the cavity resonance frequency and the coherences $\langle a_L^\dagger a_R \rangle_{ss}$ and $\langle a_L a_R^\dagger \rangle_{ss}$, as well. By a proper choice of the atomic state, $\langle\sigma_z\rangle$ can be tuned from $+1$ corresponding to the atom in its excited state to $-1$, *i.e.*, the atom is in its ground state. This feature can be used to control the energy flow (current) between the reservoirs.\ If the cavities are resonant, *i.e.*, $\omega_L=\omega_R=\omega $, equivalently, $\Delta_c=0$. In the absence of the atom, the total coherence is zero $\langle a_L^\dagger a_R \rangle_{ss}+\langle a_L a_R^\dagger \rangle_{ss}=0,$ as can be seen from Eqn. \[SteadyMean\]$d$. The current through the cavities is $$\begin{aligned} \label{ResonantCurrentAbsentAtom} I_L=\frac{4\omega J^2\Gamma_L\Gamma_R}{(4J^2+\Gamma_L\Gamma_R)(\Gamma_L+\Gamma_R)}(\bar{n}_L-\bar{n}_R).\end{aligned}$$ The current is proportional to the difference in the mean photon numbers; equivalently, the current is proportional to the temperature difference of the two reservoirs for a fixed system size, which is like the Fourier’s law.\ If the temperatures of the two reservoirs are equal ($\bar{n}_L=\bar{n}_R=\bar{n}$), the system equilibrates with the reservoirs and no current flows through the system. The mean number of photons in the cavities are $ \langle a_L^\dagger a_L \rangle_{ss}=\langle a_R^\dagger a_R \rangle_{ss}=\bar{n}$. Also, the states of the cavity fields satisfy the zero coherence condition, namely, $ \langle a_L^\dagger a_R \rangle_{ss}=\langle a_R^\dagger a_L \rangle_{ss}=0$. To know the states of the fields in the cavities, the fidelity $F(\rho_{th},\rho_x)$ $$\begin{aligned} F(\rho_{th},\rho_x)=\text{Tr}\left(\sqrt{\sqrt{\rho_{th}}\rho_x\sqrt{\rho_{th}}}\right),\end{aligned}$$ between the thermal field and the cavity field is calculated. Here $$\begin{aligned} \label{thermalstate} \rho_{th}=\frac{1}{1+\bar{n}}\sum_{n=0}^\infty \left(\frac{\bar{n}}{1+\bar{n}}\right)^n \ket{n}\bra{n},\end{aligned}$$ is the single mode Gibbs thermal state; $\rho_x (x=L,R)$ are the steady state reduced density matrices for the left- and right-cavities respectively. The steady state fidelity $F(\rho_{th},\rho_x)$ is unity. Therefore, the cavity fields are also Gibbs thermal state. The second order correlation function $$\begin{aligned} g^{(2)}_x(0)=\frac{\text{Tr}(\rho_x a_x^{\dagger 2} a_x^2)}{\left[\text{Tr}(\rho_x a_x^{\dagger} a_x)\right]^2},\end{aligned}$$ in the steady state $\rho_x$ is $2$ same as that of the thermal state. This confirms that the cavity states are thermal states $\rho_{th}$.\ If a temperature difference is maintained between the reservoirs, the high temperature reservoir is the source of energy to the system and the low temperature reservoir is the sink for the energy to establish a steady state. As a consequence, heat continuously flows from the high temperature reservoir to the low temperature reservoir. The system reaches a non-equilibrium steady state with effective mean photon numbers $\langle a_L^\dagger a_L \rangle_{ss}$ and $\langle a_R^\dagger a_R \rangle_{ss}$ in the left- and right- cavities respectively. Analytical expressions for these mean photon numbers are given in Eqn. \[SteadyMean\]$a$ and Eqn. \[SteadyMean\]$b$. A non-equilibrium steady state is not necessarily the Gibbs thermal state.\ In the presence of an atom in one of the cavities, as shown in Fig. \[CoupledCavity\], the current through the system is $$\begin{aligned} \label{Current} I_L=\Theta\frac{\bar{c}}{\gamma} \left(\gamma\omega+\frac{\Gamma_L}{2}\chi\langle \sigma_z \rangle\right)(\bar{n}_L-\bar{n}_R),\end{aligned}$$ where $$\begin{aligned} \Theta=\frac{\Gamma_L\Gamma_R}{\bar{c}(\Gamma_L+\Gamma_R)+\Gamma_L\Gamma_R},\end{aligned}$$ and $\bar{c}=2J^2\gamma/(\chi^2+\gamma^2)$.\ If $\Gamma_L=\Gamma_R=\Gamma$, then $$\begin{aligned} \label{CurrentEqualGamma} I_L=2J^2\frac{\Gamma}{4J^2+\chi^2+\Gamma^2}\left(\omega+\frac{\chi}{2}\langle\sigma_z\rangle\right)(\bar{n}_L-\bar{n}_R).\end{aligned}$$ Scaled current $I_L/\omega^2$ as a function of $\Gamma/\omega$ is shown in Fig. \[Equalbathcouplings\]. Maximum current flows through the system if $\Gamma=\sqrt{4J^2+\chi^2}$. This special value $\sqrt{4J^2+\chi^2}$ corresponds to the Rabi frequency of the oscillation of the mean number of photon when the cavity detuning is $\chi$ and the cavities are not coupled to the reservoirs. The detuning between the cavity frequencies arises due to the atom in one of the cavities. The competition between the cavity-reservoir energy exchange rate $\Gamma$ and the cavity-cavity energy exchange rate $\sqrt{4J^2+\chi^2}$ affects the current through the system. If the two rates are equal, then $$\begin{aligned} I_L=\frac{J^2}{\sqrt{4J^2+\chi^2}}\left(\omega+\frac{\chi}{2}\langle\sigma_z\rangle\right)(\bar{n}_L-\bar{n}_R),\end{aligned}$$ which is the maximum current. If $\Gamma>>\sqrt{4J^2+\chi^2}$, the cavities and their respective reservoirs exchange energy faster than the inter-cavity exchange. In the opposite limit, both the cavities exchange energy with each other faster than with their respective reservoirs. This mismatch between the energy exchange rates reduces the current. From Eqn. \[CurrentEqualGamma\], it is seen that for small $\Gamma$, $I_L \propto \Gamma$ and for large $\Gamma$, $I_L \propto \Gamma^{-1}$. It may be noted that a system of three cavities containing two 3-level atoms has also been shown to allow control of magnitude of heat current [@Manzano2]. ![Current $I_L/\omega^2$ shown as a function of reservoir coupling strength $\Gamma/\omega$ for different atom-cavity coupling strengths $\chi/\omega=0$ (continuous), $0.03$ (dashed) and $0.05$ (dot-dashed). The system-reservoir parameters are $J/\omega=0.02, \bar{n}_L-\bar{n}_R=0.5 $ and $\langle \sigma_z\rangle=1$.[]{data-label="Equalbathcouplings"}](Fig2.eps){width="9cm" height="6cm"} Negative thermal conductivity {#NTC} ============================= According to the Fourier’s law given in Eqn. \[Fourier\], current is proportional to temperature gradient. Using the fact that $\delta N\propto (\bar{n}_L-\bar{n}_R)$ as given in Eqn. \[SteadyMean\]$c$, the expression for $I_L$ in Eqn. \[nonresonantCurrent\] can be written in the form $$\begin{aligned} I_L=\tilde{\kappa}(\bar{n}_L-\bar{n}_R),\end{aligned}$$ for comparing with the Fourier’s law. Here $\tilde{\kappa}$ is the effective thermal conductivity. It is to be noted that thermal conductivity can be tuned by suitably choosing the atomic state. Two important cases corresponding to the atom being in the excited state and the ground state are considered, *i.e.*, $\langle \sigma_z\rangle=\pm 1$. The corresponding currents are $$\begin{aligned} \label{Currentpm} I_{L}=J^2\delta N\frac{\Omega}{(\chi^2-\Delta_c^2+\gamma^2)^2+4\gamma^2\Delta_c^2}((\Delta_c \pm\chi)^2+\gamma^2).\end{aligned}$$ where $\Omega=\omega_L\Gamma_R+\Gamma_L(\omega_R\pm\chi)$. We assume $\bar{n}_L> \bar{n}_R$, *i.e.*, $\delta N>0$ for subsequent discussion. In this assumption, $I_L$ and $\Omega$ have the same sign. If the atom is in its ground state, sign of $\Omega$ is changeable by properly choosing the ratios $\Gamma_R/\Gamma_L$ and $(\chi-\omega_R)/\omega_L$. Consequently, direction of current can also be changed. It is to be pointed out that $\omega_R-\chi$ is the resonance frequency of the right cavity modified by the atom. If the atom is in its excited state, *i.e.*, $\langle \sigma_z\rangle=+1$, $I_L$ is always positive, meaning the thermal current flows from the high temperature reservoir to the low temperature reservoir (conventional flow) and reversal of current is not possible.\ In order to exhibit the switching action by the atom, we choose $\chi>\omega_R$. If the system-reservoir parameters satisfy $$\begin{aligned} \label{HightoLow} \frac{\Gamma_R}{\Gamma_L}> \frac{(\chi-\omega_R)}{\omega_L},\end{aligned}$$ thermal current flows from the high temperature reservoir to the low temperature reservoir, independent of the atomic state.\ If the ratios are equal, *i.e.*, $$\begin{aligned} \label{Zerocurrent} \frac{\Gamma_R}{\Gamma_L}=\frac{(\chi-\omega_R)}{\omega_L},\end{aligned}$$ and the atom is in the ground state, the thermal current through the system is zero even if the reservoirs are at different temperatures. The system completely blocks the heat flow like a thermal insulator. By switching the atom to its excited state, the system changes from a thermal-insulator to a thermal-conductor.\ If the atom is in its ground state and the system-reservoir parameters are such that $$\begin{aligned} \label{LowtoHigh} \frac{\Gamma_R}{\Gamma_L}< \frac{(\chi-\omega_R)}{\omega_L},\end{aligned}$$ then $\Omega<0$ and the direction of thermal current reverses, *i.e.*, current flows from low temperature reservoir to high temperature reservoir (unconventional flow). In such case, thermal conductivity of the system can be interpreted to be negative in which case heat flows from the low temperature to high temperature. Emergence of this negative current may be a result of coupling the system to Markovian baths [@Zoller]. By switching the atom from its ground state to excited state, the unconventional flow of thermal current switches to the conventional flow. Thus, the atom acts as a thermal switch which brings about a controllable current flow through the cavities.\ To summarize, we define $$\begin{aligned} \alpha=\frac{\Gamma_R/\Gamma_L}{(\chi-\omega_R)/\omega_L}.\end{aligned}$$ The three conditions given in Eqns. (\[HightoLow\]-\[LowtoHigh\]) correspond to $\alpha$ becoming greater than, equal to or less than unity respectively. The signs of the respective currents established in the system are indicated in Table. \[AtomStateCurrent\].\ [ | c | c | c | ]{} & $~~\langle \sigma_z\rangle =+1~~$ & $~~\langle \sigma_z\rangle =-1~~$\ \[1ex\] $~~\alpha>1~~$ & $I_L>0$ & $I_L>0$\ \[1ex\] $\alpha=1$ & $I_L>0$ & $I_L=0$\ \[1ex\] $\alpha<1$ & $I_L>0$ & $I_L<0$\ \[AtomStateCurrent\] Scaled current $I_{L}/I_0$ for the case of the atom in its ground state is shown as a function of $\chi/\omega_L$ in Fig. \[negativecurrent\] for cavity-reservoir coupling ratios $\Gamma_R/\Gamma_L=0.1$ (continuous), $0.3$ (dashed) and $0.6$ (dot-dashed). Here $I_0$ is the amount of current flowing through the system when $\chi=0$. The inset figure shows the scaled current in the system when the atom is in excited state for the same values of $\Gamma_R/\Gamma_L$. Note that the current is always positive if the atom is in the excited state (inset figure). If the atom is in its ground state, current vanishes if the system and reservoir parameters satisfy Eqn. \[Zerocurrent\]. Negative current occurs at different values of $\chi/\omega_L$ required to satisfy Eqn. \[LowtoHigh\]. ![Normalized current $I_{L}/I_0$ shown as a function of $\chi/\omega_L$ for different cavity-reservoir coupling ratio $\Gamma_R/\Gamma_L=0.1$ (continuous), $0.3$ (dashed) and $0.6$ (dot-dashed). The system-reservoir parameters are $J/\omega_L=0.05, \bar{n}_L-\bar{n}_R=0.5, \omega_R/\omega_L=0.8$ and $\langle \sigma_z\rangle=-1$. The current in the system is shown in the inset for the same values of the parameters and $\langle \sigma_z\rangle=+1$.[]{data-label="negativecurrent"}](Fig3.eps){width="9cm" height="5.5cm"} Negative current arises because the contribution from the coherence part $I_{coh}$ is more than the current due to mean excitation number difference $I_{nd}$, which makes $I_L$ negative (refer Eqn. \[CurrentExpression\]). Dimensionless quantities $I_{nd}/\omega_L^2,I_{coh}/\omega_L^2$ and $I_{L}/\omega_L^2$ are shown in Fig. \[Currentcontribution\] as a function of the atom-field coupling strength $\chi/\omega_L$. It is to be noted that if the parameters are chosen to satisfy Eqn. \[Zerocurrent\], in which case $I_{nd}=I_{coh}$, the system completely blocks the current. Current reverses its direction from the low temperature reservoir to the high temperature reservoir when $I_{coh}>I_{nd}$. In this sense, the coherence in the system drives energy to flow to the high temperature reservoir. ![Dimensionless currents $I_{L}/\omega_L^2$(continuous), $I_{nd}/\omega_L^2$ (dashed) and $I_{coh}/\omega_L^2$ (dot-dashed) shown as function of $\chi/\omega_L$. Here $\Gamma_R/\Gamma_L=0.3$, $J/\omega_L=0.05, \bar{n}_L-\bar{n}_R=0.5, \omega_R/\omega_L=1$ and $\langle \sigma_z\rangle=-1$.[]{data-label="Currentcontribution"}](Fig4.eps){width="9cm" height="5cm"} \ Thermal Rectification {#RectificationSection} ===================== ![Reverse configuration of system-reservoirs. reservoir temperatures and the system-reservoir coupling strengths are interchanged.[]{data-label="CoupledCavityReverse"}](Fig5.eps){width="8cm" height="4.5cm"} A system exhibits thermal rectification if thermal current depends on the direction of heat flow, $$\begin{aligned} I(\Delta n)\neq -I(-\Delta n), \end{aligned}$$ where $\Delta n=\bar{n}_L-\bar{n}_R$ is the difference in the average photon number of the left and right reservoirs. This means that by swapping the thermal reservoirs, current changes both sign and magnitude.\ If the system is symmetric under the exchange of cavities, rectification is not possible. In the system under discussion, assymmetry is due to presence of the atom in one of the cavities. Thermal rectification is to be established by studying the transport of photon in the reverse configuration realized by interchanging the reservoirs and system-reservoir coupling strengths. The reverse configuration is shown in Fig. \[CoupledCavityReverse\]. The relevant Lindblad operators for the reverse configuration are $$\begin{aligned} \mathcal{D}_L(\rho)=&\frac{\Gamma_R(\bar n_R+1)}{2}(2a_L\rho a_L^\dagger-a_L^\dagger a_L \rho-\rho a_L^\dagger a_L)\nonumber\\ &+\frac{\Gamma_R \bar n_R}{2}(2a_L^\dagger\rho a_L-a_L a_L^\dagger \rho-\rho a_L a_L^\dagger),\nonumber\\ \mathcal{D}_R(\rho)=&\frac{\Gamma_L(\bar n_L+1)}{2}(2a_R\rho a_R^\dagger-a_R^\dagger a_R \rho-\rho a_R^\dagger a_R)\nonumber\\ &+\frac{\Gamma_L \bar n_L}{2}(2a_R^\dagger\rho a_R-a_R a_R^\dagger \rho-\rho a_R a_R^\dagger).\nonumber\end{aligned}$$ The atom is taken to be in its ground state. Steady state solutions for the expectation values of operators for the reverse configuration can be obtained by the transformations $\Gamma_L\longrightarrow \Gamma_R$, $\bar n_L\longrightarrow \bar n_R$ and vice-versa in Eqns. \[SteadyMean\]$a$-\[SteadyMean\]$d$. Current from the left reservoir $R_L$ to the right reservoir $R_R$ in the system shown in Fig. \[CoupledCavity\] is called forward current. The expression for the forward current is $$\begin{aligned} \label{forward} I_{f}(\Delta n,\Gamma_L,\Gamma_R)=J^2\delta N\frac{(\omega_L\Gamma_R+\Gamma_L(\omega_R-\chi))}{(\chi^2-\Delta_c^2+\gamma^2)^2+4\gamma^2\Delta_c^2}((\Delta_c-\chi)^2+\gamma^2).\end{aligned}$$\ On exchanging ($\bar{n}_L, \Gamma_L)$ and $(\bar{n}_R,\Gamma_R)$, reverse current from the right reservoir $R_R$ to the left reservoir $R_L$ in the configuration given in Fig. \[CoupledCavityReverse\] is $$\begin{aligned} \label{reverse} I_{r}(-\Delta n,\Gamma_R,\Gamma_L)=-J^2\delta N\frac{(\omega_L\Gamma_L+\Gamma_R(\omega_R-\chi))}{(\chi^2-\Delta_c^2+\gamma^2)^2+4\gamma^2\Delta_c^2}((\Delta_c-\chi)^2+\gamma^2).\end{aligned}$$ The reverse current is taken as negative as the direction of flow is opposite to the forward current. The currents $I_{f}$ and $I_{r}$, normalized with their corresponding values for $\chi=0$ and $\Delta_c=0$, are shown as a function of the atom-field coupling strength $\chi$ in Fig. \[JfJr\]. For non-zero $\chi$, the magnitudes of the forward and reverse currents are different. Therefore, the system shows thermal rectification. Importantly, if the parameters satisfy the condition given in Eqn. \[LowtoHigh\], the forward current changes the sign. As a result, $I_{f}$ and $I_{r}$ flow in same direction.\ ![Normalized forward current $I_{f}$ (continuous line) and reverse current $I_{r}$ (dashed line) as a function of $\chi/\omega_L$ for $\Gamma_R/\Gamma_L=0.3$. Currents are normalized with their respective values at $\chi=0$. The system-reservoir parameters are $J/\omega_L=0.05, \bar{n}_L-\bar{n}_R=0.5, \omega_R/\omega_L=1$ and $\langle \sigma_z\rangle=-1$. []{data-label="JfJr"}](Fig6.eps){width="9cm" height="5.5cm"} Thermal rectification is quantified by rectification coefficient $R$ defined as $$\begin{aligned} \label{DefRectification} R=-\frac{I_{f}}{I_{r}}.\end{aligned}$$ If $R=1$, there is no rectification. For the system under consideration $$\begin{aligned} \label{SystemRect} R=\frac{\omega_L\Gamma_R+\Gamma_L(\omega_R-\chi)}{\omega_L\Gamma_L+\Gamma_R(\omega_R-\chi)}.\end{aligned}$$ ![Rectification $R$ as a function of $\Gamma_L/\omega$ for $\Gamma_R/\omega=0.2$. Here $\chi/\omega=1.5$ and $\langle \sigma_z\rangle=-1$. []{data-label="Rectification"}](Fig7.eps){width="9cm" height="6cm"} If $\Gamma_L=\Gamma_R$ or $\omega_L=|\omega_R-\chi|$, rectification coefficient $R$ becomes unity. Rectification coefficient $R$ is shown as a function of $\Gamma_L/\omega$ in Fig. \[Rectification\] for the resonant case ($\Delta_c=0$). Rectification is positive, zero, or negative depending on the parameters. The system shows large rectification if $$\begin{aligned} \label{ZeroReverseCurrent} \frac{\Gamma_L}{\Gamma_R}=\frac{\chi-\omega_R}{\omega_L},\end{aligned}$$ as seen in Fig. \[Rectification\]. This originates from the fact that the atom completely blocks the current in one direction (thermally insulating) and allows in the other direction (thermally conducting). Even though the system size is finite, rectification becomes infinity theoretically. If $\Gamma_L/\omega$ is increases from values less than that satisfying Eqn. \[ZeroReverseCurrent\] to higher values, $R$ jumps from negative value of large magnitude to large positive value. Thus $R$ is very sensitive to changes in the parameters in that region. Asymmetry can also be introduced with non-resonant cavities without an atom $(\chi=0)$ in any of the cavities. However, as seen from Eqn. \[SystemRect\], large rectification is not possible. Generalization to $N$-cavities {#NcavitySection} ============================== It would be interesting to study the steady state heat transfer in $N$ coupled cavities containing a two-level atom in one of the cavities. The Hamiltonian for the system is $$\begin{aligned} \label{Ncavity} \tilde{H}=\frac{\omega_0}{2}\sigma_z+\omega \sum_{j=1}^N a_j^\dagger a_j & +J\sum_{j=1}^{N-1} (a_j^\dagger a_{j+1}+a_j a_{j+1}^\dagger) \nonumber\\ & +\chi(\sigma_+\sigma_-+a^\dagger_m a_m\sigma_z).\end{aligned}$$ The atom is embedded in the $m$th cavity and dispersively interacts with the cavity-field. The right most and the left most cavities in the array are coupled with two reservoirs $R_L$ and $R_R$ respectively. The density matrix $\tilde{\rho}$ of the system obeys $$\begin{aligned} \label{MasterN} \frac{\partial \tilde{\rho}}{\partial t}=-i[\tilde{H},\tilde{\rho}]+\mathcal{D}_L(\tilde{\rho})+\mathcal{D}_R(\tilde{\rho}),\end{aligned}$$ where $$\begin{aligned} \mathcal{D}_L(\tilde{\rho})=&\frac{\Gamma_L(\bar{n}_L+1)}{2}(2a_1\tilde{\rho} a_1^\dagger-a_1^\dagger a_1 \tilde{\rho}-\tilde{\rho} a_1^\dagger a_1)\nonumber\\ &+\frac{\Gamma_L \bar{n}_L}{2}(2a_1^\dagger\tilde{\rho} a_1-a_1 a_1^\dagger \tilde{\rho}-\tilde{\rho} a_1 a_1^\dagger),\\ \mathcal{D}_R(\tilde{\rho})=&\frac{\Gamma_R(\bar{n}_R+1)}{2}(2a_N\tilde{\rho} a_N^\dagger-a_N^\dagger a_N \tilde{\rho}-\tilde{\rho} a_N^\dagger a_N)\nonumber\\ &+\frac{\Gamma_R \bar{n}_R}{2}(2a_N^\dagger\tilde{\rho} a_N-a_N a_N^\dagger \tilde{\rho}-\tilde{\rho} a_N a_N^\dagger),\end{aligned}$$ Here $\bar{n}_L$ and $\bar{n}_R$ are the mean number of photons in the reservoirs $R_L$ and $R_R$ respectively. Without loss of generality, we assume $\bar{n}_L > \bar{n}_R$.\ Using Eqn. \[MasterN\], the equation of motion is $$\begin{aligned} \frac{d \langle G\rangle}{dt}=\frac{d}{dt} \text{Tr}(\tilde{\rho} G)=i[M_1,\langle G\rangle]+\{M_2,\langle G\rangle\}+M_3,\end{aligned}$$ where $\langle G\rangle=\langle A^\dagger A\rangle$ is the matrix whose elements are the expectation values of the operator elements of $A^\dagger A$. Here $$\begin{aligned} A&=\text{Row}(a_1,a_2,...a_N,a_1\sigma_z,...,a_N\sigma_z),\\ A^\dagger&=\text{Column}(a_1^\dagger,a_2^\dagger,...a_N^\dagger,a_1^\dagger\sigma_z,...,a_N^\dagger\sigma_z).\end{aligned}$$ Further $[M_1,\langle G\rangle]=M_1\langle G\rangle-\langle G\rangle M_1$ and $\{M_2,\langle G\rangle\}=M_2\langle G\rangle+\langle G\rangle M_2$. The transformation matrices are $$\begin{aligned} M_1=&I_{2\times 2} \otimes H_c+\sigma_x \otimes X,\\ M_2=&I_{2\times 2} \otimes \text{Diag}\left(-\frac{1}{2}\Gamma_L,0,...0,-\frac{1}{2}\Gamma_R\right)_{N\times N},\\ M_3=&I_{2\times 2} \otimes \text{Diag}\left(\Gamma_L \bar{n}_L,0,...0,\Gamma_R\bar{n}_R\right)_{N\times N}\\ &~~~~~~+ \sigma_x \otimes \text{Diag}(\Gamma_L \bar{n}_L\langle\sigma_z\rangle,0,...0,\Gamma_R\bar{n}_R\langle \sigma_z\rangle)_{N\times N},\end{aligned}$$ where $I_{2\times 2}$ is the identity matrix of dimension $2$ and $\sigma_x$ is the Pauli matrix. The matrix $$H_c=\left(\begin{matrix} \omega & J & 0 & \ldots & 0\\ J & \omega & J& \ldots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 &\ldots &\omega & J\\ 0 & 0 &\ldots &J & \omega \end{matrix}\right)_{N\times N},$$ and the matrix elements of $(X)_{N\times N}$ are zero except $X_{m,m}=\chi$.\ ![Ratio of currents $I_L/I_0$ as a function of $N$ for $\chi/\omega=0.15$ (circle) and $0.1$ (triangle). The atom is embedded in the last cavity ($m=N$). Here $J/\omega=0.05, \bar{n}_L-\bar{n}_R=0.5, \langle\sigma_z\rangle=-1, \Gamma_L/\omega=\Gamma_R/\omega=0.15$. []{data-label="Sizedependent"}](Fig8.eps){width="9cm" height="6.2cm"} Using $\tilde{H}$ given in Eqn. \[Ncavity\] in the continuity equation (refer Eqn. \[Cont\]), the current in the system is $$\begin{aligned} \label{CurrentNcavities} I_L=\Gamma_L\left[(\bar{n}_L- \right. &\langle a_1^\dagger a_1\rangle_{ss})(\omega+\chi\langle \sigma_z \rangle\delta_{m,1})\nonumber\\ &\left.-\frac{J}{2}(\langle a_1^\dagger a_2\rangle_{ss}+\langle a_1 a_2^\dagger \rangle_{ss})\right].\end{aligned}$$ Here $\delta_{m,1}$ is Kronecker delta. If there is no atom in the array, the coherence term $\langle a_j^\dagger a_{j+1}\rangle$ is purely imaginary [@Asad]. The contribution of the coherence term to the current vanishes as $I_{coh}=\frac{J}{2}(\langle a_1^\dagger a_2\rangle_{ss}+\langle a_1 a_2^\dagger \rangle_{ss})=0$. Consequently, current in the cavity array is $$\begin{aligned} \label{AbsentAtom} I_L(\chi=0)=I_0=\frac{4\omega J^2\Gamma_L\Gamma_R}{(4J^2+\Gamma_L\Gamma_R)(\Gamma_L+\Gamma_R)}(\bar{n}_L-\bar{n}_R).\end{aligned}$$ Note that the current $I_0$ is independent of the size of the array, in violation of Fourier’s law. This feature is similar to the system-size independent current in the case of ballistic transport [@Zurcher; @Gaul; @Asad]. This comparison indicates that the mean free path of the photons scales in proportion to the number of cavities $N$.\ ![Steady state mean photon number in the intermediate cavities for arrays of length $(a)N=6$ and $(b)N=12$. The atom-field coupling strength is chosen to be $\chi/\omega= 0.1$. Here $J/\omega=0.05, \bar{n}_L-\bar{n}_R=0.5, \langle\sigma_z\rangle=-1, \Gamma_L/\omega=\Gamma_R/\omega=0.15$.[]{data-label="temperatureprofile"}](Fig9.eps){width="9cm" height="5cm"} Mean free path is different from the array size if an atom is embedded in one of the cavities. The atom is considered to be in the last cavity of the array, *i.e.*, $m=N$, to keep the mean free path as close to the size of the array. This helps to understand the emergence of diffusive character if there is a single scatterer. The normalized current $I_L/I_0$ as a function of size of the array $N$ is shown in Fig. \[Sizedependent\] for a fixed temperature difference $(\bar{n}_L-\bar{n}_R)$. It is to be noted that by increasing the size of the array, the steady state current significantly decreases and asymptotically approaches a constant value. The current is size dependent for smaller array. Thus the atom is able to introduce diffusive character. It saturates with further increase in size and becomes nearly size independent, which is at odds with the Fourier’s law. Thus, the transport is of ballistic type. If many cavities in the array contain atoms, the heat transport may be expected to be diffusive. This is plausible as the effect of dephasing by all the atoms effectively reduces the mean free path for the photons [@Brune; @Asad].\ The transition from diffusive to ballistic transport as size of the array increases, can be understood by calculating the mean photon numbers $\langle n_j\rangle=\langle a_j^\dagger a_j\rangle$ (known as local temperature [@Asad]) of the respective cavities in the array. The steady state mean photon number $\langle n_j\rangle$ in the intermediate cavities for arrays containing $6$ and $12$ cavities are shown in Fig. \[temperatureprofile\]$(a)$ and $(b)$ respectively. Gradient in the mean photon number is noticed in Fig. \[temperatureprofile\]$(a)$. This implies that the transport is diffusive [@Reider; @Hu]. For larger size array, for instance $N=12$, the gradient in mean photon number approaches zero and the current is independent of the system size. Essentially, the change in mean free path in the presence of a scatterer at the end of the array is insignificant for large array. Consequently, the photon transport is not diffusive. Summary {#Summary} ======= Mesoscopic systems offer interesting possibilities when it comes to thermal properties. A system of two coupled cavities connected between thermal reservoirs provides a conduit for heat flow between the reservoirs. If a dispersively interacting atom is placed in one of the cavities, thereby providing an asymmetry in the system, many of the thermal transport properties can be tailored. The present system switches from a thermally insulating state to a conducting one, depending on whether the atom is in its ground state or excited state. If the atomic state changes from the excited state to the ground state, current through the system becomes zero or reversed depending on the system-reservoir coupling strengths and the cavity frequencies. The reversal of current implies that the effective thermal conductivity is negative.\ The presence of the atom changes the magnitude of the current on exchange of the reservoirs along with the coupling strengths, which leads to thermal rectification. Large rectification is possible if the parameters are chosen to make the system thermally resistive either for the forward current or the reverse current.\ If a cavity array contains a two-level atom in one of the cavities, the magnitude of current depends on the number of cavities. This size-dependence indicates that the thermal current through the array is analogous to the diffusive heat transport. If there is only a single atom in a large array, it is not possible to completely recover the diffusive transport. Single atom does not provide enough dephasing to recover the diffusive character.\ \ **Data accessibilities**\ This paper has no data.\ **Competing interest**\ We have no competing interest.\ **Authors’ contribution**\ Both the authors formulated and analyzed the problem. Both contributed to the interpretation of the results.\ **Funding statement**\ There is no funding.\ **Ethics statement**\ It does not apply. [64]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [****,  ()](http://dx.doi.org/10.1038/nphoton.2014.53) [****, ()](https://doi.org/10.1038/s41598-017-08569-8) [****,  ()](https://doi.org/10.1038/s41598-016-0016-1) [****,  ()](http://stacks.iop.org/1367-2630/15/i=11/a=115003) [****,  ()](\doibase 10.1103/PhysRevB.86.195312) [****, ()](\doibase 10.1103/PhysRevA.93.032310) [****,  ()](\doibase 10.1103/PhysRevA.91.053815) [****,  ()](http://stacks.iop.org/1367-2630/17/i=1/a=013032) [****,  ()](\doibase 10.1103/PhysRevA.81.042304) [****,  ()](\doibase 10.1103/PhysRevA.85.050301) [****,  ()](\doibase 10.1103/PhysRevLett.79.1467) [****,  ()](\doibase 10.1103/PhysRevA.89.013853) [****,  ()](\doibase 10.1103/PhysRevA.93.032337) [****,  ()](\doibase 10.1103/PhysRevLett.101.100501) [****,  ()](\doibase 10.1103/PhysRevLett.77.4887) [****,  ()](\doibase 10.1103/PhysRevA.45.5193) [****,  ()](\doibase 10.1103/PhysRevB.82.100507) [****,  ()](\doibase 10.1364/JOSAB.33.001233) [****,  ()](http://dx.doi.org/10.1038/srep31161) [****,  ()](http://stacks.iop.org/1367-2630/17/i=5/a=055013) [****,  ()](\doibase 10.1103/PhysRevLett.86.5486) @noop [****,  ()]{} [****,  ()](http://stacks.iop.org/0295-5075/61/i=1/a=034) [****,  ()](\doibase 10.1103/PhysRevLett.95.180602) [****,  ()](\doibase 10.1103/PhysRevE.79.042101) [****,  ()](\doibase 10.1103/PhysRevE.54.2404) [****,  ()](\doibase 10.1103/PhysRevE.90.042142) [****,  ()](\doibase 10.1103/PhysRevLett.114.196403) [****,  ()](\doibase 10.1103/PhysRevE.91.012143) [****,  ()](\doibase 10.1103/PhysRevE.86.061118) [****,  ()](\doibase 10.1103/PhysRevB.85.195452) [****,  ()](\doibase 10.1103/PhysRevB.94.155411) [****,  ()](\doibase 10.1103/PhysRevE.87.012109) [****,  ()](\doibase 10.1103/PhysRevA.94.052134) [****,  ()](\doibase 10.1103/PhysRevLett.93.184301) @noop [****,  ()]{} [****,  ()](http://stacks.iop.org/0295-5075/84/i=4/a=40009) [****, ()](\doibase 10.1103/PhysRevLett.99.177208) [****, ()](\doibase 10.1103/PhysRevLett.101.267203) [****,  ()](\doibase 10.1063/1.1705319) [****,  ()](\doibase 10.1103/PhysRevB.55.11029) [****,  ()](\doibase 10.1103/PhysRevE.57.2992) [****,  ()](http://stacks.iop.org/1367-2630/19/i=11/a=115006) [****,  ()](http://stacks.iop.org/0253-6102/59/i=3/a=19) @noop [****,  ()]{} [****,  ()](\doibase 10.1126/science.1132898) [****,  ()](\doibase 10.1103/PhysRevB.80.172301) [****,  ()](\doibase 10.1103/PhysRevB.79.014207) [****,  ()](\doibase 10.1103/PhysRevE.89.062109) [****,  ()](\doibase 10.1103/PhysRevLett.88.094302) [****, ()](\doibase 10.1103/PhysRevLett.94.034301) [****,  ()](\doibase doi.org/10.1007/BF01608499) @noop [**]{} (, ) [****, ()](\doibase 10.1103/PhysRevA.53.2857) [****,  ()](\doibase 10.1103/PhysRevLett.67.1716) @noop [**]{} (, ) @noop [**]{} (, ) @noop [****,  ()]{} [****,  ()](\doibase 10.1364/JOSAB.30.000700) [****,  ()](http://stacks.iop.org/1367-2630/12/i=11/a=113032) [****,  ()](\doibase 10.1103/PhysRevA.93.062114) [****, ()](\doibase 10.1103/PhysRevE.94.062143) [****, ()](\doibase 10.1103/PhysRevA.42.3278) [****,  ()](\doibase 10.1103/PhysRevE.76.011111)

--- abstract: 'First-principles alloy theory is used to establish the $\gamma$-surface of Fe-Cr-Ni alloys as function of chemical composition and temperature. The theoretical stacking fault energy (SFE) versus chemistry and temperature trends agree well with experiments. Combining our results with the recent plasticity theory based on the $\gamma$-surface, the stacking fault formation is predicted to be the leading deformation mechanism for alloys with effective stacking fault energy below $\sim 18$ mJm$^{-2}$. Alloys with SFE above this critical value show both twinning and full slip at room temperature and twinning remains a possible deformation mode even at elevated temperatures, in line with observations.' address: - 'Department of Materials Science and Engineering, Royal Institute of Technology, Stockholm SE-100 44, Sweden' - 'Department of Physics and Astronomy, University of Turku, FI-20014 Turku, Finland' - 'Graduate Institute of Ferrous Technology, Pohang University of Science and Technology, Pohang 37673, Korea' - 'Department of Physics and Astronomy, Division of Materials Theory, Uppsala University, Box 516, SE-751210, Uppsala, Sweden' - 'Research Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, Budapest H-1525, P.O. Box 49, Hungary' author: - Wei Li - Song Lu - Dongyoo Kim - Se Kyun Kwon - Kalevi Kokko - Levente Vitos date: 17 November 2015 title: 'First-Principles prediction of the deformation modes in austenitic Fe-Cr-Ni alloys' --- Austenitic steels ,$\gamma$-surface ,First-principles theory ,Plastic deformation mode The stacking fault energy (SFE) of austenitic steels is an important physical parameter closely related to the dislocation-mediated plastic behaviors. Especially, in the so-called transformation-induced plasticity (TRIP) and twinning-induced plasticity (TWIP) steels, SFE is recognized as the fundamental parameter that determines the transition of plastic deformation mode from the $\gamma$- $\epsilon$/$\alpha'$ martensite phase transformation to twinning. Extensive studies have been performed to establish the SFEs in various alloys and the effect of composition, temperature, grain size, strain rate, etc. on the SFE (see Ref.[@Saeed2009] and references therein). It was observed that the deformation-induced martensitic transformation is characteristic for alloys with negative or low SFE. Twinning is the effective deformation mode for intermediate SFE values placed roughly between 18 and 45 mJm$^{-2}$. For high SFEs, plasticity and strain hardening are controlled merely by the glide of full dislocation. [@Cooman2011] However, the upper limit for the TRIP mechanism is diverse in various studies. Sato *et al.* [@1989868] and Allain *et al.* [@Allain2004158] suggested SFE values of 20 and 18 mJ m$^{-2}$, respectively, as the critical values in high-Mn steels separating the TRIP and TWIP mechanisms. Frommeyer *et al.* [@frommeyer2003supra] reported that SFEs larger than about 25 mJm$^{-2}$ lead to twinning in a stable $\gamma$ phase, whereas SFEs smaller than about 16 mJm$^{-2}$ result in $\epsilon$-martensite formation. The SFE may be connected to the stability of the face-centered cubic (fcc) structure with respect to the hexagonal-closed packed (hcp) structures. Within the thermodynamic approach, the SFE is calculated based on the Olson-Cohen model [@Olson19761897], which separates the stacking fault formation energy into contributions from the Gibbs energy difference $\Delta G^{\rm hcp-fcc}$ and the interfacial energy $\sigma$ between the fcc and hcp phases, $viz.$, $$\gamma = 2 \rho \Delta G^{\rm hcp-fcc}+2\sigma. \label{gibbs}$$ where $\rho$ is the molar surface density of the fcc (111) plane. In practice, the interfacial energies are often obtained as the difference between the measured SFE and the thermodynamically calculated $\Delta G^{\rm hcp-fcc}$.[@Olson19761897] In this sense, the resulted interfacial energy includes all the errors between the measured SFE and its first-order approximation. The interfacial energy in Eq. (\[gibbs\]) is somewhat ill-defined and should be distinguished from the real interphase boundary energy or the coherent fcc/hcp interfacial energy, due to their different reference structures. [@Ruihuan2015] Under these circumstance, $\sigma$ has a large uncertainty and is normally accepted in the range of 5$-$27 mJm$^{-2}$. In particular, Pierce *et al.* reported that the interfacial energy ranges from 8 to 12 mJm$^{-2}$ in the TRIP/TWIP steels and from 15 to 33 mJm$^{-2}$ in the binary Fe-Mn alloys. [@Pierce2014238] From Eq. (\[gibbs\]), one may expect that the upper limit for the occurrence of $\epsilon$ phase is when $\Delta G^{\rm hcp-fcc}$ is zero and $\gamma=2\sigma$. Assuming $\sigma$=15 mJm$^{-2}$, SFE of 30 mJm$^{-2}$ was therefore taken as the thermodynamical upper limit of the strain-induced martensite transformation in the thermodynamic studies by Saeed *et al.* [@Saeed2009] In reality, due to the driving force required for the martensitic transformation, smaller critical SFE is expected than the above thermodynamical upper limit. The current development within the quantum-mechanical simulation enables one to access intrinsic material information beyond the experimental ones. In addition, the recent progress in the plasticity theory based on the so-called generalized stacking fault (GSF) energy ($\gamma$-surface) provides fundamentals to fully describe the mechanisms associated with plastic deformations. [@Kibey2006; @Kibey2006fen; @Kibey2007; @Kibey2007a; @Jo2014] The GSF energy comprises several intrinsic energy barriers (IEBs) such as the intrinsic stacking fault energy $\gamma_{\rm isf}$, the unstable stacking fault energy $\gamma_{\rm usf}$, the unstable twinning fault energy $\gamma_{\rm utf}$, the extrinsic stacking fault energy $\gamma_{\rm esf}$, and their combinations. There are laboratory techniques to measure the SFE, but today an experimental determination of the IEBs is not yet feasible. On the other hand, advanced *ab initio* methods have been used to compute the IEBs of metals and simple solid-solutions [@Kibey2006; @Kibey2007; @Kibey2007a; @Kibey2006fen] and most recently also of concentrated alloys [@Li2014]. It was demonstrated that the critical twinning stress ($\tau_{\rm crit}$) in fcc metals and alloys can be quantitatively predicted with these IEBs utilizing a dislocation-based model.[@Kibey2007a] Furthermore, $\tau_{\rm crit}$ was shown to have better correlation with $\gamma_{\rm utf}$ than with $\gamma_{\rm isf}$. In particular, the generalized stacking fault energy of $\gamma$-Fe and Fe alloys at non-magnetic state have been studied using *ab initio* methods. [@Kibey2006fen; @Gholizadeh2013341; @Medvedeva2014475; @Guvenc2015] Non-magnetic $\gamma$-Fe was shown to have negative SFE but a large positive $\gamma_{\rm usf}$. It was shown within the Peierls-Nabarro model that $\gamma_{\rm usf}$ is a critical parameter governing the stacking fault width. A large $\gamma_{\rm usf}$ can result in a finite stacking fault width even when $\gamma_{\rm isf}$ is negative. [@Kibey2006fen] Therefore, theories based on the $\gamma$-surface provide deep insight about the plastic deformation mechanism beyond the classical phenomenological model [@Cooman2011]. However, there were only a few studies on the dependence of GSF energies on the concentration of substitutional elements. [@Medvedeva2014475] Pioneering *ab initio* investigations for Fe-Cr-Ni alloys showed that a proper description of the paramagnetic state is crucial for an accurate description of the SFE of austenitic steels. [@Vitos2006] Recently using first-principles alloy theory, we established the GSF energies of paramagnetic $\gamma$-Fe as a function of temperature. The IEBs allowed us to predict deformation twins in $\gamma$-Fe even at very high temperatures, in spite of the very high SFE. [@wei2015] Following these previous efforts [@Vitos2006; @Li2014; @wei2015], here we present results of *ab initio* calculations for the GSF energy of Fe-Cr-Ni alloys as function of temperature and chemical composition. The results are used to establish the leading deformation modes in austenitic steels. The generalized stacking fault energy was calculated as a total energy change caused by a rigid shift of a part of the fcc structure along the $<$11$\bar{2}$$>$ direction in the (111) slip plane. The calculations were performed with a nine-layer supercell. The local layer relaxation at the SF was considered for all the stacking fault structures. The paramagnetic state was described by the disorder local magnetic moment method. [@Gyorffy1985] The magnetic entropy contribution to the SFE at finite temperature was included in a mean-field manner. The total energies were calculated using the exact muffin-tin orbitals method [@Andersen1994; @Andersen1998; @Vitos2007; @Vitos2001b; @Vitos2000] in combination with the coherent potential approximation. [@Soven1967; @Vitos2001] The one-electron Kohn-Sham equations were solved within the scalar-relativistic approximation and the soft-core scheme. The self-consistent calculations were performed within the generalized gradient approximation proposed by Perdew, Burke and Ernzerhof. [@Perdew1996] For more calculation details, readers are referred to Ref. [@wei2015]. At room temperature (300 K), the experimental lattice parameter ($a$=3.590 Å for Fe$_{71.6}$Cr$_{20}$Ni$_{8.4}$[@Hojjatphd]) was assumed for all the Fe-Cr-Ni alloys studied here (Fe$_{80-x}$Cr$_{20}$Ni$_{x}$, 8$\leq x\leq$20). Hence, we included no lattice parameter dependence on the Ni concentration, considering the fact that Ni has negligible effect on the lattice constant (about -0.0002 Å  per wt.%[@Babu2005; @Vitos2007]). The experimental linear thermal expansion coefficient, $\alpha \approx 15\times10^{-6}$ per K [@Hojjatphd], was adopted to estimate the lattice parameters at different temperatures for all the alloys studied here. ![(Color online) Theoretical $\gamma_{\rm isf}$ (a), $\gamma_{\rm usf}$ (b) and $\gamma_{\rm utf}$ (c) for Fe$_{80-x}$Cr$_{20}$Ni$_{x}$ alloys as function of Ni content at 300 K. The lattice parameters for all alloys are fixed at the experimental value of Fe$_{71.6}$Cr$_{20}$Ni$_{8.4}$ at 300 K, $a=3.590$ Å. [@Hojjatphd][]{data-label="sfe300"}](figure1){width="7cm"} In Fig. \[sfe300\], we show the calculated $\gamma_{\rm isf}$, $\gamma_{\rm usf}$ and $\gamma_{\rm utf}$ for Fe$_{80-x}$Cr$_{20}$Ni$_{x}$ alloys at 300 K with respect to Ni concentration. It is found that $\gamma_{\rm isf}$ increases with Ni addition and the concentration dependence is predicted to be $\sim$1.25 mJm$^{-2}$ per at.% Ni. This theoretical slope is in nice agreement with experimental observations. The linear regression fitting based on experimental SFE values gave the concentration dependence of the SFE in the range of 1.4-2.4 mJm$^{-2}$ per wt.% Ni in austenitic stainless steels.[@Brofman1978; @schramm1975] In absolute value, the present theoretical results also agree well with the experimental data. In particular, the very recent measurements by Lu *et al.* give $18.1\pm1.9$ mJm$^{-2}$ for Fe-20.2Cr-10.8Ni (at.%) and $24.3\pm3.1$ mJm$^{-2}$ for Fe-20.2Cr-19.6Ni (at.%). [@JunLu2015] The calculated $\gamma_{\rm usf}$ and $\gamma_{\rm utf}$ also increase with Ni, however the effect is much weaker than that for $\gamma_{\rm isf}$. Namely, the present $\gamma_{\rm usf}$ and $\gamma_{\rm utf}$ increase with Ni content by $\sim$0.42 and 0.83 mJm$^{-2}$ per at.%, respectively. In Fig. \[sfetem\], we present the temperature dependence of $\gamma_{\rm isf}$, $\gamma_{\rm usf}$, and $\gamma_{\rm utf}$ for Fe-Cr-Ni alloys with various Ni concentrations. It shows that $\gamma_{\rm isf}$ increases with temperature, which is in good agreement with the available experimental data [@Latanision1971] and previous theoretical results[@Vitos2006; @Hojjatphd]. We also observe that with increasing Ni concentration, the temperature slope of SFE becomes smaller. This is due to the fact that Ni addition increases the magnetic moment at the stacking fault, which results in a smaller difference in the magnetic moments between the stacking fault and the fcc matrix. [@vitos2006a] This theoretical trend is in line with the observations. Latanision and Ruff measured the SFE of Fe-18.3Cr-10.7Ni and Fe-18.7Cr-15.9Ni (wt.%) in the temperature range 300-600 K and the resulted $\delta \gamma/\delta$T between 300 and 400 K for the above two alloys were 0.10 and 0.05 mJ m$^{-2}$ K$^{-1}$, respectively. [@Latanision1971] The temperature factors of $\gamma_{\rm usf}$ and $\gamma_{\rm utf}$ for Fe-Cr-Ni alloys are both negative. This is similar to the case of $\gamma$-Fe. [@wei2015] Linear relations may be assumed for both of them with weak higher order terms. $\delta \gamma_{\rm usf}/\delta T$ is approximately 0.045 mJm$^{-2}$ K$^{-1}$ for all alloys studied here, which is comparable to $\delta \gamma_{\rm usf}/\delta T\approx 0.052$ obtained previously for $\gamma$-Fe. [@wei2015] On the other hand, $\delta \gamma_{\rm utf}/\delta T$ increases slightly from $\sim$0.019 mJm$^{-2}$K$^{-1}$ for Fe$_{72}$Cr$_{20}$Ni$_{8}$ to $\sim$0.028 mJm$^{-2}$K$^{-1}$ for Fe$_{60}$Cr$_{20}$Ni$_{20}$. ![(Color online) Theoretical $\gamma_{\rm isf}$ (a), $\gamma_{\rm usf}$ (b) and $\gamma_{\rm utf}$ (c) for Fe$_{80-x}$Cr$_{20}$Ni$_{x}$ ($x$=8-20) as function of temperature.[]{data-label="sfetem"}](figure2){width="7cm"} Utilizing the calculated GSF energies, we may discuss the favorable plastic deformation modes in Fe-Cr-Ni alloys with respect to composition and temperature according to the recently developed plasticity theory for fcc metals and alloys. [@Jo2014] It was proposed that the preferred plastic deformation is decided by the competition between the three effective deformation energy barriers defined as $$\begin{aligned} \label{eq:eebsf} \overline{\gamma}_{sf}(\theta)&=&\frac{\gamma_{usf}}{cos(\theta)},\nonumber\\ \overline{\gamma}_{tw}(\theta)&=&\frac{\gamma_{utf}-\gamma_{isf}}{cos(\theta)},\\ \overline{\gamma}_{sl}(\theta)&=&\frac{\gamma_{usf}-\gamma_{isf}}{cos(60-\theta)},\nonumber\end{aligned}$$ where $\theta$ ($\rm 0^o\leq\theta \leq 60^o$) measures the angle between the stacking fault easy direction $<$11$\bar{2}$$>$ and the applied stress. The activated deformation mode is decided by the lowest effect energy barrier. In particular, when $\overline{\gamma}_{\rm sf}\leq\overline{\gamma}_{\rm tw}$, stacking fault formation (TRIP) is preferred over twinning (TWIP). Notice that the competition between twinning and stacking fault formation is not influenced by the actual value of $\theta$. ![(Color online) Effective Energy barrier difference between $\rm \overline{\gamma}_{tw}(\theta)$ and $\rm \overline{\gamma}_{sf}(\theta)$ at $\rm \theta=0~^o$. The blue “cross" are the original data points, and the contour map is plot with linear interpolation. Negative value indicates the stacking fault mode is preferred, while positive one means the twinning is more favorable.[]{data-label="deform"}](fig3a) In Fig. \[deform\], we plot the difference between $\rm \overline{\gamma}_{\rm tw}$ and $\overline{\gamma}_{\rm sf}$ as a function of Ni concentration and temperature. It is observed that in the left-lower corner of the map (corresponding approximately to $T<$340 K and 8$\leq c_{\rm Ni} \leq$18 at.%), ($\overline{\gamma}_{sf}-\overline{\gamma}_{tw}$) is negative. This indicates that for these systems the activated deformation mode is stacking fault. Twinning becomes the favored deformation mode with increasing temperature and with increasing Ni concentration. Note that twinning always occurs together with the slip mechanism. [@Jo2014] It is interesting that the upper limit for the stacking fault mode at 300 K locates at 8$\lesssim$$c_{\rm Ni}$$\lesssim$11 (at.%). These alloys have SFE around 10-14 mJm$^{-2}$ (see Fig.\[sfetem\] (a)). Hence the theoretical critical SFE ($\gamma_{isf}^{crit}$) is around 14 mJm$^{-2}$, where the deformation mode changes from the $\gamma-\epsilon$/$\alpha'$ martensitic transformation to twinning. We recall that the present fault energies correspond to ideal faults without considering the strain contribution. [@ferreira1998thermodynamic; @Pierce2014238] For parallel partial dislocations in Fe-20Cr-10Ni, the strain contribution is estimated to be around 4 mJm$^{-2}$.[@Pierce2014238] Including this contribution, we arrive to 18 mJm$^{-2}$ for the critical value of the effective stacking fault energy separating the TRIP and TWIP mechanisms. The above theoretical predictions are in line with observations. The chemical composition of Fe$_{72}$Cr$_{20}$Ni$_{8}$ is close to the type 304 stainless steels. It is well documented that under low temperature deformation in the 304 stainless steels $\gamma$-austenite transforms to $\epsilon$-martensite which is usually considered as an intermediate phase before transforming to the more stable $\alpha'$ phase. [@Suzuki19771151] To further understand the disclosed TRIP/TWIP transition, we make use the so-called “universal scaling law" which describes the relations between the IBEs, $viz.$[@Jo2014; @Jin2011], $$\label{eq:univ} \gamma_{utf}=\gamma_{usf}+1/2~\gamma_{isf}+\delta.$$ For the elemental fcc metals (except e.g., Pt) and solid solutions (e.g. Cu-X (X=Al, Zn, Ni, and Ga) Pd-Ag, etc.), $\delta$ is close to zero.[@Jin2011; @Li2014] Combining the criteria for activating the stacking fault mode ($\overline{\gamma}_{sf} \le \overline{\gamma}_{tw}$) with Eqs. (\[eq:eebsf\]) and (\[eq:univ\]), we arrive at $$\gamma_{isf}^{crit} \le 2~\delta.$$ This expression implies that the upper limit for the martensite transformation is in fact given by the deviation from the “universal scaling law" expressed in terms of $\delta$. Using Eq. (\[eq:univ\]), one can derive an explicit expression for the critical SFE value by means of the energy barriers, *viz.* $$\gamma_{isf}^{crit} \le \gamma_{utf}-\gamma_{usf}.$$ That is, when the intrinsic stacking fault energy is below the difference between the two leading unstable energy barriers then stacking fault formation is the preferred deformation mode against twinning. The explanation is that twinning is always activated from a pre-existing stacking fault situation and thus the effective twinning barrier is reduced by the stacking fault energy. When this reduction is not enough to lower $\overline{\gamma}_{\rm tw}$ below $\overline{\gamma}_{\rm sf}$ then stacking fault formation survives in spite of the positive SFE energy. That can be realized in low SFE materials. Finally, in order to study the competition between twinning and full-slip modes, we calculated the difference between $\overline{\gamma}_{\rm tw}$ and $\overline{\gamma}_{\rm sl}$ for the studied ranges of composition and temperature (not shown). We found that at $\theta$=0$^{\rm o}$, ($\overline{\gamma}_{\rm tw}$ - $\overline{\gamma}_{\rm sl}$) is negative which indicates that twinning is always one of the active deformation modes when the stacking fault mode is suppressed for the present steels, irrespectively of the temperature. We notice that slip is activated for non-zero $\theta$. [@Jo2014] In other words, twinning is predicted to be the second deformation mode for the present alloys at all temperatures considered here. As far as we rely on the empirical relationship between the SFE and deformation mode [@Cooman2011], it is quite unexpected to detect deformation twins in Fe-Cr-Ni alloys at temperature as high as 1000 K where the SFE is rather high ($\approx 50$ mJm$^{-2}$ in Fig.\[sfetem\] (a)). The microstructure in the type 304L stainless steel deformed at 1473 K for various strain rates was studied by Sundararaman *et al.* [@Sundararaman19931077; @SUNDARARAMAN19941617] Deformation twins were actually observed in the entire range of strain rates and slip was also reported to coexist with twins. Similarly, a large density of deformation twins was also found in the type 316L stainless steels deformed at 10$^{-2}$ s$^{-1}$ in the temperature range of 873-1473 K. [@SUNDARARAMAN19941617] In conclusion, the present study clarifies that it is not the low SFE at room temperature that actually ensure the occurrence of deformation twins at high temperature but rather the competition between various intrinsic energy barriers. To summarize, we have used ab initio alloy theory to study the deformation modes in Fe-Cr-Ni alloys. We have predicted that $\epsilon$ phase formation is present in alloys with effective stacking fault energy as large as 18 mJm$^{-2}$, whereas twinning remains an active deformation mode even at high temperatures. The nice agreement between our theoretical prediction and the experimental observations emphasizes the importance of the intrinsic energy barriers for a better account for the deformation mechanisms in austenitic steels. Valuable discussion with Staffan Hertzman and Erik Schedin at Outokumpu Aversta Research Center is highly appreciated. This work was supported by the Swedish Research Council, the Swedish Foundation for Strategic Research, the Carl Tryggers Foundation, the Chinese Scholarship Council and the Hungarian Scientific Research Fund (OTKA 84078 and 109570). Se Kyun Kwon acknowledges the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (NRF-2014R1A2A1A12067579). Song Lu acknowledges Magnus Ehrnrooth foundation for providing a Postdoc. Grant. We acknowledge the Swedish National Supercomputer Centre in Linköping for computer resources. [40]{} natexlab\#1[\#1]{}\[1\][`#1`]{} \[2\][\#2]{} \[1\][\#1]{} \[1\][[](http://dx.doi.org/#1)]{} \[1\][[](pmid:#1)]{} \[2\][\#2]{} , , , , () . , , , () . , , , , () . , , , , , () . , , , () . , , () . , , , , , (). . , , , , , , () . , , , , (). , , , , , () . , , , , () . , , , , () . , , , , , , () . , , , , , , () . , , , () . , , , , () . , , , , () . , , , () . , , , , , , , (). . , , , , , () . , , , , , . , , , , , , , () . , , , , . , () . , , , , () . , () . , , , () . , , , () . , Ph.D. thesis, Montanuniversität Leoben, . , , , , , , , () . , , () . , , () . , , , , , , , , . . , , () . , , , () . , , () . , , , , , () . , , , , , () . , , , () . , , , () .

--- abstract: 'Ferrogels consist of magnetic colloidal particles embedded in an elastic polymer matrix. As a consequence, their structural and rheological properties are governed by a competition between magnetic particle–particle interactions and mechanical matrix elasticity. Typically, the particles are permanently fixed within the matrix, which makes them distinguishable by their positions. Over time, particle neighbors do not change due to the fixation by the matrix. Here we present a classical density functional approach for such ferrogels. We map the elastic matrix-induced interactions between neighboring colloidal particles distinguishable by their positions onto effective pairwise interactions between indistinguishable particles similar to a “pairwise pseudopotential”. Using Monte-Carlo computer simulations, we demonstrate for one-dimensional dipole-spring models of ferrogels that this mapping is justified. We then use the pseudopotential as an input into classical density functional theory of inhomogeneous fluids and predict the bulk elastic modulus of the ferrogel under various conditions. In addition, we propose the use of an “external pseudopotential” when one switches from the viewpoint of a one-dimensional dipole-spring object to a one-dimensional chain embedded in an infinitely extended bulk matrix. Our mapping approach paves the way to describe various inhomogeneous situations of ferrogels using classical density functional concepts of inhomogeneous fluids.' author: - 'P. Cremer' - 'M. Heinen' - 'A. M. Menzel' - 'H. Löwen' bibliography: - 'references.bib' title: A density functional approach to ferrogels --- Introduction {#Sec.Introduction} ============ Classical density functional theory for inhomogeneous fluids is nowadays used for many-body systems governed by a pair potential (such as hard or soft spheres) and has found widespread applications for phase separation, freezing and interfacial phenomena, for reviews see Refs. . In a one-component system, though classical, these particles are indistinguishable in principle according to standard statistical mechanics [@Hansen2006_book], which implies that the interaction between two particles is the same for any pair of particles provided they are at the same separation. This standard assumption breaks down for particles embedded in an elastic polymeric gel, if the particles are anchored to the surrounding gel matrix and/or cannot diffuse or propagate through it. In this case, the particles can be labeled according to their position in the matrix with their interaction energy persistently depending on the labeling. Thus, they are distinguishable. As a basic example, this situation is encountered for a simple bead-spring model, where the springs represent the elasticity and connectivity provided by the matrix and the beads represent the particles. Particle distinguishability leads to a different combinatorial prefactor in the classical partition function and therefore affects the entropy [@Gibbs1957_book]. However, at high density, a fluid of indistinguishable particles typically undergoes a freezing transition into a crystal. At low temperature, this crystal can be modeled by a harmonic solid [@Chaikin2000_book; @Espanol1996_PhysRevE; @Jancovici1967_PhysRevLett], where the neighboring particles are connected by springs. In fact, this effective model of distinguishable particles provides a good approximation as the free energy of the system is dominated by the particle interactions that overwhelm the combinatorial contribution. In the present paper, we exploit this idea to introduce a density functional approach for ferrogels and related systems. Such ferrogels consist of magnetic colloidal particles that are embedded in a polymeric gel matrix [@filipcsei2007magnetic; @ilg2013stimuli; @menzel2015tuned; @odenbach2016microstructure; @lopez2016mechanics]. Examples of similar materials are given by magnetic elastomers, magnetorheological elastomers, or magnetosensitive elastomers, where often these terms are used interchangeably. Remarkably, the structural properties and rheological behavior of these materials are governed by a competition between the magnetic particle–particle interactions and the mechanical elasticity of the embedding polymeric gel matrix. As a consequence, it is possible to tune their properties during application by modifying the magnetic interactions via external magnetic fields. Therefore, these magnetorheological systems have many prospective and promising applications, such as tunable dampers [@sun2008study] or vibration absorbers [@deng2006development]. The theoretical description of these materials is challenging. While the specific properties arise on the mesoscopic colloidal particle scale, for practical applications one is interested in the overall macroscopic response. To connect these scales in simulations, large numbers of individual particles need to be covered. For this purpose, recent work has focused on simplified minimal models. Starting on the microscale, at most a few individual polymer chains are resolved by coarse-grained bead-spring models [@weeber2012deformation; @ryzhkov2015coarse; @weeber2015ferrogels]. In still more reduced mesoscopic dipole-spring models, the elasticity of the matrix is directly represented by effective spring-like interactions between the particles, combined with long-ranged magnetic dipolar interactions between them [@annunziata2013hardening; @pessot2014structural; @tarama2014tunable; @ivaneyko2015dynamic; @pessot2016dynamic]. More explicit approaches treat the matrix directly by continuum elasticity theory, yet at the price of reduced accessible overall particle numbers [@han2013field; @cremer2015tailoring; @cremer2016superelastic; @metsch2016numerical]. A kind of compromise between the two concepts can be found in Refs.  and . Previous analytical approaches to link the different scales often relied on substantially simplifying idealizations concerning the positional particle configurations [@ivaneyko2011magneto; @menzel2014bridging; @zubarev2016towards]. Therefore, it is desirable to develop statistical means that allow for a more profound connection between the different scales in the future. As a step in this direction, we now suggest to employ the framework of classical density functional theory for a characterization of these complex materials. Here we mainly follow the dipole-spring concept of distinguishable particles often used for the description of ferrogels. In order to keep the models simple, we study effective one-dimensional set-ups. Such a situation is realized, for instance, for elongated magnetic particle chains embedded into an elastic matrix [@huang2016buckling], but also for magnetic filaments [@sanchez2013effects] made, *e.g.*, of magnetic colloidal particles connected by DNA polymer strands [@dreyfus2005microscopic]. We map this system with its particle-distinguishing connectivity onto another one with an effective connectivity and indistinguishable particles [@Denton2007_inbook]. Based on the considerations above, one expects a good agreement between real and effective connectivity at least for strong particle–particle interactions. We use Monte-Carlo computer simulations of both situations and confirm that the results agree at high packing fractions and/or strong particle interactions. This opens the way to employ statistical-mechanical theories like classical density functional theory to also describe systems of particles that are, in principle, distinguishable. For the one-dimensional model, we use the exact Percus free-energy functional for hard rods [@Percus1976_JStatPhys] combined with a mean-field theory for the elastic and dipolar interactions and minimize the resulting grand canonical free energy functional with respect to the equilibrium one-body density field. We study two different models. In the first one, the elastic matrix is represented by harmonic springs between nearest-neighboring particles. Including thermal fluctuations, such a simple one-dimensional bead-spring model cannot show a phase transition [@VanHove1952_Physica; @Cuesta2004_JStatPhys]. Thermal fluctuations have a strong impact in one spatial dimension and fuel the Landau-Peierls instability [@Landau2008_UkrJPhys; @Peierls1934_HelvPhysActa; @Chaikin2000_book], which impedes periodic ordering. This fact is captured by our Monte-Carlo simulations, which take all contributions by thermal fluctuations into account exactly. Our mean-field density functional theory, however, introduces an artificial crystallization at low temperatures. Still, at higher temperatures we can obtain qualitative agreement between density functional theory and Monte-Carlo simulations for the pressure and compressibility of the system. These provide key material properties for the practical use of ferrogels. Later in this paper, we turn to an extended model, including an additional external elastic pinning potential for the colloidal magnetic particles. Such pinning potentials arise when the particles are embedded in a three-dimensional elastic bulk matrix [@huang2016buckling]. Similarly, the displacement of one embedded particle results in a matrix-mediated force on all other particles, so that we have additional long-ranged elastic particle–particle interactions. In combination with the pinning potential, this suppresses the Landau-Peierls instability even though our model is effectively one-dimensional. Consequently, our density functional theory immediately shows much better agreement with Monte-Carlo simulations for both, the model with real connectivity and the version mapped towards indistinguishable particles. During the synthesis of ferrogels such permanent straight one-dimensional magnetic particle chains embedded within three-dimensional ferrogel blocks are readily generated by applying external magnetic fields during the manufacturing process [@collin2003frozen; @varga2003smart; @gunther2012xray; @borbath2012xmuct; @gundermann2013comparison; @huang2016buckling; @Gundermann2017_SmartMaterStruct]. We remark at this stage that the problem of mapping from distinguishable to indistinguishable particles also occurs in density functional descriptions of polymeric bead models [@Chandler1986_JChemPhys_1; @Chandler1986_JChemPhys_2]. Typically, in tangential bead models for hard spheres [@Gu2003_JChemPhys; @Slyk2016_JPhysCondensMatter], one neglects the linking constraints of the chain and maps the excess free energy of the system onto an unconstrained hard-sphere fluid. Our analysis paves the way to a future application of density functional theory of freezing also to two- and three-dimensional ferrogel models. There, we anticipate thermal fluctuation effects to be of less influence, leading to a better agreement with simulations. It will further be useful to characterize other particulate systems embedded in a permanent elastic matrix such as electrorheological elastomers [@an2003actuating; @allahyarov2015simulation] or possibly even drug carriers and compartments within biological tissue [@tietze2013efficient]. This paper is organized as follows. In Sec. \[Sec.Dipole-spring\_model\], we describe our first one-dimensional dipole-spring model and offer a method to map the real connectivity to an effective one. Next, we describe the various methods used to study our systems in Sec. \[Sec.Methods\]. These methods are mainly mean-field density functional theory and canonical Monte-Carlo simulations. The supplemental material [@supplemental] contains an additional treatment using the Zerah-Hansen liquid-integral equation. Then we complete the discussion of the first dipole-spring model in Sec. \[Sec.Results\], where we present results from our density functional theory and compare them to Monte-Carlo simulations. Subsequently, we proceed to our extended model in Sec. \[Sec.Embedding\_into\_the\_elastic\_matrix\]. Following a motivation of this extended model, we again compare results from density functional theory and Monte-Carlo simulations, showing their improved agreement. Finally, in Sec. \[Sec.Conclusions\], we revisit our overall approach and discuss prospective uses and extensions beyond the one-dimensional models discussed here. Dipole-spring model {#Sec.Dipole-spring_model} =================== We consider the following one-dimensional dipole-spring model, which is sketched in Fig. \[Fig.dipole\_spring\_model\]. There are two outer particles at a fixed distance $L$, forming the system boundary, and $N$ mobile particles in between. All particles have a hard core of diameter $d$, which limits the closest approach of two particle centers to this distance. Additionally, all particles carry magnetic dipole moments of magnitude $m$ that all point in the same direction aligned with the system axis. Finally, each particle is connected to its nearest neighbors by a harmonic spring of spring constant $k$ and equilibrium length $\ell$. As we will discuss below, the connectivity introduced by the harmonic springs renders the particles distinguishable. We label the particles with indices $i = 0, \dots, N+1$ according to their position $x_i$ in ascending order. The indices $i = 0$ and $i = N+1$ are used for the left and right boundary particles, respectively. The total potential energy of the system consists of three contributions $$U = U_\textrm{h} + U_\textrm{m} + U_\textrm{e}, \label{Eq.dipole_spring_model_potential}$$ *i.e.*, the hard core repulsion $U_\textrm{h}$, the magnetic dipolar interaction $U_\textrm{m}$, and the elastic interaction $U_\textrm{e}$. We can write the former two as sums over the interactions between all particle pairs $i,j$ with $j > i$ $$U_\textrm{h} = \sum_{i = 0}^{N+1} \sum_{j > i} u_\textrm{h}(x_{ij}) \, ; \quad u_\textrm{h}(x) = \begin{cases} \infty & \textnormal{for} \,\, x < d \, , \\ 0 & \textnormal{for} \,\, x \geq d \, , \end{cases} \label{Eq.hard_interaction}$$ $$U_\textrm{m} = \sum_{i = 0}^{N+1} \sum_{j > i} u_\textrm{m}(x_{ij}) \, ; \quad u_\textrm{m}(x) = -2 \frac{\mu_0}{4\pi} \frac{m^2}{x^3} \, , \label{Eq.magnetic_interaction}$$ where $\mu_0$ is the vacuum permeability and $x_{ij} := |x_j - x_i|$ is the distance between a pair of particles. Since the pair interactions $u_\textrm{h}(x)$ and $u_\textrm{m}(x)$ do not depend on any particular labeling of the particles, the total interactions $U_\textrm{h}$ and $U_\textrm{m}$ are invariant under a relabeling of all particles. In contrast to that, the elastic interactions between nearest-neighbors $$U_\textrm{e} = \frac{k}{2} \sum_{i = 0}^N (x_{i,i+1} - \ell)^2 \label{Eq.elastic_interaction}$$ persistently depend on the labeling and therefore render the particles distinguishable. ![Sketch of our one-dimensional dipole-spring model for a ferrogel. Two outer particles (blue) form the system boundary and are at a fixed distance $L$. Additionally, there are $N$ mobile particles (dark gray) in between. Each particle carries a magnetic dipole moment of magnitude $m$, all of which point into the same direction along the system axis. Finally, harmonic springs of spring constant $k$ and equilibrium length $\ell$ connect each particle to its nearest neighbors.[]{data-label="Fig.dipole_spring_model"}](dipole_spring_model){width="1.0\columnwidth"} To facilitate a description of these systems with the tools of statistical mechanics, we map it onto a system of indistinguishable particles. This can be achieved by replacing the elastic interaction with an approximative potential $\tilde{U}_\textrm{e}$ that can be decomposed into pairwise interactions $\tilde{u}_\textrm{e}(x)$. Ideally, such an approximative potential should still affect only nearest neighbors and provide the same result as Eq.  under realistic circumstances. We make the choice $$\begin{gathered} \tilde{U}_\textrm{e} = \sum_{i = 0}^{N+1} \sum_{j > i} \tilde{u}_\textrm{e}(x_{ij}) \, ; \\ \tilde{u}_\textrm{e}(x) = \begin{cases} \frac{k}{2} \left[ (x - \ell)^2 - (2d - \ell)^2 \right] & \textrm{for} \,\, x < 2d \, , \\ 0 & \textnormal{for} \,\, x \geq 2d \, . \end{cases} \end{gathered} \label{Eq.pseudoelastic_interaction}$$ The “pseudo-spring” pair potential $\tilde{u}_\textrm{e}(x)$ is illustrated in Fig. \[Fig.Vpair\] and consists of a harmonic well of spring constant $k$ centered around a distance $\ell < 2d$. The harmonic well is cut and shifted to zero potential strength at a distance $x \geq 2d$. For two particles at a distance $x < 2d$, this potential acts as a common harmonic spring. Beyond this distance, the spring “breaks” leading to zero interaction. The combination with the hard-core repulsion $u_\textrm{h}(x)$ in Eq.  limits the possible harmonic interaction to pairs of nearest-neighbors. Only nearest neighbors can be at a distance $x < 2d$. Next-nearest neighbors are always at a greater distance and, thus, excluded from the interaction. ![Illustration of how the pseudo-spring pair potential $\tilde{u}_\textrm{e}(x)$ combined with the hard-core repulsion $u_\textrm{h}(x)$ serves to approximate the effect of harmonic springs between nearest neighbors. A harmonic well of spring constant $k$ (here $k = 40 k_BT / d^2$) is centered around a distance $\ell = 1.5d$. It is cut at a distance $x = 2d$ and shifted to zero in order to confine the interaction to nearest neighbors only. The sketch on the left depicts a situation where a particle (blue) interacts with its nearest neighbor (red), as both particles are at a distance $x < 2d$. In the sketch on the right no interaction takes place since the distance is $x \geq 2d$. In any case, only nearest neighbors can ever interact as we always have $x \geq 2d$ for all other pairs of particles.[]{data-label="Fig.Vpair"}](Vpair){width="1.0\columnwidth"} In the following, we refer to this potential as “pseudo-spring” interaction as opposed to the “real-spring” permanent connectivity between nearest neighbors. Good agreement between the real-spring system and its mapped version using pseudo-springs can be expected in situations where the pseudo-springs do not break. First, this is the case at high packing fraction, when the confinement enforces small distances between nearest neighbors. The packing fraction in our finite system is defined as $$\phi = \frac{N d}{L - d} \, , \label{Eq.packing_fraction}$$ because $L - d$ is the system length enclosed between the two hard boundary particles. At $\phi > (L-2d) / (L-d)$, the distance between nearest neighbors is smaller than $2d$ everywhere, such that breaking of pseudo-springs becomes impossible. Another limit is reached at high potential strength (large value of $k$) and moderate packing fraction of $\phi \gtrsim d / \ell$. Under these conditions, the harmonic well, as illustrated in Fig. \[Fig.Vpair\], is deep compared to the thermal energy $k_BT$ and the system is sufficiently filled such that all particles are effectively trapped in the harmonic wells created by their nearest neighbors. For fixed values of $L$, $N$, and $d$, the physical input parameters determining all interactions are the magnetic moment $m$, the spring constant $k$, and the spring equilibrium length $\ell$. From now on, we measure all energies in units of $k_BT$ and all lengths in units of the particle diameter $d$. This implies to measure the spring constant in units of $k_0 = k_BT / d^2$ and the magnetic moment in units of $m_0 = \sqrt{\frac{4\pi}{\mu_0} k_BT d^3}$, while the pressure and the compression modulus are given in units of $p_0 = K_0 = k_BT / d$. Methods {#Sec.Methods} ======= We use three different methods to study our dipole-spring model. The first and most notable one is our density functional theory (DFT) description, for which we use the pseudo-spring approximation to make particles indistinguishable. Second, we perform canonical Monte-Carlo (MC) simulations for real springs as well as for pseudo-springs as a benchmark to test our DFT results. Finally, we have also solved the Zerah-Hansen liquid-integral equation to show that our pseudo-spring approximation is meaningful beyond the scope of DFT, see the supplemental material [@supplemental] for results and a description of the method. Density functional theory ------------------------- The central statement of classical DFT is that for a fixed temperature $T$ and interparticle pair potential $u(x)$, the Helmholtz free energy $\mathcal{F}[\rho]$ is a unique functional of the one-body density distribution $\rho(x)$. Likewise, there is a unique grand canonical free energy functional $\Omega[\rho]$ describing the system when it is exposed to an external potential $U_\textrm{ext}(x)$ and a particle reservoir at chemical potential $\mu$. This grand canonical free energy functional has the form [@Tarazona2008_incollection] $$\Omega[\rho] = \mathcal{F}[\rho] + \int_0^L \rho(x) \big( U_\textrm{ext}(x) - \mu \big) \, dx \label{Eq.DFT_grand_functional}$$ and is minimized by the equilibrium one-body density profile $\rho_\textrm{eq}(x)$. The minimum $\Omega[\rho_\textrm{eq}]$ corresponds to the thermodynamic grand canonical free energy in equilibrium. Unfortunately, the exact free energy functional $\mathcal{F}[\rho]$ is usually unknown, so that one has to resort to approximations. These approximations usually start by splitting the free energy functional $\mathcal{F}[\rho] = \mathcal{F}_\textrm{id}[\rho] + \mathcal{F}_\textrm{ex}[\rho]$ into the exact free energy for the ideal gas $$\mathcal{F}_\textrm{id}[\rho] = k_BT \int_0^L \rho(x) \bigg( \ln\big( \Lambda \rho(x) \big) - 1 \bigg) \, dx \, \label{Eq.DFT_ideal_functional}$$ with $\Lambda$ the thermal de Broglie wavelength, plus an excess contribution $\mathcal{F}_\textrm{ex}[\rho]$. For some special problems in one spatial dimension, the exact excess contribution can be derived [@Tutschka2000_PhysRevE]. One such example is the Percus excess functional [@Percus1976_JStatPhys] for the one-dimensional hard-rod fluid, $$\begin{gathered} \begin{aligned} \mathcal{F}_\textrm{ex}^\textrm{P}[\rho] = -k_BT \int_{0}^L & \frac{\rho(x + d/2) + \rho(x - d/2)}{2} \\ &\quad\times\, \ln\big( 1 - \eta(x) \big) \, dx \, , & \end{aligned} \\ \textrm{with } \eta(x) = \int_{x-d/2}^{x+d/2} \rho(x') \, dx' \, . \end{gathered} \label{Eq.DFT_Percus_functional}$$ It takes one-dimensional hard repulsions exactly into account and, thus, provides a good starting point for the construction of a functional describing our dipole-spring model. Here, we combine it with an approximate mean-field excess functional accounting for the soft pair interactions consisting of our pseudo-spring pair potential $\tilde{u}_\textrm{e}(x)$ and the magnetic dipolar pair interaction $u_\textrm{m}(x)$, $$\begin{aligned} \mathcal{F}_\textrm{ex}^\textrm{MF}[\rho] &= \int_0^L \int_0^L \, \big( \tilde{u}_\textrm{e}(|x - x'|) + u_\textrm{m}(|x - x'|) \big) \\ &\qquad \times g(|x - x'|) \rho(x) \rho(x') \, dx' \, dx \, , \end{aligned} \label{Eq.DFT_meanfield_functional}$$ where the distribution function $g(x)$ satisfies the no-overlap condition $g(x) = 0$ for $x < d$. The mean-field approximation assumes that the pair potentials are soft enough to regard the particle positions as basically uncorrelated [@Tarazona2008_incollection]. Here we make the simplifying assumption that $g(x) = 1$ for all distances $x > d$. In total, our free energy functional is given by $$\mathcal{F}[\rho] = \mathcal{F}_\textrm{id}[\rho] + \mathcal{F}_\textrm{ex}^\textrm{P}[\rho] + \mathcal{F}_\textrm{ex}^\textrm{MF}[\rho] \, . \label{Eq.DFT_dipole-spring_functional}$$ The boundary of our finite systems consists of the leftmost and rightmost particles, which are fixed but otherwise identical to the enclosed particles, see Fig. \[Fig.dipole\_spring\_model\]. Their influence on the enclosed density profile enters via an external potential $$U_\textrm{ext}(x) = u(L - x) + u(x) \, , \label{Eq.DFT_external_potential}$$ where $u(x) = u_\textrm{h}(x) + u_\textrm{m}(x) + \tilde{u}_\textrm{e}(x)$. This completes our grand canonical free energy functional $\Omega[\rho]$. Functional derivation of Eq.  leads to the Euler-Lagrange equation $$\frac{\delta \Omega[\rho]}{\delta \rho(x)} = \frac{\delta \mathcal{F}[\rho]}{\delta \rho(x)} + U_\textrm{ext}(x) - \mu \overset{!}{=} 0 \, , \label{Eq.DFT_Euler-Lagrange}$$ which can be used to determine the equilibrium density profile minimizing $\Omega[\rho]$. In practice, however, we numerically calculate our equilibrium density profile $\rho(x)$ by performing a dynamical relaxation of $\Omega[\rho]$ [@Loewen1993_JChemPhys]. This scheme fixes the average particle number $\langle N \rangle$ instead of the chemical potential $\mu$ and is described in detail in the appendix. After the relaxation, we have access to the grand canonical free energy $\Omega$. This enables us to calculate a pressure $p = -\frac{\partial \Omega}{\partial L}\big\vert_{\langle N \rangle, T}$ and a compression modulus $K = -L \frac{\partial p}{\partial L}\big\vert_{\langle N \rangle, T}$ by varying the system length $L$ at fixed average particle number $\langle N \rangle$ and probing the corresponding change in $\Omega$. Monte-Carlo simulation ---------------------- We perform canonical Monte-Carlo (MC) simulations at fixed particle number $N$, system length $L$, and temperature $T$ [@Frenkel2001_book]. After equilibrating the systems, we sample the pressure $p$, the compression modulus $K$, and the equilibrium density profile $\rho(x)$. To sample the pressure, we affinely deform the system by a factor $(L + \Delta L)/L$ and probe the corresponding change in Helmholtz free energy $F(N,L,T)$. $\Delta L$ is a small change in system length. It can be shown that the pressure is related to the acceptance ratio of such volume moves by [@Harismiadis1996_JChemPhys] $$\begin{aligned} p &= -\frac{\partial F(N,L,T)}{\partial L} \approx -\frac{F(N,L + \Delta L,T) - F(N,L,T)}{\Delta L} \\ &= \frac{k_BT}{\Delta L} \ln\left\langle \left(\frac{L + \Delta L}{L}\right)^N \exp(-\Delta U / k_BT) \right\rangle \, . \end{aligned} \label{Eq.Widom_pressure}$$ $\Delta U$ is the change in system energy associated with the volume move and $\langle \cdot \rangle$ denotes the ensemble average. In order to capture the pressure contributions of the hard repulsions in our systems, the volume moves must be compressive ($\Delta L < 0$). Given the pressure, the compression modulus can be calculated using $K = -L \frac{\partial p}{\partial L}\big\vert_{N,T}$. Results {#Sec.Results} ======= In the following, we present results for our one-dimensional dipole-spring model. First we concentrate on a non-magnetic system to test the feasibility of the mapping onto indistinguishable particles. Then we add the magnetic interaction and discuss how this affects the density profile and the pressure in our systems. Finally, we turn to the thermodynamic compression modulus, which is a key quantity to characterize ferrogel systems as it can be controlled by changing the magnetic properties. Non-magnetic system ------------------- First of all, we confirm within our MC-simulations that the pseudo-spring pair potential is an appropriate replacement for real springs between nearest neighbors. Figure \[Fig.real\_vs\_pseudo\_springs\] compares the equations of state $p(\phi)$ for both situations in a system of length $L = 51 d$. The spring parameters $k = 40 k_0$ and $\ell = 1.5d$ are the same as in Fig. \[Fig.Vpair\]. ![Comparison of the equations of state from MC-simulations of the real-spring system with the mapped version using pseudo-springs instead. In both systems we have $L = 51 d$, $k = 40 k_0$ and $\ell = 1.5d$. At a packing fraction $\phi \gtrsim d / \ell = \frac{2}{3}$, the pseudo-spring system is filled with particles trapped in the harmonic wells of their nearest neighbors and, thus, behaves essentially identical to the system featuring real springs. For lower packing fractions in the real-spring system, the springs between nearest neighbors are stretched on average, so that the system would contract if the boundaries were not fixed. Thus, the pressure is negative for these packing fractions.[]{data-label="Fig.real_vs_pseudo_springs"}](real_vs_pseudo_springs){width="1.0\columnwidth"} Using these parameters, we can confirm that at packing fractions $\phi \gtrsim d / \ell = \frac{2}{3}$ the mapping to indistinguishable particles using pseudo-springs works well. Let us now compare MC and DFT results using the same parameters. Figure \[Fig.MC\_vs\_DFT\_densityprofile\_k40\] shows three density profiles $\rho(x)$ at a packing fraction $\phi = \frac{2}{3}$, one from the real-spring MC, one from the pseudo-spring MC, and one from DFT. While the two MC density profiles expectedly agree with each other and display a liquid-like behavior near rigid boundaries, the DFT density profile is qualitatively different and resembles a crystal. ![Density profiles $\rho(x)$ obtained from real-spring and pseudo-spring MC as well as from DFT calculations at a packing fraction $\phi = \frac{2}{3}$ and otherwise with the same parameters as in Fig. \[Fig.real\_vs\_pseudo\_springs\]. For this packing fraction, the two MC-simulations are in good agreement and show a liquid-like behavior as expected in one spatial dimension. However, the density profile obtained from DFT is qualitatively different and displays an artificial crystalline behavior.[]{data-label="Fig.MC_vs_DFT_densityprofile_k40"}](MC_vs_DFT_densityprofile_k40){width="1.0\columnwidth"} This crystalline appearance displayed by the DFT is unphysical. Our system is one-dimensional, all particle interactions are short-ranged and there are no external fields. For such systems, the existence of a phase transition can be ruled out [@Cuesta2004_JStatPhys; @Dyson1969_CummunMathPhys; @Ruelle1968_CommunMathPhys; @VanHove1952_Physica]. This fact is accurately captured by our MC-simulations that display a liquid phase even for this high value of $k = 40 k_0$, as they explicitly include all effects of thermal fluctuations. In one spatial dimension, thermal fluctuations have a particularly strong effect. They can escalate into long-ranged fluctuations scaling in amplitude with the system size, capable of destroying periodic ordering. This is the well-known Landau-Peierls instability [@Landau2008_UkrJPhys; @Peierls1934_HelvPhysActa; @Chaikin2000_book]. Within the DFT, some thermal fluctuations are introduced by the ideal gas term \[see Eq. \], which pushes the system towards disorder. However, the mean-field term \[see Eq. \] excludes other contributions by fluctuations. We conclude that this term is responsible for the unphysical crystallization. Our conjecture is supported by setting $k = 0$ and $m = 0$, *i.e.* setting the mean-field term to zero. Then, we recover the hard-rod fluid also for the DFT and find perfect agreement with MC-simulations, see Fig. \[Fig.MC\_vs\_DFT\_hardrods\]. ![When setting $k = 0$ and $m = 0$, we recover the hard-rod fluid and observe perfect agreement between MC and DFT. In this case, both density profiles $\rho(x)$ show liquid-like behavior and the equations of state (inset) match the exact result $p(\phi) = \frac{\phi}{1 - \phi} p_0$ [@Percus1976_JStatPhys]. This demonstrates that the mean-field term in Eq.  is responsible for the disagreement between DFT and MC, as it disregards some of the contributions by thermal fluctuations.[]{data-label="Fig.MC_vs_DFT_hardrods"}](MC_vs_DFT_hardrods){width="1.0\columnwidth"} The Landau-Peierls instability is well-known to be most prominent in one spatial dimension. In future studies in two and three dimensions, we therefore expect a significantly weaker effect of the thermal fluctuations, which should lead to a better agreement between simulations and mean-field DFT. For now, we achieve qualitative agreement between DFT and MC by raising the temperature (which means decreasing $k$) until the DFT system enters the liquid state. Figure \[Fig.MC\_vs\_DFT\_k4\_mm0\] shows a comparison between density profiles as well as equations of state for the same systems as in Fig. \[Fig.MC\_vs\_DFT\_densityprofile\_k40\], but with a ten times lower spring constant $k = 4 k_0$. ![Density profiles as in Fig. \[Fig.MC\_vs\_DFT\_densityprofile\_k40\] but using a ten times lower spring constant $k = 4 k_0$. Now the depth of the harmonic well of the pseudo-spring pair potential is of the order of $k_BT$ such that pseudo-springs frequently break. As a result, DFT and pseudo-spring MC both show liquid-like behavior and are in much better agreement. However, this comes at the price of worse agreement between the pseudo-spring MC and the real-spring MC. The inset shows equations of state $p(\phi)$ for these three systems which confirm these observations. There is agreement between DFT and the pseudo-spring MC at least in the range around $\phi \approx \frac{2}{3}$ but the pseudo-spring and real-spring MC only agree at very high packing fractions.[]{data-label="Fig.MC_vs_DFT_k4_mm0"}](MC_vs_DFT_k4_mm0){width="1.0\columnwidth"} The depth of the harmonic well in the pseudo-spring potential is now of the order of $k_BT$ so that breaking of pseudo-springs is a common event. This renders the density profile in DFT more liquid-like, which improves the agreement with the pseudo-spring MC substantially. At the same time though, the pseudo-spring mapping becomes a bad approximation for the real connectivity. Only at high packing fractions, where the confinement prevents pseudo-spring breaking, we can reach agreement between the pseudo-spring and real-spring MC again. {width="100.00000%"} Influence of magnetic interactions ---------------------------------- We now activate the magnetic dipolar interactions and discuss the resulting changes for our systems. Figure \[Fig.MC\_vs\_DFT\_k4\_mm\] demonstrates that increasing $m$ increases the amplitudes of all peaks in the DFT, whereas in the pseudo-spring MC only the first peak is affected. Again, our mean-field DFT seems to overestimate the tendency to form a patterned structure because of its incomplete representation of thermal fluctuations. The reason is that, effectively, the particles do not fluctuate as much around their average positions and do not come as close to each other, where the pseudo-spring interaction and the dipolar interaction increase (the latter with inverse cubic distance). As a consequence, the DFT underestimates the averaged strength of the pair interactions in the system. This becomes apparent in the equation of state, where the MC predicts a much stronger downwards shift when increasing the magnetic moments (see the insets of Fig. \[Fig.MC\_vs\_DFT\_k4\_mm\]). As a liquid-state approach alternative to DFT, we have also solved the Zerah-Hansen liquid-integral equation. Corresponding results in comparison to MC-simulations can be found in the supplemental material [@supplemental]. Thermodynamic compression moduli -------------------------------- {width="100.00000%"} Finally, we evaluate the elastic moduli of the DFT and pseudo-spring MC systems for various magnitudes $m$ of the magnetic moment. We present them as a function of packing fraction $\phi$ in Fig. \[Fig.MC\_vs\_DFT\_k4\_compressionModulus\_mm\]. The DFT predicts only a very slight downward shift of the compression modulus when increasing the magnetic moment. In contrast to that, the shift is significantly more pronounced in the pseudo-spring MC. Additionally, the overall value of the compression modulus at high packing fractions is lower in the DFT. These observations are in line with our earlier results. The mean-field DFT overestimates the tendency to form patterned structures. It therefore underestimates both, contributions by magnetic and elastic pair potentials. If the fluctuations of the particle positions were more pronounced, there would be more emphasis on configurations with strong elastic and magnetic interactions and their influence on the compression modulus would be stronger. Embedding into the elastic matrix {#Sec.Embedding_into_the_elastic_matrix} ================================= So far, we have considered a simple one-dimensional dipole-spring model. There, the elastic matrix is solely represented by springs between nearest-neighbor magnetic particles. Now we turn to an extended model, explicitly describing a single linear chain of magnetic particles that is embedded in a three-dimensional elastic matrix. Dipole-spring model for a linear embedded chain ----------------------------------------------- We begin by constructing an effective pinning potential $U_\textrm{mp}$ for the embedded particles within the three-dimensional matrix as well as an effective pair interaction $u_\textrm{pp}$ between two embedded particles *mediated* by the matrix. Subsequently, we translate these potentials into a network of springs describing the overall elastic interactions. If a single spherical particle of diameter $d$ embedded in an infinitely extended homogeneous elastic matrix is displaced by a vector $\Delta \mathbf{R}$, it distorts the elastic environment. Then the restoring force $\mathbf{F}_\textrm{mp}$ that the matrix exerts onto the particle is given by [@Phan-Thien1993_JElasticity; @Phan-Thien1994_ZAngewMathPhys; @Puljiz2016_PhysRevLett; @Puljiz2016_arXiv] $$\mathbf{F}_\textrm{mp}(\mathbf{U}) = - \frac{12\pi(1 - \nu) G d}{5 - 6\nu} \Delta\mathbf{R} \, , \label{Eq.force_matrix-particle}$$ where $\nu$ is the Poisson ratio, that equals $\nu=1/2$ for incompressible matrices, and $G$ is the shear modulus. The force can be connected via $\mathbf{F}_\textrm{mp} = -\nabla U_\textrm{mp}$ to a harmonic potential $$U_\textrm{mp}(\Delta\mathbf{R}) = \frac{1}{2} k_\textrm{mp} (\Delta\mathbf{R})^2 \, \label{Eq.potential_matrix-particle}$$ with the spring constant $k_\textrm{mp} := \big(12\pi(1 - \nu) G d \big) / \big(5 - 6\nu \big)$. Now we consider two embedded particles, labeled as “1” and “2”, respectively. Upon displacing these particles by vectors $\Delta \mathbf{R}_1$ and $\Delta \mathbf{R}_2$, they experience the forces $\mathbf{F}_1$ and $\mathbf{F}_2$. In our one-dimensional set-up, we only consider forces and displacements along the particle center-to-center vector $\mathbf{r}$. To first order in the particle distance, i.e. to order $1/r$ with $r=|\mathbf{r}|$ we then obtain [@Puljiz2016_PhysRevLett; @Puljiz2016_arXiv; @Phan-Thien1994_ZAngewMathPhys; @Phan-Thien1993_JElasticity]: $$\left( \begin{array}{c}\mathbf{F}_1\\[.1cm] \mathbf{F}_2\end{array} \right) = \left( \begin{array}{cc} -k_\textrm{mp} & \frac{k_\textrm{mp}^2}{4\pi G}\frac{1}{r}\\[.1cm] \frac{k_\textrm{mp}^2}{4\pi G}\frac{1}{r} & -k_\textrm{mp} \end{array} \right) \cdot \left( \begin{array}{c}\Delta \mathbf{R}_1\\[.1cm] \Delta \mathbf{R}_2\end{array} \right) \, . \label{Eq.force-displacement-relation}$$ Here, entries on the diagonal represent the restoring pinning forces . The off-diagonal contributions result from the matrix distortions that are caused by the displacement of one particle but affect the other embedded particle. To construct an effective pair potential, we here only consider symmetric situations where $\Delta \mathbf{R}_1 = -\Delta \mathbf{R}_2$. Then, the change in distance between the two particles is $\Delta \mathbf{r} = \Delta \mathbf{R}_1 - \Delta \mathbf{R}_2 = 2\Delta \mathbf{R}_1$. Per particle, we can thus rewrite the effective matrix-mediated inter-particle interaction as a function of $\Delta \mathbf{r}$ in the form of an effective potential $$u_\textrm{pp}(\pm \Delta \mathbf{r}) = \frac{1}{2} k_\textrm{pp}(r)(\Delta \mathbf{r})^2 \, , \label{Eq.matrix-mediated_pair-potential}$$ with a distance-dependent spring constant $$k_\textrm{pp}(r) := \frac{k_\textrm{mp}^2}{8\pi G}\frac{1}{r} = \frac{3}{2} \frac{1 - \nu}{5 - 6\nu} k_\textrm{mp} \frac{d}{r}. \label{Eq.kpp_distance_dependent}$$ Using the two potentials $U_\textrm{mp}$ and $u_\textrm{pp}$ as an input, we now motivate an extended dipole-spring model for a linear, initially homogeneous chain of $N$ particles embedded into an elastic matrix, see Fig. \[Fig.dipole\_spring\_model\_embedded\]. We label the particles from left to right by $i = 1, \dots, N$, according to their equilibrium positions $x_i^0 := i \ell$ within the chain. The total pinning potential based on Eq.  then becomes $$U_\textrm{mp} = \sum_{i = 1}^N \frac{1}{2} k_\textrm{mp} (x_i - i \ell)^2 \, . \label{Eq.potential_matrix-particle_total}$$ ![Sketch of our extended dipole-spring model for a one-dimensional chain of magnetic particles of diameter $d$ embedded into a three-dimensional elastic matrix. The elastic embedding is represented by a harmonic potential with spring constant $k_\textrm{mp}$, pinning each particle $i$ (labeled from left to right) to its initial position $x_i^0 := i \ell$. The elastic particle–particle interaction *mediated* by the matrix is represented by the springs connecting the particles. Between nearest neighbors, there are springs of spring constant $k_\textrm{pp}$ and equilibrium length $\ell$. Next-nearest neighbors are connected by springs of spring constant $k_\textrm{pp}/2$ and equilibrium length $2\ell$, thereafter the parameters are $k_\textrm{pp}/3$ and $3\ell$, and so forth. Finally, all particles carry a fixed magnetic dipolar moment of magnitude $m$ aligned with the system axis.[]{data-label="Fig.dipole_spring_model_embedded"}](dipole_spring_model_embedded){width="1.0\columnwidth"} To approximate the matrix-mediated particle–particle interactions between two particles $i,j$ we replace the $1/r$ dependence of the spring constant by $1 / |j-i| \ell$. Thus, we have for the total interaction between all pairs of particles $$U_\textrm{pp} = \sum_{i = 1}^N \sum_{j > i} \frac{1}{2} \frac{k_\textrm{pp}}{|j-i|} \big(x_{ij} - |j-i|\ell \big)^2\, , \label{Eq.potential_particle-particle_total}$$ where $k_\textrm{pp} := \frac{3}{2} \frac{1 - \nu}{5 - 6\nu} \frac{d}{\ell} k_\textrm{mp}$. Essentially, this means that each particle $i$ is connected to all other particles $j$ with harmonic springs of spring constant $k_\textrm{pp} / |j-i|$ and spring equilibrium length $|j - i|\ell$, see Fig. \[Fig.dipole\_spring\_model\_embedded\]. From now on, we assume incompressibility of the elastic matrix and set $\nu = 1/2$. Of course, in this extended dipole-spring model the particles are again distinguishable by their positions. As before, it needs to be mapped to the use in our DFT. For this purpose, we replace the harmonic springs in the model by “pseudo-springs”, following the ideas outlined in Sec. \[Sec.Dipole-spring\_model\]. To include the pinning potential $U_\textrm{mp}$, we use an external potential consisting of a series of $N$ harmonic wells $$U_\textrm{ext}(x) = \min_{i = 1, \dots, N} \left\lbrace \frac{1}{2} k_\textrm{mp} (x - i\ell)^2 \right\rbrace \label{Eq.embedded_Vext}$$ as depicted in Fig. \[Fig.embedded\_Vext\]. ![Illustration of the pseudo-spring external potential in Eq.  representing the pinning effect of the embedding elastic matrix in our DFT. There are $N$ harmonic wells at a spacing $\ell$ with spring constant $k_\textrm{mp} = \frac{8\ell}{3d} k_\textrm{pp}$ (here $\ell = 2d$, $\nu = 1/2$, and $k_\textrm{pp} = 4 k_0$, which translates to $G = \frac{8\ell}{9 \pi d^2} k_\textrm{pp} = \frac{64}{9\pi} \frac{k_0}{d}$). The corresponding harmonic potentials of the individual wells are cut where they overlap with the potential of another well. This leaves the leftmost and rightmost wells unbounded to the sides and, therefore, the whole particle chain remains confined.[]{data-label="Fig.embedded_Vext"}](embedded_Vext){width="1.0\columnwidth"} ![Effective elastic pair potential in Eq.  to represent the network of springs in Eq.  in our DFT calculations. The harmonic wells with width $\ell = 2d$ have a spring constant $k_\textrm{pp}/i$, where $i$ is the index of the well and $k_\textrm{pp}= 4 k_0$.[]{data-label="Fig.embedded_Vpair"}](embedded_Vpair){width="1.0\columnwidth"} To represent the network of springs in Eq.  between one particle and all other particles by “pseudo-spring” interactions, we use $$\begin{gathered} \begin{aligned} \tilde{u}_\textrm{e}(x) = &\frac{k_\textrm{pp}}{2} \left[ (x - \ell)^2 - \frac{{\ell}^2}{4} \right] \mathds{1}_{[0, \frac{3}{2} \ell]}(x) \\ + \sum_{i=2}^\infty &\frac{k_\textrm{pp}}{2i} \left[ (x - i\ell)^2 - \frac{{\ell}^2}{4} \right] \mathds{1}_{[(i - \frac{1}{2})\ell, \, (i + \frac{1}{2})\ell]}(x) \, , \end{aligned} \\ \textrm{with } \mathds{1}_{[a,b]}(x) = \begin{cases} 1 & \textnormal{for} \,\, x \in [a,b] \, , \\ 0 & \textnormal{else} . \end{cases} \end{gathered} \label{Eq.embedded_Vpair}$$ This pair potential is illustrated in Fig. \[Fig.embedded\_Vpair\] and consists of a series of harmonic wells. The spring constants of the wells decay with the neighbor number $i$ from the origin just like the individual springs connecting a particle to all other particles in the dipole-spring model, see Fig. \[Fig.dipole\_spring\_model\_embedded\]. Furthermore, the boundaries of the wells are shifted to zero so that we have a vanishing pair potential at infinite distance. Since the depth of the wells roughly decays as $1/x$, the interaction is long-ranged. The width of the wells in both, the external and the pair potential, is given by $\ell$. This width should be larger than $d$ and here we choose $\ell = 2d$. Thus, both potentials in principle allow more than one particles to occupy a single well. However, the external potential pinning the particles to their equilibrium positions in the matrix is relatively strong. The particles should, therefore, remain centered in their respective wells on average. Results {#results} ------- {width="100.00000%"} {width="100.00000%"} We now discuss our results for our extended dipole-spring model for a magnetic particle chain embedded in a three-dimensional elastic matrix. To this end, we consider a chain of $N = 40$ particles, with an equilibrium interparticle distance $\ell = 2d$ and spring constants $k_\textrm{pp} = 4 k_0$, $k_\textrm{mp} = \frac{8\ell}{3d} k_\textrm{pp} = \frac{16}{3} k_\textrm{pp}$. To evaluate the contribution of the magnetic chain to the pressure and compression modulus as well as to evaluate the DFT numerically, we address one part of the elastic matrix of length $L = 100d$ that contains the magnetic chain. This choice of $L$ is arbitrary, the only requirement for $L$ is to be larger than the total equilibrium length $N\ell = 80d$ of the chain by a reasonable amount. Figure \[Fig.MC\_vs\_DFT\_embedded\_densityprofile\_mm0\] shows density profiles obtained from DFT, pseudo-spring MC, and real-spring MC when setting the magnetic moment to $m = 0$. In contrast to our former dipole-spring model, periodic structures appear here in the density profiles resulting from all three methods, even though this model is still effectively one-dimensional. The reason is, first, that the elastic particle–particle interaction decays only slowly with the distance and is effectively long-ranged. This applies to both, the real-spring and the pseudo-spring version. Second, we have a pinning potential suppressing large amplitude fluctuations of the particles around their pinning positions. Together, both contributions counteract the Landau-Peierls instability and can facilitate periodic structures also in one spatial dimension [@Cuesta2004_JStatPhys; @Wong1997_Nature; @Dyson1969_CummunMathPhys; @Ruelle1968_CommunMathPhys; @VanHove1952_Physica]. In this way, the role of thermal fluctuations is substantially reduced, and our mean-field DFT performs much better when compared to the MC-simulations. Let us now address the pressure and compression modulus. As before, they can be determined by probing the energetic change of the system upon deformation. However, we keep in mind that we have one system (the chain of particles) embedded into another system (the surrounding matrix, where here we present our results for one part of length $L$ of this infinitely extended matrix). When we perform a small affine deformation $\Delta L$ of the part of the matrix containing the chain, we alter the properties of the embedded system. In particular, the equilibrium distance $\ell$ between the embedded particles changes by a factor $(L + \Delta L)/L$. In our approach, this affects the pinning positions $x_i^{0} = i\ell$ of the particles as well as the spring constant $k_\textrm{pp} = \frac{3d}{8\ell} k_\textrm{mp}$, which is accounted for in the energetic change upon deformation. Furthermore, what we can calculate from this energetic change are only the contributions $\Delta p$ and $\Delta K$ of the embedded chain to the overall pressure and compression modulus of the composite. To obtain the overall pressure or compression modulus of the whole composite, the energetic change associated with the macroscopic deformation of the three-dimensional matrix would need to be included as well, which is beyond our particle-based approach. Figure \[Fig.MC\_vs\_DFT\_embedded\_p\_and\_K\] shows the contribution $\Delta p$ to the pressure as well as the contribution $\Delta K$ to the compression modulus as a function of the magnetic dipole moment. We can observe a linear decrease with $m^2$ in both quantities with good agreement between the DFT and MC calculations. The linear behavior is expected, because the magnetic interaction energy scales with $m^2$ and the particle chain remains relatively homogeneous while increasing the magnetic moment. As a final result, we present the contribution of the embedded chain to the stress-strain behavior of the composite material. For this purpose, we compress the surrounding matrix by $\Delta L$ and measure the pressure contribution of the embedded chain as a function of this compression. The results are shown in Fig. \[Fig.MC\_vs\_DFT\_embedded\_stressstrain\] for values of the squared magnetic moment in the range $m^2 = 0.0\,{m_0}^2 - 1.0\,{m_0}^2$. At vanishing magnetic moment, the pressure contribution slightly increases when compressing the system. This is probably due to entropic effects that favor an elongated chain and, thus, work against a compression combined with a slight increase in the spring constant $k_\textrm{pp}$ in our description. Increasing $m^2$, however, leads to a stronger magnetic attraction between the particles. This renders an overall compression more favorable. Since decreasing the particle distance also enhances the magnetic attraction, we have a negative pressure contribution that increases in magnitude when compressing the system further. ![Pressure contribution $\Delta p$ as a function of an overall matrix compression $\Delta L$ for squared magnetic moments in the range of $m^2 = 0.0\,{m_0}^2 - 1.0\,{m_0}^2$ in steps of $0.1\,{m_0}^2$. These results constitute the contribution of the embedded chain to the overall stress-strain behavior in one part of the composite material. At vanishing magnetic moment, the pressure increases with the compression. Increasing the magnetic moment, however, leads to stronger magnetic attractions so that the pressure contribution decreases when the surrounding matrix is compressed. We find good agreement between DFT and MC results, especially for low compressions and magnetic moments. The deviations for high compressions and magnetic moments appear, presumably, for the reasons already described in Fig. \[Fig.MC\_vs\_DFT\_k4\_mm\]. Agreement between the pseudo-spring and real-spring MC is excellent under all conditions.[]{data-label="Fig.MC_vs_DFT_embedded_stressstrain"}](MC_vs_DFT_embedded_stressstrain){width="1.0\columnwidth"} Again, we can observe good agreement between DFT and MC. The best agreement is observed at small compressions and low magnetic moments. Remarkably, the pseudo-spring and the real-spring MC agree exceptionally well at all considered values of $\Delta L$ and $m^2$. This demonstrates, that our approach to map the spring network to effective interactions between indistinguishable particles is a promising approach also beyond the scope of our mean-field DFT. Conclusions {#Sec.Conclusions} =========== In summary, we have proposed a density functional theory to address ferrogel model systems, here evaluated in one spatial dimension. These systems are in principle non-liquid, because the particles are arrested by the elastic matrix surrounding them. To enable the investigation with statistical-mechanical theories, we map the elastic interactions onto effective pairwise interactions and, thus, make the particles indistinguishable. The one-dimensional nature of the ferrogel model systems investigated here poses a challenge, because thermal fluctuations have a special impact in one dimension. Fluctuations can become long-ranged and destroy periodic structural order. These fluctuations, driving the Landau-Peierls instability, are not resolved within our mean-field density functional theory. Therefore, within our first dipole-spring model we observe deviations from Monte-Carlo simulations where these fluctuations are included. In a second, more advanced approach, we explicitly model a linear particle chain embedded into a three-dimensional matrix. Within this model, the Landau-Peierls instability is counteracted by a stronger long-ranged coupling between the particles and a pinning potential that localizes the particles within the elastic matrix. Since the role of the fluctuations is therefore reduced, our density functional theory now provides results that are in good agreement with Monte-Carlo simulations. Numerous experimental realizations of such a systems exist [@collin2003frozen; @varga2003smart; @gunther2012xray; @borbath2012xmuct; @gundermann2013comparison; @huang2016buckling; @Gundermann2017_SmartMaterStruct], see particularly the set-up in Ref. . For the future, it would be promising to extend the concept proposed here to higher spatial dimensions, that is to two-dimensional sheets of ferrogels or full three-dimensional samples. In those dimensions, the Landau-Peierls instability will be less relevant. We expect that especially for regular crystal-like particle arrangements, where the one-body density is regularly peaked, density functional theory is reliable and provides a useful framework to study the properties of these promising materials. It will be challenging to extend the present analysis to include the dynamics of the colloidal particles by using the concept of dynamical density functional theory [@Archer2004_JChemPhys; @Marconi1999_JChemPhys; @Espanol2009_JChemPhys; @Loewen2017_incollection]. For particles of different sizes, or different dipole moments, the same ideas can be used to map the system onto binary and multicomponent systems. Moreover, orientational degrees of freedom, such as rotating dipole moments or anisotropic particle shapes, can be tackled by density functional theory as well, both in statics [@Groh1996_PhysRevE; @Klapp2000_JChemPhys] and dynamics [@Rex2007_PhysRevE; @Menzel2016_JChemPhys], and can therefore be treated within the same framework proposed here. Our results show that density functional theory can be used to describe non-liquid systems like ferrogels, still leading to reasonable results. More generally, we have established that mapping bead-spring models to effective potentials is a feasible approach to make them accessible to statistical-mechanical theories. These theories often make use of correlation functions as an input [@Denton1991_PhysRevA], which here are related to the particle distribution in the ferrogel. Experimental extraction of the particle distribution and the corresponding correlation functions is still challenging [@Gundermann2017_SmartMaterStruct]. However, this route could be explored in the future once the available experimental techniques for particle detection in ferrogel materials are more advanced and particularly can address larger system sizes. These correlation functions could then help to construct effective pair potentials representing the real connectivity in the gel [@Hansen-Goos2006_EurophysLett], providing a formal route for the mapping onto systems of indistinguishable particles. Numerical relaxation scheme to obtain the equilibrium density profile from our DFT ================================================================================== Here, we describe in detail our numerical relaxation scheme to obtain the equilibrium density profile $\rho(x)$ minimizing $\Omega[\rho]$ within our DFT. Instead of directly solving Eq. , we perform a dynamical relaxation of the Lagrange functional $$\begin{aligned} \mathcal{L}[\alpha] &= \int_{0}^{L} \frac{1}{2} \dot{\alpha}(x)^2\, dx - \Omega[\alpha] \\ &\qquad - \lambda \left( \int_{0}^{L} \exp\big(\alpha(x) \big) \,dx - \langle N \rangle \right) \, \end{aligned} \label{Eq.DFT_grand_functional_dynamical}$$ with respect to the logarithmic density profile $\alpha(x) = \ln\big( \rho(x) \big)$ [@Ohnesorge1994_PhysRevE]. Minimizing with respect to the logarithmic density profile ensures that $\rho(x) = \exp\big( \alpha(x) \big)$ remains positive during the relaxation. The artificial kinetic term $\int_{0}^{L} \frac{1}{2} \dot{\alpha}(x)^2 \, dx$ drives $\alpha(x)$ and thus $\rho(x)$ towards the minimum in the grand canonical free energy $\Omega[\rho]$. The Lagrange multiplier $\lambda$ with the corresponding constraint $\langle N \rangle = \int_{0}^{L} \exp\big( \alpha(x) \big) \, dx = \int_{0}^{L} \rho(x) \,dx$ allows us to set the average particle number $\langle N \rangle$ instead of the chemical potential $\mu$. This is more convenient for evaluating the pressure and the compression modulus defined as $p = -\frac{\partial \Omega}{\partial L}\big\vert_{\langle N \rangle, T}$ and $K = -L \frac{\partial p}{\partial L}\big\vert_{\langle N \rangle, T}$, respectively. Solving the Euler-Lagrange equation $$\frac{d}{dt} \frac{\delta \mathcal{L}[\alpha]}{\delta \dot{\alpha}(x)} - \frac{\delta \mathcal{L}[\alpha]}{\delta \alpha(x)} \overset{!}{=} 0 \\ \label{Eq.DFT_Euler-Lagrange_dynamical}$$ then leads to the equation of motion for $\alpha(x)$ $$\ddot{\alpha}(x) = -\rho(x) \left( \frac{\delta \Omega[\rho]}{\delta \rho(x)} + \lambda \right) \, . \label{Eq.DFT_equation_of_motion}$$ The Lagrange multiplier $\lambda$ is determined by $$\begin{gathered} \frac{d^2}{dt^2} \int_{0}^{L} \exp\big( \alpha(x) \big) \, dx = 0 \quad\Leftrightarrow \\ \int_{0}^{L} \rho(x) \left( \dot{\alpha}(x)^2 + \ddot{\alpha}(x) \right) \, dx = 0 \quad\Leftrightarrow \\ \lambda = \int_{0}^{L} \rho(x) \left( \dot{\alpha}(x)^2 - \rho(x) \frac{\delta \Omega[\rho]}{\delta \rho(x)} \right) dx \, \bigg/ \int_{0}^{L} \rho(x)^2 \, dx \, . \end{gathered} \label{Eq.DFT_Lagrange_multiplier}$$ In order to perform the numerical relaxation, we discretize the system into $n$ equally spaced sampling points such that one particle diameter $d$ is represented by $100$ points. The density profile, the potentials, and the radial distribution function are all defined on this numerical grid. All integrals appearing in the calculation of the “acceleration” $\ddot{\alpha}(x)$ can then be solved numerically, making use of Fast-Fourier-Transforms in the case of convolution integrals. We iterate the equation of motion for $\alpha(x)$ forward in time using the standard Velocity-Verlet scheme, obtaining the “velocity” $\dot{\alpha}(x)$ and an update for the density profile $\rho(x)$ in each time step $\Delta t$. To ensure that the constraint $\langle N \rangle = \int_{0}^{L} \rho(x) \,dx$ remains fulfilled, we renormalize $\rho(x)$ after each time step. The time step is variable and increases by a factor $1.1$ up to a maximum $\Delta t_\textrm{max} = 0.01$ when the grand canonical energy has decreased for $5$ consecutive time steps. The decrease in energy is monitored by the “power” $P = \int_{0}^{L} \dot{\alpha}(x) \ddot {\alpha}(x) \, dx$, which is supposed to be positive. If $P \leq 0$ occurs, we set $\dot{\alpha}(x) = 0$ and halve the time step. We consider the density profile $\rho(x)$ sufficiently close to equilibrium when our measure for the error $\varepsilon := \sqrt{\int_{0}^{L} \ddot{\alpha}(x)^2 \, dx}$ becomes smaller than $10^{-6}$. At that stage, the left hand side of Eq.  as well as $\lambda$ are close to zero, so that we have $\frac{\delta \Omega[\rho]}{\delta \rho}(x) \approx 0$ as required by Eq. . We thank Mate Puljiz for a stimulating discussion concerning the matrix-mediated interactions. PC, AMM, and HL thank the Deutsche Forschungsgemeinschaft for support of this work through the priority program SPP 1681. **Supplemental material to:\ A density functional approach to ferrogels** In the main article, we have mapped our one-dimensional dipole-spring model containing magnetic particles that are distinguishable by their positions onto a description using pairwise interactions in terms of effective “pseudo-springs”, such that the particles can be treated as indistinguishable. Using this mapping, we have then compared results from density functional theory to those of Monte-Carlo simulations. Here, in a similar fashion, we compare results obtained from solving the Zerah-Hansen liquid-integral equation in comparison to those of Monte-Carlo simulations for various magnitudes of the magnetic moments of the particles. These results demonstrate, that other liquid-state theories using our pseudo-spring approximation lead to reasonable predictions as well. Let us first describe the method of using the Zerah-Hansen (ZH) liquid-integral equation to obtain the equation of state, before we proceed to the comparison with Monte-Carlo (MC) simulations. We use a numerical spectral method \[S1-S3\] to solve the Ornstein-Zernike equation \[S4\] $$\label{eq:O-Z} \gamma(x) = \bar{\rho} \int\limits_{-\infty}^{\infty} dx' \big( \gamma(x') + c(x') \big) c(x-x')$$ for a bulk liquid of mean density $\bar{\rho}$. The functions $\gamma(x) = g(x) - 1 - c(x)$ and $c(x)$ are the indirect and direct correlation functions, respectively. The latter is approximated here by the thermodynamically partially self-consistent Zerah-Hansen (ZH) closure \[S5\] $$\label{eq:ZH} c(x) = \dfrac{e^{-\beta u_\textrm{r}(x)} \left[ f(x) - 1 + e^{f(x)\left(\gamma(x) -\beta u_\textrm{a}(x)\right)} \right]}{f(x)} - \gamma(x) - 1,$$ where $\beta = 1 / k_BT$ and the interaction potential $u(x) = u_\textrm{h}(x) + u_\textrm{m}(x) + \tilde{u}_\textrm{e}(x)$ between indistinguishable particles is split into the sum $u(x) = u_\textrm{r}(x) + u_\textrm{a}(x)$ of a repulsive part $$\label{eq:u_repulsive} u_{r}(x) = \begin{cases} 0 & \textrm{for} \,\, x > x_\textrm{min}\, ,\\ u(x) - u_\textrm{min} & \textrm{otherwise,} \end{cases}$$ and an attractive part $$\label{eq:u_attractive} u_\textrm{a}(x) = \begin{cases} u(x) & \textrm{for} \,\, x > x_\textrm{min}\, ,\\ u_\textrm{min} & \textrm{otherwise.} \end{cases}$$ Here, $u_\textrm{min} = u(x_\textrm{min})$ denotes the minimum of $u(x)$. The mixing function $f(x) = 1 - e^{-\alpha x}$ depends on the non-negative mixing parameter $\alpha$ which is adjusted to achieve thermodynamic consistency with respect to the compression modulus: At the numerically determined value of $\alpha$, the fluctuation-route expression \[S4,S6\] $$\label{eq:fluct_inv_compr} K = \bar{\rho} k_B T \left( 1 - 2 ~ \bar{\rho} \int\limits_0^\infty c(x)~dx \right)$$ gives the same result as the virial-route expression \[S4,S6\] $$\label{eq:virial_inv_compr} K = \bar{\rho} {\left. \dfrac{\partial p_\textrm{v}}{\partial \bar{\rho}} \right|}_{T},$$ in which $p_\textrm{v}$ is the virial pressure. The derivative in Eq.  is numerically approximated by a finite difference and the virial pressure is calculated by numerical integration and solution of $$\label{eq:virial_pressure} \dfrac{p_\textrm{v}}{\bar{\rho} k_B T} = 1 + \bar{\rho} d g(d^{+}) - \bar{\rho} \beta \int\limits_{d}^{\infty} x ~ \dfrac{d u(x)}{dx} ~ g(x) ~ dx.$$ Here, $g(d^{+}) = \lim_{x \to d} g(x > d)$ is the contact value of the radial distribution function. We compare the equations of state obtained from this method with MC-simulations of a bulk pseudo-spring system. For the latter we use a periodic box of width $L = 500d$ and otherwise proceed as for our finite systems in the main article. Figure \[Fig.MC\_vs\_ZH\_k4\_mm\] shows the equations of state obtained from both methods, ZH and MC, for various values of the magnetic moment. ![Equations of state obtained from bulk MC-simulation and ZH liquid-integral theory for various magnitudes of the magnetic moment. Thermodynamic consistency cannot be found for many parameter combinations, explaining the lack of ZH data points. Where it can be found, however, there is good agreement with the MC, especially at high packing fractions. Also the magnitude of the downward shift when increasing the magnetic moment seems to be accurately captured.[]{data-label="Fig.MC_vs_ZH_k4_mm"}](figures/MC_vs_ZH_k4_mm){width="1.0\columnwidth"} Unfortunately, there are many parameter combinations for which the numerical ZH equation solver does not converge, because thermodynamic consistency with respect to the compression modulus is not found for any value of the mixing parameter $\alpha$. Nevertheless, for those parameters at which ZH solutions are available they agree remarkably well with the MC results. Especially at high packing fractions, the equation of state is accurately predicted by the ZH equation. Furthermore, the ZH solution captures correctly the decrease in pressure when the magnetic moment is increased. In contrast to density functional theory (DFT), the employed ZH equation is a theory for homogeneous, isotropic bulk liquids that is not expected to overestimate the tendency of the system to form regular structures and, in fact, does not even include the possibility of symmetry-breaking phase transitions. This might explain why the agreement between MC and ZH in Fig. \[Fig.MC\_vs\_ZH\_k4\_mm\] is better than between MC and DFT in Fig. 7 of the main article. However, the missing feature of reproducing localized density peaks at pre-set equilibrium positions, as required for the extended model in the main article, is an obvious drawback when compared to the DFT approach. We have also solved the hypernetted chain (HNC) integral equation \[S7\] and the Percus-Yevick (PY) integral equation \[S8\] (results not shown). Both the HNC and the PY equations can be readily solved numerically in the complete parameter range of Fig. \[Fig.MC\_vs\_ZH\_k4\_mm\]. However, the (virial and fluctuation-route) equations of state predicted by the HNC and PY equation solution exhibit distinctive disagreement with our MC-simulation results. We conclude that thermodynamic consistency with respect to the compression modulus — which is satisfied in the ZH equation solution, and which is lacking in both the HNC and the PY equations — is a crucial feature. In order to overcome the problem of missing solutions of the ZH equation in extended physical parameter ranges, it might be worthwhile to test alternative thermodynamically partially self-consistent closures of the Ornstein-Zernike equation in future work. [8]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , , , ****, (). , , , , ****, (). , , , ****, (). , ** (, , ), ed. , ****, (). , ****, (). , ****, (). , ****, ().

--- abstract: 'In this paper we study structural properties of the Cuntz semigroup and its functionals for continuous fields of C$^*$-algebras over finite dimensional spaces. In a variety of cases, this leads to an answer to a conjecture posed by Blackadar and Handelman. Enroute to our results, we determine when the stable rank of continuous fields of C$^*$-algebras over one dimensional spaces is one.' address: 'Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain' author: - Ramon Antoine - Joan Bosa - Francesc Perera - Henning Petzka title: Geometric structure of Dimension functions of certain Continuous fields --- Introduction {#introduction .unnumbered} ============ Recent years of research aiming at the classification of C$^*$-algebras demonstrated the Cuntz semigroup $\mathrm{W}(A)$, an invariant built out of the positive elements in matrix algebras over a C$^*$-algebra $A$, to be an important ingredient. Cuntz [@Cu] initiated the study of the functionals on this ordered semigroup, i.e. the normalized semigroup homomorphisms into the positive reals respecting the order. These functionals are referred to as dimension functions of $A$. Blackadar and Handelman posed two conjectures on the geometry of the set of dimension functions on a given C$^*$-algebra $A$ in their 1982 paper [@BH]. Firstly, they conjectured that the set of dimension functions forms a simplex. Secondly, they conjectured that the set of lower semicontinuous dimension functions is dense in the set of all dimension functions. The relevance of the latter conjecture lies in the fact that the set of lower semicontinuous dimension functions, being in correspondence with the quasitraces in $A$, is more tractable than the set of all dimension functions. The second named conjecture was proved in [@BH] for commutative C$^*$-algebras, while the first conjecture was left completely unanswered. In [@Per], the first conjecture was proved for unital C$^*$-algebras with stable rank one and real rank zero. Both conjectures were shown to hold for all unital, simple, separable, exact and $\mathcal{Z}$-stable C$^*$-algebras in [@BPT], by giving a suitable representation of their Cuntz semigroup. Further, the first of the above conjectures was strengthened in [@BPT] to asking the set of dimension functions to form a Choquet simplex. Taking advantage of recent advances in the computation of $\mathrm{W}(A)$ for certain C$^*$-algebras $A$, we show in this paper how we were able to answer the Blackadar-Handelman conjectures affirmatively for certain continuous fields of C$^*$-algebras. Along the way, we obtain independently interesting results on the stable rank of continuous fields completing accomplishments by Nagisa, Osaka and Phillips in [@nop]. We would like to highlight at this point the key steps of how we were able to prove the set of dimension functions to be a Choquet simplex for certain continuous fields of C$^*$-algebras. Performing Grothendieck’s construction on $\mathrm{W}(A)$ gives an ordered abelian group $\mathrm{K}_0^*(A)$. By a result in [@Goo], the set of dimension functions forms a Choquet simplex provided $\mathrm{K}_0^*(A)$ has Riesz interpolation. An affirmative answer to the first conjecture is then linked to certain structural properties of the Cuntz semigroup. Thus, it appears to be interesting to determine when $\mathrm{K}_0^*(A)$ has interpolation for a general C$^*$-algebra $A$. It is known that $\mathrm{K}_0^*(A)$ has interpolation if $\mathrm{W}(A)$ does, and, as a general strategy in our setting, we concentrate on proving the latter. We show that $\mathrm{W}(A)$ has interpolation if and only if its stabilized version ${\mathrm{Cu}}(A)=\mathrm{W}(A\otimes\mathbb{K})$ (see [@CEI]) has interpolation, provided that $\mathrm{W}(A)$ lies naturally as a hereditary subsemigroup in ${\mathrm{Cu}}(A)$. We prove for certain continuous fields of C$^*$-algebras $A$ that ${\mathrm{Cu}}(A)$ has interpolation. Since it is known that the inclusion $\mathrm{W}(A)\subseteq {\mathrm{Cu}}(A)$ is hereditary whenever $A$ has stable rank one, we are led at this point to the question on when continuous fields of C$^*$-algebras have stable rank one. Building on work of [@nop], we settle this question in great generality: The algebra $\mathrm{C}(X,D)$ of continuous functions from a one-dimensional compact metric space into a C$^*$-algebra $D$ has stable rank one if and only if the stable rank of $D$ is one and every hereditary subalgebra $B$ of $D$ has trivial $\mathrm{K}_1$. For general continuous fields of C$^*$-algebras $A$ the condition requiring each fiber to have no $\mathrm{K}_1$-obstructions (in the above sense) is still a sufficient condition for $\mathrm{sr}(A)=1$, but it is not a necessary condition in the general case. Therefore, for certain continuous fields $A$ of stable rank one, we prove that $\mathrm{K}_0^*(A)$ has interpolation and this confirms the first conjecture. Our approach to the second conjecture is based on representing $\mathrm{K}_0^*(A)$ sufficiently well into the group of affine and bounded functions on the trace space of $A$. The paper is organized as follows. Section 1 contains our results on the stable rank of continuous fields, which can be read independently of the rest of the paper. It follows a short section on hereditariness of $\mathrm{W}(A)$ in ${\mathrm{Cu}}(A)$ for continuous fields of C$^*$-algebras $A$. Interpolation results are proved in Section 3, before we apply our results in Section 4 to answer the Blackadar-Handelman conjectures affirmatively for the C$^*$-algebras under consideration. Continuous fields of stable rank one {#sr} ==================================== Let $X$ be a compact metric space of dimension one. We will prove in this section that the algebra $\mathrm{C}(X,A)$ of continuous functions from $X$ into a C$^*$-algebra $A$, has stable rank one, if $A$ has no $\mathrm{K}_1$-obstructions as defined below. We also prove the converse direction in a setting of great generality and prove, as an application, corresponding results for continuous fields. Recall from [@ABP2], [@APS], that a C$^*$-algebra $A$ has *no $\mathrm{K}_1$-obstructions* provided that $A$ has stable rank one and $\mathrm{K}_1(B)=0$ for every hereditary subalgebra $B$ of $A$ (equivalently, $\mathrm{sr}(A)=1$ and $\mathrm{K}_1(I)=0$ for every closed two-sided ideal of $A$). In the case that $A$ is simple or, as proved by Lin, if $A$ has real rank zero (see [@Li Lemma 2.4]), $A$ has no $\mathrm{K}_1$-obstructions if and only if $\mathrm{K}_1(A)=0$. We start by considering the case where $X$ is the closed unit interval. It was already noted in [@nop] that $\mathrm{sr}(A)=1$ and $\mathrm{K}_1(A)=0$ are necessary conditions for $\mathrm{sr}(C([0,1],A))=1$. The first condition follows from the fact that $A$ is a quotient of $C([0,1],A)$, the second is [@nop Proposition 5.2]. We show that also every hereditary subalgebra $B$ must have trivial $\mathrm{K}_1$. \[NecessaryConditions\] Let $A$ be any C$^*$-algebra. If $\mathrm{sr}(C([0,1],A))=1$, then $A$ has no $\mathrm{K}_1$-obstructions. As already noted previous to the proposition, it is clear that $\mathrm{sr}(A)=1$ is a necessary condition. Let $B\subseteq A$ be a hereditary subalgebra. Let $I$ denote the ideal generated by $B$. Then $C([0,1],I)$ is an ideal of $C([0,1],A)$ and therefore has stable rank one. It follows from [@nop Proposition 5.2] that $\mathrm{K}_1(I)=0$. Since $B$ is a full hereditary subalgebra of $I$, we conclude that $\mathrm{K}_1(B)=0$. We will show that conversely, for any C$^*$-algebra $A$ with no $\mathrm{K}_1$-obstructions the stable rank of $C([0,1],A)=1$. Our proof will follow the proof of [@nop Theorem 4.3], where the same result was shown to hold for C$^*$-algebras $A$ with $\mathrm{sr}(A)=1$, $RR(A)=0$, and $\mathrm{K}_1(A)=0$. The proof of [@nop Theorem 4.3] refers to Lemma 4.2 of the same paper. Our contribution is to prove the corresponding lemma to hold in a more general setting. \[InvertiblePath\] Let $A$ be a unital C$^*$-algebra with no $\mathrm{K}_1$-obstructions. For any given $\epsilon>0$ there is some $\delta>0$ such that whenever $a$ and $b$ are two invertible contractions in $A$ with $\|a-b\|<\delta$ then there is a continuous path $(c_t)_{t\in[0,1]}$ in the invertible elements of $A$ such that $c_0=a$, $c_1=b$, and $\|c_t-a\|<\epsilon$ for all $t\in[0,1]$. For given $\epsilon >0$ we choose $\delta_0>0$ satisfying the conclusion of [@Lsant Lemma 3.4] for $\frac{\epsilon}{2}$, i.e., for any positive contraction $a$ and any unitary $u$ with $\|ua-a\|<\delta_0$ there is a path of unitaries $(u_t)_{t\in[0,1]}$ in $A$ such that $u_0=u$, $u_1=1_A$, and $\|u_ta-a\|<\frac{\epsilon}{2}$ for all $t\in[0,1]$. (It follows from our assumptions and [@Rie Theorem 2.10] that $U(B^\sim)$ is connected for each hereditary subalgebra $B$ of $A$, which is needed for the application of [@Lsant Lemma 3.4].) Find $0<\delta\leq \frac{\delta_0}{2}$ such that whenever $\|a-b\|<\delta$, then $\| |a|-|b| \|<\frac{\delta_0}{2}$. (This is possible by Lemma 2.8 of [@Li2].) Take two invertible contractions $a,b$ in $A$ with $\|a-b\|<\delta$ and write $a=u|a|$ and $b=v|b|$ with unitaries $u,v\in A$. We first connect $a$ and $u|b|$ by a path of invertible elements. To do this, define a continuous path $(w_t)_{t\in[0,1]}$ by $$w_t:=u(t|b|+(1-t)|a|),\ t\in[0,1].$$ Then $w_0=a$, $w_1=u|b|$ and, for any $t\in [0,1]$, $w_t$ is invertible and $$\|w_t-a\|=\|ut|b|-ut|a|\|= t\| |b|-|a|\|<\frac{\delta_0}{2}<\epsilon.$$ Next, we connect $u|b|$ and $b$ by a path of invertible elements. Since $$\|v^*u|b|-|b|\|=\|u|b|-v|b|\|\leq \|u|b|-u|a|\|+\|u|a|-v|b|\|=\||a|-|b|\|+\|a-b\|<\delta_0,$$ an application of [@Lsant Lemma 3.4] provides us with a path of unitaries $(u_t)_{t\in[0,1]}$ in $A$ such that $u_0=v^*u$, $u_1=1$, and $\|u_t|b|-|b|\|<\frac{\epsilon}{2}$ for all $t\in[0,1]$. Define a continuous path $(z_t)_{t\in[0,1]}$ by $$z_t:=vu_t|b|,\ t\in[0,1].$$ Then $z_0=u|b|$, $z_1=b$ and, for each $t\in[0,1]$, $z_t$ is invertible and $$\|z_t-a\|\leq \|vu_t|b|-v|b|\|+\|v|b|-a\|=\|u_t|b|-|b|\|+\|b-a\|<\epsilon.$$ Hence $$c_t:=\left \{\begin{array}{ll}w_{2t},&t\in[0,\frac{1}{2}]\\ z_{2t-1},&t\in[\frac{1}{2},1]\end{array}\right.$$ is the continuous path with the desired properties. \[StableRankOfInterval\] Let $A$ be any C$^*$-algebra with $\mathrm{sr}(A)=1$. Then $$\mathrm{sr}(C([0,1],A))=\left \{ \begin{array}{ll} 1,&\mbox{ if $A$ has no $\mathrm{K}_1$-obstructions}\\ 2,& else.\end{array}\right.$$ It is known that $\mathrm{sr}(C([0,1],A))\leq 1+\mathrm{sr}(A)\leq 2$ ([@Su]). From Proposition \[NecessaryConditions\] we know that for $\mathrm{sr}(C([0,1],A))=1$ it is a necessary condition that $\mathrm{K}_1(B)=0$ for all hereditary subalgebras $B$ of $A$. To show that this condition is also sufficient we follow the lines of the proof of [@nop Theorem 4.3], applying Lemma \[InvertiblePath\] instead of [@nop Lemma 4.2]. \[StableRankForSimple\] Let $A$ be a simple C$^*$-algebra with $\mathrm{sr}(A)=1$ and $\mathrm{K}_1(A)=0$. Then $$\mathrm{sr}(C([0,1],A))=1\,.$$ The previous corollary answers positively a question raised in [@nop 5.9], and also allows for a much simpler proof of [@nop Theorem 5.7] for Goodearl algebras, since these are always simple and have stable rank one (see [@Goo3]), as follows: Let $A$ be a Goodearl algebra with $\mathrm{K}_1(A)=0$. Then $\mathrm{sr}(\mathrm{C}([0,1],A))=1$. Another application of Theorem \[StableRankOfInterval\] is the computation of the stable rank of tensor products $A\otimes\mathcal Z$ of C$^*$-algebras $A$ with no $\mathrm{K}_1$-obstructions with the Jiang-Su algebra $\mathcal{Z}$. This was proved by Sudo in [@sudo2 Theorem 1.1] assuming that $A$ has real rank zero, stable rank one, and trivial $\mathrm{K}_1$. Let $A$ be a C$^*$-algebra with no $\mathrm{K}_1$-obstructions. Then the stable rank of $A\otimes\mathcal{Z}$ is one. It is well-known that we can write $A\otimes \mathcal{Z}$ as an inductive limit $$A\otimes\mathcal{Z}=\lim_{i\rightarrow \infty}A\otimes Z_{p_i,q_i},$$ with pairs of co-prime numbers $(p_i,q_i)$ and prime dimension drop algebras $$Z_{p_i,q_i}=\{f\in C([0,1],M_{p_i}\otimes M_{q_i}\ |\ f(0)\in I_{p_i}\otimes M_{q_i},\ f(1)\in M_{p_i} \otimes I_{q_i} \}.$$ Since the stable rank of inductive limit algebras satisfies that $\mathrm{sr}(\lim_{i\rightarrow\infty}(A_i))\leq \liminf \mathrm{sr}(A_i)$ ([@Rie2]) it suffices to show that the stable rank of each $Z_{p_i,q_i}\otimes A$ is one. Fix two co-prime numbers $p$ and $q$ and write $Z_{p,q}\otimes A$ as a pullback $$\xymatrix{Z_{p,q}\otimes A\ar@{-->}[r]\ar@{-->}[d] & M_p(A)\oplus M_q(A) \ar[d]^\phi \\ C([0,1], M_{pq}(A)) \ar@{->>}[r]^{(\lambda_0,\lambda_1)} & M_{pq}(A)\oplus M_{pq}(A)}$$ with maps $\lambda_i(f)=f(i)$ and $\phi(A,B)=\left ( A\otimes I_q, I_p\otimes B\right )$. Our assumptions together with Theorem \[StableRankOfInterval\] imply that $\mathrm{sr}(C([0,1],M_{pq}(A)))=1$. Further, $\mathrm{sr}(M_{m}(A))=1$ for all $m\in\mathbb{N}$, and the map from left to right in the pullback diagram is surjective. An application of [@bp Theorem 4.1 (i)] implies that $\mathrm{sr}(Z_{p,q}\otimes A)=1$. We now turn our attention to $\mathrm{C}(X,A)$ for compact metric spaces $X$ with $\dim(X)=1$. From Theorem \[StableRankOfInterval\] it follows that the stable rank of $\mathrm{C}(X,A)$ is one whenver the stable rank of $C([0,1],A)$ is one. \[CorDimOne\] Let $A$ be a separable C$^*$-algebra $A$ with no $\mathrm{K}_1$-obstructions, and let $X$ be a compact metric space of dimension one. Then $\mathrm{sr}(\mathrm{C}(X,A))=1$. If $X$ is a finite graph with $m$ edges and $V$ is its set of vertices, then $\mathrm{C}(X,A)$ can be written as a pullback $$\xymatrix{\mathrm{C}(X,A)\ar@{-->}[r]\ar@{-->}[d] & C([0,1],A^m) \ar[d] \\ A^n \cong C(V,A) \ar[r] & A^m\oplus A^m}$$ with maps $\mathrm{C}([0,1],A^m)\to A^m\oplus A^m$ given by evaluation at vertices (hence surjective) and $\mathrm{C}(V,A)\to A^m\oplus A^m$ suitably defined (see, e.g. [@APS Section 3]). It is clear that the two entries in the bottom of the diagram have stable rank one. By Theorem \[StableRankOfInterval\] the entry in the upper right corner has stable rank one. It then follows from [@bp Theorem 4.1 (i)] that $\mathrm{sr}(\mathrm{C}(X,A))=1$. Finally, if $X$ is one dimensional, we may write $X$ as a (countable) inverse limit of finite graphs, and so $\mathrm{C}(X,A)$ is an inductive limit of algebras that have stable rank one. Therefore the stable rank of $\mathrm{C}(X,A)$ is also one. To prove the corresponding result to Theorem \[StableRankOfInterval\] for more general spaces of dimension one, we need to generalize Proposition \[NecessaryConditions\]. The proof is inspired by [@nop Proposition 5.2]. We thank Hannes Thiel for providing us with an argument that allows us to drop unnecessary assumptions in an earlier draft. Recall that, given a compact metric space $X$, a continuous map $f\colon X\to [0,1]$ is *essential* if whenever a continuous map $g\colon X\to [0,1]$ that agrees with $f$ on $f^{-1}(\{0,1\})$ must be surjective. A classical result of Alexandroff shows that if $X$ is one-dimensional space, then there is an essential map from $X$ to $[0,1]$. (A suitable generalization of the above definition can be used to characterize when a space has dimension $\geq n$, see [@Eng].) Let $A$ be any C$^*$-algebra and $X$ be a compact metric space with $\dim(X)=1$. If $\mathrm{sr}(\mathrm{C}(X,A))=1$, then $A$ has no $\mathrm{K}_1$-obstructions. Since $A$ is a quotient of $\mathrm{C}(X,A)$ it is clear that the stable rank of $A$ must be one. Further, arguing as in the proof of Proposition \[NecessaryConditions\], it suffices to show that $\mathrm{K}_1(A)=0$. Suppose, to reach a contradiction, that there is a unitary $u$ in $A$ not connected to $1$. Let $\mathrm{d}$ be the metric that induces the topology on $X$. Since $X$ is one-dimensional, there is an essential map $f\colon X\to [0,1]$. Let $S=f^{-1}(\{0\})$ and $T=f^{-1}(\{1\})$, which are disjoint closed sets and hence $\mathrm{d}(S,T)>0$. We may assume that $\mathrm{d}(S,T)=1$. Now define a continuous function $v\colon X\to A$ as follows: $$v(x)=(1-\mathrm{d}(x,T))_+\cdot u+(1-\mathrm{d}(x,S))_+\cdot 1\,.$$ Notice that, by definition, $v_{|S}=1$ and $v_{|T}=u$. As $\mathrm{C}(X,A)$ has stable rank one, there is a map $w\colon X\to A^{-1}$ such that $||v-w||<1$. Denote by $A^{-1}_0$ the connected component of $A^{-1}$ containing the identity. We have that $S\subseteq w^{-1}(A^{-1}_0)$ and $T\subseteq w^{-1}(A^{-1}\setminus A^{-1}_0)$ as $u\notin A^{-1}_0$ by assumption. Note that $A^{-1}_0$ is both open and closed in $A^{-1}$, so by continuity of $w$ we obtain that $S':=w^{-1}(A^{-1}_0)$ and $T':=w^{-1}(A^{-1}\setminus A^{-1}_0)$ form a partition of $X$ consisting of clopen sets. Thus we can define a (non-surjective) continuous function $h\colon X\to [0,1]$ such that $h(S')=0$ and $h(T')=1$, and this contradicts the essentiality of $f$. We collected everything for a repetition of the arguments of the proof of Theorem \[StableRankOfInterval\] in a more general setting. \[StableRankOfDimOne\] Let $A$ be any C$^*$-algebra with $\mathrm{sr}(A)=1$ and $X$ be a compact metric space of dimension one. Then $$\mathrm{sr}(\mathrm{C}(X,A))=\left \{ \begin{array}{ll} 1,&\mbox{ if $A$ has no $\mathrm{K}_1$-obstructions}\\ 2,& \text{else.}\end{array}\right.$$ Although we won’t need it in the following, we would like to point out that Theorem \[StableRankOfDimOne\] determines the real rank of certain algebras by an application of the well-known inequality stating that $RR(A)\leq 2\mathrm{sr}(A)-1$ and [@nop Proposition 5.1]. Let $A$ be C$^*$-algebra with no $\mathrm{K}_1$-obstructions, and let $X$ be a compact metric space of dimension one. Then $RR(\mathrm{C}(X,A))=1$. In view of Theorem \[StableRankOfDimOne\], it is natural to ask if the same can be obtained for continuous fields over $X$. We recall the main definitions (see e.g. [@nilsen; @Dixmier; @dadarlat]). If $X$ is a compact Hausdorff space, a *$\mathrm{C}(X)$-algebra* is a C$^*$-algebra together with a unital $^*$-homomorphism $\mathrm{C}(X)\to Z(\mathcal {M}(A))$. If $Y\subseteq X$ is a closed set, let $A(Y)=A/\mathrm{C}_0(X\setminus Y)A$, which also becomes a $\mathrm{C}(X)$-algebra. Denote by $\pi_Y\colon A\to A(Y)$ the natural quotient map. In the case that $Y=\{x\}$, we then write $A_x$ and $\pi_x$. The algebra $A_x$ is referred to as the *fiber* of $A$ at $x$. For a $\mathrm{C} (X)$-algebra $A$, the map $x\mapsto ||a(x)||$ is upper semicontinuous, and if it is continuous, we then say that $A$ is a *continuous field*. The natural question is then whether a continuous field of C$^*$-algebras $A$ over a one-dimensional space $X$, all of whose fibers have no $\mathrm{K}_1$-obstructions, is necessarily of stable rank one. And, conversely, if $\mathrm{sr}(A)=1$ for a continuous field $A$ over a one-dimensional space $X$ implies $\mathrm{K}_1(A_x)=0$ for all $x\in X$. We attain a positive answer to the first named question, but the second question can be answered in the negative, even for $X=[0,1]$ as we show below. \[StableRankOfCtsFields\] Let $X$ be a one-dimensional, compact metric space, and let $A$ be a continuous field over $X$ such that each fiber $A_x$ has no $\mathrm{K}_1$-obstructions. Then $\mathrm{sr}(A)=1$. As $X$ is metrizable and one-dimensional, we can apply [@ngsudo Theorem 1.2] to obtain that $\mathrm{sr}(A)\leq \sup_{x\in X} \mathrm{sr}(\mathrm{C}([0,1],A_x))$. Now the result follows immediately from Theorem \[StableRankOfDimOne\]. The previous result yields: [(cf. [@dadarlatelliottniu Lemma 3.3])]{} Let $X$ be a one-dimensional, compact metric space, and let $A$ be a continuous field of AF algebras. Then $\mathrm{sr}(A)=1$. Let $X$ be a one-dimensional, compact metric space, and let $A$ be a continuous field of simple AI algebras. Then $\mathrm{sr}(A)=1$. If $A$ is a locally trivial field of C$^*$-algebras with base space the unit interval then it is clear by the methods above that $\mathrm{sr}(A)=1$ implies that $\mathrm{K}_1(A_x)$ must be trivial for all $x$. For general continuous fields, this implication is false. \[lem:counterexample\] Let $B\subset C$ be C$^*$-algebras with stable rank one such that $C$ has no $\mathrm{K}_1$ obstructions. Let $$A=\{f\in\mathrm{C}([0,1],C)\mid f(0)\in B\}\,.$$ Then $A$ is a continuous field over $[0,1]$ with stable rank one. It is clear that $A$ is a $C([0,1])$-algebra, which is moreover a continuous field. Observe that $A$ can be obtained as the pullback of the diagram $$\xymatrix{ A\ar@{-->}[r]\ar@{-->}[d] & B\ar@{^{(}->}[d]^i \\ \mathrm{C}([0,1],C)\ar@{->>}[r]^>>>>>>{\mathrm{ev}_0} & C}$$where ev$_0$ is the map given by evaluation at 0. Since the rows are surjective, we have by [@bp Theorem 4.1] that $\mathrm{sr}(A)\leq \text{max}\{\mathrm{sr}(B),\mathrm{sr}(\mathrm{C}([0,1],C))\}$. As $C$ has no $\mathrm{K}_1$ obstructions and $\mathrm{sr}(B)=1$, we have $\mathrm{sr}(A)=1$ by Theorem \[StableRankOfInterval\]. There exists a (nowhere trivial) continuous field $A$ over $[0,1]$ such that $\mathrm{sr}(A)=1$ and $\mathrm{K}_1(A_x)\neq 0$ for a dense subset of $[0,1]$. Let $C=\mathrm{C}(X)$, and $B=\mathrm{C}(\mathbb T)$ where $X$ denotes the cantor set and $\mathbb T$ the unit circle. There exists a surjective map $\pi\colon X \to \mathbb T$ and hence there is an embedding $i\colon B \to C$. Choose a dense sequence $\{x_n\}_n\subset [0,1]$ and define $$C_n:=\{f\in \mathrm{C}([0,1],C)\mid f(x_n)\in i(B)\}\,.$$ Since $X$ is zero dimensional, $C$ is an AF-algebra and hence has no $\mathrm{K}_1$ obstructions. Therefore $C_n$ is a continuous field over $[0,1]$ of stable rank one by Proposition \[lem:counterexample\]. Note that $C_n(x_n)\cong B$ which has non trivial $\mathrm{K}_1$. We now proceed as in the proof of [@DadarlatElliott Corollary 8.3] to obtain a dense subset of such singularities. Let $A_1=C_1$, $A_{n+1}=A_n\otimes_{\mathrm{C}[0,1]} C_{n+1}$ and $A=\varinjlim (A_n,\theta_n)$ where $\theta_n(a)=a\otimes 1$ (see [@blanchard]). Note that $A_n$ can be described as $$A_n=\{f\in \mathrm{C}([0,1],C^{\otimes n})\mid f(x_i)\in C^{\otimes i-1}\otimes i(B) \otimes C^{\otimes n-i}, i=1,\dots, n\}\,,$$ and now $\theta_n(f)(x)=f(x)\otimes 1$. Hence $A_n$ is clearly a continuous field which can moreover be described by the following pullback diagram $$\xymatrix{ A_n\ar@{-->}[r]\ar@{-->}[d] & B\otimes C^{\otimes n-1}\oplus \cdots \oplus C^{\otimes n-1}\otimes B \ar@{^{(}->}[d] \\ \mathrm{C}([0,1],C^{\otimes n})\ar@{->>}[r]^>>>>>>>>>{\mathrm{ev}_{x_1,\dots,x_n}} & C^{\otimes n}\oplus \stackrel{n}{\dots} \oplus C^{\otimes n}}$$ Again, since $C^{\otimes n}$ is an AF algebra it has no $K_1$ obstructions. Then, a similar argument as that in the proof of Proposition \[lem:counterexample\] applies to conclude that $A_n$ has stable rank one. Moreover, $A$ has stable rank one since it is an inductive limit of stable rank one algebras $A_n$ (and is moreover commutative). Now, for any $x\in [0,1]$, the fiber $A(x)$ can be computed as $\varinjlim A_n(x)$. Hence, if $x\not\in\{x_n\}_n$, $A(x)\cong\varinjlim C^{\otimes n}\cong \varinjlim \mathrm{C}(X^n)\cong \mathrm{C}(\varprojlim X^n)$. Since $\varprojlim X^n$ is also zero dimensional, $A(x)$ is an AF-algebra and thus has trivial $K_1$. Assume $x=x_k\in \{x_n\}_n$. Now for any $n\geq k$, $$A_n(x_k)\cong C^{\otimes k-1}\otimes B \otimes C^{\otimes n-k}\cong \mathrm{C}(X^{k-1}\times \mathbb T \times X^{n-k})\,.$$ An application of the Künneth formula shows that $K_1(A_n(x_k))\cong \mathrm{C}(X^{n-1},\mathbb Z)$ and thus $K_1(A(x_k))\cong \varinjlim \mathrm{C}(X^{n-1},\mathbb Z)\cong \mathrm{C}(\prod_{i=1}^\infty X,\mathbb Z)\neq 0$. Hereditariness {#SectionHer} ============== In this short section we study the hereditary character of certain continuous fields. This will be used in the sequel as, since mentioned earlier, in this setting the classical and the stabilized Cuntz semigroup carry the same information. We start by recalling the definitions. Let $A$ be a C$^\ast$-algebra, and $a,b\in A_+$. We say that $a$ is *Cuntz-subequivalent* to $b$, in symbols $a\preceq b$, if there is a sequence $(v_n)$ in $A$ such that $a=\lim_n v_nbv^*_n$. We say that $a$ is *Cuntz-equivalent* to $b$, and we write $a\sim b$, if both conditions $a\preceq b$ and $b\preceq a$ are satisfied. Upon extending this relation to $M_\infty (A)_+$, one obtains an ordered set $\mathrm{W}(A)=M_\infty (A)_+ /\!\!\sim$. We denote the equivalence class of $a\in M_{\infty}(A)_+$ by ${\langle}a{\rangle}$, and then the above set becomes a partially ordered abelian semigroup when it is equipped with the operation ${\langle}a{\rangle}+{\langle}b{\rangle}={\langle}\left(\begin{smallmatrix} a&0\\ 0&b \end{smallmatrix}\right){\rangle}={\langle}a\oplus b{\rangle}$, and order given by ${\langle}a{\rangle}\leq {\langle}b{\rangle}$ if $a\preceq b$. The semigroup $\mathrm{W}(A)$ is referred to as the *Cuntz semigroup*. In [@CEI], Coward, Elliott and Ivanescu introduced a category of partially ordered semigroups ${\mathrm{Cu}}$, to which the Cuntz semigroup of a stable C$^\ast$-algebra belongs. Furthermore, they proved that ${\mathrm{Cu}}(A):=\mathrm{W}(A\otimes \mathcal K)$ defines a sequentially continuous functor from the category of C$^*$-algebras to ${\mathrm{Cu}}$. The semigroup ${\mathrm{Cu}}(A)$ is sometimes called the *stabilized Cuntz semigroup* in order to distinguish it from $\mathrm{W}(A)$. Semigroups in the category ${\mathrm{Cu}}$ have a rich ordered structure not always present in $\mathrm{W}(A)$. Hence, and in order to fit $\mathrm{W}(A)$ into this categorical description, a new category called ${\mathrm{PreCu}}$ was introduced in [@ABP], where $\mathrm{W}(A)$ belongs in a number of instances. It is shown in [@ABP Proposition 4.1] that there is a functor from ${\mathrm{PreCu}}$ to ${\mathrm{Cu}}$ which is left-adjoint to the identity. This functor is basically a completion of semigroups and, for a wide class of C$^*$-algebras, it sends $\mathrm{W}(A)$ to ${\mathrm{Cu}}(A)$. We recall some of the main facts below. Recall that, for a partially ordered semigroup $M$ and elements $a$, $b\in M$, we say that $a$ is *compactly contained* in $b$, in symbols $a\ll b$, if for any increasing sequence $(b_n)$ in $M$ such that $\sup(b_n)$ exists and $b\leq \sup(b_n)$ there exists $n_0$ such that $a\leq b_{n_0}$. When an increasing sequence $(b_n)$ satifies that $b_n\ll b_{n+1}$, then we say that $(b_n)$ is a *rapidly increasing* sequence. Let ${\mathrm{PreCu}}$ be the category whose objects are those partially ordered abelian semigroups $M$ satisfying the following properties: 1. Every element in $M$ is the supremum of a rapidly increasing sequence. 2. The relation $\ll$ and suprema are compatible with addition. Maps of PreCu are semigroup maps preserving suprema of increasing sequences (when they exist), and the relation $\ll$. In this light, ${\mathrm{Cu}}$ may be defined as the full subcategory of ${\mathrm{PreCu}}$ whose objects are those partially ordered abelian semigroups (in ${\mathrm{PreCu}}$) for which every increasing sequence has a supremum. Given a semigroup $M$ in ${\mathrm{PreCu}}$, we say that a pair $(N,\iota)$ is a *completion* of $M$ if 1. $N$ is an object of ${\mathrm{Cu}}$, 2. $\iota\colon M\to N$ is an order-embedding in ${\mathrm{PreCu}}$, and 3. for any $x\in N$, there is a rapidly increasing sequence $(x_n)$ in $M$ such that $x=\sup \iota(x_n)$. It was shown in [@ABP Theorem 5.1] that, for $M\in{\mathrm{PreCu}}$, there exists a (unique) object $\overline{M}$ in ${\mathrm{Cu}}$ and an order-embedding $\iota\colon M\to \overline{M}$ in ${\mathrm{PreCu}}$, satisfying that $(\overline{M},\iota)$ is the completion of $M$. If $M$ and $N$ are partially ordered semigroups, an order-embedding $\iota\colon M\to N$ is called *hereditary* if, whenever $x\in N$ and $y\in \iota(M)$ satisfy $x\leq y$, then $x\in \iota(M)$. If the order-embedding $\iota\colon \mathrm{W}(A)\to \mathrm{W}(A\otimes\mathcal K)={\mathrm{Cu}}(A)$ is hereditary, then we will say that $\mathrm{W}(A)$ is hereditary. In this case, $\mathrm{W}(A)\in{\mathrm{PreCu}}$ and its completion is $(\mathrm{W}(A\otimes\mathcal K),\iota)$ (see [@ABP Theorem 6.1]). There are no examples known of C$^*$-algebras $A$ for which $\mathrm{W}(A)$ is not hereditary. Recall ([@BRTTW]) that if $A$ is a unital C$^*$-algebra, the *radius of comparison* of $({\mathrm{Cu}}(A),[1_A])$, denoted by $r_A$, is defined as the infimum of $r\geq 0$ satisfying that if $x,y\in {\mathrm{Cu}}(A)$ are such that $(n+1)x+m[1_A]\leq ny$ for some $n,m$ with $\frac{m}{n}>r$, then $x\leq y$. In general, $r_A\leq \mathrm{rc}(A)$, where $\mathrm{rc}(A)$ is the radius of comparison of the algebra (see, e.g. [@toms:plms]) and equality holds if $A$ is residually stably finite (i.e. all quotients of $A$ are stably finite) (see Proposition 3.2.3 in [@BRTTW]). It is known that, if $A$ has stable rank one or finite radius of comparison, then $\mathrm{W}(A)$ is hereditary ([@ABP], [@BRTTW]). In particular, this holds if $A$ is a continuous field over a one dimensional, compact metric space such that each fiber has no $\mathrm{K}_1$-obstructions by Theorem \[StableRankOfCtsFields\]. The following is probably well known. We include a proof for completeness. \[lem:stfinite\] Let $X$ be a compact Hausdorff space, and let $A$ be a $\mathrm{C}(X)$-algebra such that $A_x$ has stable rank one for all $x$. Then $A$ is residually stably finite. Let $I$ be an ideal of $A$, which is also a $\mathrm{C}(X)$-algebra, as well as is the quotient $A/I$, with fibers $(A/I)_x\cong A/(\mathrm{C}_0(X\setminus\{x\})A+I)$. As these are quotients of $A_x$, they have stable rank one, so in particular they are stably finite, and this clearly implies $A/I$ is stably finite. \[prop:ctsfieldhered\] Let $X$ be a finite dimensional compact Hausdorff space, and let $A$ be a continuous field over $X$ whose fibers are simple, finite, and $\mathcal Z$-stable. Then $\mathrm{W}(A)$ is hereditary. We know from Lemma \[lem:stfinite\] that $A$ is residually stably finite, so $\mathrm{rc}(A)=\mathrm{r}_A$. We also know from [@hrw Theorem 4.6] that $A$ itself is $\mathcal Z$-stable, whence ${\mathrm{Cu}}(A)$ is almost unperforated ([@Rorijm Theorem 4.5]). Thus $r_A=0$. This implies that $A$ has radius of comparison zero and then [@BRTTW Theorem 4.4.1] applies to conclude that $\mathrm{W}(A)$ is hereditary. \[rem:hereditary\][ In the previous proposition, finite dimensionality is needed to ensure $\mathcal{Z}$-stability of the continuous field. Notice that in the case of a trivial continuous field $A=C(X,D)$ where $D$ is simple, finite, and $\mathcal Z$-stable, the same argument can be applied for arbitrary (infinite dimensional) compact Hausdorff spaces.]{} Let $X$ be a topological space, let $M$ be a semigroup in ${\mathrm{PreCu}}$, and let $f\colon X\to M$ be a map. We say that $f$ is *lower semicontinuous* if, for all $a\in M$, the set $\{t\in X\mid a\ll f(t)\}$ is open in $X$. We shall denote the set of lower semicontinuous functions by $\mathrm{\mathrm{Lsc}}(X,M)$ and the set of bounded lower semicontinuous functions by $\mathrm{\mathrm{Lsc}}_{\mathrm{b}}(X,M)$. Note that, if $M\in {\mathrm{Cu}}$, then $\mathrm{Lsc}(X,M)=\mathrm{Lsc}_{\mathrm{b}}(X,M)$. Furthermore, the sets just defined become ordered semigroups when equipped with pointwise order and addition. Recall that a compact metric space $X$ is termed *arc-like* provided $X$ can be written as the inverse limit of intervals. Note that arc-like spaces include non-trivial examples, such as the pseudo-arc which is a one dimensional space that does not contain an arc (see e.g. [@nadler]). \[lem:cuntzse\] Let $X$ be an arc-like compact metric space, and let $A$ be a unital, simple C$^*$-algebra with stable rank one, and finite radius of comparison. Then $\mathrm{W}(\mathrm{C}(X,A))$ is hereditary. We will prove that $\mathrm{C}(X,A)$ has finite radius of comparison, and then appeal to [@BRTTW Theorem 4.4.1]. Since ${\mathrm{Cu}}$ is a continuous functor and $X$ is an inverse limit of intervals, we can combine [@adps Theorem 2.6] and [@APS Proposition 5.18] to obtain ${\mathrm{Cu}}(\mathrm{C}(X,A))\cong \mathrm{Lsc}(X,{\mathrm{Cu}}(A))$. Now, by Lemma \[lem:stfinite\], $\mathrm{C}(X,A)$ is residually stably finite, and hence by [@BRTTW Proposition 3.3] $\mathrm{rc}(\mathrm{C}(X,A))=r_{\mathrm{C}(X,A)}$. Since the order in $\mathrm{Lsc}(X, {\mathrm{Cu}}(A))$ is the pointwise order, it is easy to verify that $r_{\mathrm{C}(X,A)}\leq r_A$. Note that this is in fact an equality as $A$ is a quotient of $\mathrm{C}(X,A)$ (see condition (i) in [@BRTTW Proposition 3.2.4]). Lower semicontinuous functions, continuous sections, and Riesz interpolation {#sec:lsc} ============================================================================ In this section we prove that the Grothendiek group of the Cuntz semigroup of certain continuous fields has Riesz interpolation. In some cases we apply the results on hereditariness from Section \[SectionHer\] together with results in [@APS], [@adps] and [@ABP2], and then if $A$ is simple, unital, ASH, with slow dimension growth, we apply the description of $\mathrm{W}(\mathrm{C}(X,A))$ given in [@Tiku]. We recall the necessary definitions. Let $(M,\leq)$ be a partially ordered semigroup. We say that $M$ is an *interpolation semigroup* if it satisfies the *Riesz interpolation property*, that is, whenever $a_1,a_2,b_1,b_2\in M$ are such that $a_i\leq b_j$ for $i,j=1,2$, there exists $c\in M$ such that $a_i\leq c\leq b_j$ for $i,j=1,2$. If the order is algebraic and $M$ is cancellative, then this property is well known to be equivalent to the Riesz decomposition property and also to the Riesz refinement property (see, e.g. [@Goo]). Define $\mathcal{C}$ as the full subcategory of ${\mathrm{PreCu}}$ whose objects are those semigroups $M$ such that $\iota\colon M\to \overline M$ is hereditary. \[interpolation\] Let $M$ be a semigroup in $\mathcal{C}$. Then $M$ is an interpolation semigroup if and only if its completion $\overline{M}$ is. Assume that $M$ satisfies the Riesz interpolation property and let $a_i\leq b_j$ be elements in $\overline{M}$ for $i,j\in\{1,2\}$. Denote by $\iota\colon M\to\overline{M}$ the corresponding order-embedding completion map. We may write $a_i=\sup(\iota(a^n_i))$ and $b_j=\sup(\iota(b^n_j))$ for $i,j\in \{1,2\}$, where $(a^n_i)$ and $(b^n_j)$ are rapidly increasing sequences in $M$. Find $m_1\geq 1$ such that $\iota(a_i^1)\leq\iota(b_j^ {m_1})$. Then $a_i^1\leq b_j^{m_1}$ and by the Riesz interpolation property there is $c_1\in M$ such that $a_i^1\leq c_1\leq b_j^{m_1}$. Suppose we have constructed $c_1\leq\dots\leq c_n$ in $M$ and $m_1<\dots <m_n$ such that $a_i^k\leq c_k\leq b_j^{m_k}$ for each $k$. Find $m_{n+1}>m_n$ such that $a_i^{n+1},c_n\leq b_j^{m_{n+1}}$, and by the interpolation property there exists $c_{n+1}\in M$ with $a_i^{n+1},c_n\leq c_{n+1}\leq b_j^{m_{n+1}}$. Now let $\bar{c}=\sup\iota(c_n)\in\overline M$, and it is clear that $a_i\leq\bar{c}\leq b_j$ for all $i,j$. Since $\iota$ is a hereditary order-embedding, the converse implication is immediate. \[lemaprevi\] Let $M\in{\mathrm{Cu}}$ satisfy the property that, whenever $a_i, b_j$ ($i,j=1,2$) are elements in $M$ and $a_i\ll b_j$ for all $i$ and $j$, then, for every $a_i'\ll a_i$, there is $c\in M$ such that $a_i'\ll c\ll b_j$. Then $M$ is an interpolation semigroup. Suppose that $a_i\leq b_j$ in $M$ (for $i,j=1,2$). Write $a_i=\sup a_i^n$ and $b_j=\sup b_j^m$, where $(a_i^n)$ and $(b_j^m)$ are rapidly increasing sequences in $M$. Since $a_i^1\ll a_i^2\ll b_j$, there is $m_1\geq 1$ such that $a_i^2\ll b_j^{m_1}$. By assumption, there are elements $c_1\ll c_1'$ in $M$ such that $a_i^1\ll c_1\ll c_1'\ll b_j^{m_1}$. Now, there is $m_2>m_1$ such that $c_1', {a_i^3}\ll b_j^{m_2}$, so a second application of the hypothesis yields elements $c_2\ll c_2'$ with $c_1,a_i^2\ll c_2\ll c_2'\ll b_j^{m_2}$. Continuing in this way we find an increasing sequence $(c_n)$ in $M$ whose supremum $c$ satisfies $a_i\leq c\leq b_j$. \[remarca\] [ It is proved in [@APS Theorem 5.15] that, if $X$ is a finite dimensional, compact metric space and $M\in {\mathrm{Cu}}$ is countably based, then $\mathrm{Lsc}(X,M)$ is also in ${\mathrm{Cu}}$. As it turns out from the proof of this fact, every function in $\mathrm{Lsc}(X,M)$ is a supremum of a rapidly increasing sequence of functions, each of which takes finitely many values.]{} \[tres\] Let $M\in \mathcal{C}$ be countably based, let $(\overline{M},\iota)$ be its completion, and let $X$ be a finite dimensional, compact metric space. Then $\mathrm{Lsc}_{\mathrm{b}}(X,M)$ is an object of $\mathcal{C}$ and $(\mathrm{Lsc}(X,\overline{M}),i)$ is its completion, where $i$ is induced by $\iota$. Notice that $\mathrm{Lsc}(X,\overline{M})\in{\mathrm{Cu}}$ and that $i(f)=\iota\circ f$ defines an order-embedding. Given $f\in \mathrm{Lsc}(X,\overline{M})$, write $f=\sup f_n$, where $(f_n)$ is a rapidly increasing sequence of functions taking finitely many values. Since $f_n\ll f$ and thus $f_n(x)\ll f(x)$ for every $x\in X$, the range of $f_n$ is a (finite) subset of $\iota(M)$. Therefore each $f_n$ belongs to $\mathrm{Lsc}_{\mathrm{b}}(X,M)$. \[inter\] Let $M$ be a countably based, interpolation semigroup in ${\mathrm{Cu}}$, and let $X$ be a finite dimensional, compact metric space. Then $\mathrm{Lsc}(X,M)$ is an interpolation semigroup. We apply Lemma \[lemaprevi\], so assume $f_i\ll f_i'\ll g_j$ for $i,j=1,2$. Since $f_i\ll f_i'$, given $x\in X$ there is a neighborhood $U_x'$ of $x$ and $c_{i,x}\in M$ such that $f_i(y)\ll c_{i,x}\ll f_i'(y)$ for all $y\in U_x'$ (by [@APS Proposition 5.5]). Now $c_{i,x}\ll f_i'(y)\ll g_j(y)$ for each $y\in U_x'$, so in particular it will hold for $x$. Since $M$ is an interpolation semigroup, there is $d_x\in M$ such that $c_{i,x}\ll d_x\ll g_j(x)$ and, by lower semicontinuity of $g_j$, there is a neighborhood $U_x''$ such that $d_x\ll g_j(y)$ for every $y\in U_x''$. Thus, if $U_x=U_x'\cap U_x''$, we have $f_i(y)\ll d_x\ll g_j(y)$ for all $y\in U_x$. We may now run the argument in [@APS Proposition 5.13] to patch the values $d_x$ into a function $h\in \mathrm{Lsc}(X,M)$ that takes finitely many values and $f_i\ll h\ll g_j$, as desired. \[interC\] Let $M$ be a countably based semigroup in $\mathcal{C}$ and let $X$ be a finite dimensional, compact metric space. Then, if $M$ is an interpolation semigroup, so is $\mathrm{Lsc}_{\mathrm{b}}(X,M)$. By Lemma \[interpolation\] followed by Proposition \[inter\], the semigroup $\mathrm{Lsc}(X,\overline M)$ is an interpolation semigroup, where $\overline M$ is the completion of $M$. On the other hand, by Proposition \[tres\], $\mathrm{Lsc}(X,\overline M)$ is the completion of $\mathrm{Lsc}_{\mathrm{b}}(X,M)\in\mathcal C$, whence another application of Lemma \[interpolation\] yields the conclusion. Let $(M,\leq)$ be a partially ordered semigroup. We denote by $\mathrm{G}(M)$ its Grothendieck group, and order $\mathrm{G}(M)$ by setting $\mathrm{G}(M)^+=\{[a]-[b]\mid b\leq a\}$ as its positive cone. This defines a partial order on $\mathrm{G}(M)$ and, for $a,b,c,d\in M $ $$[a]-[b]\leq [c]-[d]\text{ in }\mathrm{G}(M)\iff a+d+e\leq b+c+e \text{ in } M \text { for some }e\in M.$$ If $A$ is a C$^*$-algebra, we denote by $\mathrm{K}_0^*(A)$ the Grothendieck group of $\mathrm{W}(A)$ and by $[a]-[b]$ the elements of this group, where $a,b\in M_\infty(A)_+$ (see [@Cu]). It is easy to see that the set of states on a semigroup can be naturally identified with the set of states on its Grothendieck group. Condition (i) in the result below is a special case of Theorem \[th:ctsfields\], but the proof in this case is easier. \[interpolacio\] Let $X$ be a compact metric space, and let $A$ be a separable, C$^*$-algebra of stable rank one. Then $\mathrm{K}_0^*(\mathrm{C}(X,A))$ is an interpolation group in the following cases: 1. $\dim X\leq 1$, $\mathrm{K}_1(A)=0$ and has either real rank zero or is simple and $\mathcal Z$-stable. 2. $X$ is arc-like, $A$ is simple and either has real rank zero and finite radius of comparison, or else is $\mathcal Z$-stable. 3. $\dim X\leq 2$ with vanishing second Čech cohomology group $\check{\mathrm{H}}^2(X,\mathbb Z)$, and $A$ is an infinite dimensional AF-algebra. (i): If $A$ has real rank zero, it was proved in [@Per Theorem 2.13] that $\mathrm{W}(A)$ satisfies the Riesz interpolation property, and then so does ${\mathrm{Cu}}(A)$ by Lemma \[interpolation\]. In the case that $A$ is simple and $\mathcal Z$-stable, ${\mathrm{Cu}}(A)$ is an interpolation semigroup by [@Tiku Proposition 5.4]. Since, by [@APS Theorem 3.4], ${\mathrm{Cu}}(\mathrm{C}(X,A))$ is order-isomorphic to $\mathrm{Lsc}(X,{\mathrm{Cu}}(A))$ we obtain, using Proposition \[inter\], that ${\mathrm{Cu}}(\mathrm{C}(X,A))$ is an interpolation semigroup in both cases. By Corollary \[CorDimOne\], $\mathrm{C}(X,A)$ has stable rank one, and so $\mathrm{W}(\mathrm{C}(X,A))$ is hereditary, hence also an interpolation semigroup by Lemma \[interpolation\]. Thus $\mathrm{K}_0^*(\mathrm{C}(X,A))$ is an interpolation group (using [@Per Lemma 4.2]). (ii): By Proposition \[lem:cuntzse\] and its proof we see that $\mathrm{W}(\mathrm{C}(X,A))$ is hereditary and that ${\mathrm{Cu}}(\mathrm{C}(X,A))$ is order-isomorphic to $\mathrm{Lsc}(X,{\mathrm{Cu}}(A))$. Now the proof follows the lines of the previous case. (iii): This follows as above, using [@APS Corollary 3.6], so that ${\mathrm{Cu}}(\mathrm{C}(X,A))$ is order-isomorphic to $\mathrm{Lsc}(X,{\mathrm{Cu}}(A))$, and the proof of Proposition \[lem:cuntzse\], so that $\mathrm{W}(\mathrm{C}(X,A))$ is hereditary. We now turn our consideration to algebras of the form $\mathrm{C}(X,A)$ where $A$ is a unital, simple, non-type I ASH-algebra with slow dimension growth. In this setting we are able to obtain the same conclusion as above without the necessity to go over proving interpolation of ${\mathrm{Cu}}(\mathrm{C}(X,A))$. We first need a preliminary result. \[Grot\] Let $N$ be a partially ordered abelian semigroup and let $M$ be an ordered subsemigroup of $N$ such that $M+N\subseteq M$. Then $\mathrm{G}(M)$ and $\mathrm{G}(N)$ are isomorphic as partially ordered abelian groups. Let us denote by $\gamma\colon M\to \mathrm{G}(M)$ and $\eta\colon N\to \mathrm{G}(N)$ the natural Grothendieck maps. Fix $c\in M$, and define $\alpha\colon N\to \mathrm{G}(M)$ by $\alpha(a):= \gamma(a+c)-\gamma(c)$. Using that $M+N\subseteq M$, it is easy to verify that the definition of $\alpha$ does not depend on $c$. Now, if $a$, $b\in N$, we have $$\begin{aligned} \alpha(a+b) & =\gamma(a+b+c)-\gamma(c) =\gamma(a+b+c)+\gamma(c)-2\gamma(c) \\&=\gamma(a+c+b+c)-2\gamma(c) =(\gamma(a+c)-\gamma(c))+(\gamma(b+c)-\gamma(c))\\ &=\alpha(a)+\alpha(b)\,,\end{aligned}$$ so that $\alpha$ is a homomorphism. It is clear that $\alpha(N)\subseteq \mathrm{G}(M)^+$. By the universal property of the Grothendieck group, there exists a group homomorphism $\alpha'\colon \mathrm{G}(N)\to \mathrm{G}(M)$ such that $\alpha'(\eta(a)-\eta(b))=\alpha(a)-\alpha(b) $. Note that $\alpha'$ is injective. Indeed, if $\alpha(a)-\alpha(b)=0$, then $\gamma(a+c)=\gamma(b+c)$ and so $a+c+c'=b+c+c'$ for some $c'\in M$, and thus $\eta(a)=\eta(b)$. If $\eta(a)-\eta(b) \in \mathrm{G}(N)^+$ with $b\leq a$ in $N$, then $b+c\leq a+c$ in $M$ and so $\gamma(b+c)-\gamma(a+c)\in \mathrm{G}(M)^+$. Therefore $$\begin{aligned} \alpha'(\eta(a)-\eta(b)) & =\alpha(a)-\alpha(b)\\ & =\gamma(a+c)-\gamma(c)-(\gamma(b+c)-\gamma(c))=\gamma(a+c)-\gamma(b+c)\,,\end{aligned}$$ which shows that $\alpha' (\mathrm{G}(N)^+)\subseteq \mathrm{G}(M)^+$. Observe that, if $a\in M\subseteq N$, then $\alpha(a)=\gamma(a+c)-\gamma(c)=\gamma(a)$. This implies that any element in $\mathrm{G}(M)$ has the form $$\gamma(a)-\gamma(b)=\gamma(a+c)-\gamma(b+c)=\alpha'(\eta(a+c)-\eta(b+c))\,$$ and so $\alpha'$ is surjective and $\alpha'(\mathrm{G}(N)^+)=\alpha(\mathrm{G}(M)^+)$. Given semigroups $N$ and $M$ as above, we will say that $M$ *absorbs* $N$. For a C$^*$-algebra $A$, let us denote by $\mathrm{W}(A)_+$ the classes of those elements in $M_{\infty}(A)_+$ which are not Cuntz equivalent to a projection. Note that, if $A$ has stable rank one, then $\mathrm{W}(A)_+$ absorbs $\mathrm{W}(A)$ (see, e.g. [@APT]). If now $X$ is a finite dimensional compact metric space, define $$\mathrm{Lsc}_{\mathrm{b}}(X,\mathrm{W}(A)_+)=\{f\in \mathrm{Lsc}_{\mathrm{b}}(X,\mathrm{W}(A))\mid f(X)\subseteq \mathrm{W}(A)_+\}\,.$$ It is clear that $\mathrm{Lsc}_{\mathrm{b}}(X,\mathrm{W}(A)_+)$ absorbs $\mathrm{Lsc}_{\mathrm{b}}(X,\mathrm{W}(A))$. \[cor:tiku\] Let $X$ be a finite dimensional, compact metric space, and let $A$ be a unital, simple, non-type I, ASH algebra with slow dimension growth. Then $\mathrm{K}_0^*(\mathrm{C}(X,A))$ is an interpolation group. A description of $\mathrm{W}(\mathrm{C}(X,A))$ for the algebras in the hypothesis is given in [@Tiku Corollary 7.1] by means of pairs $(f,P)$ consisting of a lower semicontinuous function $f\in\mathrm{Lsc}_{\mathrm{b}}(X,\mathrm{W}(A))$, and a collection $P$, indexed over $[p]\in V(A)$, of projection valued functions in $\mathrm{C}(f^{-1}([p]),A\otimes \mathcal K)$ modulo a certain equivalence relation. If $f\in \mathrm{Lsc}_{\mathrm{b}}(X,\mathrm{W}(A)_+)$, then clearly $f^{-1}([p])=\emptyset$ for all $[p]\in V(A)$ thus notably simplifying the description of these elements. Namely, there is only one pair of the form $(f,P_0)$, where $P_0$ does not depend on $f\in\mathrm{Lsc}_{\mathrm b}(X,\mathrm{W}(A)_+)$. In particular, the assignment $f\mapsto (f,P_0)$ defines an order-embedding $\mathrm{Lsc}_{\mathrm{b}}(X,\mathrm{W}(A)_+)\to \mathrm{W}(\mathrm{C}(X,A))$ whose image absorbs $\mathrm{W}(\mathrm{C}(X,A))$. As we also have that $\mathrm{Lsc}_{\mathrm{b}}(X,\mathrm{W}(A)_+)$ absorbs $\mathrm{Lsc}_{\mathrm{b}}(X,\mathrm{W}(A))$, we have by Proposition \[Grot\] that $$\mathrm{K}_0^*(\mathrm{C}(X,A))\cong \mathrm{G}(\mathrm{Lsc}_{\mathrm{b}}(X,\mathrm{W}(A)_+))\cong \mathrm{G}(\mathrm{Lsc}_{\mathrm{b}}(X,\mathrm{W}(A)))\,,$$ as partially ordered abelian groups. Since $\mathrm{W}(A)$ is an interpolation semigroup ([@Tiku Proposition 5.4]) we conclude, using Corollary \[interC\] and [@Per Lemma 4.2], that $\mathrm{K}_0^*(\mathrm{C}(X,A))$ is an interpolation group. We close this section by analysing continuous fields over one-dimensional spaces. As in the previous results, we will need a representation of the Cuntz semigroup of such algebras, which in this case can be done in terms of continuous sections over a topological space. We recall the main definitions below (see [@ABP2] for a fuller account). Let $X$ be a compact Hausdorff space, and denote by $\mathcal V_X$ the category of closed sets with non-empty interior, with morphisms given by inclusion. A *${\mathrm{Cu}}$-presheaf* over $X$ is a contravariant functor $\mathcal S\colon \mathcal V_X\to{\mathrm{Cu}}$. We denote by $S=\mathcal S(X)$ and by $\pi_V^W\colon \mathcal S(W)\to \mathcal S(V)$ the restriction maps (where $V\subseteq W$). A ${\mathrm{Cu}}$-presheaf is a *${\mathrm{Cu}}$-sheaf* if, whenever $V$ and $V'\in\mathcal V_X$ are such that $V\cap V'\in\mathcal{V}_X$, the map $$\pi_{V}^{V\cup V'}\times\pi_{V'}^{V\cup V'}\colon \mathcal{S}(V\cup V')\to \{(f,g)\in \mathcal{S}(V)\times\mathcal{S}(V')\mid \pi_{V\cap V'}^V(f)=\pi_{V\cap V}^{V'}(g)\},$$ is bijective. We say that a ${\mathrm{Cu}}$-presheaf (respectively a ${\mathrm{Cu}}$-sheaf) is *continuous* if for any decreasing sequence of closed subsets $(V_i)_{i=1}^\infty$ with $\cap_{i=1}^\infty V_i=V\in\mathcal V_X$, the limit $\lim \mathcal{S}(V_i)$ is isomorphic to $\mathcal{S}(V)$. We will assume from now on that all ${\mathrm{Cu}}$-presheaves and all ${\mathrm{Cu}}$-sheaves are continuous. Given a ${\mathrm{Cu}}$-presheaf $\mathcal{S}$ over $X$, and $x\in X$, we define the *fiber* of $\mathcal{S}$ at $x$ as $S_x:=\lim_{x\in \mathring{V}}\mathcal{S}(V)$. We will denote by $\pi_x\colon S\to S_x$ the natural maps, and also by $\pi_U$ the maps $\pi_U^X$. To ease the notation in the sequel, we shall refer to ${\mathrm{Cu}}$-presheaves or ${\mathrm{Cu}}$-sheaves as presheaves or sheaves, respectively. A (pre)sheaf is *surjective* if all the restriction maps are surjective. For a presheaf $\mathcal S$, we define $F_S=\sqcup_{x\in X} S_x$ and $\pi\colon F_S\to X$ by $\pi(s)=x$ if $s\in S_x$. A *section* of $F_S$ is a map $f\colon X\to F_S$ such that $\pi f=\mathrm{id}_X$. Elements of $S$ induce sections as follows: given $s\in S$, we denote by $\hat s(x)=\pi_x(s)$, and it is clear that $\hat s$ defines a section of $F_S$. A section $f\colon X\to F_S$ is *continuous* if the following holds: - For all $x\in X$ and $a_x\in S_x$ such that $a_x\ll f(x)$, there exist a closed set $V$ with $x\in\mathring{V}$ and $s\in S$ such that $\hat{s}(x)\gg a_{x}$ and $\hat{s}(y)\ll f(y)$ for all $y\in V$. We denote by $\Gamma(X,F_S)$ the set of all continuous sections, which becomes a partially ordered abelian semigroup when equipped with the pointwise order and addition. As shown in [@ABP2 Theorem 3.10], if $X$ is one dimensional and $\mathcal S\colon \mathcal V_X\to {\mathrm{Cu}}$ is a surjective sheaf over $X$ with $S$ countably based, then $\Gamma(X,F_S)$ belongs to the category ${\mathrm{Cu}}$. Our main interest is in the ${\mathrm{Cu}}$-sheaf determined by continuous fields. If $A$ is a continuous field over $X$ whose fibers have no $\mathrm{K}_1$-obstructions, and ${\mathrm{Cu}}_A$ denotes the sheaf given by ${\mathrm{Cu}}_A(V)={\mathrm{Cu}}(A(V))$, then the natural map ${\mathrm{Cu}}(A)\to\Gamma(X,F_{{\mathrm{Cu}}(A)})$ defined by $s\mapsto\hat s$ is an order-isomorphism in ${\mathrm{Cu}}$ (by [@ABP2 Theorem 3.12]). \[prop:interpsheaf\] Let $X$ be a one dimensional, compact metric space, and let $\mathcal S\colon\mathcal V_X\to {\mathrm{Cu}}$ be a surjective sheaf such that $S_x$ is an interpolation semigroup for each $x\in X$. Then $\Gamma(X,F_S)$ is also an interpolation semigroup. We apply Lemma \[lemaprevi\], and so suppose that $f_i'\ll f_i\ll g_j$, for $i,j=1,2$. Given $x\in X$, there are elements $a_{i,x}\in S_x$ such that $f_i'(x)\ll a_{i,x}\ll f_i(x)$ for each $i$, and that satisfy condition (ii) in [@ABP2 Proposition 3.2]. As $a_{i,x}\ll f_i(x)$, there are by continuity a closed neighborhood $V_x$ of $x$ with $x\in\mathring{V}_x$ and $s_i\ll s'_i\ll s''_i\in S$ (depending on $x$) such that $a_{i,x}\ll \hat{s}_i$ and $\hat{s}_i''(y)\ll f_i(y)$ for all $y\in V_x$. Now apply [@ABP2 Proposition 3.2 (ii)] so there is a closed neighborhood $W_x\subseteq V_x$ (whose interior contains $x$) such that $f_i'(y)\leq \hat{s}_i(y)\ll \hat{s}_i'(y)\ll \hat{s}_i''(y)\ll f_i(y)$ for all $y\in W_x$. At $x$, we have that $\hat{s}_i'(x)\ll g_j(x)$, so by the interpolation property assumed on $S_x$ and [@ABP2 Lemma 3.3], there are elements $c_x\ll c_x'$ in $S$ such that $$f_i'(x)\leq \hat{s}_i(x)\ll \hat{s}_i'(x)\ll \hat{c}_x(x)\ll \hat{c}_x'(x)\ll g_j(x)\,.$$ Since $c_x\ll c_x'$, we may apply [@ABP2 Corollary 3.4] to find a closed subset $W'_x\subseteq\mathring{W}_x$ such that $\pi_{W'_x}(c_x)\ll {g_j}_{|W'_x}$ for each $j$. Since $s_i\ll s'_i$ for each $i$, another application of [@ABP2 Corollary 3.4] yields a closed subset $W_x''\subseteq\mathring{W}_x$ such that $\pi_{W''_x}(s_i)\ll \pi_{|W''_x}(c_x)$ for each $i$. We therefore conclude that $f_i'(y)\ll \hat{c}_x(y)\ll g_j(y)$ for all $y\in W_x'\cap W''_x$ and for all $i,j\in\{1,2\}$. By compactness we obtain a finite cover $W_1,\ldots, W_n$ of $X$ and elements $c_1,\ldots,c_n\in S$ such that $f_i'(y)\ll\hat{c_i}(y)\ll g_j(y)$ for all $y\in W_i$. We now run the argument in [@ABP2 Proposition 3.8] to patch the sections $\hat{c}_i$ into a continuous section $h\in\Gamma(X, F_S)$ such that $f_i\ll h\ll g_j$. \[th:ctsfields\] Let $X$ be a one dimensional, compact metric space. Let $A$ be a continuous field over $X$ such that, for all $x\in X$, $A_x$ has stable rank one, trivial $K_1$, and is either of real rank zero, or simple and $\mathcal Z$-stable. Then $\mathrm{K}_0^*(A)$ is an interpolation group. As mentioned above, by [@ABP2 Theorem 3.12] we have an order-isomorphism between ${\mathrm{Cu}}(A)$ and $\Gamma(X,F_{{\mathrm{Cu}}(A)})$, and the latter is an interpolation semigroup by Proposition \[prop:interpsheaf\]. Furthermore, $A$ has stable rank one by Theorem \[StableRankOfCtsFields\], and so $\mathrm{W}(A)$ is hereditary. Hence, $\mathrm{W}(A)$ will also be an interpolation semigroup (Lemma \[interpolation\]) and $\mathrm{K}_0^*(A)$ is an interpolation group. Structure of dimension functions ================================ In this section we apply the above results to confirm the conjectures of Blackadar and Handelman for certain continuous fields of C$^*$-algebras. If $A$ is unital, the maps $d\colon \mathrm{W}(A)\to \mathbb{R}^+$ that respect addition, order, and satisfy $d({\langle}1_A{\rangle})=1$ are called *dimension functions*, and we denote the set of them by $\mathrm{DF}(A)$. In other words, $\mathrm{DF}(A)$ equals the set of states $\mathrm{St}(\mathrm{W}(A),{\langle}1_A{\rangle})$ on the semigroup $\mathrm{W}(A)$, which clearly agrees with $\mathrm{St}(\mathrm{K}_0^*(A),\mathrm{K}_0^*(A)^+,[1_A])$. \[cuntzse\] Let $X$ be a finite dimensional, compact metric space, and let $A$ be a separable, unital C$^*$-algebra. Then $\mathrm{DF}(A)$ is a Choquet simplex in the following cases: 1. $\dim X\leq 1$ and $A$ is a continuous field such that, for all $x\in X$, $A_x$ has stable rank one, trivial $\mathrm{K}_1$ and is either of real rank zero or else simple and $\mathcal Z$-stable. 2. $X$ is an arc-like space and $A=\mathrm{C}(X,B)$ where $B$ is simple, with real rank zero, and has finite radius of comparison, or else $B$ is simple and $\mathcal Z$-stable. 3. $\dim X\leq 2$, $\check{\mathrm{H}}^2(X,\mathbb Z)=0$, and $A=\mathrm{C}(X,B)$ with $B$ an AF-algebra. 4. $A=\mathrm{C}(X,B)$, where $B$ is a non-type I, simple, ASH algebra with slow dimension growth. By the results of Section \[sec:lsc\], $\mathrm{K}_0^*(A)$ is an interpolation group in all the cases. Then, by [@Goo Theorem 10.17], $\mathrm{DF}(A)$ is a Choquet simplex. \[lsc\] Let $X$ and $Y$ be compact Hausdorff spaces. Put $$\mathrm{G}_{\mathrm{b}}(X,Y)=\{f\colon X\times Y\to \mathbb R\mid f=g-h\text{ with }g,h\in \mathrm{Lsc}_{\mathrm{b}}(X\times Y)^{++}\}\,.$$ Then 1. $\mathrm{G}_{\mathrm{b}}(X,Y)$, equipped with the pointwise order, is a partially ordered abelian group. 2. For any $f\in \mathrm{\mathrm{Lsc}}_{\mathrm{b}}(X,\mathrm{Lsc}_{\mathrm{b}}(Y)^{++})$, the map $\tilde{f}\colon X\times Y\to \mathbb{R}^+$, defined by $\tilde{f}(x,y)=f(x)(y)$, is lower semicontinuous. 3. The map $\beta\colon \mathrm{G}(\mathrm{Lsc}_{\mathrm{b}}(X,\mathrm{Lsc}_{\mathrm{b}}(Y)^{++}))\to \mathrm{G}_{\mathrm{b}}(X,Y)$ defined by $$\beta([f]-[g])=\tilde{f}-\tilde{g}$$ is an order-embedding. (i): This is trivial. (ii): We have to show that the set $U_{\alpha}=\{(x,y)\mid f(x)(y)>\alpha\}$ is open for all $\alpha>0$. Fix $(x_0,y_0)\in U_{\alpha}$. Since $f(x_0)(y_0)>\alpha$, we may consider $f(x_0)(y_0)>\alpha+\epsilon'>\alpha+\epsilon>\alpha$ for some $\epsilon,\epsilon'> 0$. Since $f(x_0)$ is lower semicontinuous, there exists an open set $V'_{y_0} \subseteq Y$ containing $y_0$ such that $f(x_0)(y)> \alpha+\epsilon$ for all $y\in V'_{y_0}$. Now, as $Y$ is compact, $f(x_0)$ is bounded away from zero and we find $0<\epsilon_0< \alpha$ such that $f(x_0)(y)>\epsilon_0$ for all $y\in Y$. Let $V_{y_0}$ be an open neighborhood of $y_0$ such that $V_{y_0}\subseteq \overline{V}_{y_0}\subseteq V'_{y_0}$. Define $g\in \mathrm{Lsc}_{\mathrm{b}}(Y)^{++}$ by $g(y)=\alpha+\epsilon\text{ when } y\in V_{y_0}$ and $g(y)=\epsilon_0<\alpha$ otherwise. Observe that, by the way we have chosen $V_{y_0}$ and the construction of $g$, for every $y\in Y$, there exists $U_y$ containing $y$ and $\lambda_y\in\mathbb{R}^+$ such that $g(y')\leq \lambda_y <f(x_0)(y')$ whenever $y'\in U_y$. This implies that $g\ll f(x_0)$ in $\mathrm{Lsc}_{\mathrm{b}}(Y)$. Since $f$ is lower semicontinuous, $\{x\in X\mid f(x)\gg g\}$ is an open set containing $x_0$. Thus, we may find an open set $U_{x_0}$ such that $x_0\in U_{x_0}$ and $f(x)\gg g$ for all $x\in U_{x_0}$. Now, for $(x,y)$ in the open set $U_{x_0}\times V_{y_0}\subseteq X\times Y$ we have $\tilde{f}(x,y)=f(x)(y)>g(y)=\alpha+\epsilon>\alpha$. (iii): We first need to check that $\beta$ is well-defined. Suppose that $[f]-[g]= [f']-[g']$ in $\mathrm{G}(\mathrm{Lsc}_{\mathrm{b}}(X,\mathrm{Lsc}_{\mathrm{b}}(Y)^{++}))$. Then there is $h$ such that $f+g'+h=f'+g+h$. Since $h(x)$ is bounded for every $x$, we obtain $f(x)(y)+g'(x)(y)=f'(x)(y)+g(x)(y)$ for all $x$ and $y$, and so $f(x)(y)-g(x)(y)=f'(x)(y)-g'(x)(y)$. By (ii), it is clear that $\beta([f]-[g])\in G_\mathrm{b}(X,Y)$, and that it is a group homomorphism. If $[f]-[g]\in \mathrm{G}(\mathrm{Lsc}_{\mathrm{b}}(X,\mathrm{Lsc}_{\mathrm{b}}(Y)^{++}))$, then $g\leq f$, if and only if $g(x)(y)\leq f(x)(y)$ for each $x$ and $y$, proving that $\beta$ is an order-embedding. As customary, for a unital C$^*$-algebra $A$ we we denote the set of normalized quasi traces on $A$ by $\mathrm{QT}(A)$, and the set of normalized tracial states by $\mathrm{T}(A)$. Recall that for an exact C$^*$-algebra $\mathrm{QT}(A)=\mathrm{T}(A)$ ([@haag]). We also denote the set of extreme points of a convex set $K$ by $\partial_eK$. Although the result below might be well-known to experts, we provide a proof for completeness (with thanks to Nate Brown). \[nate\] Let $X$ be a compact Hausdorff space and let $A$ be a unital C$^*$-algebra. Then there exists a homeomorphism between $\partial_e\mathrm{T}(\mathrm{C}(X,A))$ and $X\times \partial_e\mathrm{T}(A)$. Moreover, if $\tau\in\partial_e\mathrm{T}(\mathrm{C}(X,A))$ corresponds to $(x,\tau_A)$, then $d_{\tau}(b)=d_{\tau_A}(b(x))$ for any $b\in M_{\infty}(\mathrm{C}(X,A))_+$. Recall that a normalized trace on a unital C$^*$-algebra is extremal if, and only if, the weak closure of its corresponding GNS-representation is a factor (i.e. has trivial center), see, e.g. [@Dixmier Theorem 6.7.3]. Now identify $\mathrm{C}(X,A)$ with $B:=\mathrm{C}(X)\otimes A$. Let $\tau\in\partial_e\mathrm{T}(B)$ and let $(\pi_\tau,\mathcal H_\tau,v)$ be the GNS-triple associated to $\tau$, and we know that $\pi_\tau(B)''$ is a factor. Since $\mathrm{C}(X)\otimes 1_A$ is in the center of $B$, we have that $\pi_\tau(\mathrm{C}(X)\otimes 1_A)$ is in the center of $\pi_\tau(B)''$, whence $\pi_\tau(\mathrm{C}(X)\otimes 1_A)=\mathbb{C}$. Thus, the restriction of $\pi_\tau$ to $\mathrm{C}(X)\otimes 1_A$ corresponds to a point evaluation $\mathrm{ev}_{x_0}$ for some $x_0\in X$. Next, $$\tau(f\otimes a)=\langle \pi_\tau(f\otimes a)v,v\rangle=\langle \mathrm{ev}_{x_0}(f)\pi_\tau(1\otimes a)v,v\rangle=f(x_0)\langle \pi_\tau(1\otimes a)v,v\rangle=f(x_0)\tau(1\otimes a)\,,$$ for all $f\in \mathrm{C}(X)$ and $a\in A$. Therefore $\tau=ev_{x_0}\otimes \tau_A$ where $\tau_A$ is the restriction of $\tau$ to $1\otimes A$. Note that $\tau_A$ is extremal as $\tau$ is. We thus have a map $ \psi\colon \partial_e\mathrm{T}(B)\to \partial_e\mathrm{T}(\mathrm{C}(X))\times \partial_e\mathrm{T}(A)$ defined by $\psi(\tau)=(ev_{x_0}, \tau_A)$, which is easily seen to be a homeomorphism. Now identify $M_n(\mathrm{C}(X,A))$ with $\mathrm{C}(X,M_n(A))$ and let $b\in \mathrm{C}(X,M_n(A))_+$. Let $\tau\in\partial_e\mathrm{T}(B)$ and $\psi(\tau)=(x,\tau_A)$. Then $$d_{\tau}(b)=\lim_{k\to\infty}\tau(b^{1/k})=\lim_{k\to\infty}\tau_A(b^{1/k}(x))=d_{\tau_A}(b(x))\,.$$ Given $\tau\in \mathrm{QT}(A)$ and $a\in M_\infty(A)_+$, we may construct $$d_\tau(a)=\lim_{n\to\infty}\tau(a^{1/n})\,.$$ It turns out that the above map only depends on the Cuntz equivalence class of $a$, and that it defines a lower semicontinuous state on $\mathrm{W}(A)$ (see [@BH; @Cu]). These states are called *lower semicontinuous dimension functions*, and we denote them by $\textrm{LDF}(A)$. If $K$ is a compact convex set, we shall denote by $\mathrm{LAff}_{\mathrm{b}}(K)^{++}$ the semigroup of (real-valued) bounded, strictly positive, lower semicontinuous and affine functions on $K$. This is a subsemigroup of the group $\mathrm{Aff}_{\mathrm{b}}(K)$ of all real-valued, bounded affine functions defined on $K$. Now, given a C$^*$-algebra $A$, we may define a semigroup homomorphism $$\varphi\colon \mathrm{W}(A)_+\to \mathrm{LAff}_{\mathrm{b}}(\mathrm{QT}(A))\,,$$ by $\varphi(\langle a\rangle)(\tau)=d_{\tau}(a)$ (see, e.g. [@APT], [@pertoms]). For ease of notation, we shall denote $\varphi(\langle a\rangle)=\hat{a}$. Notice that, if $A$ is simple, then $\hat{a}\in\mathrm{LAff}_{\mathrm{b}}(\mathrm{QT}(A))^{++}$ if $a$ is non-zero. Observe also that there is an ordered morphism $\alpha\colon \mathrm{W}(\mathrm{C}(X,A))\to \mathrm{Lsc}_{\mathrm b}(X,\mathrm{W}(A))$, given by $\alpha(\langle b\rangle)(x)=\langle b(x)\rangle$. \[prop:embedding\] Let $X$ be a compact Hausdorff space, and let $A$ be a separable, exact, infinite dimensional, simple, unital, C$^*$-algebra with strict comparison and such that $\mathrm{T}(A)$ is a Bauer simplex. Then there is an order-embedding $$\mathrm{G}(\mathrm{Lsc}_{\mathrm{b}}(X, \mathrm{W}(A)))\to\mathrm{Aff}_{\mathrm{b}}(\mathrm{T}(\mathrm{C}(X,A)))\,.$$ Moreover, given $b\in \mathrm{C}(X,M_n(A))_+$, this map sends the class of the function $\alpha(\langle b\rangle)$ to $\hat b$. Since $\mathrm{Lsc}_{\mathrm{b}}(X,\mathrm{W}(A)_+)$ absorbs $\mathrm{Lsc}_{\mathrm{b}}(X,\mathrm{W}(A))$, there is by Lemma \[Grot\] an order-isomorphism between $\mathrm{G}(\mathrm{Lsc}_{\mathrm{b}}(X,\mathrm{W}(A)))$ and $\mathrm{G}(\mathrm{Lsc}_{\mathrm{b}}(X,\mathrm{W}(A)_+))$. In fact, if we take $\langle a\rangle\in \mathrm{W}(A)_+$, and let $v\colon X\to \mathrm{W}(A)_+$ be the function defined as $v(x)=\langle a\rangle$, the previous isomorphism takes $[\alpha(\langle b\rangle)]$ to $[\alpha(\langle b\rangle)+v]-[v]$. Next, as $A$ has strict comparison, the semigroup homomorphism $\varphi$ defined previous to this proposition is an order-embedding (see [@pertoms Theorem 4.4]) and thus induces $$\mathrm{G}(\mathrm{Lsc}_{\mathrm{b}}(X,\mathrm{W}(A)_+))\to \mathrm{G}(\mathrm{Lsc}_{\mathrm{b}}(X,\mathrm{LAff}_{\mathrm{b}}(\mathrm{T}(A))^{++}))\,,$$ which is also an order-embedding, and takes $[\alpha(\langle b\rangle)+v]-[v]$ to $[\tilde{\varphi}(\alpha(\langle b\rangle))+\hat{a}]-[\hat{a}]$, where we identify $\hat{a}$ with a constant function and $\tilde{\varphi}(\alpha(\langle b\rangle))(x)=\widehat{b(x)}$. Now, since $\mathrm{T}(A)$ is a Bauer simplex, the restriction to the extreme boundary yields a semigroup isomorphism $r\colon \mathrm{LAff}_{\mathrm{b}}(\mathrm{T}(A))^{++}\cong \mathrm{Lsc}_{\mathrm{b}}(\partial_e \mathrm{T}(A))^{++}$ (see, e.g. [@Goo2 Lemma 7.2]). Combining these observations with condition (iii) in Proposition \[lsc\], we obtain an order-embedding $$\mathrm{G}(\mathrm{Lsc}_{\mathrm{b}}(X,\mathrm{Lsc}_{\mathrm{b}}(\partial_e\mathrm{T}(A))^{++}))\to \mathrm{G}_{\mathrm{b}}(X,\partial_e\mathrm{T}(A))\,,$$ that sends $[r(\tilde{\varphi}(\alpha(\langle b\rangle))+\hat{a})]-[r(\hat{a})]$ to $r(\tilde{\varphi}(\alpha(\langle b\rangle))+\hat{a})^{\sim}-r(\hat{a})^{\sim}$, which equals the function $(x,\tau_A)\mapsto d_{\tau_A}(b(x))$. Finally, upon identifying the compact space $X\times\partial_e\mathrm{T}(A)$ with $\partial_e\mathrm{T}(\mathrm{C}(X,A))$ (by Lemma \[nate\]), a second usage of [@Goo2 Lemma 7.2] allows us to order-embed $\mathrm{G}_{\mathrm{b}}(X,\partial_e\mathrm{T}(A))$ into $\mathrm{Aff}_{\mathrm{b}}(\mathrm{T}(\mathrm{C}(X,A)))$, and the map $(x,\tau_A)\mapsto d_{\tau_A}(b(x))$ is sent to $\hat{b}$, as desired. \[A\] Let $X$ be a finite dimensional, compact metric space, and let $A$ be a unital, separable, infinite dimensional and exact C$^*$-algebra of stable rank one such that $\mathrm{T}(A)$ is a Bauer simplex. Then $\mathrm{LDF}(\mathrm{C}(X,A))$ is dense in $\mathrm{DF}(\mathrm{C}(X,A))$ in the following cases: 1. $\dim X\leq 1$ and $A$ is simple, $\mathrm{K}_1(A)=0$ and $A$ has strict comparison. 2. $X$ is arc-like, $A$ is simple, has real rank zero, and strict comparison. 3. $\dim X\leq 2$ and $\check{\mathrm{H}}^2(X,\mathbb Z)=0$, with $A$ an AF-algebra. 4. $A$ is a non-type I, simple, unital ASH algebra with slow dimension growth. (i): By [@APS Theorem 3.4], we have that ${\mathrm{Cu}}(\mathrm{C}(X,A))$ and $\mathrm{Lsc}(X,{\mathrm{Cu}}(A))$ are order-isomorphic, and $\mathrm{W}(\mathrm{C}(X,A))$ is hereditary as it has stable rank one by Theorem \[StableRankOfDimOne\]. It follows easily from this that $\mathrm{W}(\mathrm{C}(X,A))$ is order-isomorphic to $\mathrm{Lsc}_{\mathrm{b}}(X,\mathrm{W}(A))$. We may apply Proposition \[prop:embedding\] so that $\mathrm{K}_0^*(\mathrm{C}(X,A))$ is order-isomorphic to a (pointwise ordered) subgroup $G$ of $\mathrm{Aff}_{\mathrm b}(\mathrm{T}(\mathrm{C}(X,A)))$ in such a way that $[b]$ is mapped to $\hat{b}$, and in particular $[1]$ is sent to the constant function $1$. We apply now the same argument as in [@BPT Theorem 6.4], which we sketch for convenience. If $d\in \mathrm{DF}(\mathrm{C}(X,A))$, then it can be identified with a normalized state (at $1$) on $G$. By [@BPT Lemma 6.1], there is a net of traces $(\tau_i)$ in $\mathrm{T}(\mathrm{C}(X,A))$ such that $d(s)=\lim_i s(\tau_i)$ for any $s\in G$. In particular, $d([b])=\lim_i \hat{b}(\tau_i)=d_{\tau_i}(b)$ for $b\in M_{\infty}(\mathrm{C}(X,A))_+$. (ii): This case uses the same arguments as (i), replacing [@APS Theorem 3.4] by Proposition \[lem:cuntzse\] and its proof. (iii): Proceed as in case (i), using [@APS Corollary 3.6] instead of [@APS Theorem 3.4] and Remark \[rem:hereditary\]. (iv): As in the proof of Theorem \[cor:tiku\], we see that $\mathrm{K}_0^*(\mathrm{C}(X,A))\cong \mathrm{G}(\mathrm{Lsc}_{\mathrm b}(X,\mathrm{W}(A)))$ as ordered groups, and then we may use the same argument as in case (i). Acknowledgements {#acknowledgements .unnumbered} ================ This work has been partially supported by a MICIIN grant (Spain) through Project MTM2011-28992-C02-01, and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya. The fourth named author has received support from the DFG (Germany), PE 2139/1-1. [99]{} R. Antoine, J. Bosa, F. Perera, *Completion of monoids with applications to the Cuntz semigroup*, Internat. J. Math. 6 (2011), pp. 837–861. R. Antoine, J. Bosa, F. Perera, *The Cuntz semigroup of continuous fields*, to appear in Indiana Univ. Math. J. R. Antoine, M. Dadarlat, F. Perera, L. Santiago, *Recovering the Elliott invariant from the Cuntz semigroup*, to appear in Trans. Amer. Math. Soc. R. Antoine, F. Perera, L. Santiago,*Pullbacks, $\mathrm{C}(X)$-algebras, and their Cuntz semigroup*, J. Funct. Anal. 260:10 (2011), pp. 2844–2880. P. Ara, F. Perera, A. Toms, *K-theory for operator algebras. Classification of C$^*$-algebras. Aspects of operator algebras and applications*, 1-71, Contemp. Math. 534, Amer. Math. Soc., Providence, RI, 2011. B. Blackadar, D. Handelman, *Dimension functions and traces on C$^*$-algebras*, J. Funct. Anal. 45 (1982), pp. 297–340. B.Blackadar, L. Robert, A. Tikuisis, A. S. Toms, W. Winter, *An algebraic approach to the radius of comparison*, Trans. Amer. Math. Soc. 364 (2012), pp. 3657–3674 É. Blanchard *Tensor products of [C]{}$(X)$-algebras over [C]{}$(X)$.* In: Recent advances in operator algebras, Astérisque 232, (1995), pp 81-–92. L. G. Brown, G. K. Pedersen, *Limits and C$^*$-algebras of low rank or dimension*, J. Operator Theory 61:2 (2009), pp. 381–-417. N. Brown, F. Perera, A. S. Toms,*The Cuntz semigroup, the Elliott conjecture and dimension functions on C$^*$-algebras*, J. Reine Angew. Math 621 (2008), pp. 191–211. K.T. Coward, G.A. Elliott and C. Ivanescu, *The Cuntz semigroup as an invariant for C$^*$-algebras*, J. Reine Angew. Math. 623 (2008), pp. 161–193. J. Cuntz, *Dimension functions on simple C$^*$-algebras*, Math. Ann. 233 (1978), pp. 145–153. M. Dadarlat, *Continuous fields of [C]{}$^*$-algebras over finite dimensional spaces*, Adv. Math. 222 (2009), pp. 1850–1881. M. Dadarlat, G.A. Elliott, *One-parameter continuous fields of Kirchberg algebras*, Comm. Math. Phys. 274 (3) (2007), pp 795–-819. M. Dadarlat, G.A. Elliott, Z. Niu, *One-parameter continuous fields of Kirchberg algebras. II.* Canad. J. Math. 63 (3) (2011), pp. 500–-532. J. Dixmier, *C$^ *$-algebras*, volume 15 of [*North-Holland Mathematical Library*]{}, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. R. Engelking, *Dimension Theory*, North-Holland Publishing Co., Amsterdam-Oxford-New York, 1978. K. R. Goodearl, *Partially ordered abelian groups with interpolation*, Am. Math. Soc, Math. surveys and monographs, no 20, 1986. K. R. Goodearl, *Notes on a class of simple C$^*$-algebras with real rank zero*, Publ. Mat. 36 (1992), pp. 637–654. K. R. Goodearl, *$\mathrm{K}_0$ of multiplier algebras of C$^*$-algebras with real rank zero*, K-Theory 10:5 (1996), pp. 419–489. U. Haagerup, *Quasi traces on exact C$^*$-algebras are traces*, preprint (1991). I. Hirshberg, M. R[ø]{}rdam and W. Winter, *[C$_0(X)$]{}-algebras, stability and strongly self-absorbing [C$^*$]{}-algebras*, Math. Ann. 339:3 (2007), pp. 695–732. E. Kirchberg, N.C. Phillips, *Embedding of exact C$^*$-algebras in the Cuntz algebra $\mathcal{O}_2$*, J. Reine Angew. Math. 525 (2000), pp. 55–94. H. Lin, *Generalized Weyl-von Neumann theorem (II)*, Math. Scand 77 (1995), pp. 129–147. H. Lin, *Approximation by normal elements with finite spectra in C$^*$-algebras of real rank zero*, Pacific J. Math. 173:2 (1996), pp. 443–489. S. Nadler, *Continuum theory. An introduction*, Monographs and Textbooks in Pure and Applied Mathematics, 158. Marcel Dekker, Inc., New York, (1992). M. Nagisa, H. Osaka, N. C. Phillips, *Ranks of algebras of continuous C$^*$-algebra valued functions*, Canad. J. Math. 53:5 (2001), pp. 979–1030. P. W. Ng, T. Sudo, *On the stable rank of algebras of operator fields over metric spaces*, J. Funct. Anal. 220 (2005), pp. 228–236. M. Nilsen, *C$^*$-bundles and C$_0$(X)-algebras*, Indiana Univ. Math. J. 45:2 (1996), pp. 463–477. F. Perera, *The structure of positive elements of C$^*$-algebras with real rank zero*, Internat. J. of Math. 8:3 (1997), pp. 383–405. F. Perera, A. S. Toms, *Recasting the Elliott conjecture*, Math. Ann. 338:3 (2007), pp. 669–702. M. A. Rieffel, *Dimension and stable rank in the K-theory of C$^*$-algebras*, Proc. London Math. Soc. 46 (1983), pp. 301–333. M. A. Rieffel, *The homotopy groups of the unitary groups of non-commutative tori*, J. Operator Theory 17 (1987), pp. 237–254. M. Rørdam, *The stable and the real rank of $\mathcal Z$-absorbing C$^*$-algebras*, Internat. J. Math. 15:10 (2004), pp. 1065–1084. L. Santiago, *Reduction of the dimension of nuclear C$^*$-algebras*, preprint arXiv:1211.7159 \[math.OA\]. T. Sudo, *Stable rank of C$^*$-tensor products with the algebras of continuous functions on product spaces of intervals*, Sci. Math. Jpn 7 (2002), pp. 575–579. T. Sudo, *Stable rank for C$^*$-tensor products with the Jiang-Su algebra*, Tokyo J. Math. 32:1 (2009), pp. 19–26. A. Tikuisis, *The Cuntz semigroup of continuous functions into certain simple C$^*$-algebras*, Internat. J. Math. 22:8 (2011), pp. 1051–1087. A. S. Toms, *Stability in the Cuntz semigroup of a commutative C$^*$-algebra*, Proc. London Math. Soc. 96:1 (2008), pp. 1-–25.

--- abstract: 'The magnetism in SrFe$_2$As$_2$ can be suppressed by electron doping through a small substitution of Fe by Co or Ni, giving way to superconductivity. We demonstrate that a massive substitution of Fe by isovalent ruthenium similarly suppresses the magnetic ordering in SrFe$_{2-x}$Ru$_x$As$_2$ and leads to bulk superconductivity for $0.6 \leq x \leq 0.8$. Magnetization, electrical resistivity, and specific heat data show $T_c$ up to $\approx 20$K. Detailed structural investigations reveal a strong decrease of the lattice parameter ratio $c/a$ with increasing $x$. DFT band structure calculations are in line with the observation that the magnetic order in SrFe$_{2-x}$Ru$_x$As$_2$ is only destabilized for large $x$.' author: - 'W. Schnelle' - 'A. Leithe-Jasper' - 'R. Gumeniuk' - 'U. Burkhardt' - 'D. Kasinathan' - 'H. Rosner' title: ' Superconductivity induced by ruthenium substitution in an iron arsenide: investigation of SrFe$_{2-x}$Ru$_x$As$_2$ ($0 \leq x \leq 2$) ' --- Introduction ============ Soon after the discovery of superconductivity (SC) in doped $R$FeAsO ($R$ = rare-earth element) materials,[@Kamihara08a] investigations also focused on the structurally related $A$Fe$_2$As$_2$ compounds ($A$ = alkaline, alkaline-earth, or rare-earth metal). In the latter structures, the Fe$_2$As$_2$ slabs are separated only by single elemental $A$ layers.[@Rotter08a] The compounds become superconductors if appropriately modified by substitutions on the $A$ site by alkali metals[@Rotter08b; @Sasmal08a; @GFChen08a] or direct substitution within the Fe$_2$As$_2$ slab by Co[@LeitheJasper08b; @Sefat08b; @CWang08a; @Kumar08b] or Ni.[@LJLi08a] Recently, also the appearance of SC upon substitution of As by P was reported.[@SJiang09a] Controlled tuning of the electronic structure by selective substitutions provides an opportunity to test and refine theoretical models since these substitutions can introduce charge carriers, modify the lattice parameters, and may significantly suppress the structural/magnetic transitions observed in the ternary parent compounds.[@Rotter08a] Application of external pressure has been understood as a “clean” alternative to substitutions in tuning the electronic state. $A$Fe$_2$As$_2$ compounds indeed show crossover to SC at pressures as low as 0.4 GPa for CaFe$_2$As$_2$,[@HLee08a; @Park08a; @Torikachvili08a] and high $T_c$ were observed (27K at 3GPa for SrFe$_2$As$_2$; 29K at 3.5GPa for BaFe$_2$As$_2$).[@Alireza08a] Pressure reduces the antiferromagnetic (AFM) phase transition temperature ($T_0$) in SrFe$_2$As$_2$ and an abrupt loss of resistivity hints for the onset of SC.[@Kumar08a; @Igawa08a; @Torikachvili08b] However, the nature of pressure (hydrostatic vs. anisotropic strains) in these experimental procedures is currently up for debate[@WYu08a]. A feature common to both approaches (chemical substitution or external pressure) is the correlation of anisotropic changes in the crystal lattice with the suppression of the spin density wave (SDW) type of AFM. Transition-metal ($T$) substitution studies in $A$Fe$_{2-x}T_x$As$_2$ carried out so far show a significant contraction of the tetragonal $c$ axis length[@LeitheJasper08b; @Sefat08b; @LJLi08a] and SC upon partial substitution of Fe by Co or Ni (electron doping) and an opposite trend if Fe is replaced by Mn (hole doping).[@Kasinathan09a] For the latter substitution series the SDW transition temperature $T_0$ is not suppressed with increasing $x$ and no SC is observed.[@Kasinathan09a] In contrast, in indirectly-doped Sr$_{1-x}$K$_x$Fe$_2$As$_2$ and Ba$_{1-x}$K$_x$Fe$_2$As$_2$[@Sasmal08a; @GFChen08a; @Rotter08b] the $a$ lattice parameter decreases with $x$ while $c$ increases, keeping the unit cell volume almost constant. For BaFe$_2$As$_{2-x}$P$_x$, an isovalent substitution where SC is observed for $x > 0.5$, both $a$ and $c$ decrease with $x$.[@SJiang09a] The fact that substitutions within the Fe$_2$As$_2$ slab with other $d$-metals lead to superconductivity, albeit with lower $T_c$ than for indirect doping, favors an itinerant electronic theory, in contrast to the strongly correlated cuprates.[@LeitheJasper08b] A point to note is that the modification of the electron count by substitutions is inevitably connected with structural changes in the Fe$_2$As$_2$ slabs. Unfortunately, for most substitution series only the unit cell dimensions are reported while further crystallographic data (bonding angles) are unknown. For pressure studies on Sr/BaFe$_2$As$_2$ compounds aiming at physical properties it is difficult to connect the results to variations of the crystal structure since corresponding compressibility studies are largely missing. Thus, currently, it is not known how the valence electron concentration *and/or* the structural parameters have to be modified by substitution in order to suppress the magnetism and eventually generate SC in $A$Fe$_2$As$_2$ compounds. Recently, Nath *et al*.[@Nath09a] reported on the physical properties of SrRu$_2$As$_2$ and BaRu$_2$As$_2$, which are isostructural to SrFe$_2$As$_2$ and do not show SC.[@Jeitschko87a] On the other hand, the isostructural LaRu$_2$P$_2$ is a long-known superconductor with $T_c$ = 4.1K.[@Jeitschko87a] In this communication we present results of a study of the solid solution SrFe$_{2-x}$Ru$_x$As$_2$. A massive substitution of Fe by nominally isoelectronic Ru suppresses the SDW-ordered state and bulk superconductivity is observed for $0.6 \leq x \leq 0.8$. Characterization of the SC state by magnetic susceptibility, electrical resistivity, and specific heat measurements shows a maximum $T_c$ of $\approx 20$K. Detailed crystallographic information of all samples of the series are obtained from full profile refinement of powder X-ray diffraction data. Band structure calculations confirm that AFM order is only suppressed for *large* $x$, giving way to SC in SrFe$_{2-x}$Ru$_x$As$_2$. In this study we obtain the pertinent crystallographic details for a complete substitution series with magnetically ordered, superconducting, and normal-metallic ground states. Such data are a prerequisite for a future comparative study of substitution systems (with or without superconducting phases) based on the SrFe$_2$As$_2$ parent compound. Another aspect is that substitution with nominally isovalent Ru – in contrast to the supposed electron doping by Co or Ni – is expected not to change the charge. In this way also no charge disorder is generated within the Fe$_2$As$_2$ slab. This is in contrast to substitutions with $d$ metals from the cobalt or nickel group. On the other hand, Ru substitution possibly will generate disorder due to its heavier mass and larger size. Both scattering mechanisms seem to limit the achievable $T_c$ of materials with in-plane substitutions, in contrast to SC material obtained through indirect doping of the Fe$_2$As$_2$ slabs by substitution on the $A$ site. Experimental and crystal structure ================================== Samples were prepared by powder metallurgical techniques. Blended and compacted mixtures of precursor alloys SrAs, Fe$_2$As together with As and Ru powder were placed in glassy-carbon crucibles, welded into tantalum containers, and sealed into evacuated quartz tubes for heat treatment at 900$^\circ$C for 24h to 7d followed by several regrinding and densification steps. Samples were obtained in the form of sintered pellets. Details of powder XRD procedures and electron-probe microanalysis (EPMA) are given in previous publications.[@LeitheJasper08b; @Kasinathan09a] The magnetic susceptibility was measured in a SQUID magnetometer (MPMS) and heat capacity by a relaxation method (PPMS, Quantum Design). The electrical resistivity was determined by a four-point dc method (current density $<3$Amm$^{-2}$). Due to the contact geometry the absolute resistivity could be determined only with an inaccuracy of $\pm$30[%]{}. Band structure calculations were performed within the local density approximation (LDA) using the full potential local orbital code FPLO (v. 8.00) with a $k$-mesh of 24$\times$24$\times$24 $k$-points and the Perdew-Wang parameterization of the exchange-correlation potential. The used structural parameters were those from Table \[thetable\]. [@FeConote] Results and discussion ====================== ------ ------------ ------------ -------- ----------- ----------------- ------------------- ------ ------------- -------------- -------------------- -------------------- ------------------------------- ---------- ------------ $x$ $a$ $c$ $c/a$ $V$ $z_\mathrm{As}$ $d_\mathrm{T-As}$ $x$ $R_I$/$R_P$ $T_0$ $T_c^\mathrm{mag}$ $T_c^\mathrm{cal}$ $\Delta c_p/T_c^\mathrm{cal}$ $\gamma$ $\Theta_D$ nom. \[Å\] \[Å\] \[Å$^3$\] \[Å\] ref. [%]{} \[K\] \[K\] \[K\] \[K\] 0.0 3.9243(1) 12.3644(1) 3.1507 190.5 0.3600(1) 2.388(1) 0 - - 203 - - - - - 0.1 3.93210(3) 12.3446(1) 3.1347 190.9 0.3603(1) 2.3911(3) 0.12 3.7/6.3 190 - - - - - 0.2 3.94387(2) 12.2905(1) 3.1125 191.2 0.3602(1) 2.3922(3) 0.21 3.6/6.8 165 - - - - - 0.3 3.95145(3) 12.2551(2) 3.1014 191.4 0.3601(1) 2.3924(4) 0.31 3.8/10.3 $\approx$140 - - - - - 0.4 3.96689(2) 12.1772(1) 3.0697 191.6 0.3599(1) 2.3925(4) 0.42 4.6/8.4 $\approx$100 fil. - - - - 0.5 3.97720(4) 12.1300(2) 3.0499 191.9 0.3597(1) 2.3927(5) 0.50 5.3/14.1 - fil. $<$2.0 - - - 0.6 3.99178(2) 12.0635(1) 3.0221 192.2 0.3599(1) 2.3959(4) 0.61 4.2/7.8 - 19.3 19.8 13.4 6.2 232 0.7 4.00507(2) 12.0087(1) 2.9983 192.6 0.3598(1) 2.3976(3) 0.71 3.3/6.3 - 19.3 20.1 11.6 7.3 229 0.8 4.01096(2) 11.9835(1) 2.9877 192.8 0.3598(1) 2.3983(4) 0.81 4.9/7.5 - 17.6 17.2 13.6 6.7 231 1.0 4.04437(6) 11.8097(3) 2.9200 193.2 0.3597(1) 2.4015(5) 1.03 4.5/10.2 - fil. $<$2.0 0 6.9 243 1.5 4.09818(2) 11.5301(1) 2.8135 193.6 0.3593(1) 2.4056(4) 1.50 3.8/7.4 - - - - - - 2.0 4.16911(2) 11.1706(1) 2.6794 194.2 0.3591(1) 2.4148(4) 2 4.2/7.6 - - - - 4.1 270 ------ ------------ ------------ -------- ----------- ----------------- ------------------- ------ ------------- -------------- -------------------- -------------------- ------------------------------- ---------- ------------ For all samples the crystal structure of ThCr$_2$Si$_2$ type (space group $I4/mmm$)[@Tegel08a] was refined from powder XRD data by full profile methods (Sr in 2$a$ (0, 0, 0), Fe/Ru in 4$d$ (0, 1/2, 1/4), As in 4$e$ (0, 0, $z$)) (see Table \[thetable\]). In the powder X-ray diffractograms of samples with $x \geq 0.5$ broadening of (00$l$) reflections are observed suggesting some local disorder along \[001\]. Nevertheless, they give no evidence for superstructure formation due to long-range Ru ordering. The refined lattice parameters, the refined Ru occupancies, as well as EPMA unambiguously reveal the substitution of Fe by Ru. The nominal Ru contents are in good agreement with both the Ru occupancies from XRD and the EPMA data. The samples contained as minor impurity Ru$_{1-x}$Fe$_x$As. Upon exchange of Fe by Ru a strong linear decrease of the $c$ parameter of the unit cell is observed ($-$9.7[%]{} for $x = 2$). The tetragonal $a$,$b$ plane and with it the transition-metal distance $d_{T-T} = a/\sqrt{2}$ expands by the substitution (by +6.2[%]{} for $x = 2$). The increase of the distances $d_{T-\mathrm{As}}$ with $x$ on the other hand is small (+1.1[%]{} for $x = 2$) and the $z$ parameter of As decreases by only 0.25[%]{}. Surprisingly, the exchange of Fe by the larger Ru atoms[@Emsley92] results only in a 2.0[%]{} increase in the unit cell volume $V$. The by far largest structural effect of Ru (or Co)[@LeitheJasper08b] substitution is the strong decrease of the $c/a$ ratio, i.e. a strong strain-like deformation with respect to the crystal lattice of the ternary Fe parent compound. Correspondingly, the tetrahedral bonding angles $\epsilon_{1,2}$ As–(Fe,Ru)–As depart from each other with increasing $x$. There are no visible discontinuous changes in these room-temperature structure data which could be connected to the various electronic ground state of SrFe$_{2-x}$Ru$_x$As$_2$. ![(Color online) Magnetic susceptibility $\chi(T)$ of SrFe$_{2-x}$Ru$_x$As$_2$ samples in a magnetic field $\mu_0H$ = 2mT. \[figchilo\]](SchnelleFig1.ps){height="3.4in"} ![(Color online) Corrected high-field magnetic susceptibility $\chi(T)$ of selected SrFe$_{2-x}$Ru$_x$As$_2$ samples. \[figchihi\]](SchnelleFig2.ps){height="3.4in"} In Fig. \[figchilo\] the low-field magnetic susceptibility of Sr(Fe$_{2-x}$Ru$_x$)As$_2$ samples is plotted. Weak diamagnetic signals for $T < 16$K in warming after zero-field cooling (zfc) are already visible for a Ru concentration $x = 0.4$. However, the diamagnetic shielding signal increases dramatically from $x = 0.5$ to $x = 0.6$. For $x = 0.6$, 0.7, and 0.8 the shielding comprises the full volume of the sample suggesting bulk SC. On the other hand, the Meissner effect (measured during field cooling) is extremely small. This peculiarity is also observed for other $A$(Fe$_{2-x}T_x$)As$_2$ materials[@LeitheJasper08b; @Sefat08b] and is probably due to strong pinning in materials with substitutions on the iron site, i.e. within the superconducting slab. The sample with $x = 1.0$ also does not show bulk SC but only a very small diamagnetism after zfc. The superconducting transition temperatures are listed in Table \[thetable\], however the transitions are rather broad. This observation of diamagnetic traces and the appearance of superconducting filaments for low Ru concentrations is certainly due to microscopic inhomogeneities of the Ru distribution. The high-temperature susceptibility (high-field data eventually corrected for small ferromagnetic impurities, Fig.\[figchihi\]) of SrFe$_{2-x}$Ru$_x$As$_2$ with $x \leq 1.5$ is generally paramagnetic and shows a typical linear increase with $T$ for $T > 100$K, similar to that of compounds of the Co substituted system.[@LeitheJasper08b; @XFWang09a] The absolute values decrease systematically with the Ru content $x$ as does the linear $T$ dependence. SrRu$_2$As$_2$ finally is diamagnetic and shows no significant slope of $\chi(T)$ above 100K, in agreement with Ref. . No phase transitions (except for SC) are observed for $x > 0.5$. The samples with $x \leq 0.4$ display anomalies at around 200K, 190K, and 150–170K for $x$ = 0, 0.1, and 0.2, respectively. These temperatures are close to the temperatures $T_0$ in Table \[thetable\] which were identified from the anomalies in the resistivity $\rho(T)$ (see below) which mark the SDW transition. ![(Color online) Electrical resistivity (normalized at 300K) of SrFe$_{2-x}$Ru$_x$As$_2$ samples ($x$ as indicated on the curves). Data for $x = 0$ from Ref.. \[figrho\]](SchnelleFig3.ps){height="3.4in"} ![(Color online) Specific heat $c_p/T$ vs. $T^2$ of SrFe$_{2-x}$Ru$_x$As$_2$ ($x$ = 0.6, 0.7, 0.8, 1.0). Data for zero magnetic field (red circles) are shown with the corresponding two-fluid (full black line) and BCS (orange dashed line) type fits (see text). For a field $\mu_0H = 9$T only data (blue diamonds) are shown. Data for $x$ = 0.7, 0.8, 1.0 are shifted by 0.04, 0.08, and 0.12 units upwards, respectively. Inset: difference of the specific heats ($\Delta c_p$) of the SC sample with $x = 0.8$ and the sample $x = 1.0$ (no bulk SC). Both the difference curves for zero and 9T fields are given. \[figcp\]](SchnelleFig4.ps){width="3.4in"} Due to the inaccuracy of the contact geometry we prefer to plot normalized electrical resistivity in Fig. \[figrho\]. The room temperature resistivity values $\rho$(300K) are 500–800 $\mu\Omega$cm for samples with $x = 0$ and low Ru content $x$. The $\rho$(300K) values decrease slightly with increasing $x$ and for SrRu$_2$As$_2$ we find $\rho$(300K) of only $\approx$ 230 $\mu\Omega$cm. The temperature dependence of $\rho(T)/\rho$(300K) corroborates the superconducting transitions. Zero resistivity is observed for Ru concentrations of $x = 0.5$, 0.6, 0.7, and 0.8 at 7.0K, 17.5K, 16.0K, and 18.0K, respectively. The sample with $x = 1.0$ also shows a drop in $\rho(T)$ below 18K, however $\rho = 0$ is not reached at 4K, indicating only filamentary SC. Also, a small drop in $\rho(T)$ at low $T$ is already visible for $x = 0.4$. The resistivity of the sample $x$ = 0.4 however also shows a kink at $T_0 \approx 100$K, the $x = 0.3$ sample at $\approx 140$K. For lower Ru concentration $T_0$ increases continuously (see Table \[thetable\]). The kink in $\rho(T)$, which roughly coincides with anomalies in the high-field susceptibility (see above), is the signature of the SDW transition[@Tegel08a; @Krellner08a; @LeitheJasper08b; @Kasinathan09a] which is found at $T_0 = 203$K in SrFe$_2$As$_2$.[@Tegel08a] Interestingly, in Sr/BaFe$_{2-x}$Co$_x$As$_2$ crystals the resistivity for all $x > 0$ increases below $T_0$ while for unsubstituted Sr/BaFe$_2$As$_2$ $\rho(T)$ decreases below $T_0$ (see, e.g., Refs. ). In contrast, for SrFe$_{2-x}$Ru$_x$As$_2$ the resistivity *decreases* below $T_0$ for all concentrations of Ru. Generally, for a full opening of a gap due to an SDW ordering an increase of $\rho(T)$ would be expected. Instead, it seems that the behavior of $\rho(T)$ below $T_0$ is connected to the presence or absence of charge disorder caused by electron doping. Obviously, for SrFe$_{2-x}$Ru$_x$As$_2$ and undoped materials, in absence of such disorder the mechanism leading to a decrease of $\rho(T)$ below $T_0$ is dominating. The specific heat of selected samples is given in Fig.\[figcp\]. Clear albeit broadened anomalies are seen close the transitions temperatures $T_c^\mathrm{mag}$ indicated by the low-field magnetization (see table \[thetable\]). The sample with $x = 0.7$ shows a less pronounced transition than the samples with neighboring compositions. In agreement with the reduced resistive $T_c$ we conclude that this sample is inhomogeneous and of lower quality than the other samples. The sample with $x = 0.8$ displays a pronounced transition at 17.2K. The composition SrFeRuAs$_2$ shows no bulk SC above 2K. Since the specific heats of the two latter samples are very similar for $T > 20$K the data for $x = 1$ may serve as a reference for phonon and normal electronic contributions to the $x = 0.8$ data. The inset in Fig. \[figcp\] shows the difference of the specific heat of the samples. The size of the resulting step $\Delta c_p/T_c$ as evaluated by the usual entropy-conserving construction (equal areas in $c_p/T$) for $H$ = 0 is 18.4(1.8) mJmol$^{-1}$ K$^{-2}$ and $T_c$ = 17.3(5)K. In order to obtain further electronic and phononic properties from the specific heat, the data between 5.0K and 25.5K were fitted with a model including a phonon contribution (harmonic lattice approach $c_\mathrm{ph} = \beta T^3 + \delta T^5$) and an electronic term according to the weak coupling BCS theory or the phenomenological two-fluid model. The latter model is a good approximation for the thermodynamic properties of some strong coupling superconductors. Folding with a Gaussian simulates the broadening of the transitions due to chemical inhomogeneities. The parameters resulting from the fits are given in Table \[thetable\]. The relative specific heat step $\Delta c_p/T_c$ at $T_c^\mathrm{cal}$ is quite similar for the three investigated SC samples. It has to be remarked that these values are smaller than the value obtained from the difference of the samples with $x = 0.8$ and $x = 1.0$. Nevertheless, the size of the specific heat step at $T_c$ is small and comparable to values observed for superconducting compositions of SrFe$_{2-x}$Co$_x$As$_2$ (10–13 mJmol$^{-1}$K$^{-2}$; hole doping within the Fe$_2$As$_2$ slab),[@LeitheJasper08b] but much smaller than for, e.g., Ba$_{0.6}$K$_{0.4}$Fe$_2$As$_2$ ($\approx$ 98 mJmol$^{-1}$K$^{-2}$).[@GMu08a] The fits with the two-fluid models (shown in Fig. \[figcp\]) are superior to those with the BCS model. The ratio $\Delta c_p/(\gamma T_c^\mathrm{cal})$ is $\approx 2.1$, significantly larger than the weak coupling BCS limit (1.43). Both these findings indicate strongly coupled SC, in agreement with $\mu$SR measurements (see, e.g., Ref. ). ![(Color online) FSM curves for SrFe$_{2-x}$Ru$_x$As$_2$ for $x$ = 0, 0.5, 1 and 2. The inset shows a comparison of the total and partial DOS for SrFeRuAs$_2$ and SrFeCoAs$_2$. The dashed perpendicular line shows the Fermi level for SrFe$_2$As$_2$. \[figfsm\]](SchnelleFig5.ps){width="3.4in"} To understand the changes in the electronic structure upon substitution of Fe by Ru, we carried out band structure calculations for $x$ = 0, 0.5, 1, 2. The partial Ru substitution was modeled by supercells. Since any SDW pattern would be strongly influenced by the choice of the particular supercell, especially for the larger Ru content, we decided to compare the stability with respect to magnetism applying the fixed spin moment (FSM) approach (cf. Fig. 3 in Ref.). The resulting curves are shown in Fig. \[figfsm\]. As one may expect, the magnetic moment and the energy gain due to magnetic ordering are reduced with increasing Ru content $x$. The destabilization is especially strong between $0.5 \le x \le 1$, the fully Ru substituted compound is clearly nonmagnetic. In accord with the classical Stoner picture for itinerant magnets, the suppression of magnetism originates from a strong decrease of the density of states (DOS) at the Fermi level $N(\varepsilon_F)$ = 4.9, 4.4, 3.5 and 1.9 states/(eV f.u.) for $x$ = 0, 0.5, 1, 2, in agreement with experimental $\gamma$ values (Table \[thetable\]). In the real system, the magnetism will be further destabilized due to the Fe-Ru disorder. In contrast to the Co substituted compound, where the Co and Fe 3$d$ states are almost indistinguishable (see inset of Fig. \[figfsm\]), the Ru 4$d$ states differ considerably in the region close to $\varepsilon_F$. This difference has its origin in the different potential and lower site energy of Ru with respect to Fe. In turn, the distinct potential will be responsible for an enhanced scattering and therefore destabilize magnetic ordering. In the Co substituted compound, however, only the shift of the Fermi level due to the additional electron (see Fig. \[figfsm\]) leads to a nonmagnetic state. Although our calculation illustrate semi-quantitatively the suppression of magnetism (clearing the scene for incipient SC) in SrFe$_{2-x}$Ru$_x$As$_2$, the real interplay between magnetism and the slight volume expansion for increasing Ru content, accompanied by a reduction of the $c$ axis and small changes of the As $z$ position is rather complex: the volume expansion leads to narrower Fe 3$d$ bands that would stabilize magnetic order, but due to the $c$ axis contraction these states are simultaneously shifted to lower energy resulting in a reduced DOS at $\varepsilon_F$. However, the Fe 3$d$ states are pushed up in energy upon substitution of Ru due to the lower site energy of the Ru 4$d$ states, compensating the reduction of $N(\varepsilon_F)$ partially. Furthermore, the Stoner factor $I$ for Ru is significantly smaller than for Fe, disfavoring magnetic order further. Discussion and conclusions ========================== To sum up, we have shown that the partial isovalent substitution of Ru for Fe in SrFe$_2$As$_2$ suppresses the SDW transition and gives rise to bulk superconductivity. Both end members of the series, SrFe$_2$As$_2$ as well as SrRu$_2$As$_2$, crystallize in the ThCr$_2$Si$_2$ type structure ($I4/mmm$) and there is no indication of a change or of a superstructure for intermediate Ru concentrations at the applied experimental conditions. This observation indicates that Fe and Ru are isovalent. The lattice parameters $a$ and $c$ vary linearly but in opposite directions with $x$. As seen from the ratio $c/a$ (decreases by 15[%]{} for $x$ between 0 and 2) a strong anisotropic modification of the lattice (a compression of the tetragonal cell along $c$) is introduced by the exchange of Fe by Ru. While $z_\mathrm{As}$ does not change significantly with $x$ this implies a large change of the two different As–Fe–As bonding angles. While for SrFe$_2$As$_2$ these angles are quite similar (110.50 and 108.90$^\circ$) they deviate quite strongly in SrRu$_2$As$_2$ (119.43 and 104.73$^\circ$).[@Tegel08a] Bulk superconductivity exists in this series for a $c/a$ ratio (at room temperature) around 3. Electronic properties like the high-temperature susceptibility also show a smooth variation with $x$. Whether any of these structural modifications is of special importance for the occurrence of superconductivity in SrFe$_{2-x}T_{x}$As$_2$-type substitution series remains to be determined. Currently, it can be stated that electron doping by Co (or Ni) is much more efficient in order to suppress the SDW and allowing for a superconducting ground state than a compression of the cell along $c$. The fact that Ru substitutions do not suppress the SDW state for as low concentrations as for Co, strongly indicates that Ru is isovalent to Fe in SrFe$_{2-x}$Ru$_x$As$_2$. In (optimally superconducting) SrFe$_{1.80}$Co$_{0.20}$As$_2$ the $c/a$ ratio is still quite large ($c/a$ = 3.132), however 0.2 electrons were added per formula unit. In order to compare all these parameters further detailed structural studies on various SrFe$_{2-x}T_x$As$_2$ systems are required. For the BaFe$_{2-x}$(Co,Ni,Cu)$_x$As$_2$ systems a first attempt in this direction was presented recently.[@Canfield09a] Also, it is desirable to measure the compressibility of the compounds under hydrostatic pressure. Only such data would allow a comparison of pressure experiments with chemical substitutions studies. During revision of this work, superconductivity in several systems Sr/BaFe$_{2-x}T_x$As$_2$ with $T$ = Ru,[@Paulraj09a; @YQi09a] Rh,[@FHan09b] Pd,[@XZhu09a], and Ir[@FHan09a] was reported. For some of these systems the electronic state was already addressed by DFT methods.[@LJZhang09a] Experimentally, it appears that the critical temperatures are generally limited to $\approx 20$K and that only for isovalent substitution (*viz.* Ru) the required level of substitution is such high as reported here. All these new $3d$, $4d$, and $5d$-metal substitutions deserve both an experimental as well as a sophisticated theoretical treatment, including the investigation of the influence of disorder and charge doping. [38]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , , , , ****, (). , , , , , , ****, (). , , , ****, (). , , , , , , , ****, (). , , , , , , , , , , ****, (). , , , , ****, (). , , , , , , ****, (). , , , , , , , , (), . , , , , , , (), . , , , , , , , , , , (), . , , , , , (), . , , , , , (), . , , , , , , , ****, (). , , , , ****, (). , , , , , , , ****, (). , , , , , , , , , , ****, (). , , , , , , , , (), . , , , , ****, (). , , , , , , (), . , , , , , , , ****, (). , , (), . , , , ****, (). , , , , , , ****, (). , , , , , , ****, (). , ** (, ). , , , , , , , ****, (). , , , , , , ****, (). , , , , , , (), . , , , , , , (), . , , , , , , , , , , (), . , , , , (), . , , , , , , (), . , , , , , (), . , , , , (), . , , (), . , , , , , , , (), . (), .

--- author: - Sanjeev Arora - Yuanzhi Li - Yingyu Liang - Tengyu Ma - 'Andrej Risteski [^1]' bibliography: - 'polysemy.bib' title: 'Linear Algebraic Structure of Word Senses, with Applications to Polysemy' --- [^1]: Princeton University, Computer Science Department. `{arora,yuanzhil,yingyul,tengyu,risteski}@cs.princeton.edu`. This work was supported in part by NSF grants CCF-0832797, CCF-1117309, CCF-1302518, DMS-1317308, Simons Investigator Award, and Simons Collaboration Grant. Tengyu Ma was also supported by Simons Award for Graduate Students in Theoretical Computer Science.

--- abstract: 'We present quenched lattice QCD results for the contribution of higher-twist operators to the lowest non-trivial moment of the pion structure function. To be specific, we consider the combination $F_2^{\pi^+} + F_2^{\pi^-} - 2 F_2^{\pi^0}$ which has $I = 2$ and receives contributions from 4-Fermi operators only. We introduce the basis of lattice operators. The renormalization of the operators is done perturbatively in the $\overline{\rm{MS}}$ scheme using the ’t Hooft-Veltman prescription for $\gamma_5$, taking particular care of mixing effects. The contribution is found to be of $O(f_\pi^2/Q^2)$, relative to the leading contribution to the moment of $F_2^{\pi^+}$.' address: - 'Deutsches Elektronen-Synchrotron DESY, D-22603 Hamburg, Germany' - 'Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany' - 'Institut für Physik, Humboldt-Universität zu Berlin, D-10115 Berlin, Germany' - | Deutsches Elektronen-Synchrotron DESY,\ John von Neumann-Institut für Computing NIC, D-15738 Zeuthen, Germany - 'Institut für Theoretische Physik, Freie Universität Berlin, D-14195 Berlin, Germany' author: - 'S. Capitani' - 'M. Göckeler' - 'R. Horsley' - 'B. Klaus' - 'V. Linke' - 'P.E.L. Rakow' - 'A. Schäfer' - 'G. Schierholz' date: title: | **Higher–Twist Contribution to Pion\ Structure Function: 4–Fermi Operators** --- [DESY 99-098]{}\ [HLRZ 99-31]{}\ [HUB-EP-99/33]{}\ [TPR-99-14]{}\ , , , , , , and Structure Functions, Higher Twist, Lattice QCD. Introduction ============ The deviations from Bjorken scaling seen in deep-inelastic structure functions are usually interpreted as logarithmic scaling violations, as predicted by perturbative QCD. There is mounting evidence [@Liuti], however, that part of the deviations are due to power corrections induced by higher-twist effects. This is good news: higher-twist operators probe the non-perturbative features of hadronic bound states beyond the parton model, and they may provide valuable information on the interface between perturbative and non-perturbative physics, at least in those cases where the operator product expansion (OPE) allows a clean separation between short- and long-distance phenomena. The OPE expresses the moments of the structure function as a series of forward hadron matrix elements of local operators with coefficients decreasing as powers of $1/Q^2$. In general the expansion takes the form $$\begin{aligned} \lefteqn{M_n(Q^2) = \int_0^1 \mbox{d}x\,x^{n-2}F_2(x,Q^2)} \nonumber \\ & &= C^{(2)}_n(Q^2/\mu^2,g(\mu))\,A^{(2)}_n(\mu) + \frac{C^{(4)}_n (Q^2/\mu^2,g(\mu))}{Q^2} \, A^{(4)}_n (\mu) + O\Big(\frac{1}{(Q^2)^2}\Big) \nonumber \\ & &\equiv M_n^{(2)}(Q^2) + M_n^{(4)}(Q^2) + O\Big(\frac{1}{(Q^2)^2}\Big), \label{eq:ope} \end{aligned}$$ where $A(\mu)$ and $C(Q^2/\mu^2, g(\mu))$ are the operator matrix elements and Wilson coefficients, respectively, renormalized at a scale $\mu$, with the superscript (2 or 4) denoting the twist. Under ideal circumstances the Wilson coefficients can be calculated perturbatively, which leaves only the matrix elements to be computed on the lattice. The leading-twist contribution can be written as $$\label{eq:m2} M_n^{(2)}=\sum_f Q_f^2 {\langle}x_f^{n-1}{\rangle},$$ where $x_f$ is the energy fraction carried by the quark of flavor [$f$]{}, and $Q_f$ is the charge of the quark. The higher-twist contributions usually have no parton model interpretation. In this paper we will consider the $I=2$ pion structure function [@Gottlieb:1978; @Morelli:1993] $$\label{eq:i2sf} F_2^{I=2} \equiv F_2^{\pi^+}+F_2^{\pi^-}-2 F_2^{\pi^0}.$$ This belongs to a flavor 27-plet, is purely higher twist, and receives contributions from 4-Fermi operators only.[^1] We restrict ourselves to the lowest non-trivial moment $M_2$. Thus we will have to compute $A_2^{(4)}(\mu)$. The problem splits into two separate tasks. The first task obviously is to compute the pion matrix elements of the appropriate lattice 4-Fermi operators. The second task is to renormalize these operators at some finite scale $\mu$. The paper is organized as follows. In sect. 2 we identify the (renormalized) operator whose matrix elements we have to compute according to the OPE. In sect. 3 we perform the renormalization of the 4-Fermi operators. This is done to 1-loop order in perturbation theory. In sect. 4 we present the results of the lattice calculation. Finally, in sect. 5 we collect our results and conclude. 4–Fermi Contribution ==================== The leading twist-2 matrix element $A_2^{(2)}$ that enters $M_2$ is given by $$\label{eq:ltwist} \langle\vec{p}|O_{\{\mu\nu\}}|\vec{p}\rangle = 2 A_2^{(2)}[p_{\mu}p_{\nu} - \mbox{trace}] ,$$ $$\label{eq:ltop} O_{\mu\nu} = \frac{\mbox{i}}{2}{\bar{\psi}}\gamma_{\mu} G^2 \overleftrightarrow{D}_{\nu} {\psi}- \mbox{trace} ,$$ where $\{\cdots\}$ means symmetrization, and $\overleftrightarrow{D}=\overrightarrow{D}-\overleftarrow{D}$. The fermion fields are 2-component vectors in flavor space, corresponding to $u$ and $d$ quarks, and the charge matrix is $$\label{eq:charge} G = \begin{pmatrix} e_u & 0 \\ 0 & e_d \end{pmatrix} = \begin{pmatrix} 2/3 & 0 \\ 0 & -1/3 \end{pmatrix} .$$ The states are normalized as $$\label{minkonorm} \langle\vec{p}\,|\vec{p}\,'\rangle = (2 \pi)^3 \, 2 E_{\vec{p}} \, \delta (\vec{p} -\vec{p}\,') .$$ The Wilson coefficient has the form $$\label{eq:ltcoeff} C_2^{(2)} = 1+O(g^2) .$$ This contribution is the same for charged and neutral pions, and so vanishes when considering the structure function $F_2^{I=2}$. The twist-4 matrix element $A_2^{(4)}$ receives, in general, contributions from a large variety of operators. Here we shall only be interested in 4-Fermi operators, because these are the only operators that contribute to $F_2^{I=2}$. Following [@Jaffe:1981; @Jaffe:1982] we have $$\label{eq:htwist} \langle\vec{p}|A^c_{\{\mu\nu\}}|\vec{p}\rangle = 2 A_2^{(4)}[p_{\mu}p_{\nu} - \mbox{trace}] ,$$ $$\label{eq:htop} A^c_{\mu\nu} = {\bar{\psi}}G\gamma_{\mu}\gamma_5 t^a{\psi}{\bar{\psi}}G\gamma_{\nu}\gamma_5 t^a{\psi}- \mbox{trace}.$$ The corresponding Wilson coefficient is given by [@Jaffe:1981; @Jaffe:1982; @Shuryak:1982b] $$\label{eq:htcoeff} C_2^{(4)} = g^2(1+O(g^2)).$$ The operator (\[eq:htop\]) is understood to be the renormalized, continuum operator. Perturbative Renormalization ============================ In the following we will be working in Euclidean space-time. The 4-Fermi operators that we need to consider on the lattice are $$\label{eq:6ops} \begin{split} V^c_{\mu\nu} &= {\bar{\psi}}G\gamma_{\mu} t^a {\psi}\, {\bar{\psi}}G\gamma_{\nu} t^a {\psi}-\mbox{trace},\\[0.5ex] A^c_{\mu\nu} &= {\bar{\psi}}G\gamma_{\mu} \gamma_5 t^a {\psi}\, {\bar{\psi}}G\gamma_{\nu} \gamma_5 t^a {\psi}-\mbox{trace}, \\[0.5ex] T^c_{\mu\nu} &= {\bar{\psi}}G\sigma_{\mu\rho} t^a {\psi}\, {\bar{\psi}}G\sigma_{\nu\rho} t^a {\psi}-\mbox{trace},\\[0.5ex] V_{\mu\nu} &= {\bar{\psi}}G\gamma_{\mu} {\psi}\, {\bar{\psi}}G\gamma_{\nu} {\psi}-\mbox{trace},\\[0.5ex] A_{\mu\nu} &= {\bar{\psi}}G\gamma_{\mu} \gamma_5 {\psi}\, {\bar{\psi}}G\gamma_{\nu} \gamma_5 {\psi}-\mbox{trace},\\[0.5ex] T_{\mu\nu} &= {\bar{\psi}}G\sigma_{\mu\rho} {\psi}\, {\bar{\psi}}G\sigma_{\nu\rho} {\psi}-\mbox{trace}.\\ \end{split}$$ Summation over repeated indices and symmetrization in $\mu$, $\nu$ is understood. In our conventions $\sigma_{\mu\nu}=\frac{\rm i}{2} [\gamma_\mu,\gamma_\nu]$. The $I=2$ parts of the operators are related by Fierz transformations: $$\label{eq:fierz} \begin{split} V^c_{\mu\nu} &= -\frac{N_c+2}{4N_c} V_{\mu\nu} -\frac{1}{4} A_{\mu\nu} +\frac{1}{4} T_{\mu\nu}, \\ A^c_{\mu\nu} &= -\frac{1}{4} V_{\mu\nu} -\frac{N_c+2}{4N_c} A_{\mu\nu} -\frac{1}{4} T_{\mu\nu}, \\ T^c_{\mu\nu} &= \frac{1}{2} V_{\mu\nu} -\frac{1}{2} A_{\mu\nu} -\frac{1}{2N_c} T_{\mu\nu} . \\ \end{split}$$ The reason for considering all six operators is that they will mix under renormalization. In principle, the operators (\[eq:6ops\]) could also mix with gauge variant, BRST invariant operators. But there are no such 4-Fermi operators with dimension six, and 2-Fermi operators do not contribute. A 1-loop calculation in the continuum and on the lattice gives for the respective operator matrix elements $$\begin{split} \langle p, p'|O_i^{\rm cont}(\mu)| p, p' \rangle &= \sum_j \Big(\delta_{ij} +\frac{g_0^2}{16\pi^2} R_{ij}^{\rm cont} \Big) \langle p, p' | O_j^{\rm tree}| p, p' \rangle , \\ \langle p, p' |O_i^{\rm lat}(a)| p, p' \rangle &= \sum_j \Big(\delta_{ij} +\frac{g_0^2}{16\pi^2} R_{ij}^{\rm lat} \Big) \langle p, p' | O_j^{\rm tree}| p, p' \rangle, \\ \end{split}$$ where $|p,p'\rangle$ are quark states in some covariant gauge and $g_0$ is the bare coupling constant. The continuum matrix elements are understood to be renormalized in the $\overline{\rm{MS}}$ scheme at the scale $\mu$, while the lattice matrix elements are unrenormalized. Note that the tree-level matrix elements $\langle p, p' | O_j^{\rm tree}| p, p' \rangle$ are the same in both cases. The lattice and continuum matrix elements are then connected by $$\label{eq:contmlatt1} \begin{split} \langle p, p' | O_i^{\rm cont}(\mu)| p, p' \rangle = \sum_j \Big( \delta_{ij} - &\frac{g_0^2}{16 \pi^2} \big( R_{ij}^{\rm lat} - R_{ij}^{\rm cont} \big) \Big) \\[0.7ex] &\times \,\langle p, p' |O_j^{\rm lat}(a)| p, p' \rangle .\\ \end{split}$$ Let us write $$\label{eq:contmlatt2} \Delta R_{ij} = R_{ij}^{\rm lat} - R_{ij}^{\rm cont}.$$ While $R^{\rm lat}$ and $R^{\rm cont}$ depend on the state, $\Delta R$ is independent of the state and depends only on $a\mu$. The renormalization constants, that take us from bare lattice to renormalized continuum numbers, are given by $$Z_{ij} (a\mu, g) = \delta_{ij} - \frac{g_0^2}{16\pi^2} \Delta R_{ij} .$$ We have found it convenient to take $p=p'$. The algebraic manipulations have been done with the help of FORM, using the ’t Hooft-Veltman prescription for dealing with the $\gamma_5$ matrices. This is the only prescription that has proven to give consistent results. Integrating the ’t Hooft-Veltman prescription into FORM was a non-trivial task. The algebraic workload increased by about an order of magnitude. To check the results, many of the symbolic calculations were repeated by hand. We follow the method of [@Kawai] for regularizing the infrared divergences. The diagrams that we have calculated are shown in fig. \[fig:diagrams\]. We have not made use of Fierz transformations to reduce the number of diagrams, as we could do the numerical integration fast and with high precision, and also to avoid possible problems with $d$-dimensional Fierz transformations. \[fig:diagrams\] The 1-loop result for the renormalized 4-Fermi operators in the $\overline{\rm{MS}}$ scheme is $$\begin{aligned} \label{eq:1-loop} V^c_{\mu\nu}(\mu) &=& V^c_{\mu\nu} -\frac{g_0^2}{16\pi^2} \Big[ \Big( -\frac{1}{2N_c} c_1 + \frac{N_c^2-1}{2N_c} 2S - N_c c_2 \Big) V^c_{\mu\nu} \nonumber \\[0.95ex] &+& c_4 \Big(\frac{N_c^2-1}{N_c^2} A_{\mu\nu} +\frac{N_c^2-4}{N_c} A^c_{\mu\nu} \Big) - N_c c_3 T^c_{\mu\nu} \Big], \nonumber \\[0.95ex] V_{\mu\nu}(\mu) &=& V_{\mu\nu} -\frac{g_0^2}{16\pi^2} \Big[ \frac{N_c^2-1}{2N_c} \Big(c_1 + 2S\Big) V_{\mu\nu} +4 c_4 A^c_{\mu\nu} \Big], \\[0.95ex] A^c_{\mu\nu}(\mu) &=& A^c_{\mu\nu} -\frac{g_0^2}{16\pi^2} \Big[ \Big( -\frac{1}{2N_c} c^{(5)}_1 + \frac{N_c^2-1}{2N_c} 2S - N_c c_2 \Big) A^c_{\mu\nu} \nonumber \\[0.95ex] &+& c_4 \Big(\frac{N_c^2-1}{N_c^2} V_{\mu\nu} +\frac{N_c^2-4}{N_c} V^c_{\mu\nu} \Big) - c_3 \Big(\frac{N_c^2-1}{N_c^2} T_{\mu\nu} +\frac{N_c^2-4}{N_c} T^c_{\mu\nu} \Big)\Big], \nonumber \\[0.95ex] A_{\mu\nu}(\mu) &=& A_{\mu\nu} -\frac{g_0^2}{16\pi^2} \Big[ \frac{N_c^2-1}{2N_c} \Big(c^{(5)}_1 +2S\Big) A_{\mu\nu} +4 c_4 V^c_{\mu\nu} - 4 c_3 T^c_{\mu\nu} \Big], \nonumber \end{aligned}$$ where $$S = \log a^2\mu^2 +12.852404 + (1-\alpha)\, (-\log a^2\mu^2 +3.792010)$$ is the difference between the (leg) self-energy on the lattice, including the tadpole diagram, and in the continuum, and $$\begin{split} c_1 &= -2\log a^2\mu^2 +15.530790 + (1-\alpha)\, (2\log a^2\mu^2 -7.584020), \\[0.5ex] c^{(5)}_1 &= -2\log a^2\mu^2 +5.887758 + (1-\alpha)\, (2\log a^2\mu^2 -7.584020), \\[0.5ex] c_2 &= \frac{1}{2}\log a^2\mu^2 -4.260157 + (1-\alpha)\, (-\log a^2\mu^2 +3.792010), \\[0.5ex] c_3 &= 1.205379, \\[0.5ex] c_4 &= -\frac{1}{2}\log a^2\mu^2 +0.094480 . \\ \end{split}$$ Here $\alpha$ is the covariant gauge parameter with $\alpha = 1$ corresponding to Feynman gauge and $\alpha = 0$ to Landau gauge. The renormalization constants are gauge invariant. Furthermore, they are independent of $\mu$ and $\nu$, and hence of the particular representation the operators reduce to [@rep]. The anomalous dimensions of $V+A$ and $V^c + A^c$, which are eigenvalues of the mixing matrix, agree with the result found in [@Okawa]. Later on we will be interested in (\[eq:1-loop\]) for $N_c=3$ and $\mu = \displaystyle{1/a}$. For this case we have $$\label{eq:renormres} \begin{split} V^c_{\mu\nu}(\mu\!=\!1/a) &= V^c_{\mu\nu} -g_0^2 \big( 0.281578 \,\, V^c_{\mu\nu} + 0.000532 \,\, A_{\mu\nu} \\ &+ 0.000997 \,\, A^c_{\mu\nu} - 0.022899 \,\, T^c_{\mu\nu}\big), \\[0.95ex] V_{\mu\nu}(\mu\!=\!1/a) &= V_{\mu\nu} -g_0^2 \big( 0.348170 \,\, V_{\mu\nu} + 0.002393 \,\, A^c_{\mu\nu} \big), \\[0.95ex] A^c_{\mu\nu}(\mu\!=\!1/a) &= A^c_{\mu\nu} -g_0^2 \big( 0.291756 \,\, A^c_{\mu\nu} + 0.000532 \,\, V_{\mu\nu} \\ &+ 0.000997 \,\, V^c_{\mu\nu} - 0.006785 \,\, T_{\mu\nu} - 0.012722 \,\, T^c_{\mu\nu} \big), \\[0.95ex] A_{\mu\nu}(\mu\!=\!1/a) &= A_{\mu\nu} -g_0^2 \big( 0.266750 \,\, A_{\mu\nu} + 0.002393 \,\, V^c_{\mu\nu} \\ & - 0.030533 \,\, T^c_{\mu\nu} \big) .\\ \end{split}$$ Our results will be of use to other applications of 4-Fermi operators as well. For a similar calculation in the context of weak matrix elements, involving a different set of operators, see [@Rajan]. Lattice Calculation =================== General Formalism ----------------- To obtain the pion matrix elements of the operators (\[eq:6ops\]), we need to compute the 3- and 2-point functions $$\label{eq:corr} \begin{split} C^{(3)}_O(t,\tau)&= \frac{1}{V_S}{\langle}{}^S\!\pi^F(t) \,O(\tau)\,{}^S\!\pi^{F^\dagger}(0){\rangle}, \\ C^{(2)}(t)&= -\frac{1}{V_S}{\langle}{}^S\!\pi^F(t){}^S\!\pi^{F^\dagger}(0){\rangle},\\ \end{split}$$ where $V_S$ is the spatial volume of the lattice. The pion field is $$\label{eq:pistat} {}^S\!\pi^F(t) = \sum_{x;x_4=t} {}^S\!{\bar{\psi}}_{f\alpha}^a(x ) F_{ff'}({\gamma_5})_{\alpha\beta}{}^S\!{\psi}_{f'\beta}^a(x),$$ where $F$ is the flavor matrix. For the different pion states the flavor matrix takes the form $$\label{eq:tau} \begin{split} \pi^+ &: F = \begin{pmatrix} 0 &1 \\ 0 &0 \end{pmatrix} , \\ \pi^- &: F = \begin{pmatrix} 0 &0 \\ 1 &0 \end{pmatrix} , \\ \pi^0 &: F= \frac{1}{\sqrt{2}}\begin{pmatrix} 1 &0 \\ 0 &-1 \end{pmatrix} .\\ \end{split}$$ The superscript $S$ in (\[eq:pistat\]) denotes Jacobi-smeared quark fields: $$\label{eq:squark} \begin{split} {}^S\!{\psi}_{f\alpha}^a(x) &= \sum_{y;x_4=y_4=t} H^{ab}(x,y;U){\psi}_{f\alpha}^b(y), \\ {}^S\!{\bar{\psi}}_{f\alpha}^a(x) &= \sum_{y;x_4=y_4=t}{\bar{\psi}}_{f\alpha}^b(y) H^{ba}(y,x;U), \\ \end{split}$$ with $H$ being given in [@Best:1997qp], and $U$ denoting the link variables. The 4-Fermi operators in (\[eq:6ops\]) can be expressed as $$\label{eq:gen4f} O(\tau) = \sum_{x;\,x_4=\tau}{\bar{\psi}}_{f\alpha}^a(x) G_{ff'}\Gamma_{\alpha\beta}^{ab}\, {\psi}_{f'\beta}^b(x) {\bar{\psi}}_{g\gamma}^c(x) G_{gg'}{\Gamma'}_{\gamma \delta}^{\,cd}\, {\psi}_{g'\delta}^d(x),$$ where $\Gamma$ and $\Gamma'$ are products of $\gamma$ and $t^a$ matrices. A transfer matrix calculation gives for the time dependence of the 2-point function $$\label{eq:2tdep} \begin{split} {\langle}{}^S\!\pi^F(t){}^S\!\pi^{F^\dagger}(0){\rangle}&= {\operatorname{Tr}}[{\hat{S}}^{T-t}{\hat{\pi}}^F{\hat{S}}^t{\hat{\pi}}^{F^\dagger}] \\ &={\langle}0 |{\hat{\pi}}^{F}|\pi{\rangle}{\langle}\pi|{\hat{\pi}}^{F^\dagger}|0{\rangle}e^{-m_\pi t} +{\langle}0 |{\hat{\pi}}^{F^\dagger}|\pi{\rangle}{\langle}\pi|{\hat{\pi}}^{F}|0{\rangle}e^{-m_\pi (T-t)},\\ \end{split}$$ where $|\pi{\rangle}$ is the pion ground state, $T$ is the time extent of the lattice, and ${\hat{S}}$ is the transfer matrix. All contributions from excited states have been suppressed. The pion states are normalized according to $$\label{latnorm} {\langle}\pi(\vec{p}) | \pi(\vec{p}\,') {\rangle}= \delta_{\vec{p}\,\vec{p}\,'} .$$ The same calculation gives for the 3-point function $$\label{eq:3tdep} \begin{split} {\langle}{}^S\!\pi^F(t)O(\tau){}^S\!\pi^{F^\dagger}(0){\rangle}&= \begin{cases} {\operatorname{Tr}}[{\hat{S}}^{T-t}{\hat{\pi}}^F{\hat{S}}^{t-\tau}O{\hat{S}}^\tau{\hat{\pi}}^{F^\dagger}], & T\ge t\ge\tau\ge 0, \\ {\operatorname{Tr}}[{\hat{S}}^{T-\tau}O{\hat{S}}^{\tau-t}{\hat{\pi}}^F{\hat{S}}^t{\hat{\pi}}^{F^\dagger}], & T\ge\tau\ge t\ge 0 \end{cases} \\[1.5ex] &= \begin{cases} {\langle}0 |{\hat{\pi}}^F|\pi{\rangle}{\langle}\pi|O|\pi{\rangle}{\langle}\pi|{\hat{\pi}}^{F^\dagger}|0{\rangle}e^{-m_\pi t}, & T\ge t\ge\tau\ge 0, \\ {\langle}0 |{\hat{\pi}}^{F^\dagger}|\pi{\rangle}{\langle}\pi|O|\pi{\rangle}{\langle}\pi|{\hat{\pi}}^F|0{\rangle}e^{-m_\pi (T-t)}, & T\ge\tau\ge t\ge 0 . \end{cases}\\ \end{split}$$ Thus for the ratio of 3- and 2-point functions we may expect to see a plateau in $\tau$ at $0\ll\tau\ll t$, and a plateau at $t\ll\tau\ll T$: $$\label{eq:ratio} R_O(t,\tau) \equiv \frac{C^{(3)}_O(t,\tau)}{C^{(2)}(t)} = \begin{cases} R_O^{t\geqslant\tau}, & 0\ll\tau\ll t, \\ R_O^{t\leqslant\tau}, & t\ll\tau\ll T , \end{cases}$$ with $$\label{eq:lrplateaux} \begin{split} R_O^{t\geqslant\tau} &= - {\langle}\pi|O|\pi{\rangle}\frac{ e^{-m_\pi t}} { e^{-m_\pi t} + e^{-m_\pi (T-t)}},\\ R_O^{t\leqslant\tau} &= - {\langle}\pi|O|\pi{\rangle}\frac{ e^{-m_\pi (T-t)}} {e^{-m_\pi t} + e^{-m_\pi (T-t)}}.\\ \end{split}$$ The sum is independent of $t$, and finally we obtain $$\label{eq:opval} {\langle}\pi|O|\pi{\rangle}= - R_O^{t\geqslant\tau} - R_O^{t\leqslant\tau}.$$ Technical Details ----------------- We will now re-write the 3-point function (\[eq:corr\]) in terms of quark propagators. Two different kinds of propagators are emerging, depending on whether we start from a smeared quark field and end on a local one, or vice versa. The local-smeared and smeared-local propagators, ${}^{SL}\!{\Delta}$ and ${}^{LS}\!{\Delta}$, are given by $$\label{eq:fprop} \begin{split} {\langle}{}^S\!{\psi}_{f\alpha}^a (x){\bar{\psi}}_{f'\beta}^b(y){\rangle}_{\bar{\psi},\psi} &= \delta_{ff'}{}^{SL}\!{\Delta}_{\alpha\beta}^{ab}(x,y) \\ {\langle}{\psi}_{f\alpha}^a (x){}^S\!{\bar{\psi}}_{f'\beta}^b(y){\rangle}_{\bar{\psi},\psi} &= \delta_{ff'}{}^{LS}\!{\Delta}_{\alpha\beta}^{ab}(x,y) , \end{split}$$ with $$\label{eq:proprel} {}^{LS}\!{\Delta}(x,y) ={\gamma_5}{}^{SL}\!{\Delta}(y,x)^\dagger{\gamma_5}.$$ (We have suppressed the dependence on $U$.) For $I=2$, and after having summed over flavor indices, we obtain $$\begin{aligned} C^{(3)}_O(t,\tau) &=& (e_u - e_d)^2 \frac{1}{V_S} \sum_{x;\,x_4=t}\sum_{y;\,y_4=\tau}\sum_{z;\,z_4=0} \big\{ \nonumber \\ & &{\langle}\,{\operatorname{Tr}}[{}^{LS}\!{\Delta}(y,z){\gamma_5}{}^{SL}\!{\Delta}(z,y)\Gamma' {}^{LS}\!{\Delta}(y,x){\gamma_5}{}^{SL}\!{\Delta}(x,y)\Gamma]\,{\rangle}_U \nonumber \\ &+& {\langle}\,{\operatorname{Tr}}[{}^{LS}\!{\Delta}(y,z){\gamma_5}{}^{SL}\!{\Delta}(z,y)\Gamma {}^{LS}\!{\Delta}(y,x){\gamma_5}{}^{SL}\!{\Delta}(x,y)\Gamma']\,{\rangle}_U \label{eq:gen4f2} \\ &-&{\langle}\,{\operatorname{Tr}}[{}^{LS}\!{\Delta}(y,z){\gamma_5}{}^{SL}\!{\Delta}(z,y)\Gamma'] {\operatorname{Tr}}[{}^{LS}\!{\Delta}(y,x){\gamma_5}{}^{SL}\!{\Delta}(x,y)\Gamma]\,{\rangle}_U \nonumber \\ &-&{\langle}\,{\operatorname{Tr}}[{}^{LS}\!{\Delta}(y,z){\gamma_5}{}^{SL}\!{\Delta}(z,y)\Gamma] {\operatorname{Tr}}[{}^{LS}\!{\Delta}(y,x){\gamma_5}{}^{SL}\!{\Delta}(x,y)\Gamma']\,{\rangle}_U \big\} . \nonumber \end{aligned}$$ One of the spatial sums can be eliminated by making use of translational invariance, finally giving $$\begin{aligned} C^{(3)}_O(t,\tau) &=& (e_u - e_d)^2 \sum_{x;\,x_4=t-\tau}\sum_{z;\,z_4=-\tau} \big\{ \nonumber \\ & &{\langle}\,{\operatorname{Tr}}[{}^{LS}\!{\Delta}(0,z){\gamma_5}{}^{SL}\!{\Delta}(z,0)\Gamma' {}^{LS}\!{\Delta}(0,x){\gamma_5}{}^{SL}\!{\Delta}(x,0)\Gamma]\,{\rangle}_U \nonumber \\ &+& {\langle}\,{\operatorname{Tr}}[{}^{LS}\!{\Delta}(0,z){\gamma_5}{}^{SL}\!{\Delta}(z,0)\Gamma {}^{LS}\!{\Delta}(0,x){\gamma_5}{}^{SL}\!{\Delta}(x,0)\Gamma']\,{\rangle}_U \\ &-&{\langle}\,{\operatorname{Tr}}[{}^{LS}\!{\Delta}(0,z){\gamma_5}{}^{SL}\!{\Delta}(z,0)\Gamma'] {\operatorname{Tr}}[{}^{LS}\!{\Delta}(0,x){\gamma_5}{}^{SL}\!{\Delta}(x,0)\Gamma]\,{\rangle}_U \nonumber \\ &-&{\langle}\,{\operatorname{Tr}}[{}^{LS}\!{\Delta}(0,z){\gamma_5}{}^{SL}\!{\Delta}(z,0)\Gamma] {\operatorname{Tr}}[{}^{LS}\!{\Delta}(0,x){\gamma_5}{}^{SL}\!{\Delta}(x,0)\Gamma']\,{\rangle}_U \big\} . \nonumber \end{aligned}$$ All 3-point functions can be built from $$\label{eq:Q} Q_{\alpha\beta}^{ab}(t)=\sum_{x;\,x_4=t} {}^{LS}\!{\Delta}_{\alpha\gamma}^{ac}(0,x)({\gamma_5})_{\gamma\delta} {}^{SL}\!{\Delta}_{\delta\beta}^{cb}(x,0) ,$$ where the first propagator can be obtained from the second one by means of (\[eq:proprel\]). In terms of (\[eq:Q\]) the 3-point function reads $$\label{eq:qform} \begin{split} C^{(3)}_O(t,\tau) &= (e_u - e_d)^2 \big\{ \langle \,{\operatorname{Tr}}[ Q(-\tau) \Gamma' Q(t-\tau) \Gamma ] \,\rangle_U \\ &+\langle {\operatorname{Tr}}[ Q(-\tau) \Gamma Q(t-\tau) \Gamma'] \,\rangle_U \\ &-\langle\, {\operatorname{Tr}}[ Q(-\tau) \Gamma'] {\operatorname{Tr}}[ Q(t-\tau) \Gamma ] \,\rangle_U \\ &-\langle\, {\operatorname{Tr}}[ Q(-\tau) \Gamma ] {\operatorname{Tr}}[ Q(t-\tau) \Gamma'] \,\rangle_U \big\} . \\ \end{split}$$ By computing the propagators from a local source at $t=0$ to all $t<T$ using sink-smearing we obtain the 3-point functions for all $t$ and $\tau$. Results of the Simulation ------------------------- The numerical calculations are done on a $16^3\times 32$ lattice at $\beta \equiv 6/g_0^2 = 6.0$ in the quenched approximation. We use Wilson fermions. To be able to extrapolate our results to the chiral limit, the calculations are done at three values of the hopping parameter, $\kappa = 0.1550$, 0.1530 and 0.1515. This corresponds to physical quark masses of about 70, 130 and 190 MeV, respectively. For the gauge update we use a 3-hit Metropolis sweep followed by 16 overrelaxation sweeps, and we repeat this 50 times to generate a new configuration. Our data sample consists of 400 configurations. In the following we shall restrict ourselves to zero momentum pion states and operators with $\mu = \nu =4$. In fig. \[fig:plat\] we show the ratio (\[eq:ratio\]) for the operator $A$ at three different values of $t$. We find very good plateaus in $\tau$. In our final fits we have averaged over $t$ values around $T/2$. To obtain continuum results we have to multiply each quark field by $\sqrt{2\kappa}$, and a factor of $2m_\pi$ is needed to convert to the continuum normalization of states: $$\label{eq:opnorm} \langle \pi|O|\pi \rangle^{\rm cont} = (2\kappa)^2\, 2m_\pi \, \langle \pi|O|\pi \rangle^{\rm lat} .$$ For the reduced matrix element $A_2^{(4)}$ (in (\[eq:htwist\])) we then find $$\label{eq:a4ar} A_2^{(4)} = \frac{4}{3}\,\frac{(2\kappa)^2}{m_\pi}\,\langle \pi|A^c_{44}|\pi \rangle^{\rm lat}, \; \langle \pi|A^c_{44}|\pi \rangle^{\rm lat} = - R_{A^c_{44}}^{t\geqslant\tau} - R_{A^c_{44}}^{t\leqslant\tau},$$ where $A^c_{44}$ is the renormalized operator, as given in (\[eq:1-loop\]). $\kappa$ $af_{\pi B}$ $am_\pi$ ---------- -------------- ------------ 0.1515 0.122(2) 0.5037(8) 0.1530 0.113(2) 0.4237(8) 0.1550 0.098(2) 0.3009(10) : Pion masses and unrenormalized (bare) decay constants.[]{data-label="tab:fpimpi"} ----------------------------- ------------ ------------ ------------ -------------------- -------------- $\;0.1515$ $\;0.1530$ $\;0.1550$ $\kappa_c=0.15717$ $\;\;\chi^2$ $\bar{A}\,f^{-2}_{\pi B}$ 0.561(13) 0.546(17) 0.514(25) 0.490(37) 0.0593 $\bar{V}\,f^{-2}_{\pi B}$ -0.139(9) -0.154(13) -0.212(23) -0.237(31) 0.897 $\bar{T}\,f^{-2}_{\pi B}$ -0.207(21) -0.200(30) -0.197(47) -0.190(67) 0.00397 $\bar{A}^c\,f^{-2}_{\pi B}$ -0.147(11) -0.139(16) -0.111(27) -0.098(38) 0.111 $\bar{V}^c\,f^{-2}_{\pi B}$ -0.134(12) -0.122(17) -0.089(29) -0.071(40) 0.113 $\bar{T}^c\,f^{-2}_{\pi B}$ -0.315(30) -0.317(43) -0.330(68) -0.334(96) 0.00880 ----------------------------- ------------ ------------ ------------ -------------------- -------------- : The unrenormalized, reduced matrix elements $\bar{O}$, together with their extrapolations to the chiral limit.[]{data-label="tab:latres"} The lattice pion masses are given in table \[tab:fpimpi\]. In the following we shall express the dimensionful matrix elements in terms of the pion decay constant $f_\pi$, whose unrenormalized values are also given in the table. The pion masses and the decay constants are taken from [@Gockeler:1998fn], where we had a slightly higher statistics. In table \[tab:latres\] we give our results for the unrenormalized, reduced matrix elements $\bar{O} = - ((2\kappa)^2/m_\pi)\, \langle\pi|O_{44}|\pi\rangle^{\rm lat}$ of the various $I=2$ operators. The reader can easily check that the Fierz identities (\[eq:fierz\]) hold identically at each value of $\kappa$. In fig. \[fig:chifit1\] we plot $\bar{O}$ for the operators without color matrices, and in fig. \[fig:chifit2\] for the operators with color matrices, as a function of $\kappa^{-1}$. The data suggest a linear extrapolation to the chiral limit. The result of the extrapolation is given in table \[tab:latres\]. The critical hopping parameter is $\kappa_c=0.15717(3)$. Results and Conclusions ======================= We are now ready to give results for the structure function. To minimize effects of higher order contributions to the Wilson coefficient $C_2^{(4)}$ in $g^2$, we shall take $$Q^2 = \mu^2 = a^{-2}.$$ The $g^2$ in (\[eq:htcoeff\]) is therefore replaced by $4\pi \alpha_s(Q^2)$. If we fix the scale by adjusting the $\rho$ mass to its physical value, we have [@Gockeler:1998fn] $a^{-2} \approx 5 \,\mbox{GeV}^2$. If, instead, we take the string tension or the force parameter to set the scale, we have $a^{-2} \approx 4 \,\mbox{GeV}^2$. We calculate the renormalization constants from (\[eq:renormres\]) with $g_0=1$. Combining these with the unrenormalized lattice results in table \[tab:latres\], we find for the $I=2$ structure function $$A_2^{(4)\,I=2} = 0.100(38)\, f^{\,2}_{\pi B}.$$ Multiplying this number with the Wilson coefficient and the kinematical factor, and expressing the result in terms of the renormalized decay constant $f_{\pi} = Z_A f_{\pi B}$ with $Z_A = 0.867$, computed in perturbation theory, we finally obtain $$\label{final} M_2^{I=2} = 1.67(64)\, \frac{f_{\pi}^{2}\,\alpha_s(Q^2)}{Q^2} + O(\alpha_s^2).$$ All numbers refer to the chiral limit. An early calculation [@Morelli:1993], based on 15 configurations, found a negative value[^2] for $M_2^{I=2}$. It is instructive to compare (\[final\]) with the twist-2 contribution to the structure function of (say) the $\pi^+$. In [@Best:1997qp] we found $$M_2^{(2)\,\pi^+} = 0.152(7).$$ Relative to this number (\[final\]) is only a small correction, except perhaps at very small values of $Q^2$. In case of the nucleon we may expect similar numbers, but with $f_\pi$ being replaced by the nucleon mass. This would then result in a significant correction. Calculations of 4-Fermi contributions to the nucleon structure function are in progress. Acknowledgment {#acknowledgment .unnumbered} ============== The numerical calculations have been done on the Quadrics computers at DESY-Zeuthen. We thank the operating staff for support. This work was supported in part by the Deutsche Forschungsgemeinschaft. [10]{} S. Liuti, Nucl. Phys. B (Proc. Suppl.) 74 (1999) 380 ([hep-ph/9809248]{}). S. Gottlieb, Nucl. Phys. B139 (1978) 125. A. Morelli, Nucl. Phys. B392 (1993) 518. S. Capitani, M. Göckeler, R. Horsley, H. Oelrich, D. Petters, P. Rakow and G. Schierholz, Nucl. Phys. B (Proc. Suppl.) 73 (1999) 288 ([hep-lat/9809171]{}). S. Capitani, M. Göckeler, R. Horsley, D. Petters, D. Pleiter, P. Rakow, G. Schierholz, DESY preprint DESY 99-069 ([hep-ph/9906320]{}). R. L. Jaffe and M. Soldate, Phys. Lett. 105B (1981) 467. R. L. Jaffe and M. Soldate, Phys. Rev. D26 (1982) 49. E. V. Shuryak and A. I. Vainshtein, Nucl. Phys. B199 (1982) 451. H. Kawai, R. Nakayama and K. Seo, Nucl. Phys. B189 (1981) 40. M. Göckeler, R. Horsley, E.-M. Ilgenfritz, H. Perlt, P. Rakow, G. Schierholz and A. Schiller, Phys. Rev. D54 (1996) 5705. M. Okawa, Nucl. Phys. B187 (1981) 71. M. Gupta, T. Bhattacharya and S. R. Sharpe, Phys. Rev. D55 (1997) 4036. C. Best, M. Göckeler, R. Horsley, E.-M. Ilgenfritz, H. Perlt, P. Rakow, A. Schäfer, G. Schierholz, A. Schiller and S. Schramm, Phys. Rev. D56 (1997) 2743. M. Göckeler, R. Horsley, H. Perlt, P. Rakow, G. Schierholz, A. Schiller and P. Stephenson, Phys. Rev. D57 (1998) 5562. [^1]: It thus evades mixing with operators of lower dimensions. In the general case where we do have mixing, and Wilson coefficients and higher-twist matrix elements are afflicted with renormalon ambiguities, calculations are much more difficult. In particular, one will also have to compute the Wilson coefficients non-perturbatively. For a first, fully non-perturbative calculation of higher-twist contributions to the nucleon structure function see [@Capitani:1998lm; @Capitani:1999]. So far it is, however, not clear how 4-Fermi operators can be incorporated in such a calculation. [^2]: Due to several misprints and inconsistencies in this paper [@Morelli:1993] we were not able to trace the origin of the discrepancy.

--- author: - 'M. Salz' - 'S. Czesla' - 'P. C. Schneider' - 'E. Nagel' - 'J. H. M. M. Schmitt' - 'L. Nortmann' - 'F. J. Alonso-Floriano' - 'M. López-Puertas' - 'M. Lampón' - 'F. F. Bauer' - 'I. A. G. Snellen' - 'E. Pallé' - 'J. A. Caballero' - 'F. Yan' - 'G. Chen' - 'J. Sanz-Forcada' - 'P. J. Amado' - 'A. Quirrenbach' - 'I. Ribas' - 'A. Reiners' - 'V. J. S. Béjar' - 'N. Casasayas-Barris' - 'M. Cortés-Contreras' - 'S. Dreizler' - 'E. W. Guenther' - 'T. Henning' - 'S. V. Jeffers' - 'A. Kaminski' - 'M. Kürster' - 'M. Lafarga' - 'L. M. Lara' - 'K. Molaverdikhani' - 'D. Montes' - 'J. C. Morales' - 'A. Sánchez-López' - 'W. Seifert' - 'M. R. Zapatero Osorio' - 'M. Zechmeister' bibliography: - 'hd189\_he10830.bib' date: 'Received 21 June 2018; accepted 2 November 2018' title: | Detection of [ $\uplambda10830$ Å]{} absorption on HD 189733b\ with CARMENES high-resolution transmission spectroscopy --- =1 Introduction {#SectIntroduction} ============ The atmospheres of close-in planets are exposed to intense high-energy irradiation by their host stars. Stellar extreme-UV and X-ray photons deposit large amounts of energy high up in the planetary atmosphere, capable of powering planetary evaporation with supersonic wind speeds of around 10 [kms$^{-1}$]{} [e.g., @Lammer2003; @Watson1981; @Salz2016]. Such radiation-induced planetary mass loss may be strong enough to completely evaporate the gaseous envelopes of small planets [@Lecavelier2004], which would explain the detected population of hot super-Earths [@Lundkvist2016; @Fulton2017]. An extended hydrogen atmosphere around KELT-9b was recently detected via optical H$\alpha$ transit spectroscopy with CARMENES [@Yan2018]. To date, the strongest observational evidence for the existence of planetary evaporation winds comes from [Ly$\alpha$]{} and UV transit spectroscopy. Prominent examples are the planets [HD209458b]{}, HD189733b, [WASP-12b]{}, and GJ436b [@Vidal2003; @Vidal2004; @Vidal2008; @Vidal2013; @Ehrenreich2008; @BenJaffel2010; @Linsky2010; @Ballester2015; @Lecavelier2010; @Lecavelier2012; @Bourrier2013; @BenJaffel2013; @Fossati2010; @Haswell2012; @Kulow2014; @Ehrenreich2015; @Lavie2017]. [Ly$\alpha$]{} observations can only be obtained using space-borne instrumentation and, more importantly, interstellar material absorbs the [Ly$\alpha$]{} line core even for the nearest stars. This absorption suppresses any signal at velocities of around 10 [kms$^{-1}$]{}, which is the characteristic speed of supersonic evaporation close to the planet. Therefore, alternative diagnostics for planetary winds are highly desirable. [@Seager2000] were the first to emphasize the potential of the [ $\uplambda10830$ Å]{} triplet lines to study upper atmospheric layers, which are the launching region of the planetary wind. The triplet is composed of two closely spaced lines with central wavelengths[^1] of 10830.33 and 10830.25 Å and a third weaker line at 10829.09 Å. These lines are accessible from the ground and are not affected by interstellar absorption. For example, along one line of sight, @Indriolo2009 derived an upper limit of $3.2\times 10^9$ cm$^{-2}$ for the interstellar triplet state column density, which is two to three orders of magnitude lower than the column densities derived in the following. Recently, @Oklopcic2018 used 1D models of the escaping atmospheres of [GJ436b]{} and [HD209458b]{}, including all sources and sinks of the triplet ground state, and showed that planetary absorption in the [ $\uplambda10830$ Å]{} lines could reach several percent. Absorption in the triplet lines in the stellar atmosphere is related to stellar activity features [e.g., @Zarro1986; @Sanz2008], which can cause serious complications for exoplanet transit observations. In the solar context, the lines have been studied in detail by [@Avrett1994] and [@Mauas2005]. The Sun shows a highly inhomogeneous surface distribution of [ $\uplambda10830$ Å]{} absorption [see Fig. 1 of @Andretta2017], with absorption practically restricted to active regions [@Avrett1994]. Accordingly, the disk-integrated absorption reveals 10% rotational variation and 30% variation over activity cycles [@Harvey1994]. While a search with the VLT/ISAAC instrument for atmospheric [ $\uplambda10830$ Å]{} absorption of the hot Jupiter orbiting the inactive host star [HD209458]{} resulted in an upper limit of 0.5% in a 3 Å wide window [@Moutou2003], the signal has now been detected in WASP-107b with Wide Field Camera 3 on board the Hubble Space Telescope. Here, 0.05% excess absorption was observed over a 98 Å wide window [@Spake2018]. In @Lisa2018, we report $3.6\pm0.2$% absorption observed during the transit of WASP-69b at high spectral resolution with the CARMENES spectrograph. Here, we present the detection and analysis of absorption in the [HD189733]{} system. The [HD189733]{} exoplanetary system {#SectTarget} ==================================== [l c l]{}\ Parameter & Value & Reference\ \ \ $\alpha$ \[J2015.5\] & 20:00:43.70 & [*Gaia*]{} DR2\ \ $\delta$ \[J2015.5\] & +22:42:35.3 & [*Gaia*]{} DR2\ \ $d$ & $19.775\pm 0.013$ pc & [*Gaia*]{} DR2\ \ $R_{\rm S}$ & $0.805\,(16)~R_\sun$ & @Boyajian2015\ \ $M_{\rm S}$ & $0.846^{+0.06}_{-0.049}~M_\sun$ & @deKok2013\ \ $P_{\rm rot}$ & $11.953\,(9)$ d & @Henry2008\ \ $K_{\rm S}$ & $201.96^{+1.07}_{-0.63}$ ms$^{-1}$ & @Triaud2009\ \ $v_{\rm sys}$ & $-2.361$(3) [kms$^{-1}$]{} & @Bouchy2005\ \ $T_0$ \[BJD$_{\rm TDB}$\] & 2453955.5255511(88) & @Baluev2015\ \ $P_{\rm orb}$ & 2.218575200(77) d & @Baluev2015\ \ $b$ & 0.6636(19) & @Baluev2015\ \ $R_{\rm p}/R_{\rm s}$ & 0.15712(40) & @Baluev2015\ \ $M_{\rm p}$ & $1.162^{+0.058}_{-0.039}$ [M$_{\rm jup}$]{} & @deKok2013\ \ $a/R_{\rm s}$ & 8.863(20) & @Agol2010\ \ $i$ & 85.710(24) & @Agol2010\ \ $K_{\rm p}$ & 162.2(3.3) [kms$^{-1}$]{} & this work\ \ The hot Jupiter [HD189733b]{} is among the best-studied planets to date. The 1.16 [M$_{\rm jup}$]{} mass planet orbits an active K dwarf with a period of 2.2 days [@Bouchy2005]; see Table \[TabPara\] for details. Its atmospheric transmission is likely dominated by Rayleigh scattering in the wavelength range from 3000 to 10000 Å [@Pont2008; @Pont2013; @Lecavelier2008; @Sing2009; @Sing2011; @Sing2016; @Gibson2012]. However, the contribution of unocculted stellar spots to the alleged Rayleigh scattering slope remains uncertain [@McCullough2014]. In lower atmospheric layers, absorption of carbon monoxide [@deKok2013; @Rodler2013; @Brogi2016] and water [@Birkby2013; @McCullough2014; @Brogi2018] has been detected. Measurements of sodium absorption were used to reconstruct the atmospheric temperature-pressure profile up to the lower thermosphere [@Redfield2008; @Huitson2012; @Wyttenbach2015; @Louden2015; @Sara2017]. A reported detection of planetary H$\alpha$ absorption by @Jensen2012 remains difficult to interpret due to the confounding spectral effects of stellar variability [@Barnes2016; @Cauley2017]. [HD189733]{} is an active star with strong emission cores in the H&K lines, resulting in a high value of $0.508$ for the Mount Wilson S-index [@Baliunas1995; @Knutson2010]. The star shows frequent flaring at optical and X-ray wavelengths [@Pilliteri2014; @Klocova2017] and an overall X-ray luminosity of $\approx$ 2 $\times$ 10$^{28}$  [@Huensch1999], which places it in the top decile of the X-ray luminosity distribution function [@schmitt1995]. @Sanz2011 reconstructed an extreme UV luminosity of $3\times10^{28}$ [ergs$^{-1}$]{}, which implies substantial high-energy irradiation levels on [HD189733b]{} that ought to trigger an evaporative wind in the upper planetary atmosphere [@Salz2016]. The escape of this upper atmosphere has been detected through hydrogen [Ly$\alpha$]{} and oxygen absorption [@Lecavelier2010; @Lecavelier2012; @Bourrier2013; @BenJaffel2013], and @Poppenhaeger2013 proposed a tentative $6-8$% deep X-ray transit. These findings make [HD189733b]{} a promising candidate to search for [ $\uplambda10830$ Å]{} absorption. Observations and data analysis {#SectCarmenes} ============================== We analyzed three spectral transit time series[^2] of [HD189733]{} taken on 8 Aug 2016, 17 Sept 2016, and 7 Sept 2017, in the following referred to as night 1, 2, and 3. The observations were taken with the CARMENES spectrograph, mounted at the 3.5 m telescope at the Calar Alto Observatory; see [@Quirrenbach2016] for a detailed description. The CARMENES spectrographs simultaneously cover the visual and near-infrared (NIR) ranges ($5500 - 9600$ Å and $9600 - 17\,200$ Å) with a nominal resolution of $94\,600$ and $80\,400$, respectively, and a sampling of 2.3 pixel per resolution element around the triplet. The two independent channels are housed in vacuum tanks to optimize radial velocity precision. The NIR spectrograph is fed by two fibers. In our configuration, fiber A carried the light from the target and fiber B was used to obtain simultaneous sky spectra. The data reduction was carried out with the pipeline [CARACAL v2.10, @Caballero2016]. ------- ------------ ------------- ----------------- --------------- ---------- -------- ------------- ------------------ Night Date Proposal ID Principal Calar Alto Nr. of S/N at Pre-/post- Mid-transit time investigator Archive ID spectra 10830Å transit (h) (BJD TDB) 1 2016-08-08 H16-3.5-024 P. J. Amado 246903-246952 45 (1/1) 160 1.0 / 1.0 2457609.51891 2 2016-09-17 H16-3.5-024 P. J. Amado 249745-249798 50 (4/2) 210 2.3 / 1.2 2457649.45326 3 2017-09-07 H16-3.5-022 J. A. Caballero 263103-263156 46 (1/1) 240 1.7 / 1.0 2458004.42529 ------- ------------ ------------- ----------------- --------------- ---------- -------- ------------- ------------------ The exposure time of our spectra was 198s throughout the campaign. Further details on the observations and the observing conditions are provided in Table \[tab:obs\_details\] and Fig. \[fig:obscond\]. The airmass was mostly below 1.5 and the typical seeing was better than 1. During night 3 the column of water vapor was higher, but this night offered the best seeing conditions and resulted in the highest signal-to-noise ratio (S/N) per spectrum. The S/N during the first two nights may have been affected by stability issues with the NIR channel, which is evident in the NIR radial velocity measurements (see Appendix \[Sect:RM\]). Scrutinizing the spectral time series, we identified a total of four spectral regions that exhibited excessive spectral variations in all three nights. These spectral anomalies were likely caused by bad detector pixels and were, therefore, discarded in our analysis. Fortunately, none of the affected regions overlapped with the lines (shaded regions in Fig. \[fig:tsp\]). Telluric correction ------------------- Although the line cores of the [ $\uplambda10830$ Å]{} lines are not blended with telluric lines, the spectral region is contaminated with water vapor absorption originating from the Earth’s atmosphere. We used the [molecfit](molecfit) software in version 1.2.0 [@Smette2015; @Kausch2015] to remove the telluric contribution from each individual CARMENES spectrum. A nightly mid-latitude ($45^{\circ}$) reference model atmosphere[^3] and the Global Data Assimilation System (GDAS) profiles for the location of the Calar Alto Observatory were used to create atmospheric temperature-pressure profiles for the three nights, which are required for the line-by-line radiative transfer in [molecfit](molecfit). We included O$_2$, CO$_2$, and CH$_4$ in our transmission model with fixed abundances, taken from the reference atmosphere, and fitted the precipitable water vapor. As reported by @Allart2017, the choice of the optimization ranges is crucial to derive precise transmission models with [molecfit](molecfit). We selected nine 50–100$\AA$ broad wavelength intervals evenly distributed over the CARMENES NIR channel, taking full advantage of the large amount of telluric lines contained in this spectral region. These intervals exhibit few stellar lines, a well determined continuum level, and comprise various deep but unsaturated telluric absorption lines. Stellar features were identified and masked using a high-resolution synthetic stellar spectrum [PHOENIX, @Husser2013] with $T_{\mathrm{eff}}=4700\,$K, $\log g=4.5\,$dex, and solar metallicity. In addition, we excluded wavelength ranges with sky emission features. The instrumental line spread function was determined using hollow cathode lamp spectra. Based on the best-fit parameters derived by [molecfit](molecfit), we finally generated a transmission model for the entire wavelength range of CARMENES and corrected each science spectrum. Telluric emission lines were removed using the sky spectrum from fiber B. We fitted and subtracted the continuum in the sky spectra and subtracted the remaining emission line spectrum from the science spectrum. The comparison of Figs. \[fig:tsp\] and \[fig:tsp\_ntc\] shows that the strongest emission line in the red wing of the triplet at 10830.91 Å was successfully removed to within the noise level. At any rate, this line can only affect the egress phase, where its position partially overlaps with the [ $\uplambda10830$ Å]{} lines. Two further telluric emission lines at 10829.01 Å and 10828.73 Å are located such that they could affect the analysis of the weaker triplet component at 10829 Å. However, these lines are a factor of 11 weaker than the line at 10830.91 Å. As we found no significant differences in our results when correcting or masking them, we consider these lines irrelevant. Continuum normalization and stellar rest frame alignment \[tab:cont\_norm\] --------------------------------------------------------------------------- The continuum was normalized with a third-order polynomial and the spectra were shifted into the stellar rest frame by correcting for a systematic radial velocity of $-2.361$ [kms$^{-1}$]{} [@Bouchy2005], Earth’s barycentric velocity, and the stellar orbital motion. The barycentric velocity correction was computed using the [helcorr]{} routine[^4], and mid-exposure time stamps were converted into Barycentric Julian Dates (BJD) in Barycentric Dynamic Time (TDB) using the [Astropy]{} time package [@Astropy2013]. For each night, the spectral alignment was controlled using the seven strongest and isolated stellar lines in the vicinity of the triplet (see Fig. \[fig:star\_rf\]): $\lambda$10811.084 Å, $\lambda$10818.276 Å, $\lambda$10827.091 Å, $\lambda$10838.970 Å, $\lambda$10843.854 Å, $\lambda$10849.467 Å, and $\lambda$10863.520 Å. We found an average redshift of 360 [ms$^{-1}$]{} with respect to the radial velocity of @Bouchy2005, which was corrected. During night 2, the instrument showed an apparent drift of $\approx$200 [ms$^{-1}$]{}. This drift was modeled for the out-of-transit phases with a second-order polynomial, which was then used to correct the alignment of all spectra during this night. Radial velocity drifts or offsets can occur because our observations were not obtained with a setup optimized for radial velocity measurements, i.e., the instrumental radial velocity drift was not monitored through the Fabry Perot in the calibration fiber. In the end, we found all seven stellar lines within $\approx$200 [ms$^{-1}$]{} of their nominal central wavelengths and interpolated all spectra onto a common wavelength grid. \ \ \ Residual spectra {#SectAnalysis} ---------------- For each night, we combined the out-of-transit spectra to construct a master reference spectrum, discarding the first and last few spectra during each night for technical reasons (see Table \[tab:obs\_details\]). All spectra were then divided by this master out-of-transit spectrum to obtain a time series of residual spectra (see Fig. \[fig:tsp\]). In all three observed transits, a pronounced absorption feature attributable to the [ $\uplambda10830$ Å]{} lines is detected. These results are independent of our telluric emission correction (see Fig. \[fig:tsp\_ntc\]). The highest S/N was reached on night 3, during which the radial velocity of the absorption signals clearly shifts along with the orbital radial velocity of the planet in the mid-transit phase. This indicates that the absorption is indeed associated with the hot Jupiter. The association is less clear on the first two nights. This is could be caused by the reduced instrumental stability and S/N (see Appendix \[Sect:RM\]), but could also be related to stellar pseudo-signals interfering with the planetary [ $\uplambda10830$ Å]{} absorption signals. This possibility is further investigated in Sect. \[SectSpots\]. The Rossiter–McLaughlin effect (RME) in the [ $\uplambda10830$ Å]{} lines is also superimposed on the residual spectra, but we calculate its amplitude to be smaller than 0.07%, which is negligible during the mid-transit phase (see Sect. \[SectSpots\]). The out-of-transit stellar [ $\uplambda10830$ Å]{} lines do not show detectable variation across the three observing nights or within any of them (see Fig. \[fig:tsp\]). The cores of the infrared triplet lines, which are well-known activity indicators, show an activity trend during night 1 along with small activity fluctuations, but no clear signature of flaring (see Appendix \[Sect:APP\_IRT\]). Figure \[fig:tsp\] exhibits a weaker feature associated with the line at 10827.1 Å that exhibits a radial velocity of only about 100 [ms$^{-1}$]{} in the stellar rest frame. This feature can be nicely described by center-to-limb variations in combination with the RME in the stellar line [@Czesla2015]. There is also a slight activity trend in this line that correlates with the trend seen in the infrared triplet lines. Using the ephemeris of @Baluev2015, we shifted the residual spectra into the planetary rest frame and computed nightly means for the ingress, mid-transit, and egress phases. Observations that start after the first contact and have a mid-exposure time before the second contact were included in the ingress phase. An equivalent procedure was used for the egress phase. In total, we have 10 spectra during the ingress, 34 during mid-transit, and 13 during the egress phases. Because of the different data quality, the three nightly means were combined weighting them by the inverse variance in the surrounding continuum. The obtained mean transmission spectra are displayed in Fig. \[fig:mean\], and the nightly residual spectra and their variation with respect to the mean transmission spectrum are shown in Fig. \[fig:var+lc2\]. Bootstrap analysis of the absorption depth {#Sect:APP_absdepth} ------------------------------------------ For each phase, the mean absorption depth was calculated in a $\pm10$ [kms$^{-1}$]{} window centered on the shifted signal position determined by our model fitting in Sect. \[SectModel\]: $+6.5$ [kms$^{-1}$]{} for ingress, $-3.5$ [kms$^{-1}$]{} for mid-transit, and $-12.6$ [kms$^{-1}$]{} for egress. Errors were determined by a bootstrap method. On average, we randomly drew half of the spectra from each phase, computed the mean absorption spectrum, and determined the average absorption level. This was performed 1000 times and the resulting histograms are shown in Fig. \[fig:APP\_bootstrap\]. We find average absorption levels of $0.24\pm0.12$% for the ingress phase, $0.88\pm0.04$% for the mid-transit phase, and $0.46\pm0.06$% for the egress phase. We also randomly drew half of the out-of-transit residual spectra, averaged in the center of the two strong helium lines at 10830 Å in the stellar rest frame, and find zero absorption in this control sample. The ingress signal is formally only a 2$\sigma$ result and exhibits the largest scatter of the three transit phases. This cannot be caused by residual telluric contamination since the ingress signal does not overlap with any telluric emission or absorption line. In the line core of the strong triplet component, the absorption depth reaches its maximum of $1.04\pm0.09$%, using the standard deviation of the surrounding continuum as error estimate. If this absorption feature is caused by the planet, its atmosphere must exhibit a radial extent of at least 0.2 [R$_{\rm p}$]{}. The absorption signal has a total equivalent width of $12.7\pm0.6$ mÅ and we further derive a ratio of the [ $\uplambda10830$ Å]{} to $\uplambda10829$ Å triplet components of $2.8\pm0.2$ by fitting Gaussians with free relative strengths. Here the errors are propagated from the bootstrap analysis. To investigate nightly variability in the absorption depth, we repeated the bootstrap analysis for the mid-transit signals of the individual nights averaging over the full signal ($-70$ to $+30$ [kms$^{-1}$]{}). We measure absorption levels of $0.41\pm0.04$%, $0.39\pm0.04$%, and $0.35\pm0.04$% for nights 1, 2, and 3 respectively. While the nightly mean transmission spectra suggest some variation at different epochs (see Fig. \[fig:var+lc2\]), they do not exceed the rednoise level. Temporal evolution of the signal {#sec:TempEvol} -------------------------------- We computed light curves of the signal by averaging over a broad velocity range from $-$20 to $+$15 [kms$^{-1}$]{} in the planetary rest frame, corresponding to 10829.57 to 10830.84 Å. This range covers the observed velocity shifts at ingress and egress. The light curves are depicted in Fig. \[fig:lc\]. A light curve centered on the weak triplet component is provided in Fig. \[fig:var+lc2\]. We detect no significant inter-night variations. While some apparent in-transit outliers could be associated with spot crossing events, we consider the evidence immaterial. A mean light curve was constructed by binning the three phased light curves with a temporal resolution of 10 min. The nightly weights were maintained in this procedure. Errors are obtained from the variation in the out-of-transit phase. The resulting time series is nearly symmetric with respect to the transit center. Some pre-transit absorption may be present, but the evidence remains inconclusive; we find no evidence for post-transit absorption. Discussion {#SectDiscussion} ========== Our data show a clear in-transit signal in the triplet lines, consistently present in all three spectral transit time series. However, this signal is not necessarily caused only through absorption in a planetary atmosphere because the transit of the opaque planetary disk across an inhomogeneous stellar surface can produce pseudo-absorption and pseudo-emission signals. As a first step toward a better understanding of the presented signal, we study the two extreme cases: a heterogeneous stellar surface and no planetary absorption, and only planetary absorption. Impact of a spotted stellar surface \[SectSpots\] ------------------------------------------------- Even without atmosphere, the transit of the planetary disk over stellar surface regions with below-average absorption (bright stellar surface patches) produces an apparent absorption signal in the residual spectra (pseudo-absorption). Similarly, pseudo-emission is produced when the planet traverses dark stellar surface regions with strong stellar absorption. The impact of these pseudo-signals can be investigated considering the limiting scenarios. When the planet transits stellar surface patches completely lacking the absorption line, the maximum amount of pseudo-absorption is observed. While the amplitude of this signal only depends on the average stellar [ $\uplambda10830$ Å]{} spectrum and the optical transit depth, it is necessary to specify the stellar spectrum in the eclipsed section of the disk to compute the expected pseudo-emission signal. To that end, we applied a two-component disk model, consisting only of regions that are either entirely free of absorption or show strong absorption. ### Filling factor of dark patches {#Sect:stellar_he} An approximation of the filling factor of dark patches can be determined through the equivalent width (EW) ratio of the [ $\uplambda10830$ Å]{} lines and the optical [ $\uplambda5876$ Å]{} line [@Andretta2017]. To measure these lines, we combined all out-of-transit spectra from all nights to obtain a master spectrum. The helium [ $\uplambda10830$ Å]{} lines are located in the wing of the line at 10827.091 Å (see Fig \[fig:stellar\_ew\]). As noted by @Andretta2017, this line is poorly reproduced with a single Voigt profile, so we fitted the line with two superposed Voigt profiles with the same central wavelength. The [ $\uplambda10830$ Å]{} lines were fitted with Gaussians with fixed relative wavelength but free relative strengths. The line is found within the stellar rest frame velocity to an accuracy of 140 [ms$^{-1}$]{}, but for the [ $\uplambda10830$ Å]{} lines we find a redshift of 0.94 [kms$^{-1}$]{}. The main component has an equivalent width of 323 mÅ and the minor component of 52 mÅ, resulting in an EW ratio of 6.2. For the optical [ $\uplambda5876$ Å]{} line, we also fitted a Gaussian profile and derived an equivalent width of 21.5 mÅ along with a redshift of 1.3 [kms$^{-1}$]{}. In the fit, we neglected some minor line blends identified by @Andretta2017. The resulting radial velocity shift is similar in the optical and infrared helium lines. According to the EWs of the helium lines, [HD189733]{} is located above the theoretical curve for a helium spot filling factor of 100% adopted by [@Andretta2017 see their Fig. 10], which can likely be attributed to insufficient stellar atmosphere models. Among the stellar sample studied by @Andretta2017, $\epsilon$ Eri shows properties comparable to [HD189733]{}. In particular, this active K dwarf shows values of 258 mÅ and 51 mÅ for the EWs of the stellar infrared triplet components and 18.1 mÅ for the optical line along with an X-ray luminosity of $2.1\times10^{28}$ [ergs$^{-1}$]{}. For $\epsilon$ Eri, @Andretta2017 derived a minimum filling factor of $59$% for dark patches. By analogy, we adopt a high helium spot filling factor of $75$% for our pseudo-signal analysis in [HD189733]{}. Such a large filling factor is also consistent with other aspects of our data, viz, the absence of both significant inter-transit variability in the stellar line and detectable spot crossing events in our data (Sects. \[Sect:APP\_absdepth\] and \[sec:TempEvol\]). We reconstructed the average stellar spectrum shown in the top panel of Fig. \[fig:tsp\] by assuming that the complete stellar [ $\uplambda10830$ Å]{} absorption EW is produced by 75% dark patches on the stellar surface. For the remaining 25% of bright regions we assumed negligible absorption. This procedure provides estimates for the spectra of bright and dark patches on the stellar surface. ### Quantifying the pseudo-signal To compute model spectral time series, we adopted a discretized stellar surface, rigidly rotating with a projected rotation velocity, $v\sin i$, of 3.5 [kms$^{-1}$]{}. In the planetary rest frame, the stellar surface is Doppler shifted, most pronouncedly during ingress and egress. The amount of this shift depends on the relative motion of the star and the planet and the motion of the rotating stellar surface elements. We used the two-component stellar surface model to calculate spectral time series with the planet occulting only dark or bright patches. The occulted spectrum was removed from the in-transit spectrum and division by the out-of-transit spectrum provided the model residual spectra, which were averaged for the ingress, mid-transit, and egress phases to obtain estimates for the pseudo-signals. The resulting pseudo-signals are shown in Fig. \[fig:mock\]. The pseudo-absorption signal is on a par in strength with the observed signal at all phases, but several features of the observations are not reproduced: ($i$) the model predicts too little absorption during mid-transit, 10.6 mÅ compared to the observed $12.7\pm0.6$ mÅ; ($ii$) the line ratio of the pseudo absorption reflects that of the host star of 6.2, which is inconsistent with the observed value of $2.8\pm0.2$; ($iii$) the pseudo-absorption line is redshifted by 4.5 [kms$^{-1}$]{} with respect to the observations at mid-transit; ($iv$) the model predicts symmetric red- and blueshifted absorption at ingress and egress, which is not observed. Furthermore, such strong pseudo-absorption signals need a very special geometric configuration as the planet traverses about 10% of the stellar disk during the mid-transit phase and these signals would have to cover only bright stellar surface patches that also cover only 25% of the stellar disk. Moreover, this configuration would have to be very similar for all three transits, which were observed more than one year apart. These shortcomings make it unlikely that the observed transit signals are exclusively pseudo-signals. However, the observed radial velocities of the absorption line during ingress and egress, place the planetary signal close to the stellar rest frame, which hinders a clear distinction between planetary absorption and stellar pseudo-signals. At any rate, an active star can produce strong features in the residual spectra with an amplitude anywhere between the limiting cases presented in Fig. \[fig:mock\]. A special case of the above assumptions that is very likely to interfere with the planetary signal are center-to-limb variations in the stellar [ $\uplambda10830$ Å]{} lines. The Sun shows limb-darkening in the [ $\uplambda10830$ Å]{} lines, i.e., the chromospheric absorption profile is deeper at the solar rim than in the center of the disk [@deJager1966]. This modulation produces phase-dependent pseudo-signals. If the [ $\uplambda10830$ Å]{} lines of [HD189733]{} behave as those of much less active Sun, we would expect pseudo-emission signals during ingress and egress, and a pseudo-absorption signal during the mid-transit phase. Such a stellar contribution to the supposed planetary absorption signal would place the radial velocity of the absorption signal between the planetary and stellar rest frames during the mid-transit phase. This may have contributed to deflections from the planetary rest frame that are suggested in Fig. \[fig:tsp\], but since they only affected the first two nights, where the instrumental stability was suboptimal, we refrain from further interpretations. In this context, we also investigated the impact of the RME in the [ $\uplambda10830$ Å]{} lines. Assuming that the lines are homogeneous on the stellar disk, the RME causes signals with an amplitude of less than 0.07% in the residual spectra (see Fig. \[fig:mock\]). The effect is negligible during the mid-transit phase and it is smaller than the possible impact of center-to-limb variations during the ingress and egress phases. Therefore, we do not include the RME in the following analysis. Transmission spectrum modeling {#SectModel} ------------------------------ We now assume the other extreme, namely a homogeneous stellar disk that causes no pseudo-signals. Neglecting the structure of the planetary atmosphere as well, we modeled the wavelength-dependent transmission, $T(\lambda)$, with a single absorption component: $$\begin{aligned} T(\lambda) = (1-f)+f\exp{[-N^*_{\ion{He}{i}}\sigma(\lambda)]} \, . \label{eq:transmission}\end{aligned}$$ Here, $f$ denotes the fraction of the stellar disk covered by the cloud, $N^*_{\ion{He}{i}}$ the column density of excited helium, and $\sigma(\lambda)$ the wavelength-dependent absorption cross section, which we parameterized by three Gaussians with their central wavelengths and oscillator strengths fixed to the known values from atomic physics [see National Institute of Standards and Technology, NIST; @Drake2006]. The free parameters in our model are therefore the covering fraction, the  column density in the triplet state, and a common velocity shift and line width. Since the covering fraction and the column density are highly correlated, we adopted two opposing extreme values of $f=1.1$% and 20% in our analysis. Assuming that the material is distributed in an annulus surrounding the opaque planet body, the covering fractions correspond to an atmosphere extending to 1.2 and 3.0 [R$_{\rm p}$]{}[^5] The former represents the minimum atmospheric extent that can produce the observed 1% absorption signal and the latter is the effective Roche lobe radius of [HD189733b]{} [Eq. 2 of @Eggleton1983]. In the following we refer to them as the compact and the extended assumption. For the ingress and egress phases, we derive the average covering fraction for an atmospheric ring at the exact observing times. To that end, we use analytic light curve models [@Mandel2002] and subtract the solid body light curve from that with an opaque atmosphere with the given expansion. Adopting uniform priors for all parameters, we explored the posterior probability distributions with the Markov chain Monte Carlo technique[^6] (MCMC). The chains were run over $10^5$ steps with a burn-in of $10^4$ steps. Our results for the ingress, mid-transit, and egress phase are summarized in Table \[TabFit\]. There we also provide $\chi^2$ values and p-values for the null hypothesis that the maximum likelihood solution is true. Our maximum likelihood models are shown in Fig. \[fig:mean\]. From our spectral modeling, we find average radial velocities of $6.5\pm3.1$ [kms$^{-1}$]{} for ingress, $-3.5\pm0.4$ [kms$^{-1}$]{} for mid-transit, and $-12.6\pm1.0$ [kms$^{-1}$]{} for egress, independent of the adopted atmospheric extent. We stress once more that these velocities measurements are potentially affected by stellar pseudo-signals (Sect. \[SectSpots\]). At mid-transit, the ratio of the [ $\uplambda10830$ Å]{} to $\uplambda10829$ Å components deviates from the optically thin ratio of 8 [NIST, @Drake2006]. This can be explained by sufficiently large column densities, because saturation in the stronger component of the triplet increases the relative depth of the weaker component when the optically thin approximation breaks down. The observed ratio of $2.8\pm0.2$ corresponds to an optical depth of about 3.2 in the main component [@deJager1966]. While the signal during the ingress and egress phases is equivalently reproduced using either assumptions (see Table \[TabFit\]), the compact case provides a superior approximation for the mid-transit phase. In particular, the extended assumption yields a larger than observed line ratio of 7.9 and a total equivalent width of 11.4 mÅ, whereas the compact assumption results in an equivalent width of 12.0 mÅ and a line ratio of 4.6, which better reproduces the observed [ $\uplambda10829$ Å]{} absorption. Formally, this is reflected by a decrease from 1.34 to 1.05 in the reduced $\chi^2$ statistics (see Table \[TabFit\]) and a p-value of 0.3 for the compact assumption, which provides no evidence against the null. Finally, we note that the line ratio is also not fully recovered under the compact assumption because the main component becomes too broad before the depth of the minor component is reproduced. Nevertheless, the mid-transit line ratio strongly favors a small covering fraction, which corresponds to a compact atmosphere. While we do not fit a proper atmosphere model here, the derived triplet state column density is to be understood as an effective value, which can be compared to theoretical models. From the evaporation model of @Oklopcic2018 for [HD209458b]{}, we derived a weighted mean absorption height of 1.6 [R$_{\rm p}$]{} with a column of about $7.9\times10^{11}$ cm$^{-2}$. We used $N^*_{\ion{He}{i}} R_{\rm p}$ as weights, which accounts for the increasing geometric weight of higher atmospheric layers. Although the model is for the [HD209458]{} system, the column density lies between the values derived for our compact and extended cases considered above, which shows that absorption in a planetary atmosphere is a viable origin of the observed signals. It is not unlikely that evaporation models like those of @Oklopcic2018 can reproduce the mid-transit signal. Our finding that a compact atmosphere better reproduces the data than a compact atmosphere does not exclude that the atmosphere of [HD189733b]{} is evaporating, but could simply mean that the helium triplet state is not significantly populated in atmospheric layers above 0.2 [R$_{\rm p}$]{}. In the end, neither of our fits is fully satisfactory, particularly in the region of the weaker triplet component at $\uplambda10829$ Å. We attribute this to shortcomings of the adopted model. A more comprehensive model should consider both pseudo-signals and dedicated atmospheric transmission models. [l c c c c c c c]{}\ Phase & $f$ & $N^*_{\ion{He}{i}}$ & $b$ & Rad. vel. & Line ratio & $\chi^2_{\rm red}$ & p-value\ & (%) & ($10^{11}$ cm$^{-2}$) & ([kms$^{-1}$]{}) & ([kms$^{-1}$]{}) & & (174 DOF) &\ \ \ \ \ T$_1$ - T$_2$ & 9.5 & $0.62\pm 0.11$ & $15.0\pm 5.0$ & $6.4\pm 3.2$ & & 1.15 & 0.09\ T$_2$ - T$_3$ & 20& $1.04\pm 0.03$ & $15.1\pm 0.6$ & $-3.7\pm 0.4$ & 7.9 & 1.34 & 0.002\ T$_3$ - T$_4$ & 9.5 & $1.07\pm 0.07$ & $13.7\pm 1.0$ & $-12.2\pm 0.9$ & & 1.17 & 0.06\ \ \ \ T$_1$ - T$_2$ & 0.58 & $12.9\pm 2.2$ & $13.0\pm 4.9$ & $6.6\pm 3.0$ & & 1.14 & 0.10\ T$_2$ - T$_3$ & 1.1 & $35.7\pm 2.0$ & $11.3\pm 0.5$ & $-3.3\pm 0.4$ & 4.6 & 1.05 & 0.30\ T$_3$ - T$_4$ & 0.58 & $29.8\pm 3.7$ & $11.7\pm 0.9$ & $-13.0\pm 0.9$ & & 1.17 & 0.07\ \ Velocity structure and ingress-egress asymmetry {#SectVelo} ----------------------------------------------- The observed absorption is about twice as strong at egress as at the ingress phase, and it exhibits velocity shifts from the planetary rest frame at all phases. If [HD189733b]{} was tidally locked and its atmosphere rotated as a solid body, we would expect symmetric radial velocities of the signals at ingress and egress of $\pm$3.5 [kms$^{-1}$]{} if the absorption arises in atmospheric layers at a height of 0.2 [R$_{\rm p}$]{}. However, the observed shifts are asymmetric and significantly exceed this value at egress. We see two plausible hypotheses explaining the observed features, viz, atmospheric circulation in a dense helium atmosphere or an additional upper, low-density, and asymmetrically expanding atmosphere. ### Atmospheric circulation Hydrodynamic models of the irradiated atmospheres of synchronized hot Jupiters predict rather complex circulation patterns. For the specific case of [HD189733b]{}, the atmospheric circulation model by @Showman2013, which covers a pressure range in the atmosphere from 2 to $200\times10^{-6}$ bar, predicts the presence of a superrotating equatorial jet, where the bulk velocity increases with height in the atmosphere. Additionally, the model shows a general day-to-night side flow across the poles in high altitude layers, which is the main cause for the predicted net blueshift of around $-3$ [kms$^{-1}$]{} of molecular absorption signals in transmission spectra [see Fig. 12 of @Showman2013]. Molecular absorption of CO and H$_2$O indicate such a net blueshift [$-1.7\pm1.2$ [kms$^{-1}$]{} and $-1.6^{+3.2}_{-2.7}$ [kms$^{-1}$]{}, respectively; @Brogi2016; @Brogi2018]. These signals are sensitive to pressure levels between 0.1 and $10^{-6}$ bar. In contrast, ground-based high-resolution transmission spectroscopy in the sodium lines is sensitive to the lower pressure levels ($<$$10^{-6}$ bar) that are reached in the lower planetary thermosphere [@Pino2018]. Despite their origin in higher atmospheric layers, the sodium absorption signals of [HD189733b]{} indicate a net blueshift of $-1.9\pm0.7$ [kms$^{-1}$]{} during mid-transit [@Louden2015]. The helium absorption presented here likely probes even lower pressure levels in the planetary atmosphere. Particularly, the peak column density occurs at a pressure level of around $10^{-9}$ bar in the evaporation models of @Oklopcic2018. Nevertheless, our mid-transit absorption signal also exhibits a significant blueshift of $-3.5\pm0.4$ [kms$^{-1}$]{}, which is consistent with the previous results for the lower atmosphere. At ingress and egress, the sodium signal indicates radial velocities of $+2.3^{+1.3}_{-1.5}$ [kms$^{-1}$]{} and $-5.3^{+1.0}_{-1.4}$ [kms$^{-1}$]{}, which are believed to be caused by an equatorial superrotating jet. At ingress the absorption signal is dominated by the leading limb where superrotating material moves away from the observer, which would explain the observed redshift. At egress the situation is reversed. The amplitude of the observed bulk velocities in the sodium signal are about a factor of 2.5 smaller than our results, but they do exhibit the same red- to blueshifted asymmetry [@Louden2015]. While our ingress and egress velocities are also larger than those predicted by the circulation models of @Showman2013, the observed pressure levels are clearly beyond the modeled atmospheric range. If the advection timescale is comparable to the de-excitation timescale of the triplet state, the equatorial jet transports ground level helium atoms from the night side to the leading atmospheric limb and excited helium atoms from the dayside to the trailing limb. This naturally causes stronger [ $\uplambda10830$ Å]{} absorption at egress compared to the ingress phase. If we approximate the advection timescale by dividing the observed average ingress/egress radial velocities by the planetary radius, we derive a value of $10^{-4}$ s$^{-1}$. This is on the same order as the radiative transition rate to the ground level $A_{31} = 1.272\times10^{-4}$ s$^{-1}$ [@Drake2006]. Depending on the local conditions other processes could be faster in depopulating the metastable state, but it seems reasonable that the superrotating jet can also cause strong egress absorption through advection of excited helium atoms from the dayside. Overall, the observations are consistent with previous observations and with the models of @Showman2013 if the equatorial superrotating jet continues to exhibit increasing bulk velocities in higher atmospheric layers. ### Asymmetric expanding atmosphere The observed ingress signal is significant only at the 2$\sigma$ level. If we attribute the redshift of the ingress signal to rednoise or another source unrelated to the planetary atmosphere, we are left with slightly blueshifted absorption at mid-transit and a larger blueshift at egress. The mid-transit signal is consistent with being caused by dense material in a compact atmosphere as shown in Sect. \[SectModel\], but the density of the material that dominates the egress signal is not confined by our data. The blueshifted radial velocities could be explained by material that evaporates from the planet and is subsequently being pushed backward, perhaps as a result of the stellar wind pressure creating an asymmetrically expanding atmosphere. The mechanism would be similar to the Type I interaction studied by @Matsakos2015. In this case, the compact atmosphere that is observed during mid-transit causes only a small contribution to the observed egress signal similar to that during ingress. If the asymmetrically expanding atmosphere trails the planet, it would still cover the stellar disk at egress and dominate the observed absorption at this phase. The observed egress radial velocity would then be a measure of the bulk radial velocity of the evaporating material streaming away from the planet with velocity components pointing out of the system and in the reverse direction of orbit motion. At mid-transit the trailing material would be superposed onto that of the compact atmosphere causing the observed blueshift. The nondetection of post-transit absorption constrains the (projected) extent of the hypothesized distribution of trailing material observed by means of in the triplet state. The lack of post-transit absorption could be explained by a tail structure that is nearly aligned with the star-planet axis. In fact, the 3D model of @Spake2018 for WASP-107b demonstrates that radiation pressure can create such a tail. However, we do not detect the strong blueshifts of the tail material predicted by the authors. A better explanation comes from the the spherical evaporation models of @Oklopcic2018. The average absorption height for [HD209458b]{} was 1.6 [R$_{\rm p}$]{} (Sect. \[SectModel\]), and the triplet state density quickly decreased at higher atmospheric levels. If this characteristic height also applies to an asymmetric extended atmosphere, it is consistent with our nondetection of post-transit absorption because excited helium atoms are not expected at large distances from the planet. We therefore find the observations consistent with signals from a superposition of a dense, symmetric helium atmosphere and an asymmetrically expanding component that streams away from the planet and slightly trails it. Comparison to [Ly$\alpha$]{} absorption {#SectLya} --------------------------------------- [Ly$\alpha$]{} observations have revealed variable absorption during the transit of [HD189733b]{} at high blueshifts between $-$230 and $-$140 [kms$^{-1}$]{} [@Lecavelier2012; @Bourrier2013]. Applying the same bootstrap method as in Sect. \[Sect:APP\_absdepth\], we determine a mean absorption level of $-0.017\pm0.018$% in this range, which is consistent with no absorption. In the optically thin limit, the absorption EW is proportional to the cross section , $\sigma$, times the column density $${\rm EW} = N\sigma = \frac{N \lambda^2 \pi e^2 f}{m_{\rm e} c} \sim \lambda^2 f,$$ where $\lambda$ is the central wavelength and $f$ the oscillator strength. The two blended lines of the [ $\uplambda10830$ Å]{} triplet are by a factor of 90 more strongly absorbed than the hydrogen [Ly$\alpha$]{} line [NIST, @Drake2006]. The relative abundance of neutral hydrogen to that of helium in the metastable triplet state is about $10^5$ in the simulations of @Oklopcic2018. Thus, the [Ly$\alpha$]{} line traces different atmospheric layers that can be a factor $10^3$ more rarefied. We note that @Bourrier2013B interpreted the [Ly$\alpha$]{} absorption signal of [HD189733b]{} in terms of the presence of energetic neutrals created through charge exchange of stellar wind protons with neutrals in the planetary atmosphere. Energetic neutrals comprise a different population and their density depends on parameters like the stellar wind density and velocity. Currently, we cannot assess whether charge exchange can also create substantial amounts of helium in the excited triplet state. The variability of the [Ly$\alpha$]{} signal, with absorption detected in only about half of the observations [@Bourrier2013], is in contrast with the stability of the helium absorption signal. This fosters the picture that the two signals arise in populations that are decoupled to a larger degree, i.e., that the [Ly$\alpha$]{} absorption at large blueshifts originates from the interaction of the stellar wind with the planetary atmosphere, and that the helium absorption occurs in the thermosphere closer to the planetary body. Conclusions {#SectConclusions} =========== We present the detection of spectrally resolved absorption signals in the near-infrared during three individual transit observations of [HD189733b]{} with CARMENES. The mid-transit signal in the [ $\uplambda10830$ Å]{} main component has a depth of $0.88\pm0.04$%. It exhibits a net blueshift of $-3.5\pm0.4$ [kms$^{-1}$]{}, and shows no detectable variation in strength between the three transits. The ingress and egress signals show red- and blueshifts of $+6.5\pm3.1$ [kms$^{-1}$]{} and $-12.6\pm1.0$ [kms$^{-1}$]{}, respectively. Our analysis reveals that pseudo-signals induced by the stellar surface structure in the [ $\uplambda10830$ Å]{} lines might interfere with the atmospheric signal of the planet, but do not reproduce all features of the data. We consider it unlikely that pseudo-signals can exclusively explain the transit signal. In the worst-case, we estimate that they could account for up to 80% of the detected signal strength. Additionally, pseudo-signals might also affect the measured radial velocities at all phases. When interpreted in terms of planetary atmospheric absorption, a compact atmosphere is favored with an extent of 0.2 planetary radii, which is easily contained within the planetary Roche lobe. This is consistent with the lack of both clear pre- or post-transit absorption and at radial velocities exceeding the planetary escape velocity. We discuss two hypotheses to explain the observed radial velocity signature, namely, atmospheric circulation in the upper planetary atmosphere and an asymmetric extended atmosphere of evaporating material. Atmospheric circulation with equatorial superrotation has been indicated in observations of lower atmospheric layers, which it is also predicted by models, and here we propose that it might persist throughout the higher layers responsible for the absorption. The superrotation hypothesis hinges on the signals and radial velocity shifts during the ingress and egress phases. While we consider the latter significant, the result for the ingress phase is more uncertain. If we attribute the observed redshift during ingress to an unrelated source, such as red-noise interference or unaccounted for stellar effects, the case for atmospheric superrotation wanes. In this case, the hypothesis of an asymmetrically evaporating atmosphere that accounts for the blueshifted egress signal becomes more attractive. Indeed, models of planetary evaporation predict such structures and [Ly$\alpha$]{} observations have demonstrated the existence of material at large blueshifts exceeding $-$140 [kms$^{-1}$]{}. However, the lower radial velocity of the helium signal shows that we observe a different region of the planetary atmosphere. Our analysis shows that transit spectroscopy of the line is a highly promising tool for the study of planetary atmospheric physics. Although, the atmosphere of [HD189733b]{} is almost certainly escaping from the planet, as evidenced by the [Ly$\alpha$]{} observations, it remains uncertain how well the mass-loss rate can be determined from [ $\uplambda10830$ Å]{} absorption and to what degree the atmospheric absorption can be distinguished from stellar pseudo-signals. Detailed modeling is needed to investigate the physical plausibility of the two sketched interpretations, and only new observations will allow us to distinguish between them by a confirmation or rejection of the ingress signal. CARMENES is an instrument for the Centro Astronómico Hispano-Alemán de Calar Alto (CAHA, Almería, Spain). CARMENES is funded by the German Max-Planck-Gesellschaft (MPG), the Spanish Consejo Superior de Investigaciones Científicas (CSIC), the European Union through FEDER/ERF FICTS-2011-02 funds, and the members of the CARMENES Consortium (Max-Planck-Institut für Astronomie, Instituto de Astrofísica de Andalucía, Landessternwarte Königstuhl, Institut de Ciències de l’Espai, Insitut für Astrophysik Göttingen, Universidad Complutense de Madrid, Thüringer Landessternwarte Tautenburg, Instituto de Astrofísica de Canarias, Hamburger Sternwarte, Centro de Astrobiología and Centro Astronómico Hispano-Alemán), with additional contributions by the Spanish Ministerio de Ciencia, Innovación y Universidades through projects ESP2013-48391-C4-1-R, ESP2014-54062-R, ESP2014-54362-P, ESP2014-57495-C2-2-R, AYA2015-69350-C3-2-P, AYA2016-79425-C3-1/2/3-P, ESP2016 76076-R, ESP2016-80435-C2-1-R, ESP2017-87143-R, and AYA2018-84089; the German Science Foundation through the Major Research Instrumentation Programme and DFG Research Unit FOR2544 “Blue Planets around Red Stars”; the Klaus Tschira Stiftung; the states of Baden-Württemberg and Niedersachsen; and by the Junta de Andalucía.\ We also acknowledge support from the Deutsche Forschungsgemeinschaft through projects SCHM 1032/57-1 and SCH 1382/2-1, the Deutsches Zentrum für Luft- und Raumfahrt through projects 50OR1706 and 50OR1710, the European Research Council through project No 694513, the Fondo Europeo de Desarrollo Regional, and the Generalitat de Catalunya/CERCA programme. This work has made use of data from the European Space Agency (ESA) mission [*Gaia*]{} (<https://www.cosmos.esa.int/gaia>), processed by the [*Gaia*]{} Data Processing and Analysis Consortium (DPAC, <https://www.cosmos.esa.int/web/gaia/dpac/consortium>). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the [*Gaia*]{} Multilateral Agreement. Finally, we thank the referee for constructive comments that helped to improve this publication. Complementary figures {#Sect:APP} ===================== \ \ Figure \[fig:star\_rf\] shows the stellar lines used to correct the alignment of all spectra in the stellar rest frame. Figure \[fig:tsp\_ntc\] shows the residual spectra as depicted in Fig. \[fig:tsp\], but without the removal of telluric emission lines. In the top panel of Fig. \[fig:var+lc2\], we display the average residual spectra during the mid-transit phase of the three individual transit observations. Finally, the bottom panel of Fig. \[fig:var+lc2\] depicts light curves centered on the minor component. \ Calcium IRT lines {#Sect:APP_IRT} ================= To monitor the activity evolution of the host star, we checked the time evolution of the three IRT lines. We averaged the residual flux in a $\pm$25 [kms$^{-1}$]{} range in the line cores and then averaged the three individual lines. The resulting light curves are shown in Fig. \[fig:cairt\] for nights 1 and 2. The visual channel failed on night 3. Night 2 shows a slight activity trend that was fit linearly and subtracted before computing the mean IRT transit light curve. The light curve has the usual shape expected from center-to-limb variation in the stellar lines [@Czesla2015]. We find no indications of flaring activity, and in particular, we do not find evidence that the planet crossed unusually strong active regions during the two transit nights, which would cause larger excursions from the average light curve. This is consistent with our analysis. Rossiter–McLaughlin effect {#Sect:RM} ========================== We derived relative radial velocity measurements for the NIR channel of CARMENES using SERVAL [@Zechmeister2018]. As described in Sect. \[tab:cont\_norm\], the instrument can exhibit nightly drifts with the configuration used. Therefore, we corrected linear trends from the radial velocity measurements. The resulting RVs are shown in Fig. \[fig:shifts\]. The first two nights exhibit a large scatter. These observations were taken shortly after the instrument commissioning and the NIR spectrograph appears not to be fully stabilized yet [@Quirrenbach2018]. We used only night 3 to fit the Rossiter–McLaughlin effect (RME) following the prescription of @Ohta2005. The posterior distributions were explored with an MCMC chain of $10^5$ steps with a burn-in of $10^4$ steps. The reference mid-transit time, period, and semimajor axis were set-up with Gaussian priors according to the values in Table. \[TabPara\]. These parameters were not further confined by our data. The linear limb-darkening parameter was fixed to 0.43, which is the value calculated for the $J$ band and a [HD189733]{}-like star by @Claret2011. The inclination of the stellar rotation axis was fixed to 92 [@Cegla2016]. The remaining free parameters were the radius ratio, the rotation velocity of the host $\Omega$, the inclination of the orbit $i$, and the obliquity $\lambda$. For the obliquity and the inclination, we find $\lambda=-1.6 \pm 1.1$ and $i=85.77 \pm 0.11$, consistent with literature values [e.g., @Triaud2009; @Cegla2016; @CasasayasBarris2017]. With a value of $0.162\pm0.024$, the radius ratio is only weakly confined by our data and consistent with all the literature values. For the angular rotation we find $\Omega = 0.40 \pm 0.11$ rad/day, which converts to a rotation period of $15.8\pm4.6$ d. This is somewhat larger than but still consistent with the well-determined value of $11.953\pm0.009$ [@Henry2008]. The slight discrepancy of the stellar rotation rate could be caused by differential rotation as noted by @Cegla2016. Also, a wavelength dependence of the radius ratio [@DiGloria2015] would affect the measured stellar rotation rate. We further note that our RVs were determined with a cross-correlation technique, while the model assumed a line centroid method. A detailed analysis of the chromatic RME including the visual channel of CARMENES is beyond the scope of this paper. Neither the stellar rotation period nor the radius ratio have any impact on our main conclusions here. The RME confirms the ephemeris used. The large scatter of the RVs during the first two observation nights shows that the instrument was less stable, and supports our analysis procedure to put the greatest weight on the observations during night 3. [^1]: We use air wavelengths throughout the manuscript. [^2]: Reduced spectra are available at the Calar Alto Archive (<http://caha.sdc.cab.inta-csic.es/calto/>). [^3]: <http://eodg.atm.ox.ac.uk/RFM/atm/> [^4]: Implemented in PyAstronomy <https://github.com/sczesla/PyAstronomy> and adapted from the REDUCE package [@Piskunov2002]. [^5]: The atmosphere starts at 1 [R$_{\rm p}$]{} and has a radial extent of 0.2 or 2.0 [R$_{\rm p}$]{}. [^6]: See the PyAstronomy wrapper for PyMC (<https://github.com/pymc-devs/pymc>)

--- abstract: 'Using seven-dimensional Sasaki-Einstein spaces we construct solutions of $D=11$ supergravity that are holographically dual to superconductors in three spacetime dimensions. Our numerical results indicate a new zero temperature solution dual to a quantum critical point.' author: - 'Jerome P. Gauntlett, Julian Sonner and Toby Wiseman' title: 'Holographic superconductivity in M-Theory' --- Introduction ============ The AdS/CFT correspondence provides a powerful framework for studying strongly coupled quantum field theories using gravitational techniques. It is an exciting possibility that these techniques can be used to study classes of superconductors which are not well described by more standard approaches [@Gubser:2008px][@H31][@H32]. The basic setup requires that the CFT has a global abelian symmetry corresponding to a massless gauge field propagating in the $AdS$ space. We also require an operator in the CFT that corresponds to a scalar field that is charged with respect to this gauge field. Adding a black hole to the $AdS$ space describes the CFT at finite temperature. One then looks for cases where there are high temperature black hole solutions with no charged scalar hair but below some critical temperature black hole solutions with charged scalar hair appear and moreover dominate the free energy. Since we are interested in describing superconductors in flat spacetime we consider black holes with planar symmetry. In order to obtain a critical temperature, conformal invariance then implies that another scale needs to be introduced. This is achieved by considering electrically charged black holes which corresponds to studying the dual CFT at finite chemical potential. Precisely this set up has been studied using a phenomenological theory of gravity in $D=4$ coupled to a single charged scalar field and it has been shown that, for certain parameters, the system manifests superconductivity in three spacetime dimensions, in the above sense [@H32]. It is important to go beyond such models and construct solutions in the context of string/M-theory so that there is a consistent underlying quantum theory and CFT dual. Also, as we shall see, the behaviour of the string/M-theory solutions will differ substantially from that of the phenomenological model [@H32] at low temperature. It was shown in [@dh] that the $D=4$ phenomenological models of [@H32] arise, at the linearised level, after Kaluza-Klein (KK) reduction of $D=11$ supergravity on a seven-dimensional Sasaki-Einstein space $SE_7$. Here we go beyond this linearised approximation by working with a consistent truncation of the $D=4$ KK reduced theory presented in [@Gauntlett:2009zw]. The truncation is consistent in the sense that any solution of this $D=4$ theory, combined with a given $SE_7$ metric, gives rise to an exact solution of $D=11$ supergravity. Here we shall use this $D=4$ theory to construct exact solutions of $D=11$ supergravity that correspond to holographic superconductivity. The KK truncation ================= We begin by recalling that any $SE_7$ metric can, locally, be written as a fibration over a six-dimensional Kähler-Einstein space, $KE_6$: ds\^2(SE\_7)ds\^2(KE\_6)+Here $\eta$ is the one-form dual to the Reeb Killing vector satisfying $d\eta=2J$ where $J$ is the Kähler form of $KE_6$. We denote the $(3,0)$ form defined on $KE_6$ by $\Omega$. For a regular or quasi-regular $SE_7$ manifold, the orbits of the Reeb vector all close, corresponding to compact $U(1)$ isometry, and the $KE_6$ is a globally defined manifold or orbifold, respectively. For an irregular $SE_7$ manifold, the Reeb-vector generates a non-compact ${\mathbb{R}}$ isometry and the $KE_6$ is only locally defined. In the KK ansatz of [@Gauntlett:2009zw] the $D=11$ metric is written \[KKmetT\] ds\^2=e\^[-6U-V]{}ds\^2\_4+e\^[2U]{}ds\^2(KE\_6)+e\^[2V]{}(+A\_1)(+A\_1) while the four-form is written \[KKG4T\] G\_4 &= 6 e\^[-18U-3V]{}( +h\^2+||\^2)\_4 + H\_3 (+A\_1)[\ ]{} &+ H\_2 J + dh J (+A\_1) + 2h J J[\ ]{} &+ [3]{} where $ds^2_4$ is a four-dimensional metric (in Einstein frame), $U,V,h$ are real scalars, and $\chi$ is a complex scalar defined on the four-dimensional space. Furthermore, also defined on this four-dimensional space are $A_1$ a one-form potential, with field strength $F_2\equiv dA_1$, two-form and three-form field strengths $H_2$ and $H_3$, related to one-form and two-form potentials via $H_3=dB_2$ and $H_2= dB_1+2B_2+hF_2$. Finally $D\chi\equiv d\chi-4iA_1\chi$. This is a consistent KK truncation of $D=11$ supergravity in the sense that if the equations of motion for the 4d-fields $ds^2_4,U,V,A_1,H_2,H_3,h,\chi$ as given in [@Gauntlett:2009zw] are satisfied then so are the $D=11$ equations. The $D=4$ equations of motion admit a vacuum solution with vanishing matter fields which uplifts to the $D=11$ solution: \[d11vac\] ds\^2=ds\^2(AdS\_4)+ds\^2(SE\_7),G\_4=Vol(AdS\_4) where $ds^2(AdS)_4$ is the standard unit radius metric. When $\epsilon=+1$, this $AdS_4\times SE_7$ solution is supersymmetric and describes $M2$-branes sitting at the apex of the Calabi-Yau four-fold ($CY_4$) cone whose base space is given by the $SE_7$. When $\epsilon=-1$ the solution is a “skew-whiffed” $AdS_4\times SE_7$ solution, which describes anti-$M2$-branes sitting at the apex of the $CY_4$ cone. These solutions break all of the supersymmetry except for the special case when the $SE_7$ is the round seven-sphere, $S^7$, in which case it is maximally supersymmetric. Note that the skew-whiffed solutions with $SE_7\ne S^7$ are perturbatively stable [@Duff:1984sv], despite the absence of supersymmetry. Thus such backgrounds should be dual to three-dimensional CFTs at least in the strict $N=\infty$ limit. We are most interested in the skew-whiffed case because it is for that case that the operator dual to $\chi$ has scaling dimensions $\Delta=1$ or $2$ [@Gauntlett:2009zw] and, based on the work of [@dh], is when we expect holographic superconductivity. The $D=4$ equations of motion can be derived from a four-dimensional action given in [@Gauntlett:2009zw]. It is convenient to work with an action that is obtained after dualising the one-form $B_1$ to another one form $\tilde B_1$ and the two-form $B_2$ to a scalar $a$ as explained in section 2.3 of [@Gauntlett:2009zw]. The dual fields are related to the original fields via H\_3&=&-e\^[-12U]{}\*[\ ]{}H\_2&=&(4h\^2+e\^[4U+2V]{})\^[-1]{}(H\_2+h\^2 F\_2) where $\tilde H_2\equiv d\tilde B_1$. We now restrict to the (skew-whiffed) case $\epsilon=-1$. For this case we can make the following additional truncation of the $D=4$ theory: \[truncansatz\] a&=&h=0,V=-2U,[\ ]{}A\_1 &=& -B\_1,e\^[6U]{}=1-||\^2 One can show that provided that we restrict to configurations satisfying ${F_2}\wedge {F_2}=0$ we obtain equations of motion that can be derived from the action \[lageinfull\] S = d\^4 xwhere $D\hat\chi\equiv d\hat\chi-2i\hat A_1 \hat\chi$, and we have defined $\hat{A_1}\equiv 2A_1$, $\hat{\chi}\equiv (3/2)^{1/2}\chi$. Linearizing in the complex scalar $\hat\chi$, this gives the action considered in [@H32] (with their $L = 1/2$ and their $q=2$). This non-linear action is in the class considered in [@Franco:2009yz] and in addition to the $AdS_4$ vacuum with $\hat A_1=0$ and $\hat \chi=0$, which uplifts to [(\[d11vac\])]{}, it also admits $AdS_4$ vacuua with $\hat A_1=0$ and constant $|\hat\chi|=1$, which uplift to the $D=11$ solutions[^1] found in [@pw]. Black Hole Solutions ==================== The key result of the last section is that any solution to the $D=4$ equations of motion of the action [(\[lageinfull\])]{} with $\hat{F}\wedge \hat{F}=0$, gives an exact solution of $D=11$ supergravity for any $SE_7$ metric. To find solutions relevant for studying superconductivity via holography we consider the following ansatz ds\^2&=&-g e\^[-]{}dt\^2+g\^[-1]{}dr\^2+r\^2(dx\^2+dy\^2)[\ ]{}\_1&=&dt,where $g,\beta,{{\hat{\phi}}}$ and $\sigma$ are all functions of $r$ only. Being purely electrically charged this satisfies the $\hat{F} \wedge \hat{F} = 0$ condition. After substituting into the equations of motion arising from [(\[lageinfull\])]{}, we are led to ordinary differential equations which can also be obtained from the action obtained by substituting the ansatz directly into [(\[lageinfull\])]{}: S&=&cdr r\^2e\^[-/2]{}where $c=(16\pi G)^{-1}\int dt dx dy$. We next observe that the system admits the following exact AdS Reissner-Nordström type solution $\sigma=\beta=0$ \[asoln\] g=4r\^2-(4r\_+\^3+)+,=(-) for some constants $\alpha,r_+$. The horizon is located at $r=r_+$ and for large $r$ it asymptotically approaches 1/4 of a unit radius $AdS_4$ (see [(\[d11vac\])]{}). This solution should describe the high temperature phase of the superconductor. We are interested in finding more general black hole solutions with charged scalar hair, $\sigma\ne 0$. Let us examine the equations at the horizon and at infinity. At the horizon $r=r_+$ we demand that $g(r_+)={{\hat{\phi}}}(r_+)=0$. One then finds that the solution is specified by 4 parameters at the horizon $r_+$, $\beta(r_+)$, ${{\hat{\phi}}}'(r_+)$, $\sigma(r_+)$. At $r=\infty$ we have the asymptotic expansion, $$\begin{aligned} \beta = \beta_a + \ldots, \; \frac{\sigma}{\sqrt{8\pi G}} = \frac{ \sigma_1 }{r} + \frac{ \sigma_2}{r^2} + \ldots , {\nonumber \\}\frac{{{\hat{\phi}}}}{\sqrt{16\pi G}} = e^{-\beta_a/2} ( { {\hat{\mu}} }- \frac{{{\hat{q}} }}{r}) + \ldots \nonumber \end{aligned}$$ $$\begin{aligned} \label{aexpc} e^{-\beta} g = e^{-\beta_a} ( 4 r^2 - \frac{8\pi G(m+\frac{4}{3}\sigma_1\sigma_2)}{r} ) + \ldots \qquad \qquad \qquad \qquad \qquad\end{aligned}$$ determined by the data $\beta_a, \sigma_{1,2}, m, \hat{\mu}, \hat{q}$. The scaling \[sc1\] rar,(t,x,y)a\^[-1]{}(t,x,y), ga\^2g, aleaves the metric, $A_1$, and all equations of motion invariant. Action and thermodynamics ------------------------- We analytically continue by defining $\tau\equiv i t$. The temperature of the black hole is $T = e^{\beta_a/2}/\Delta \tau$ where $\Delta\tau$ is fixed by demanding regularity of the Euclidean metric at $r=r_+$. We find: T= \_[r=r\_+]{} Defining $I\equiv -iS$, we can calculate the on-shell Euclidean action $I_{OS}$ \[actone\] I\_[OS]{} &=&\_[r\_+]{}\^dr ’[\ ]{}&=&\_[r\_+]{}\^dr ’ where $vol_2\equiv \int dx dy$. The latter expression only gets contributions from the on-shell functions at $r=\infty$ since $g(r_+)=0$, while the former expression gets contributions from $r=r_+$ and $r=\infty$. The on-shell action diverges and we need to regulate by adding appropriate counter terms. We define $I_{Tot}\equiv I+I_{ct}$ and, for simplicity, we will focus on the following counter-term action $I_{ct}$: I\_[ct]{}=dd\^2x where $K=g^{{ {\hat{\mu}} }\nu}_\infty\nabla_{ {\hat{\mu}} }n_\nu$ is the trace of the extrinsic curvature. For our class of solutions we find $$\begin{aligned} I_{ct}&=&\frac{\Delta \tau vol_2}{16\pi G} \lim_{r \rightarrow \infty}e^{-\beta/2}\Big[-r^2g' +r^2g\beta'-4gr {\nonumber \\}&& \qquad \qquad \qquad \qquad \qquad +r^2g^{1/2}(8+2\sigma^2)\Big]\end{aligned}$$ Notice that under a variation of the action $I_{Tot}$ with respect to $\beta,g,{{\hat{\phi}}}$ yields the equations of motion together with surface terms. For an on-shell variation the only terms remaining are these surface terms, and after substituting the asympototic boundary expansion [(\[aexpc\])]{} (higher order terms are also required) we find $$\begin{aligned} \label{osvary} [ \delta I_{Tot} ]_{OS} & = & \frac{\Delta \tau vol_2}{16 \pi} e^{-\beta_a/2} \Big[ ( - \tfrac{1}{2}m + \tfrac{1}{2} { {\hat{\mu}} }\, {{\hat{q}} }) \delta \beta_a {\nonumber \\}&& \qquad \qquad - {{\hat{q}} }\, \delta { {\hat{\mu}} }- 4\sigma_2 \delta \sigma_1 \Big] $$ Note that we are keeping $\Delta\tau$ fixed in this variation. Hence we see that $I_{Tot}$ is stationary for fixed temperature and chemical potential (ie. $\delta \beta_a = \delta { {\hat{\mu}} }= 0$) and for either $\sigma_2 = 0$ or fixed $\sigma_1$. We also find that the on-shell total action is given by $$\begin{aligned} \label{osact} [ I_{Tot} ]_{OS} & = & \frac{vol_2}{T} \Big[ m - \, { {\hat{\mu}} }\, {{\hat{q}} }- Ts \Big] {\nonumber \\}& = & \frac{vol_2}{T} \Big[ - \frac{1}{2} m - 2\sigma_1 \sigma_2 \Big]\end{aligned}$$ where $s = \frac{r_+^2}{4 G}$ is the entropy density of the solution and $m$ is the energy density. The two forms of the on-shell action come from the two ways of writing the action as a total derivative given above. We note that the equality of these expressions imply a Smarr-like relation. Also note that after using $\delta \beta_a = 2 \, \delta T / T$ (since $\Delta\tau$ is held fixed) the equality of and the variation of the first line of imply a first law, $$\begin{aligned} \delta m = T \delta s + \, { {\hat{\mu}} }\, \delta {{\hat{q}} }- 4\sigma_2 \delta \sigma_1 \, .\end{aligned}$$ Both this Smarr relation and the first law were used to confirm the accuracy of our numerical solutions below. For simplicity we restrict discussion to solutions with boundary condition $\sigma_1=0$ [^2] and we interpret $TI_{Tot}=(vol_2)(-m/2)$ as a thermodynamic potential, $\Omega(T,\mu)$. Note also that $\sigma_2$ then determines the vev of the operator dual to $\chi$. Recall from [@Gauntlett:2009zw] that writing $U=-u+v/3$, $V=6u+v/3$, the fields $u,v$ are dual to operators $\mathcal{O}_{u,v}$ with dimensions $\Delta_u=4$, $\Delta_v=6$. The truncation [(\[truncansatz\])]{} implies that the vevs of these dual operators are fixed by $\sigma_2$. The asympotic expansion of $u$ to $o(1/r^4)$ and $v$ to $o(1/r^6)$ gives $\langle \, \mathcal{O}_{u} \, \rangle \propto \sigma_2^2$ and $ \langle \mathcal{O}_{v} \, \rangle\propto \mu^2 \sigma_2 ^2 $. Numerical Results ----------------- Following [@H32] we solved the differential equations numerically using a shooting method. We used [(\[sc1\])]{} to fix the scale $\hat{\mu}=1$. At high temperatures the black hole solutions have no scalar hair ($\sigma_2=0$) and are just the solutions given in [(\[asoln\])]{}. At a critical temperature $T_c\sim 0.042$ a new branch of solutions with $\sigma_2\ne 0$ appears and moreover dominates the free energy. We refer to these as the unbroken and broken phase solutions, corresponding to normal and superconducting phases, respectively. In the figures we have plotted some features of our solutions and compared them with the solutions of the phenomenological model considered in [@H32]. While the results are in agreement near the critical temperature, as expected, we see marked differences as the temperature goes to zero. We have calculated the Ricci scalar and curvature invariant $\sqrt{R_{\alpha\beta\mu\nu} R^{\alpha\beta\mu\nu}}$ at $r=r_+$ which indicate that the solutions of [@H32] are becoming singular but our solutions are approaching a regular zero temperature solution, without horizon, holographically dual to a quantum critical point. Indeed as $r\to r_+$ we find $\sigma \sim 1$, $\beta \sim $const, $\hat\phi \sim 0$ and $g \sim \tfrac{16}{3} \left( r^2 - r_+^3 / r \right)$ and fixing $\hat{\mu} = 1$ gives $r_+ \rightarrow 0$ in the extremal limit. In particular, the geometry near $r=r_+$ is consistent with being the exact $AdS_4$ solution with $\sigma= 1$, mentioned earlier, which uplifts to the $D=11$ solution found in [@pw]. For such a solution $R = -64$ and $\sqrt{R_{\alpha\beta\mu\nu} R^{\alpha\beta\mu\nu}} = 32$, agreeing with the low temperature limit seen in the figures. The full zero temperature solution thus appears to be a charged domain wall, of the type considered in [@Gubser:2008wz], connecting two $AdS_4$ vacua of , one with $\sigma=0$ and the other with $\sigma=1$. Interestingly this implies the entropy of the solutions vanish in the low temperature limit, unlike for the Reissner-Nordström solution .[^3] The asymptotic charge appears to be derived from the scalar hair, with the region near $r=r_+$ carrying no flux. ![ Plot showing $-\tfrac{1}{2} Gm$ (proportional to the thermodynamic potential $\Omega(T,\mu)$) against $T$ with fixed $\hat{\mu} = 1$, for unbroken phase solutions (long dashed red), broken phase (blue) and solutions of [@H32] (with their $L=1/2$ and their $q=2$) (dashed blue). ](FreeErgplt.eps){height="2.2in" width="3.0in"} {height="2.4in" width="3.0in"} {height="2.4in" width="3.0in"} Concluding remarks ================== For any seven-dimensional Sasaki-Einstein space we have constructed solutions of $D=11$ supergravity corresponding to holographic superconductors in three spacetime dimensions. We have studied electric black holes using the action whose solutions lift to $D=11$ when $\hat{F} \wedge \hat{F} = 0$. One may consider adding magnetic charge using the full consistent truncation of [@Gauntlett:2009zw]. Our results indicate the existence of a regular zero temperature solution which is a charged domain wall connecting two $AdS_4$ vacua of and dual to a new quantum critical point. An important open issue is whether or not there are additional unstable charged modes for skew-whiffed $AdS_4\times SE_7$ solutions, which condense at higher temperatures. If they exist, and dominate the free energy, then the corresponding supergravity solutions would be the appropriate ones to describe the superconductivity and not the ones that we have constructed. However, it is plausible that we have found the dominant modes for large classes of $SE_7$, if not all. For the specific class of deformations of the four-form that were considered in [@dh], it was proven that the modes that we consider are in fact the only condensing modes. It would be worthwhile extending this result to cover other bosonic and/or fermionic modes. [*Note added:*]{} after this work was completed we received [@Gubser:2009qm] which constructs solutions of string theory that are dual to superconductors in four spacetime dimensions. We are supported by EPSRC (JG,JS), the Royal Society (JG) and STFC (TW). We thank R. Emparan, S. Hartnoll, G. Horowitz, V. Hubeny, M. Rangamani, O. Varela and D. Waldram for helpful discussions. [99]{} S. S. Gubser, Phys. Rev.  D [**78**]{} (2008) 065034 \[arXiv:0801.2977 \[hep-th\]\]. S. A. Hartnoll, C. P. Herzog and G. T. Horowitz, Phys. Rev. Lett.  [**101**]{} (2008) 031601 \[arXiv:0803.3295 \[hep-th\]\]. S. A. Hartnoll, C. P. Herzog and G. T. Horowitz, JHEP [**0812**]{}, 015 (2008) \[arXiv:0810.1563 \[hep-th\]\]. F. Denef and S. A. Hartnoll, arXiv:0901.1160 \[hep-th\]. J. P. Gauntlett, S. Kim, O. Varela and D. Waldram, JHEP [**0904**]{} (2009) 102 \[arXiv:0901.0676 \[hep-th\]\]. M. J. Duff, B. E. W. Nilsson and C. N. Pope, Phys. Lett.  B [**139**]{} (1984) 154. S. S. Gubser, C. P. Herzog, S. S. Pufu and T. Tesileanu, arXiv:0907.3510 \[hep-th\]. S. Franco, A. Garcia-Garcia and D. Rodriguez-Gomez, arXiv:0906.1214 \[hep-th\] . C. N. Pope and N. P. Warner, Phys. Lett.  B [**150**]{} (1985) 352; C. N. Pope and N. P. Warner, Class. Quant. Grav.  [**2**]{} (1985) L1. L. J. Romans, Phys. Lett.  B [**153**]{} (1985) 392. S. S. Gubser and F. D. Rocha, Phys. Rev. Lett.  [**102**]{} (2009) 061601 \[arXiv:0807.1737 \[hep-th\]\]. [^1]: There are analogous $AdS_5$ solutions of the theory considered in [@Gubser:2009qm] which uplift to IIB solutions found in [@romans]. [^2]: One may also consider fixing $\sigma_2 = 0$ [@H32] with similar results. Non-zero $\sigma_1$ is less interesting as we want the scalar to condense without being sourced. [^3]: We thank Gary Horowitz for a discussion on this point.

--- author: - | The HERMES Collaboration\ A. Airapetian$^{13,16}$, N. Akopov$^{27}$, Z. Akopov$^{6}$, W. Augustyniak$^{26}$, A. Avetissian$^{27}$, H.P. Blok$^{18,25}$, A. Borissov$^{6}$, V. Bryzgalov$^{20}$, M. Capiluppi$^{10}$, G.P. Capitani$^{11}$, E. Cisbani$^{22}$, G. Ciullo$^{10}$, M. Contalbrigo$^{10}$, P.F. Dalpiaz$^{10}$, W. Deconinck$^{6}$, R. De Leo$^{2}$, E. De Sanctis$^{11}$, M. Diefenthaler$^{15,9}$, P. Di Nezza$^{11}$, M. Düren$^{13}$, M. Ehrenfried$^{13}$, G. Elbakian$^{27}$, F. Ellinghaus$^{5}$, E. Etzelmüller$^{13}$, R. Fabbri$^{7}$, L. Felawka$^{23}$, S. Frullani$^{22}$, D. Gabbert$^{7}$, G. Gapienko$^{20}$, V. Gapienko$^{20}$, F. Garibaldi$^{22}$, G. Gavrilov$^{19,6,23}$, V. Gharibyan$^{27}$, M. Hartig$^{6}$, D. Hasch$^{11}$, Y. Holler$^{6}$, I. Hristova$^{7}$, A. Ivanilov$^{20}$, H.E. Jackson$^{1}$, S. Joosten$^{15,12}$, R. Kaiser$^{14}$, G. Karyan$^{27}$, T. Keri$^{13}$, E. Kinney$^{5}$, A. Kisselev$^{19}$, V. Korotkov$^{20}$, V. Kozlov$^{17}$, P. Kravchenko$^{19}$, V.G. Krivokhijine$^{8}$, L. Lagamba$^{2}$, L. Lapikás$^{18}$, I. Lehmann$^{14}$, P. Lenisa$^{10}$, W. Lorenzon$^{16}$, B.-Q. Ma$^{3}$, D. Mahon$^{14}$, S.I. Manaenkov$^{19}$, Y. Mao$^{3}$, B. Marianski$^{26}$, H. Marukyan$^{27}$, A. Movsisyan$^{10,27}$, M. Murray$^{14}$, Y. Naryshkin$^{19}$, A. Nass$^{9}$, W.-D. Nowak$^{7}$, L.L. Pappalardo$^{10}$, R. Perez-Benito$^{13}$, A. Petrosyan$^{27}$, P.E. Reimer$^{1}$, A.R. Reolon$^{11}$, C. Riedl$^{15,7}$, K. Rith$^{9}$, A. Rostomyan$^{6}$, D. Ryckbosch$^{12}$, A. Schäfer$^{21}$, G. Schnell$^{4,12}$, K.P. Schüler$^{6}$, B. Seitz$^{14}$, T.-A. Shibata$^{24}$, M. Stahl$^{13}$, M. Stancari$^{10}$, M. Statera$^{10}$, E. Steffens$^{9}$, J.J.M. Steijger$^{18}$, S. Taroian$^{27}$, A. Terkulov$^{17}$, R. Truty$^{15}$, A. Trzcinski$^{26}$, M. Tytgat$^{12}$, Y. Van Haarlem$^{12}$, C. Van Hulse$^{4,12}$, V. Vikhrov$^{19}$, I. Vilardi$^{2}$, S. Wang$^{3}$, S. Yaschenko$^{6,9}$, S. Yen$^{23}$, D. Zeiler$^{9}$, B. Zihlmann$^{6}$, P. Zupranski$^{26}$ date: 'DESY Report 14-116 / Compiled:   / Version: 6.0 (final, Erratum incorporated)' title: 'Spin density matrix elements in exclusive $\omega$ electroproduction on $^1$H and $^2$H targets at 27.5 GeV beam energy' --- Introduction {#intro} ============ Exclusive electroproduction of vector mesons on nucleons offers a rich source of information on the mechanisms that produce these mesons, see e.g., Refs. [@Fran; @diehl1]. This process can be considered to consist of three subprocesses: i) the incident lepton emits a virtual photon $\gamma^*$, which dissociates into a $q\bar{q}$ pair; ii) this pair interacts strongly with the nucleon; iii) from the scattered $q\bar{q}$ pair the observed vector meson is formed. In Regge phenomenology, the interaction of the $q\bar{q}$ pair with the nucleon proceeds through the exchange of a pomeron or (a combination of) the exchanges of other reggeons (e.g., $\rho$, $\omega$, $\pi$, ...). If the quantum numbers of the particle lying on the Regge trajectory are $J^P=0^+,\;1^-$, ..., the process is denoted Natural Parity Exchange (NPE). Alternatively, the case of $J^P=0^- ,\;1^+$, ... is denoted Unnatural Parity Exchange (UPE). In perturbative quantum chromodynamics (pQCD), the interaction of the $q\bar{q}$ pair with the nucleon can proceed via two-gluon exchange or quark-antiquark exchange, where the former corresponds to the exchange of a pomeron and the latter to the exchange of a (combination of) reggeon(s). Spin density matrix elements (SDMEs) describe the final spin states of the produced vector meson. In this work, SDME values will be determined and discussed in the formalism that was developed in Ref. [@Schill] for the case of an unpolarized or longitudinally polarized beam and an unpolarized target. For completeness, we also present SDME values in the more general formalism of Ref. [@Diehl]. The SDMEs can be expressed in terms of helicity amplitudes that describe the transitions from the initial helicity states of virtual photon and incoming nucleon to the final helicity states of the produced vector meson and the outgoing nucleon. The values of SDMEs will be used to establish a hierarchy of helicity amplitudes, to test the hypothesis of $s$-channel helicity conservation, to investigate UPE contributions, and to determine the longitudinal-to-transverse cross-section ratio. In the framework of pQCD, the nucleon structure can also be studied through hard exclusive meson production as the process amplitude contains Generalized Parton Distributions (GPDs) [@gpd1; @gpd2; @gpd3]. For longitudinal virtual photons, this amplitude is proven to factorize rigorously into a perturbatively calculable hard-scattering part and two soft parts (collinear factorization) [@Radyushkin:1996ru; @Collins:1996fb]. The soft parts of the convolution contain GPDs and a meson distribution amplitude. At leading twist, the chiral-even GPDs $H^f$ and $E^f$ are sufficient to describe exclusive vector-meson production on a spin-$1/2$ target such as a proton or a neutron, where $f$ denotes a quark of flavor $f$ or a gluon. These GPDs are of special interest as they are related to the total angular momentum carried by quarks or gluons in the nucleon [@Ji:1996e]. Although there is no such rigorous proof for transverse virtual photons, phenomenological models use the modified perturbative approach [@Botts:1989kf] instead, which takes into account parton transverse momenta. The latter are included at subleading twist in the subprocess $\gamma^* f \rightarrow {\cal M} f$, where ${\cal M}$ denotes the meson, while the partons are still emitted and reabsorbed by the nucleon collinear to the nucleon momentum. By using this approach, the pQCD-inspired phenomenological “GK model" can describe existing data on cross sections, SDMEs and spin asymmetries in exclusive vector-meson production for values of Bjorken-$x$, $x_{B}$, below about 0.2 [@Goloskokov:2005sd; @Goloskokov:2007nt; @Goloskokov:2013mba]. It can also describe exclusive leptoproduction of pseudoscalar mesons by including the full contribution to the electromagnetic form factor from the pion, in contrast to earlier studies at leading-twist, which took into account only the relatively small perturbative contribution to this form factor (see Ref. [@Goloskokov:2009ia] and references therein). The GK model also applies successfully to the description of deeply virtual Compton scattering [@k_mo_sa]. The results of the most recent variant of the GK model, in which the unnatural-parity contributions due to pion exchange are included to describe exclusive $\omega$ leptoproduction [@thcal], will be compared in this paper to the HERMES proton data in terms of SDMEs and certain combinations of them. Early papers on exclusive $\omega$ electroproduction are summarized in Ref. [@Bauer], which particularly contains results on SDMEs obtained at DESY for 0.3 GeV$^2$ $< Q^{2} <$ 1.4 GeV$^2$ and 0.3 GeV $< W <$ 2.8 GeV. The symbol $Q^{2}$ represents the negative square of the virtual-photon four-momentum and $W$ is the invariant mass of the photon-nucleon system. Recently, SDMEs in exclusive $\omega $ electroproduction were studied for 1.6 GeV$^2$ $< Q^{2}<$ 5.2 GeV$^2$ by CLAS [@clas] and it was found that the exchange of the pion Regge trajectory dominates exclusive $\omega$ production, even for Q$^2$ values as large as 5 GeV$^2$. Formalism ========= Spin density matrix elements ---------------------------- The $\omega$ meson is produced in the following reaction: $$\begin{aligned} e + p \to e + p + \omega, \label{omprod}\end{aligned}$$ with a branching ratio $Br = 89.1 \%$ for the $\omega$ decay: $$\begin{aligned} \omega \to \pi^+ + \pi^- + \pi^0,~\pi^0 \to2\gamma. \label{omdecay}\end{aligned}$$ The angular distribution of the three final-state pions depends on SDMEs. The first subprocess of vector-meson production, the emission of a virtual photon ($ e \rightarrow e+ \gamma^{*}$), is described by the photon spin density matrix [@Schill], $$\varrho^{U+L}_{\lambda_{\gamma} \lambda '_{\gamma }} = \varrho^{U}_{\lambda_{\gamma} \lambda '_{\gamma }} + P_{b}\varrho^{L}_{\lambda_{\gamma} \lambda '_{\gamma }}, \label{phspden}$$ where U and L denote unpolarized and longitudinally polarized beam, respectively, and $P_{b}$ is the value of the beam polarization. The photon spin density matrix can be calculated in quantum electrodynamics. The vector-meson spin density matrix $\rho_{\lambda_{V}\lambda_{V}^{'}}$ is expressed through helicity amplitudes [$F_{\lambda _{V} \lambda '_{N}\lambda _{\gamma} \lambda_{N}}$]{}. These amplitudes describe the transition of a virtual photon with helicity $\lambda _{\gamma}$ to a vector meson with helicity $ \lambda _{V}$, while $\lambda_{N}$ and $\lambda '_{N}$ are the helicities of the nucleon in the initial and final states, respectively. Helicity amplitudes depend on $W$, $Q^{2}$, and $t'=t - t_{min}$, where $t$ is the Mandelstam variable and $-t_{min}$ represents the smallest kinematically allowed value of $-t$ at fixed virtual-photon energy and $Q^{2}$. The quantity $\sqrt{-t'} $ is approximately equal to the transverse momentum of the vector meson with respect to the direction of the virtual photon in the $\gamma^{*}N$ centre-of-mass (CM) system. In this system, the spin density matrix of the vector meson is given by the von Neumann equation [@Schill], $$\begin{aligned} \rho_{\lambda_{V} \lambda '_{V}}= \frac{1}{2 \mathcal{N} } \sum_{\lambda_{\gamma} \lambda '_{\gamma}\lambda_N \lambda '_N} F_{\lambda_{V}\lambda '_N\lambda_{\gamma}\lambda _N} \varrho^{U+L}_{\lambda_{\gamma} \lambda '_{\gamma }} F_{\lambda '_{V} \lambda '_N\lambda '_{\gamma}\lambda _N}^{*} \, ,\end{aligned}$$ where $\mathcal{N}$ is a normalization factor, see Refs. [@Schill; @DC-24]. After the decomposition of $\varrho^{U+L}_{\lambda_{\gamma} \lambda'_{\gamma}}$ into the standard set of $3\times 3$ Hermitian matrices $ \Sigma^{\alpha}$, the vector-meson spin density matrix is expressed in terms of a set of nine matrices $\rho^{\alpha}_{\lambda_V\lambda'_V}$ related to various photon polarization states: transversely polarized photon ($\alpha$=0,...,3), longitudinally polarized photon ($\alpha$=4), and terms describing their interference ($\alpha$=5,...,8) [@Schill]. When contributions of transverse and longitudinal photons cannot be separated, the SDMEs are customarily defined as $$\begin{aligned} r^{04}_{\lambda_{V}\lambda '_{V}} &= (\rho^{0}_{\lambda_{V}\lambda '_{V}} + \epsilon R \rho^{4}_{\lambda_{V}\lambda '_{V}})( 1 + \epsilon R )^{-1}, \nonumber\\ r^{\alpha}_{\lambda_{V}\lambda'_{V}} &= \begin{cases} { \rho^{\alpha}_{\lambda_{V}\lambda'_{V}}}{( 1 + \epsilon R )^{-1}}, \; \alpha = 1,2,3,\\ { \sqrt{R} \rho^{\alpha}_{\lambda_{V}\lambda '_{V}}} {(1 + \epsilon R )^{-1}}, \; \alpha = 5,6,7,8. \end{cases} \hspace*{0.25cm} \label{rmatr}\end{aligned}$$ The quantity $R= d\sigma_{L}/ d\sigma_{T}$ is the longitudinal-to-transverse virtual-photon differential cross-section ratio and $\epsilon$ is the ratio of fluxes of longitudinal and transverse virtual photons.\ Helicity amplitudes {#hel_ampli} ------------------- A helicity amplitude can be decomposed into a sum of a NPE amplitude T and a UPE amplitude U, $$F_{\lambda_{V} \lambda '_{N} \lambda_{\gamma} \lambda_{N} } = T_{\lambda_{V} \lambda '_{N} \lambda_{\gamma} \lambda_{N} }+ U_{\lambda_{V}\lambda '_{N} \lambda_{\gamma} \lambda_{N}}, \label{nu}$$ for details see Refs. [@Schill; @DC-24]. The relations between the amplitudes $F$, $T$, and $U$ are the following [@Schill]: $$\begin{aligned} T_{\lambda_V \lambda'_N \lambda_{\gamma} \lambda_N}=\frac{1}{2}[ F_{\lambda_V \lambda'_N \lambda_{\gamma} \lambda_N}\nonumber ~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ +(-1)^{\lambda_V-\lambda_{\gamma}}F_{-\lambda_V \lambda'_N -\lambda_{\gamma}\lambda_N}], \label{fnat}\\ U_{\lambda_V \lambda'_N \lambda_{\gamma} \lambda_N}=\frac{1}{2}[ F_{\lambda_V \lambda'_N \lambda_{\gamma} \lambda_N} \nonumber~~~~~~~~~~~~~~~~~~~~~~~~~~~ \\ -(-1)^{\lambda_V-\lambda_{\gamma}}F_{-\lambda_V \lambda'_N -\lambda_{\gamma} \lambda_N}]. \label{funnnat}\end{aligned}$$ The asymptotic behaviour of amplitudes $F$ at small $-t'$ [@Diehl], $$F_{\lambda_V \lambda'_N \lambda_{\gamma} \lambda_N} \propto \Bigl (\frac{\sqrt{-t'}}{M}\Bigr )^{|(\lambda_V-\lambda'_N)-(\lambda_{\gamma}-\lambda_N)|}, \label{asytpr}$$ follows from angular-momentum conservation. Equations - show that the double-helicity-flip amplitudes with $|\lambda _V-\lambda_{\gamma}|=2$ are suppressed at least by a factor of $\sqrt{-t'}/M$, and the contributions of these double-helicity-flip amplitudes to the SDMEs are suppressed by $-t'/M^2$. Therefore they will be neglected throughout the paper. For an unpolarized target, there exists no interference between NPE and UPE amplitudes and there is no linear contribution from nucleon-helicity-flip amplitudes to SDMEs. For brevity, the following notations will be used: $$\widetilde{\sum}T_{\lambda_V \lambda_{\gamma}} T^*_{\lambda'_V \lambda'_{\gamma}}\equiv \frac{1}{2} \sum_{\lambda_N \lambda'_N} T_{\lambda_V \lambda'_N\lambda_{\gamma}\lambda_N} T^*_{\lambda'_V \lambda'_N\lambda'_{\gamma}\lambda_N}. \label{tilde-sum}$$ Using the symmetry properties [@Schill; @DC-24] of the amplitudes $T$, Eq. (\[tilde-sum\]) can be rewritten as $$\begin{aligned} \widetilde{\sum}T_{\lambda_V \lambda_{\gamma}} T^*_{\lambda'_V \lambda'_{\gamma}}=~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\ T_{\lambda_V \frac{1}{2}\lambda_{\gamma}\frac{1}{2}} T^*_{\lambda'_V \frac{1}{2}\lambda'_{\gamma}\frac{1}{2}}+ T_{\lambda_V -\frac{1}{2}\lambda_{\gamma}\frac{1}{2}} T^*_{\lambda'_V -\frac{1}{2}\lambda'_{\gamma}\frac{1}{2}}.\label{sum-two}\end{aligned}$$ Here, the first and second product on the right-hand side gives the contribution of NPE amplitudes without and with nucleon-helicity flip, respectively. Analogous relations hold for UPE amplitudes. An additional abbreviated notation in the text will be the omission of the nucleon-helicity indices when discussing the amplitudes with $\lambda_N=\lambda'_N$, i.e., $$\begin{aligned} T_{\lambda_V \lambda_{\gamma}}&\equiv T_{\lambda_V \frac{1}{2}\lambda_{\gamma}\frac{1}{2}}= T_{\lambda_V -\frac{1}{2}\lambda_{\gamma}-\frac{1}{2}} \nonumber\\ U_{\lambda_V \lambda_{\gamma}}&\equiv U_{\lambda_V \frac{1}{2}\lambda_{\gamma}\frac{1}{2}}= -U_{\lambda_V -\frac{1}{2}\lambda_{\gamma}-\frac{1}{2}}. \label{abrr}\end{aligned}$$ The dominance of diagonal $\gamma^{*} \to V$ transitions ($\lambda_{V}=\lambda_{\gamma}$) is called $s$-channel helicity conservation (SCHC). Angular distribution -------------------- The SDMEs in exclusive electroproduction of $\omega$ mesons are determined using the process in Eq. (\[omprod\]). They are fitted as parameters of $\mathcal{W}^{U+L}(\Phi,\phi,\cos\Theta)$, which is the three-dimensional angular distribution, to the corresponding experimental distribution of the three pions originating from the $\omega$-meson decay. The angular distribution $\mathcal{W}^{U+L}(\Phi,\phi,\cos\Theta)$ is decomposed into $\mathcal{W}^{U}$ and $\mathcal{W}^{L}$, see Eq. , which are the respective distributions for unpolarized and longitudinally polarized beams. From the fit, 15 “unpolarized” SDMEs (see Eq. ) are extracted and additionally 8 “polarized" SDMEs (see Eq. ) from data collected with a longitudinally polarized beam. $$\begin{aligned} \mathcal{W}^{U+L}(\Phi,\phi,\cos{\Theta}) =& \ \mathcal{W}^{U}(\Phi,\phi,\cos{\Theta}) + P_{b}\mathcal{W}^{L}(\Phi,\phi,\cos{\Theta}), \label{eqang1}\\ \mathcal{W}^{U}(\Phi,\phi,\cos{\Theta}) =& \ \frac{3} {8 \pi^{2}} \Bigg[ \frac{1}{2} (1 - r^{04}_{00}) + \frac{1}{2} (3 r^{04}_{00}-1) \cos^2{\Theta} - \sqrt{2} \mathrm{Re} \{ r^{04}_{10} \} \sin 2\Theta \cos \phi - r^{04}_{1-1} \sin ^{2} \Theta \cos 2 \phi \hspace*{1.0cm} \nonumber \\ &- \epsilon \cos 2 \Phi \Big( r^{1}_{11} \sin ^{2} \Theta + r^{1}_{00} \cos^{2}{\Theta} - \sqrt{ 2} \mathrm{Re} \{r^{1}_{10}\} \sin 2 \Theta \cos \phi - r^{1}_{1-1} \sin ^{2} \Theta \cos 2 \phi \Big) \nonumber \\ &- \epsilon \sin 2 \Phi \Big( \sqrt{2} \mathrm{Im} \{r^{2}_{10}\} \sin 2 \Theta \sin \phi + \mathrm{Im} \{ r^{2}_{1-1} \} \sin ^{2} \Theta \sin 2 \phi \Big) \nonumber \\ &+ \sqrt{ 2 \epsilon (1+ \epsilon)} \cos \Phi \Big( r^{5}_{11} \sin ^2 {\Theta} + r^{5}_{00} \cos ^{2} \Theta - \sqrt{2} \mathrm{Re} \{r^{5}_{10}\} \sin 2 \Theta \cos \phi - r^{5}_{1-1} \sin ^{2} \Theta \cos 2 \phi \Big) \nonumber \\ &+ \sqrt{ 2 \epsilon (1+ \epsilon)} \sin \Phi \Big( \sqrt{ 2} \mathrm{Im} \{ r^{6}_{10} \} \sin 2 \Theta \sin \phi + \mathrm{Im} \{r^{6}_{1-1} \} \sin ^{2} \Theta \sin 2 \phi \Big) \Bigg], \label{eqang2} \\ \mathcal{W}^{L}(\Phi,\phi,\cos \Theta) =& \ \frac{3}{8 \pi^{2}} \Bigg[ \sqrt{ 1 - \epsilon ^{2} } \Big( \sqrt{ 2} \mathrm{Im} \{ r^{3}_{10} \} \sin 2 \Theta \sin \phi + \mathrm{Im} \{ r^{3}_{1-1}\} \sin ^{2} \Theta \sin 2 \phi \Big) \nonumber \\ &+ \sqrt{ 2 \epsilon (1 - \epsilon)} \cos \Phi \Big( \sqrt{2} \mathrm{Im} \{r^{7}_{10}\} \sin 2 \Theta \sin \phi + \mathrm{Im} \{ r^{7}_{1-1} \} \sin ^{2} \Theta \sin 2 \phi \Big) \nonumber \\ &+ \sqrt{ 2 \epsilon (1 - \epsilon)} \sin \Phi \Big( r^{8}_{11} \sin ^{2} \Theta + r^{8}_{00} \cos ^{2} \Theta - \sqrt{2} \mathrm{Re}\{ r^{8}_{10}\} \sin 2 \Theta \cos \phi - r^{8}_{1-1} \sin ^{2} \Theta \cos 2\phi \Big) \Bigg]. \label{eqang3}\end{aligned}$$ Definitions of angles and reference frames are shown in Fig. \[defang\]. The directions of the axes of the hadronic CM system and of the $\omega$-meson rest frame follow the directions of the axes of the helicity frame [@Schill; @DC-24; @joos]. The angle $\Phi$ between the $\omega $ production and the lepton scattering plane in the hadronic CM system is given by $$\begin{aligned} \cos \Phi &= \frac{ (\vec{q} \times \vec{v}) \cdot (\vec{k} \times \vec{k}')} { | \vec{q} \times \vec{v} | \cdot |\vec{k} \times \vec{k}'| } , \\ \sin \Phi &= \frac{ [ (\vec{q} \times \vec{v} )\times (\vec{k} \times \vec{k}' )] \cdot \vec{q} } { |\vec{q} \times \vec{v}| \cdot |\vec{k} \times \vec{k}'| \cdot |\vec{q}| }. \label{phicap-def}\end{aligned}$$ Here $\vec{k}$, $\vec{k'}$, $\vec{q}=\vec{k}-\vec{k'}$, and $\vec{v}$ are the three-momenta of the incoming and outgoing leptons, virtual photon, and $\omega $ meson respectively. The unit vector normal to the decay plane in the $\omega$ rest frame is defined by $$\vec n = \frac{\vec{p}_{\pi^+} \times \vec{p}_{\pi^-}}{|\vec{p}_{\pi^+} \times\vec{p}_{\pi^-}| }, \label{normal}$$ where $\vec{p}_{\pi^+}$ and $\vec{p}_{\pi^-}$ are the three-momenta of the positive and negative decay pions in the $\omega$ rest frame. The polar angle $\Theta$ of the unit vector $\vec{n}$ in the $\omega$-meson rest frame, with the $z$-axis aligned opposite to the outgoing nucleon momentum $\vec{p}'$ and the $y$-axis directed along $\vec{p}' \times \vec{q}$, is defined by $$\begin{aligned} \cos \Theta &= -\frac{ \vec{p}' \cdot \vec n }{| \vec{p}' |}, \label{theta-def}\end{aligned}$$ while the azimuthal angle $\phi$ of the unit vector $\vec n$ is given by $$\begin{aligned} \cos \phi &= \frac{ (\vec{q} \times \vec{p}' )\cdot (\vec{p}' \times \vec n) } { | \vec{q} \times \vec{p}'| \cdot |\vec{p}' \times \vec{n} | }, \\ \sin \phi &= - \frac{[ (\vec{q} \times \vec{p}' )\times \vec{p}' ] \cdot ( \vec n \times \vec{p}' ) } { | (\vec{q} \times \vec{p}' )\times \vec{p}' | \cdot |\vec n \times \vec{p}' | } . \label{phismall-def}\end{aligned}$$ ![ Definition of angles in the process $e N \to e N \omega$, where $\omega \to \pi^+ \pi^- \pi^0$. Here, $\Phi$ is the angle between the $\omega$ production plane and the lepton scattering plane in the center-of-mass system of the virtual photon and the target nucleon. The variables $\Theta$ and $\phi$ are respectively the polar and azimuthal angles of the unit vector normal to the decay plane in the $\omega$-meson rest frame.[]{data-label="defang"}](./pl/figure1.pdf){width=".5\textwidth"} Data analysis ============= HERMES experiment ----------------- The data analyzed in this paper were accumulated with the HERMES spectrometer during the running period of 1996 to 2007 using the 27.6 GeV longitudinally polarized electron or positron beam of HERA, and gaseous hydrogen or deuterium targets. The HERMES forward spectrometer, which is described in detail in Ref. [@identif], was built of two identical halves situated above and below the lepton beam pipe. It consisted of a dipole magnet in conjunction with tracking and particle identification detectors. Particles were accepted when their polar angles were in the range $\pm 170$ mrad in the horizontal direction and $\pm(40-140)$ mrad in the vertical direction. The spectrometer permitted a precise measurement of charged-particle momenta, with a resolution of $1.5\%$. A separation of leptons was achieved with an average efficiency of $98\%$ and a hadron contamination below $1\%$. Selection of exclusively produced $\omega$ mesons ------------------------------------------------- ![Two-photon invariant mass distribution after application of all criteria to select exclusively produced $\omega$ mesons. The Breit–Wigner fit to the mass distribution is shown as a continuous line and the dashed line indicates the PDG value of the $\pi^{0}$ mass.[]{data-label="pi0"}](./pl/mass_pi0_1.pdf){width=".45\textwidth"} The following requirements were applied to select exclusively produced $\omega$ mesons from reaction (\[omprod\]):\ i) Exactly two oppositely charged hadrons, which are assumed to be pions, and one lepton with the same charge as the beam lepton are identified through the analysis of the combined responses of the four particle-identification detectors [@identif].\ ii) A $\pi^{0}$ meson that is reconstructed from two calorimeter clusters as explained in Ref. [@arny] is selected requiring the two-photon invariant mass to be in the interval $0.11$ GeV $ < M({\gamma\gamma})<$ 0.16 GeV. The distribution of $M ({\gamma\gamma})$ is shown in Fig. \[pi0\]. This distribution is centered at $m_{\pi^{0}}=134.69\pm19.94$ MeV, which agrees well with the PDG [@pdg] value of the $\pi^{0}$ mass.\ iii) The three-pion invariant mass is required to obey 0.71 GeV$\le M(\pi^+ \pi^- \pi^0) \le$ 0.87 GeV.\ iv) The kinematic requirements for exclusive production of $\omega$ mesons are the following:\ a) The scattered-lepton momentum lies above $3.5$ GeV.\ b) The constraint $-t' <$ 0.2 GeV$^{2}$ is used.\ c) For exclusive production the missing energy $ \Delta E $ must vanish. Here, the missing energy is calculated both for proton and deuteron as $ \Delta E = \frac{ M^{2}_{X} -M^{2}_{p}}{2 M_{p}}$, with $M_{p}$ being the proton mass and $ M^{2}_{X}=({p} + {q}- {p}_{\pi^+} - {p}_{\pi^-} - {p}_{\pi^0})^{2}$ the missing mass squared, where ${p}$, ${q}$, ${p}_{\pi^+}$, $ {p}_{\pi^-}$, and ${p}_{\pi^0}$ are the four-momenta of target nucleon, virtual photon, and each of the three pions respectively. In this analysis, taking into account the spectrometer resolution, the missing energy has to lie in the interval $-1.0$ GeV $<\Delta E < 0.8$ GeV, which is referred to as “exclusive region" in the following.\ d) The requirement $ Q^2> $ 1.0 GeV$^2$ is applied in order to facilitate the application of pQCD.\ e) The requirement $W > 3.0$ GeV is applied in order to be outside of the resonance region, while an upper cut of $W < 6.3$ GeV is applied in order to define a clean kinematic phase space. ![Breit-Wigner fit (solid line) of $ \pi^+ \pi^- \pi^0$ invariant mass distributions after application of all criteria to select $\omega$ mesons produced exclusively from proton (top) and from deuteron (bottom). The dashed line represents the PDG value of the $\omega$ mass.[]{data-label="omega"}](./pl/momega_mass.pdf){width=".45\textwidth"} After application of all these constraints, the proton sample contains 2260 and the deuteron sample 1332 events of exclusively produced $\omega$ mesons. These data samples are referred to in the following as data in the “entire kinematic region". The invariant-mass distributions for exclusively produced $\omega$ mesons are shown in Fig. \[omega\]. Note the reasonable agreement of the fit result, $m_{\omega} = 784.8\pm55.8$ MeV for proton data and $m_{\omega} = 784.6\pm58.2$ MeV for deuteron data, with the PDG [@pdg] value of the $\omega$ mass. The distributions of missing energy $\Delta$E, shown in Fig. \[deltae\], exhibit clearly visible exclusive peaks. The shaded histograms represent semi-inclusive deep-inelastic scattering (SIDIS) background obtained from a PYTHIA [@pythia] Monte Carlo simulation that is normalized to data in the region $2~\text{GeV} < \Delta E < 20~\text{GeV}$. The simulation is used to determine the fraction of background under the exclusive peak, which is calculated as the ratio of number of background events to the total number of events. It amounts to about 20% for the entire kinematic region and increases from 16% to 26% with increasing $-t'$. ![ The $\Delta E$ distributions of $\omega$ mesons produced in the entire kinematic region and in three kinematic bins in $-t'$ are compared with SIDIS $\Delta E$ distributions from PYTHIA (shaded area). The vertical dashed line denotes the upper limit of the exclusive region.[]{data-label="deltae"}](./pl/q17.pdf "fig:"){width=".24\textwidth"} ![ The $\Delta E$ distributions of $\omega$ mesons produced in the entire kinematic region and in three kinematic bins in $-t'$ are compared with SIDIS $\Delta E$ distributions from PYTHIA (shaded area). The vertical dashed line denotes the upper limit of the exclusive region.[]{data-label="deltae"}](./pl/t1.pdf "fig:"){width=".24\textwidth"}\ ![ The $\Delta E$ distributions of $\omega$ mesons produced in the entire kinematic region and in three kinematic bins in $-t'$ are compared with SIDIS $\Delta E$ distributions from PYTHIA (shaded area). The vertical dashed line denotes the upper limit of the exclusive region.[]{data-label="deltae"}](./pl/t2.pdf "fig:"){width=".24\textwidth"} ![ The $\Delta E$ distributions of $\omega$ mesons produced in the entire kinematic region and in three kinematic bins in $-t'$ are compared with SIDIS $\Delta E$ distributions from PYTHIA (shaded area). The vertical dashed line denotes the upper limit of the exclusive region.[]{data-label="deltae"}](./pl/t3.pdf "fig:"){width=".24\textwidth"} Comparison of data and Monte Carlo events ----------------------------------------- Distributions of experimental data in some kinematic variables are compared to those simulated by PYTHIA. The comparison is shown in Fig. \[comdatpyt\] and mostly demonstrates good agreement between experimental and simulated data. ![Distributions of several kinematic variables from experimental data on exclusive $\omega$-meson leptoproduction (black squares) in comparison with simulated exclusive events from the PYTHIA generator (dashed areas). Simulated events are normalized to the experimental data.[]{data-label="comdatpyt"}](./pl/varp1.pdf "fig:"){width=".48\textwidth"} ![Distributions of several kinematic variables from experimental data on exclusive $\omega$-meson leptoproduction (black squares) in comparison with simulated exclusive events from the PYTHIA generator (dashed areas). Simulated events are normalized to the experimental data.[]{data-label="comdatpyt"}](./pl/varp2.pdf "fig:"){width=".48\textwidth"} Extraction of $\omega$ spin density matrix elements =================================================== The unbinned maximum likelihood method -------------------------------------- The SDMEs are extracted from data by fitting the angular distribution $\mathcal{W}^{U+L}(\Phi,\phi,\cos{\Theta})$ to the experimental angular distribution using an unbinned maximum likelihood method. The probability distribution function is $\mathcal{W}^{U+L}({\cal R};\Phi,\phi,\cos{\Theta})$, where ${\cal R}$ represents the set of 23 SDMEs, i.e., the coefficients of the trigonometric functions in Eqs. (\[eqang2\], \[eqang3\]). The negative log-likelihood function to be minimized reads $$-\ln L({\cal R})= -\sum_{i=1}^{N}\ln\frac{\mathcal{W}^{U+L}({\cal R};\Phi_{i},\phi_{i},\cos{\Theta_{i}})}{\widetilde{\mathcal N}({\cal R})}, \label{loglik-def}$$ where the normalization factor $$\widetilde {\mathcal N}({\cal R})= \sum_{j=1}^{N_{MC}}\mathcal{W}^{U+L}({\cal R};\Phi_{j},\phi_{j},\cos{\Theta_{j}}) \label{loglik-def1}$$ is calculated numerically using events from a PYTHIA Monte Carlo generated according to an isotropic three-dimensional angular distribution and passed through the same analytical process as experimental data. The numbers of data and Monte Carlo events are denoted by ${N}$ and $N_{MC}$, respectively. Background treatment --------------------- In order to account for the SIDIS background in the fit, first “SIDIS-background SDMEs" are obtained using Eqs. (\[loglik-def\], \[loglik-def1\]) for the PYTHIA SIDIS sample in the exclusive region. Then, SDMEs corrected for SIDIS background are obtained as follows [@phdami]: $$\begin{aligned} -\ln L({\cal R})=\nonumber ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ -\sum_{i=1}^{N}\ln\Bigl[ \frac{ (1-f_{bg})*\mathcal W^{U+L}({\cal R};\Phi_{i},\phi_{i},\cos\Theta_{i})} {\widetilde {\mathcal N}({\cal R},\Psi)} \nonumber\\ +\frac{f_{bg}*\mathcal W^{U+L}(\Psi; \Phi_{i},\phi_{i}, \cos \Theta_{i})} {\widetilde{\mathcal N}({\cal R},\Psi)}\Bigr].~~~ \label{logbac}\end{aligned}$$ From now on, ${\cal R}$ denotes the set of SDMEs corrected for background, $\Psi$ the set of the SIDIS-background SDMEs, and $f_{bg}$ is the fraction of SIDIS background. The normalization factor reads correspondingly $$\begin{aligned} \widetilde{\mathcal N}({\cal R},\Psi)= \sum_{j=1}^{N_{MC}}\left[(1-f_{bg})*\mathcal W^{U+L}({\cal R}; \Phi_{j}, \phi_{j},\cos\Theta_{j})\right.\nonumber\\ \left. +f_{bg}*\mathcal W^{U+L}(\Psi; \Phi_{j}, \phi_{j},\cos\Theta_{j})\right].~~~ \label{logbacnor}\end{aligned}$$ Systematic uncertainties ------------------------ The total systematic uncertainty on a given extracted SDME $r$ is obtained by adding in quadrature the uncertainty from the background subtraction procedure, $\Delta r_{sys}^{bg}$, and the one due to the extraction method, $\Delta r_{sys}^{MC}$. The former uncertainty is assigned to be the difference between the SDME obtained with and without background correction. This conservative approach also covers the small uncertainty on the fraction of SIDIS background, $f_{bg}$. The uncertainty $\Delta r_{sys}^{MC}$ is estimated using the Monte Carlo data that were generated with an angular distribution determined by the set of SDMEs ${\cal R}$. The statistics of the Monte Carlo data exceed those of the experimental data by about a factor of six. The generated events were passed through a realistic model of the HERMES apparatus using GEANT [@geant] and were then reconstructed and analyzed in the same way as experimental data. These Monte Carlo data were used to extract the SDME set ${\cal R}^{MC}$. In this way, effects from detector acceptance, efficiency, smearing, and misalignment are accounted for. Two uncertainties are considered to be responsible for the difference between input and output value of a given SDME $r$, $$(r - r^{MC})^2 = (\Delta r_{sys}^{MC})^2 + (\Delta r_{stat}^{MC})^2, \label{sysextr}$$ where $\Delta r_{stat}^{MC}$ is the statistical uncertainty of $r^{MC}$ as obtained in the fitting procedure that uses MINUIT [@minuit]. From Eq. (\[sysextr\]), $\Delta r_{sys}^{MC}$ is determined, using the convention that $\Delta r_{sys}^{MC}$ is set to zero if $[(r - r^{MC})^2- (\Delta r_{stat}^{MC})^2]$ is negative. Results {#sec-2} ======= The results on SDMEs in the Schilling-Wolf [@Schill] representation are given in Tables \[tab1\]-\[tab5\] in Appendix \[sec:tables\] and in the Diehl [@Diehl] representation in Table \[tab6\] in the same Appendix. The SDMEs for the entire kinematic region are discussed in Sect. \[sec\_sdme\_fullkin\], while their dependences on $ Q^2$ and $-t'$ are discussed in Sect. \[sec\_sdme\_kindep\]. SDMEs for the entire kinematic region {#sec_sdme_fullkin} ------------------------------------- {width=".75\textwidth"} The SDMEs of the $\omega$ meson for the entire kinematic region ($\left< Q^{2} \right>= 2.42$ GeV$^2$, $\left< W \right>= 4.8$ GeV, and $\left<-t^{\prime} \right> = 0.080$ GeV$^2$) are presented in Fig. \[sdmescaled\]. These SDMEs are divided into five classes corresponding to different helicity transitions. The main terms in the expressions of class-A SDMEs correspond to the transitions from longitudinal virtual photons to longitudinal vector mesons, $\gamma^*_L \to V_L$, and from transverse virtual photons to transverse vector mesons, $\gamma^*_T \to V_T$. The dominant terms of class B correspond to the interference of these two transitions. The main terms of class-C, class-D, and class-E SDMEs are proportional to small amplitudes describing $\gamma^*_T \to V_L$, $\gamma^*_L \to V_T$, and $\gamma^*_T \to V_{-T}$ transitions respectively.\ The SDMEs for the proton and deuteron data are found to be consistent with each other within their quadratically combined total uncertainties, with a $\chi^2$ per degrees of freedom of $28/23 \approx 1.2$. In Fig. \[sdmescaled\], the eight polarized SDMEs are presented in shaded areas. Their experimental uncertainties are larger in comparison to those of the unpolarized SDMEs because the lepton beam polarization is smaller than unity ($|P_{b}| \approx 40\%$) and in the equation for the angular distribution they are multiplied by the small kinematic factor $|P_{b}|\sqrt{1- \epsilon}\approx 0.2 $, cf. Eq. (\[eqang2\]) vs. Eq. (\[eqang3\]). Test of the SCHC hypothesis {#sec_schc} --------------------------- In the case of SCHC, the seven SDMEs of class A and class B ($r_{00}^{04}$, $r_{1-1}^1$, $\mathrm{Im} \{r_{1-1}^2\}$, $\mathrm{Re}\{r_{10}^5\}$, $\mathrm{Im}\{r_{10}^{6}\}$, $\mathrm{Im} \{r_{10}^{7}\}$, $\mathrm{Re}\{r_{10}^{8}\}$) are not restricted to be zero, but six of them have to obey the following relations [@Schill]: $$\begin{aligned} r_{1-1}^1 &=&-\mathrm{Im} \{r_{1-1}^2\},\nonumber\\ \mathrm{Re}\{r_{10}^5\} &=&-\mathrm{Im}\{r_{10}^{6}\},\nonumber\\ \mathrm{Im} \{r_{10}^{7}\} &=& \mathrm{Re} \{r_{10}^{8}\}.\nonumber\end{aligned}$$ The proton data yield $$\begin{aligned} r^{1}_{1-1} +\mathrm{Im}\{r^{2}_{1-1}\}&=& -0.004 \pm 0.038 \pm0.015,\nonumber\\ \mathrm{Re}\{r^{5}_{10}\} + \mathrm{Im}\{r^{6}_{10}\}&=& -0.024 \pm 0.013 \pm0.004 ,\nonumber\\ \mathrm{Im}\{r^{7}_{10}\} -\mathrm{Re}\{r^{8}_{10}\} &=&-0.060 \pm 0.100 \pm0.018, \nonumber\end{aligned}$$ and the deuteron data yield $$\begin{aligned} r^{1}_{1-1} +\mathrm{Im}\{r^{2}_{1-1}\}&=& 0.033 \pm 0.049\pm 0.016,\nonumber\\ \mathrm{Re} \{r^{5}_{10}\} + \mathrm{Im} \{r^{6}_{10}\}&=& 0.001 \pm 0.016 \pm0.005,\nonumber\\ \mathrm{Im} \{r^{7}_{10}\} -\mathrm{Re} \{r^{8}_{10}\} &=& 0.104 \pm0.110\pm0.023. \nonumber\end{aligned}$$ Here and in the following, the first uncertainty is statistical and the second systematic. In the calculation of the statistical uncertainty, the correlations between the different SDMEs are taken into account, see correlation matrices in Tables \[tab8\] and \[tab9\]. It can be concluded that the above SCHC relations are fulfilled for class A and B. The SCHC relations for the class-A SDMEs $r^1_{1-1}$ and $\mathrm{Im} \{r^2_{1-1}\}$ can be violated only by the quadratic contributions of the double-helicity-flip amplitudes $T_{1\pm\frac{1}{2}-1 \frac{1}{2}}$ and $U_{1 \pm\frac{1}{2}-1 \frac{1}{2}}$ with $|\lambda_V-\lambda_{\gamma}|=2$. The observed validity of SCHC means that their possible contributions are smaller than the experimental uncertainties. Also for class-B SDMEs, to which the same small double-helicity-flip amplitudes contribute linearly, no SCHC violation is observed. In addition, class-B SDMEs contain the contribution of the two small products $T_{0 \pm\frac{1}{2}1 \frac{1}{2}} T^*_{1 \pm\frac{1}{2}0 \frac{1}{2}}$ ($U_{0 \pm\frac{1}{2}1 \frac{1}{2}}U^*_{1 \pm\frac{1}{2}0 \frac{1}{2}}$). As the SCHC hypothesis is fulfilled, all these contributions are concluded to be negligibly small compared to the experimental uncertainties. This validates the assumption made in Sect. \[hel\_ampli\] that the double-helicity-flip amplitudes can be neglected. All SDMEs of class C to E have to be zero in the case of SCHC. The class-C SDME $r^5_{00}$ deviates from zero by about three standard deviations for the proton and two standard deviations for the deuteron (see Fig. \[sdmescaled\]). Since the numerator of the equation for $r_{00}^{5}$ [@DC-24], $$r^{5}_{00}=\frac{\mathrm{Re} \left\{T_{0-\frac{1}{2}1 \frac{1}{2}} T^*_{0-\frac{1}{2}0\frac{1}{2}}+ T_{0\frac{1}{2}1 \frac{1}{2}} T^*_{0\frac{1}{2}0 \frac{1}{2}}\right\}}{\cal{N}}, \label{r500}$$ contains two amplitude products, at least one product is nonzero. However, without an amplitude analysis of the presented data it cannot be concluded which contribution to $r^5_{00}$ dominates. Both amplitudes $T_{0-\frac{1}{2}1 \frac{1}{2}}$ and $T_{0\frac{1}{2}1 \frac{1}{2}}$ have to be zero if the SCHC hypothesis holds. Figure \[sdmescaled\] shows that out of the six SDMEs of class D three, i.e., $r_{11}^5$, $r_{1-1}^5$, and $\mathrm{Im}\{r_{1-1}^6\}$, slightly differ from zero (see Table \[tab1\]). As will be discussed in Sections \[sec:UPE\] and \[sec:hierarchy\], the largest UPE amplitudes in $\omega$ production are $U_{11}$ and $U_{10}$, and $|U_{11}|\gg|U_{10}|$. The main term of the first two SDMEs is proportional to $\mathrm{Re}[U_{10}U_{11}^*]$, while $\mathrm{Im}\{r_{1-1}^6\}$ is proportional to $-\mathrm{Re}[U_{10}U_{11}^*]$. The calculated linear combination of these three SDMEs, $r_{11}^5+r_{1-1}^5-\mathrm{Im}\{r_{1-1}^6\}$, is $-0.14 \pm 0.03 \pm 0.04$ for the proton and $-0.10 \pm 0.03 \pm 0.03$ for the deuteron. These values differ from zero by about three standard deviations of the total uncertainty for the proton. This, together with the experimental information on measured class-C and class-D SDMEs, indicates a violation of the SCHC hypothesis in exclusive $\omega$ production. Dependences of SDMEs on $Q^{2}$ and $-t'$ and comparison to a phenomenological model {#sec_sdme_kindep} ------------------------------------------------------------------------------------ {width=".8\textwidth"} In the following sections, kinematic dependences of the measured SDMEs and certain combinations of them are presented and interpreted wherever possible. In particular, the proton data presented in this paper are compared to the calculations of the phenomenological GK model described in Sect. \[intro\]. In each case, model calculations are shown with and without inclusion of the pion-pole contribution. In order to stay in the framework of handbag factorization and to avoid large $1/Q^2$ corrections, model calculations are only shown for $Q^2 > 2$ GeV$^2$, which leaves for the $Q^2$ dependence only two data points that can be compared to the model calculation. This paucity of comparable points makes it sometimes difficult to draw useful conclusions about the data-model comparison. The kinematic dependences of SDMEs on $Q^{2}$ and $-t'$ are presented in three bins of $Q^{2}$ with $\langle Q^2 \rangle = 1.28$ GeV$^2$, $\langle Q^2 \rangle = 2.00$ GeV$^2$, $\langle Q^2 \rangle = 4.00$ GeV$^2$, and $t'$ with $\langle -t' \rangle = 0.021$ GeV$^2$, $\langle -t' \rangle = 0.072$ GeV$^2$, $\langle -t' \rangle = 0.137$ GeV$^2$. Table \[tab7\] shows the average value of $Q^2$ and $-t'$ for bins in $-t'$ and $Q^2$, respectively. {width=".8\textwidth"} The $Q^2$ and $-t'$ dependences of class-A SDMEs are shown and compared to the model calculations in Fig. \[Aq2t\]. All three SDMEs clearly show the need for the unnatural-parity contribution of the pion pole and the measured $-t'$ dependence is well reproduced both in shape and magnitude. The same holds for the two unpolarized class-B SDMEs that are shown in Fig. \[Bq2t\]. For the polarized class-B SDMEs as well as for all class-C SDMEs, which are shown in Fig. \[Cq2t\], the pion-pole contribution has little or no effect, and the model describes the magnitude of the data reasonably well. The class-D and E SDMEs are shown in Figs. \[Dq2t\] and \[Eq2t\], respectively. These SDMEs are expected to be zero if the pion-pole contribution is not included. When comparing the $-t'$ dependences of the three unpolarized class-D SDMEs to the model calculation, also here the unnatural-parity pion-exchange contribution seems to be required. The two unpolarized class-E SDMEs are measured with reasonable precision, and agreement with the model calculation can be seen. Within experimental uncertainties, the SDMEs measured on the proton are seen to be very similar to those measured on the deuteron. This can be understood by considering the different contributions to exclusive omega production. The pion-pole contribution is seen to be substantial [@thcal]. For the NPE amplitudes, the dominant contribution comes from gluons and sea quarks, which are the same for protons and neutrons, while the valence-quark contribution is different. Thus altogether, only small differences between the proton and deuteron SDMEs are expected for incoherent scattering. As coherence effects are difficult to estimate, one can not exclude that they are of the size of the valence-quark effects. Therefore, the deuteron SDMEs are presently difficult to calculate reliably. {width=".8\textwidth"} {width=".8\textwidth"} {width=".8\textwidth"} {width=".8\textwidth"} {width=".8\textwidth"} UPE in $\omega$-meson production {#sec:UPE} -------------------------------- In Fig. \[sdmeomrho\], the comparison of $\omega$ and $\rho^{0}$ [@DC-24] SDMEs is shown. One can see that the SDMEs $r^{1}_{1-1}$ and $\mathrm{Im} \{r^{2}_{1-1}\} $ of class A have opposite sign for $\omega$ and $\rho^{0}$. The SDME $r^{1}_{1-1}$ is negative for the $\omega$ meson and positive for $\rho^{0}$, while $\mathrm{Im}\{r^{2}_{1-1}\} $ is positive for $\omega$ and negative for $\rho^{0}$. In terms of helicity amplitudes, these two SDMEs are written [@DC-24] as $$\begin{aligned} r_{1-1}^1 = \frac {1}{2\mathcal{N}}\widetilde{\sum} & \left( |T_{11}|^2+|T_{1-1}|^2 \right. \nonumber \\ & \left. -|U_{11}|^2-|U_{1-1}|^2 \right), \label{sdprz} \\ \mathrm{Im} \{r_{1-1}^2 \} =\frac {1}{2\mathcal{N}} \widetilde{\sum} & \left( -|T_{11}|^2+|T_{1-1}|^2 \right. \nonumber\\ & \left. +|U_{11}|^2 -|U_{1-1}|^2 \right) .\label{sdprz1}\end{aligned}$$ The difference between Eqs. (\[sdprz1\]) and (\[sdprz\]) reads $$\mathrm{Im} \{r_{1-1}^2\} - r_{1-1}^1 = \frac {1}{\mathcal{N}}\widetilde{\sum}(-|T_{11}|^2+|U_{11}|^2). \label{difff}$$ For the entire kinematic region, this difference is clearly positive for the $\omega$ meson, hence $\widetilde{\sum}|U_{11}|^2 >\widetilde{\sum}|T_{11}|^2$, while for the $\rho^{0}$ meson $\widetilde{\sum}|T_{11}|^2 > \widetilde{\sum}|U_{11}|^2$ [@DC-24]. This suggests a large UPE contribution in exclusive $\omega$-meson production. Applying Eq. (\[sum-two\]) to relation (\[difff\]), the latter can be rewritten as $$\begin{aligned} \mathrm{Im} \{r^2_{1-1}\}-r^1_{1-1} =\frac{1}{\mathcal{N}} (&-|T_{1 \frac{1}{2}1 \frac{1}{2}}|^2-|T_{1 -\frac{1}{2}1 \frac{1}{2}}|^2 \nonumber \\ &+|U_{1 \frac{1}{2}1 \frac{1}{2}}|^2+|U_{1 -\frac{1}{2}1\frac{1}{2}}|^2). \label{appl}\end{aligned}$$ The amplitudes with nucleon helicity flip, $T_{1 -\frac{1}{2}1 \frac{1}{2}}$ and $U_{1 -\frac{1}{2}1 \frac{1}{2}}$, should be zero at $t'=0$ and are proportional to $\sqrt{-t'}$ at small $t'$ (see Eq. (\[asytpr\]) and Ref. [@Diehl]). The small contribution of $|T_{1-\frac{1}{2}1\frac{1}{2}}|^2$ will be neglected from now on. As it was established above, the UPE contribution is larger than the NPE one. This means that if the dominant UPE helicity-flip amplitude is $U_{1 -\frac{1}{2}1 \frac{1}{2}}$, expression (\[appl\]) would increase proportionally to $-t'$. However, the experimental values of ($\mathrm{Im} \{r^2_{1-1}\}-r^1_{1-1}$) (see Tables \[tab3\] and \[tab5\]) do not demonstrate such an increase; the values for the proton data even decrease smoothly with $-t'$. Hence the dominant UPE amplitude is $U_{1\frac{1}{2}1\frac{1}{2}}$, and it holds $|U_{11}|^2>|T_{11}|^2$. The existence of UPE in $\omega$ production on the proton and deuteron can also be tested with linear combinations of SDMEs such as $$u_1=1-r^{04}_{00}+2r^{04}_{1-1}-2r^{1}_{11}-2r^{1}_{1-1}, \label{uu1}$$ $$u_2=r^{5}_{11}+r^{5}_{1-1}, \label{uu2}$$ $$u_3=r^{8}_{11}+r^{8}_{1-1}. \label{uu3}$$ The quantity $u_1$ can be expressed in terms of helicity amplitudes as $$u_1=\frac{1}{\mathcal{N}} \ \widetilde{\sum}{\left( 4\epsilon|U_{10}|^2+2|U_{11}+U_{-11}|^2\right)}. \label{u1u}$$ A non-zero result for $u_1$, implying that at least one of the four amplitudes $U_{1\pm\frac{1}{2}0\frac{1}{2} }$ or ($U_{1\pm\frac{1}{2}1\frac{1}{2} }+U_{-1\pm\frac{1}{2}1\frac{1}{2} } $) is nonzero, indicates the existence of UPE contributions. In the entire kinematic region, $u_1$ is 1.15 $\pm$ 0.09 $\pm$ 0.12 and 1.47 $\pm$ 0.12 $\pm$ 0.18 for proton and deuteron data, respectively. In Fig. \[u1\], the $Q^{2}$ and $-t'$ dependences of $u_1$ for proton and deuteron data are presented. It can be seen that $u_1$ is larger than unity, which implies the existence of large contributions from UPE transitions. {width=".8\textwidth"} The expression for the quantities $u_2$ and $u_3$ in terms of helicity amplitudes is $$u_2 + iu_3 =\frac{\sqrt2}{\mathcal{N}} \ \widetilde{\sum} {(U_{11}+U_{-11})U^{*}_{10}}, \label{u2u3n}$$ showing that these quantities are nonzero if at least one of the products $U^*_{1\frac{1}{2}0\frac{1}{2}}(U_{1\frac{1}{2}1\frac{1}{2}}+U_{-1\frac{1}{2}1\frac{1}{2}})$ or $U^*_{1-\frac{1}{2}0\frac{1}{2}}(U_{1-\frac{1}{2}1\frac{1}{2}}+U_{-1-\frac{1}{2}1\frac{1}{2}})$ is nonzero. Therefore $u_2$ and $u_3$ provide information complementary to that given by $u_1$. In Fig. \[u1\], also the quantities $u_2$ and $u_3$ versus $Q^2$ and $-t'$ are presented both for proton and deuteron data. As seen from this figure, there are no clear dependences on $Q^2$ and $-t'$, but $u_2$ for the proton data is definitely nonzero and there is some evidence that it is also nonzero for the deuteron data. Note that $u_2$ and $u_3$ are compatible with zero in $\rho^0$-meson electroproduction [@DC-24]. Figure \[u1\] also demonstrates good agreement between proton data and the model calculation. It appears that including the pion-pole into the model fully accounts for the unnatural-parity contribution measured through $u_1$ and $u_2$, both in $-t'$ shape and magnitude. Conclusions on $u_3$ are prevented by the considerable experimental uncertainties. Phase difference between amplitudes ----------------------------------- {width=".75\textwidth"} Taking the amplitude without helicity flip, $U_{1 \frac{1}{2}1 \frac{1}{2}}$, as the dominant UPE one, Eq. (\[u2u3n\]) can be simplified as $$u_2+iu_3=\frac{\sqrt{2}}{\mathcal{N}}U_{1 \frac{1}{2}1 \frac{1}{2}} U^*_{1 \frac{1}{2}0 \frac{1}{2}} \equiv \frac{\sqrt{2}}{\mathcal{N}} U_{11}U^*_{10}. \label{u2u3simpl}$$ The expressions for the phase difference $\delta_U$ between the UPE amplitudes $U_{11}$ and $U_{10}$ follow immediately from Eq. (\[u2u3simpl\]): $$\cos \delta_{U} =u_2/\sqrt{(u_2)^2+(u_3)^2},$$ $$\sin \delta_{U} =u_3/\sqrt{(u_2)^2+(u_3)^2},$$ $$\tan \delta_{U} =u_3/u_2 = \frac{r^{8}_{11}+r^{8}_{1-1}}{r^{5}_{11}+r^{5}_{1-1}}.$$ The phase differences obtained for the entire kinematic region are $\delta_{U}$ = (-126 $\pm$ 12 $\pm$ 2) degrees for proton and $\delta_{U}$ = (-100 $\pm$ 61 $\pm$ 3) degrees for deuteron data. The phase difference $\delta_{N}$ between the NPE amplitudes $T_{11}$ and $T_{00}$ can be calculated as follows [@DC-24]: $$\cos \delta_{N}= \frac{ 2 \sqrt{\epsilon} (\mathrm{Re}\{r^{5}_{10}\}-\mathrm{Im}\{r^{6}_{10} \})} {\sqrt{ r^{04}_{00}(1-r^{04}_{00}+r^{1}_{1-1}-\mathrm{Im}\{r^{2}_{1-1}\} )} }. \label{eq:cosdelta}$$ The phase differences obtained for the entire kinematic region are $|\delta_{N}|$ = (51 $\pm$ 5 $\pm$ 14) degrees and $|\delta_{N}|$ = (50 $\pm$ 7 $\pm$ 16) degrees for proton and deuteron data, respectively. Using polarized SDMEs, in principle also the sign of $\delta_{N}$ can be determined from the following equation: $$\sin \delta_{N}= \frac{ 2 \sqrt{\epsilon} (\mathrm{Re}\{r^{8}_{10}\}+\mathrm{Im}\{r^{7}_{10}\} ) } {\sqrt{ r^{04}_{00}(1-r^{04}_{00}+r^{1}_{1-1}-\mathrm{Im}\{r^{2}_{1-1} \}) }}, \label{eq:sindelta}$$ which is given in Ref. [@DC-24]. For the present data, the large experimental uncertainties of the polarized SDMEs make it impossible to determine the sign of $\delta_{N}$. Longitudinal-to-transverse cross-section ratio ---------------------------------------------- {width=".75\textwidth"} Usually, the longitudinal-to-transverse virtual-photon differential cross-section ratio $$R= \frac{d\sigma_{L}(\gamma^{*}_{L} \to V)}{ d\sigma_{T}(\gamma^{*}_{T} \to V)}\nonumber$$ is experimentally determined from the measured SDME $r^{04}_{00}$ using the approximated equation [@DC-24] $$R \approx \frac{1}{\epsilon}\frac{r^{04}_{00}}{1-r^{04}_{00}}. \label{sigto}$$ This relation is exact in the case of SCHC. The $Q^{2}$ dependence of $R$ for the $\omega$ meson is shown in the left panel of Fig. \[R\_sigma\], where also for comparison the same dependence for the $\rho^{0}$ meson [@DC-24] is shown. For $\omega$ mesons produced in the entire kinematic region, it is found that $R$ = 0.25 $\pm$ 0.03 $\pm$ 0.07 for the proton and $R$ = 0.24 $\pm$ 0.04 $\pm$ 0.07 for the deuteron data. Compared to the case of exclusive $\rho^{0}$ production, this ratio is about four times smaller, and for the $\omega$ meson this ratio is almost independent of Q$^2$. The $-t'$ dependence of $R$ is shown in the right panel of Fig. \[R\_sigma\]. The comparison of the proton data to the GK model calculations with and without inclusion of the pion-pole contribution demonstrates the clear need to include the pion pole. The data are well described by the model and appear to follow the $-t'$ dependence suggested by the model when the pion-pole contribution is included. This implies that transverse and longitudinal virtual-photon cross sections have different $-t'$ dependences. Hence the usual high-energy assumption that their ratio can be identified with the corresponding ratio of the integrated cross sections does not hold in exclusive $\omega$ electroproduction at HERMES kinematics, due to the pion-pole contribution. The GK model appears to fully account for the unnatural-parity contribution to $R$ and shows rather good agreement with the data. The UPE-to-NPE asymmetry of the transverse cross section -------------------------------------------------------- The UPE-to-NPE asymmetry of the transverse differential cross section is defined as [@sigmat] $$\begin{aligned} P=\frac{d\sigma^N_T-d\sigma^U_T}{d\sigma^N_T+d\sigma^U_T} &\equiv \frac{d\sigma^N_T/d\sigma^U_T-1}{d\sigma^N_T/d\sigma^U_T+1} \nonumber \\ &=(1+\epsilon R)(2r^1_{1-1}-r^1_{00}), \label{asymm} \end{aligned}$$ where $\sigma^N_T$ and $\sigma^U_T$ denote the part of the cross section due to NPE and UPE, respectively. Substituting Eq. (\[sigto\]) in Eq. (\[asymm\]) leads to the approximate relation $$\begin{aligned} P \approx \frac{2r^{1}_{1-1}-r^{1}_{00}}{1-r^{04}_{00}}.\end{aligned}$$ The value of $P$ obtained in the entire kinematic region is $-0.42 \pm 0.06 \pm 0.08$ and $-0.64 \pm 0.07 \pm 0.12$ for proton and deuteron, respectively. This means that a large part of the transverse cross section is due to UPE. In Fig. \[P\_sigma\], the $Q^{2}$ and $-t'$ dependences of the UPE-to-NPE asymmetry of the transverse differential cross section for exclusive $\omega$ production are presented. Again, the GK model calculation appears to fully account for the unnatural-parity contribution and shows very good agreement with the data both in shape and magnitude. ![ The $Q^{2}$ and $-t'$ dependences of the UPE-to-NPE asymmetry $P$ of the transverse differential cross section for exclusive $\omega$ electroproduction at HERMES. The open symbols represent the values over the entire kinematic region. Otherwise as for Fig. \[Aq2t\].[]{data-label="P_sigma"}](./pl/p_assym_q2tprime_newest4.pdf){width=".57\textwidth"} Hierarchy of amplitudes {#sec:hierarchy} ----------------------- In order to develop a hierarchy of amplitudes, in the following a number of relations between individual helicity amplitudes is considered. The resulting hierarchy is given in Eqs. (\[hierarchy\]) and (\[hierarchy-2\]) below. ### $U_{10}$ versus $U_{11}$ From Eqs. (\[u1u\]) and (\[u2u3simpl\]), the relation $$\begin{aligned} \nonumber \frac{\sqrt{2(u_2^2+u_3^2)}}{u_1} \approx \frac{|U_{11} U^*_{10}|}{|U_{11}|^2+2 \epsilon |U_{10}|^2}\\ =\frac{|U_{10}/U_{11}|}{1+2 \epsilon |U_{10}/U_{11}|^2} \label{u23u1}\end{aligned}$$ is obtained. Using the measured values of those SDMEs that determine $u_1$, $u_2$, and $u_3$, the following amplitude ratio is estimated: $$\begin{aligned} \frac{|U_{10}|}{|U_{11}|} \approx \frac{\sqrt{2(u_2^2+u_3^2)}}{u_1} \approx 0.2. \label{u10u11}\end{aligned}$$ In order to reach the best possible accuracy for such estimates, the mean values of SDMEs for the proton and deuteron are used and preference will be given to quantities that do not contain polarized SDMEs, which have much less experimental accuracy than the unpolarized SDMEs. The relatively large value for the ratio $|U_{10}/U_{11}|$ is due to the large measured value of $u_3$. However, as this value is compatible with zero within about one standard deviation of the total uncertainty, the contribution of $u_3$ in Eq. (\[u10u11\]) can be neglected, which leads to the value of $0.06$ as lower bound on $|U_{10}/U_{11}|$. ### $T_{11}$ versus $U_{11}$ With the above considerations, it follows from Eq. (\[u1u\]) that the contribution of $|U_{10}/U_{11}|^2$ is only a few percent and hence will be neglected everywhere. Then, in particular, the relation $$\begin{aligned} u_1 \approx 2|U_{11}|^2/\mathcal{N} \label{appru1}\end{aligned}$$ is valid with a precision of a few percent. Equations (\[fnat\]-\[asytpr\]) show that the nucleon-helicity-flip amplitudes $T_{1\pm\frac{1}{2}1\mp\frac{1}{2}}$ ($T_{0\pm\frac{1}{2}0\mp\frac{1}{2}}$) are suppressed by a factor of about $\sqrt{-t'}/M$ compared to the amplitude $T_{11}$ ($T_{00}$) with diagonal transitions ($\lambda'_N=\lambda_N$). Therefore, the second-order contributions of the amplitudes $T_{\lambda\pm\frac{1}{2}\lambda \mp\frac{1}{2}}$ for any $\lambda$ will be neglected compared to any bilinear product of $T_{00}$ and $T_{11}$. In this approximation, the relation $$\begin{aligned} \frac{2[\rm{Im} \{r^2_{1-1}\}-r^2_{1-1}]}{u_1}=1-\Bigl |\frac{T_{11}}{U_{11}} \Bigr |^2 \label{t11u11}\end{aligned}$$ follows from Eqs. (\[appl\]) and (\[appru1\]). Substituting numerical values for the SDMEs in Eq. (\[t11u11\]) leads to the estimate $|T_{11}/U_{11}| \approx 0.6$. ### $T_{00}$ versus $U_{11}$ Using Eq. (\[appru1\]) and the expression for $r^{04}_{00}$ from Refs. [@Schill; @DC-24] yields $$\begin{aligned} \frac{2r^{04}_{00}}{u_1}=\frac{\widetilde{\sum}[\epsilon |T_{00}|^2+|T_{01}|^2+|U_{01}|^2]}{|U_{11}|^2}. \label{r04u1}\end{aligned}$$ Neglecting in the numerator of the right-hand side of Eq. (\[r04u1\]) all positive terms except $|T_{00}|^2$, the inequality of interest is obtained: $$\begin{aligned} \frac{2r^{04}_{00}}{u_1}>\frac{\epsilon |T_{00}|^2}{|U_{11}|^2}. \label{inr04u1}\end{aligned}$$ Using for the estimate $\epsilon =0.8$ and values of SDMEs from Table \[tab1\] yields the result $|T_{00}/U_{11}| < 0.6$. The same ratio can be estimated from other SDMEs. Using expressions for the SDMEs from [@Schill; @DC-24], the following equations can be written: $$\begin{aligned} \nonumber \rm{Re}\{r^5_{10}\}-\rm{Im}\{r^6_{10}\}=~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ \frac{1}{\mathcal{N}\sqrt{2}}\widetilde{\sum}{\rm Re}[T_{11}T^*_{00}+T_{01}T^*_{10}-U_{01}U^*_{10}], \label{r510-r610}\\ \nonumber \rm{Im}\{r^7_{10}\}+\rm{Re}\{r^8_{10}\}=~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ \frac{1}{\mathcal{N}\sqrt{2}}\widetilde{\sum}{\rm Im}[T_{11}T^*_{00}+T_{10}T^*_{01}-U_{10}U^*_{01}]. \label{r710-r810}\end{aligned}$$ From Eqs. (\[fnat\]-\[asytpr\]), it follows that the terms $\widetilde{\sum}T_{01}T^*_{10}$ and $\widetilde{\sum}U_{01}U^*_{10}$ on the right-hand side of Eqs. (\[r510-r610\], \[r710-r810\]) are suppressed by a factor $(-t')/M^2$ compared to $T_{11}T^*_{00}$ and will be neglected. The simplest consequence of Eqs. (\[r510-r610\], \[r710-r810\]) is the relation $$\begin{aligned} \nonumber [\rm{Re}\{r^5_{10}\}-\rm{Im}\{r^6_{10}\}]^2+[\rm{Im}\{r^7_{10}\}+\rm{Re}\{r^8_{10}\}]^2= \\ \frac{1}{2\mathcal{N}^2}|T_{11}|^2|T_{00}|^2. \label{a2b2}\end{aligned}$$ Dividing this relation by $u_1^2/8$ and using Eq. (\[appru1\]), one gets the formula of interest: $$\begin{aligned} \nonumber \frac{[\rm{Re}\{r^5_{10}\}-\rm{Im}\{r^6_{10}\}]^2+[\rm{Im}\{r^7_{10}\}+\rm{Re}\{r^8_{10}\}]^2}{u^2_1/8}\approx\\ \frac{|T_{11}|^2|T_{00}|^2}{|U_{11}|^4}.~~~ \label{r510-r610-r710-r810-u1}\end{aligned}$$ Using numerical SDME values from Table \[tab1\] and $|T_{11}/U_{11}|=0.6$, the estimate $|T_{00}/U_{11}|\approx0.5$ is obtained, which is in agreement with the previous estimate. However, as the polarized SDMEs $\rm{Im}\{r^7_{10}\}$ and $\rm{Re}\{r^8_{10}\}$ have very large uncertainties, the latter result is less reliable than the former. Omitting the contribution of the polarized SDMEs in Eq. (\[r510-r610-r710-r810-u1\]) leads to the inequality $$\begin{aligned} \frac{8[\rm{Re}\{r^5_{10}\}-\rm{Im}\{r^6_{10}\}]^2}{u^2_1}< \frac{|T_{11}|^2|T_{00}|^2}{|U_{11}|^4}, \label{r510-r610-u1}\end{aligned}$$ which provides the lower limit of $0.3$ for the same ratio $|T_{00}/U_{11}|$. This result combined with the former estimate leads to the boundaries $0.3 < |T_{00}/U_{11}| < 0.6$.\ ### $T_{00}$ versus $T_{01}$ In order to estimate the value of $|T_{01}|$, the quantity $$\begin{aligned} \frac{\sqrt{(r^5_{00})^2+(r^8_{00})^2}}{r^{04}_{00}}=\frac{\sqrt{2}|\widetilde{\sum}T_{01}T^*_{00}|} { \widetilde{\sum}[\epsilon|T_{00}|^2+|T_{01}|^2+|U_{01}|^2]}~~ \label{r5r8r04}\end{aligned}$$ can be formed. Neglecting in the denominator of the right-hand side of Eq. (\[r5r8r04\]) all the terms except $\epsilon |T_{00}|^2$, the inequality $$\begin{aligned} \frac{\sqrt{(r^5_{00})^2+(r^8_{00})^2}}{r^{04}_{00}}< \frac{\sqrt{2}|\widetilde{\sum}T_{01}T^*_{00}|}{\epsilon|T_{00}|^2} \label{int01}\end{aligned}$$ is obtained. The sum in the numerator of the right-hand side of Eq. (\[int01\]) is $$\begin{aligned} \widetilde{\sum}T_{01}T^*_{00}=T_{0\frac{1}{2}1 \frac{1}{2}}T^*_{0\frac{1}{2}0\frac{1}{2}} +T_{0-\frac{1}{2}1 \frac{1}{2}}T^*_{0-\frac{1}{2}0\frac{1}{2}} \label{gensum}\end{aligned}$$ according to Eq. (\[sum-two\]). If the first product on the right-hand side of Eq. (\[gensum\]) dominates, then inequality (\[int01\]) becomes simpler: $$\begin{aligned} \frac{\sqrt{(r^5_{00})^2+(r^8_{00})^2}}{r^{04}_{00}} < \frac{\sqrt{2}}{\epsilon}\frac{|T_{01}|}{|T_{00}|}. \label{estt01}\end{aligned}$$ Numerically, this yields the estimate $|T_{01}/T_{00}| \simeq 0.3$. The dominant contribution to this number comes from the polarized SDME $r^8_{00}$ that is compatible with zero within about one standard deviation of the total uncertainty. Retaining only the contribution of the unpolarized SDME $r^5_{00}$ in Eq. (\[estt01\]) gives the following result: $|T_{01}/T_{00}|>0.1$. The experimental accuracy of the presented data is not sufficient to provide a reliable estimate for the upper bound to the ratio $|T_{01}/T_{00}|$. As shown in Appendix A, the upper limits for $$\begin{aligned} \mathcal{A} \equiv \frac{\widetilde{\sum}(|T_{01}|^2+|U_{01}|^2)}{|T_{00}|^2} \label{definA}\end{aligned}$$ are $1.3 \pm 0.7$ for the proton and $1.1 \pm 1.2$ for the deuteron. In the below consideration the estimate based on Eq. (\[estt01\]), namely $|T_{01}/T_{00}| \simeq 0.3$, is assumed to be realistic. The numerator in the definition of $r^1_{00}$ is $\widetilde{\sum}[|U_{01}|^2-|T_{01}|^2]$. The values of $r^1_{00}$ are compatible with zero within two standard deviations of the total experimental uncertainty, hence $|U_{01}|$ cannot be much larger than $|T_{01}|$.\ Considering the SDME combinations ($r^5_{11}-r^5_{1-1}$) and ($\rm{Im}\{r^8_{1-1}\}-r^8_{11}$), which are proportional to the real and imaginary parts of $\widetilde{\sum}T_{10}(T_{11}-T_{1-1})^*$, respectively, it is possible in principle to estimate the value of $|T_{10}|$. Since these combinations are compatible with zero within one standard deviation of the total uncertainty, it can be concluded that $|T_{10}|$ is negligibly small compared to the large amplitude moduli $|U_{11}|$, $|T_{11}|$, and $|T_{00}|$. ### Resulting hierarchy of amplitudes As a result, the following hierarchy is obtained: $$\begin{aligned} & |U_{11}|^2>|T_{00}|^2 \sim |T_{11}|^2 \nonumber\\ \gg\ & |U_{10}|^2 \sim |T_{01}|^2 \sim |U_{01}|^2 \nonumber \\ \gg\ & |T_{10}|^2,|T_{1-1}|^2,|U_{1-1}|^2, \label{hierarchy}\end{aligned}$$ where negligibly small amplitudes are neglected. However, there exists a possible alternative for the hierarchy presented on the second line of Eq. (\[hierarchy\]), if the helicity-flip amplitudes $T_{0-\frac{1}{2}1\frac{1}{2}}$ and $U_{0-\frac{1}{2}1\frac{1}{2}}$ are of the same order of magnitude as the helicity-conserving amplitudes $T_{00}$ and $T_{11}$. Indeed, the sum $\widetilde{\sum}T_{01}T^*_{00}$ in Eq. (\[int01\]) is the sum of two products, $T_{0\frac{1}{2}1\frac{1}{2}}T^*_{0\frac{1}{2}0\frac{1}{2}}$ and $T_{0-\frac{1}{2}1\frac{1}{2}}T^*_{0-\frac{1}{2}0\frac{1}{2}}$, according to Eq. (\[gensum\]). In order to obtain Eq. (\[estt01\]) from Eq. (\[int01\]), the dominance of the first product was assumed. If instead the second product is assumed to be dominant, Eq. (\[estt01\]) has to be replaced by $$\begin{aligned} \nonumber \frac{\sqrt{(r^5_{00})^2+(r^8_{00})^2}}{r^{04}_{00}} &\leq\frac{\sqrt{2}}{\epsilon} \frac{|T_{0-\frac{1}{2}1\frac{1}{2}} T^{*}_{0-\frac{1}{2}0\frac{1}{2}}|}{|T_{00}|^2}\\ &=\frac{\sqrt{2}}{\epsilon}\frac{|T_{0-\frac{1}{2}1\frac{1}{2}}|}{|T_{00}|} \frac{|T_{0-\frac{1}{2}0\frac{1}{2}}|}{|T_{00}|}. \label{eq54}\end{aligned}$$ The nucleon-helicity-flip amplitude $T_{0-\frac{1}{2}0\frac{1}{2}}$ is smaller than the helicity-conserving amplitude $T_{00} \equiv T_{0\frac{1}{2}0\frac{1}{2}}$ by a factor of about $\sqrt{-t'}/M \approx 0.3$ (see Eq. (\[asytpr\])). Substituting this factor for $|T_{0-\frac{1}{2}0\frac{1}{2}}/T_{00}|$, using $\epsilon=0.8$ and the measured SDME values, the final estimate $|T_{0-\frac{1}{2}1\frac{1}{2}}| \simeq |T_{00}|$ is obtained. This result shows that the nucleon-helicity-flip amplitude $T_{0-\frac{1}{2}1\frac{1}{2}}$ could be of the same order of magnitude as $T_{00}$, while the values of $T_{01}$ and $U_{01}$ could be as given in the previous estimates. As the SDME $r^1_{00}$, which is proportional to $\widetilde{\sum}[|U_{01}|^2-|T_{01}|^2]$, was measured to be compatible with zero, the value of $|U_{0-\frac{1}{2}1\frac{1}{2}}|$ should be about the same as that of $|T_{0-\frac{1}{2}1\frac{1}{2}}|$. Then, the values of $|T_{0-\frac{1}{2}1\frac{1}{2}}|$, $|U_{0-\frac{1}{2}1\frac{1}{2}}|$, and $|T_{00}|$ are of the same order of magnitude, so that the hierarchy of amplitudes becomes $$\begin{aligned} |U_{11}|^2 &>|T_{00}|^2 \sim |T_{11}|^2 \sim |T_{0-\frac{1}{2}1\frac{1}{2}}|^2 \sim |U_{0-\frac{1}{2}1\frac{1}{2}}|^2 \nonumber\\ &\gg |U_{10}|^2 \sim |T_{01}|^2 \sim |U_{01}|^2\nonumber\\ &\gg |T_{10}|^2, |T_{1-1}|^2,|U_{1-1}|^2, \label{hierarchy-2}\end{aligned}$$ where again negligibly small amplitudes are neglected. Note that the usually used Eq. (\[sigto\]) for $R$ is not applicable in this case. The estimation performed in Appendix A shows that the accuracy of the presented data is not sufficient to decide between hierarchies (\[hierarchy\]) and (\[hierarchy-2\]). The best way to get information on the amplitudes $T_{0-\frac{1}{2}1\frac{1}{2}}$ and $U_{0-\frac{1}{2}1\frac{1}{2}}$ is to study electroproduction of $\omega$ mesons on transversely polarized protons, where these amplitudes contribute linearly to the angular distribution. Summary ======= Exclusive $\omega $ electroproduction is studied at HERMES using a longitudinally polarized lepton beam and unpolarized hydrogen and deuterium targets in the kinematic region $Q^2>1.0$ GeV$^2$, $3.0$ GeV $< W < 6.3$ GeV, and $ - t' < 0.2$ GeV$^2$. The average kinematic values are $\langle Q^2 \rangle = 2.42$ GeV$^2$, $\langle W\rangle = 4.8$ GeV, and $\langle -t' \rangle = 0.080$ GeV$^2$. Using an unbinned maximum likelihood method, 15 unpolarized and, for the first time, 8 polarized spin density matrix elements are extracted. The kinematic dependences of all 23 SDMEs are presented for proton and deuteron data. No significant differences between proton and deuteron results are seen. The SDMEs are presented in five classes corresponding to different helicity transitions between the virtual photon and the $\omega$ meson. While the values of class-A and B SDMEs agree with the hypothesis of $s$-channel helicity conservation, the class-C SDME $r^{5}_{00}$ indicates a violation of this hypothesis. The values of those class-D SDMEs that correspond to the transition $\gamma^*_L\to \omega_T$ also indicate a small violation of the hypothesis of $s$-channel helicity conservation. Using the SDMEs $r_{1-1}^{1}$ and $\mathrm{Im} \{r_{1-1}^2\}$, it is shown that for exclusive $\omega$-meson production the amplitude of the UPE transition $\gamma^{*}_{T} \to \omega_{T}$ is larger than the NPE amplitude for the same transition, i.e., $|U_{11}|^2 >|T_{11}|^2$. The importance of UPE transitions is also shown by a combination of SDMEs denoted $u_1$. This suggests that at HERMES energies in exclusive $\omega$ electroproduction the quark-exchange mechanism, or $\pi^{0}$, $a_1$... exchanges in Regge phenomenology, plays a significant role. The phase shift between those UPE amplitudes that describe transverse $\omega$ production by transverse and longitudinal virtual photons, $U_{11}$ for $\gamma^{*}_{T} \to \omega_{T}$ and $U_{10}$ for $\gamma^{*}_{L} \to \omega_{T}$, respectively, as well as the magnitude of the phase difference between the NPE amplitudes $T_{11}$ and $T_{00}$ is determined for the first time. The ratio $R$ between the differential longitudinal and transverse virtual-photon cross-sections is determined to be $R$ = 0.25 $\pm$ 0.03 $\pm$ 0.07 for the $\omega$ meson, which is about four times smaller than in the case of the $\rho^{0}$ meson. In contrast to the case of the $\rho^{0}$ meson, $R$ shows only a weak dependence on $Q^{2}$ for the $\omega $ meson. The UPE-to-NPE asymmetry of the transverse virtual-photon cross section is determined to be $P= -0.42 \pm 0.06 \pm 0.08$ and $P= -0.64 \pm 0.07 \pm 0.12$ for the proton and deuteron data, respectively. From the extracted SDMEs, two slightly different hierarchies of helicity amplitudes can be derived, which remain indistinguishable for the given experimental accuracy of the presented data. Both hierarchies consistently mean that the UPE amplitude describing the $\gamma^*_T \rightarrow \omega_T$ transition dominates over the two NPE amplitudes describing the $\gamma^*_L \rightarrow \omega_L$ and $\gamma^*_T \rightarrow \omega_T$ transitions, with the latter two being of similar magnitude. Good agreement between the presented proton data and results of a pQCD-inspired phenomenological model is found only when including pion-pole contributions, which are of unnatural parity. The distinct $-t'$ dependence of the pion-pole contribution leads to a $-t'$ dependence of $R$. This invalidates for exclusive $\omega$ production at HERMES energies the common high-energy assumption of identifying $R$ with the ratio of the integrated longitudinal and transverse cross sections. [**Acknowledgements  **]{} We are grateful to Sergey Goloskokov and Peter Kroll for fruitful discussions on the comparison between our data and their model calculations. We gratefully acknowledge the DESY management for its support and the staff at DESY and the collaborating institutions for their significant effort. This work was supported by the Ministry of Education and Science of Armenia; the FWO-Flanders and IWT, Belgium; the Natural Sciences and Engineering Research Council of Canada; the National Natural Science Foundation of China; the Alexander von Humboldt Stiftung, the German Bundesministerium für Bildung und Forschung (BMBF), and the Deutsche Forschungsgemeinschaft (DFG); the Italian Istituto Nazionale di Fisica Nucleare (INFN); the MEXT, JSPS, and G-COE of Japan; the Dutch Foundation for Fundamenteel Onderzoek der Materie (FOM); the Russian Academy of Science and the Russian Federal Agency for Science and Innovations; the Basque Foundation for Science (IKERBASQUE) and the UPV/EHU under program UFI 11/55; the U.K. Engineering and Physical Sciences Research Council, the Science and Technology Facilities Council, and the Scottish Universities Physics Alliance; as well as the U.S. Department of Energy (DOE) and the National Science Foundation (NSF). [50]{} L. Frankfurt et al., Phys. Rev. D [**54**]{}, 3194 (1996) M. Diehl, Physics Reports [**388**]{}, 41 (2003) K. Schilling, G. Wolf, Nucl. Phys. B [**61**]{}, 381 (1973) M. Diehl, J. High Energy Phys. [**0709**]{}, 064 (2007) D. Müller et al., Fortschr. Phys. [**42**]{}, 101 (1994) X. Ji, Phys. Rev. Lett. [**78**]{}, 610 (1997) X. Ji, Phys. Rev. D [**55**]{}, 7114 (1997) A.V. Radyushkin, Phys. Rev. D [**56**]{}, 5524 (1997) J.C. Collins, L. Frankfurt, M.S. Strikman, Phys. Rev. D [**56**]{}, 2982 (1997) X. Ji, Phys. Rev. Lett [**74**]{}, 610, (1997) J. Botts, G.F. Sterman, Nucl. Phys. B [**325**]{}, 62 (1989) S.V. Goloskokov, P. Kroll, Eur. Phys. J. C [**42**]{}, 02298 (2005) S.V. Goloskokov, P. Kroll, Eur. Phys. J. C [**50**]{}, 829 (2007) S.V. Goloskokov, P. Kroll, Eur. Phys. J. C [**74**]{}, 2725 (2014) S.V. Goloskokov, P. Kroll, Eur. Phys. J. C [**65**]{}, 137 (2010) P. Kroll, H. Moutarde, F. Sabatié, Eur. Phys. J. C [**73**]{}, 2278 (2013) S.V. Goloskokov, P. Kroll, Eur. Phys. J. A [**50**]{}, 146 (2014) T.H. Bauer, R.D. Spital, D.R. Yenni, F.M. Pipkin, Rev. Mod. Phys. [**50**]{}, 261 (1978) L. Morand et al., (CLAS Collaboration), Eur. Phys. J. A [**24**]{}, 445 (2005) A. Airapetian et al., (HERMES Collaboration), Eur. Phys. J. C [**62**]{}, 659 (2009) P. Joos et al., Nucl. Phys. B [**122**]{}, 365 (1977) K. Ackerstaff et al., (HERMES Collaboration), Nucl. Instr. and Meth. A [**417**]{}, 230 (1998) A. Vandenbroucke Ph. D. Thesis, [*Exclusive $\pi^0$ Production at HERMES. Detection-Simulation-Analysis*]{}, Ghent University, Belgium, February 2007; DESY-THESIS-2007-003 J. Beringer et al. (Particle Data Group), Phys. Rev. D [**86**]{} 010001 (2012) T. Sjöstrand, L. Lonnblad, S. Mrenna, P. Skands, [*PYTHIA 6.3: Physics and Manual*]{}, hep-ph/0308153 (2003) A.A. Rostomyan, Ph. D. Thesis, [*Exclusive $\rho^0$ production at HERMES*]{}, Hamburg University, DESY-THESIS-2008-042 (2008) R. Brun, R. Hagelberg, M. Hansroul, J. Lassalle, [*“Geant: Simulation Program for Particle Physics Experiments. User Guide and Reference Manual”*]{}, CERN Report CERN-DD-78-2-REV (1978) CERN-CN Division, CERN Program Library Long Writeup D [**506**]{} (1992) S. Donnachie, G. Dosch, P. Landshoff, O. Nachtmann,[ *Pomeron Physics and QCD*]{}, Cambridge University press (2005) Estimate of $T_{0-\frac{1}{2}1\frac{1}{2}}$ and $U_{0-\frac{1}{2}1\frac{1}{2}}$ values {#app_a} ====================================================================================== The normalization factor $\mathcal{N}$ is given by (see, e.g., [@Schill; @DC-24]) $$\mathcal{N}=\mathcal{N}_T+\epsilon \mathcal{N}_L, \label{nor1}$$ with $$\begin{aligned} \mathcal{N}_T =\widetilde{\sum}( &|T_{11}|^2+|T_{01}|^2+|T_{-11}|^2 \nonumber\\ &+|U_{11}|^2+|U_{01}|^2+|U_{-11}|^2),\label{nor2} \\ \mathcal{N} _L= \widetilde{\sum}( &|T_{00}|^2+2|T_{10}|^2+2|U_{10}|^2 ).\label{nor3}\end{aligned}$$ Using Eqs. (\[nor1\]-\[nor3\]) and the expression defining $r^{04}_{00}$ [@Schill; @DC-24], $$r^{04}_{00}=\frac{1}{\mathcal{N}} \widetilde{\sum}( \epsilon|T_{00}|^2+|T_{01}|^2+|U_{01}|^2), \label{defr04}$$ the exact relation $$\begin{aligned} 1-r^{04}_{00}=\frac{1}{\mathcal{N}}\widetilde{\sum}[& |T_{11}|^2 + |U_{11}|^2 +|T_{1-1}|^2 +|U_{1-1}|^2 \nonumber \\ & + 2 \epsilon(|T_{10}|^2+|U_{10}|^2)] \label{1-r04}\end{aligned}$$ is obtained. Neglecting, as usual, $\widetilde{\sum}[|T_{1-1}|^2+|U_{1-1}|^2+|T_{10}|^2+|U_{10}|^2]$ in this expression, we get the approximate relation $$\begin{aligned} 1-r^{04}_{00} \approx \frac{1}{\mathcal{N}}\widetilde{\sum}(|T_{11}|^2+|U_{11}|^2). \label{appr1-r04}\end{aligned}$$ Neglecting also the small nucleon-helicity-flip amplitudes $T_{1-\frac{1}{2}1\frac{1}{2}}$ and $U_{1-\frac{1}{2}1\frac{1}{2}}$ in Eq. (\[difff\]) and then subtracting it from Eq. (\[appr1-r04\]), the relation $$\begin{aligned} 1-r^{04}_{00} +r^{1}_{1-1}- \rm{Im}\{r^2_{1-1}\} \approx \frac{2}{\mathcal{N}} |T_{11}|^2 \label{cc}\end{aligned}$$ is obtained. After neglecting in Eq. (\[defr04\]) only the nucleon-helicity-flip amplitude $T_{0-\frac{1}{2}0\frac{1}{2}}$, it can be rewritten as $$r^{04}_{00} \approx \frac{1}{\mathcal{N}} \left[ \epsilon |T_{00}|^2+\widetilde{\sum}(|T_{01}|^2+|U_{01}|^2) \right]. \label{dd}$$ Multiplying this equation by Eq. (\[cc\]) and dividing it by Eq. (\[a2b2\]) with a factor of four, the equation of interest reads $$\begin{aligned} \epsilon + \mathcal{A} \approx \nonumber ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ \frac{r^{04}_{00}(1-r^{04}_{00} +r^{1}_{1-1}- \rm{Im}\{r^2_{1-1}\})/4} {[\rm{Re}\{r^5_{10}\}-\rm{Im}\{r^6_{10}\}]^2+[\rm{Im}\{r^7_{10}\}+\rm{Re}\{r^8_{10}\}]^2},~~~~ \label{cda2b2}\end{aligned}$$ where the quantity $\mathcal{A}$ is defined in Eq. (\[definA\]). The value of $\mathcal{A}$ is close to zero, if $|T_{0-\frac{1}{2}1\frac{1}{2}}|^2$ and $|U_{0-\frac{1}{2}1\frac{1}{2}}|^2$ are much smaller than $|T_{00}|^2$, and it should be of the order of one if they are comparable to $|T_{00}|^2$. Since the uncertainties of the polarized SDMEs $\rm{Im}\{r^7_{10}\}$ and $\rm{Re}\{r^8_{10}\}$ are large, the use of Eq. (\[cda2b2\]) for the present data is not very successful. Indeed, using for numerical calculations $\epsilon=0.8$ and the values for the SDMEs in Eq. (\[cda2b2\]) from Table \[tab1\] we get $\mathcal{A}=-0.56 \pm 0.20 $ and $\mathcal{A}=0.50 \pm 1.8$ for the proton and deuteron data, respectively. In contrast, in $\rho^{0}$-meson production, the corresponding values of $\mathcal{A}$ [@DC-24], $-0.031 \pm 0.084$ and $-0.064 \pm 0.068$, exclude practically the possibility that the amplitudes $T_{0-\frac{1}{2}1\frac{1}{2}}$ and $U_{0-\frac{1}{2}1\frac{1}{2}}$ are comparable to the dominant amplitudes $U_{11}$, $T_{00}$ and $T_{11}$.\ If the contribution of $[\rm{Im}\{r^7_{10}\}+\rm{Re}\{r^8_{10}\}]$ in the denominator of the right-hand side of Eq. (\[cda2b2\]) is neglected, the useful inequality $$\begin{aligned} \mathcal{A} \leq \frac{r^{04}_{00}(1-r^{04}_{00} +r^{1}_{1-1}- \rm{Im}\{r^2_{1-1}\})} {4[\rm{Re}\{r^5_{10}\}-\rm{Im}\{r^6_{10}\}]^2} -\epsilon \label{ineqcda2}\end{aligned}$$ can be obtained. The numerical estimates $\mathcal{A} \leq 1.3 \pm 0.7$ and $\mathcal{A} \leq 1.1 \pm 1.2$ for the proton and deuteron data, respectively, show that the possibility for the values of $|T_{0-\frac{1}{2}1\frac{1}{2}}|^2$ and $|U_{0-\frac{1}{2}1\frac{1}{2}}|^2$ to be of the same order of magnitude as $|T_{00}|^2$ is not excluded by the presented results on $\omega$ SDMEs. For comparison, when applying Eq. (\[ineqcda2\]) to the results on proton and deuteron data in exclusive $\rho^0$-meson production [@DC-24], one obtains $\mathcal{A} \leq 0.22 \pm 0.09$ and $\mathcal{A} \leq 0.28 \pm 0.09$, respectively. This shows that in this case the probability for the amplitudes $T_{0-\frac{1}{2}1\frac{1}{2}}$ and $U_{0-\frac{1}{2}1\frac{1}{2}}$ to be of the same order of magnitude as $T_{00}$ is small. SDMEs for proton and deuteron {#sec:tables} ============================= element proton deuteron ------------------ -------------------------------- -------------------------------- $r^{04}_{00}$ 0.168 $\pm$ 0.018 $\pm$ 0.036 0.160 $\pm$ 0.024 $\pm$ 0.038 $r^{1}_{1-1}$ -0.175 $\pm$ 0.029 $\pm$ 0.039 -0.215 $\pm$ 0.036 $\pm$ 0.047 Im $r^{2}_{1-1}$ 0.171 $\pm$ 0.029 $\pm$ 0.023 0.248 $\pm$ 0.037 $\pm$ 0.039 Re $r^{5}_{10}$ 0.037 $\pm$ 0.009 $\pm$ 0.012 0.045 $\pm$ 0.010 $\pm$ 0.014 Im $r^{6}_{10}$ -0.061 $\pm$ 0.008 $\pm$ 0.012 -0.043 $\pm$ 0.010 $\pm$ 0.009 Im $r^{7}_{10}$ 0.109 $\pm$ 0.075 $\pm$ 0.021 0.021 $\pm$ 0.087 $\pm$ 0.004 Re $r^{8}_{10}$ 0.169 $\pm$ 0.075 $\pm$ 0.035 -0.083 $\pm$ 0.083 $\pm$ 0.017 Re $r^{04}_{10}$ -0.010 $\pm$ 0.012 $\pm$ 0.002 0.020 $\pm$ 0.014 $\pm$ 0.005 Re $r^{1}_{10}$ -0.014 $\pm$ 0.019 $\pm$ 0.005 0.016 $\pm$ 0.022 $\pm$ 0.009 Im $r^{2}_{10}$ 0.039 $\pm$ 0.018 $\pm$ 0.007 -0.003 $\pm$ 0.023 $\pm$ 0.002 $r^{5}_{00}$ 0.042 $\pm$ 0.015 $\pm$ 0.012 0.036 $\pm$ 0.019 $\pm$ 0.014 $r^{1}_{00}$ 0.006 $\pm$ 0.029 $\pm$ 0.008 0.107 $\pm$ 0.036 $\pm$ 0.023 Im $r^{3}_{10}$ 0.059 $\pm$ 0.047 $\pm$ 0.012 0.038 $\pm$ 0.056 $\pm$ 0.008 $r^{8}_{00}$ -0.142 $\pm$ 0.110 $\pm$ 0.029 -0.017 $\pm$ 0.131 $\pm$ 0.004 $r^{5}_{11}$ -0.059 $\pm$ 0.012 $\pm$ 0.022 -0.025 $\pm$ 0.015 $\pm$ 0.015 $r^{5}_{1-1}$ -0.043 $\pm$ 0.014 $\pm$ 0.006 -0.021 $\pm$ 0.018 $\pm$ 0.001 Im $r^{6}_{1-1}$ 0.036 $\pm$ 0.014 $\pm$ 0.008 0.056 $\pm$ 0.019 $\pm$ 0.013 Im $r^{7}_{1-1}$ -0.092 $\pm$ 0.117 $\pm$ 0.018 0.113 $\pm$ 0.135 $\pm$ 0.028 $r^{8}_{11}$ -0.079 $\pm$ 0.089 $\pm$ 0.017 -0.097 $\pm$ 0.103 $\pm$ 0.020 Im $r^{8}_{1-1}$ -0.060 $\pm$ 0.110 $\pm$ 0.012 -0.150 $\pm$ 0.125 $\pm$ 0.034 $r^{04}_{1-1}$ -0.004 $\pm$ 0.018 $\pm$ 0.004 0.060 $\pm$ 0.023 $\pm$ 0.016 $r^{1}_{11}$ 0.014 $\pm$ 0.024 $\pm$ 0.004 -0.037 $\pm$ 0.030 $\pm$ 0.007 $r^{3}_{1-1}$ 0.023 $\pm$ 0.076 $\pm$ 0.010 -0.122 $\pm$ 0.089 $\pm$ 0.025 element $\langle$Q$^{2}$$\rangle$ = 1.28 GeV$^{2}$ $\langle$Q$^{2}$$\rangle$ = 2.00 GeV$^{2}$ $\langle$Q$^{2}$$\rangle$ = 4.00 GeV$^{2}$ ------------------ -------------------------------------------- -------------------------------------------- -------------------------------------------- $r^{04}_{00}$ 0.164 $\pm$ 0.034 $\pm$ 0.022 0.166 $\pm$ 0.030 $\pm$ 0.044 0.179 $\pm$ 0.031 $\pm$ 0.036 $r^{1}_{1-1}$ -0.032 $\pm$ 0.050 $\pm$ 0.032 -0.175 $\pm$ 0.049 $\pm$ 0.037 -0.314 $\pm$ 0.053 $\pm$ 0.090 Im $r^{2}_{1-1}$ 0.172 $\pm$ 0.048 $\pm$ 0.027 0.133 $\pm$ 0.050 $\pm$ 0.043 0.163 $\pm$ 0.057 $\pm$ 0.029 Re $r^{5}_{10}$ 0.038 $\pm$ 0.016 $\pm$ 0.018 0.022 $\pm$ 0.015 $\pm$ 0.010 0.053 $\pm$ 0.015 $\pm$ 0.022 Im $r^{6}_{10}$ -0.062 $\pm$ 0.015 $\pm$ 0.012 -0.069 $\pm$ 0.012 $\pm$ 0.014 -0.046 $\pm$ 0.014 $\pm$ 0.013 Im $r^{7}_{10}$ 0.163 $\pm$ 0.139 $\pm$ 0.030 -0.006 $\pm$ 0.125 $\pm$ 0.009 0.170 $\pm$ 0.128 $\pm$ 0.042 Re $r^{8}_{10}$ 0.088 $\pm$ 0.143 $\pm$ 0.021 0.078 $\pm$ 0.137 $\pm$ 0.028 0.280 $\pm$ 0.119 $\pm$ 0.067 Re $r^{04}_{10}$ 0.005 $\pm$ 0.021 $\pm$ 0.004 -0.060 $\pm$ 0.020 $\pm$ 0.011 0.016 $\pm$ 0.019 $\pm$ 0.022 Re $r^{1}_{10}$ -0.005 $\pm$ 0.032 $\pm$ 0.013 -0.090 $\pm$ 0.031 $\pm$ 0.012 0.073 $\pm$ 0.034 $\pm$ 0.016 Im $r^{2}_{10}$ 0.012 $\pm$ 0.030 $\pm$ 0.012 0.042 $\pm$ 0.030 $\pm$ 0.003 0.036 $\pm$ 0.034 $\pm$ 0.016 $r^{5}_{00}$ 0.031 $\pm$ 0.029 $\pm$ 0.001 0.029 $\pm$ 0.025 $\pm$ 0.012 0.068 $\pm$ 0.027 $\pm$ 0.016 $r^{1}_{00}$ 0.009 $\pm$ 0.049 $\pm$ 0.011 0.039 $\pm$ 0.049 $\pm$ 0.013 -0.032 $\pm$ 0.053 $\pm$ 0.015 Im $r^{3}_{10}$ 0.044 $\pm$ 0.096 $\pm$ 0.008 0.047 $\pm$ 0.076 $\pm$ 0.009 0.073 $\pm$ 0.076 $\pm$ 0.018 $r^{8}_{00}$ -0.147 $\pm$ 0.210 $\pm$ 0.039 0.035 $\pm$ 0.196 $\pm$ 0.026 -0.197 $\pm$ 0.171 $\pm$ 0.045 $r^{5}_{11}$ -0.074 $\pm$ 0.020 $\pm$ 0.021 -0.050 $\pm$ 0.020 $\pm$ 0.012 -0.070 $\pm$ 0.021 $\pm$ 0.029 $r^{5}_{1-1}$ -0.047 $\pm$ 0.024 $\pm$ 0.007 -0.078 $\pm$ 0.025 $\pm$ 0.021 0.008 $\pm$ 0.025 $\pm$ 0.009 Im $r^{6}_{1-1}$ 0.070 $\pm$ 0.025 $\pm$ 0.013 -0.015 $\pm$ 0.024 $\pm$ 0.017 0.043 $\pm$ 0.026 $\pm$ 0.026 Im $r^{7}_{1-1}$ -0.326 $\pm$ 0.223 $\pm$ 0.058 -0.161 $\pm$ 0.198 $\pm$ 0.030 0.046 $\pm$ 0.204 $\pm$ 0.023 $r^{8}_{11}$ 0.276 $\pm$ 0.171 $\pm$ 0.049 -0.120 $\pm$ 0.155 $\pm$ 0.021 -0.312 $\pm$ 0.144 $\pm$ 0.080 Im $r^{8}_{1-1}$ -0.507 $\pm$ 0.212 $\pm$ 0.093 -0.026 $\pm$ 0.188 $\pm$ 0.005 0.185 $\pm$ 0.178 $\pm$ 0.063 $r^{04}_{1-1}$ -0.004 $\pm$ 0.032 $\pm$ 0.000 -0.023 $\pm$ 0.031 $\pm$ 0.003 0.008 $\pm$ 0.031 $\pm$ 0.014 $r^{1}_{11}$ 0.063 $\pm$ 0.040 $\pm$ 0.015 -0.037 $\pm$ 0.041 $\pm$ 0.012 0.003 $\pm$ 0.044 $\pm$ 0.012 $r^{3}_{1-1}$ 0.074 $\pm$ 0.153 $\pm$ 0.013 -0.110 $\pm$ 0.131 $\pm$ 0.021 0.088 $\pm$ 0.124 $\pm$ 0.024 element $\langle-t'\rangle$ = 0.021 GeV$^{2}$ $\langle-t'\rangle$ = 0.072 GeV$^{2}$ $\langle-t'\rangle$ = 0.147 GeV$^{2}$ ------------------ --------------------------------------- --------------------------------------- --------------------------------------- $r^{04}_{00}$ 0.136 $\pm$ 0.027 $\pm$ 0.036 0.197 $\pm$ 0.032 $\pm$ 0.027 0.212 $\pm$ 0.036 $\pm$ 0.032 $r^{1}_{1-1}$ -0.239 $\pm$ 0.043 $\pm$ 0.023 -0.141 $\pm$ 0.048 $\pm$ 0.043 -0.120 $\pm$ 0.060 $\pm$ 0.048 Im $r^{2}_{1-1}$ 0.220 $\pm$ 0.045 $\pm$ 0.033 0.138 $\pm$ 0.050 $\pm$ 0.015 0.111 $\pm$ 0.057 $\pm$ 0.012 Re $r^{5}_{10}$ 0.015 $\pm$ 0.013 $\pm$ 0.008 0.032 $\pm$ 0.015 $\pm$ 0.010 0.081 $\pm$ 0.018 $\pm$ 0.025 Im $r^{6}_{10}$ -0.051 $\pm$ 0.012 $\pm$ 0.012 -0.077 $\pm$ 0.013 $\pm$ 0.013 -0.058 $\pm$ 0.015 $\pm$ 0.018 Im $r^{7}_{10}$ -0.143 $\pm$ 0.121 $\pm$ 0.037 0.340 $\pm$ 0.123 $\pm$ 0.071 0.277 $\pm$ 0.146 $\pm$ 0.073 Re $r^{8}_{10}$ 0.151 $\pm$ 0.125 $\pm$ 0.039 0.232 $\pm$ 0.127 $\pm$ 0.044 0.151 $\pm$ 0.136 $\pm$ 0.039 Re $r^{04}_{10}$ -0.022 $\pm$ 0.018 $\pm$ 0.004 0.010 $\pm$ 0.020 $\pm$ 0.006 0.006 $\pm$ 0.023 $\pm$ 0.002 Re $r^{1}_{10}$ -0.020 $\pm$ 0.030 $\pm$ 0.007 -0.013 $\pm$ 0.032 $\pm$ 0.001 -0.029 $\pm$ 0.035 $\pm$ 0.011 Im $r^{2}_{10}$ 0.017 $\pm$ 0.029 $\pm$ 0.008 -0.003 $\pm$ 0.029 $\pm$ 0.005 0.125 $\pm$ 0.033 $\pm$ 0.023 $r^{5}_{00}$ -0.016 $\pm$ 0.023 $\pm$ 0.029 0.059 $\pm$ 0.027 $\pm$ 0.011 0.100 $\pm$ 0.031 $\pm$ 0.012 $r^{1}_{00}$ 0.032 $\pm$ 0.047 $\pm$ 0.033 0.067 $\pm$ 0.050 $\pm$ 0.024 -0.106 $\pm$ 0.053 $\pm$ 0.067 Im $r^{3}_{10}$ 0.063 $\pm$ 0.073 $\pm$ 0.010 0.076 $\pm$ 0.082 $\pm$ 0.018 0.121 $\pm$ 0.090 $\pm$ 0.036 $r^{8}_{00}$ 0.155 $\pm$ 0.179 $\pm$ 0.033 -0.138 $\pm$ 0.197 $\pm$ 0.026 -0.442 $\pm$ 0.191 $\pm$ 0.115 $r^{5}_{11}$ -0.059 $\pm$ 0.018 $\pm$ 0.012 -0.051 $\pm$ 0.020 $\pm$ 0.015 -0.068 $\pm$ 0.024 $\pm$ 0.048 $r^{5}_{1-1}$ -0.034 $\pm$ 0.022 $\pm$ 0.002 -0.060 $\pm$ 0.024 $\pm$ 0.007 -0.052 $\pm$ 0.030 $\pm$ 0.011 Im $r^{6}_{1-1}$ 0.010 $\pm$ 0.022 $\pm$ 0.000 0.090 $\pm$ 0.024 $\pm$ 0.020 0.020 $\pm$ 0.028 $\pm$ 0.009 Im $r^{7}_{1-1}$ -0.027 $\pm$ 0.176 $\pm$ 0.004 0.244 $\pm$ 0.197 $\pm$ 0.046 -0.601 $\pm$ 0.233 $\pm$ 0.165 $r^{8}_{11}$ -0.136 $\pm$ 0.145 $\pm$ 0.023 -0.155 $\pm$ 0.150 $\pm$ 0.029 0.038 $\pm$ 0.169 $\pm$ 0.010 Im $r^{8}_{1-1}$ -0.182 $\pm$ 0.181 $\pm$ 0.046 0.085 $\pm$ 0.180 $\pm$ 0.017 -0.055 $\pm$ 0.210 $\pm$ 0.025 $r^{04}_{1-1}$ -0.006 $\pm$ 0.029 $\pm$ 0.003 -0.007 $\pm$ 0.030 $\pm$ 0.006 -0.023 $\pm$ 0.036 $\pm$ 0.008 $r^{1}_{11}$ 0.009 $\pm$ 0.037 $\pm$ 0.005 0.023 $\pm$ 0.040 $\pm$ 0.006 0.033 $\pm$ 0.047 $\pm$ 0.029 $r^{3}_{1-1}$ -0.016 $\pm$ 0.111 $\pm$ 0.006 0.160 $\pm$ 0.134 $\pm$ 0.036 -0.154 $\pm$ 0.156 $\pm$ 0.054 element $\langle$Q$^{2}$$\rangle$ = 1.28 GeV$^{2}$ $\langle$Q$^{2}$$\rangle$ = 2.00 GeV$^{2}$ $\langle$Q$^{2}$$\rangle$ = 4.00 GeV$^{2}$ ------------------ -------------------------------------------- -------------------------------------------- -------------------------------------------- $r^{04}_{00}$ 0.148 $\pm$ 0.043 $\pm$ 0.025 0.132 $\pm$ 0.041 $\pm$ 0.053 0.186 $\pm$ 0.040 $\pm$ 0.034 $r^{1}_{1-1}$ -0.045 $\pm$ 0.063 $\pm$ 0.030 -0.347 $\pm$ 0.058 $\pm$ 0.075 -0.258 $\pm$ 0.072 $\pm$ 0.070 Im $r^{2}_{1-1}$ 0.232 $\pm$ 0.063 $\pm$ 0.045 0.216 $\pm$ 0.065 $\pm$ 0.063 0.313 $\pm$ 0.073 $\pm$ 0.056 Re $r^{5}_{10}$ 0.059 $\pm$ 0.020 $\pm$ 0.021 0.056 $\pm$ 0.017 $\pm$ 0.015 0.025 $\pm$ 0.020 $\pm$ 0.014 Im $r^{6}_{10}$ -0.034 $\pm$ 0.018 $\pm$ 0.006 -0.039 $\pm$ 0.016 $\pm$ 0.009 -0.055 $\pm$ 0.021 $\pm$ 0.015 Im $r^{7}_{10}$ -0.174 $\pm$ 0.160 $\pm$ 0.032 0.225 $\pm$ 0.150 $\pm$ 0.044 -0.068 $\pm$ 0.156 $\pm$ 0.015 Re $r^{8}_{10}$ -0.026 $\pm$ 0.154 $\pm$ 0.005 -0.197 $\pm$ 0.148 $\pm$ 0.039 0.020 $\pm$ 0.140 $\pm$ 0.004 Re $r^{04}_{10}$ -0.004 $\pm$ 0.027 $\pm$ 0.007 0.020 $\pm$ 0.024 $\pm$ 0.011 0.040 $\pm$ 0.025 $\pm$ 0.012 Re $r^{1}_{10}$ -0.039 $\pm$ 0.037 $\pm$ 0.019 0.052 $\pm$ 0.037 $\pm$ 0.015 0.025 $\pm$ 0.046 $\pm$ 0.008 Im $r^{2}_{10}$ 0.014 $\pm$ 0.037 $\pm$ 0.013 0.003 $\pm$ 0.036 $\pm$ 0.012 -0.028 $\pm$ 0.049 $\pm$ 0.004 $r^{5}_{00}$ 0.074 $\pm$ 0.033 $\pm$ 0.007 0.050 $\pm$ 0.032 $\pm$ 0.012 -0.006 $\pm$ 0.035 $\pm$ 0.031 $r^{1}_{00}$ 0.079 $\pm$ 0.061 $\pm$ 0.028 0.077 $\pm$ 0.059 $\pm$ 0.012 0.143 $\pm$ 0.073 $\pm$ 0.048 Im $r^{3}_{10}$ 0.124 $\pm$ 0.107 $\pm$ 0.031 0.009 $\pm$ 0.095 $\pm$ 0.002 0.016 $\pm$ 0.096 $\pm$ 0.004 $r^{8}_{00}$ 0.186 $\pm$ 0.248 $\pm$ 0.041 -0.024 $\pm$ 0.242 $\pm$ 0.005 -0.088 $\pm$ 0.211 $\pm$ 0.019 $r^{5}_{11}$ -0.027 $\pm$ 0.026 $\pm$ 0.013 -0.054 $\pm$ 0.025 $\pm$ 0.018 -0.001 $\pm$ 0.030 $\pm$ 0.011 $r^{5}_{1-1}$ -0.040 $\pm$ 0.031 $\pm$ 0.005 -0.049 $\pm$ 0.031 $\pm$ 0.010 0.021 $\pm$ 0.036 $\pm$ 0.009 Im $r^{6}_{1-1}$ 0.062 $\pm$ 0.031 $\pm$ 0.016 0.050 $\pm$ 0.032 $\pm$ 0.004 0.057 $\pm$ 0.035 $\pm$ 0.021 Im $r^{7}_{1-1}$ 0.399 $\pm$ 0.250 $\pm$ 0.079 -0.053 $\pm$ 0.236 $\pm$ 0.011 -0.003 $\pm$ 0.234 $\pm$ 0.001 $r^{8}_{11}$ -0.332 $\pm$ 0.193 $\pm$ 0.059 -0.103 $\pm$ 0.184 $\pm$ 0.020 -0.022 $\pm$ 0.164 $\pm$ 0.005 Im $r^{8}_{1-1}$ -0.260 $\pm$ 0.234 $\pm$ 0.075 -0.051 $\pm$ 0.216 $\pm$ 0.033 -0.129 $\pm$ 0.200 $\pm$ 0.029 $r^{04}_{1-1}$ 0.043 $\pm$ 0.040 $\pm$ 0.013 0.005 $\pm$ 0.039 $\pm$ 0.008 0.150 $\pm$ 0.040 $\pm$ 0.040 $r^{1}_{11}$ 0.009 $\pm$ 0.048 $\pm$ 0.003 -0.027 $\pm$ 0.051 $\pm$ 0.011 -0.104 $\pm$ 0.060 $\pm$ 0.012 $r^{3}_{1-1}$ -0.006 $\pm$ 0.174 $\pm$ 0.001 -0.337 $\pm$ 0.157 $\pm$ 0.071 0.021 $\pm$ 0.141 $\pm$ 0.005 element $\langle-t'\rangle$ = 0.021 GeV$^{2}$ $\langle-t'\rangle$ = 0.071 GeV$^{2}$ $\langle-t'\rangle$ = 0.147 GeV$^{2}$ ------------------ --------------------------------------- --------------------------------------- --------------------------------------- $r^{04}_{00}$ 0.153 $\pm$ 0.034 $\pm$ 0.031 0.147 $\pm$ 0.041 $\pm$ 0.036 0.215 $\pm$ 0.050 $\pm$ 0.028 $r^{1}_{1-1}$ -0.167 $\pm$ 0.054 $\pm$ 0.029 -0.298 $\pm$ 0.063 $\pm$ 0.074 -0.238 $\pm$ 0.074 $\pm$ 0.083 Im $r^{2}_{1-1}$ 0.281 $\pm$ 0.056 $\pm$ 0.044 0.198 $\pm$ 0.064 $\pm$ 0.036 0.309 $\pm$ 0.070 $\pm$ 0.067 Re $r^{5}_{10}$ 0.030 $\pm$ 0.015 $\pm$ 0.010 0.043 $\pm$ 0.018 $\pm$ 0.012 0.070 $\pm$ 0.024 $\pm$ 0.024 Im $r^{6}_{10}$ -0.050 $\pm$ 0.016 $\pm$ 0.008 -0.045 $\pm$ 0.017 $\pm$ 0.010 -0.030 $\pm$ 0.022 $\pm$ 0.011 Im $r^{7}_{10}$ -0.067 $\pm$ 0.130 $\pm$ 0.010 0.041 $\pm$ 0.150 $\pm$ 0.008 0.201 $\pm$ 0.179 $\pm$ 0.055 Re $r^{8}_{10}$ 0.062 $\pm$ 0.136 $\pm$ 0.015 -0.406 $\pm$ 0.153 $\pm$ 0.078 -0.011 $\pm$ 0.143 $\pm$ 0.003 Re $r^{04}_{10}$ 0.032 $\pm$ 0.022 $\pm$ 0.004 -0.020 $\pm$ 0.025 $\pm$ 0.006 0.050 $\pm$ 0.030 $\pm$ 0.014 Re $r^{1}_{10}$ 0.028 $\pm$ 0.035 $\pm$ 0.002 0.007 $\pm$ 0.038 $\pm$ 0.009 0.001 $\pm$ 0.045 $\pm$ 0.001 Im $r^{2}_{10}$ -0.060 $\pm$ 0.034 $\pm$ 0.012 0.082 $\pm$ 0.038 $\pm$ 0.022 -0.020 $\pm$ 0.048 $\pm$ 0.016 $r^{5}_{00}$ 0.007 $\pm$ 0.027 $\pm$ 0.021 0.036 $\pm$ 0.033 $\pm$ 0.018 0.089 $\pm$ 0.043 $\pm$ 0.012 $r^{1}_{00}$ 0.092 $\pm$ 0.057 $\pm$ 0.043 0.117 $\pm$ 0.055 $\pm$ 0.039 0.145 $\pm$ 0.080 $\pm$ 0.005 Im $r^{3}_{10}$ -0.009 $\pm$ 0.081 $\pm$ 0.001 0.160 $\pm$ 0.099 $\pm$ 0.033 0.059 $\pm$ 0.119 $\pm$ 0.016 $r^{8}_{00}$ 0.029 $\pm$ 0.209 $\pm$ 0.004 -0.302 $\pm$ 0.223 $\pm$ 0.063 0.211 $\pm$ 0.256 $\pm$ 0.058 $r^{5}_{11}$ -0.030 $\pm$ 0.022 $\pm$ 0.008 -0.032 $\pm$ 0.027 $\pm$ 0.011 -0.022 $\pm$ 0.032 $\pm$ 0.038 $r^{5}_{1-1}$ -0.029 $\pm$ 0.027 $\pm$ 0.000 -0.025 $\pm$ 0.032 $\pm$ 0.004 0.014 $\pm$ 0.042 $\pm$ 0.013 Im $r^{6}_{1-1}$ 0.077 $\pm$ 0.028 $\pm$ 0.022 0.063 $\pm$ 0.033 $\pm$ 0.014 0.008 $\pm$ 0.035 $\pm$ 0.009 Im $r^{7}_{1-1}$ -0.157 $\pm$ 0.208 $\pm$ 0.023 0.411 $\pm$ 0.238 $\pm$ 0.085 0.087 $\pm$ 0.267 $\pm$ 0.024 $r^{8}_{11}$ 0.005 $\pm$ 0.163 $\pm$ 0.001 0.018 $\pm$ 0.182 $\pm$ 0.007 -0.325 $\pm$ 0.186 $\pm$ 0.089 Im $r^{8}_{1-1}$ -0.165 $\pm$ 0.193 $\pm$ 0.024 -0.100 $\pm$ 0.228 $\pm$ 0.040 -0.172 $\pm$ 0.229 $\pm$ 0.047 $r^{04}_{1-1}$ 0.021 $\pm$ 0.034 $\pm$ 0.001 0.052 $\pm$ 0.041 $\pm$ 0.022 0.140 $\pm$ 0.048 $\pm$ 0.052 $r^{1}_{11}$ 0.009 $\pm$ 0.045 $\pm$ 0.005 -0.013 $\pm$ 0.053 $\pm$ 0.005 -0.145 $\pm$ 0.059 $\pm$ 0.038 $r^{3}_{1-1}$ 0.030 $\pm$ 0.132 $\pm$ 0.011 -0.083 $\pm$ 0.165 $\pm$ 0.029 -0.247 $\pm$ 0.177 $\pm$ 0.068 element proton deuteron ---------------------------------------------------------------- -------------------------------- -------------------------------- $u^{00}_{++} + \epsilon \cdot u^{00}_{00}$ 0.168 $\pm$ 0.018 $\pm$ 0.036 0.160 $\pm$ 0.024 $\pm$ 0.038 Re $u^{00}_{0+}$ -0.010 $\pm$ 0.012 $\pm$ 0.002 0.020 $\pm$ 0.014 $\pm$ 0.005 $u^{00}_{-+}$ -0.004 $\pm$ 0.018 $\pm$ 0.004 0.060 $\pm$ 0.023 $\pm$ 0.016 Re $(u^{0+}_{0+} - u^{-0}_{0+})$ 0.014 $\pm$ 0.024 $\pm$ 0.004 -0.037 $\pm$ 0.030 $\pm$ 0.007 Re $(u^{0+}_{++} - u^{-0}_{++} + 2\epsilon \cdot u^{0+}_{00})$ 0.006 $\pm$ 0.029 $\pm$ 0.008 0.107 $\pm$ 0.036 $\pm$ 0.023 Re $u^{0+}_{-+}$ -0.014 $\pm$ 0.019 $\pm$ 0.005 0.016 $\pm$ 0.022 $\pm$ 0.009 Re $(u^{0-}_{0+} - u^{+0}_{0+})$ -0.175 $\pm$ 0.029 $\pm$ 0.039 -0.215 $\pm$ 0.036 $\pm$ 0.047 Re $u^{0+}_{-+}$ 0.039 $\pm$ 0.018 $\pm$ 0.007 -0.003 $\pm$ 0.023 $\pm$ 0.002 $u^{-+}_{-+}$ 0.171 $\pm$ 0.029 $\pm$ 0.023 0.248 $\pm$ 0.037 $\pm$ 0.039 Re $(u^{++}_{0+} + u^{--}_{0+})$ -0.059 $\pm$ 0.012 $\pm$ 0.022 -0.025 $\pm$ 0.015 $\pm$ 0.015 Re $u^{-+}_{0+}$ 0.042 $\pm$ 0.015 $\pm$ 0.012 0.036 $\pm$ 0.019 $\pm$ 0.014 Re $(u^{-+}_{++} + \epsilon \cdot u^{-+}_{00})$ 0.037 $\pm$ 0.009 $\pm$ 0.012 0.045 $\pm$ 0.010 $\pm$ 0.014 Re $u^{++}_{-+}$ -0.043 $\pm$ 0.014 $\pm$ 0.006 -0.021 $\pm$ 0.018 $\pm$ 0.001 Re $u^{+-}_{0+}$ -0.061 $\pm$ 0.008 $\pm$ 0.012 -0.043 $\pm$ 0.010 $\pm$ 0.009 $u^{+-}_{-+}$ 0.036 $\pm$ 0.014 $\pm$ 0.008 0.056 $\pm$ 0.019 $\pm$ 0.013 Im $u^{00}_{0+}$ 0.059 $\pm$ 0.047 $\pm$ 0.012 0.038 $\pm$ 0.056 $\pm$ 0.008 Im $(u^{0+}_{0+} - u^{-0}_{0+})$ 0.023 $\pm$ 0.076 $\pm$ 0.010 -0.122 $\pm$ 0.089 $\pm$ 0.025 Im $(u^{0+}_{++} - u^{-0}_{++})$ 0.109 $\pm$ 0.075 $\pm$ 0.021 0.021 $\pm$ 0.087 $\pm$ 0.004 Im $(u^{0-}_{0+} - u^{+0}_{0+})$ -0.092 $\pm$ 0.117 $\pm$ 0.018 0.113 $\pm$ 0.135 $\pm$ 0.028 Im $(u^{++}_{0+} + u^{--}_{0+})$ -0.079 $\pm$ 0.089 $\pm$ 0.017 -0.097 $\pm$ 0.103 $\pm$ 0.020 Im $u^{-+}_{0+}$ -0.142 $\pm$ 0.110 $\pm$ 0.029 -0.017 $\pm$ 0.131 $\pm$ 0.004 Im $u^{-+}_{++}$ 0.169 $\pm$ 0.075 $\pm$ 0.035 -0.083 $\pm$ 0.083 $\pm$ 0.017 Im $u^{+-}_{0+}$ -0.060 $\pm$ 0.110 $\pm$ 0.012 -0.150 $\pm$ 0.125 $\pm$ 0.034 bin $\langle Q^{2} \rangle$ \[GeV$^2$\] $\langle-t' \rangle$ \[GeV$^2$\] $\langle W \rangle$ \[GeV\] $\langle x_{B} \rangle$ ----------------------------------- ------------------------------------- ---------------------------------- ----------------------------- ------------------------- “overall” 2.42 0.080 4.80 0.097 1.00 GeV$^2 <Q^{2}< 1.57$ GeV$^2$ 1.28 0.082 4.87 0.059 1.57 GeV$^2 <Q^{2}< 2.55$ GeV$^2$ 2.00 0.079 4.78 0.085 $Q^{2}>2.55$ GeV$^2$ 4.00 0.078 4.91 0.147 0.000 GeV$^2 <-t'< 0.044$ GeV$^2$ 2.38 0.021 4.73 0.097 0.044 GeV$^2 <-t'< 0.105$ GeV$^2$ 2.49 0.072 4.78 0.099 0.105 GeV$^2 <-t'< 0.200$ GeV$^2$ 2.39 0.147 4.85 0.095 SDME $r^{04}_{00}$ $r^{04}_{10}$ $r^{04}_{1-1}$ $r^1_{11}$ $r^1_{00}$ $r^1_{10}$ $r^1_{1-1}$ $r^2_{10}$ $r^2_{1-1}$ $r^5_{11}$ $r^5_{00}$ $r^5_{10}$ $r^5_{1-1}$ $r^6_{10}$ $r^6_{1-1}$ $r^3_{10}$ $r^3_{1-1}$ $r^7_{10}$ $r^7_{1-1}$ $r^8_{11}$ $r^8_{00}$ $r^8_{10}$ $r^8_{1-1}$ ------------------ --------------- --------------- ---------------- ------------ ------------ ------------ ------------- ------------ ------------- ------------ ------------ ------------ ------------- ------------ ------------- ------------ ------------- ------------ ------------- ------------ ------------ ------------ ------------- $r^{04}_{00}$ 1.00 Re $r^{04}_{10}$ 0.07 1.00 $r^{04}_{1-1}$ 0.03 -0.06 1.00 $r^1_{11}$ 0.02 -0.01 -0.37 1.00 $r^1_{00}$ -0.07 -0.08 0.06 -0.38 1.00 Re $r^1_{10}$ -0.04 0.17 0.02 0.00 0.08 1.00 $r^1_{1-1}$ 0.20 0.01 0.01 -0.02 -0.02 -0.03 1.00 Im $r^2_{10}$ 0.09 -0.20 0.08 0.01 0.00 -0.04 0.04 1.00 Im $r^2_{1-1}$ -0.23 -0.06 -0.03 -0.00 0.01 -0.01 -0.13 -0.05 1.00 $r^5_{11}$ -0.18 0.03 -0.08 -0.16 0.11 -0.07 0.00 0.07 0.06 1.00 $r^5_{00}$ 0.48 0.12 0.01 0.12 -0.37 -0.08 0.08 0.09 -0.10 -0.38 1.00 Re $r^5_{10}$ 0.11 0.34 -0.07 -0.04 -0.13 -0.24 0.10 -0.02 -0.13 0.03 0.15 1.00 $r^5_{1-1}$ 0.04 -0.07 0.18 0.01 0.03 0.07 -0.15 0.13 0.03 0.23 -0.03 -0.06 1.00 Im $r^6_{10}$ -0.14 -0.05 -0.12 -0.04 -0.14 -0.15 -0.14 -0.10 0.15 -0.01 -0.03 0.14 -0.09 1.00 Im $r^6_{1-1}$ 0.02 0.12 -0.03 0.15 -0.00 0.13 0.01 0.07 -0.10 -0.32 0.05 0.02 -0.07 0.01 1.00 Im $r^3_{10}$ 0.00 -0.01 0.04 -0.02 -0.01 -0.00 0.02 -0.05 0.00 0.04 -0.00 -0.01 0.03 -0.01 -0.06 1.00 Im $r^3_{1-1}$ 0.03 -0.01 0.03 -0.00 0.00 0.03 0.01 0.01 -0.08 -0.04 0.02 -0.02 0.01 -0.05 -0.01 -0.06 1.00 Im $r^7_{10}$ 0.00 -0.01 0.02 0.00 -0.01 0.00 0.01 -0.01 0.01 0.03 0.01 -0.01 0.03 0.03 -0.03 0.34 -0.08 1.00 Im $r^7_{1-1}$ 0.03 -0.03 0.01 0.01 -0.00 0.01 0.02 0.03 0.00 -0.02 0.02 -0.03 0.02 -0.02 0.05 -0.06 0.32 -0.07 1.00 $r^8_{11}$ 0.02 0.03 -0.01 -0.02 0.03 0.01 -0.04 -0.04 0.05 0.04 -0.02 0.01 0.01 0.01 -0.01 -0.07 0.07 -0.02 -0.26 1.00 $r^8_{00}$ -0.07 0.04 0.00 0.03 -0.09 0.03 -0.01 0.01 -0.01 -0.03 0.06 0.01 -0.01 0.00 0.01 -0.11 0.03 -0.08 0.05 -0.38 1.00 Re $r^8_{10}$ 0.02 -0.02 -0.01 0.02 0.03 -0.02 0.01 -0.00 0.01 -0.01 -0.00 0.01 -0.00 -0.01 0.01 -0.10 0.12 0.13 0.04 0.00 -0.01 1.00 $r^8_{1-1}$ 0.00 -0.01 -0.03 -0.02 0.00 -0.00 -0.00 -0.02 -0.02 -0.00 0.00 0.00 0.01 0.03 0.02 -0.08 -0.01 -0.07 0.11 -0.26 0.03 -0.03 1.00 SDME $r^{04}_{00}$ $r^{04}_{10}$ $r^{04}_{1-1}$ $r^1_{11}$ $r^1_{00}$ $r^1_{10}$ $r^1_{1-1}$ $r^2_{10}$ $r^2_{1-1}$ $r^5_{11}$ $r^5_{00}$ $r^5_{10}$ $r^5_{1-1}$ $r^6_{10}$ $r^6_{1-1}$ $r^3_{10}$ $r^3_{1-1}$ $r^7_{10}$ $r^7_{1-1}$ $r^8_{11}$ $r^8_{00}$ $r^8_{10}$ $r^8_{1-1}$ \[tab8\] SDME $r^{04}_{00}$ $r^{04}_{10}$ $r^{04}_{1-1}$ $r^1_{11}$ $r^1_{00}$ $r^1_{10}$ $r^1_{1-1}$ $r^2_{10}$ $r^2_{1-1}$ $r^5_{11}$ $r^5_{00}$ $r^5_{10}$ $r^5_{1-1}$ $r^6_{10}$ $r^6_{1-1}$ $r^3_{10}$ $r^3_{1-1}$ $r^7_{10}$ $r^7_{1-1}$ $r^8_{11}$ $r^8_{00}$ $r^8_{10}$ $r^8_{1-1}$ ------------------ --------------- --------------- ---------------- ------------ ------------ ------------ ------------- ------------ ------------- ------------ ------------ ------------ ------------- ------------ ------------- ------------ ------------- ------------ ------------- ------------ ------------ ------------ ------------- $r^{04}_{00}$ 1.00 Re $r^{04}_{10}$ 0.11 1.00 $r^{04}_{1-1}$ -0.03 -0.09 1.00 $r^1_{11}$ -0.00 -0.01 -0.38 1.00 $r^1_{00}$ 0.07 -0.06 0.07 -0.40 1.00 Re $r^1_{10}$ -0.02 0.22 0.09 0.00 0.12 1.00 $r^1_{1-1}$ 0.21 0.09 -0.03 0.02 0.01 -0.07 1.00 Im $r^2_{10}$ 0.01 -0.23 0.06 0.04 0.01 -0.04 0.02 1.00 Im $r^2_{1-1}$ -0.20 -0.06 -0.04 0.01 -0.04 -0.05 -0.11 -0.05 1.00 $r^5_{11}$ -0.21 0.03 -0.02 -0.20 0.13 -0.09 0.02 0.05 -0.05 1.00 $r^5_{00}$ 0.51 0.13 -0.03 0.15 -0.40 -0.09 0.08 0.10 -0.07 -0.40 1.00 Re $r^5_{10}$ 0.13 0.38 -0.14 -0.07 -0.11 -0.23 0.17 -0.00 -0.12 0.05 0.16 1.00 $r^5_{1-1}$ -0.01 -0.13 0.26 -0.02 0.04 0.10 -0.16 0.11 -0.02 0.32 -0.06 -0.15 1.00 Im $r^6_{10}$ -0.09 -0.02 -0.09 -0.07 -0.06 -0.12 -0.09 -0.24 0.14 -0.01 -0.00 0.16 -0.08 1.00 Im $r^6_{1-1}$ -0.01 0.08 -0.02 0.20 -0.03 0.13 0.01 0.09 -0.27 -0.32 0.04 -0.02 -0.08 -0.08 1.00 Im $r^3_{10}$ 0.02 0.02 -0.00 0.01 0.04 0.05 0.02 0.04 0.02 -0.02 0.01 0.00 -0.00 -0.05 0.03 1.00 Im $r^3_{1-1}$ -0.01 -0.00 0.01 -0.01 -0.02 -0.03 0.01 0.03 0.04 0.03 -0.00 0.02 -0.00 0.02 -0.05 -0.10 1.00 Im $r^7_{10}$ 0.03 0.02 -0.00 0.03 -0.03 -0.03 0.03 -0.00 0.01 0.01 -0.04 -0.03 0.00 -0.03 -0.02 0.44 -0.09 1.00 Im $r^7_{1-1}$ -0.02 0.01 0.02 -0.02 0.01 0.01 -0.01 0.01 0.00 -0.03 0.02 0.03 -0.01 -0.01 -0.02 -0.08 0.43 -0.05 1.00 $r^8_{11}$ 0.02 -0.03 -0.03 -0.03 0.00 -0.03 0.02 0.01 -0.01 0.04 -0.02 0.00 0.00 0.01 -0.03 -0.05 0.14 -0.05 -0.24 1.00 $r^8_{00}$ -0.03 0.02 -0.01 0.01 -0.00 -0.02 -0.01 -0.05 0.02 -0.01 -0.00 -0.02 -0.00 0.02 -0.02 -0.06 -0.02 0.03 0.02 -0.40 1.00 Re $r^8_{10}$ 0.02 0.03 0.01 -0.01 -0.03 -0.01 -0.00 -0.02 0.01 0.02 -0.01 -0.04 0.04 0.03 -0.02 -0.08 0.08 0.17 0.00 0.01 0.06 1.00 $r^8_{1-1}$ 0.00 0.02 0.00 0.01 -0.01 0.00 -0.05 -0.02 0.03 -0.01 0.02 0.03 0.00 0.02 0.03 -0.04 -0.03 -0.05 0.11 -0.21 0.03 -0.00 1.00 SDME $r^{04}_{00}$ $r^{04}_{10}$ $r^{04}_{1-1}$ $r^1_{11}$ $r^1_{00}$ $r^1_{10}$ $r^1_{1-1}$ $r^2_{10}$ $r^2_{1-1}$ $r^5_{11}$ $r^5_{00}$ $r^5_{10}$ $r^5_{1-1}$ $r^6_{10}$ $r^6_{1-1}$ $r^3_{10}$ $r^3_{1-1}$ $r^7_{10}$ $r^7_{1-1}$ $r^8_{11}$ $r^8_{00}$ $r^8_{10}$ $r^8_{1-1}$ \[tab9\]

--- abstract: 'Since long back, scientists have been putting enormous effort to understand earthquake dynamics -the goal is to develop a successful prediction scheme which can provide reliable alarm that an earthquake is imminent. Model studies sometimes help to understand in some extend the basic dynamics of the real systems and therefore is an important part of earthquake research. In this report, we review several physics models which capture some essential features of earthquake phenomenon and also suggest methods to predict catastrophic events being within the range of model parameters.' address: 'Department of Physics, Norwegian University of Science and Technology, N–7491 Trondheim, Norway' author: - Srutarshi Pradhan title: '[<span style="font-variant:small-caps;">Physics Models of Earthquake</span>]{}' --- ***Introduction*** .1in ‘Earthquake’ is a long standing problem to the scientist community as it is a subject of many unknowns. Strong earthquakes often causes huge devastation in terms of lives and properties and therefore this subject always demands high priority. But so far, little advancement has been made to understand the entire earthquake-dynamics and the challenge still remains -to develop a successful prediction scheme which can provide exact space-time information of future earthquakes and their expected magnitudes. However, several models and hypothesis [@book-1; @book-2; @book-3; @book-4; @GR54; @PB06] have been proposed and some recent studies suggest potential methods to predict catastrophic events in some earthquake models. In this short report we discuss such physics models of earthquake, giving importance to their capability of prediction. .1in ***Geological facts*** .1in Plate-tectonic theory explains the origin of earthquake through a stick-slip dynamics: Earth’s solid outer crust (about 20 km thick) rests on a tectonic shell which is divided into numbers (about 12) of mobile plates, having relative velocities of the order of few centimeters per year. This motion of the plates arises due to the powerful convective flow of the earth’s mantle at the inner core of earth. On the other hand solid-solid frictional force arises at the crust-plate boundary and it sticks them together. This kind of sticking develops elastic strains and the strain energy gradually increases because of the uniform motion of the tectonic plates. Therefore a competition comes to play between the sticking frictional force and the restoring elastic force (stress). When the accumulated stress exceeds the frictional force, a slip (earthquake) occurs and it releases the stored elastic energy in the form of sound, heat and mechanical vibrations. It has been observed that generally a series of small earthquakes appear before (foreshocks) and after (aftershocks) a big quake (main shock). {width="3in" height="2in"} [Figure.]{} **1**[:]{} ****[The magnitude distribution of earthquakes in Japan (from JUNEC catalog) [@Kawamura-06]. The straight line with slope $-0.9$ is the best fit to the data points for $m>3$ and supports Gutenberg-Richter law. ]{} .2in ***Physics models*** .1in The overall frequency distribution of earthquakes including foreshocks, mainshocks and aftershocks, seems to follow the empirical Gutenberg-Richter law [@GR54]: $$\ln N(m)=constant-bm,$$ where $N(m)$ is the number of earthquakes having magnitude (in Richter scale) greater than or equal to $m$, $b$ is the power exponent. The observed value of $b$ ranges between $0.7$ and $1.0$ (Fig.1). The amount of energy $\epsilon$ released in an earthquake is related to the magnitude as $$\ln\epsilon=constant+am.$$ Therefore Gutenberg-Richter law can be expressed as an alternative form: $$N(\epsilon)=\epsilon^{-\alpha},$$ where $N(\epsilon)$ is the number of earthquakes releasing energy greater than or equal to $\epsilon$ and $\alpha=b/a$. The Gutenberg-Richter law is being considered as one of the most fundamental observations by the physicists. Several models have been proposed to study the nature of the earthquake phenomenon. The main intension is to capture the Gutenberg-Richter type power law for the frequency distribution of failures (quakes) by modelling different aspects of faults –structure, material properties, geometry etc. These models can be classified into four groups according to their basic assumptions: (A) Friction models, incorporate the stick-slip dynamics through the collective motion of an assembly of locally connected elements subject to slow driving force (B) Fracture models, look at earthquake phenomena as a fracture-failure process of deformable materials that break under external loading through slow build-up of stress (C) Self-organised critical (SOC) models, consider earthquake as a self-driven slow process and (D) Fractal models, give importance to fractal nature of the crust-plate interfaces and address the phenomenon as a two-fractal overlap problem. .1in **(A)** ***Friction models*** .1in In 1967 Burridge and Knopoff [@BK67] introduced model studies in earthquake research. They proposed a spring-block model to mimic the typical stick-slip dynamics of earthquake phenomena, which has been extended later by Carlson and Langer [@CL89]. The Burridge-Knopoff type model contains a linear array of blocks of mass $m$ coupled to each other by identical harmonic springs of strength $k_{c}$ and also attached to a fixed surface at the top by a different set of identical springs having strength $k_{p}$. The blocks are kept on a horizontal platform (rough surface) which moves with a uniform velocity $V$ (Fig.2). Here qualitatively the blocks can be thought of as the points of contact between two plates moving at a relative speed $V$, where the spring constants $k_{c}$ and $k_{p}$ represents the linear elastic response of the contact region to compression and shear respectively. [Figure]{} **2**[:]{} ****[The Burridge-Knopoff model (above) and two different forms of velocity weakening friction force (below) [@rumi].]{} .1in Starting from unstrained condition, when the system is pulled slowly (at constant rate), the blocks initially remain stuck to the surface due to friction between surfaces. A slip of a block occurs when the corresponding spring force overcomes the threshold value (maximum static frictional force between that block and the rough surface). The dynamics of each block is basically the resultant of two phases: a static phase and a dynamic phase. During static phase the block remains stuck on the rough surface and the elastic strain continuously grows up. The static phase comes to a sudden end when driving force on that block attains the threshold value and dynamic phase begins. Due to the presence of frictional force this dynamic phase is definitely dissipative in nature. This dissipation reduces the relative velocity of the block and the system again goes back to the static phase. The dynamic friction is assumed as a velocity weakening function $f$ (shown in the Fig.2 ). Presence of spring force and velocity weakening friction force leads the system to a complex dynamical state. At the initial stage, individual small slip occurs. But due to constant pulling, the springs are gradually stretched and attain the limit of total frictional stability of the blocks where the collective slips of almost all the blocks occur. Clearly this is a big event and a large amount of elastic energy is released here. The equation of motion of the $j^{th}$ block of the system is $$m\ddot{x}_{j}=k_{c}(x_{j+1}-2x_{j}+x_{j-1})-k_{p}x_{j}-f(\dot{x}_{j}-V)$$ where dots denote differentiation with respect to $t$, $m$ is the mass of a block, $k_{p}$ and $k_{c}$ are spring constants of the connecting springs and $f$ represents the nonlinear velocity weakening friction force. The position coordinate $x$ is measured along the chain length and the velocity $V$ of the platform is in the increasing $x$ direction. The total energy of the system at time $t$ can be calculated by solving the above equation. A sudden drops in the energy of the system is identified as the released energy during a slip event and the distribution of such energies follows power-law: $$N(\epsilon)\sim\epsilon^{-c},c\sim1,$$ where $N(\epsilon)$ is the number of slips releasing energy greater than or equal to $\epsilon$. .1in *Prediction possibility of major events (slips)* Recently Dey et al. [@rumi] developed a method to predict the major slip event in Burridge-Knopoff type spring-block model. Introducing an additional dissipative force in the spring-block arrangement they identified the dissipative functional $R(t)$ as energy bursts similar to the acoustic emission signals [@AE-1; @AE-2] observed in experiments. The distribution of $R(t)$ shows power laws if one records all slip events including the major slips. {width="2.5in" height="1.5in"} [Figure]{} **3**[:]{} ****[The dissipative functional $R(t)$ vs. time ($t$) in Burridge-Knopoff model [@rumi].]{} .1in The plot of $R(t)$ vs. $t$ shows a gradual increase in activity (Fig.3) prior to the occurrence of a major slip. But as $R(t)$ is noisy it can not help much to predict the major slip event, rather the cumulative energy dissipated $E_{ae}(t)\sim\int_{0}^{t}R(t^{'})dt^{'}$ grows in steps and it seems to diverge as a major slip event is approached (Fig.4). From such divergence one can predict the occurrence time ($t_{c}$) of a major slip through proper extrapolation. {width="2.5in" height="1.8in"} [Figure]{} **4**[:]{} ****[Cumulative energy $E_{ae}$ versus time ($t$) plot. Inset shows the time series of $R(t)$ including two major slip events [@rumi]. ]{} .1in **(B)** ***Fracture models*** .1in Earthquake can be considered as a fracture-failure phenomenon through slow build up of stress at the plate-crust boundary with the movement of the plate as external driving force. The deformation properties of the materials sitting at the boundary play crucial role on spatial redistribution of the stress around a broken region and this actually decides in which direction the crack front should propagate. Fiber bundle model and random fuse model address such scenario in material breakdown and sometimes they are treated as models of earthquake since they produce time series of avalanches which follow power law distribution similar to Gutenberg-Richter law. {#section .unnumbered} Fiber bundle (RFB) model consists of many ($N_{0}$) fibers connected in parallel to each other and clamped at their two ends and having randomly distributed strengths. The model exhibits a typical relaxational dynamics when external load is applied uniformly at the bottom end (Fig.5). In the global load-sharing approximation [@FT; @Dan; @HH-92], surviving fibers share equally the external load. Initially, after the load $F$ is applied on the bundle, fibers having strength less than the applied stress $\sigma=F/N_{0}$ fail immediately. After this, the total load on the bundle redistributes globally as the stress is transferred from broken fibers to the remaining unbroken ones. This redistribution causes secondary failures which in general causes further failures and produces an “avalanche” which denotes simultaneous failure of several elements. With steady increase of external load, avalanches of different size appear before the global breakdown where the bundle collapses. The scaling properties of such mean-field dynamics and the avalanche statistics are expected to be extremely useful in analysing fracture and breakdown in real materials, including earthquakes [@D92; @More94; @KHH-97; @JV-97]. [Figure]{} **5**[:]{} ****[The fiber bundle model . ]{} .1in When external load is applied, the surviving fraction of total fibers follows a simple recursion relation $$U_{t+1}(\sigma)=1-P(\sigma/U_{t}),$$ where $U_{t}=N_{t}/N_{0}$, $\sigma$ is the external stress and $P$ is cumulative probability function. The recursion relation has the form of an iterative map $U_{t+1}=Y(U_{t})$ and finally the dynamics stops at a fixed point where $U_{t+1}=U_{t}$. If external load is increased in steps by equal amount $\Delta F$, then the entire failure process can be formulated through the recursive dynamics [@SPB-02] mentioned above and the fixed point solution gives the value of critical stress $\sigma_{c}$ above which the bundle collapses. It can be shown that the bundle undergoes a phase transition from partially broken state to completely broken state. The order parameter ($O$), susceptibility ($\chi$) and relaxation time ($\tau$) follow robust power laws with universal exponent values [@PSB-03]. In case of quasi-static load increment, only the weakest fiber (among the intact fibers) fails after loading and then the bundle undergoes load redistribution till a fixed point is reached. The fluctuations in strength distributions produces different size of avalanches during the entire failure process and the avalanche distribution follows power law (Fig.6) with exponent $-5/2$. Hemmer and Hansen [@HH-92] has analytical proved that this exponent is universal under mild restriction on strength distributions. [Figure]{} **6**[:]{} ****[Avalanche time series in a fiber bundle model (left) and the corresponding avalanche distribution (right). ]{} .2in {#section-1 .unnumbered} [Figure]{} **7**[:]{} ****[The random fuse model. ]{} .1in The fuse model **[@fuse-94; @PS97; @fuse-05] consists of a lattice in which each bond is a fuse, i.e., an ohmic resistor as long as the electric current it carries is below a threshold value. If the threshold is passed, the fuse burns out irreversibly. The threshold $t$ of each bond is drawn from an uncorrelated distribution $p(t)$. The lattice is placed at $45{}^{\circ}$ with regards to the electrical bus bars (Fig.7) and an increasing current is passed through it. Numerically, the Kirchhoff equations are solved at each node. When a bond breaks, current value on the neighboring bonds increases and sometimes it triggers secondary failures. Finally the system comes to a state where no current passes through the lattice; that means there is a crack which separates the lattice in two pieces. With gradual increase of current/voltage a series of intermediate avalanches appear before the final breakdown. The distribution of such avalanches follows power law with exponent close to $3$ (Fig.8). [Figure]{} **8**[:]{} ****[Avalanche time series in a fuse model (left) and the corresponding avalanche distribution (right). ]{} .1in ### {#section-2 .unnumbered} A robust crossover behavior has been observed [@PH-04; @PHH-05; @PHH-06] recently in the two very different models described above where the system gradually approaches the global failure through several intermediate failure events. If intermediate avalanches are recorded, avalanche distribution follows a power law with an exponent that crosses over from one value to a very different value when the system is close to the global failure or breakdown point (Fig.9). Therefore, this crossover is a signature of imminent breakdown. [Figure]{} **9**[:]{} ****[Crossover signature in avalanche power law. In fiber bundle model: $x_{0}$ is the starting position of recording avalanches and $x_{c}$ is the global failure point; exponent value changes from $5/2$ to $3/2$ (left) and in random fuse model: exponent value changes from $3$ to $2$ when the system comes closer to breakdown point (right). ]{} .1in Recently Kawamura [@Kawamura-06] observed similar crossover behavior for the local magnitude distribution of earthquakes in Japan (Fig.10). This observation has strengthened the possibility of using crossover signal as a tool of predicting catastrophic events. {width="2in" height="1.8in"} [Figure]{} **10**[:]{} ****[Crossover signature in the magnitude distribution of earthquakes within Japan [@Kawamura-06].]{} .2in **(C)** ***Self-organised critical models*** .1in In various thermodynamics systems, there is a “critical” point where the systems become totally correlated and show scale free (power law) behaviour. Apart from the critical point the average microscopic quantities of the systems follow scaling behaviour. Generally such critical states are achieved through the fine-tuning of physical parameters, such as temperature, pressure etc. and the power law behavior is considered to be a signature of the “critical” state of the system. However, it has been observed that some complex systems evolve collectively to such critical state only through mutual interactions and show power law behavior there. These systems do not need any fine tuning of physical parameter and therefore are considered as “self-organised critical” (SOC) systems. The term SOC was first introduced by Bak, Tang and Wiesenfeld in 1987. The magnitude distribution of earthquake shows power-law (Gutenberg-Richter law), therefore it is tempting to assume that earthquake happens through a self-organised dynamics: the build up of stress due to tectonic motion is a self organised slow process; gradually the critical state is achieved where the stress releases in bursts of various sizes. Extensive research have been going on to establish relation between earthquake and SOC systems, for which several models have been proposed. So far, sandpile models are the best example of SOC system. {#section-3 .unnumbered} The first attempt to study SOC through model systems was made by Bak, Tang and Wiesenfeld [@BTW]. This is a model of sandpile whose natural dynamics drives it towards the critical state. The model can be described on a two dimensional square lattice. At each lattice site $(i,j)$, there is an integer variable $h_{i,j}$ which represents the height of the sand column at that site. A unit of height (one sand grain) is added at a randomly chosen site at each time step and the system evolves in discrete time. The dynamics starts as soon as any site $(i,j)$ has got a height equal to the threshold value ($h_{th}$= $4$): that site topples, i.e., $h_{i,j}$ becomes zero there, and the heights of the four neighbouring sites increase by one unit $$h_{i,j}\rightarrow h_{i,j}-4,h_{i\pm1,j}\rightarrow h_{i\pm1,j}+1,h_{i,j\pm1}\rightarrow h_{i,j\pm1}+1.$$ [Figure]{} **11**[:]{} ****[BTW model on square lattice [@manna].]{} .1in The process continues till all sites become stable (Fig.11). In case of toppling at the boundary of the lattice ($4$ nearest neighbours are not available), grains falling outside the lattice are removed and considered to be absorbed/collected at the boundary. Gradually the average height $h_{av}$ attains a critical value $h_{c}$, beyond which it does not increase at all - on an average the additional sand grains are flown away from the lattice . Total number of toppling between two successive stable states, determines the size of an avalanche and at the critical state avalanche size distribution follows power laws: $n_{s}\sim s^{-\Gamma}$, where $n_{s}$ denotes the density of $s$ size avalanches. The exponent $\Gamma$ has the value $\Gamma\simeq1.15\pm0.10$ in 2D [@manna]. {#section-4 .unnumbered} Manna proposed the stochastic sand-pile model [@manna] by introducing randomness in the dynamics of sand-pile growth. Here, the critical height is $2$. Therefore at each toppling, the two rejected grains choose their host among the four available neighbours randomly with equal probability. After constant adding of sand grains, the system ultimately settles at a critical state having height $h_{c}$ and exhibits scale free behavior in terms of avalanche and life time distributions. But the power law exponents are different compared to those in BTW model and therefore Manna model belongs to a different universality class [@dhar]. {width="1.7in" height="1.8in"}{width="1.7in" height="1.8in"} [Figure]{} **12**[:]{} ****[Avalanche time series in a BTW model (left) and in a Manna model (right). ]{} {#section-5 .unnumbered} On constant adding of grains, sandpile models gradually attains critical state from sub-critical states. Now the question is: can one predict the critical state from the sub-critical response of the pile? A pulse perturbation method [@Acharyya; @SB01] gives the answer: At an average height $h_{av}$, a fixed number of height units $h_{p}$ (pulse of sand grains) is added at any central point of the system. Just after this addition, the local dynamics starts and it takes a finite time or iterations to return back to the stable state after several toppling events. which is the distance of the furthest toppled site from the site where $h_{p}$ has been dropped. .1in [Figure]{} **13**[:]{} ****[Sub-critical response: Correlation length ($\xi$) versus average hight ($h_{av}$) in BTW model (left) and in Manna model (right). Inset shows the plot of inverse $\xi$ versus $h_{av}$ and predicts the critical hight through extrapolation.]{} .1in Therefore, although BTW and Manna models belong to different universality classes with respect to their properties at the critical state, both the models show similar sub-critical response or precursors. A proper extrapolation method can estimate the respective critical heights of the models quite accurately. .1in **(D)** ***Fractal models*** .1in The surfaces of earth’s crust and tectonic plate at the fault zone are not compact rather fractal in nature. In fact, these surfaces are the results of the large scale fracture separating the crust from the moving tectonic plate. It has been observed that these surfaces are self similar fractals [@BS..] having the self-affine scaling property $h(\lambda x,\lambda y)\sim\lambda^{\zeta}h(x,y)$, where $h(x)$ denotes the height of the crack surface at the point $x$ and $\zeta$ is the roughness exponent. It has been claimed recently that since the fractured surfaces have got well-characterized self-affine properties, the distribution of the elastic energies released during the slips (earthquake events) between two rough surfaces (crust and plate) may follow the overlap distribution of two fractal surfaces [@V96; @CS99] {width="3in" height="1.5in"} [Figure]{} **14**[: A typical self-affine fracture surface]{} ****[in ($2+1$) dimension with roughness exponent $\zeta=0.8$.]{} {#section-6 .unnumbered} V. De Rubies et. al. [@V96] has proposed a new model for earthquakes where the scale invariance of the Gutenberg-Richter law is claimed to come from the fractal geometry of the fault surfaces. In this model the sliding fault surfaces have been represented by fractional Brownian surfaces, whose height scales as $|h(x+r)-h(x)|\sim r^{\zeta}$. The roughness of the surfaces are determined from the value of the exponent $\zeta$ which lies between $0$ and $1$. Two surfaces are simulated by two statistically self-affine profiles, say, $h_{1}(x)$ and $h_{2}(x)$, one drifting over other with a constant speed $v$ such that $h_{1}(x,t)=h_{2}(x-vt)$. An interaction between two profiles represents a single seismic event and the energy released is assumed to be proportional to the breaking area of the asperities. At the contact point of the surfaces the ‘surface roughness’ prevents slipping and the stress is accumulated there. When the stress exceeds a certain threshold value, breaking (earthquake) occurs. This model produces Gutenberg-Richter type power law: $P(E)\sim E^{-\beta-1}$, where $P(E)dE$ is the probability that an earthquake releases energy between $E$ and $E+dE$ and the exponent $\beta$ is directly related to the roughness exponent $\zeta$ as : $\beta=1-\zeta/(d-1)=(D_{f}-1)/(d-1)$, where $D_{f}$ and $d$ are respectively the fractal dimension and the embedding dimension of the surfaces. {#section-7 .unnumbered} This is an analytical model, proposed by Chakrabarti and Stinchcombe [@CS99], which incorporates the self-similar nature of both the crust and the tectonic plate. They used self-similar fractals to represent fault surfaces (Fig.15). [Figure]{} **15**[:]{} ****[Schematic representation of the rough surfaces of earth’s crust and moving tectonic plate [@CS99]. ]{} .1in The total contact area between the surfaces is assumed to be proportional to the elastic strain energy that can be grown during the sticking period, as the solid-solid friction force arises from the elastic strain at the contacts between the asperities. This energy is considered to be released as one surface slips over the other and sticks again to the next contact between the rough surfaces. Chakrabarti and Stichcombe have shown analytically through renormalization group calculations that for regular fractals (Cantor sets and carpets) the contact area follows power law distribution: $$\rho(s)\sim s^{-\gamma};\gamma=1,$$ which is comparable to that of Gutenberg-Richter law. .1in {#section-8 .unnumbered} The claim of Chakrabarti-Stichcombe model has been verified by extensive numerical simulations [@PCRD-03] taking different type of synthetic fractals: regular or non-random Cantor sets, random Cantor sets (in one dimension), regular and random gaskets on square lattice and percolating clusters embedded in two dimensions. {width="1.5in" height="1.5in"}.2in{width="1.5in" height="1.5in"} .1in [Figure]{} **16**[:]{} ****[(a) A regular Cantor set of dimension $\ln2/\ln3$; only three finite generations are shown. (b) The overlap of two identical (regular) Cantor sets, at $n=3$, when one slips over other; the overlap sets are indicated within the vertical lines, where periodic boundary condition has been used. ]{} .4in The contact area distributions $P(m,L)$ seem to follow a universal scaling: $$P(m,L)\sim L^{\alpha}P^{\prime}(m^{\prime});m^{\prime}=mL^{\alpha},$$ where $L$ denotes the size of the fractal and $\alpha=2(d-d_{f})$; $d_{f}$ being the mass dimension of the fractal and $d$ is the embedding dimension. Also the overlap distribution $P(m)$, and hence the scaled distribution $P^{\prime}(m^{\prime})$, decay with $m$ or $m^{\prime}$ following a power law (Fig.17) for both regular and random Cantor sets and gaskets: $$P(m)=m^{-\beta};\beta=d.$$ A very recent report [@pratip05] analytically explains the origin of such asymptotic power laws in case of Cantor set overlap. .1in [Figure]{} **17**[: The plot of $P^{\prime}(m^{\prime})$ against $m^{\prime}$ for Cantor sets with $d_{f}=\ln2/\ln3$ at finite generations: $n=10$ (square), $n=11$ (plus), $n=12$ (cross) and $n=13$ (star) and the dotted lines indicate the best fit curves of the form $a(x-b)^{-d}$; where $d=1$. Inset shows $P(m)$ vs. $m$ plots. ]{} {#section-9 .unnumbered} If one cantor set moves uniformly over other, the overlap between the two fractals change quasi-randomly with time and produces a time series of overlaps $m(t)$. Such a time series is shown in Fig.18, for Cantor sets of dimensions $\ln2/\ln3$. While most of the overlaps are small in magnitude, some are really big, where as the cumulative overlap size $Q(t)=\int_{o}^{t}mdt$ ‘on average’ grows linearly with time. Is it possible to predict a large future overlap analysing the time series data? A recent study [@PCC] suggests a method: {width="2.5in" height="2.2in"} [Figure]{} **18**[:]{} ****[The time ($t$) series data of overlap size ($m$) for regular Cantor sets: of dimension $\ln2/\ln3$, at $8$th generation. ]{} .1in One can identify the ‘large events’ occurring at time $t_{i}$ in the $m(t)$ series, where $m(t_{i})\geq M$, a pre-assigned number, then calculate the cumulative overlap size $Q(t)=\int_{t_{i}}^{t_{i+1}}mdt$, where the successive large events occur at times $t_{i}$ and $t_{i+1}$. Obviously $Q(t)$ is reset to $0$ value after every large event. The behavior of $Q_{i}$ with time is shown in Fig.19 for regular cantor sets . It appears that there are discrete values up to which $Q_{i}$ grows with time. .1in {width="1.5in" height="1.7in"}{width="1.5in" height="1.7in"} .1in [Figure]{} **19**[: The cumulative overlap size variation with time (for regular Cantor sets of dimension $\ln2/\ln3$, at $8$th generation), where the cumulative overlap has been reset to $0$ value after every big event (of overlap size $\geq M$ where $M=128$ and $32$ respectively).]{} .6in Therefore, if one fixes a magnitude $M$ of the overlap sizes $m$, so that overlaps with $m\geq M$ are called ‘events’ (or earthquake), then the cumulative overlap $Q_{i}$ grows linearly with time up to some discrete levels $Q_{i}\cong lQ_{0}$, where $Q_{0}$ is the minimal overlap size, dependent on $M$ and $l$ is an integer. This information certainly does not help to predict a future large event accurately, but it gives some hints by identifying discrete levels of $Q_{i}$ where a large overlap is likely to happen. .1in ***Discussions and concluding remarks*** .1in The Gutenberg-Richter law is a well established law in earthquake research. It should be mentioned that Gutenberg and Richter obtained this law from the statistics of earthquake events observed throughout the world. Also, earthquakes within a tectonically active region (Japan, California etc) follow similar power law. It is still a controversial issue whether the exponent of the Gutenberg-Richter power law is an universal constant or it varies in a narrow range ($0.8$ to $1.2$) depending upon the nature of the fault zone. As the motion of the tectonic plates is surely an observed fact, the stick-slip process should be a major ingredient of earthquake models. Although Burridge-Knopoff type spring-block model successfully captures the stick-slip dynamics and reproduces Gutenberg-Richter type power law in the size distribution of events, it is far from the accurate representation of earthquake dynamics -as it does not contain a mechanism for aftershocks which are observed facts. But this model has some important features: The dynamics is inherently chaotic -therefore technically unpredictable which agrees well with the occurrence of earthquakes. However, the method proposed by Dey et. al [@rumi] to predict a major slip event by monitoring the cumulative energy function, may open up a wide scope of future research in this field. On the other hand, fiber bundle model and fuse model have been developed basically to study breakdown phenomena in composite materials. As earthquake is a major breakdown phenomenon, these models can be used as earthquake models due to their inherent mean-field nature. Avalanche distributions show power laws in both the models and a crossover in exponent value appears [@PHH-05] near breakdown point, which can be treated as a criterion for imminent breakdown. SOC models assume a self-driven slow dynamics and reproduces Gutenberg-Richter law at the critical point. Although the critical state can be predicted accurately from the sub-critical response of the systems, the behavior remains unpredictable at the critical state. Fractal overlap models are different (from the other three types) modelling approaches in the sense that they focus on the fractal nature of the fault interfaces and not on the dynamics. The contact area distribution follows asymptotic power law and this suggests a possibility that fractal geometry of the faults might be the true origin of Gutenberg-Richter law. There are several difficulties in studying earthquake phenomenon, as it is a “N body” complex problem. While exact solution of a $3$-body problem needs rigorous mathematical calculations and it does not work for a mutually interacting system having more than $3$ bodies, naturally, theoretical physics can not help to formulate the entire earthquake dynamics. Again, as the dynamics of earthquake happens at a depth more than $20$ km from earth surface -experimental observation of such dynamics is almost impossible. Moreover, earthquake dynamics involves crust and plates which are highly heterogeneous at many scales from the atomic scale to the scale of tectonic plates, with the presence of dislocation, impurities, grains, water etc and the scenario becomes very much complicated. In this situation, model studies are very important in earthquake research in the sense that they can produce synthetic earthquake events, allow us to monitor the dynamics and analyse the event statistics to compare with the real earthquake data. Moreover, such studies suggest potential methods to predict a major event. It will be a real breakthrough if any of such methods can help a little to predict a future earthquake. .1in ***Acknowledgment:*** .1in We are grateful to Bikas. K. Chakrabarti for important collaborations and useful suggestions. Thanks to Research Council of Norway (NFR) for financial support through grant No. 166720/V30. [10]{} H. J. Herrmann and S. Roux (Eds), *Statistical Models for the Fracture of Disordered Media*, North Holland, Amsterdam (1990). B. K. Chakrabarti and L. G. Benguigui, *Statistical Physics of Fracture and Breakdown in Disorder Systems*, Oxford Univ. Press, Oxford (1997). D. Sornette, *Critical Phenomena in Natural Sciences*, Springer-Verlag, Berlin Heidelberg (2000). M. Sahimi, *Heterogeneous Materials II: Nonlinear and Breakdown Properties*, Springer-Verlag, Berlin (2003). B. Gutenberg and C. F. Richter, *Seismicity of the Earth and Associated phenomena*, Princeton University Press, Princeton, N.J. (1954). P. Bhattacharyya and B. K. Chakrabarti (Eds), *Modelling Critical and Catastrophic Phenomena in Geoscience*, Springer, Berlin (2006). R. Burridge, L. Knopoff, Bull. Seis. Soc. Am. **57** 341 (1967). J. M. Carlson, J. S. Langer, Phys. Rev. Lett. **62** (1989) 2632-2635. R. Dey and G. Ananthakrishna, Europhys. Lett. **66**, 715 (2004). A. Petri, G. Paparo, A. Vespignani, A. Alippi and M. Costantini, Phys. Rev. Lett **73** 3423 (1994). A. Garcimartin, A. Guarino, L. Bellon and S. Ciliberto, Phys. Rev. Lett. **79** 3202 (1997). F. T. Peirce, J. Textile Inst. **17**, T355 (1926). H. E. Daniels, Proc. R. Soc. London A **183** 405 (1945). P. C. Hemmer and A. Hansen, ASME J. Appl. Mech. **59**, 909 (1992). D. Sornette, J. Phys. I (France) **2** 2089 (1992). D. Sornette, J. Phys. A **22** L243 (1989); A. T. Bernardes and J. G. Moreira, Phys. Rev. B **49** 15035 (1994). M. Kloster, A. Hansen and P. C. Hemmer, Phys. Rev. E **56** 2615 (1997). J. V. Andersen, D. Sornette and K. T. Leung, Phys. Rev. Lett. **78** 2140 (1997). S. Pradhan, P. Bhattacharyya and B. K. Chakrabarti, Phys. Rev. E **66** 016116 (2002). P. Bhattacharyya, S. Pradhan and B. K. Chakrabarti, Phys. Rev. E **67** 046122 (2003). A. Hansen and P. C. Hemmer, Phys. Lett. A **184**, 394 (1994). S. Zapperi, P. Ray, H. E. Stanley and A. Vespignani, Phys. Rev. Lett. **78** 1408 (1997). S. Zapperi, P. Nukala and S. Simunovic, Phys. Rev. E **71**, 026106 (2005). S. Pradhan and A. Hansen Phys. Rev. E **72**, 026111 (2005). S. Pradhan, A. Hansen and P. C. Hemmer Phys. Rev. Lett. **95**, 125501 (2005). S. Pradhan, A. Hansen and P. C. Hemmer Phys. Rev. E **74**, 016122 (2006). H. Kawamura, arXiv:cond-mat/0603335 (2006). P. Bak, C. Tang, K. Weisenfeld, Phys. Rev. Lett. **59** 381 (1987); Phys. Rev. A **38** 364 (1988). S. S. Manna, J. Stat. Phys. **59**, 509 (1990); P. Grassberger and S. S. Manna, J. Phys. France **51**, 1077(1990); S. S. Manna, J. Phys. A: Math. Gen **24**, L363 (1991). D. Dhar, Physica A **186**, 82 (1992); Physica A **263**, 4 (1999);Physica A **270**, 69 (1999). M. Acharyya and B. K. Chakrabarti, Physica A **224**, 254 (1996); Phys. Rev. E **53**, 140 (1996). S. Pradhan and B. K. Chakrabarti, Phys. Rev. E **65** 016113 (2002). A. L. Barabasi and H. E. Stanley, (1995). *Fractal Concepts in Surface Growth*, Cambridge University Press, Cambridge. V. De Rubeis, R. Hallgass, V. Loreto, G. Paladin, L. Pietronero and P. Tosi, Phys. Lett. **76** 2599 (1996). B. K. Chakrabarti, R. B. Stinchcombe, Physica A, **270** 27 (1999). S. Pradhan, B. K. Chakrabarti, P. Ray and M. K. Dey, Phys. Scr. T **106** 77 (2003). P. Bhattacharyya, Physica A **348**, 199 (2005). S. Pradhan, P. Chaudhuri and B. K. Chakrabarti, in Continuum Models and Discrete Systems, Ed. D. Bergman, E. Inan, Nato Sc, Series, Kluwer Academic Publishers (Dordrecht) 245 (2004).

--- abstract: 'We show an It\^ o’s formula for nondegenerate Brownian martingales $X_t=\int_0^t u_s dW_s$ and functions $F(x,t)$ with locally integrable derivatives in $t$ and $x$. We prove that one can express the additional term in Itô’s s formula as an integral over space and time with respect to local time.' --- and [**Carles Rovira$^2$**]{} *$^1$ Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193-Bellaterra (Barcelona), Spain.* $^2$ Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, 08007-Barcelona, Spain. [*E-mail addresses*]{}: Xavier.Bardina@uab.cat, Carles.Rovira@ub.edu [$^{*}$corresponding author]{} [**Keywords:**]{} Martingales; Integration wrt local time; It[ô]{}’s formula; Local time; [**Running title:**]{} It[ô]{}’s formula for nondegenerate martingales Introduction {#introduction .unnumbered} ============ We consider a continuous nondegenerate martingale $X=\{X_t, \,t \in [0,1]\}$ of the form $X_t = \int_0^t u_s dW_s$ where $W=\{W_t, t \in [0,1]\}$ is a standard Brownian motion and $u$ is an adapted stochastic process. Let $F: {\mathbb{R}}\times [0,1] \rightarrow {\mathbb{R}}$ be an absolutely continuous function with partial derivatives satisfying some local integrability properties. The main aim of this paper is to obtain an Itô’s formula for $F(X_t,t)$ where the term corresponding usually to the second order derivative is expressed as an integral over space and time with respect to local time. We will prove this results when $u$ satisfies (locally) the assumptions - For all $t \in [0,1]$, $u_t$ belongs to the space ${\mathbb D}^{3,2}$ and for all $p \ge 2$ $$E \vert u_t \vert^p + E \vert D_s u_t \vert^p + E \left( \int_{r \vee s}^1 \vert D_rD_s u_\theta \vert^2 d\theta \right)^{p/2} + E \left( \int_{r \vee s \vee v }^1 \vert D_v D_rD_s u_\theta \vert^2 d\theta \right)^{p/2} \le K_p,$$ - $\vert u_t \vert \ge \rho >0$ for some constant $\rho$ and for all $t \in [0,1].$ Moret and Nualart (2000) consider an Itô’s formula for this class of nondegenerate martingales. Their main result reads as follows: \[teoMN\] Let $u$ be a process satisfying [**(H1)**]{} and [**(H2)**]{}. Set $X=\int_0^t us dW_s$. Then for any funcion $f \in L^2_{\rm loc} ({\mathbb{R}})$ the quadratic covariation $[f(X),X]$ exists and the following Itô’s formula holds $$F(X_t)=F(0)+ \int_0^t f(X_s) dX_s + \frac12 [f(X),X]_t,$$ for all $t \in [0,1]$, where $F(x)=F(0)+\int_0^x f(y) dy.$ Moret (1999), gave an extension of this last result for functions $F$ depending also on $t$. They consider a new hypothesis on functions $f$: - $f(\cdot,t) \in L^2_{loc}({\mathbb{R}})$ and for all compact set $K\subset{\mathbb{R}}$ $f(x,t)$ is continuous in $t$ as a function of $[0,T]$ to $L^2(K)$ Then, their result is the following: \[teoM\] Let $u$ be a process satisfying [**(H1)**]{} and [**(H2)**]{}. Set $X=\int_0^t us dW_s$. Let $F(x,t)$ be an absolutely continuous function in $x$ such that the partial derivative $f(\cdot,t)$ satisfies [**(C)**]{}. Then, the quadratic covariation $\left[f(X,\cdot),X\right]$ exists and the following Itô’s formula holds $$F(X_t,t)=F(0,0)+\int_0^tf(X_s,s)dX_s+\frac12\left[f(X,\cdot),X\right]+\int_0^tF(X_s,ds),$$ where $$\int_0^tF(X_s,ds)\equiv \lim_{n\to+\infty}\sum_{t_i\in D_n, t_i\leq t} \left(F(X_{t_{i+1}},t_{i+1})-F(X_{t_{i+1}},t_i)\right),$$ exists uniformly in probability for $(D_n)_n$ a sequence of smooth partitions of $[0,1]$. In these two results, following the ideas of F[ö]{}llmer, Protter and Shiryayev (1995) for the Brownian motion, the additional term is written as a quadratic covariation. Bardina and Jolis (1997, 2002) extended the results of F[ö]{}llmer [*et al.*]{} (1995) to the case of the elliptic and hypoelliptic diffusions. Nevertheless, it is important to point out the differences between the work of Moret and Nualart (2000) and F[ö]{}llmer [*et al.*]{} (1995). One of the keys of their proofs is to obtain some a priori estimates on the Riemann sums. In F[ö]{}llmer [*et al.*]{} (1995) these estimates are obtained using the semimartingale expression of the time-reversed Brownian motion and well-known bounds for the density of the Brownian motion. Moret and Nualart (2000) used another approach, using Malliavin calculus in order to obtain sharp estimates for the density of the process $X_t$ and avoiding the time-reversed arguments. We want to express the quadratic variation term as an integral with respect to the local time. There are several papers where the integrals with respect to local time are used in Itô’s formula. In 1981, Bouleau and Yor obtained the following extension of the It[ô]{}’s formula : \[teoBY\] Let $X=\left(X_t\right)_{t\geq0}$ be a continuous semimartingale and let $F:{\mathbb{R}}\longrightarrow{\mathbb{R}}$ be an absolutely continuous function with derivative $f$. Assume that $f$ is a mesurable locally bounded function. Then: $$F(X_t)=F(X_0)+\int_0^tf(X_s)dX_s-\frac12\int_{{\mathbb{R}}}f(x)d_xL_t^x$$ where $d_xL_t^x$ is an integral with respect to $x\longrightarrow L_t^x$. Eisenbaum (2000, 2001) defined an integral in time and space with respect to the local time of the Brownian motion. Using this integral, the quadratic covariation in the formula given in F[ö]{}llmer [*et al.*]{} can be expressed as an integral with respect to the local time. She obtained the following result: \[teoE\] Let $W=\left(W_t\right)_{0\leq t\leq1}$ be a standard Brownian motion and $F$ a function defined on ${\mathbb{R}}\times[0,1]$ such that there exist first order Radon-Nikodym derivatives $\frac{\partial F}{\partial t}$ and $\frac{\partial F}{\partial x}$ such that for every $A\in{\mathbb{R}}_{+}$, $$\int_0^1\int_{-A}^A\left|\frac{\partial F}{\partial t}(x.s)\right|\frac1{\sqrt{s}}dxds<+\infty$$ and $$\int_0^1\int_{-A}^A \left(\frac{\partial F}{\partial x}(x,s)\right)^2\frac1{\sqrt{s}}dxds<+\infty.$$ Then, $$\begin{aligned} F(W_t,t)=F(W_0,0)+\int_0^t\frac{\partial F}{\partial x}(W_s,s)dW_s+\int_0^t\frac{\partial F}{\partial t}(W_s,s)ds-\frac12\int_0^t\int_{{\mathbb{R}}}\frac{\partial F}{\partial x}(x,s)dL_s^x.\end{aligned}$$ This result has been extended by Bardina and Rovira (2007) for elliptic diffusion processes. In our papers we will follow the ideas Eisenbaum (2000,2001), assuming on the function $F$ the hypothesis considered in Theorem \[teoE\]. In the papers of Eisenbaum (2000,2001), as well as in F[ö]{}llmer [*et al.*]{} (1995) or in the extension of Bardina and Rovira (2007), one of the main ingredients is the study of the time reversed process and the relationship between the quadratic covariation and the forward and backward stochastic integrals. We show that we can adapt the methods of Eisebaum without using the time reversed process and the backward integral. We will follow the methods of Moret and Nualart (2000) and we will use Malliavin calculus to obtain the necessary estimates for the Riemann sums . In our paper, the existence of the quadratic covariation is not one of our main objectives. Nevertheless, it will be an important tool in our computations. We recall its definition. Given two stochastic processes $Y=\{Y_t, t \in [0,1]\}$ and $Z=\{Z_t, t \in [0,1]\}$ we define their quadratic covariation as the stochastic process $[Y,Z]$ given by the following limit in probability, if it exists, $$[Y,Z]_t = \lim_n \sum_{ t_i \in D_n, t_i < t } (Y_{t_{i+1}}- Y_{t_i}) (Z_{t_{i+1}}- Z_{t_i}).$$ where $D_n$ is a sequence of partitions of $[0,1]$. We will assume that the partitions $D_n$ satisfy - $\lim_n \sup_{t_i \in D_n} (t_{i+1} - t_i )=0, \qquad M:=\sup_n \sup_{t_i \in D_n} \frac{t_{i+1}}{t_i} < \infty.$ We impose this condition in order to avoid certain possibly exploding Riemann sums. Other extensions for It[ô]{}’s formula has been obtained recently. Among others, there is the paper of Dupoiron [*et al.*]{} (2004) for uniformly elliptic diffusions and Dirichet processes, the work of Ghomrasni and Peskir (2006) for continuous semimartingales, the paper of Flandoli, Russo and Wolf (2004) for a Lyons-Zheng process or the work of Di Nunno, Meyer-Brandis, [Ø]{}ksendal and Proske (2005) for Lévy processes. The paper is organized as follows. In Section 1 we give some basic definitions and results on Malliavin calculus, recalling some results obtained in Moret and Nualart (2000). In Section 2 we define the space where we are able to construct an integral in the plane with respect to the local time of a nondegenerate Brownian martingale. Finally, Section 3 is devoted to present our main result the extension of It[ô]{}’s formula. Along the paper we will denote all the constants by $C, C_p, K$ or $K_p$, unless they may change from line to line. Preliminaries ============= Let $(\Omega,{\mathcal{F}},P)$ be the canonical probability space of a standard Brownian motion $W=\{ W_t, 0 \le t \le 1 \}$, that is, $\Omega$ is the space of all continuous functions $\omega:[0,1] \to {\mathbb{R}}$ vanishing at 0, $P$ is the standard Wiener measure on $\Omega$ and ${\mathcal{F}}$ is the completion of the Borel $\sigma$-field of $\Omega$ with respect to $P$. Let $H=L^2([0,1])$. Let ${{\cal S}}$ be the set of smooth random variables of the form $$\label{A1} F=f(W_{t_1},\ldots,W_{t_n}),$$ $f\in {{\cal C}}_b^\infty({\mathbb{R}}^n)$ and $t_1, \ldots, t_n \in [0,1].$ The Malliavin derivative of a smooth random variable $F$ of the form (\[A1\]) is the stochastic process $\{ D_tF, t \in [0,T]\}$ given by $$D_t F = \sum_{i=1}^n \frac{\partial f}{\partial x_i} (W_{t_1},\ldots,W_{t_n}) I_{[0,t_i]}(t),\qquad t \in [0,1].$$ The Malliavin derivative of order $N \ge 2$ is defined by iteration, as follows. For $F \in {{\cal S}},\, t_1,\ldots,t_N \in [0,T],$ $$D^N_{t_1,\ldots,t_N} F = D_{t_1}D_{t_2} \ldots D_{t_N} F.$$ For any real number $p \ge 1$ and any integer $N\ge 1$ we denote by ${\mathbb D}^{N,p}$ the completion of the set ${{\cal S}}$ with respect to the norm $$\Vert F \Vert_{N,p} = \big[ E( \vert F \vert^p ) + \sum_{i=1}^N E ( \Vert D^i F \Vert^p_{L^2([0,T]^i)} ) \big]^\frac1p.$$ The domain of the derivative operator $D$ is the space ${\mathbb D}^{1,2}.$ The divergence operator $\delta $ is the adjoint of the derivative operator. The domain of the operator $\delta $, denoted by Dom $\delta $, is the set of processes $u \in L^{2}([0,T] \times \Omega)$ such that there exists a square integrable random variable $\delta(u)$ verifying $$E(F\delta (u))=E\left( \int_0^1 D_tF u_t dt \right),$$ for any $F \in {{\cal S}}$. The operator $\delta$ is an extension of Itô’s stochastic integral and we will make use of the notation $\delta(u)=\int_0^1 u_s dW_s.$ We will recall some useful results from Moret and Nualart (2000). We refer the reader to this paper for their proof and a detailed account of these results. We also refer to Nualart (1995, 2006) for any other property about operators $D$ and $\delta$. \[2.6MN\] Let $Y$ be a random variable in the space ${\mathbb D}^{1,2}$ such that $\int_a^b ( D_s Y)^2 ds > 0$ a.s. for some $0 \le a < b \le 1.$ Assume that $(DY/\int_a^b (D_sY)^2 ds) I_{[a,b]}$ belongs to Dom $\delta$. Then $Y$ has an absolutely continuous distribution with density $p$ that satisfies the inequality $$p(x) \le E \Big\vert \int_a^b \left( \frac{D_sY}{\int_a^b (D_s Y)^2 ds } \right) dW_s \Big\vert.$$ It follows from Proposition 1 and (2.6) in Moret and Nualart (2000). $\Box$ The following Proposition is also a slight modification of Corollary 2 of Moret-Nualart (2000). \[corol2MN\] Let $Y$ be a random variable in the space ${\mathbb D}^{1,2}$ such that $\int_0^1 ( D_s Y)^2 ds > 0$ a.s. Let $Z$ be a positive square integrable random variable such that $( Z DY / \int_0^1 (Ds Y )^2 ds ) I_{[0,1]}$ belongs to Dom $\delta$. Then, for any $f \in L^2({\mathbb{R}})$, we have $$\vert E (f(Y)^2 Z) \vert = \Vert f \Vert_2^2 E \Big\vert \delta \left( \frac{Z DY}{\Vert DY \Vert_H^2} \right) \Big\vert.$$ See Corollary 2 in Moret-Nualart (2000). The same proof works using a dominated convergence argument. $\Box$ \[lema10MN\] Fix $p \ge 1$. Suppose that $u$ satisfies hypotheses [**(H1)**]{} and [**(H2)**]{}. Let $Z \in {\mathbb D}^{1,2p}$. Then, we have, for $0 \le a < b \le 1$: $$\begin{aligned} && E \Big\vert \int_a^b Z \frac{ D_t X_b}{ \int_a^b (D_t X_b )^2 dt} dW_t \Big\vert^p \\ && \qquad \le C_0 (b-a)^{-p/2} \left( \left( E \vert Z \vert^{2p} \right)^{1/2} + \left( E \big\vert \int_a^b (D_t Z)^2 dt \big\vert^{p} \right)^{1/2} \right),\end{aligned}$$ where $C_0$ is a constant does not depend on $Z$. See Lemma 10 in Moret-Nualart (2000). $\Box$ Stochastic integration with respect to local time of the martingale =================================================================== Following the ideas of Eisenbaum (2000), we consider first the space of functions for whose elements we can define a stochastic integration with respect to local time. Let $f$ be a measurable function from ${\mathbb{R}}\times[0,1]$ into ${\mathbb{R}}$. We define the norm $\| \cdot \|$ by $$\|f\|=\left(\int_0^1\int_{{\mathbb{R}}} f^2(x,s)\frac1{s^{\frac34}}dxds\right)^{\frac12}$$ Consider the set of functions $$\mathcal H=\{f:\,\|f\|<+\infty\}.$$ It is easy to check that $\mathcal H$ is a Banach space. Let us consider $X$ a nondegenerate martingale of the type $X_t=\int_0^t u_s dW_s$ where $u$ is an adapted stochastic process satisfying hypotheses [**(H1)**]{} and [**(H2)**]{}. Let us show now how to define a stochastic integration over the plane with respect to the local time $L$ of the process $X$ for the elements of $\mathcal H$. Let $f_{\Delta}$ be an elementary function, $$f_{\Delta}(x,s):=\sum_{(x_k,s_l)\in\Delta} f_{kl}I_{(x_k,x_{k+1}]}(x)I_{(s_l,s_{l+1}]}(s),$$where $(x_k)_{1\leq k\leq m_1}$ is a finite sequence of real numbers, $(s_l)_{1\leq l\leq m_2}$ is a subdivision of $[0,1]$, $(f_{kl})_{1\leq k\leq m_1;\, 1\leq l\leq m_2}$ is a sequence of real numbers and finally, $\Delta=\{(x_k,s_l),\,1\leq k\leq m_1,\,1\leq l\leq m_2\}$. It is easy to check that the elementary functions are dense in $\mathcal H$. We define the integration for the elementary function $f_{\Delta}$ with respect to the local time $L$ of the martingale $X$ as follows $$\int_0^1\int_{{\mathbb{R}}}f_{\Delta}(x,s)dL_s^x= \sum_{(x_k,s_l)\in\Delta} f_{kl}(L_{s_{l+1}}^{x_{k+1}}-L_{s_{l}}^{x_{k+1}}-L_{s_{l+1}}^{x_{k}}+L_{s_{l}}^{x_{k}}).$$ Let $f$ be a function of $\mathcal H$. Let us consider $(f_n)_{n\in{\mathbb{N}}}$ a sequence of elementary functions converging to $f$ in $\mathcal H$. We will check that the sequence $\left(\int_0^1\int_{{\mathbb{R}}}f_{n}(x,s)dL_s^x\right)_{n\in{\mathbb{N}}}$ converges in $L^1$ and that the limit does not depend on the choice of the sequence $(f_n)_{n\in{\mathbb{N}}}$. So, we will use this limit as the definition of the integral $\int_0^1\int_{{\mathbb{R}}}f(x,s)dL_s^x$. First of all, let us see a previous lemmas. \[lema1\] For any locally bounded Borel measurable function $f$ and any $t\in(0,1]$ we have $$\int_{{\mathbb{R}}}f(a)d_aL^a_t=-\left[f(X),X\right]_t,$$ where $d_aL_t^a$ denotes the integral with respect to $a\longrightarrow L_t^a$. It follows easily from Theorem \[teoBY\] and Theorem \[teoMN\]. $\Box$ \[lemafita\] Consider $f_1(x):=I_{(a,b]}(x)$ and $f_2(x):=I_{(c,d]}(x)$, where $a<b$ and $c<d$ are real numbers. Then for all $t_i<t_j\leq t$, $$\begin{aligned} &&E\left[f_1(X_{t_{i+1}})f_2(X_{t_{j+1}})(X_{t_{i+1}}-X_{t_{i}})(X_{t_{j+1}}-X_{t_{j}})\right]\\ &\leq&E\left[f_1(X_{t_{i+1}})f_2(X_{t_{j+1}})C_{ij}\right],\end{aligned}$$ where $$\|C_{ij}\|_2\leq C\frac{(t_{i+1}-t_i)(t_{j+1}-t_j)}{\sqrt{t_{i+1}(t_{j+1}-t_{i+1})}},$$ and $C$ does not depend on $f_1$ and $f_2$. When $f_1=f_2=f\in\mathcal C_K^{\infty}({\mathbb{R}})$, this inequality is checked in the proof of Proposition 14 in Moret and Nualart (2000). The same proof also works when $f_1\neq f_2$ with $f_1,f_2\in\mathcal C_K^{\infty}({\mathbb{R}})$. Now, fixed our functions $f_1,f_2$ let us consider sequences $f_1^n\uparrow f_1$ and $f_2^n\uparrow f_2$ with $f_n^i\in\mathcal C_K^{\infty}({\mathbb{R}})$ for all $n$ and $i\in\{1,2\}$. Then, the result can be obtained by a dominated convergence argument. $\Box$ \[teo2\] Let $f$ be a function of $\mathcal H$. Then, there exists the integral $\int_0^t\int_{{\mathbb{R}}}f(x,s)dL_s^x$ for any $t\in[0,1]$. Let $f_{\Delta}$ be an elementary function. From Theorem \[teoM\] and Lemma \[lema1\] it is easy to get that the quadratic covariation $[f(X,.),X]_t$ exists and that $$\int_0^t\int_{{\mathbb{R}}}f_{\Delta}(x,s)dL_s^x=-\left[f_\Delta(X,\cdot),X\right]_t.$$ The key of the proof is to check that for all elementary function $f_\Delta$ $$\label{simpli} E\left(\left|\int_0^t\int_{{\mathbb{R}}}f_{\Delta}(x,s)dL_s^x\right|\right)\leq C\|f_{\Delta}\|,$$ where the constant does not depend on $f_\Delta$. Notice that, $$\begin{aligned} &&E\left(\left|\int_0^t\int_{{\mathbb{R}}}f_{\Delta}(x,s)dL_s^x\right|\right)\nonumber\\ &=&E\left(\left|\left[f_\Delta(X,\cdot),X\right]_t\right|\right)\nonumber\\ &=& E \left(\left|\lim_{n\to\infty}\sum_{t_i\in D_n,t_i\leq t}\left(f_{\Delta}(X_{t_{i+1}},t_{i+1})-f_{\Delta}(X_{t_{i}},t_{i})\right)\left(X_{t_{i+1}}-X_{t_i} \right) \right| \right) \nonumber\\&\leq& \left\{ 2 \liminf_{n\to\infty} E \left( \left|\sum_{t_i\in D_n,t_i\leq t}f_{\Delta}(X_{t_{i+1}},t_{i+1})\left(X_{t_{i+1}}-X_{t_i} \right) \right|^2 \right) \right.\nonumber\\&& +\left. 2 \liminf_{n\to\infty} E \left( \left|\sum_{t_i\in D_n,t_i\leq t}f_{\Delta}(X_{t_{i}},t_{i})\left(X_{t_{i+1}}-X_{t_i} \right) \right|^2 \right) \right\}^{\frac12} \nonumber\\ &:=& \left( 2 \left(\liminf_{n\to\infty}I_1\right)+2 \left(\liminf_{n\to\infty}I_2\right)\right)^{\frac12},\label{aaa1}\end{aligned}$$ where in the last inequality we have used Fatou’s lemma. Along the study of $I_1$ and $I_2$ we will make use of the methods presented in the proofs of Propositions 13 and 14 in Moret and Nualart (2000). For the sake of completeness, we will give the main steps of our proofs in the study of $I_2$. For the other terms, we will refer the reader to the paper of Moret and Nualart (2000). By the isometry, and using Propositions \[corol2MN\] and \[lema10MN\] $$\begin{aligned} I_2&=&E \left(\sum_{t_i\in D_n,t_i\leq t}f_{\Delta}^2(X_{t_{i}},t_{i})\int^{t_{i+1}}_{t_i}u_s^2ds\right)\\ &=&\sum_{t_i\in D_n,t_i\leq t}E\left(f_{\Delta}^2(X_{t_{i}},t_{i})\int^{t_{i+1}}_{t_i}u_s^2ds\right)\\ &\leq&\sum_{t_i\in D_n,t_i\leq t}\int_{{\mathbb{R}}}f_{\Delta}^2(x,t_{i})dx E\Big\vert \delta \left( \frac{ (\int_{t_i}^{t_{i+1}} u_s^2 ds ) DX_{t_i}}{ \Vert DX_{t_i} \Vert_H^2} \right) \Big\vert\\ &\leq&C\sum_{t_i\in D_n,t_i\leq t}\int_{{\mathbb{R}}}f_{\Delta}^2(x,t_{i})dxt_i^{-\frac12} \left( \sqrt{ E \big\vert \int_{t_i}^{t_{i+1}} u_s^2 ds \big\vert^2} + \sqrt{ E \int_0^{t_i} \left( D_t \left( \int_{t_i}^{t_{i+1}} u_s^2 ds \right) \right)^2 dt} \right)\\ &=&C\sum_{t_i\in D_n,t_i\leq t}\int_{{\mathbb{R}}}\sum_{k=1}^{m_1}\sum_{l=1}^{m_2} f^2_{kl}I_{(x_k,x_{k+1}]}(x)I_{(s_l,s_{l+1}]}(t_i) t_i^{-\frac12}(t_{i+1}-t_{i})dx\\ &=&C\int_{{\mathbb{R}}}\sum_{k=1}^{m_1}\sum_{l=1}^{m_2} f^2_{kl}I_{(x_k,x_{k+1}]}(x)\left(\sum_{t_i\in D_n,t_i\leq t}I_{(s_l,s_{l+1}]}(t_i)t_i^{-\frac12}(t_{i+1}-t_{i})\right)dx\\ &=&C\int_{{\mathbb{R}}}\sum_{k=1}^{m_1}\sum_{l=1}^{m_2} f^2_{kl}I_{(x_k,x_{k+1}]}(x)\left(\int_0^t\sum_{t_i\in D_n,t_i\leq t}I_{(s_l,s_{l+1}]}(t_i)t_i^{-\frac12}I_{(t_{i},t_{i+1}]}(s)ds\right)dx.\end{aligned}$$ Using the condition [**(M)**]{} over the partitions, we have that, by bounded convergence, $$\lim_{n\to\infty}\int_0^t\sum_{t_i\in D_n,t_i\leq t} I_{(s_l,s_{l+1}]}(t_i)t_i^{-\frac12}I_{(t_{i},t_{i+1}]}(s)ds= \int_0^t I_{(s_l,s_{l+1}]}(s)s^{-\frac12}ds,$$ and then, $$\begin{aligned} \liminf_{n\to\infty}I_2&\leq& C\int_{{\mathbb{R}}}\sum_{k=1}^{m_1}\sum_{l=1}^{m_2} f^2_{kl}I_{(x_k,x_{k+1}]}(x)\int_0^t I_{(s_l,s_{l+1}]}(s)s^{-\frac12}dsdx\nonumber\\ &=&C\int_0^t\int_{{\mathbb{R}}}f^2_{\Delta}(x,s)\frac1{\sqrt{s}}dxds\nonumber\\ &\leq&C\|f_{\Delta}\|^2.\label{aaa2}\end{aligned}$$ On the other hand, $$\begin{aligned} I_1&=&E \left( \vert \sum_{t_i\in D_n,t_i\leq t}f_{\Delta}(X_{t_{i+1}},t_{i+1})\left(X_{t_{i+1}}-X_{t_i}\right) \vert^2 \right)\nonumber\\ &=&E \left( \sum_{t_i\in D_n,t_i\leq t}f_{\Delta}^2(X_{t_{i+1}},t_{i+1})\left(X_{t_{i+1}}-X_{t_i}\right)^2 \right) \nonumber\\ &&+2 E \left( \sum_{t_i,t_j\in D_n,t_i<t_j\leq t}f_{\Delta}(X_{t_{i+1}},t_{i+1})f_{\Delta}(X_{t_{j+1}},t_{j+1})\left(X_{t_{i+1}}-X_{t_i}\right) \left(X_{t_{j+1}}-X_{t_j}\right) \right) \nonumber\\ &:=&I_{1,1}+2I_{1,2}.\label{aaa3}\end{aligned}$$ Following now the methods of Proposition 14 of Moret and Nualart (1999) and using again Propositions \[corol2MN\] and \[lema10MN\] as we did in the study of $I_2$, we get that $$I_{1,1}\leq C\sum_{t_i\in D_n,t_i\leq t}\int_{{\mathbb{R}}}f_{\Delta}^2(x,t_{i})dxt_{i+1}^{-\frac12}(t_{i+1}-t_{i}).$$ By similar computations to those of the term $I_2$ we obtain that $$\liminf_{n\to\infty}I_{1,1}\leq C\|f_{\Delta}\|^2.\label{aaa4}$$ Let us study now $I_{1,2}$. Using Lemma \[lemafita\], notice that $$\begin{aligned} I_{1,2}&=&E \left( \sum_{t_i,t_j\in D_n,t_i<t_j\leq t}f_{\Delta}(X_{t_{i+1}},t_{i+1})f_{\Delta}(X_{t_{j+1}},t_{j+1})\left(X_{t_{i+1}}-X_{t_i}\right) \left(X_{t_{j+1}}-X_{t_j}\right) \right) \\ &\leq&\sum_{t_i,t_j\in D_n,t_i<t_j\leq t}E\left(f_{\Delta}(X_{t_{i+1}},t_{i+1})f_{\Delta}(X_{t_{j+1}},t_{j+1})C_{ij}\right).\end{aligned}$$ Following again the methods of the proof of Proposition 14 of Moret and Nualart (1999) -more precisely, the proof of inequalities (5.36) and (5.37)- the last expression is bounded by $$\begin{aligned} &&\sum_{t_i,t_j\in D_n,t_i<t_j\leq t}E\left(f_{\Delta}^2(X_{t_{i+1}},t_{i+1})f_{\Delta}^2(X_{t_{j+1}},t_{j+1})\right)^{\frac12}E\left(C^2_{ij}\right)^{\frac12}\nonumber\\ &\leq&C\sum_{t_i,t_j\in D_n,t_i<t_j\leq t}\frac{(t_{i+1}-t_i)(t_{j+1}-t_j)}{(t_{i+1}(t_{j+1}-t_{i+1}))^{\frac34}}\left(\int_{{\mathbb{R}}}f^2_{\Delta}(x,t_{i+1})dx\right)^{\frac12} \left(\int_{{\mathbb{R}}}f^2_{\Delta}(x,t_{j+1})dx\right)^{\frac12}\nonumber\\ &\leq &C\sum_{t_i,t_j\in D_n,t_i<t_j\leq t} \frac{(t_{i+1}-t_i)(t_{j+1}-t_j)}{(t_{i+1}(t_{j+1}-t_{i+1}))^{\frac{3}{4}}} \left(\int_{{\mathbb{R}}}f^2_{\Delta}(x,t_{i+1})dx+ \int_{{\mathbb{R}}}f^2_{\Delta}(x,t_{j+1})dx\right)\nonumber\\ &:=&C(I_{1,2,1}+I_{1,2,2}).\label{aaa5}\end{aligned}$$ Since $$\sum_{t_j\in D_n,t_i<t_j\leq t}\frac{(t_{j+1}-t_j)}{(t_{j+1}-t_{i+1})^{\frac34}}=\int_{t_{i+1}}^t \sum_{t_j\in D_n,t_i<t_j\leq t}\frac{1}{(t_{j+1}-t_{i+1})^{\frac34}}I_{(t_j,t_{j+1}]}(s)ds,$$ we get that $$\begin{aligned} I_{1,2,1}&\leq&\sum_{t_i\in D_n,t_i<t_j\leq t} \frac{(t_{i+1}-t_i)}{t_{i+1}^{\frac{3}{4}}} \int_{t_{i+1}}^t \frac{1}{(s-t_{i+1})^{\frac34}}ds \left(\int_{{\mathbb{R}}}f^2_{\Delta}(x,t_{i+1})dx\right)\\ &\leq&C\sum_{t_i\in D_n,t_i<t_j\leq t} \frac{(t_{i+1}-t_i)}{t_{i+1}^{\frac{3}{4}}} \int_{{\mathbb{R}}}f^2_{\Delta}(x,t_{i+1})dx.\end{aligned}$$ And this clearly yields that $$\liminf_{n\to\infty}I_{1,2,1}\leq C\int_0^t\int_{{\mathbb{R}}}f^2_{\Delta}(x,s)\frac1{s^{\frac34}}ds =C\|f_{\Delta}\|^2.\label{aaa6}$$ Finally we have to consider $I_{1,2,2}$. First of all, notice that $$\begin{aligned} I_{1,2,2}&:=&\sum_{t_i,t_j\in D_n,t_i<t_j\leq t} \frac{(t_{i+1}-t_i)(t_{j+1}-t_j)}{(t_{i+1}(t_{j+1}-t_{i+1}))^{\frac{3}{4}}} \left(\int_{{\mathbb{R}}}f^2_{\Delta}(x,t_{j+1})dx\right)\\ &=&\sum_{t_i,t_j\in D_n,t_i<t_j\leq t} \frac{(t_{i+1}-t_i)(t_{j+1}-t_j)}{(t_{i+1}(t_{j+1}-t_{i+1}))^{\frac{3}{4}}} \left(\int_{{\mathbb{R}}} \sum_{k=1}^{m_1}\sum_{l=1}^{m_2} f^2_{kl}I_{(x_k,x_{k+1}]}(x)I_{(s_l,s_{l+1}]}(t_{j+1}) dx\right)\\ &\leq&\int_{{\mathbb{R}}} \sum_{k=1}^{m_1}\sum_{l=1}^{m_2}f^2_{kl}I_{(x_k,x_{k+1}]}(x)\sum_{t_i\in D_n,t_i< t} \frac{(t_{i+1}-t_i)}{t_{i+1}^{\frac{3}{4}}} \\&&\times\sum_{t_j\in D_n,t_i<t_j\leq t}\int_{t_{i+1}}^t\frac1{(s-t_{i+1})^{\frac34}} I_{(s_l,s_{l+1}]}(t_{j+1})I_{(t_{j},t_{j+1}]}(s)dsdx\end{aligned}$$ From the obvious inequality $$I_{(s_l,s_{l+1}]}(t_{j+1})I_{(t_{j},t_{j+1}]}(s)\leq I_{(s_l,s_{l+1}]}(s)I_{(t_{j},t_{j+1}]}(s) +I_{(t_j,s_l]}(s)I_{(t_{j},t_{j+1}]}(s_l),$$ we obtain the bound $$I_{1,2,2}\leq I_{1,2,2,1}+I_{1,2,2,2},\label{aaa9}$$ where $$\begin{aligned} I_{1,2,2,1}&=&\int_{{\mathbb{R}}} \sum_{k=1}^{m_1}\sum_{l=1}^{m_2}f^2_{kl}I_{(x_k,x_{k+1}]}(x) \\ & & \qquad \times \sum_{t_i\in D_n,t_i< t} \frac{(t_{i+1}-t_i)}{t_{i+1}^{\frac{3}{4}}} \sum_{t_j\in D_n,t_i<t_j\leq t}\int_{t_{i+1}}^t\frac1{(s-t_{i+1})^{\frac34}} I_{(s_l,s_{l+1}]}(s)I_{(t_{j},t_{j+1}]}(s)dsdx\\ I_{1,2,2,2}&=&\int_{{\mathbb{R}}} \sum_{k=1}^{m_1}\sum_{l=1}^{m_2}f^2_{kl}I_{(x_k,x_{k+1}]}(x)\\ & & \qquad \times \sum_{t_i\in D_n,t_i< t} \frac{(t_{i+1}-t_i)}{t_{i+1}^{\frac{3}{4}}} \sum_{t_j\in D_n,t_i<t_j\leq t}\int_{t_{i+1}}^t\frac1{(s-t_{i+1})^{\frac34}} I_{(t_j,s_{l}]}(s)I_{(t_{j},t_{j+1}]}(s_l)dsdx.\end{aligned}$$ Now, since we can write $$I_{1,2,2,1} = \int_{{\mathbb{R}}} \sum_{t_i\in D_n,t_i< t} \frac{(t_{i+1}-t_i)}{t_{i+1}^{\frac{3}{4}}} \int_{t_{i+1}}^tf^2_{\Delta}(x,s)\frac1{(s-t_{i+1})^{\frac34}} dsdx,$$ using an argument of bounded convergence we have that $$\begin{aligned} \liminf_{n\to\infty}I_{1,2,2,1}& \leq &\int_{{\mathbb{R}}}\int_0^t \frac{1}{u^{\frac{3}{4}}} \int_{u}^tf^2_{\Delta}(x,s)\frac1{(s-u)^{\frac34}} dsdudx\nonumber\\ &=&\int_{{\mathbb{R}}}\int_0^t f^2_{\Delta}(x,s)\int_{0}^s\frac{1}{u^{\frac{3}{4}}} \frac1{(s-u)^{\frac34}} dudsdx\nonumber\\ &\leq&C\int_{{\mathbb{R}}}\int_0^t f^2_{\Delta}(x,s)\frac{1}{s^{\frac12}} dsdx\nonumber\\ &\leq&C\|f_{\Delta}\|^2\label{aaa10}.\end{aligned}$$ On the other hand, observe that fixed $l$, there exists only one $j$ (that we will denote by $j(l)$) such that $t_{j(l)}<s_l\leq t_{j(l)+1}$. So, $$\begin{aligned} &&\sum_{t_i,t_j\in D_n,t_i<t_j\leq t} \frac{(t_{i+1}-t_i)}{t_{i+1}^{\frac{3}{4}}} \int_{t_{i+1}}^t\frac1{(s-t_{i+1})^{\frac34}} I_{(t_{j},s_{l}]}(s)I_{(t_{j},t_{j+1}]}(s_l)ds\\ &\leq&\sum_{t_i\in D_n,t_i<t_{j(l)}\leq t} \frac{(t_{i+1}-t_i)}{t_{i+1}^{\frac{3}{4}}} \int_{t_{j(l)}}^{t_{j(l)+1}}\frac1{(s-t_{i+1})^{\frac34}}ds.\end{aligned}$$ Now, using that for $i<j(l)$ $$\int_{t_{j(l)}}^{t_{j(l)+1}}\frac1{(s-t_{i+1})^{\frac34}}ds \leq \int_{t_{j(l)}}^{t_{j(l)+1}}\frac1{(s-t_{j(l)})^{\frac34}}ds \leq4|D_n|^{\frac14},$$ and that $$\sum_{t_i\in D_n,t_i<t_{j(l)}\leq t} \frac{(t_{i+1}-t_i)}{t_{i+1}^{\frac{3}{4}}}\leq\int_0^1\frac1{s^{\frac34}}ds<\infty,$$ we obtain easily that $$\lim_{n\to\infty}I_{1,2,2,2}=0.\label{aaa7}$$ So, putting together (\[aaa1\])-(\[aaa7\]), we have proved (\[simpli\]). Now, given $f\in\mathcal H$, let us consider $\{f_n\}_{n\in{\mathbb{N}}}$ a sequence of elementary functions converging to $f$ in $\mathcal H$, and we define $$\int_0^t\int_{{\mathbb{R}}}f(x,s)dL_s^x=L^1-\lim_{n\to\infty}\left(\int_0^t\int_{{\mathbb{R}}}f_{n}(x,s)dL_s^x\right).$$ Clearly, this limit exists. Indeed, for any $\varepsilon>0$ there exists $n_0$ such that for any $n,m\geq n_0$, $\|f_n-f_m\|<\varepsilon$ and using inequality (\[simpli\]) we obtain that $$\begin{aligned} E\left|\int_0^t\int_{{\mathbb{R}}}f_n(x,s)dL_s^x-\int_0^t\int_{{\mathbb{R}}}f_{m}(x,s)dL_s^x\right| =E\left|\int_0^t\int_{{\mathbb{R}}}(f_n(x,s)-f_m(x,s))dL_s^x\right|\leq\|f_n-f_m\|<\varepsilon.\end{aligned}$$ Moreover, using again inequality (\[simpli\]), it is clear that the definition does not depend on the choice of the sequence $(f_n)$. Indeed, given $(f_n^1)_{n\in{\mathbb{N}}}$ and $(f_n^2)_{n\in{\mathbb{N}}}$ two sequences converging to $f$ in $\mathcal H$, we have $$\begin{aligned} E\left(\left|\int_0^t\int_{{\mathbb{R}}}f_n^1(x,s)dL_s^x-\int_0^t\int_{{\mathbb{R}}}f_{n}^2(x,s)dL_s^x\right|\right)\leq\|f_n^1-f_n^2\|\leq\|f_n^1-f\|+\|f-f_n^2\|,\end{aligned}$$ that goes to zero when $n$ tends to infinity. $\Box$ \[obsC\] If $f$ satisfies condition [**(C)**]{}, from Theorem \[teoM\] we know that the quadratic covariation $\left[f(X,\cdot),X\right]$ exists. Moreover, if $f\in\mathcal H$, from the uniqueness of the extension in the construction of the integral in Theorem \[teo2\] we get that $$\int_0^t\int_{{\mathbb{R}}}f(x,s)dL_s^x=-\left[f(X,\cdot),X\right]_t.$$ The following results is an obvious consequence of Theorem \[teoM\] and Remark \[obsC\]. \[elcor\] Let $u$ be a process satisfying [**(H1)**]{} and [**(H2)**]{}. Set $X=\int_0^t us dW_s$. Consider a sequence of partitions $D_n$ of partitions of $[0,1]$ verifying conditions [**(M)**]{}. Let $F(x,t)$ be an absolutely continuous function in $x$ such that the partial derivative $f(\cdot,t)$ satisfies [**(C)**]{}. Then, if $f\in\mathcal H$, we have the following extension for the It[ô]{}’s formula: $$F(X_t,t)=F(0,0)+\int_0^tf(X_s,s)dX_s-\frac12\int_0^t\int_{{\mathbb{R}}}f(x,s)dL_s^x+\int_0^tF(X_s,ds).$$ It[ô]{}’s formula extension =========================== Now we can state the main result of this paper. \[teo3\] $ $ Hypotheses over the martingale: 1. Let $u$ be an adapted process satisfying [**(H1)**]{} and [**(H2)**]{}. Set $X=\int_0^t u_s dW_s$. Hypotheses over the function: 1. Let $F$ be a function defined on ${\mathbb{R}}\times[0,1]$ such that $F$ admits first order Radon-Nikodym derivatives with respect to each parameter. 2. Assume that these derivatives satisfy that for every $A\in{\mathbb{R}}$, $$\begin{aligned} \int_0^1\int_{-A}^{A}\left|\frac{\partial F}{\partial t}(x,s)\right|dx \frac1{\sqrt{s}} ds&<&+\infty \\ \int_0^1\int_{-A}^{A}\left(\frac{\partial F}{\partial x}(x,s)\right)^2dx \frac1{\sqrt{s}} ds&<&+\infty.\end{aligned}$$ Then, for all $t\in[0,1]$, $$\begin{aligned} F(X_t,t)=F(0,0)+\int_0^t\frac{\partial F}{\partial x}(X_s,s)dX_s+\int_0^t\frac{\partial F}{\partial t}(X_s,s)ds-\frac12\int_0^t\int_{{\mathbb{R}}}\frac{\partial F}{\partial x}(x,s)dL_s^x.\end{aligned}$$ Using localization arguments we can assume that $F$ has compact support and $$\begin{aligned} \int_0^1\int_{{\mathbb{R}}}\left|\frac{\partial F}{\partial t}(x,s)\right|dx \frac1{\sqrt{s}} ds&<&+\infty \\ \int_0^1\int_{{\mathbb{R}}}\left(\frac{\partial F}{\partial x}(x,s)\right)^2dx \frac1{\sqrt{s}} ds&<&+\infty.\end{aligned}$$ Let $g\in\mathcal C^{\infty}$ be a function with compact support from ${\mathbb{R}}$ to ${\mathbb{R}}^{+}$ such that $\int_{{\mathbb{R}}}g(s)ds=1$. We define, for any $n\in{\mathbb{N}}$, $$g_n(s)=ng(ns)$$ and $$F_n(x,t)=\int_0^1\int_{{\mathbb{R}}}F(y,s)g_n(t-s)g_n(x-y)dyds.$$ Then $F_n\in\mathcal C^{\infty}({\mathbb{R}}\times[0,1])$. Hence, by the usual It[ô]{}’s formula, for every $\varepsilon>0$, we can write $$\begin{aligned} \label{nova} F_n(X_t,t)=F_n(X_{\varepsilon},\varepsilon)+\int_{\varepsilon}^t\frac{\partial F_n}{\partial x}(X_s,s)dX_s+\int_{\varepsilon}^t\frac{\partial F_n}{\partial t}(X_s,s)ds+\frac12\int_{\varepsilon}^t u_s^2\frac{\partial^2 F_n}{\partial x^2}(X_s,s)ds.\end{aligned}$$ Using the arguments of Az[é]{}ma [*et al.*]{} (1998) we will study the convergence of (\[nova\]). Since $F$ is a continuous function with compact support, it is easy to check that $(F_n(X_t,t))_{n\in{\mathbb{N}}}$ converges in probability to $F(X_t,t)$. On the other hand $$\int_0^1\int_{{\mathbb{R}}} \left| \frac{\partial F}{\partial t}(x,s) \right| dxds\leq\int_0^1\int_{{\mathbb{R}}}\left|\frac{\partial F}{\partial t}(x,s)\right|dx \frac1{\sqrt{s}} ds<+\infty.$$ Hence, $\frac{\partial F}{\partial t}\in L^1({\mathbb{R}}\times[0,1])$. Under our hypothesis over the martingale $X$, it follows from Proposition \[2.6MN\] and Lemma \[lema10MN\] that for any $t \in [0,1]$, the random variable $X_t$ is absolutely continuous with density $p_t$ satisfying the estimate $$p_t(x)\leq\frac{C}{\sqrt{t}}.$$ Then, it is easy to see that $\left(\int_{\varepsilon}^t\frac{\partial F_n}{\partial t}(X_s,s)ds\right)_{n\in{\mathbb{N}}}$ converges in probability to $\left(\int_{\varepsilon}^t\frac{\partial F}{\partial t}(X_s,s)ds\right)$. Indeed, $$\begin{aligned} E\left(\left|\int_{\varepsilon}^t\left(\frac{\partial F_n}{\partial t}(X_s,s)-\frac{\partial F}{\partial t}(X_s,s)\right)ds\right|\right) &\leq&\int_{\varepsilon}^t\int_{{\mathbb{R}}}\left|\frac{\partial F_n}{\partial t}(x,s)-\frac{\partial F}{\partial t}(x,s) \right|p_s(x)dxds\\ &\leq&C\int_{\varepsilon}^t\int_{{\mathbb{R}}}\left|\frac{\partial F_n}{\partial t}(x,s)-\frac{\partial F}{\partial t}(x,s) \right|\frac1{\sqrt{s}}dxds\\ &\leq&\frac{C}{\sqrt{\varepsilon}}\int_{\varepsilon}^t\int_{{\mathbb{R}}}\left|\frac{\partial F_n}{\partial t}(x,s)-\frac{\partial F}{\partial t}(x,s) \right|dxds,\end{aligned}$$ that goes to zero, when $n$ tends to infinity, since $\frac{\partial F}{\partial t}\in L^1({\mathbb{R}}\times[0,1])$ and $$\frac{\partial F_n}{\partial t}(x,t)=\int_0^1\int_{{\mathbb{R}}}\frac{\partial F}{\partial t} (y,s)g_n(t-s)g_n(x-y)dyds.$$ Similarly, we can prove that $\left(\int_{\varepsilon}^t\frac{\partial F_n}{\partial x}(X_s,s)dX_s\right)_{n\in{\mathbb{N}}}$ converges in probability to $\left(\int_{\varepsilon}^t\frac{\partial F}{\partial x}(X_s,s)dX_s\right)$. Indeed, using the same arguments we get that $\frac{\partial F}{\partial x}\in L^2({\mathbb{R}}\times[0,1]).$ Then, $$\begin{aligned} &&E\left(\left|\int_{\varepsilon}^t\left(\frac{\partial F_n}{\partial x}(X_s,s)-\frac{\partial F}{\partial x}(X_s,s)\right)dX_s\right|^2\right) \\&&\quad = E\left(\left|\int_{\varepsilon}^t\left(\frac{\partial F_n}{\partial x}(X_s,s)-\frac{\partial F}{\partial x}(X_s,s)\right)u_sdW_s\right|^2\right)\\ &&\quad = E\left(\int_{\varepsilon}^t\left(\frac{\partial F_n}{\partial x}(X_s,s)-\frac{\partial F}{\partial x}(X_s,s)\right)^2u_s^2ds\right).\end{aligned}$$ Following the same ideas of Proposition 12 in Moret and Nualart (2000), Proposition \[corol2MN\] and Lemma \[lema10MN\] yield the following bound for the last expression $$\begin{aligned} &&C \int_{\varepsilon}^t\int_{{\mathbb{R}}}\left(\frac{\partial F_n}{\partial x}(x,s)-\frac{\partial F}{\partial x}(x,s) \right)^2\frac1{\sqrt{s}}dxds\nonumber\\ &\leq&\frac{C}{\sqrt{\varepsilon}} \int_{\varepsilon}^t\int_{{\mathbb{R}}}\left(\frac{\partial F_n}{\partial x}(x,s)-\frac{\partial F}{\partial x}(x,s) \right)^2dxds\label{ladarrera}\end{aligned}$$ that goes to zero when $n$ tends to infinity, since $\frac{\partial F}{\partial x}\in L^2({\mathbb{R}}\times[0,1])$ and $$\frac{\partial F_n}{\partial x}(x,t)=\int_0^1\int_{{\mathbb{R}}}\frac{\partial F}{\partial x} (y,s)g_n(t-s)g_n(x-y)dyds.$$ So, letting $n$ to infinity in (\[nova\]), we get that the sequence $$\left(\frac12\int_{\varepsilon}^tu_s^2\frac{\partial^2 F_n}{\partial x^2}(X_s,s)ds\right)_{n\in{\mathbb{N}}}$$ converges in probability to $$F(X_t,t)-F(X_{\varepsilon},\varepsilon)-\int_{\varepsilon}^t\frac{\partial F}{\partial x}(X_s,s)dX_s-\int_{\varepsilon}^t\frac{\partial F}{\partial t}(X_s,s)ds.$$ But, since $\frac{\partial F_n}{\partial x}(x,s)I_{(\varepsilon,t)}(s)\in\mathcal H$, from Theorem \[teoM\] and Corollary \[elcor\], we get that $$\begin{aligned} \int_{\varepsilon}^tu_s^2\frac{\partial^2 F_n}{\partial x^2}(X_s,s)ds&=&\left[\frac{\partial F_n}{\partial x}(X,\cdot),X\right]_t-\left[\frac{\partial F_n}{\partial x}(X,\cdot),X\right]_{\varepsilon}\\ &=&-\int_0^1\int_{{\mathbb{R}}}\frac{\partial F_n}{\partial x}(x,s)I_{(\varepsilon,t)}(s)dL_s^x.\end{aligned}$$ The next step of the proof is to check that $\left( \frac{\partial F_n}{\partial x}(x,s)I_{(\varepsilon,t)}(s),x\in{\mathbb{R}},s\in[0,1]\right)_{n\in{\mathbb{N}}}$ converges in $\mathcal H$ to $\big( \frac{\partial F}{\partial x}(x,s)I_{(\varepsilon,t)}(s),x\in{\mathbb{R}}, s\in[0,1]\big)$. It suffices to notice that, $$\int_{\varepsilon}^t\int_{{\mathbb{R}}}\left(\frac{\partial F_n}{\partial x}(x,s)-\frac{\partial F}{\partial x}(x,s)\right)^2\frac1{s^{\frac34}}dxds\leq \frac1{{\varepsilon}^{\frac34}}\int_{\varepsilon}^t\int_{{\mathbb{R}}}\left(\frac{\partial F_n}{\partial x}(x,s)-\frac{\partial F}{\partial x}(x,s)\right)^2dxds$$ that converges to zero when $n$ tends to infinity. Then, we clearly have proved that $$\left(\int_0^1\int_{{\mathbb{R}}}\frac{\partial F_n}{\partial x}(x,s)I_{(\varepsilon,t)}(s)dL_s^x\right)_{n\in{\mathbb{N}}}$$ converges in $L^1$ to $\int_0^1\int_{{\mathbb{R}}}\frac{\partial F}{\partial x}(x,s)I_{(\varepsilon,t)}(s)dL_s^x$. So, we have that for any $\varepsilon>0$ $$\begin{aligned} \label{eq4} F(X_t,t)=F(X_{\varepsilon},\varepsilon)+\int_{\varepsilon}^t\frac{\partial F}{\partial x}(X_s,s)dX_s+\int_{\varepsilon}^t\frac{\partial F}{\partial t}(X_s,s)ds-\frac12\int_0^1\int_{{\mathbb{R}}}\frac{\partial F}{\partial x}(x,s)I_{(\varepsilon,t)}(s)dL_s^x.\end{aligned}$$ The last steep is to let $\varepsilon$ to zero. But we need to check that the limit of the stochastic integral exists. Actually, it is enough to show that $$E\left(\left|\int_0^t\frac{\partial F}{\partial t}(X_s,s)ds\right|\right)<\infty$$ and that $$E\left(\int_0^t\frac{\partial F}{\partial x}(X_s,s)dX_s\right)^2<\infty.$$ But, $$E\left(\left|\int_0^t\frac{\partial F}{\partial t}(X_s,s)ds\right|\right)\leq C \int_0^1\int_{{\mathbb{R}}}\left|\frac{\partial F}{\partial t}(x,s)\right|dx \frac1{\sqrt{s}} ds<+\infty.$$ On the other hand, following the same type of arguments that in (\[ladarrera\]), we are able to write $$\begin{aligned} &&E\left(\int_0^t\frac{\partial F}{\partial x}(X_s,s)dX_s\right)^2 = E \left(\int_0^t\left(\frac{\partial F}{\partial x}(X_s,s)\right)^2u_s^2ds\right)\\ \quad & & \leq C \int_{\mathbb{R}}\int_0^t\left(\frac{\partial F}{\partial x}(x,s)\right)^2 \frac1{\sqrt{s}}dsdx <\infty.\end{aligned}$$ Letting $\varepsilon$ to zero, the proof is finished. $\Box$ \[obs6\] Notice that under the hypotheses of Theorem \[teo3\], it is possible that $\frac{\partial F}{\partial x}$ does not belong to the space $\mathcal H$. In this case, using the localization arguments, we can always assume that $\left( \frac{\partial F}{\partial x}(x,s)I_{(\varepsilon,t)}(s),\, x\in{\mathbb{R}}, s\in[0,1]\right)$ belongs to $\mathcal H$ for any $\varepsilon>0$ and we can define $$\int_0^t\int_{{\mathbb{R}}}\frac{\partial F}{\partial x}(x,s)dL_s^x=\lim_{\varepsilon\to 0}\int_0^1\int_{{\mathbb{R}}}\frac{\partial F}{\partial x}(x,s)I_{(\varepsilon,t)}(s)dL_s^x.$$ This limit exists in probability since all the other limits in (\[eq4\]) exist. Acknowledgements {#acknowledgements .unnumbered} ================ This work was partially supported by DGES Grants MTM2006-01351 (Carles Rovira) and MTM2006-06427 (Xavier Bardina). [99]{} : [Az[é]{}ma, J., Jeulin, T., Knight, F., Yor, M. (1998)]{} [Quelques calculs de compensateurs impliquant l’injectivit[é]{} de certains processus croissants.]{} [*S[é]{}minaire de Probabilit[é]{}s, XXXII, Lecture Notes in Math., [**1686**]{},*]{} [316-327.]{} : [Bardina, X., Jolis, M. (1997)]{} [An extension of Itô’s formula for elliptic martingale processes.]{} [*Stochastic Process. Appl.*]{} [**69**]{} [(1), 83-109.]{} : [Bardina, X., Jolis, M. (1997)]{} [Estimations of the density of hypoelliptic diffusion processes with applications to an extension of Itô’s formula.]{} [*J. Theoret. Probab.*]{} [**15**]{} [(1), 223-247.]{} : [Bardina, X., Rovira, C. (2007)]{} [On Itô’s formula for elliptic diffusion processes.]{} [*Bernoulli*]{} [**13**]{} [(3), 820-830.]{} : [Bouleau, N., Yor, M. (1981)]{} [Sur la variation quadratique des temps locaux de certaines semimartingales.]{} [*C. R. Acad. Sci. Paris S[é]{}r. I Math.*]{} [**292**]{} [(9), 491-494.]{} : [Di Nunno, G., Meyer-Brandis, T., [Ø]{}ksendal, B., Proske, F. (2005)]{} Malliavin calculus and anticipative Itô formulae for Lévy processes. [*Infin. Dimens. Anal. Quantum Probab. Relat. Top.*]{} [**8**]{} (2), 235–258. : [Dupoiron, K., Mathieu, P., San Martin, J. (2004)]{} [Formule d’It[ô]{} pour des martingales uniform[é]{}ment elliptiques, et processus de Dirichlet.]{} [*Potential Anal.*]{} [**21**]{} [(1), 7-33.]{} : [Eisenbaum, N. (2000)]{} [Integration with respect to local time.]{} [*Potential Anal.*]{} [**13**]{} [(4), 303-328.]{} : [Eisenbaum, N. (2001)]{} [On It[ô]{}’s formula of F[ö]{}llmer and Protter.]{} [*S[é]{}minaire de Probabilit[é]{}s, XXXV, Lecture Notes in Math. [**1755**]{},*]{} [390-395.]{} : Flandoli, F., Russo, F., Wolf, J. (2004) Some SDEs with distributional drift. II. Lyons-Zheng structure, Itô’s formula and semimartingale characterization. [*Random Oper. Stochastic Equations*]{} [**12**]{} (2), 145–184. : [F[ö]{}llmer, H., Protter, P., Shiryayev, A.N. (1995)]{} [Quadratic covariation and an extension of It[ô]{}’s formula.]{} [*Bernoulli*]{} [**1**]{} [(1-2), 149-169.]{} : [Ghomrasni, R., Peskir, G. (2003)]{} [Local time-space calculus and extensions of It[ô]{}’s formula.]{} [*High dimensional probability, III (Sandjberg, 2002), Progr. Probab.,*]{} [**55**]{}, [177-192.]{} : [Moret, S. (1999)]{} [Ph.D. thesis: “Generalitzacions de la Formula d’Itô i estimacions per martingales".]{} [*Universitat de Barcelona*]{}. : [Moret, S., Nualart, D. (2000)]{} [Quadratic Covariation and Itô’s Formula for Smooth Nondegenerate Martingales.]{} [*Journal of Theoretical Probability*]{} [**13**]{}, [193-224.]{} : [Nualart, D. (2006)]{} [*Malliavin Calculus and Related Topics. Second edition.*]{} Springer-Verlag. : [Nualart, D. (1998)]{} Analysis on Wiener space and anticipating stochastic calculus and Related Topics. In: École d’été de Saint-Flour XXV. [*Lect. Notes in Math.*]{} [**1690**]{}, [123-227.]{}

--- author: - Daniel Dietsch - Matthias Heizmann - Jochen Hoenicke - | \ Alexander Nutz - Andreas Podelski bibliography: - 'main.bib' subtitle: | A Program Transformation for\ Intermediate Verification Languages title: Different Maps for Different Uses --- Introduction ============ Example {#chap_heapsep_sec_example} ======= Preliminaries ============= Dependency Analysis {#chap_heapsep_sec_dataflow} =================== Computing Dependencies {#chap_heapsep_sec_preprocessing} ====================== Program Transformation {#chap_heapsep_sec_transformation} ====================== Implementation in Ultimate ========================== Experiments on a Scalable Benchmark Suite ========================================= Related Work ============ Discussion {#chap_heapsep_sec_discussion} ========== Conclusion ==========

--- abstract: 'Lamellar gratings are widely used diffractive optical elements; gratings etched into Si can be used as structural or prototypes of structural elements in integrated electronic circuits. For the control of the lithographic manufacturing process, a rapid in-line characterization of nanostructures is indispensable. Numerous studies on the determination of regular geometry parameters of lamellar gratings from optical and Extreme Ultraviolet (EUV) scattering highlight the impact of roughness on the optical performance as well as on the reconstruction of these structures. Thus, a set of nine lamellar Si-gratings with a well-defined line edge roughness or line width roughness was designed. The investigation of these structures using EUV small angle scattering reveals a strong correlation between the type of line roughness and the angular scattering distribution. These distinct scatter patterns open new paths for the unequivocal characterization of such structures by EUV scatterometry.' author: - 'A. Fernández Herrero' - 'M. Pflüger' - 'J. Probst' - 'F. Scholze' - 'V. Soltwisch' bibliography: - 'main.bib' title: Characteristic diffuse scattering from distinct line roughnesses --- introduction ============ Lithographically manufactured nanostructures play an important role as structural elements of integrated electronic circuits. With shrinking structure sizes, the impact of the roughness has gained more influence on their performance. The demand for better manufactured nanostructures has motivated the development of methods and techniques for structure and roughness analysis. For high resolution surface analysis, scanning techniques have been widely used. These methods have the advantage of measuring in real space and they have already been used for the characterization of the roughness in nanopatterned structures  [@fouchier_atomic_2013; @patsis_roughness_2003]. Despite the advantages of these direct techniques, long measuring times are needed if one is to obtain relevant statistical information. However, for the control of the lithographic manufacturing process, a rapid in-line and alteration-free characterization of such structures is indispensable  [@bunday_hvm_2016]. Indirect optical methods are non-destructive techniques with a much lower acquisition time. Besides the well-established optical methods, also X-ray methods are investigated for future metrology solutions  [@Lemaillet_intercomparison_2013]. The measurements can be performed in transmission geometry, such as in Critical Dimension Small-Angle X-ray Scattering (cd-SAXS)  [@thiel_advances_2011] or in reflection geometry, where the enhancement of the surface signal can be obtained by illuminating the sample at a grazing incident angle, close to the critical angle of reflection. Grazing incidence small angle scattering is a well-established  [@muller-buschbaum_grazing_2003; @muller-buschbaum_basic_2009; @hofmann_grazing_2009], rapid and non-destructive technique used for the characterization of nanopatterned surfaces. The scattering pattern provides statistical information about the illuminated area. Nevertheless, it is an indirect technique which requires a non-straightforward data interpretation  [@renaud_probing_2009], i.e. the structures and their uncertainties must be reconstructed from the scattered intensities. In this regard, reasonable understanding has been achieved when laterally periodic samples are investigated  [@baumbach_grazing_1999; @yan_intersection_2007; @wernecke_direct_2012]. Significant diffuse scattering, caused by rough lamellar gratings, has been constantly reported  [@rueda_grazing-incidence_2012; @suh_characterization_2016; @soltwisch_correlated_2016]. Recently, also all dominant diffuse scatter contributions could be attributed to basic scattering processes  [@soltwisch_correlated_2016] similar to scattering in multilayer systems. Roughness in multilayer systems is usually understood as surface or interface roughness, which has been widely studied by X-ray scattering and reflectivity  [@mikulik_x-ray_1999; @haase_multiparameter_2016]. In latterally nanopatterned structures the equivalent to interface roughness is the unevenness of the sidewalls. In an idealized figure, in a two-dimensional representation of the nanostructured surface, this can be described by the edge roughness, i.e. the deviation of the actual edge contour from a straight line. For line-and-space structures, it is usually classified into two types: line edge roughness (LER) where the line centre position varies along the line and line width roughness (LWR), where the width of the line changes along the line. Although there is no pure LER nor LWR in real structures, this distinction has been made for simplification and the separate study of the impact of the roughness in several analyses  [@wang_characterization_2007; @kato_effect_2010]. These calculations use Fourier optics and a binary grating, and conclude that a Debye-Waller-like factor can describe the impact of the roughness on the diffraction efficiency. Following these results one could assume that for the description of real samples the superposition of periodic functions might be sufficient. But this assumption has not been corroborated yet or exploited to characterize the roughness-induced scatter of real samples. Considering those previous reports, we have designed a set of nine gratings, comprising LWR and LER with different distributions to be investigated by Extreme Ultraviolet (EUV) scatterometry. EUV small angle scattering exploits the high sensitivity of grazing incidence techniques while reducing the beam footprint, due to the larger incidence angles; which allows the investigation of smaller samples. The study presented here aims to provide a better understanding of scattering caused by line roughness. The samples were illuminated at several grazing incidence angles and the specular reflection and the discrete diffraction orders as well as the diffuse scattering distributions were recorded. The distinct distribution of the scattering patterns opens new paths for the unequivocal characterization of such structures by EUV scatterometry. Experimental Details ==================== EUV-Small Angle Scattering -------------------------- The experiments were conducted at the soft X-ray beamline  [@scholze_high-accuracy_2001] of the Physikalisch-Technische Bundesanstalt (PTB), which covers the photon energy range from $50$ eV to $1700$ eV, at the electron storage ring BESSY II. The experimental set-up is illustrated in Fig. \[figure\_sketch\]. A monochromatic X-ray beam with a wavevector $\vec{k}_i$ impinges on the sample surface at an incidence angle $\alpha_i$. The elastically scattered wavevector $\vec{k}_f$ propagates along **the direction with** exit angle $\alpha_f$ and the azimuthal angle $\theta_f$. The sample is illuminated in a conical diffraction mounting with the incidence plane parallel to the grating lines, $\varphi=0$, and placed in a goniometer, which allows us to rotate and move the sample, inside a vacuum chamber. The detector is a $2048 \times 2048$ pixel Andor CCD camera with a pixel size of $13.5$ $\mu$m, which is placed at $14^\circ$ to the incoming beam and at $740$ mm off the sample, covering a field of view of approximately $2^\circ \times 2^\circ$ with the specular reflection of $\alpha_i = 7 ^\circ$ centered at the CCD. The orders of diffraction are given by the intersection of the Ewald sphere of elastic scattering with the reciprocal lattice. The coordinates in reciprocal space correspond to the momentum transfer: $$\begin{pmatrix} q_x \\ q_y \\ q_z \end{pmatrix} = \frac{2\pi}{\lambda} \begin{pmatrix} \cos(\theta_f) \cos(\alpha_f) - cos(\alpha_i)\\ \sin(\theta_f) \cos(\alpha_f)\\ \sin(\alpha_f) + \sin(\alpha_i) \end{pmatrix} \label{eq_q}$$ If the projection of the incidence plane is parallel to the grating lines $\varphi = 0$; the diffraction orders describe a semicircle in the detector plane [@mikulik_coplanar_2001; @yan_intersection_2007] with its center at the intersection of the sample horizon and the specular reflection plane. Therefore, the azimuthal angle $\varphi$ was aligned such that this condition was met, with the elevation angle from the sample horizon of the respective positive and negative diffraction orders being equal. The achieved angular uncertainty in $\varphi$ was $0.02 ^\circ$, which is sufficient for this experiment. The experiment is analogous to the well-known Grazing Incidence Small Angle X-ray Scattering (GISAXS) technique but working with lower photon energy which allows to use a larger incidence angle, significantly reducing the beam footprint on the sample. ![Experimental set-up. The red frame shows the area covered by our CCD. []{data-label="figure_sketch"}](exp_sketch_FINAL.pdf){width="40.00000%"} Sample Design ------------- Nine Si-gratings with different types of roughness were prepared by e-beam lithography and reactive ion etching at the Helmholtz-Zentrum Berlin. Each grating has a size of by , with the lines parallel to the long direction. The first grating is a reference grating with no artificial roughness added. The other eight gratings were designed to accomplish a well-defined line roughness, introducing perturbations to the ideal grating. The parameters of the gratings, such as pitch and linewidth, were chosen to be compatible with the roughness amplitudes. The width of the trenches between the lines influences the etching rate into the silicon during the reactive ion etching process such that for too narrow trenches the etch depth is reduced. Therefore, the pitch and linewidth, and the perturbations introduced were chosen to keep the trench width above a threshold of to assure constant etch depth along the gratings. Thus, the pitch is , the nominal linewidth and the nominal etch depth, i.e. line height . Following the previous studies on the impact of the line roughness, we have designed four samples with a periodic roughness distribution. However, in order to study the effect of the roughness in real samples we have completed the set with another four gratings with a stochastic roughness distribution. Two samples of each distribution correspond to a different type of line roughness: line edge roughness or line width roughness. For the LER, the size of the pitch and of the linewidth are maintained constant, but the line centre position along the line is changed. On the other hand, for the LWR the centre position is kept constant, and also the pitch, while the width of the line is varied along the line. For the reference grating, each line can be understood as a chain of juxtaposed boxes long and wide, centred at $x_0$ in a nominal pitch of ; for the other eight gratings a perturbation is introduced to each of the boxes (see Fig. \[periodic\_sem\_sketch\] a) or Fig. \[stochastic\_sem\_sketch\] a)).\ For the periodic roughness, we consider a basis cell composed of two adjacent boxes centred in the pitch, i.e. a size of by . A positive or negative perturbation, $\delta$, is introduced alternatively to the centre position of each box, in the LER frame, or to the width of the box, for the samples with LWR (see Fig. \[periodic\_sem\_sketch\] (a)). This basis cell is repeated until the sample area of by is completed. Thus there are two periodic dimensions: the intrinsic periodicity of the grating, i.e. the pitch, $p$, in the *y*-direction and an artificial periodicity caused by the periodic roughness distribution, $p_r$, long along the line (in *x*-direction) and identical lines are placed next to each other (see Fig. \[periodic\_sem\_sketch\] (b)). For each type of line roughness, two samples, each of them with a different perturbation amplitude $\delta_j$, were prepared, $\delta_1 = \SI{5}{\nm}$ and $\delta_2 = \SI{10}{\nm}$. ![Periodic roughness: a) Basis cell for the construction of the periodic LER (up) and LWR (bottom). b) SEM images of a sample with periodic LER (up) and LWR (bottom) with pitch of and a period of the roughness of .[]{data-label="periodic_sem_sketch"}](periodic_sem_sketchIII.pdf){width="48.00000%"} Thus, adjacent blocks in the LER structure have a constant centre displacement and adjacent blocks of the LWR structure have a constant difference in width of 2$\delta_j$.\ For the stochastic roughness a squared basis cell of side length was used. The size of the basis cell was chosen as a compromise to, on the one hand, limit the amount of data required for the e-beam writing process and on the other hand, to be large enough that, when repeated to fill the length of the grating, no observable periodicity is expected. Every a new line of juxtaposed boxes is built, each of these lines follows a uniform discrete distribution. As was done for the periodic roughness, two samples with different amplitudes are designed with LER and another two with LWR. For each type of line roughness, one samples has a maximum perturbation amplitude of $\SI{10}{\nm}$ and the other of . The e-beam writer’s resolution was set to be because smaller values would have increased the writing time drastically, which means all line boundaries are located on this discrete grid. Therefore, the distribution of the actual perturbation amplitude is in discrete steps of for the LER samples but of for the LWR samples, as a perturbation in the latter type involves the displacement of two boundaries in opposite directions. In these samples, there is no constant displacement or width difference between juxtaposed boxes but rather a stepwise interval from a difference to a maximum of or , respectively. ![Stochastic roughness: a) Design of a basis cell of , where each line follows a uniform distribution. b) SEM images of a sample with LER (top) and LWR (bottom).[]{data-label="stochastic_sem_sketch"}](stochastic_sem_sketchIV.pdf){width="48.00000%"} Results ======= The scatter distributions presented here were measured at a wavelength of , or a photon energy of 1 keV, and three different grazing angles. The CCD camera was positioned at $\ang{14}$ to the direct beam. The first incidence angle, $\alpha_i=\ang{7}$ was chosen to have specular reflection centred at the CCD camera. For further images, the sample was rocked until the orders of diffraction were out of the CCD image area ($\alpha_i=\ang{6.45},\ang{5.54}$) to enable long exposures of the diffuse scatter distributions without saturating the CCD camera. ![2D scattering pattern from the reference grating. No significant diffuse scattering is observable.[]{data-label="reference"}](reference.png){width="35.00000%"} {width="75.00000%"}\ The data are represented in the $(q_x, q_y)$ momentum space; the discrete orders of diffraction for the regular grating are found at $q_x = 0$ (see Fig. \[reference\]). Due to the larger footprint of the beam compared to the grating size, we have the contribution of the specular reflection coming from the substrate, which is superposed to the zero order of diffraction, see the larger spot size of the zero order shown in Fig.\[reference\]. The distance between the orders of diffraction in $q_y$ corresponds to the pitch size, $\Delta q_y=\frac{2\pi}{p}$. Then the sample was rocked and measured for the other angles of incidence. We acquired these images with a longer exposure time. All rough samples show intense contributions which are not observable at the reference grating due to the high quality of the structure (compare Fig. \[reference\] and Fig. \[figure\_diffuse\_regular\]). Note that just the samples with a stochastic distribution present diffuse scattering patterns. Periodic Line Roughness ----------------------- The four samples with a periodic roughness have a two-dimensional periodicity: one is the pitch of the grating, $p$, along the *y*-axis and the second one is the roughness periodicity, $p_r$, in the *x*-direction. Due to this particularity, satellite orders are observable at frequencies $q_x= n \frac{2\pi}{p_r}$ with integer n, see Fig. \[figure\_diffuse\_regular\] a) and b). These periodicities dominate the intensity distribution of the out-of-plane scattering pattern. Thus, for the samples with a periodic roughness no diffuse scatter is observed but the light scattering is fully governed by the two-directional periodicity of the sample. This case of periodic roughness was already studied using Fourier optics. Several authors have focused their work on the behaviour and the impact of the roughness on the diffraction orders, for instance @kato_analytical_2012 and @wang_characterization_2007. Both studies consider a binary model and distinguish between LER and LWR. Their proposed models and their applicability are discussed here. The distribution of the satellite orders is qualitatively similar to the results obtained by  @kato_analytical_2012. Here the authors showed that for a grating with a sinusoidal line-width or -position variation, the intensity of the satellite orders is given by Bessel functions of the first kind. For LER the intensity of the satellite peaks at $q_y= 0$ diminishes while for LWR only the first order satellite peak at $q_y= 0$ exists. This is clearly confirmed by our measurements, where the satellite peak at $q_y=0$ for samples with LER is only very weak as compared to the LWR samples, see Fig. \[figure\_diffuse\_regular\] b) and c) for $\alpha_{i,3} = 5.59^\circ$. The remaining low intensity for the LER sample can be explained by the structures having a small amount of LWR also present in the LER structures. However, we observe that the two-dimensional model considered by @kato_analytical_2012 is not sufficient for the characterization of the roughness amplitudes. In the same way, the model described by @wang_characterization_2007, which uses a unit cell with a two-dimensional lattice formed by the stack of two blocks, and carried out in the frame of transmission cd-SAXS, is not applicable to a grazing incident measurement. By considering a binary grating, the influence of the height of the structure on the intensity distribution, i.e. the possible effect in the $q_z$ component of the scattering vector, is underestimated. In the frame of cd-SAXS, in the study of @wang_characterization_2007 this effect is totally disregarded as the $q_z$ changes are not observable in these measurements. In the same way, other studies performed in transmission cd-SAXS, for instance @freychet_study_2016, can not be directly compared to the results discussed here. Therefore, we can conclude that the off-specular scattering intensity distribution from samples with periodic roughness is dominated by the two-dimensional pitch. However, the existing studies do not consider the corresponding changes in the $q_z$ component of the scattering vector which strongly influences the intensity of the satellite orders. This effect is clearly visible in the diffuse scattering patterns given by the samples with stochastic roughness in the next section. Stochastic Line Roughness ------------------------- In contrast to the patterns from samples with periodic roughness, samples with stochastic line roughness just have the periodicity given by the pitch $p$, and therefore no satellite peaks are observed but a purely diffuse scattering pattern. In multilayer systems, the resonant diffuse scattering (RDS) is well known. It appears due to the correlation of the roughness of the interfaces  [@kaganer_bragg_1995; @holy_nonspecular_1994]. For lamellar gratings these effects were already reported  [@soltwisch_correlated_2016; @hlaing_nanoimprint-induced_2011] in the form of palm-like diffuse scattering sheets, caused by interference within the *effective layer* of the grating. In our case, depending on the type of line roughness, LER or LWR, the diffuse scattering pattern shows a different angular intensity distribution (see Fig. \[figure\_diffuse\_regular\] c) (LER) and d) (LWR)). A combination of the LER and LWR contributions corresponds to the palm-like pattern usually observed when real samples are investigated. Here each type of roughness leads to a characteristic diffuse scattering pattern. In line with the observation for the periodic samples, no intensity is observable for the off-specular scattering of the LER structures at $q_y=0$. The maxima of the diffuse scatter are shifted by a half-period with respect to the discrete diffraction orders. This is particularly notable because the diffraction intensities for the quasi-periodic samples are, naturally, aligned with the discrete diffraction orders of the regular grating. It is therefore not possible to obtain the scattering pattern of the non-regular LER sample from a superposition of quasi-peroidic solutions. In contrast, for LWR diffuse scattering along $q_y=0$ is observed and the diffuse scatter distributions are in phase with the discrete diffraction orders. This difference allows distinguishing between samples with LER and LWR. This difference on the behaviour of the constructive interference for samples with LER and LWR also implies that the rigorous calculations, applicable for periodic roughness distributions, cannot be applied to a sample with a stochastic roughness distribution. ![The curves in the lower panel show the integration of the signal from the $(q_y,q_z)$ maps (shown above) for one amplitude of the LER (LWR) and the two incident angles where the regular orders are out of field ($\alpha_{i,2}$ and $\alpha_{i,3}$). For the sample with LER (red) the integration was done around $q_y = - \pi/p$, and for the LWR (blue) around $q_y = 0$. In the $(q_y,q_z)$ maps, the integration areas (red or blue, respectively) are shaded.[]{data-label="cut_plot"}](cutplot_qyz.eps){width="45.00000%"} Likewise the influence of the amplitude of the roughness does not influence the angular distribution of the diffuse scattering, just the intensity of the contributions. For a better illustration we also projected the scatter intensities in the $(q_y,q_z)$ plane. Fig. \[cut\_plot\] shows a linear cuts of these projections along $q_z$. The observed modulation is well in phase for both roughness amplitudes in both cases, LER and LWR. We conclude that the palm like modulation is caused by the above mentioned interference effects in the grating *effective layer*, i.e. a $q_z$ effect which is naturally not covered by the previous studies based on Fourier optics  [@kato_analytical_2012]. Conclusion ========== We investigated a set of nine lamellar Si-gratings which comprises samples with different types of line roughness (LER and LWR), roughness distributions (periodic and stochastic) and amplitudes. The periodic and stochastic roughness samples present a different off-specular scatter distribution. Periodic roughness leads to a distribution of satellite orders given by the periodicity of the roughness and the periodicity of the original structure, the pitch. On the other hand, for samples with stochastic roughness a pure diffuse scattering pattern is observable. We have observed a correlation between the type of roughness and the scatter distribution. The intensity of the satellite peaks from samples with periodic roughness provides information on the type of the roughness encountered. The satellite peaks, in $q_x \neq 0$, at $q_y=0$ are suppressed for samples with LER in contrast to samples with a predominant LWR. This finding confirms the previous rigorous calculations where it was stated that the samples with a predominant LER lead to the extinction of the zero-order satellites. However, the existing models do not take into account the $q_z$ effect which is explored in our geometry. For samples with stochastic roughness the distribution of the diffuse scattering strongly depends on the type of roughness. For the LER, no non-specular intensity is observable for $q_y=0$ and the diffuse scattering is phase shifted with respect to the regular diffraction orders of the periodic sample. For the LWR, non-specular intensity is observed at $q_y=0$ and the diffuse scattering is in phase with the regular diffraction orders of the periodic sample. This fact questions previous analytical studies which proposed to use a superposition of solutions for periodic roughness to describe the behaviour of a real sample. Our finding for LER structures, in particular, shows that this is not feasible because of the phase shift in the diffuse scatter pattern. We could show that the diffuse scatter is periodic in $q_z$. The interpretation of the diffuse scattering distributions from these samples, where the parameters are known, opens new perspectives to the characterization of the roughness using EUV scatterometry.

--- abstract: 'In my commentary, I will argue that the conclusions drawn in the paper *Noncommutative causality in algebraic quantum field theory*[^1] by Gábor Hofer-Szabó are incorrect. As proven by J.S. Bell, a local common causal explanation of correlations violating the Bell inequality is impossible.' author: - | Dustin Lazarovici\ Ludwig-Maximilians-University Munich, Department of Mathematics\ dustin.lazarovici@math.lmu.de date: title: | **Why “noncommutative common causes”\ don’t explain anything\ A comment on\ Gábor Hofer-Szabó’s *Noncommutative causality in algebraic quantum field theory*** --- > *“Let me guess. He pulled a lost in translation on you?”*\ > \ What is the meaning of Bell’s theorem? What are its implications for which it was dubbed, and rightfully so, “the most profound discovery of science” (Stapp 1975)? In brief, Bell’s theorem tells us that certain statistical correlations between space-like separated events that are predicted by quantum mechanics and observed in experiment imply that *our world is non-local*. More precisely, it tells us that those correlations are *not locally explainable*, meaning that they cannot be accounted for by any local candidate theory since the frequencies predicted by a local account would have to satisfy a certain inequality – the Bell, respectively the CHSH inequality – that is empirically violated in the pertinent scenarios. Any candidate theory that correctly predicts the violation of the Bell inequality must therefore describe some sort of direct influence between the correlating events, even if they are so far apart that they cannot be connected by a signal moving at maximum the speed of light. Hence we say that the principle of *locality* or *local causality* is violated in nature. The genius of Bell’s argument lies in its simplicity and in its generality. Bell’s theorem is not about quantum mechanics or quantum field theory or *any* theory in particular, it is not confined to the “classical” domain or the quantum domain or a relativistic or non-relativistic domain, it is a *meta-theoretical* claim, excluding (almost) all possibilities of a local explanation for the statistical correlations observed in EPR-type experiments.[^2] Admittedly, a statement about nature can never reach the same degree of rigor as a theorem of pure mathematics for there is always an issue of connecting formal concepts to “real-world” concepts. Bell, however, was one of the clearest thinkers of the 20th century and his analysis, unobscured by the misunderstandings of some of his later commentators, is perfectly precise and conclusive in this respect.[^3] It is against this background that contributions to the subject have to be evaluated.\ The paper *Noncommutative causality in algebraic quantum field theory* by Gábor Hofer-Szabó, that I was gratefully given the opportunity to comment on, seems to be an offspring of a research project started about one and a half decades ago by Miklos Rédei (Rédei 1997) and concerned with the question whether in Algebraic Quantum Field Theory[^4] correlations between space-like separated events (in particular such violating the Bell inequality) have local explanations in terms of “common causes”. You see, what worries me about this research program is that it seems to suggest that the status of Bell’s theorem is not yet clear, that the issue of non-locality is not yet settled, because somehow the technical details of Algebraic Quantum Field Theory could turn out to matter, and type III von-Neumann algebras could turn out to matter, and noncommutativity could turn out to matter. But that would be incorrect; none of this really matters.\ Contrary to what I’ve just so emphatically stated, Mr. Hofer-Szabó makes quite an astonishing announcement. He claims that by committing ourselves to the framework of AQFT and by “embracing noncommuting common causes” we can achieve what Bell’s theorem would seem to exclude, namely to provide a “local (joint common causal) explanation for a set of correlations violating the Bell inequalities”. Although such a statement will certainly make a huge impression on people who believe that noncommutativity holds the one great mystery of quantum physics, we should pause for an instant to assess its plausibility. Physical events, or “causes” and “effects”, whatever we might mean by that, are certainly not the kind of thing that can either commute or not commute. Operators, I grant, can commute or not commute, and so can perhaps elements of lattices with respect to certain set-theoretic operations. Bell’s theorem, however, doesn’t care about any of this. Bell’s argument is solely concerned with the predictions that a candidate theory makes for the probabilities (relative frequencies) of certain physical events, not with the mathematical structure that it posits to make those predictions or represent those events. So how could it be possible to avoid the consequences of Bell’s theorem by denying “commutativity”, which hasn’t been among its premises in the first place?\ Let me try to explain what I think the result presented in the paper of Hofer-Szabó actually consists in and why I think it’s completely missing the point as concerning the issue of local causality. Contrary to what is being suggested in the paper, the existence of a “commuting/noncommuting (weak/strong) (joint) common cause system” according to its definitions 2 and 3 is *not sufficient* for a local (common causal) explanation of correlations between space-like separated events. Such an explanation would at least be required to *reproduce* the statistical correlations that it was set out to explain. The kind of “explanation” that the author provides *doesn’t do this*. As his paper correctly states, the statistics for the events $A_i, B_i$ are different whether the state is first projected on the possible “common causes” (since that’s what happens when we compute $\phi \circ E_c$) or not. Most notably, the probabilities for the correlated events after the “occurrence” (more correctly: measurement) of “noncommuting common causes” (the right-hand-side of eq. below) satisfy the Bell inequality – in accordance with Bell’s theorem – whereas the statistical correlations that the author *claims to explain* violate Bell’s inequality. Note that in the case where $A_i, B_j$ don’t commute with $C_k$ we will generally find that $$\label{noteq} \phi(AB) \neq \sum_{k} \frac{\phi(C_k AB C_k)}{\phi(C_k)} \phi(C_k).$$ This is, I assume, a familiar fact (if you find the notation confusing, write $\langle \psi | C_k A_i C_k | \psi \rangle$ instead of $\phi(C_k A_i C_k)$, and so on). In particular, there is nothing deep or mysterious or metaphysically interesting about it, if only we appreciate the fact that the right-hand-side of does not describe the same physical situation in which the system remains undisturbed in the common past of $A$ and $B$, but that the projection on the common cause system (indeed one could think of a measurement of an observable $C$ with spectral decomposition $\lbrace C_k \rbrace$) affects (decoheres) the quantum state in a way that can influence subsequent events. In our case, it will simply destroy the EPR-correlations, so that violations if the Bell inequality don’t occur at all. In particular, the correlations described by the left-hand-side of are *not explained* by the right-hand-side of since the two probability distributions are different. In (Hofer-Szabó and Vecsernyés 2012), the authors explicitly acknowledge this point, yet respond by saying that “*the definition of the common cause does *not* contain the requirement (which our classically informed intuition would dictate) that the conditional probabilities, when added up, should give back the unconditional probabilities \[...\] or, in other words, that the probability of the correlating events should be built up from a finer description of the situation provided by the common cause.*” (p.20)[^5]\ Although this doesn’t strike me as a particularly strong argument, it may be a good starting point for adding a few remarks and highlighting some of the disagreements between the Hofer-Szabó and myself. 1. As I see it, the problem here is not with probability theory (“that the conditional probabilities, when added up, should give back the unconditional probabilities”), but rather with the assumption that the “common cause system” provides a “finer description” of the *same* physical situation. The fact that $C + C^\perp = \mathds{1}$ in terms of operators does not imply that it makes no *physical* difference whether *any* of the “events” occur, or *none*. There is a difference between a physical situation in which a photon can either pass or not pass a polarization filter and a physical situation with no polarization filter at all. 2. Even if we accepted the premise of the answer, it would not resolve the issue. The interesting question concerning correlations between space-like separated events is whether they can be explained by some sort of local “mechanism” (I’m using “mechanism” broadly here). Bell’s theorem states that this is impossible if the Bell inequality is violated. The fact that Hofer-Szabó presents us a local mechanism that produces *different* correlations that do *not* violate Bell’s inequality seems quite irrelevant in this context. 3. Despite the tone of the paper suggesting a certain naturalness or inevitability to the concepts it explores, we should keep in mind that it was the author himself who has chosen to *redefine* the “common cause principle” for the needs and purposes of AQFT (or rather, who has chosen to follow Rédei 1997 while admitting noncommuting operators). Hence, when confronted with the objection that the very concept he defined is unsubstantial because it lacks a certain crucial property, he can hardly defend himself by pointing out that the concept lacks this property by definition. As far as I was able to understand from this and other publications (e.g. Rédei 1997, Hofer-Szabó and Vecsernyés 2013a, 2013b), the whole reasoning behind the concept of a “common cause in AQFT”, on which the author’s work is crucially based, is that the common cause principle formulated by Hans Reichenbach is somehow “classical” and that there is a canonical way to translate or generalize it to the framework of algebraic quantum field theory. However, leaving aside whether Reichenbach’s common cause principle is at all the relevant concept in this context (since his discussion had a different focus), to characterize it as “classical” strikes me as a rather confused and confusing statement. The common cause principle is a meta-physical concept, formulated in terms of (what some people call) “classical probability theory”. However, the word “classical” in “classical probability theory” shouldn’t be confused with the same adjective in the term “classical (i.e. Newtonian) mechanics”. It doesn’t refer to a particular *physical theory* that can be empirically tested and falsified, but to a well-founded mathematical framework expressing a certain *way of reasoning* about nature. It is possible, of course, to borrow Reichenbach’s definition and replace the probabilistic events, assumed to be modeled on a classical probability space, by projections in local algebras and the (classical) probability measure by a “quantum state”, yielding a value between $0$ and $1$ when evaluated on such projections; but there’s no reason to believe that this procedure must yield a meaningful notion of “common causes” or “common cause explanations” in the context of *any* theory. Of course, there are people who believe that quantum theory is and has always been about replacing “classical probability spaces” by so-called “quantum probability spaces”. Nevertheless, I would think that, for the sake of a meaningful and enlightening philosophical discussion, we will have to do better. In any case, I would insist that, if we work under this general hypothesis, the fact that certain results we obtain may strike us as counterintuitive or even logically inconsistent need not necessarily reflect some sort of quantum weirdness in nature; it may just as well reflect a lack of imagination or understanding on our side to appreciate that quantum physics is *not* always about putting little hats on capital A’s and B’s and C’s to turn them into operators.[^6] In my opinion, much of the confusion in the paper stems from its commitment to a particular jargon that insists on using familiar and intuitive terms (“events”, “common causes”, etc.) with a non-standard meaning (usually referring to operators). Thus, I find it very helpful to drop this jargon altogether and discuss the situation in the framework of good old-fashioned quantum mechanics to see what the result obtained by Hofer-Szabó actually consists in. Let’s consider the usual spin-singlet state $$\frac{1}{\sqrt{2}}\bigl(\lvert \uparrow \rangle_1 \otimes \lvert\downarrow \rangle_2 - \lvert\downarrow \rangle_1\otimes \lvert \uparrow \rangle_2\bigr),$$ giving rise to EPR-correlations between the results of spin-measurements on two entangled particles, which can be performed simultaneously by Alice and Bob. In the sense promoted in the paper, a “noncommuting joint common causal explanation” would, for instance, consist in the following: after the particles leave the EPR-source, we perform a z-spin measurement on particle 1, taking place in the common past of the measurements of Alice and Bob, who are free to choose between certain orientations other than the z-direction. This (chronologically) first measurement now provides a (non-trivial) “noncommuting joint common cause system” in the sense of Hofer-Szabó, namely $$\Bigl\lbrace C= \lvert \uparrow \rangle\langle \uparrow\rvert \otimes \mathds{1}, \; C^\perp = \lvert \downarrow \rangle\langle \downarrow\rvert \otimes \mathds{1}\Bigr\rbrace.$$ Obviously, the probabilities for the outcomes of the succeeding measurements will now split (which, the author insist, is the defining property of a “common cause”). Just as obviously, the first measurement would simply destroy (decohere) the singlet-state so that the outcomes of the spin-measurements by Alice and Bob will no longer be correlated in a way that violates the Bell- or CHSH-inequality. The result presented in the paper *Noncommutative causality in algebraic quantum field theory*, although more general and technically more sophisticated, doesn’t do anything more than this. I leave it to the reader to judge its explanatory value. References {#references .unnumbered} ========== [ ]{}[ ]{} [Bell, J.S. (1964), “On the Einstein-Podolsky-Rosen paradox”, reprinted in Bell, J.S. (2004) *Speakable and Unspeakable in Quantum Mechanics*. Cambridge: Cambridge Univ. Press, pp. 14–21.]{} [Bell, J.S. (1981), “Bertlmann’s socks and the nature of reality”, reprinted in Bell, J.S. (2004) *Speakable and Unspeakable in Quantum Mechanics*. Cambridge: Cambridge Univ. Press, pp. 139–158.]{} [Bell, J.S. (1990), “La nouvelle cuisine”, reprinted in Bell, J.S. (2004) *Speakable and Unspeakable in Quantum Mechanics*. Cambridge: Cambridge Univ. Press, pp. 233–248.]{} [Stapp, H.P. (1975), “Bell’s theorem and world process”, *Il Nuovo Cimento* 40B(29):270-276.]{} [Goldstein, S. et.al. (2011) “Bell’s theorem”, *Scholarpedia, 6(10):8378*,\ http://www.scholarpedia.org/article/Bell’s\_theorem.]{} [Norsen, T. (2006) “Bell Locality and the Nonlocal Character of Nature”, *Found. Phys. Letters*, 19(7): 633-655.]{} [Maudlin, T. (2011) *Quantum Non-Locality and Relativity*, Third Ed., Malden, Oxford: Wiley-Blackwell.]{} [Hofer-Szabó, G. and Vecsernyés, P. (2012) ”Bell Inequality and common causal explanation in algebraic quantum field theory.” *Preprint: http://philsci-archive.pitt.edu/9101*]{}. [Hofer-Szabó, G. and Vecsernyés, P. (2013a) “Bell inequality and common causal explanation in algebraic quantum field theory,” *Studies in the History and Philosophy of Modern Physics* 44 (4), 404–416.]{} [Hofer-Szabó, G. and Vecsernyés, P. (2013b) “Noncommutative Common Cause Principles in algebraic quantum field theory”, *Journal of Mathematical Physics* 54.]{} [Rédei, M. (1997) ”Reichenbach’s Common Cause Principle and Quantum Field Theory,” *Found. Phys.* 27: 1309–1321.]{} [^1]: The paper, alongside this commentary, will appear in *New Directions in the Philosophy of Science* (proceedings of the PSE workshops held in 2012), edited by Maria Carla Galavotti, Dennis Dieks, Wenceslao J. Gonzalez, Stephan Hartmann, Thomas Uebel, Marcel Weber, Springer (2014). The same results have been published, in greater detail, by Hofer-Szabó and collaborators in a series of other papers, for instance Hofer-Szabó and Vecsernyés (2013a, 2013b). [^2]: The only additional assumption entering the derivation of the Bell inequalities is the so-called *no-conspiracy* assumption, meaning that certain parameter choices involved in the experiments are assumed to be “free” or random, that is, not predetermined in precisely such a way as to arrange for apparently non-local correlations. See (Bell 1990) for details. [^3]: A beautiful presentation of his analysis can be found in (Bell 1981) and (Bell 1990), the original version of the theorem is (Bell 1964). For a more recent discussion, see (Goldstein et. al 2011) or (Maudlin 2011). The most common misunderstandings are addressed, for instance, in (Norsen 2006) or (Goldstein et.al. 2011). [^4]: Algebraic Quantum Field Theory is often referred to as *Local Quantum Field theory*, which is a quite unfortunate double-use of terminology. “Locality” in quantum field theory usually refers to the postulate of “microcausality” or “local commutativity”, requiring that operators localized in space-like separated regions of space-time commute. This, however, is very different from the concept of Bell-locality as discussed above. In AQFT, local commutativity assures the impossibility of faster-than-light signaling, the theory nevertheless contains *non-local* correlations between space-like separated events due to the non-local nature of the quantum state which is defined as a functional on the entire “net” of operator algebras, all over space-time. [^5]: I have to emphasize that the corresponding statement is slightly different – and slightly more plausible – in the published version (Hofer-Szabó and Vecsernyés 2013a, p. 414), possibly as a result of our exchange on the matter. I leave here the quote from the preprint, which was available to me by the time my commentary was written, and which I still believe to reflect a fundamental misconception on part of Hofer-Szabó and collaborators. [^6]: About common controversies or misconceptions regarding the relevance of “quantum logic”, “quantum probabilities” or “noncommutativity” for the issue of non-locality, see also (Goldstein et.al. 2011).

--- abstract: 'We show that spin-orbit coupling (SOC) in InSe enables the optical transition across the principal band gap to couple with in-plane polarized light. This transition, enabled by $p_{x,y}\leftrightarrow p_z$ hybridization due to intra-atomic SOC in both In and Se, can be viewed as a transition between two dominantly $s$- and $p_z$-orbital based bands, accompanied by an electron spin-flip. Having parametrized $\mathbf{k\cdot p}$ theory using first principles density functional theory we estimate the absorption for $\sigma^{\pm}$ circularly polarized photons in the monolayer as $\sim 1.5\%$, which saturates to $\sim 0.3\%$ in thicker films ($3-5$ layers). Circularly polarized light can be used to selectively excite electrons into spin-polarized states in the conduction band, which permits optical pumping of the spin polarization of In nuclei through the hyperfine interaction.' author: - 'S. J. Magorrian' - 'V. Zólyomi' - 'V. I. Fal’ko' title: 'Spin-orbit coupling, optical transitions, and spin pumping in mono- and few-layer InSe' --- Introduction ============ Two-dimensional (2D) materials, such as atomic layers of transition metal dichalcogenides [@Mak2010; @Splendiani2010; @Korn2011; @Wang2012; @Xu2014; @Jones2013; @Gan2013; @Wu2014; @Sie2015; @Wang2015; @Liu2015] and metal chalcogenides GaSe [@zhou2015strong; @jie2015layer; @hu2012synthesis; @late2012gas; @lei2013synthesis; @aziza2017tunable; @Pozo2015; @jung2015], GaTe [@liu2014gate; @hu2014highly; @huang2016plane], and InSe [@Lei2014; @Mudd2013; @Mudd2014; @Mudd2015; @Tamalampudi2014; @Balakrishnan2014; @yan2017fast] are attracting a lot of attention due to their promise for applications in optoelectronics. This is based on observations of their optical properties, which can differ strongly from those of their parent bulk materials, and which have demonstrated room-temperature electroluminesence[@Balakrishnan2014], strong photoresponsivity[@Lei2014] with a broad spectral response[@Mudd2015; @Tamalampudi2014; @yan2017fast], and band gap tunability[@Mudd2013]. Recent studies of luminescence [@bandurin2017high] and magnetoluminescence [@SciRep_2016] in InSe have shown a strong dependence of the band gap on the number of layers, from $\sim2.8$ eV for the monolayer to $\sim1.3$ eV for thick films. These experiments have identified two main photoluminescence lines, interpreted[@bandurin2017high] as a lower energy transition between bands dominated by $s$ and $p_z$ orbitals (A-line) and hot luminescence, involving holes in a deeper valence band based on $p_x$ and $p_y$ orbitals (B-line). The band structure analysis of mono- and few-layer InSe [@zhuang2013single; @rybkovskiy2014transition; @magorrian2016electronic; @*tb_erratum; @Zolyomi2014; @zhou2017multiband; @debbichi2015two; @jalilian2017electronic; @xiao2017defects; @demirci2017structural; @wick2015electronic; @cai2017charge; @hung2017thermo; @Do2015; @Li2015; @Nissim2017; @Ayadi2017; @Li2017] has revealed that the conduction and valence band edges near the $\Gamma$-point are non-degenerate, being dominated by $s$ and $p_z$ orbitals of both metal and chalcogen atoms. Combined with the opposite $z\rightarrow -z$ (mirror reflection) symmetry of conduction and valence bands, this determines that the transition across the principal band gap has a dominantly electric dipole-like character, coupled to out-of-plane polarized photons. In contrast, the B-line is found to be related to the recombination of hot holes in a twice-degenerate valence band based on $p_x$ and $p_y$ orbitals, and this transition is strongly coupled to in-plane polarized light. These selection rules are important in understanding experimental observations of these transitions, in particular where the experiments are carried out at normal incidence or emission, since the polarization of the incident/emitted light is then necessarily parallel to the plane of the 2D crystal. Spin-orbit coupling (SOC) in indium and selenium is capable of igniting additional transitions, accompanied by an electron spin-flip, with polarization properties and selection rules which differ from the selection rules for transitions in the absence of SOC, and which are determined by the angular momentum transfer from the photon to the spin of the $e$-$h$ excitation. Here, we use $\mathbf{k\cdot p}$ theory to show how SOC ignites spin-flip transitions between the conduction and valence band edges in monolayer InSe, coupled to in-plane polarized light, and we evaluate the optical oscillator strength in mono- and few-layer InSe films, which corresponds to an absorption coefficient $\sim 1.5\%$ in the monolayer, and $\sim 0.3\%$ in thicker films ($3-5$ layers). Monolayer ========= To analyse the effect of SOC on the band-edge states and their optical properties in the vicinity of the $\Gamma$-point in monolayer InSe we amend the $\mathbf{k\cdot p}$ Hamiltonian of Ref.  by including atomic SOC, $\lambda\mathbf{L\cdot s}$, leading to inter-band ($p_x/p_y\leftrightarrow p_z$) mixing, $$\hat{H}= \left( \begin{array}{cccc} H_c\mathbb{1}_s & E_zd_z & \frac{e\beta_1}{cm_e}\mathbb{1}_s\otimes\mathbf{A} & \lambda_{c,v_2}\boldsymbol{\hat{s}}\\ E_zd_z & H_v\mathbb{1}_s&\lambda_{v,v_1}\boldsymbol{\hat{s}} & \frac{e\beta_2}{cm_e}\mathbb{1}_s\otimes\mathbf{A} \\ \frac{e\beta_1}{cm_e}\mathbb{1}_s\otimes\mathbf{A^T}& \lambda_{v,v_1}\boldsymbol{\hat{s}}^\mathbf{T} & \mathbb{1}_s\otimes \mathbf{H}_{v_1}+\lambda_{v_1}s_z\otimes\sigma_y & 0\\ \lambda_{c,v_2}\boldsymbol{\hat{s}}^\mathbf{T}& \frac{e\beta_2}{cm_e}\mathbb{1}_s\otimes\mathbf{A^T} & 0 & \mathbb{1}_s\otimes \mathbf{H}_{v_2}+\lambda_{v_2}s_z\otimes\sigma_y \\ \end{array} \right); \label{eq_h0_6x6}$$ $$H_c\approx\hbar^2 k^2/2m_c;\quad H_{v} (k)\approx E_{v} + E_{2}k^{2} + E_{4}k^{4};\quad\mathbf{H}_{v_{1(2)}}\approx\left[E_{v_{1(2)}}+\frac{\hbar^2k^2}{2m_{1(2)}}\right]\mathbb{1}_{\sigma}+\frac{\hbar^2(k_x^2-k_y^2)}{2m_{1(2)}^{\prime}}\sigma_z +\frac{2\hbar^2k_x k_y}{2m_{1(2)}^{\prime}}\sigma_x. \label{kdotp_v12}$$ $D_{3h}$ $E$ $2C_3$ $3C_2^{\prime}$ $\sigma_h$ $2S_3$ $3\sigma_v$ Band ---------------------- ----- -------- ----------------- ------------ -------- ------------- ------- $A_1^{\prime}$ 1 1 1 1 1 1 $v$ $A_2^{\prime}$ 1 1 $-1$ 1 1 $-1$ $E^{\prime}$ 2 $-1$ 0 2 $-1$ 0 $v_2$ $A_1^{\prime\prime}$ 1 1 $1$ $-1$ $-1$ $-1$ $A_2^{\prime\prime}$ 1 1 $-1$ $-1$ $-1$ $1$ $c$ $E^{\prime\prime}$ 2 $-1$ 0 $-2$ 1 0 $v_1$ : \[tab\_d3h\]Character table for irreducible representations of point group $D_3h$ of monolayer InSe, together with $\Gamma$-point classification of bands included in basis of the $\mathbf{k\cdot p}$ Hamiltonian, Eq. (\[eq\_h0\_6x6\]). Here, we use a basis of spin up/down states, $\mu\equiv\mathbf{s}\cdot\mathbf{\hat{e}_z}=\pm\frac{1}{2}$, in the low-energy bands neglecting SOC, $\{c,v,v_1^{p_x},v_1^{p_y},v_2^{p_x},v_2^{p_y}\}$, labelled in the left hand side of Fig. \[fig\_bands\] and classified according to the irreducible representations of point group $D_{3h}$ of monolayer InSe in Table \[tab\_d3h\]. Band $c$ is the lowest energy conduction band, with its quadratic $\Gamma$-point dispersion described by an effective-mass single-band Hamiltonian, $H_c$. The highest-energy occupied band, $v$, has a maximum offest from $\Gamma$ in the monolayer[@Zolyomi2014; @rybkovskiy2014transition; @zhou2017multiband; @debbichi2015two; @jalilian2017electronic; @wick2015electronic; @hung2017thermo], and we therefore describe the band with a 4th order polynomial, $H_v$. The next-highest energy valence bands, $v_{1}$ and $v_2$, are dominated by $p_x,p_y$ orbitals, and are twice-degenerate at $\Gamma$. We therefore represent the bands using 2-component Hamiltonians, $\mathbf{H}_{v_{1(2)}}$, written as matrices in a space of $p_x$ and $p_y$ orbitals, with $\mathbb{1}_\sigma$ an identity matrix, and $\sigma_{x,y,z}$ the Pauli matrices. The values of the $\mathbf{k\cdot p}$ parameters listed in Table \[tab\_params\] are determined [@magorrian2016electronic] from fitting to DFT dispersions without SOC near $\Gamma$. The dispersions of these bands coincide with the DFT-calculated $\Gamma$-point dispersion of InSe bands[@zhuang2013single; @rybkovskiy2014transition; @magorrian2016electronic; @*tb_erratum; @Zolyomi2014; @zhou2017multiband; @debbichi2015two; @jalilian2017electronic; @xiao2017defects; @demirci2017structural; @wick2015electronic; @cai2017charge; @hung2017thermo; @Do2015; @Li2015; @Nissim2017; @Ayadi2017; @Li2017], but with the band gap corrected by a ‘scissor correction’ adjustment to the bands[@magorrian2016electronic]. The factors $\dfrac{e\beta_{1(2)}}{cm_e}$, are couplings of the spin-conserving $v_1\rightarrow c$ interband transition (B-line), and of the transition between bands $v$ and $v_2$ [^1], respectively, to in-plane polarized light described by vector potential $\mathbf{A}=(A_x,A_y)$, with $\beta_{1(2)}=|\bra{c(v)}\mathbf{P}\ket{v_1(v_2)}|$ the magnitude of the interband matrix element of the momentum operator. The matrix element $E_zd_z$ accounts for electric dipole coupling of the $c\leftrightarrow v$ transition to out-of-plane polarized light, where $d_z=e\bra{c}z\ket{v}$ is the z-component of the interband dipole operator. Similar band structure properties have been found in monolayer GaSe [@Zlyomi2013; @Cao2015; @Chen2015; @Feng2016; @Si2016]. In the spin space, $\mathbb{1}_s$ is an identity matrix, while $s_{x,y,z}$ are spin operators, with the ‘vectors’, $\boldsymbol{\hat{s}}=(s_x,s_y)$ and $\boldsymbol{\hat{s}}^{\mathbf{T}}=\left(\begin{array}{c}s_x\\s_y\\\end{array}\right)$, introduced to achieve a compact form for representing $\hat{H}$. Parameters $\lambda_{v_{1(2)}}$ are intraband SOC constants for band $v_{1(2)}$, determined by the atomic orbital compositions of the bands and atomic SOC strengths, while $\lambda_{v,v_1}$ and $\lambda_{c,v_2}$ take into account SOC-induced hybridization between $v$ and $v_1$, and between $c$ and $v_2$, respectively. In calculating the latter parameters, we compare the result of diagonalization of $\hat{H}$ in Eq. (\[eq\_h0\_6x6\]) with density functional theory (DFT) calculations for InSe with SOC (VASP code [@VASP_PhysRevB.54.11169] in the local spin density approximation) and without SOC. We employed a plane-wave basis with a cutoff energy of 600 eV and the Brillouin zone was sampled by a $24 \times 24 \times 1$ grid for monolayer InSe, followed by a ‘scissor correction’ adjustment of the band gap, described in Ref. . Atomic SOC splits[@Li2015; @zhou2017multiband] the otherwise [@Zolyomi2014; @magorrian2016electronic] degenerate bands at $\Gamma$ ($v_1$ and $v_2$) into pairs of states with projections $J_z=\pm\frac{3}{2}$ and $J_z=\pm\frac{1}{2}$ of total angular momentum, $\mathbf{J}=\mathbf{L}+\mathbf{s}$. The ‘spin-flip’ part of atomic SOC, $L^{\pm}s^{\mp}$, hybridizes the lower branch of $v_1$ ($J_z=\pm\frac{1}{2}$) with $v$, to the measure determined by the matrix element $\bra{p_z}L^{\pm}\ket{p_{x,y}}$ of the angular momentum operator acting between the orbitals of the same atoms contributing to the $v$ and $v_1$ band states. This pushes $v$ higher in energy and reduces the band gap. It also hybridizes $v_2$ with $c$, but the associated shift appears to be much smaller, due to a larger energy separation between these two bands. There is no mixing between $c$ and $v_1$ or between $v$ and $v_2$ as the symmetry of these bands under $\sigma_h$ reflection requires $\bra{c}L^{\pm}\ket{v_1}=\bra{v}L^{\pm}\ket{v_2}=0$.. ![Left hand side - Schematic of low-energy bands at $\Gamma$ in monolayer InSe without SOC, with allowed optical transitions marked. Solid arrows denote coupling of optical transitions to in-plane polarized light, dashed arrows coupling to the out-of-plane dipole transition. Right hand side - Bands and allowed optical transitions in the presence of SOC. Superscripts denote the $z$ components of total angular momentum ($J_z$) of the bands. Energies in parentheses are scissor-corrected DFT values, for comparison with the bands given by the model Hamiltonian, Eq. \[eq\_h0\_6x6\]. The conduction band edge is used as a reference energy, set at 0 eV.[]{data-label="fig_bands"}](bands_1030.pdf){width="48.00000%"} By diagonalizing $\hat{H}$ in Eq. (\[eq\_h0\_6x6\]) and comparing the band energies with those found from DFT with SOC, Table \[tab\_params\], we find that $\lambda_{v_1}\approx\lambda_{v_2}\approx0.30$ eV and $\lambda_{v,v_1}\approx0.25$ eV. The SOC-induced changes to the band energies at $\Gamma$ are illustrated in Fig. \[fig\_bands\], showing good agreement between the DFT and $\mathbf{k\cdot p}$ bands. Note that inspection of the wavefunction decomposition of the conduction band edge with SOC shows a negligible ($<0.1\%$) contribution from $p_{x,y}$ orbitals (which are present due to the hybridization of $c$ and $v_2$), so we neglect the effect of $\lambda_{c,v_2}$ in our analysis, and use the conduction band edge as a reference energy, set at 0 eV. In contrast, $v-v_1$ hybridization is retained in the model as a significant effect. We find that perturbation theory overestimates the weight of $v_1$ band $p_x,p_y$ orbitals admixed into $v$, $$\label{eq_v1_v_pt} |\delta C_v(v_1)|^2=\bigg|\frac{\frac{1}{\sqrt{2}}\lambda_{v,v_1}}{E_v-(E_{v_1}-\lambda_{v_1}/2)}\bigg|^2\sim 0.2,$$ as compared to $|\delta C_v(v_1)|^2=0.13$ found from numerical exact diagonalization of the Hamiltonian in Eq. (\[eq\_h0\_6x6\]). [cc]{}\ $E_c$ & 0 eV\ $m_c$ & 0.19 $m_e$\ $E_v$ & $-2.79$ eV\ $E_2$ & $2.92$ eVÅ$^2$\ $E_4$ & $-38.06$ eVÅ$^4$\ $E_{v_1}$ & $-3.04$ eV\ $E_{v_2}$ & $-3.12$ eV\ $m_1$ & $-0.31$ $m_e$\ $m_2$ & $-0.30$ $m_e$\ $m_{1,2}^{\prime}$ & $-0.45$ $m_e$\ $\beta_1$&$1.09$ $\hbar/\mathrm{\AA}$\ $\beta_2$&$0.21$ $\hbar/\mathrm{\AA}$\ $d_z$&$1.68$ $e$Å\ [cc]{}\ $E_c^{\pm\frac{1}{2}}$ & 0 eV\ $E_v^{\pm\frac{1}{2}}$ & $-2.72$ eV\ $E_{v_1}^{\pm\frac{3}{2}}$ & $-2.89$ eV\ $E_{v_1}^{\pm\frac{1}{2}}$ & $-3.27$ eV\ $E_{v_2}^{\pm\frac{3}{2}}$ & $-2.97$ eV\ $E_{v_2}^{\pm\frac{1}{2}}$ & $-3.28$ eV\ [cc]{}\ $\lambda_{v_1}$ & 0.30 eV\ $\lambda_{v_2}$ & 0.30 eV\ $\lambda_{v,v_1}$ & 0.25 eV\ In the absence of SOC, the coupling of the principal interband transition, $v\leftrightarrow c$, to in-plane polarized light is forbidden due to the opposite symmetry of $c$ and $v$ under $\sigma_h$, and also due to the lack of an orbital angular momentum difference between the bands. The coupling of $v$ to the lower branch of $v_1$ by SOC relaxes this selection rule, allowing coupling to in-plane polarized light. The transitions will be between total angular momentum states $v^{\pm\frac{1}{2}}\leftrightarrow c^{\mp\frac{1}{2}}$, coupling to $\sigma^{\mp}$-polarized photons. Since $c$ changes negligibly on application of SOC, we can consider the spin-projection $\mu$ to remain a good quantum number in the conduction band, and make the observation that $\sigma^{+/-}$-polarized light will excite electrons into spin up/down states in the conduction band, as sketched in the right hand side of Fig. \[fig\_bands\]. For coupling of the principal interband transition to in-plane polarized light, we can estimate it as $$a_{SO}=\frac{e|\delta C_v(v_1)|\beta_1}{cm_e}\mathbf{A^{\pm}}\equiv\frac{e\beta_{\mathrm{sf}}}{cm_e}\mathbf{A^{\pm}},$$ where $\mathbf{A^{\pm}}=A(\mathbf{\hat{x}}\pm i\mathbf{\hat{y}})/\sqrt{2}$ correspond to the vector potential of $\sigma^{\pm}$ circularly polarized light, and $|C_v(v_1)|^2$ is the weight of $p_x,p_y$ orbitals from $v_1$ admixed in the dominantly $p_z$ orbital based $v$-band wavefunction. The oscillator strength of such transitions can be estimated as $\beta_{\mathrm{sf}}=|C_v(v_1)|\beta_1\approx0.4~\hbar/$Å  leading to the absorption coefficient[@magorrian2016electronic] for $\sigma^{\pm}$ light incident perpendicular to the 2D crystal, $$g_{A\mathrm{(\sigma^{\pm})}}=8 \pi \frac{e^2}{\hbar c}|\beta_{\mathrm{sf}}|^2 \frac{m_c}{|E_v^{\pm\frac{1}{2}}|m_e^2}\approx 1.5\%.$$ For comparison [@magorrian2016electronic], a photon incident at an angle $\theta\approx 45^{\circ}$ to the surface and coupled to the the principal interband transition via the interband out-of-plane electric dipole moment, $d_z$, is absorbed with $g_{A\mathrm{(E_z)}}[\theta\approx 45^{\circ}]\approx 3.7\%$, while for the B-line $g_{B(\sigma^{\pm})}\approx 10\%$. Multilayers =========== Going on from the monolayer[@zhuang2013single; @rybkovskiy2014transition; @magorrian2016electronic; @*tb_erratum; @Zolyomi2014; @zhou2017multiband; @debbichi2015two; @jalilian2017electronic; @xiao2017defects; @demirci2017structural; @wick2015electronic; @cai2017charge; @hung2017thermo; @Do2015; @Li2015; @Nissim2017; @Ayadi2017; @Li2017] to $N$-layer films, interlayer hopping between successive layers of InSe splits each band into $N$ subbands, as studied earlier using DFT and tight-binding calculations[@magorrian2016electronic; @rybkovskiy2014transition; @SciRep_2016]. At the $\Gamma$-point, $v_1$ and $v_2$ split very weakly, whereas $c$ and $v$, which are dominated by $s$ and $p_z$ orbitals on In and Se, exhibit a much stronger splitting. This moves the valence band edge (the top sub-band of $v$) further away from $v_1$, reducing the effect of SOC on the band edge states. This weakening in the effect of SOC, anticipated from a perturbation theory analysis similar to Eq. (\[eq\_v1\_v\_pt\]), reduces the oscillator strengths of the spin-flip interband transitions. To describe quantitatively the spin-flip transitions in a multilayer film we employ a hopping model for few-layer InSe, $$\begin{aligned} \begin{split} \label{full_ham} \hat{H}^{(N)}&=\sum_{n}^N\sum_{\substack{\alpha,\alpha^{\prime}\\\mu,\mu^{\prime}}}\hat{H}_{\alpha\mu,\alpha^{\prime}\mu^{\prime}}a^{\dagger}_{n\alpha\mu}a_{n\alpha^{\prime}\mu^{\prime}}+\sum_{n}^{N-1}\sum_{\alpha,\mu}\delta_{\alpha}\left(a_{n\alpha\mu}^{\dagger}a_{n\alpha\mu}+a_{(n+1)\alpha\mu}^{\dagger}a_{(n+1)\alpha\mu}\right)\\ &+\sum_{n}^{N-1}\sum_{\mu}\left[t_{c}a^{\dagger}_{(n+1)c\mu}a_{nc\mu}+t_{v}a^{\dagger}_{(n+1)v\mu}a_{nv\mu}+t_{cv}\left(a^{\dagger}_{(n+1)v\mu}a_{nc\mu}-a^{\dagger}_{(n+1)c\mu}a_{nv\mu}\right)+\mathrm{H.c.}\right]. \end{split}\end{aligned}$$ Here, the operator $a^{(\dagger)}_{n\alpha\mu}$ annihilates (creates) an electron in band $\alpha$, spin state $\mu=\pm\frac{1}{2}$, in layer $n$ of the $N$-layer crystal. The sum over $\alpha,\alpha^{\prime}$ runs twice over the bands included in the monolayer Hamiltonian $\hat{H}$ in Eq. (\[eq\_h0\_6x6\]), and $\hat{H}_{\alpha\mu,\beta\mu^{\prime}}$ are the matrix elements of $\hat{H}$ between band $\alpha$ with spin projection $\mu$ and band $\alpha^{\prime}$ with spin projection $\mu^{\prime}$. The parameters $t_c\approx0.34$ eV, $t_v\approx-0.42$ eV, and $t_{cv}\approx0.29$ eV are interlayer hoppings, and $\delta_v\approx -0.06$ eV and $\delta_c<0.01$ eV (which we neglect as it is much smaller than the interlayer hoppings) are onsite energy shifts due to interlayer potentials; all of these were determined from fitting of the $\Gamma$-point energies obtained from Eq. (\[full\_ham\]) to the DFT sub-bands in 2-5 layer InSe [@magorrian2016electronic]. Diagonalization of $\hat{H}^{(N)}$ allows us to find the wavefunction and hence the $N$-dependence of the absorption coefficient for in-plane polarized light for the optical transition across the principal band gap of $N$-layer InSe; $$g_{A\mathrm{(\sigma^{\pm})}}(N)=8 \pi \frac{e^2}{\hbar c}|\delta C^{N}_v(v_1)\beta_1|^2 \frac{m_c(N)}{\hbar \omega(N) m_e^2},$$ where $m_c(N)$ and $\hbar\omega(N)$ are the $N$-layer conduction-band effective mass and band gap, respectively, and $|\delta C^{N}_v(v_1)|^2$ is the total weight of all $p_x,p_y$ orbitals from sub-bands of $v_1$ admixed by SOC into the highest energy valence sub-band. Overall we find $$g_{A\mathrm{(\sigma^{\pm})}}(N=3-5)\sim 0.3\%,$$ see Table \[tab\_absorb\]. This compares with an increase in $g_{A\mathrm{(E_z)}}(N)[\theta\approx 45^{\circ}]$ from $3.7\%$ in the monolayer to $\sim 15\%$ for large $N$, and with a roughly constant $g_{B(\sigma^{\pm})}(N)\sim 10\%$ absorption for the B-line [@magorrian2016electronic]. $N$ $\hbar\omega$(eV) $g_{A\mathrm{(\sigma^{\pm})}}$(%) $g_{A\mathrm{(E_z)}}[\theta\approx 45^{\circ}]$(%) $g_{B(\sigma^{\pm})}$(%) ----- ------------------- ----------------------------------- ---------------------------------------------------- -------------------------- $1$ 2.72 1.5 3.7 $10.3$ $2$ 2.00 $0.5$ 6.4 $8.3$ $3$ 1.67 $0.3$ 8.2 $8.8$ $4$ 1.50 $0.3$ 9.6 $9.0$ $5$ 1.40 $0.3$ 10.7 $9.1$ : Band gaps ($\hbar\omega$) and absorption coefficients as a function of number of layers, $N$, for coupling of A-line transition to in-plane polarized light after application of SOC ($g_{A\mathrm{(\sigma^{\pm})}}$), in comparison with absorption coefficients for the B-line ($g_{B(\sigma^{\pm})}$) and for coupling to the out-of-plane dipole transition for a photon incident at an angle $\theta\approx 45^{\circ}$ to the crystal ($g_{A\mathrm{(E_z)}}$), taken from Ref. . \[tab\_absorb\] ![Schematic of spin-pumping into In nuclear spin polarization - an electron excited by $\sigma^-$-polarized light goes into a spin-down state in the conduction band, and can recombine by emission of another $\sigma^-$ photon. After the electron flips its spin via hyperfine interaction with In nuclei, it can then recombine with the photoexcited hole by emitting an out-of-plane polarized photon, leaving the angular momentum imparted in the nuclear spin of the In atoms.[]{data-label="fig_hfs"}](hyperfine.pdf){width="40.00000%"} Nuclear spin pumping ==================== The excitation of electrons into $\mu=\pm\frac{1}{2}$ states of the conduction band by $\sigma^{\pm}$-polarized light tuned to the energy of the principal interband transition allows for the optical pumping of In nuclear spins. This would occur through the mechanism sketched in Fig. \[fig\_hfs\]. An electron excited by $\sigma^{-}$-polarized light into the $\mu=-\frac{1}{2}$ conduction band state, leaving behind a hole in the $J_z=+\frac{1}{2}$ valence band, $v^{+\frac{1}{2}}$, can transfer its spin to the nucleus via the hyperfine interaction, $e^{\downarrow}+\mathrm{In^{I_z}}\rightarrow e^{\uparrow}+\mathrm{In^{I_z-1}}$, where $I_z$ is the $z$-component of the indium’s nuclear spin. After that, it can recombine without spin-flip with the photoexcited hole, with a similar cycle to be repeated following the absorption of the next $\sigma^{-}$-polarized photon. A strong contribution of an indium $s$ orbital to the conduction band states presents the possibility of a strong coupling between the electronic spin and In nuclear spins. The most common isotope of indium, $^{115}\mathrm{In}$, has a nuclear spin $I=\frac{9}{2}$, and an atomic hyperfine coupling constant $A_{\mathrm{In}}\approx 60~\mu\mathrm{eV}$ [@In_hfs] (in comparison with Se, for which the only stable nucleus with non-zero spin, $^{77}\mathrm{Se}$, $I=\frac{1}{2}$, abundance 7%, has a much smaller coupling constant $A_{\mathrm{Se}}\approx 2~\mu\mathrm{eV}$ [@Se_hfs]). Using the orbital decomposition reported in Ref.  we can estimate an effective hyperfine coupling constant for the conduction band, $A^{eff}_{c}=|C_c(\mathrm{In}_s)|^2A\approx 15~\mu\mathrm{eV}$, where $|C_c(\mathrm{In}_s)|^2$ is the weight of In $s$ orbitals in the conduction band state near the $\Gamma$-point. As a result, one can expect that monolayers and bilayers of InSe will offer a suitable material for achieving a high degree of optically induced nuclear spin polarization. Conclusions =========== In conclusion, we have used a $\mathbf{k\cdot p}$ model to show how spin-orbit coupling in mono- and few-layer InSe allows coupling between the principal interband transition and in-plane polarized light, accompanied by a spin-flip, with the coupling at its strongest in the monolayer, $g_{A(\sigma^{\pm})}\sim 1.5\%$, saturating to $g_{A(\sigma^{\pm})}\sim 0.3\%$ for $3-5$ layers. We would expect a similar effect in the other III-VI semiconductors, such as GaSe, with the strength primarily determined by the SOC strength in the group VI atoms. Electrons, excited selectively into spin up/down states in the conduction band using $\sigma^{\pm}$-polarized light, can transfer their spin angular momentum to the In nuclei , allowing for optically induced nuclear spin polarization. The $\mathbf{k\cdot p}$ model presented here can be amended to include the effect on optical transitions in few-layer InSe of an applied displacement field in a dielectric environment, through changes to the onsite band energies and coupling to states in the dielectric environment, and this will be addressed in future work. The authors thank A. Patanè, A. V. Tyurnina, M. Potemski, Y. Ye, J. Lischner, and N. D. Drummond for discussions. This work made use of the facilities the CSF cluster of the University of Manchester. SJM acknowledges support from EPSRC CDT Graphene NOWNANO EP/L01548X. VF acknowledges support from ERC Synergy Grant Hetero2D, EPSRC EP/N010345, and Lloyd Register Foundation Nanotechnology grant. VZ and VF acknowledge support from the European Graphene Flagship Project, the N8 Polaris service, the use of the ARCHER national UK supercomputer (RAP Project e547), and the Tianhe-2 Supercomputer at NUDT. Research data is available from the authors on request. [35]{} ifxundefined \[1\][ ifx[\#1]{} ]{} ifnum \[1\][ \#1firstoftwo secondoftwo ]{} ifx \[1\][ \#1firstoftwo secondoftwo ]{} ““\#1”” @noop \[0\][secondoftwo]{} sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{} @startlink\[1\] @endlink\[0\] @bib@innerbibempty @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} [****,  ()](\doibase 10.1038/nnano.2012.193) [****,  ()](\doibase 10.1038/nphys2942) [****,  ()](\doibase 10.1038/nnano.2013.151) [****,  ()](\doibase 10.1063/1.4826679) [****, ()](\doibase 10.1088/2053-1583/1/1/011001) @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} [****,  ()](\doibase 10.1021/nn300889c) @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} [****, ()](\doibase 10.1021/acsnano.5b04876) [****,  ()](\doibase 10.1088/2053-1583/2/3/035010) @noop [****,  ()]{} [****,  ()](\doibase 10.1007/s12274-014-0430-2) [****,  ()](\doibase 10.1021/acsnano.6b05002) [****,  ()](\doibase 10.1021/nn405036u) [****,  ()](\doibase 10.1002/adma.201302616) [****,  ()](\doibase 10.1063/1.4903738) [****,  ()](\doibase 10.1002/adma.201500889) [****,  ()](\doibase 10.1021/nl500817g) [****,  ()](\doibase 10.1002/adom.201400202) @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} [****,  ()](\doibase 10.1103/PhysRevB.89.205416) @noop [****,  ()]{} @noop [****,  ()]{} @noop [**** ()]{}@noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} [****,  ()](\doibase 10.1088/1361-648x/aa7b63) @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} [****, ()](\doibase 10.1021/acs.jpcc.5b09635) [****,  ()](\doibase 10.1088/2053-1583/4/1/015017) [****,  ()](\doibase 10.1088/2053-1583/4/1/015027) [****,  ()](\doibase 10.1103/PhysRevB.54.11169) @noop [****,  ()]{} [****,  ()](\doibase http://dx.doi.org/10.1016/0009-2614(72)85077-2) [^1]: Although in the absence of SOC this term is not relevant for the optical properties of InSe, it becomes important for the analysis of the effect of SOC on the principal interband transition

-.8cm **Moving sources in a ghost condensate** .2cm 1.2cm **Marco Peloso $^a$ and Lorenzo Sorbo $^b$** *$^a$ School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA* *$^b$ Department of Physics, University of California, Davis, CA 95616, USA* 1.2cm peloso@physics.umn.edu, sorbo@physics.ucdavis.edu 1.2cm > Ghost condensation has been recently proposed as a mechanism inducing the spontaneous breaking of Lorentz symmetry. Corrections to the Newton potential generated by a static source have been computed: they yield a limit $M \la 10 \, {\rm MeV}$ on the symmetry breaking scale, and - if the limit is saturated - they are maximal at a distance $L \sim 1000 \, {\rm km}$ from the source. However, these corrections propagate at a tiny velocity, $v_{\rm s} \sim 10^{-12}\,$m/s, many orders of magnitude smaller than the velocity of any plausible source. We compute the gravitational potential taking the motion of the source into account: the standard Newton law is recovered in this case, with negligible corrections for any distance from the source up to astrophysical scales. Still, the vacuum of the theory is unstable, and requiring stability over the lifetime of the Universe imposes a limit on $M$ which is not too far from the one given above. In the absence of a direct coupling of the ghost to matter, signatures of this model will have to be searched in the form of exotic astrophysical events. Introduction ============ The whole interpretation of the currently available cosmological data strongly depends upon the hypothesis that the behavior of field theories – and, in particular, of gravity – at cosmological distances is the same one that we observe at local scales. It is hard to check the validity of such a theoretical assumptions by means of sufficiently prior-independent observations. It is therefore important to explore the possibility to modify gravity at large distances [*in a theoretically consistent way*]{}. Such goal is far from being trivially achievable. The simplest way to modify gravity in the infrared probably resides in the introduction of a tiny mass term for the graviton. The introduction of a hard mass term, however, leads to a series of problems, which show some similarity with the ones encountered when introducing a mass term for a spin-one gauge boson. Consistency of the theory requires the mass term to be of the Fierz-Pauli type [@fp], and the propagator for the graviton is affected by the van Dam-Veltman-Zakharov discontinuity [@dvz], related to the different number of polarizations between the massive and the massless case. In addition, due to the specific form of the kinetic term for the longitudinal component, the perturbative description of gravity breaks down at macroscopic lengths, that are probed in everyday life [@vai]. These properties can be nicely understood with the technique of [@ags], where the gravitational counterpart of the Goldstone description of massive gauge theories was constructed. The recent years have witnessed a wide debate about the modifications of gravity at large scales in higher dimensional models, in which the four dimensional graviton emerges as a resonance, and the higher–dimensional theory shows up at large distances as well as at short ones [@extra]. In these scenarios, four-dimensional covariance is preserved both for the theory and for its vacuum state. If we instead restrict the attention to four-dimensional scenarios, and in view of the analogies of General Relativity with flat space gauge theories, it is natural to expect that a Higgs phenomenon could be a simple way to give consistently a mass to the graviton. Indeed, spontaneous breaking of Lorentz symmetry has been studied by several authors (see, for instance [@lsb]) in the past years, with a main interest in the quantum gravitational origin of this phenomenon. Recently, a further progress in this direction has been made in [@ghost], where the spontaneous breaking of (part of) the Lorentz group is achieved by giving a (timelike) expectation value to the gradient of a scalar field $\phi\,$. [^1] When close to its Lorentz invariant unstable equilibrium state, this field behaves as a ghost. For this reason, this mechanism has been named [*ghost condensation*]{}. The scale of the transition to the broken phase is set by a fundamental dimensionful parameter $M\,$. At energy larger than $M$, the theory needs a UV completion that should presumably describe the emergence of a symmetric phase. At scales below $M$, Lorentz symmetry is broken. The proposal of [@ghost] has the virtue of being clearly treatable by perturbative methods all the way up to the symmetry breaking scale. [^2] The breaking of Lorentz symmetry is associated to some very unusual features, the most striking of which is probably the nonrelativistic dispersion relation $\omega^2=k^4/M^2$ for the fluctuations of $\phi\,$. When the ghost is coupled to gravity, its dispersion relation is modified: the system develops a (Jeans-like) instability in the IR. Such effect can be used to set model independent[^3] bounds on the scale $M$, that fixes the time and the length scales of the gravitational instability of the model. Indeed, in the Newtonian limit $\omega^2\ll k^2$, the dispersion relation of the scalar degree of freedom of the system reads $$\omega^2=\frac{k^4}{M^2}-\frac{M^2}{M_p^2}\,k^2\,\,.$$ For sufficiently small $k$, $\omega$ turns out to be imaginary. The corresponding instability is maximal for wavenumbers $k\sim m\equiv M^2/M_p$ and frequencies $\left\vert \omega\right\vert\sim \Gamma\equiv M^3/M_p^2$. Going to real space, the instability is thus expected to develop on timescales $\tau \sim \Gamma^{-1}$ and on lengthscales $L \sim m^{-1}\,$. Notice that, due to the breaking of Lorentz invariance, the typical space– and time–scales for the instability can be very different: unless $M$ is close to the Planck scale, $\tau$ will be much larger than $L\,$. For the reasons we mentioned, the proposal of [@ghost] is very interesting, and it is worth to subject it to close scrutiny. In order to set constraints on the parameter space of the model, it is crucial to establish the observables over which the instability will leave its strongest imprint. A natural candidate is the gravitational force between two masses. Indeed, the computation [@ghost] of the gravitational potential generated by a static source shows, in addition to the standard Newton potential, a correction which is growing with time. In agreement with the above discussion, the growth occurs on a timescale $\tau\,$. In order to make sure that the new term has not yet grown to an observable size, one can conservatively require this time to be greater than the age of the Universe[^4]. This results in a rather stringent upper bound on the scale of the condensate, $M \la 10 \,{\rm MeV}\,$. However, after a time $\tau$, the exponential growth will have occurred only at a distance $r \la L$ from the source [@ghost]. For such a value of $M\,$, one finds $L \simeq 1000 \, {\rm km}\,$. At these scales, an oscillating modulation of the newtonian potential around a static source would be a distinctive signature of the mechanism. In the present note we will discuss how this picture is changed in the case in which the sources are in motion. The mechanism of [@ghost] breaks only a part of Lorentz symmetry. For a time-like $\langle \partial_\mu \phi \rangle \,$, we can choose a frame in which only $\langle \partial_t \phi \rangle$ is nonvanishing. We call this coordinate system the rest frame of the ghost. The presence of a privileged system allows to speak about absolute motion: $\phi-$mediated interactions between particles will depend on their velocity (even if constant) with respect to the ghost rest-frame. The standard newtonian interaction propagates instantaneously (the whole computation is performed in the nonrelativistic regime). However, the unstable modes of the ghost condensate propagate with an extremely low velocity $v_{\rm s}$. As we remarked, for $M\sim 10 \, {\rm MeV}\,$, it takes a time comparable to the age of the Universe for the modified interaction to propagate at a distance $\sim 1000 \, {\rm km}$ from the source. This gives $v_{\rm s} \sim M/M_p \sim 10^{-12}\,$m/s. Even if we do not know which the rest-frame of the condensate is, the typical velocity of celestial bodies is of the order of $10-100\,$km$/$s. This is the case, for instance, both for the motion of the Earth around the Sun, and of the Earth in the frame in which the Cosmic Microwave Background (CMB) radiation shows no dipole. As a consequence, any realistic source will be moving with respect to the ghost rest frame with a velocity that exceeds $v_{\rm s}$ by many orders of magnitude. We compute the gravitational potential for a source moving with velocity $v \gg v_{\rm s}$ in the next Section. As in [@ghost], we assume that the source has been created (for instance, by gravitational collapse) at some finite time $t=0\,$. We then specify the general expression to two simplifying situations. Eq. (\[nearsource\]) describes the potential measured by a (late time) observer at rest with respect to the source. This observer [*does not*]{} see an exponentially growing correction to the Newton potential. Both the source and the observer are moving too fast with respect to the rest frame of the condensate, and the $\phi-$mediated interactions will not be observable. The situation is different if the gravitational potential is computed at late times in the vicinity of the place where the source was created. In this case, the nonstandard term is indeed growing exponentially. Perhaps most surprisingly, the exponential growth is taking place even if the source has in the meantime moved to very faraway distances! This shows that the growth of the potential is not really due to the presence of a nearby source at all times, but rather to the instability of the theory. The source is simply triggering the initial instability, and a time $t \ga \tau\,$ is then needed for the instability to grow at exponentially large values. The massive source is required to start this growth, since the calculation performed here is classical. Most probably, quantum effects will also trigger the instability, even in the case in which classical sources were absent. Although we do not expect signatures in the form of a modified Newton law, the instability of the vacuum will lead to potentially observable effects, if the scale of the instability is not too far from to the present age of the Universe (that is, $M\sim 10$ MeV). Unstable regions will presumably collapse to a strong gravity regime, and one may expect formation of compact objects, possibly leading to exotic astrophysical events. Such structures could be located in regions which are not correlated to any visible matter (since the source has in the meantime moved away) and hence they would be hard to explain in more conventional scenarios. Modification of the gravitational potential =========================================== As in [@ghost], we compute the gravitational potential generated by a source of mass $\mu$ “nucleated” at a given time $t_{\rm in}=0\,$ at the position ${\bf r} = {\bf 0}\,$. We assume that, after it is nucleated, the source moves with a constant velocity ${\bf v}$ in the rest-frame of the condensate. This extends the computation of [@ghost] performed for $v = 0\,$. The gravitational potential generated by the source is, at a generic position $r$ and at the time $t > 0\,$, $$\begin{aligned} V &\equiv& V_{\rm n} + \Delta \, V = \frac{\mu \, G}{4 \, \pi^3} \, \int d^3 {\bf r}' \, d t' \, d^3 {\bf k} \, d \omega \times \nonumber\\ &&\times {\rm e}^{\,-i\,\omega \left( t - t' \right)+ i \, {\bf k} \cdot \left( {\bf r} - {\bf r}' \right)} \, \theta \left( t' \right) \, \delta^3 \left( {\bf r}' - {\bf v} \, t' \right) \, \Pi \,\,,\end{aligned}$$ where $G = 1/ \left( 8 \, \pi M_p^2 \right)$ is the Newton constant. $\Pi$ is the propagator (in Fourier space) of the scalar perturbations of the metric [@ghost] $$\Pi \equiv - \frac{1}{k^2} + \frac{\alpha^2 \, m^2 / M^2}{\omega^2 + \frac{\alpha^2 \, m^4}{M^2} \, \left[ \left( \frac{k}{m} \right)^2 - \left( \frac{k}{m} \right)^4 \right]} \,\,, \label{propago}$$ which is valid in the nonrelativistic limit $\omega \ll k\,$. The parameter $M$ is the scale of the ghost condensate, while $$m \equiv M^2 / \left( \sqrt{2} \, M_p \right) \simeq 3 \cdot 10^{-19} \, {\rm GeV} \left( \frac{M}{\rm GeV} \right)^2 \,\,.$$ Finally, $\alpha$ is model dependent parameter of order one (in the following we simply set $\alpha = 1\,$). Let us define ${\bf {\tilde r}} \equiv {\bf r} - {\bf v} \, t$ to be the position of a given point in the rest frame of the source. The first term in (\[propago\]) gives rise to the standard newtonian potential $V_{\rm n} = - G \, \mu / \vert {\bf {\tilde r}} \vert \,$, dependent on the (instantaneous [^5]) distance from the source only. To compute the second term, we first perform the trivial $d^3 {\bf r'}\,$ integration, $$\begin{aligned} \Delta \, V = \frac{G \, m \, \mu}{4 \, \pi^3} \int_0^\infty d T' \int d^3 {\bf u} \int d W {\rm e}^{-i W \left( T - T' \right) + i {\bf u} \cdot {\bf R} - i {\bf u} \cdot {\bf {\cal V}} T'} \frac{1}{W^2+u^2-u^4}\end{aligned}$$ where we have introduced the following adimensional quantities $$\begin{aligned} && u \equiv k / m \quad,\quad W \equiv M \, w / m^2 \,\,, \nonumber\\ && R \equiv m \, r = 1.5 \left( \frac{M}{\rm GeV} \right)^2 \left( \frac{r}{\rm km} \right) \,\,, \nonumber\\ && {\cal V} \equiv \frac{M \, v}{m} = 3.3 \cdot 10^{15} \left( \frac{\rm GeV}{M} \right) \, \left( \frac{v}{10^{-3} \, c} \right) \,\,, \nonumber\\ && T \equiv \frac{M^3}{2 \, M_p^2} \, t = 6.2 \cdot 10^{4} \left( \frac{M}{\rm GeV} \right)^3 \, \left( \frac{t}{t_0} \right) \,\,, \nonumber\\ && {\tilde R} \equiv m {\tilde r} = m \left( r - v \, t \right) = R - {\cal V} \, T \,\,, \label{para}\end{aligned}$$ where $c$ is the speed of light, while $t_0 \simeq 15$ billion years is the age of the Universe. To proceed further, we integrate over complex frequencies $\omega$ with a given prescription for the contour. This term is analogous to the propagator of a tachyonic field: for small momenta ($k < m\,$, in our case) the frequency has imaginary poles, which are related to the tachyonic instability of the vacuum; high momenta $k > m$ balance this effect, and these modes are stable. Causality arguments prescribe the use of the retarded propagator ($\omega \rightarrow \omega + i \, \epsilon$) for these modes. For low momentum modes the choice of the contour is instead more ambiguous. We deform the contour so that it never crosses the poles, when - as $k$ decreases - they go from real to imaginary. Mathematically, the case $k < m$ becomes the analytic continuation of $k > m\,$. Physically, different contours correspond to different initial conditions, and the one we choose gives $\Delta V = 0$ at the initial time $t=0\,$. Taking this into account, we find $$\begin{aligned} &&\Delta V = - \frac{G \, m \, \mu}{2 \, \pi^2} \times \nonumber\\ &&\!\!\!\!\!\!\left\{ \: \int_{u \leq 1} d^3 {\bf u} \frac{{\rm e}^{\, i \, {\bf u} \cdot {\bf R}}}{u \sqrt{1-u^2}} \frac{u \sqrt{1-u^2} \, {\cal Q}_1 + i \, {\bf u} \cdot {\bf {\cal V}} \, {\rm sinh} \left( u \sqrt{1-u^2} \, T \right)} {u^2 \left( 1 - u^2 \right) + \left( {\bf u} \cdot {\bf {\cal V}} + i \, \epsilon \right)^2} \,\,, \right. \nonumber\\ && \left. \int_{u \geq 1} d^3 {\bf u} \frac{{\rm e}^{\, i \, {\bf u} \cdot {\bf R}}}{u \sqrt{u^2-1}} \frac{u \sqrt{u^2-1} \, {\cal Q}_2 + i \, {\bf u} \cdot {\bf {\cal V}} \, {\rm sin} \left( u \sqrt{u^2-1} \, T \right)} {u^2 \left( u^2 - 1 \right) - \left( {\bf u} \cdot {\bf {\cal V}} + i \, \epsilon \right)^2} \right\} \nonumber\\ \label{preangular}\end{aligned}$$ where we have defined $$\begin{aligned} {\cal Q}_1 &\equiv& {\rm e}^{-i \, {\bf u} \cdot {\bf {\cal V}} T} - {\rm cosh } \left( u \sqrt{1-u^2} \, T \right) \,\,, \nonumber\\ {\cal Q}_2 &\equiv& {\rm e}^{-i \, {\bf u} \cdot {\bf {\cal V}} T} - {\rm cos } \left( u \sqrt{u^2-1} \, T \right) \,\,,\end{aligned}$$ and where $u \equiv \vert {\bf u} \vert \,$. Already at this stage we can appreciate the effect of the velocity of the source. In the rescaled units, the speed of propagation of $\Delta V$ (see the previous Section) corresponds to ${\cal V}_{\rm s} \simeq 1 \,$. Sources which move much faster than $v_{\rm s}$ will have ${\cal V} \gg 1\,$. In this case, we then expect the ${\cal V}-$dependent terms to be important in (\[preangular\]). The angular integration can be performed exactly when ${\bf {\tilde r}}$ and ${\bf v}$ are parallel, that is when the potential is computed along the line of motion of the source. The following explicit computation is restricted to this case; however, the arguments presented above and our conclusions do not depend on this assumption. The angular integrals then give $$\begin{aligned} \Delta V &=& \frac{G \, m \, \mu}{\pi \, {\cal V}} \left\{ \, \int_0^1 \frac{d u}{\sqrt{1-u^2}} \left[ {\rm cosh } \left( b \, u \, {\tilde R} \right) {\cal A} + {\rm sinh } \left( b \, u \, {\tilde R} \right) {\cal B} \right] + \right. \nonumber\\ && \left. \quad\quad\quad + \int_1^\infty \frac{d u}{\sqrt{u^2-1}} \left[ {\rm cos } \left( b \, u \, {\tilde R} \right) {\cal C} + {\rm sin } \left( b \,u \, {\tilde R} \right) {\cal D} \right] \; \right\} \,\,, \nonumber\\ &&\nonumber\\ {\cal A} &\equiv& \left[ \pi \, \sigma \left( {\tilde R} \right) + i \, CI \left( u {\tilde R} \left( 1 + i \, b \right) \right) - i \, CI \left( u {\tilde R} \left( 1 - i \, b \right) \right) \right] + \nonumber\\ &&- \left[ {\tilde R} \rightarrow R \right] \,\,, \nonumber\\ {\cal B} &\equiv& \left[ - SI \left( u {\tilde R} \left( 1 + i \, b \right) \right) - SI \left( u {\tilde R} \left( 1 - i \, b \right) \right) \right] - \left[ {\tilde R} \rightarrow R \right] \,\,, \nonumber\\ {\cal C} &\equiv& \left[ - i \, \pi \, \theta \left( b - 1 \right) \sigma \left( {\tilde R} \right) - CI \left( u {\tilde R} \left( 1 + b \right) + i \, \epsilon \right) + \right. \nonumber\\ && \left. + CI \left( u {\tilde R} \left( 1 - b \right) + i \, \epsilon\right) \right] - \left[ {\tilde R} \rightarrow R \right] \,\,, \nonumber\\ {\cal D} &\equiv& \left[ - SI \left( u {\tilde R} \left( 1 + b \right) \right) - SI \left( u {\tilde R} \left( 1 - b \right) \right) \right] - \left[ {\tilde R} \rightarrow R \right] \,\,, \nonumber\\ b &\equiv& \frac{\sqrt{\vert 1 - u^2 \vert}}{\cal V} \,\,, \label{pot}\end{aligned}$$ where $\sigma \left( x \right)$ is the sign of $x\,$, while SI and CI denote, respectively, the sine and cosine integral, defined as [@librosacro] $$SI \left( x \right) \equiv - \int_x^\infty d t \: \frac{{\rm sin } \, t}{t} \quad,\quad CI \left( x \right) \equiv - \int_x^\infty d t \: \frac{{\rm cos } \, t}{t} \,\,.$$ At the initial time, $R={\tilde R}$ and, as we noted, $\Delta V = 0\,$. In addition, one can verify that (\[pot\]) reproduces the analogous expression of [@ghost] as the velocity ${\cal V}$ is sent to zero. The first line of (\[pot\]) describes the tachyonic modes, and we will focus only on the calculation of this piece (we denote it by $\Delta V_1$) in the remaining of the Section. The second line describes the stable modes, and indeed it does not contain any term which grows exponentially with time. For this reason, it is not responsible for the effect we are considering, and, as done in [@ghost], we can simply ignore it. The expression (\[pot\]) is still exact [^6]. To proceed further, we can approximate the integrand for ${\cal V} \gg 1$ (we recall that ${\cal V} = 1$ corresponds to $v = v_{\rm s}\,$). $$\begin{aligned} \Delta V_1 &\simeq& \frac{2 \, G \, m \, \mu}{\pi \, {\cal V}} \int_0^1 \frac{d u}{\sqrt{1-u^2}} \, \times \nonumber\\ && \Bigg\{ b \bigg[ \Big[ {\rm cos } \left( u \vert R \vert \right) + u \vert R \vert \, SI \left( u \vert R \vert \right) \Big] \, {\rm cosh } \left( u \, b \left( R - {\tilde R} \right) \right) - {\rm cos } \left( u \vert {\tilde R} \vert \right) + \nonumber\\ && \;\;\;\;\;\;\; - u \vert {\tilde R} \vert \, SI \left( u \vert {\tilde R} \vert \right) \bigg] - \sigma \left( R \right) \, SI \left( u \vert R \vert \right) \, {\rm sinh } \left( u \left( R - {\tilde R} \right) \, b \right) + \nonumber\\ && \;\;\; + \pi \, {\rm sinh } \left( u \, b \vert {\tilde R} \vert \right) \sigma \left( {\tilde R} \right) \left[ \theta \left( - {\tilde R} \right) - \theta \left( - R \right) \right] \Bigg\} \,\,. \label{largev}\end{aligned}$$ This result holds for $b \equiv \sqrt{1-u^2} / {\cal V} \ll 1\,$, but arbitrary $b\, \vert R \vert$ and $b \, \vert {\tilde R} \vert \,$. Much simpler expressions can be obtained in the limits of very small or large $R,\; {\tilde R}\,$. In particular, we can distinguish two relevant cases, which we have already discussed in the previous Section. 1\) ${\cal V \, T} \gg 1 \;,\; \vert {\tilde R} \vert \ll {\cal V} \;,\; R \gg T \;,\; R \gg 1 \,$. In this case we find $$\begin{aligned} \Delta V_1 \simeq \left\{ \begin{array}{c} - \, \frac{2\,G\,\mu\,m}{\pi\,{\cal V}^2} \quad\quad\quad\quad\quad\;,\;\;\quad \vert {\tilde R} \vert \ll 1 \,\,,\\ \\ \frac{G \, \mu \, m}{{\cal V}^2} \left[ {\tilde R} \, \theta\left( - {\tilde R} \right) - 2\,\frac{{\rm cos } {\tilde R}}{{\tilde R}^2} \right] \quad, \quad \vert {\tilde R} \vert \gg 1 \,\,. \end{array} \right. \label{nearsource}\end{aligned}$$ this computation applies for a late time observer at a fixed distance ${\tilde R}$ (taken to be small, on cosmological scales) from the source. As we explained in the previous Section, $\Delta V$ [*does not*]{} grow exponentially in this case. To clarify why this happens, let us compute in more details $\Delta V_1$ for ${\tilde R} =0\,$, that is at the exact location of the (moving) source. Starting from (\[pot\]), and expanding CI and SI for large argument, we get $$\Delta V_1 \left( {\tilde R} = 0 \right) \! = \! \frac{2 \, G \, m \, \mu}{\pi \, {\cal V}^2} \!\! \int_0^1 \! d u \left[ \frac{{\rm cos } \left( u \, {\cal V} \, T \right) \, {\rm sinh } \left( u \sqrt{1-u^2} \, T \right)}{u \sqrt{1-u^2} T} - 1 \right] + {\rm O } \left( \frac{1}{{\cal V}^3} \right) . \label{rt0}$$ This expression contains the function ${\rm sinh } \left( u \sqrt{1-u^2} \, T \right) \,$, which is growing exponentially with time. However, it is multiplied by the much faster oscillating function ${\rm cos } \left( u \, {\cal V} \, T \right)\,$. Since ${\cal V} \gg 1\,$, positive and negative part of the integrand average to a very small value, which does not grow with time. This suppression effect can be explicitly seen to be at work already at the level of eq. (\[pot\]), unless $R$ is small (that is, close to the point where the source was nucleated, see below). The first term of (\[rt0\]) cannot be computed exactly. We obtain an approximate primitive by noticing that, for ${\cal V} \gg 1\,$, $$\frac{{\rm cos } \left( u \, {\cal V} \, T \right) \, {\rm sinh } \left( u \sqrt{1-u^2} T \right)}{u \sqrt{1-u^2}} \simeq \frac{d}{du} \left[ \frac{{\rm sin } \left( u \, {\cal V} \, T \right) \, {\rm sinh } \left( u \sqrt{1-u^2} T \right)}{{\cal V} \, T \, u \sqrt{1-u^2}} \right] \label{primitiva}$$ (in practice, we simply use the primitive of the most rapidly oscillating function; we have verified numerically that this approximation is accurate[^7]). Inserting eq. (\[primitiva\]) into (\[rt0\]) we recover the first line of (\[nearsource\]). The result (\[nearsource\]) has a few other features which is worth noting. First, it is time independent. This was expected, since the potential close to the source resents only negligibly of the previous history of the source , due to the fact that both the source and the observer are moving much faster than the speed at which the signal is propagating. [^8] Finally, we also note that, for $1 \ll \vert {\tilde R} \vert \ll {\cal V} \,$, the potential is greater at negative ${\tilde R}\,$. This is easily understood by remembering that negative ${\tilde R}$ describe points where the source has already passed, and where the instability has started to develop, although the growth is still in the linear regime. Positive ${\tilde R}$ are instead points at which the source has yet to come. 2\) The second interesting regime for the computation of (\[pot\]) is $\vert R \vert \ll {\cal V} \;,\; \vert {\tilde R} \vert \simeq {\cal V} T \, \gg {\cal V} \,$. In this case we get $$\begin{aligned} \Delta V_1 \simeq \left\{ \begin{array}{c} - \frac{G\,m\,\mu\,\sqrt{\pi}}{2\,{\cal V}\,\sqrt{T}}\,{\rm e }^{T/2} \quad\quad\quad\;,\quad \vert R \vert \ll 1\\ \\ - \frac{G\,m\,\mu\,\sqrt{\pi}}{2\,{\cal V}\,\sqrt{T}}\,{\rm e }^{T/2} \left[ 2 \, \theta \left( R \right) +{\rm O}\left(\frac{1}{R}\right) \right] , \\ \\ \quad\quad\quad\quad 1 \ll \vert R \vert \ll T \end{array} \right. \label{farsource}\end{aligned}$$ This computation applies for a late time observer which is at rest in the rest-frame of the condensate, and close to where the source was originally nucleated. In this case, $\Delta V$ is growing exponentially with time. The exponential growth is related to the instability of the vacuum triggered by the source when it was passing through these points, as discussed in the previous Section. [^9] Conclusions =========== Effects associated to the breaking of Lorentz symmetry will in general depend upon the motion of the observer with respect to the preferred frame of the theory. In particular, a phenomenologically very interesting aspect of ghost condensation is the presence of long wavelength instabilities that evolve at a very low speed $v_{\rm s}\simeq M/M_p$, and that will be triggered by classical sources (and, it is natural to expect, by quantum fluctuations). An observer at rest with respect to the preferred frame will see the growth of the instability that, after a sufficiently long time, will show up as a measurable modification of Newtonian gravity. An observer in motion, however, will not have the time to see the development of the instability. As an analogy, we can think of a overheated liquid, which, similarly to the ghost condensate, is in a metastable state. A small perturbation of the liquid – such as the introduction of a particle – will lead to the generation of a bubble, which will then expand at the speed of sound in the liquid. In this picture, the motion of a celestial body in the ghost condensate is analogous to the motion of a particle quickly traveling through this bubble chamber. Regions of modified gravity will be nucleated where the source passes, but will then expand at the much smaller velocity $v_{\rm s}\,$. An observer sitting on the particle will not have time to see the growth of the bubbles. In the absence of observability of effects in tests of gravity, we expect the signatures of the scenario to be associated to astrophysical or cosmological effects. While in conventional gravity every source is associated to a potential well, in this scenario every source will leave behind itself a potential furrow, that will keep growing even when the source is far away. Depending on the value of the parameter $M$, such unstable region will either be still in its linear regime or will have evolved to a nonlinear stage. In the latter case, it is possible to speculate about the existence of exotic compact objects whose position would not be manifestly correlated with the present distribution of matter. Even if the furrows are still in their linear regime, they could manifest themselves as irregularities in the gravitational potential. These irregularities could be felt by systems (such as the one formed by the Earth and a satellite) that go across them. The possibility to detect them through irregular motions of celestial bodies, or maybe also through lensing effects, should be presumably studied statistically. Acknowledgements ================ It is a pleasure to thank Andy Albrecht, Paolo Creminelli, Nemanja Kaloper, Manoj Kaplinghat, Markus Luty, David Mattingly, Shinji Mukohyama, Keith Olive, Erich Poppitz, and Arkady Vainshtein for very fruitful discussions. The work of L. S. was supported in part by the NSF Grant PHY-0332258. After the completion of the present manuscript, we became aware of the related work [@dub]. The analysis and the conclusions of [@dub] are in agreement with the results presented here. [99]{} M. Fierz and W. Pauli, Proc. Roy. Soc. Lond. A [**173**]{} (1939) 211. H. van Dam and M. J. G. Veltman, Nucl. Phys. B [**22**]{} (1970) 397; V. I. Zakharov, JETP Letters (Sov. Phys.) [**12**]{} (1970) 312. A. I. Vainshtein, Phys. Lett. B [**39**]{} (1972) 393. N. Arkani-Hamed, H. Georgi and M. D. Schwartz, Annals Phys.  [**305**]{} (2003) 96. C. Charmousis, R. Gregory and V. A. Rubakov, Phys. Rev. D [**62**]{} (2000) 067505; R. Gregory, V. A. Rubakov and S. M. Sibiryakov, Phys. Rev. Lett.  [**84**]{}, 5928 (2000) \[arXiv:hep-th/0002072\]. C. Csaki, J. Erlich and T. J. Hollowood, Phys. Rev. Lett.  [**84**]{} (2000) 5932; Phys. Lett. B [**481**]{} (2000) 107 L. Pilo, R. Rattazzi and A. Zaffaroni, JHEP [**0007**]{} (2000) 056; G. R. Dvali, G. Gabadadze and M. Porrati, Phys. Lett. B [**484**]{} (2000) 112, Phys. Lett. B [**484**]{} (2000) 129, Phys. Lett. B [**485**]{} (2000) 208; C. Deffayet, G. R. Dvali and G. Gabadadze, Phys. Rev. D [**65**]{} (2002) 044023; I. I. Kogan, S. Mouslopoulos, A. Papazoglou, G. G. Ross and J. Santiago, Nucl. Phys. B [**584**]{} (2000) 313; I. I. Kogan and G. G. Ross, Phys. Lett. B [**485**]{} (2000) 255; I. I. Kogan, S. Mouslopoulos, A. Papazoglou and G. G. Ross, Nucl. Phys. B [**595**]{} (2001) 225; T. Damour and I. I. Kogan, Phys. Rev. D [**66**]{} (2002) 104024. V. A. Kostelecky and S. Samuel, Phys. Rev. D [**39**]{}, 683 (1989); D. Colladay and V. A. Kostelecky, Phys. Rev. D [**58**]{}, 116002 (1998) \[arXiv:hep-ph/9809521\]; S. R. Coleman and S. L. Glashow, Phys. Rev. D [**59**]{}, 116008 (1999) \[arXiv:hep-ph/9812418\]; T. Jacobson and D. Mattingly, Phys. Rev. D [**64**]{}, 024028 (2001); V. A. Kostelecky, Ed., Proceedings of the [*“Second Meeting on CPT and Lorentz Symmetry”*]{}, Bloomington, Usa, 15-18 August 2001”. Singapore, World Scientific (2002). N. Arkani-Hamed, H. C. Cheng, M. A. Luty and S. Mukohyama, hep-th/0312099. N. Arkani-Hamed, P. Creminelli, S. Mukohyama and M. Zaldarriaga, hep-th/0312100. I. S. Gradshteyn, I. M. Ryzhik, [*Table of integrals, series, and products,*]{} Academic Press, 1980. S.L.Dubovsky, hep-ph/0403308. [^1]: The work [@ghost] was accompanied by [@paolo], which studied inflation in this theory. The primordial perturbations generated during inflation were found to have distinctive signatures with respect to the standard results of slow roll inflation. [^2]: This is due to the fact that $\phi$ has a well behaving kinetic term in its ground state in the absence of gravity. [^3]: Direct couplings of the ghost to matter are expected to emerge at least through loop effects. Such couplings have to be strongly suppressed in order not lead to disagreement with observation, and depend on the details of the theory. In order to stick only to the minimal features of the scenario, we will not consider these couplings here. [^4]: In this analysis, the effects related to the expansion of the Universe are neglected. Due to the magnitude of the scales in consideration, we do not expect these effects to change significantly the present discussion. [^5]: In the nonrelativistic approximation that we are here considering, the Newtonian interaction propagates instantaneously. [^6]: More precisely, it holds in the nonrelativistic limit $\omega \ll k\,$. For $\Delta V_1\,$, the poles are at $\vert \omega \vert \sim \vert k \vert m/M\,$. Hence, the nonrelativistic limit is correct as long as $M \ll M_p\,$. [^7]: Alternatively, one can proceed as in [@ghost], by substituting ${\rm sinh } \left( u \sqrt{1-u^2} T \right),$ with a gaussian, and by extending the integration from $0 < u < 1$ to $- \infty < u < + \infty\,$. This approximation is valid only in the limit ${\cal V}\ll 1$, and would give a result $\propto {\rm exp } \left( - {\cal V}^2 \, T / 8 \right)\,$. [^8]: Strictly speaking, this is not true in the transient $T \ll 1\,$ regime, since in this case one cannot neglect the fact that the source was nucleated at the finite time $T=0\,$. [^9]: The growth visible in (\[farsource\]) is analogous to the one computed in [@ghost] for a static source, with the difference of the strong suppression factor $1/{\cal V}\,$.

--- abstract: 'The sort transform (ST) is a modification of the Burrows-Wheeler transform (BWT). Both transformations map an arbitrary word of length $n$ to a pair consisting of a word of length $n$ and an index between $1$ and $n$. The BWT sorts all rotation conjugates of the input word, whereas the ST of order $k$ only uses the first $k$ letters for sorting all such conjugates. If two conjugates start with the same prefix of length $k$, then the indices of the rotations are used for tie-breaking. Both transforms output the sequence of the last letters of the sorted list and the index of the input within the sorted list. In this paper, we discuss a bijective variant of the BWT (due to Scott), proving its correctness and relations to other results due to Gessel and Reutenauer (1993) and Crochemore, D[é]{}sarm[é]{}nien, and Perrin (2005). Further, we present a novel bijective variant of the ST.' author: - Manfred Kufleitner title: | On Bijective Variants of the\ Burrows-Wheeler Transform --- Introduction {#sec:intro} ============ The Burrows-Wheeler transform (BWT) is a widely used preprocessing technique in lossless data compression [@bw94tr]. It brings every word into a form which is likely to be easier to compress [@man01acm]. Its compression performance is almost as good as PPM (prediction by partial matching) schemes [@cw84toc] while its speed is comparable to that of Lempel-Ziv algorithms [@lz77; @lz78]. Therefore, BWT based compression schemes are a very reasonable trade-off between running time and compression ratio. In the classic setting, the BWT maps a word of length $n$ to a word of length $n$ and an index (comprising $O(\log n)$ bits). Thus, the BWT is not bijective and hence, it is introducing new redundancies to the data, which is cumbersome and undesired in applications of data compression or cryptography. Instead of using an index, a very common technique is to assume that the input has a unique end-of-string symbol [@bk00tc; @man01acm]. Even though this often simplifies proofs or allows speeding up the algorithms, the use of an end-of-string symbol introduces new redundancies (again $O(\log n)$ bits are required for coding the end-of-string symbol). We discuss bijective versions of the BWT which are one-to-one correspondences between words of length $n$. In particular, no index and no end-of-string symbol is needed. Not only does bijectivity save a few bits, for example, it also increases data security when cryptographic procedures are involved; it is more natural and it can help us to understand the BWT even better. Moreover, the bijective variants give us new possibilities for enhancements; for example, in the bijective BWT different orders on the letters can be used for the two main stages. Several variants of the BWT have been introduced [@aa04ijcm; @mrrs07tcs]. An overview can be found in the textbook by Adjeroh, Bell, and Mukherjee [@abm08book]. One particularly important variant for this paper is the sort transform (ST), which is also known under the name Schindler transform [@schindler97dcc]. In the original paper, the inverse of the ST is described only very briefly. More precise descriptions and improved algorithms for the inverse of the ST have been proposed recently [@nz06dcc; @nz07ieee; @nzc08cpm]. As for the BWT, the ST also involves an index or an end-of-string symbol. In particular, the ST is not onto and it introduces new redundancies. The bijective BWT was discovered and first described by Scott (2007), but his exposition of the algorithm was somewhat cryptic, and was not appreciated as such. In particular, the fact that this transform is based on the Lyndon factorization went unnoticed by Scott. Gil and Scott [@gs09submitted] provided an accessible description of the algorithm. Here, we give an alternative description, a proof of its correctness, and more importantly, draw connections between Scott’s algorithm and other results in combinatorics on words. Further, this variation of the BWT is used to introduce techniques which are employed at the bijective sort transform, which makes the main contribution of this paper. The forward transform of the bijective ST is rather easy, but we have to be very careful with some details. Compared with the inverse of the bijective BWT, the inverse of the bijective ST is more involved. **Outline.**   The paper is organized as follows. In Section \[sec:prelim\] we fix some notation and repeat basic facts about combinatorics on words. On our way to the bijective sort transform (Section \[sec:lst\]) we investigate the BWT (Section \[sec:bwt\]), the bijective BWT (Section \[sec:lbwt\]), and the sort transform (Section \[sec:st\]). We give full constructive proofs for the injectivity of the respective transforms. Each section ends with a running example which illustrates the respective concepts. Apart from basic combinatorics on words, the paper is completely self-contained. Preliminaries {#sec:prelim} ============= Throughout this paper we fix the finite non-empty alphabet $\Sigma$ and assume that $\Sigma$ is equipped with a linear order $\leq$. A *word* is a sequence $a_1 \cdots a_n$ of letters $a_i \in \Sigma$, $1 \leq i \leq n$. The set of all such sequences is denoted by $\Sigma^*$; it is the free monoid over $\Sigma$ with concatenation as composition and with the empty word $\varepsilon$ as neutral element. The set $\Sigma^+ = \Sigma^* \setminus {\left\{\mathinner{\varepsilon}\right\}}$ consists of all non-empty words. For words $u,v$ we write $u \leq v$ if $u=v$ or if $u$ is lexicographically smaller than $v$ with respect to the order $\leq$ on the letters. Let $w = a_1 \cdots a_n \in \Sigma^+$ be a non-empty word with letters $a_i \in \Sigma$. The *length* of $w$, denoted by ${\left|\mathinner{w}\right|}$, is $n$. The empty word is the unique word of length $0$. We can think of $w$ as a labeled linear order: position $i$ of $w$ is labeled by $a_i \in \Sigma$ and in this case we write ${\lambda}_w(i) = a_i$, so each word $w$ induces a labeling function ${\lambda}_w$. The first letter $a_1$ of $w$ is denoted by ${\mathrm{first}}(w)$ while the last letter $a_n$ is denoted by ${\mathrm{last}}(w)$. The *reversal* of a word $w$ is $\overline{w} = a_n \cdots a_1$. We say that two words $u,v$ are *conjugate* if $u = st$ and $v = ts$ for some words $s,t$, i.e., $u$ and $v$ are cyclic shifts of one another. The $j$-fold concatenation of $w$ with itself is denoted by $w^j$. A word $u$ is a *root* of $w$ if $w = u^j$ for some $j \in {\mathbb{N}}$. A word $w$ is *primitive* if $w = u^j$ implies $j = 1$ and hence $u = w$, i.e., $w$ has only the trivial root $w$. The *right-shift* of $w = a_1 \cdots a_n$ is ${r}(w) = a_n a_1 \cdots a_{n-1}$ and the $i$-fold right shift ${r}^i(w)$ is defined inductively by ${r}^0(w) = w$ and ${r}^{i+1}(w) = {r}({r}^{i}(w))$. We have ${r}^{i}(w) = a_{n-i+1} \cdots a_n a_1 \cdots a_{n-i}$ for $0 \leq i < n$. The word ${r}^i(w)$ is also well-defined for $i \geq n$ and then ${r}^i(w) = {r}^j(w)$ where $j = i \bmod n$. We define the *ordered conjugacy class* of a word $w \in \Sigma^n$ as $[w] = (w_1, \ldots, w_n)$ where $w_i = {r}^{i-1}(w)$. It is convenient to think of $[w]$ as a cycle of length $n$ with a pointer to a distinguished starting position. Every position $i$, $1 \leq i \leq n$, on this cycle is labeled by $a_i$. In particular, $a_1$ is a successor of $a_n$ on this cycle since the position $1$ is a successor of the position $n$. The mapping ${r}$ moves the pointer to its predecessor. The (unordered) conjugacy class of $w$ is the multiset ${\left\{\mathinner{w_1, \ldots, w_n}\right\}}$. Whenever there is no confusion, then by abuse of notation we also write $[w]$ to denote the (unordered) conjugacy class of $w$. For instance, this is the case if $w$ is in some way distinguished within its conjugacy class, which is true if $w$ is a Lyndon word. A *Lyndon word* is a non-empty word which is the unique lexicographic minimal element within its conjugacy class. More formally, let $[w] = (w, w_2, \ldots, w_n)$, then $w\in \Sigma^+$ is a Lyndon word if $w < w_i$ for all $i \in {\left\{\mathinner{2, \ldots, n}\right\}}$. Lyndon words have a lot of nice properties [@Lot83]. For instance, Lyndon words are primitive. Another interesting fact is the following. \[thm:lyndon\] Every word $w \in \Sigma^+$ has a unique factorization $ w = v_s \cdots v_1 $ such that $v_1 \leq \cdots \leq v_s$ is a non-decreasing sequence of Lyndon words. An alternative formulation of the above fact is that every word $w$ has a unique factorization $w = v_s^{n_s} \cdots v_1^{n_1}$ where $n_i \geq 1$ for all $i$ and where $v_1 < \cdots < v_s$ is a strictly increasing sequence of Lyndon words. The factorization of $w$ as in Fact \[thm:lyndon\] is called the *Lyndon factorization* of $w$. It can be computed in linear time using Duval’s algorithm [@duv83ja]. Suppose we are given a multiset $V = {\left\{\mathinner{v_1, \ldots, v_s}\right\}}$ of Lyndon words enumerated in non-decreasing order $v_1 \leq \cdots \leq v_s$. Now, $V$ uniquely determines the word $w = v_s \cdots v_1$. Therefore, the Lyndon factorization induces a one-to-one correspondence between arbitrary words of length $n$ and multisets of Lyndon words of total length $n$. Of course, by definition of Lyndon words, the multiset ${\left\{\mathinner{v_1, \ldots, v_s}\right\}}$ of Lyndon words and the multiset ${\left\{\mathinner{[v_1], \ldots, [v_s]}\right\}}$ of conjugacy classes of Lyndon words are also in one-to-one correspondence. We extend the order $\leq$ on $\Sigma$ as follows to non-empty words. Let $w^{\omega} = w w w \cdots$ be the infinite sequences obtained as the infinite power of $w$. For $u,v \in \Sigma^+$ we write $u \leq^\omega v$ if either $u^{\omega} = v^{\omega}$ or $u^{\omega} = paq$ and $v^{\omega} = pbr$ for $p \in \Sigma^*$, $a,b \in \Sigma$ with $a<b$, and infinite sequences $q,r$; phrased differently, $u \leq^\omega v$ means that the infinite sequences $u^\omega$ and $v^\omega$ satisfy $u^\omega \leq v^\omega$. If $u$ and $v$ have the same length, then $\leq^\omega$ coincides with the lexicographic order induced by the order on the letters. For arbitrary words, $\leq^\omega$ is only a preorder since for example $u \leq^\omega uu$ and $uu \leq^\omega u$. On the other hand, if $u \leq^\omega v$ and $v \leq^\omega u$ then $u^{{\left|\mathinner{v}\right|}} = v^{{\left|\mathinner{u}\right|}}$. Hence, by the periodicity lemma [@FW65], there exists a common root $p \in \Sigma^+$ and $g,h \in {\mathbb{N}}$ such that $u = p^g$ and $v = p^h$. Also note that $b \leq ba$ whereas $ba \leq^{\omega} b$ for $a<b$. Intuitively, the *context of order $k$* of $w$ is the sequence of the first $k$ letters of $w$. We want this notion to be well-defined even if ${\left|\mathinner{w}\right|} < k$. To this end let ${\mathrm{context}}_k(w)$ be the prefix of length $k$ of $w^\omega$, i.e., ${\mathrm{context}}_k(w)$ consists of the first $k$ letters on the cycle $[w]$. Note that our definition of a context of order $k$ is left-right symmetric to the corresponding notion used in data compression. This is due to the fact that typical compression schemes are applying the BWT or the ST to the reversal of the input. An important construction in this paper is the *standard permutation* $\pi_w$ on the set of positions ${\left\{\mathinner{1,\ldots, n}\right\}}$ induced by a word $w = a_1 \cdots a_n \in \Sigma^n$ [@gr93jct]. The first step is to introduce a new order $\preceq$ on the positions of $w$ by sorting the letters within $w$ such that identical letters preserve their order. More formally, the linear order $\preceq$ on ${\left\{\mathinner{1,\ldots,n}\right\}}$ is defined as follows: $i \preceq j$ if $$a_i < a_j \qquad \text{or} \qquad a_i = a_j \,\text{ and }\, i \leq j.$$ Let $j_1 \prec \cdots \prec j_n$ be the linearization of ${\left\{\mathinner{1, \ldots, n}\right\}}$ according to this new order. Now, the standard permutation $\pi_w$ is defined by $\pi_w(i) = j_i$. \[exa:start\] Consider the word $w = bcbccbcbcabbaaba$ over the ordered alphabet $a < b < c$. We have ${\left|\mathinner{w}\right|} = 16$. Therefore, the positions in $w$ are ${\left\{\mathinner{1, \ldots, 16}\right\}}$. For instance, the label of position $6$ is ${\lambda}_w(6) = b$. Its Lyndon factorization is $w = bcbcc \cdot bc \cdot bc \cdot abb \cdot aab \cdot a$. The context of order $7$ of the prefix $bcbcc$ of length $5$ is $bcbccbc$ and the context of order $7$ of the factor $bc$ is $bcbcbcb$. For computing the standard permutation we write $w$ column-wise, add positions, and then sort the pairs lexicographically:  word $w$    $w$ with positions    sorted$\ $ ---------------- ----------------------- ------------- $b$ $(b,1)$ $(a,10)$ \[-1.2pt\] $c$ $(c,2)$ $(a,13)$ \[-1.2pt\] $b$ $(b,3)$ $(a,14)$ \[-1.2pt\] $c$ $(c,4)$ $(a,16)$ \[-1.2pt\] $c$ $(c,5)$ $(b,1)$ \[-1.2pt\] $b$ $(b,6)$ $(b,3)$ \[-1.2pt\] $c$ $(c,7)$ $(b,6)$ \[-1.2pt\] $b$ $(b,8)$ $(b,8)$ \[-1.2pt\] $c$ $(c,9)$ $(b,11)$ \[-1.2pt\] $a$ $(a,10)$ $(b,12)$ \[-1.2pt\] $b$ $(b,11)$ $(b,15)$ \[-1.2pt\] $b$ $(b,12)$ $(c,2)$ \[-1.2pt\] $a$ $(a,13)$ $(c,4)$ \[-1.2pt\] $a$ $(a,14)$ $(c,5)$ \[-1.2pt\] $b$ $(b,15)$ $(c,7)$ \[-1.2pt\] $a$ $(a,16)$ $(c,9)$ This yields the standard permutation $$\pi_w = \left( \begin{array}{*{16}{p{5.5mm}}} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\ 10 & 13 & 14 & 16 & 1 & 3 & 6 & 8 & 11 & 12 & 15 & 2 & 4 & 5 & 7 & 9 \end{array} \right).$$ The conjugacy class $[w]$ of $w$ is depicted in Figure \[sfg:w\]; the $i$-th word in $[w]$ is written in the $i$-th row. The last column of the matrix for $[w]$ is the reversal $\overline{w}$ of $w$.     The Burrows-Wheeler transform {#sec:bwt} ============================= The *Burrows-Wheeler transform* (BWT) maps words $w$ of length $n$ to pairs $(L,i)$ where $L$ is a word of length $n$ and $i$ is an index in ${\left\{\mathinner{1, \ldots, n}\right\}}$. The word $L$ is usually referred to as the Burrows-Wheeler transform of $w$. In particular, the BWT is not surjective. We will see below how the BWT works and that it is one-to-one. It follows that only a fraction of $1/n$ of all possible pairs $(L,i)$ appears as an image under the BWT. For instance $(bacd,1)$ where $a<b<c<d$ is not an image under the BWT. For $w \in \Sigma^+$ we define $M(w) = (w_1, \ldots, w_n)$ where ${\left\{\mathinner{w_1, \ldots, w_n}\right\}} = [w]$ and $w_1 \leq \cdots \leq w_n$. Now, the Burrows-Wheeler transform of $w$ consists of the word ${\mathrm{BWT}}(w) = {\mathrm{last}}(w_1) \cdots {\mathrm{last}}(w_n)$ and an index $i$ such that $w = w_i$. Note that in contrast to the usual definition of the BWT, we are using right shifts; at this point this makes no difference but it unifies the presentation of succeeding transforms. At first glance, it is surprising that one can reconstruct $M(w)$ from ${\mathrm{BWT}}(w)$. Moreover, if we know the index $i$ of $w$ in the sorted list $M(w)$, then we can reconstruct $w$ from ${\mathrm{BWT}}(w)$. One way of how to reconstruct $M(w)$ is presented in the following lemma. For later use, we prove a more general statement than needed for computing the inverse of the BWT. \[lem:contextretrieval\] Let $k \in {\mathbb{N}}$. Let $\bigcup_{i=1}^{s}\, [v_i] = {\left\{\mathinner{w_1, \ldots, w_n}\right\}} \subseteq \Sigma^+$ be a multiset built from conjugacy classes $[v_i]$. Let $M = (w_1, \ldots, w_n)$ satisfy ${\mathrm{context}}_k(w_1) \leq \cdots \leq {\mathrm{context}}_k(w_n)$ and let $L = {\mathrm{last}}(w_1) \cdots {\mathrm{last}}(w_n)$ be the sequence of the last symbols. Then $${\mathrm{context}}_k(w_i) = {\lambda}_{L}\pi_{L}(i) \cdot {\lambda}_{L}\pi_{L}^2(i) \,\cdots\, {\lambda}_{L}\pi_{L}^k(i)$$ where $\pi_{L}^t$ denotes the $t$-fold application of $\pi_{L}$ and ${\lambda}_L \pi_L^t (i) = {\lambda}_L\bigl(\pi_L^t(i)\bigr)$. By induction over the context length $t$, we prove that for all $i \in {\left\{\mathinner{1, \ldots, n}\right\}}$ we have ${\mathrm{context}}_t(w_i) = {\lambda}_{L}\pi_{L}(i) \,\cdots\, {\lambda}_{L}\pi_{L}^t(i)$. For $t = 0$ we have ${\mathrm{context}}_0(w_i) = \varepsilon$ and hence, the claim is trivially true. Let now $0 < t \leq k$. By the induction hypothesis, the $(t-1)$-order context of each $w_i$ is ${\lambda}_{L} \pi_{L}(i) \cdots {\lambda}_{L} \pi_{L}^{t-1}(i)$. By applying one right-shift, we see that the $t$-order context of ${r}(w_i)$ is ${\lambda}_{L}(i) \cdot {\lambda}_{L} \pi_{L}^1(i) \cdots {\lambda}_{L} \pi_{L}^{t-1}(i)$. The list $M$ meets the sort order induced by $k$-order contexts. In particular, $(w_1, \ldots, w_n)$ is sorted by $(t-1)$-order contexts. Let $(u_1, \ldots, u_n)$ be a stable sort by $t$-order contexts of the right-shifts $({r}(w_1), \ldots, {r}(w_n))$. The construction of $(u_1, \ldots, u_n)$ only requires a sorting of the first letters of $({r}(w_1), \ldots, {r}(w_n))$ such that identical letters preserve their order. The sequence of first letters of the words ${r}(w_1), \ldots, {r}(w_n)$ is exactly $L$. By construction of $\pi_L$, it follows that $(u_1, \ldots, u_n) = (w_{\pi_L(1)}, \ldots, w_{\pi_L(n)})$. Since $M$ is built from conjugacy classes, the multisets of elements occurring in $(w_1, \ldots, w_n)$ and $({r}(w_1), \ldots, {r}(w_n))$ are identical. The same holds for the multisets induced by $(w_1, \ldots, w_n)$ and $(u_1, \ldots, u_n)$. Therefore, the sequences of $t$-order contexts induced by $(w_1, \ldots w_n)$ and $(u_1, \ldots, u_n)$ are identical. Moreover, we conclude $$\begin{aligned} {\mathrm{context}}_t(w_i) = {\mathrm{context}}_t(u_i) = {\mathrm{context}}_t(w_{\pi_L(i)}) = {\lambda}_{L}\pi_{L}(i) \cdot {\lambda}_L \pi_L^2(i) \,\cdots\, {\lambda}_{L}\pi_{L}^t(i) \end{aligned}$$ which completes the induction. We note that in general $u_i \neq w_i$ since the sort order of $M$ beyond $k$-order contexts is arbitrary. Moreover, for $t=k+1$ the property ${\mathrm{context}}_t(w_i) = {\mathrm{context}}_t(u_i)$ does not need to hold (even though the multisets of $(k+1)$-order contexts coincide). Note that in Lemma \[lem:contextretrieval\] we do not require that all $v_i$ have the same length. Applying the BWT to conjugacy classes of words with different lengths has also been used for the *Extended BWT* [@mrrs07tcs]. The BWT is invertible, i.e., given $({\mathrm{BWT}}(w),i)$ where $i$ is the index of $w$ in $M(w)$ one can reconstruct the word $w$. We set $k = {\left|\mathinner{w}\right|}$. Let $M = M(w)$ and $L = {\mathrm{BWT}}(w)$. Now, by Lemma \[lem:contextretrieval\] we see that $$w = w_i = {\mathrm{context}}_k(w_i) = {\lambda}_L\pi_L^1(i) \cdots {\lambda}_L\pi_L^{{\left|\mathinner{L}\right|}}(i).$$ In particular, $w = {\lambda}_L\pi_L^1(i) \cdots {\lambda}_L\pi_L^{{\left|\mathinner{L}\right|}}(i)$ only depends on $L$ and $i$. In the special case of the BWT it is possible to compute the $i$-th element $w_i$ of $M(w)$ by using the inverse $\pi_L^{-1}$ of the permutation $\pi_L$: $$w_i = {\lambda}_L\pi_L^{-{\left|\mathinner{w_i}\right|}+1}(i) \cdots {\lambda}_L\pi_L^{-1}(i) {\lambda}_L(i).$$ This justifies the usual way of computing the inverse of $({\mathrm{BWT}}(w),i)$ from right to left (by using the restriction of $\pi_L^{-1}$ to the cycle containing the element $i$). The motivation is that the (required cycle of the) inverse $\pi_L^{-1}$ seems to be easier to compute than the standard permutation $\pi_L$. \[exa:bwt\] We compute the BWT of $w = bcbccbcbcabbaaba$ from Example \[exa:start\]. The lexicographically sorted list $M(w)$ can be found in Figure \[sfg:M\]. This yields the transform $({\mathrm{BWT}}(w),i) = (bacbbaaccacbbcbb,10)$ where $L = {\mathrm{BWT}}(w)$ is the last column of the matrix $M(w)$ and $w$ is the $i$-th row in $M(w)$. The standard permutation of $L$ is $$\pi_L = \left(\begin{array}{*{16}{p{5.5mm}}} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\ 2 & 6 & 7 & 10 & 1 & 4 & 5 & 12 & 13 & 15 & 16 & 3 & 8 & 9 & 11 & 14 \end{array}\right).$$ Now, $\pi_L^1(10) \cdots \pi_L^{16}(10)$ gives us the following sequence of positions starting with $\pi_L(10) = 15$: $$15 \stackrel{\pi_L}{\mapsto} 11 \stackrel{\pi_L}{\mapsto} 16 \stackrel{\pi_L}{\mapsto} 14 \stackrel{\pi_L}{\mapsto} 9 \stackrel{\pi_L}{\mapsto} 13 \stackrel{\pi_L}{\mapsto} 8 \stackrel{\pi_L}{\mapsto} 12 \stackrel{\pi_L}{\mapsto} 3 \stackrel{\pi_L}{\mapsto} 7 \stackrel{\pi_L}{\mapsto} 5 \stackrel{\pi_L}{\mapsto} 1 \stackrel{\pi_L}{\mapsto} 2 \stackrel{\pi_L}{\mapsto} 6 \stackrel{\pi_L}{\mapsto} 4 \stackrel{\pi_L}{\mapsto} 10.$$ Applying the labeling function ${\lambda}_L$ to this sequence of positions yields $$\begin{aligned} & \ {\lambda}_L(15) {\lambda}_L(11) {\lambda}_L(16) {\lambda}_L(14) {\lambda}_L(9) {\lambda}_L(13) {\lambda}_L(8) {\lambda}_L(12) \\ & \quad \cdot {\lambda}_L(3) {\lambda}_L(7) {\lambda}_L(5) {\lambda}_L(1) {\lambda}_L(2) {\lambda}_L(6) {\lambda}_L(4) {\lambda}_L(10) \\ &= bcbccbcbcabbaaba = w, \end{aligned}$$ i.e., we have successfully reconstructed the input $w$ from $({\mathrm{BWT}}(w),i)$. The bijective Burrows-Wheeler transform {#sec:lbwt} ======================================= Now we are ready to give a comprehensive description of Scott’s bijective variant of the BWT and to prove its correctness. It maps a word of length $n$ to a word of length $n$—without any index or end-of-string symbol being involved. The key ingredient is the Lyndon factorization: Suppose we are computing the BWT of a Lyndon word $v$, then we do not need an index since we know that $v$ is the first element of the list $M(v)$. This leads to the computation of a multi-word BWT of the Lyndon factors of the input. The bijective BWT of a word $w$ of length $n$ is defined as follows. Let $w = v_s \cdots v_1$ with $v_s \geq \cdots \geq v_1$ be the Lyndon factorization of $w$. Let ${\mathrm{LM}}(w) = (u_1, \ldots, u_n)$ where $u_1 \leq^\omega \cdots \leq^{\omega} u_n$ and where the multiset ${\left\{\mathinner{u_1, \ldots, u_n}\right\}} = \bigcup_{i=1}^s [v_i]$. Then, the bijective BWT of $w$ is ${\mathrm{BWTS}}(w) = {\mathrm{last}}(u_1) \cdots {\mathrm{last}}(u_n)$. The *S* in ${\mathrm{BWTS}}$ is for *Scottified*. Note that if $w$ is a power of a Lyndon word, then ${\mathrm{BWTS}}(w) = {\mathrm{BWT}}(w)$. In some sense, the bijective BWT can be thought of as the composition of the Lyndon factorization [@cfl58ann] with the inverse of the Gessel-Reutenauer transform [@gr93jct]. In particular, a first step towards a bijective BWT can be found in a 1993 article by Gessel and Reutenauer [@gr93jct] (prior to the publication of the BWT [@bw94tr]). The link between the Gessel-Reutenauer transform and the BWT was pointed out later by Crochemore et al. [@cdp05tcs]. A similar approach as in the bijective BWT has been employed by Mantaci et al. [@mrrs05cpm]; instead of the Lyndon factorization they used a decomposition of the input into blocks of equal length. The output of this variant is a word and a sequence of indices (one for each block). In its current form, the bijective BWT has been proposed by Scott [@scott09] in a newsgroup posting in 2007. Gil and Scott gave an accessible version of the transform, an independent proof of its correctness, and they tested its performance in data compression [@gs09submitted]. The outcome of these tests is that the bijective BWT beats the usual BWT on almost all files of the Calgary Corpus [@bwc89acm] by at least a few hundred bytes which exceeds the gain of just saving the rotation index. \[lem:cycleretrieval\] Let $w = v_s \cdots v_1$ with $v_s \geq \cdots \geq v_1$ be the Lyndon factorization of $w$, let ${\mathrm{LM}}(w) = (u_1, \ldots, u_n)$, and let $L = {\mathrm{BWTS}}(w)$. Consider the cycle $C$ of the permutation $\pi_L$ which contains the element $1$ and let $d$ be the length of $C$. Then ${\lambda}_L \pi_L^1(1) \cdots {\lambda}_L \pi_L^{d}(1) = v_1$. By Lemma \[lem:contextretrieval\] we see that $\bigl( {\lambda}_L \pi_L^1(1) \cdots {\lambda}_L \pi_L^{d}(1) \bigr)^{{\left|\mathinner{v_1}\right|}} = v_1^{d}$. Since $v_1$ is primitive it follows ${\lambda}_L \pi_L^1(1) \cdots {\lambda}_L \pi_L^{d}(1) = v_1^z$ for some $z \in {\mathbb{N}}$. In particular, the Lyndon factorization of $w$ ends with $v_1^z$. Let $U$ be the subsequence of ${\mathrm{LM}}(w)$ which consists of those $u_i$ which come from this last factor $v_1^z$. The sequence $U$ contains each right-shift of $v_1$ exactly $z$ times. Moreover, the sort-order within $U$ depends only on ${\left|\mathinner{v_1}\right|}$-order contexts. The element $v_1 = u_1$ is the first element in $U$ since $v_1$ is a Lyndon word. In particular, $\pi_L^0(1) = 1$ is the first occurrence of ${r}^{0}(v_1) = v_1$ within $U$. Suppose $\pi_L^j(1)$ is the first occurrence of ${r}^{j}(v_1)$ within $U$. Let $\pi_L^{j}(1) = i_1 < \cdots < i_z$ be the indices of all occurrences of ${r}^{j}(v_1)$ in $U$. By construction of $\pi_L$, we have $\pi_L(i_1) < \cdots < \pi_L(i_z)$ and therefore $\pi_L^{j+1}(1)$ is the first occurrence of ${r}^{j+1}(v_1)$ within $U$. Inductively, $\pi_L^j(1)$ always refers to the first occurrence of ${r}^{j}(v_1)$ within $U$ (for all $j \in {\mathbb{N}}$). In particular it follows that $\pi_L^{{\left|\mathinner{v_1}\right|}}(1) = 1$ and $z=1$. \[thm:bijbwt\] The bijective BWT is invertible, i.e., given ${\mathrm{BWTS}}(w)$ one can reconstruct the word $w$. Let $L = {\mathrm{BWTS}}(w)$ and let $w = v_s \cdots v_1$ with $v_s \geq \cdots \geq v_1$ be the Lyndon factorization of $w$. Each permutation admits a cycle structure. We decompose the standard permutation $\pi_L$ into cycles $C_1, \ldots, C_t$. Let $i_j$ be the smallest element of the cycle $C_j$ and let $d_j$ be the length of $C_j$. We can assume that $1 = i_1 < \cdots < i_t$. We claim that $t = s$, $d_j = {\left|\mathinner{v_j}\right|}$, and ${\lambda}_L \pi_L^1(i_j) \cdots {\lambda}_L \pi_L^{d_j}(i_j) = v_j$. By Lemma \[lem:cycleretrieval\] we have ${\lambda}_L \pi_L^1(i_1) \cdots {\lambda}_L \pi_L^{d_1}(i_1) = v_1$. Let $\pi_L'$ denote the restriction of $\pi_L$ to the set $C = C_2 \cup \cdots \cup C_t$, where by abuse of notation $C_2 \cup \cdots \cup C_t$ denotes the set of all elements occurring in $C_2, \ldots, C_t$. Let $L' = {\mathrm{BWTS}}(v_s \cdots v_2)$. The word $L'$ can be obtained from $L$ by removing all positions occurring in the cycle $C_1$. This yields a monotone bijection $$\alpha : C \to {\left\{\mathinner{1, \ldots, {\left|\mathinner{L'}\right|}}\right\}}$$ such that ${\lambda}_L(i) = {\lambda}_{L'}\alpha(i)$ and $\alpha\pi_L(i) = \pi_{L'}\alpha(i)$ for all $i \in C$. In particular, $\pi_{L'}$ has the same cycle structure as $\pi_L'$ and $1 = \alpha(i_2) < \cdots < \alpha(i_t)$ is the sequence of the minimal elements within the cycles. By induction on the number of Lyndon factors, $$\begin{aligned} v_s \cdots v_2 &= {\lambda}_{L'} \pi_{L'}^1\alpha(i_t) \cdots {\lambda}_{L'} \pi_{L'}^{d_t} \alpha(i_t) \;\cdots\; {\lambda}_{L'} \pi_{L'}^1\alpha(i_2) \cdots {\lambda}_{L'} \pi_{L'}^{d_2}(i_2) \\ &= {\lambda}_{L'}\alpha\pi_{L}^1(i_t) \cdots {\lambda}_{L'}\alpha\pi_{L}^{d_t}(i_t) \;\cdots\; {\lambda}_{L'}\alpha\pi_{L}^1(i_2) \cdots {\lambda}_{L'}\alpha\pi_{L}^{d_2}(i_2) \\ &= {\lambda}_{L}\pi_{L}^1(i_t) \cdots {\lambda}_{L}\pi_{L}^{d_t}(i_t) \;\cdots\; {\lambda}_{L}\pi_{L}^1(i_2) \cdots {\lambda}_{L}\pi_{L}^{d_2}(i_2). \end{aligned}$$ Appending ${\lambda}_L \pi_L^1(i_1) \cdots {\lambda}_L \pi_L^{d_1}(i_1) = v_1$ to the last line allows us to reconstruct $w$ by $$w = {\lambda}_L \pi_L^1(i_t) \cdots {\lambda}_L \pi_L^{d_t}(i_t) \;\cdots\; {\lambda}_L \pi_L^1(i_1) \cdots {\lambda}_L \pi_L^{d_1}(i_1).$$ Moreover, $t=s$ and $d_j = {\left|\mathinner{v_j}\right|}$. We note that this formula for $w$ only depends on $L$ and does not require any index to an element in ${\mathrm{LM}}(w)$. \[exa:bijbwt\] We again consider the word $w = bcbccbcbcabbaaba$ from Example \[exa:start\] and its Lyndon factorization $w = v_6 \cdots v_1$ where $v_6 = bcbcc$, $v_5 = bc$, $v_4 = bc$, $v_3 = abb$, $v_2 = aab$, and $v_1 = a$. The lists $([v_1], \ldots, [v_6])$ and ${\mathrm{LM}}(w)$ are: [r|\*[5]{}[p[1.75mm]{}]{}|]{} &\ & $a$ & & & &\ & $a$ & $a$ & $b$ & &\ & $b$ & $a$ & $a$ & &\ & $a$ & $b$ & $a$ & &\ & $a$ & $b$ & $b$ & &\ & $b$ & $a$ & $b$ & &\ & $b$ & $b$ & $a$ & &\ & $b$ & $c$ & & &\ & $c$ & $b$ & & &\ & $b$ & $c$ & & &\ & $c$ & $b$ & & &\ & $b$ & $c$ & $b$ & $c$ & $c$\ & $c$ & $b$ & $c$ & $b$ & $c$\ & $c$ & $c$ & $b$ & $c$ & $b$\ & $b$ & $c$ & $c$ & $b$ & $c$\ & $c$ & $b$ & $c$ & $c$ & $b$\ [r|\*[5]{}[p[1.75mm]{}]{}|]{} &\ & $a$ & & & &\ & $a$ & $a$ & $b$ & &\ & $a$ & $b$ & $a$ & &\ & $a$ & $b$ & $b$ & &\ & $b$ & $a$ & $a$ & &\ & $b$ & $a$ & $b$ & &\ & $b$ & $b$ & $a$ & &\ & $b$ & $c$ & & &\ & $b$ & $c$ & & &\ & $b$ & $c$ & $b$ & $c$ & $c$\ & $b$ & $c$ & $c$ & $b$ & $c$\ & $c$ & $b$ & & &\ & $c$ & $b$ & & &\ & $c$ & $b$ & $c$ & $b$ & $c$\ & $c$ & $b$ & $c$ & $c$ & $b$\ & $c$ & $c$ & $b$ & $c$ & $b$\ Hence, we obtain $L = {\mathrm{BWTS}}(w) = abababaccccbbcbb$ as the sequence of the last symbols of the words in ${\mathrm{LM}}(w)$. The standard permutation $\pi_L$ induced by $L$ is $$\pi_L = \left(\begin{array}{*{16}{p{5.5mm}}} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\ 1 & 3 & 5 & 7 & 2 & 4 & 6 & 12 & 13 & 15 & 16 & 8 & 9 & 10 & 11 & 14 \end{array}\right)$$ The cycles of $\pi_L$ arranged by their smallest elements are $C_1 = (1)$, $C_2 = (2, 3, 5)$, $C_3 = (4, 7, 6)$, $C_4 = (8, 12)$, $C_5 = (9, 13)$, and $C_6 = (10, 15, 11, 16, 14)$. Applying the labeling function ${\lambda}_L$ to the cycle $C_i$ (starting with the second element) yields the Lyndon factor $v_i$. With this procedure, we reconstructed $w = v_6 \cdots v_1$ from $L = {\mathrm{BWTS}}(w)$. The sort transform {#sec:sorttransform} ================== \[sec:st\] The *sort transform* (ST) is a BWT where we only sort the conjugates of the input up to a given depth $k$ and then we are using the index of the conjugates as a tie-breaker. Depending on the depth $k$ and the implementation details this can speed up compression (while at the same time slightly slowing down decompression). In contrast to the usual presentation of the ST, we are using right shifts. This defines a slightly different version of the ST. The effect is that the order of the symbols occurring in some particular context is reversed. This makes sense, because in data compression the ST is applied to the reversal of a word. Hence, in the ST of the reversal of $w$ the order of the symbols in some particular context is the same as in $w$. More formally, suppose $\overline{w} = x_0 c a_1 x_1 c a_2 x_2 \cdots c a_s x_s$ for $c \in \Sigma^+$ then in the sort transform of order ${\left|\mathinner{c}\right|}$ of $w$, the order of the occurrences of the letters $a_i$ is not changed. This property can enable better compression ratios on certain data. While the standard permutation is induced by a sequence of letters (i.e., a word) we now generalize this concept to sequences of words. For a list of non-empty words $V = (v_1, \ldots, v_n)$ we now define the *$k$-order standard permutation* $\nu_{k,V}$ induced by $V$. As for the standard permutation, the first step is the construction of a new linear order $\preceq$ on ${\left\{\mathinner{1,\ldots,n}\right\}}$. We define $i \preceq j$ by the condition $${\mathrm{context}}_k(v_{i}) < {\mathrm{context}}_k(v_{j}) \qquad \text{or} \qquad {\mathrm{context}}_k(v_{i}) = {\mathrm{context}}_k(v_{j}) \,\text{ and }\, i \leq j.$$ Let $j_1 \prec \cdots \prec j_n$ be the linearization of ${\left\{\mathinner{1,\ldots,n}\right\}}$ according to this new order. The idea is that we sort the line numbers of $v_1, \ldots, v_n$ by first considering the $k$-order contexts and, if these are equal, then use the line numbers as tie-breaker. As before, the linearization according to $\preceq$ induces a permutation $\nu_{k,V}$ by setting $\nu_{k,V}(i) = j_i$. Now, $\nu_{k,V}(i)$ is the position of $v_i$ if we are sorting $V$ by $k$-order context such that the line numbers serve as tie-breaker. We set $M_k(v_1, \ldots, v_n) = (w_1, \ldots, w_n)$ where $w_i = v_{\nu_{k,V}(i)}$. Now, we are ready to define the sort transform of order $k$ of a word $w$: Let $M_k([w]) = (w_1, \ldots, w_n)$; then ${\mathrm{ST}}_k(w) = {\mathrm{last}}(w_1) \cdots {\mathrm{last}}(w_n)$, i.e., we first sort all cyclic right-shifts of $w$ by their $k$-order contexts (by using a stable sort method) and then we take the sequence of last symbols according to this new sort order as the image under ${\mathrm{ST}}_k$. Since the tie-breaker relies on right-shifts, we have ${\mathrm{ST}}_0(w) = \overline{w}$, i.e., ${\mathrm{ST}}_0$ is the reversal mapping. The $k$-order sort transform of $w$ is the pair $({\mathrm{ST}}_k(w),i)$ where $i$ is the index of $w$ in $M_k([w])$. As for the BWT, we see that the $k$-order sort transform is not bijective. Next, we show that it is possible to reconstruct $M_k([w])$ from ${\mathrm{ST}}_k(w)$. Hence, it is possible to reconstruct $w$ from the pair $({\mathrm{ST}}_k(w),i)$ where $i$ is the index of $w$ in $M_k([w])$. The presentation of the back transform is as follows. First, we will introduce the *$k$-order context graph* $G_k$ and we will show that it is possible to rebuild $M_k([w])$ from $G_k$. Then we will show how to construct $G_k$ from ${\mathrm{ST}}_k(w)$. Again, the approach will be slightly more general than required at the moment; but we will be able to reuse it in the presentation of a bijective ST. Let $V = ([u_1], \ldots, [u_s]) = (v_1, \ldots, v_n)$ be a list of words built from conjugacy classes $[u_i]$ of non-empty words $u_i$. Let $M = (w_1, \ldots, w_n)$ be an arbitrary permutation of the elements in $V$. We are now describing the edge-labeled directed graph  $k$-order context graph of $M$ – which will be used later as a presentation tool for the inverses of the ST and the bijective ST. The vertices of $G_k(M)$ consist of all $k$-order contexts ${\mathrm{context}}_k(w)$ of words $w$ occurring in $M$. We draw an edge $(c_1, i, c_2)$ from context $c_1$ to context $c_2$ labeled by $i$ if $c_1 = {\mathrm{context}}_k(w_i)$ and $c_2 = {\mathrm{context}}_k({r}(w_i))$. Hence, every index $i \in {\left\{\mathinner{1, \ldots, n}\right\}}$ of $M$ defines a unique edge in $G_k(M)$. We can also think of ${\mathrm{last}}(w_i)$ as an additional implicit label of the edge $(c_1, i, c_2)$, since $c_2 = {\mathrm{context}}_k({\mathrm{last}}(w_i) c_1)$. A *configuration* $(\mathcal{C},c)$ of the $k$-order context graph $G_k(M)$ consists of a subset of the edges $\mathcal{C}$ and a vertex $c$. The idea is that (starting at context $c$) we are walking along the edges of $G_k(M)$ and whenever an edge is used, it is removed from the set of edges $\mathcal{C}$. We now define the transition $$(\mathcal{C}_1,c_1) \stackrel{u}{\to} (\mathcal{C}_2,c_2)$$ from a configuration $(\mathcal{C}_1,c_1)$ to another configuration $(\mathcal{C}_2,c_2)$ with output $u \in \Sigma^*$ more formally. If there exists an edge in $\mathcal{C}_1$ starting at $c_1$ and if $(c_1,i,c_2) \in \mathcal{C}_1$ is the unique edge with the smallest label $i$ starting at $c_1$, then we have the single-step transition $$(\mathcal{C}_1,c_1) \stackrel{a}{\to} (\mathcal{C}_1 \setminus {\left\{\mathinner{(c_1,i,c_2)}\right\}},c_2) \qquad \text{where } a = {\mathrm{last}}(w_i)$$ If there is no edge in $\mathcal{C}_1$ starting at $c_1$, then the outcome of $(\mathcal{C}_1,c_1) \stackrel{}{\to}$ is undefined. Inductively, we define $(\mathcal{C}_1,c_1) \stackrel{\varepsilon}{\to} (\mathcal{C}_1,c_1)$ and for $a \in \Sigma$ and $u \in \Sigma^*$ we have $$(\mathcal{C}_1,c_1) \stackrel{au}{\to} (\mathcal{C}_2,c_2) \quad \text{if} \quad (\mathcal{C}_1,c_1) \stackrel{u}{\to} (\mathcal{C}',c') \,\text{ and }\, (\mathcal{C}',c') \stackrel{a}{\to} (\mathcal{C}_2,c_2)$$ for some configuration $(\mathcal{C}',c')$. Hence, the reversal $\overline{au}$ is the label along the path of length ${\left|\mathinner{au}\right|}$ starting at configuration $(\mathcal{C}_1,c_1)$. In particular, if $(\mathcal{C}_1,c_1) \stackrel{u}{\to} (\mathcal{C}_2,c_2)$ holds, then it is possible to chase at least ${\left|\mathinner{u}\right|}$ transitions starting at $(\mathcal{C}_1,c_1)$; vice versa, if we are chasing $\ell$ transitions then we obtain a word of length $\ell$ as a label. We note that successively taking the edge with the smallest label comes from the use of right-shifts. If we had used left-shifts we would have needed to chase largest edges for the following lemma to hold. The reverse labeling of the big-step transitions is motivated by the reconstruction procedure which will work from right to left. \[lem:smallestedgechasing\] Let $k \in {\mathbb{N}}$, $V = ([v_1], \ldots, [v_s])$, $c_i = {\mathrm{context}}_k(v_i)$, and $G = G_k(M_k(V))$. Let $\mathcal{C}_1$ consist of all edges of $G$. Then $$\begin{aligned} (\mathcal{C}_1,c_1) &\stackrel{v_1}{\to} (\mathcal{C}_2,c_1) \\ (\mathcal{C}_2,c_2) &\stackrel{v_2}{\to} (\mathcal{C}_3,c_2) \\[-2mm] &\ \ \vdots \\ (\mathcal{C}_s,c_s) &\stackrel{v_s}{\to} (\mathcal{C}_{s+1},c_s) . \end{aligned}$$ Let $M_k(V) = (w_1, \ldots, w_n)$. Consider some index $i$, $1 \leq i \leq s$, and let $(u_1, \ldots, u_t) = ([v_1], \ldots, [v_{i-1}])$. Suppose that $\mathcal{C}_i$ consists of all edges of $G$ except for those with labels $\nu_{k,V}(j)$ for $1 \leq j \leq t$. Let $q = {\left|\mathinner{v_i}\right|}$. We write $v_i = a_1 \cdots a_q$ and $u_{t+j} = {r}^{j-1}(v_i)$, i.e., $[v_i] = (u_{t+1}, \ldots, u_{t+q})$. Starting with $(\mathcal{C}_{i,1},c_{i,1})=(\mathcal{C}_i,c_i)$, we show that the sequence of transitions $$(\mathcal{C}_{i,1},c_{i,1}) \stackrel{a_q}{\to} (\mathcal{C}_{i,2},c_{i,2}) \stackrel{a_{q-1}}{\to} \cdots\, (\mathcal{C}_{i,q},c_{i,q}) \stackrel{a_1}{\to} (\mathcal{C}_{i,q+1},c_{i,q+1})$$ is defined. More precisely, we will see that the transition $(\mathcal{C}_{i,j},c_{i,j}) \stackrel{a_{q+1-j}}{\longrightarrow} (\mathcal{C}_{i,j+1},c_{i,j+1})$ walks along the edge $\bigl(c_{i,j}, \nu_{k,V}(t+j), c_{i,j+1}\bigr)$ and hence indeed is labeled with the letter $a_{q+1-j} = {\mathrm{last}}(u_{t+j}) = {\mathrm{last}}(w_{\nu_{k,V}(t+j)})$. Consider the context $c_{i,j}$. By induction, we have $c_{i,j} = {\mathrm{context}}_k(u_{t+j})$ and no edge with label $\nu_{k,V}(\ell)$ for $1 \leq \ell < t+j$ occurs in $\mathcal{C}_{i,j}$ while all other labels do occur. In particular, $(c_{i,j}, \nu_{k,V}(t+j), c_{i,j+1})$ for $c_{i,j+1} = {\mathrm{context}}_k({r}(u_{t+j})) = {\mathrm{context}}_k(u_{t+j+1})$ is an edge in $\mathcal{C}_{i,j}$ (where ${\mathrm{context}}_k({r}(u_{t+j})) = {\mathrm{context}}_k(u_{t+j+1})$ only holds for $j<q$; we will consider the case $j=q$ below). Suppose there were an edge $(c_{i,j}, z ,c') \in \mathcal{C}_{i,j}$ with $z < \nu_{k,V}(t+j)$. Then ${\mathrm{context}}_k(w_z) = c_{i,j}$ and hence, $w_z$ has the same $k$-order context as $w_{\nu_{k,V}(t+j)}$. But in this case, in the construction of $M_k(V)$ we used the index in $V$ as a tie-breaker. It follows $\nu_{k,V}^{-1}(z) < t+1$ which contradicts the properties of $\mathcal{C}_{i,j}$. Hence, $(c_{i,j}, \nu_{k,V}(t+j), c_{i,j+1})$ is the edge with the smallest label starting at context $c_{i,j}$. Therefore, $\mathcal{C}_{i,j+1} = \mathcal{C}_{i,j} \setminus {\left\{\mathinner{(c_{i,j}, \nu_{k,V}(t+j), c_{i,j+1})}\right\}}$ and $(\mathcal{C}_{i,j},c_{i,j}) \stackrel{a_{q+1-j}}{\longrightarrow} (\mathcal{C}_{i,j+1},c_{i,j+1})$ indeed walks along the edge $(c_{i,j}, \nu_{k,V}(t+j), c_{i,j+1})$. It remains to verify that $c_{i,1} = c_{i,q+1}$, but this is clear since $c_{i,1} = {\mathrm{context}}_k(u_{t+1}) = {\mathrm{context}}_k({r}^{q}(u_{t+1})) = c_{i,q+1}$. \[lem:reconstructingG\] Let $k \in {\mathbb{N}}$, $V = ([v_1], \ldots, [v_s])$, $M = M_k(V) = (w_1, \ldots, w_n)$, and $L = {\mathrm{last}}(w_1) \cdots {\mathrm{last}}(w_n)$. Then it is possible to reconstruct $G_k(M)$ from $L$. By Lemma \[lem:contextretrieval\] it is possible to reconstruct the contexts $c_i = {\mathrm{context}}_k(w_i)$. This gives the vertices of the graph $G_k(M)$. Write $L = a_1 \cdots a_n$. For each $i \in {\left\{\mathinner{1, \ldots, n}\right\}}$ we draw an edge $(c_i, i, {\mathrm{context}}_k(a_i c_i))$. This yields the edges of $G_k(M)$. The $k$-order ST is invertible, i.e., given $({\mathrm{ST}}_k(w),i)$ where $i$ is the index of $w$ in $M_k([w])$ one can reconstruct the word $w$. The construction of $w$ consists of two phases. First, by Lemma \[lem:reconstructingG\] we can compute $G_k(M_k([w]))$. By Lemma \[lem:contextretrieval\] we can compute $c = {\mathrm{context}}_k(w)$ from $({\mathrm{ST}}_k(w),i)$. In the second stage, we are using Lemma \[lem:smallestedgechasing\] for reconstructing $w$ by chasing $$(\mathcal{C},c) \stackrel{w}{\to} (\emptyset,c)$$ where $\mathcal{C}$ consists of all edges in $G_k(M_k([w]))$. Efficient implementations of the inverse transform rely on the fact that the $k$-order contexts of $M_k([w])$ are ordered. This allows the implementation of the $k$-order context graph $G_k$ in a vectorized form [@abm08book; @nz06dcc; @nz07ieee; @nzc08cpm]. \[exa:st\] We compute the sort transform of order $2$ of $w = bcbccbcbcabbaaba$ from Example \[exa:start\]. The list $M_2([w])$ is depicted in Figure \[sfg:M2\]. This yields the transform $({\mathrm{ST}}_2(w),i) = (bbacabaacccbbcbb,8)$ where $L = {\mathrm{ST}}_2(w)$ is the last column of the matrix $M_2([w])$ and $w$ is the $i$-th element in $M_2([w])$. Next, we show how to reconstruct the input $w$ from $(L,i)$. The standard permutation induced by $L$ is $$\pi_L = \left(\begin{array}{*{16}{p{5.5mm}}} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\ 3 & 5 & 7 & 8 & 1 & 2 & 6 & 12 & 13 & 15 & 16 & 4 & 9 & 10 & 11 & 14 \end{array}\right).$$ Note that $\pi_L$ has four cycles $C_1 = (1,3,7,6,2,5)$, $C_2 = (4,8,12)$, $C_3 = (9,13)$, and $C_4 = (10,15,11,16,14)$. We obtain the context of order $2$ of the $j$-th word by $c_j = {\lambda}_L\pi_L(j) {\lambda}_L\pi_L^2(j)$. In particular, $c_1 = aa$, $c_2 = c_3 = c_4 = ab$, $c_5 = c_6 = ba$, $c_7 = bb$, $c_8 = c_9 = c_{10} = c_{11} = bc$, $c_{12} = ca$, $c_{13} = c_{14} = c_{15} = cb$, and $c_{16} = cc$. With $L$ and these contexts we can construct the graph $G = G_2(M_2([w])$. The vertices of $G$ are the contexts and the edge-labels represent positions in $L$. The graph $G$ is depicted below: (0,0) node\[circle,draw,inner sep=2.5pt\] (ba) [$ba$]{}; (3,0) node\[circle,draw,inner sep=2.5pt\] (aa) [$aa$]{}; (6,0) node\[circle,draw,inner sep=2.5pt\] (ca) [$ca$]{}; (9,0) node\[circle,draw,inner sep=2.5pt\] (cb) [$cb$]{}; (0,3) node\[circle,draw,inner sep=2.5pt\] (bb) [$bb$]{}; (3,3) node\[circle,draw,inner sep=2.5pt\] (ab) [$ab$]{}; (6,3) node\[circle,draw,inner sep=2.5pt\] (bc) [$bc$]{}; (9,3) node\[circle,draw,inner sep=2.5pt\] (cc) [$cc$]{}; (aa) – node\[above\] [$1$]{} (ba); (ab) .. controls (1.25,1.75) .. node\[sloped,above\] [$2$]{} (ba); (ab) – node\[left\] [$3$]{} (aa); (ab) – node\[above right\] [$4$]{} (ca); (ba) .. controls (1.75,1.25) .. node\[sloped,above\] [$5$]{} (ab); (ba) – node\[left\] [$6$]{} (bb); (bb) – node\[above\] [$7$]{} (ab); (bc) – node\[above\] [$8$]{} (ab); (bc) – node\[sloped,above\] [$9$]{} (cb); (bc) .. controls (8,2) .. node\[sloped,above\] [$10$]{} (cb); (bc) .. controls (8.5,2.5) .. node\[sloped,above\] [$11$]{} (cb); (ca) – node\[left\] [$12$]{} (bc); (cb) .. controls (7,1) .. node\[sloped,above\] [$13$]{} (bc); (cb) – node\[right\] [$14$]{} (cc); (cb) .. controls (6.5,0.5) .. node\[sloped,above\] [$15$]{} (bc); (cc) – node\[above\] [$16$]{} (bc); We are starting at the context $c_i = c_8 = bc$ and then we are traversing $G$ along the smallest edge-label amongst the unused edges. The sequence of the edge labels obtained this way is $$(8,2,5,3,1,6,7,4,12,9,13,10,14,16,11,15).$$ The labeling of this sequence of positions yields $\overline{w} = abaabbacbcbccbcb$. Since we are constructing the input from right to left, we obtain $w = bcbccbcbcabbaaba$. The bijective sort transform {#sec:lst} ============================ The bijective sort transform combines the Lyndon factorization with the ST. This yields a new algorithm which serves as a similar preprocessing step in data compression as the BWT. In a lot of applications, it can be used as a substitute for the ST. The proof of the bijectivity of the transform is slightly more technical than the analogous result for the bijective BWT. The main reason is that the bijective sort transform is less modular than the bijective BWT (which can be grouped into a ‘Lyndon factorization part’ and a ‘Gessel-Reutenauer transform part’ and which for example allows the use of different orders on the alphabet for the different parts). For the description of the bijective ST and of its inverse, we rely on notions from Section \[sec:sorttransform\]. The bijective ST of a word $w$ of length $n$ is defined as follows. Let $w = v_s \cdots v_1$ with $v_s \geq \cdots \geq v_1$ be the Lyndon factorization of $w$. Let $M_k([v_1], \ldots, [v_s]) = (u_1, \ldots, u_n)$. Then the bijective ST of order $k$ of $w$ is ${\mathrm{LST}}_k(w) = {\mathrm{last}}(u_1) \cdots {\mathrm{last}}(u_n)$. That is, we are sorting the conjugacy classes of the Lyndon factors by $k$-order contexts and then take the sequence of the last letters. The letter *L* in ${\mathrm{LST}}_k$ is for *Lyndon*. \[thm:bijst\] The bijective ST of order $k$ is invertible, i.e., given ${\mathrm{LST}}_k(w)$ one can reconstruct the word $w$. Let $w = v_s \cdots v_1$ with $v_s \geq \cdots \geq v_1$ be the Lyndon factorization of $w$, let $c_i = {\mathrm{context}}_k(v_i)$, and let $L = {\mathrm{LST}}_k(w)$. By Lemma \[lem:reconstructingG\] we can rebuild the $k$-order context graph $G = G_k(M_k([v_1], \ldots, [v_s])) = (w_1, \ldots, w_n)$ from $L$. Let $\mathcal{C}_1$ consist of all edges in $G$. Then by Lemma \[lem:smallestedgechasing\] we see that $$\begin{aligned} (\mathcal{C}_1,c_1) &\stackrel{v_1}{\to} (\mathcal{C}_2,c_1) \\[-2mm] &\ \ \vdots \\ (\mathcal{C}_s,c_s) &\stackrel{v_s}{\to} (\mathcal{C}_{s+1},c_s). \end{aligned}$$ We cannot use this directly for the reconstruction of $w$ since we do not know the Lyndon factors $v_i$ and the contexts $c_i$. The word $v_1$ is the first element in the list $M_k([v_1], \ldots, [v_s])$ because $v_1$ is lexicographically minimal and it appears as the first element in the list $([v_1], \ldots, [v_s])$. Therefore, by Lemma \[lem:contextretrieval\] we obtain $c_1 = {\mathrm{context}}_k(v_1) = {\lambda}_L\pi_L(1) \cdots {\lambda}_L\pi_L^k(1)$. The reconstruction procedure works from right to left. Suppose we have already reconstructed $w' v_j \cdots v_1$ for $j \geq 0$ with $w'$ being a (possibly empty) suffix of $v_{j+1}$. Moreover, suppose we have used the correct contexts $c_1, \ldots, c_{j+1}$. Consider the configuration $(\mathcal{C}',c')$ defined by $$\begin{aligned} (\mathcal{C}_1,c_1) &\stackrel{v_1}{\to} (\mathcal{C}_2,c_1) \\[-2mm] &\ \ \vdots \\ (\mathcal{C}_j,c_j) &\stackrel{v_j}{\to} (\mathcal{C}_{j+1},c_j) \\ (\mathcal{C}_{j+1},c_{j+1}) &\stackrel{w'}{\to} (\mathcal{C}',c') \end{aligned}$$ We assume that the following invariant holds: $\mathcal{C}_{j+1}$ contains no edges $(c'',\ell,c''')$ with $c'' < c_{j+1}$. We want to rebuild the next letter. We have to consider three cases. First, if ${\left|\mathinner{w'}\right|} < {\left|\mathinner{v_{j+1}}\right|}$ then $$(\mathcal{C}',c') \stackrel{a}{\to} (\mathcal{C}'',c'')$$ yields the next letter $a$ such that $a w'$ is a suffix of $v_{j+1}$. Second, let ${\left|\mathinner{w'}\right|} = {\left|\mathinner{v_{j+1}}\right|}$ and suppose that there exists an edge $(c_{j+1},\ell,c''') \in \mathcal{C}'$ starting at $c' = c_{j+1}$. Then there exists a word $v'$ in $[v_{j+2}], \ldots, [v_s]$ such that ${\mathrm{context}}_k(v') = c_{j+1}$. If ${\mathrm{context}}_k(v_{j+2}) \neq c_{j+1}$ then from the invariant it follows that ${\mathrm{context}}_k(v_{j+2}) > c_{j+1} = {\mathrm{context}}_k(v')$. This is a contradiction, since $v_{j+2}$ is minimal among the words in $[v_{j+2}], \ldots, [v_s]$. Hence, ${\mathrm{context}}_k(v_{j+2}) = c_{j+2} = c_{j+1}$ and the invariant still holds for $\mathcal{C}_{j+2} = \mathcal{C}'$. The last letter $a$ of $v_{j+2}$ is obtained by $$(\mathcal{C}',c') = (\mathcal{C}_{j+2},c_{j+2}) \,\stackrel{a}{\to}\, (\mathcal{C}'',c'').$$ The third case is ${\left|\mathinner{w'}\right|} = {\left|\mathinner{v_{j+1}}\right|}$ and there is no edge $(c_{j+1},\ell,c''') \in \mathcal{C}'$ starting at $c' = c_{j+1}$. As before, $v_{j+2}$ is minimal among the (remaining) words in $[v_{j+2}], \ldots, [v_s]$. By construction of $G$, the unique edge $(c'',\ell,c''') \in \mathcal{C}'$ with the minimal label $\ell$ has the property that $w_{\ell} = v_{j+2}$. In particular, $c'' = c_{j+2}$. Since $v_{j+2}$ is minimal, the invariant for $\mathcal{C}_{j+2} = \mathcal{C}'$ is established. In this case, the last letter $a$ of $v_{j+2}$ is obtained by $$(\mathcal{C}_{j+2},c_{j+2}) \stackrel{a}{\to} (\mathcal{C}'',c''').$$ We note that we cannot distinguish between the first and the second case since we do not know the length of $v_{j+1}$, but in both cases, the computation of the next symbol is identical. In particular, in contrast to the bijective BWT we do not implicitly recover the Lyndon factorization of $w$. We note that the proof of Theorem \[thm:bijst\] heavily relies on two design criteria. The first one is to consider $M_k([v_1], \ldots, [v_s])$ rather than $M_k([v_s], \ldots, [v_1])$, and the second is to use right-shifts rather than left-shifts. The proof of Theorem \[thm:bijst\] yields the following algorithm for reconstructing $w$ from $L = {\mathrm{LST}}_k(w)$: (1) Compute the $k$-order context graph $G = G_k$ and the $k$-order context $c_1$ of the last Lyndon factor of $w$. (2) Start with the configuration $(\mathcal{C},c)$ where $\mathcal{C}$ contains all edges of $G$ and $c {\mathrel{\mathop:}=}c_1$. (3) If there exists an outgoing edge starting at $c$ in the set $\mathcal{C}$, then - Let $(c,\ell,c')$ be the edge with the minimal label $\ell$ starting at $c$. - Output ${\lambda}_L(\ell)$. - Set $\mathcal{C} {\mathrel{\mathop:}=}\mathcal{C} \setminus {\left\{\mathinner{(c,\ell,c')}\right\}}$ and $c {\mathrel{\mathop:}=}c'$. - Continue with step (3). (4) If there is no outgoing edge starting at $c$ in the set $\mathcal{C}$, but $\mathcal{C} \neq \emptyset$, then - Let $(c',\ell,c'') \in \mathcal{C}$ be the edge with the minimal label $\ell$. - Output ${\lambda}_L(\ell)$. - Set $\mathcal{C} {\mathrel{\mathop:}=}\mathcal{C} \setminus {\left\{\mathinner{(c',\ell,c'')}\right\}}$ and $c {\mathrel{\mathop:}=}c''$. - Continue with step (3). (5) The algorithm terminates as soon as $\mathcal{C} = \emptyset$. The sequence of the outputs is the reversal $\overline{w}$ of the word $w$. We consider the word $w = bcbccbcbcabbaaba$ from Example \[exa:start\] and its Lyndon factorization $w = v_6 \cdots v_1$ where $v_6 = bcbcc$, $v_5 = bc$, $v_4 = bc$, $v_3 = abb$, $v_2 = aab$, and $v_1 = a$. For this particular word $w$ the bijective Burrows-Wheeler transform and the bijective sort transform of order $2$ coincide. From Example \[exa:bijbwt\], we know $L = {\mathrm{LST}}_2(w) = {\mathrm{BWTS}}(w) = abababaccccbbcbb$ and the standard permutation $\pi_L$. As in Example \[exa:st\] we can reconstruct the $2$-order contexts $c_1, \ldots, c_{16}$ of $M_2([v_1],\ldots,[v_6])$: $c_1 = c_2 = aa$, $c_3 = c_4 = ab$, $c_5 = c_6 = ba$, $c_7 = bb$, $c_8 = c_9 = c_{10} = c_{11} = bc$, $c_{12} = c_{13} = c_{14} = c_{15} = cb$, and $c_{16} = cc$. With $L$ and the $2$-order contexts we can construct the graph $G = G_k(M_2([v_1],\ldots,[v_6]))$: (0,0) node\[circle,draw,inner sep=2.5pt\] (ba) [$ba$]{}; (3,0) node\[circle,draw,inner sep=2.5pt\] (aa) [$aa$]{}; (9,0) node\[circle,draw,inner sep=2.5pt\] (cb) [$cb$]{}; (0,3) node\[circle,draw,inner sep=2.5pt\] (bb) [$bb$]{}; (3,3) node\[circle,draw,inner sep=2.5pt\] (ab) [$ab$]{}; (6,3) node\[circle,draw,inner sep=2.5pt\] (bc) [$bc$]{}; (9,3) node\[circle,draw,inner sep=2.5pt\] (cc) [$cc$]{}; (aa) .. controls (4,-0.5) and (4,+0.5) .. node\[right\] [$1$]{} (aa); (aa) – node\[above\] [$2$]{} (ba); (ab) – node\[left\] [$3$]{} (aa); (ab) .. controls (1.25,1.75) .. node\[sloped,above\] [$4$]{} (ba); (ba) .. controls (1.75,1.25) .. node\[sloped,above\] [$5$]{} (ab); (ba) – node\[left\] [$6$]{} (bb); (bb) – node\[above\] [$7$]{} (ab); (bc) .. controls (7,1) .. node\[sloped,above\] [$8$]{} (cb); (bc) – node\[sloped,above\] [$9$]{} (cb); (bc) .. controls (8,2) .. node\[sloped,above\] [$10$]{} (cb); (bc) .. controls (8.5,2.5) .. node\[sloped,above\] [$11$]{} (cb); (cb) .. controls (6.5,0.5) .. node\[sloped,above\] [$12$]{} (bc); (cb) .. controls (6.0,0.0) .. node\[sloped,above\] [$13$]{} (bc); (cb) – node\[right\] [$14$]{} (cc); (cb) .. controls (5.5,-0.5) .. node\[sloped,above\] [$15$]{} (bc); (cc) – node\[above\] [$16$]{} (bc); We are starting with the edge with label $1$ and then we are traversing $G$ along the smallest unused edges. If we end in a context with no outgoing unused edges, then we are continuing with the smallest unused edge. This gives the sequence $(1,2,5,3)$ after which we end in context $aa$ with no unused edges available. Then we continue with the sequences $(4,6,7)$ and $(8,12,9,13,10,14,16,11,15)$. The complete sequence of edge labels obtained this way is $$(1,2,5,3,\ 4,6,7,\ 8,12,9,13,10,14,16,11,15)$$ and the labeling of this sequence with ${\lambda}_L$ yields $\overline{w} = abaabbacbcbccbcb$. As for the ST, we are reconstructing the input from right to left, and hence we get $w = bcbccbcbcabbaaba$. Summary ======= We discussed two bijective variants of the Burrows-Wheeler transform (BWT). The first one is due to Scott. Roughly speaking, it is a combination of the Lyndon factorization and the Gessel-Reuternauer transform. The second variant is derived from the sort transform (ST); it is the main contribution of this paper. We gave full constructive proofs for the bijectivity of both transforms. As a by-product, we provided algorithms for the inverse of the BWT and the inverse of the ST. For the latter, we introduced an auxiliary graph structure—the $k$-order context graph. This graph yields an intermediate step in the computation of the inverse of the ST and the bijective ST. It can be seen as a generalization of the cycle decomposition of the standard permutation—which in turn can be used as an intermediate step in the computation of the inverse of the BWT and the bijective BWT. **Acknowledgments.**   The author would like to thank Yossi Gil and David A. Scott for many helpful discussions on this topic as well as Alexander Lauser, Antonio Restivo, and the anonymous referees for their numerous suggestions which improved the presentation of this paper. [10]{} : , Springer Publishing Company, Incorporated, 2008. : . Int. J. Comput. Math., 81(10) 2004, pp. 1213–1222. : . IEEE Trans. Computers, 49(10) 2000, pp. 1043–1053. : . ACM Comput. Surv., 21(4) 1989, pp. 557–591. : , Tech. Rep. 124, Digital SRC Research Report, 1994. : . Ann. Math., 68(1) 1958, pp. 81–95. : . IEEE Trans. Commun., 32(4) 1984, pp. 396–402. : . Theor. Comput. Sci., 332(1-3) 2005, pp. 567–572. : . J. Algorithms, 4(4) 1983, pp. 363–381. : . Proc. Amer. Math. Soc., 16 1965, pp. 109–114. : . J. Comb. Theory, Ser. A, 64(2) 1993, pp. 189–215. : , submitted. : . IEEE Trans. Inform. Theory, 23(3) 1977, pp. 337–343. height 2pt depth -1.6pt width 23pt: . IEEE Trans. Inform. Theory, 24(5) 1978, pp. 530–536. , ed., [*Combinatorics on Words*]{}, Addison-Wesley, Reading, MA, 1983. : , in Combinatorial Pattern Matching, CPM 2005, Proceedings, vol. 3537 of LNCS, Springer, 2005, pp. 178–189. height 2pt depth -1.6pt width 23pt: . Theor. Comput. Sci., 387(3) 2007, pp. 298–312. : . Journal of the ACM, 48(3) 2001, pp. 407–430. : , in Data Compression Conference, DCC 2006. Proceedings, IEEE Computer Society, 2006, p. 464. height 2pt depth -1.6pt width 23pt: . IEEE Trans. Computers, 56(11) 2007, pp. 1564–1574. : , in Combinatorial Pattern Matching, CPM 2008, Proceedings, vol. 5029 of LNCS, Springer, 2008, pp. 178–190. : , in Data Compression Conference, DCC 1997. Proceedings, IEEE Computer Society, 1997, p. 469. : , 2009.

--- abstract: 'The exact number of CNOT and single qubit gates needed to implement a Quantum Algorithm in a given architecture is one of the central problems of Quantum Computation. In this work we study the importance of concise realizations of Partially defined Unitary Transformations for better circuit construction using the case study of Dicke State Preparation. The Dicke States $(\ket{D^n_k})$ are an important class of entangled states with uses in many branches of Quantum Information. In this regard we provide the most efficient Deterministic Dicke State Preparation Circuit in terms of CNOT and single qubit gate counts in comparison to existing literature. We further observe that our improvements also reduce architectural constraints of the circuits. We implement the circuit for preparing $\ket{D^4_2}$ on the “ibmqx2” machine of the IBM QX service and observe that the error induced due to noise in the system is lesser in comparison to the existing circuit descriptions. We conclude by describing the CNOT map of the generic $\ket{D^n_k}$ preparation circuit and analyze different ways of distributing the CNOT gates in the circuit and its affect on the induced error.' author: - title: On Actual Preparation of Dicke State on a Quantum Computer --- Quantum Computing, Quantum Circuit, Dicke States, IBMQ, CNOT, Noisy Computation. Introduction {#sec:1} ============ One of the most fundamental aspects of Quantum Mechanics is Quantum Computation. Quantum Computers enable Quantum Algorithms that can perform operations with even super exponential speed-ups in time over the best known classical algorithms. Any quantum algorithm can be defined as a series of unitary transformations and can be implemented as a Quantum Circuit. A quantum circuit has a discrete set of gates such that their combinations can express any unitary transformation with any desired accuracy. Such a set of gates is called a universal set of gates. We know from the fundamental work by Barenco et.al [@barenco] that single qubit gates and the controlled NOT (CNOT) gate form a universal set of gates. We call these gates as elementary gates. Quantum State Preparation is a topic within Quantum Computation that has garnered interest in the past two decades due to applications of special quantum states in several fields of Quantum Information Theory. A $n$-qubit quantum state $\ket{\psi_n}$ can be expressed as the superposition of $2^n$ orthonormal basis states. In this work we look at $n$ qubit states as super position of the computational basis states $\ket{x_1x_2\ldots x_n}, x_i \in\{0,1\},~ 1 \leq i \leq n$. The basis states in the expression of $\ket{\psi_n}$ with non zero amplitude are called the active basis states. Starting from the state $\ket{0}^{\otimes n}$ any arbitrary quantum state can be formed using $\mathcal{O}(2^n)$ elementary gates, although for many $n$ qubit states preparation circuits with polynomial (in $n$) number of elementary gates is possible. The family of Dicke States $\ket{D^n_k}$ is one such example. $\ket{D_k^n}$ is the $n$-qubit state which is the equal superposition state of all $n \choose w$ basis states of weight $k$. For example $\ket{D^3_1}=\frac{1}{\sqrt{3}}(\ket{001}+\ket{010}+\ket{100})$. Dicke states are an interesting family of states due to the fact that they have $n \choose k$ active basis states, which can be exponential in $n$ when $k=\mathcal{O}(n)$ but need only polynomial number of elementary gates to prepare. Dicke states also have applications in the areas of Quantum Game Theory, Quantum Networking, among others. One can refer to [@dicke] for getting a more in-depth view of these applications. There has been several probabilistic and deterministic Dicke state algorithms designed in the last two decades [@dicke1; @dicke2; @dicke3]. In this paper we focus on the algorithm described by B[ä]{}rtschi et.al [@dicke] which gives a deterministic algorithm that takes $\mathcal{O}(kn)$ CNOT gates and $\mathcal{O}(n)$ depth to prepare the state $\ket{D^n_k}$. To the best of our knowledge this circuit description has the best gate count among the deterministic algorithms. Here it is important to note that the paper by Cruz et.al [@dn1] describes two algorithms for preparing the $\ket{D^n_1}$ states, also known as $W_n$ states. Both the algorithms have better gate count than the description by B[ä]{}rtschi et.al [@dicke] and one of the algorithms has logarithmic depth. However, their work is restricted to $\ket{D^n_1}$ and has no implication on the circuits for $\ket{D^n_k},~2 \leq k \leq n-2$. We further observe in Section \[sec:4\] that the circuit obtained by us after the improvements for $\ket{D^n_1}$ is same as the linear $W_n$ circuit described in [@dn1]. Because of the noisy behavior of current generation Quantum Computers the exact number of elementary gates needed and the distribution of the gates over the corresponding circuit become crucial issues which need to be optimized in order to prepare a state with high fidelity. An example of a very recent work done in this area is [@aes] which reduces the gate count of AES implementation. In this regard we discuss the following important problems in the domain of Quantum Circuit Design. A unitary transformation acting on $n$ qubits can be expressed as a $2^n \times 2^n$ unitary matrix and can be decomposed into elementary gates in several ways. Therefore finding the decomposition that needs the least amount of elementary gates is a very fundamental problem, with [@song], [@work] being examples of work done in this area. It is crucial to minimize the number of gates while decomposing a unitary matrix as every gate induces some amount of error into the result. Especially reducing the number of CNOT gates is of importance due to the well known fact that it induces more error compared to single qubit gates. In this work we first describe a fundamental problem that decomposition of matrix using a universal set of gates poses. Let there be a unitary transformation that is to be performed on a system of $n$ qubits. This task can be represented as a unitary matrix $U_n$ that works on the Hilbert Space $H_n$ of dimension $2^n$. If we know the intended transformation for all the states of any orthonormal basis of $H_n$, that completely defines the unitary matrix $U_n$. Let us consider such a transformation for $n=1$. If the transformation is defined for the two states in the computational basis $\ket{0}$ and $\ket{1}$ then the corresponding unitary matrix is completely defined. If the transformation is defined as $\ket{0} \rightarrow \frac{1}{\sqrt{2}}(\ket{0}+\ket{1})$ and $\ket{1} \rightarrow \frac{1}{\sqrt{2}}(\ket{0}-\ket{1})$ then the corresponding matrix is the Hadamard matrix, expressed as $\begin{bmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{bmatrix}$. However if the transformation is only defined for one state, $\ket{0} \rightarrow \frac{1}{\sqrt{2}}(\ket{0}+\ket{1})$ and not defined for $\ket{1}$ then there can be uncountably many unitary matrices that can perform the said transformation. Specifically, any matrix of the form $\begin{bmatrix} \frac{1}{\sqrt{2}} & \alpha\\ \frac{1}{\sqrt{2}} & -\alpha \end{bmatrix}$ can perform this task, where $\alpha \in \mathbb{C},~ \abs{\alpha}^2=\frac{1}{2}$. There exists many quantum algorithms where at a step a particular transformation on $n$ qubits is defined only for a a subset of the states of a orthonormal basis. This creates the possibility of there being uncountably many unitary matrices capable of such a transformation. The algorithm described in [@dicke] contains such transformations that are not completely defined for all basis states. We call such a transformation a partially defined unitary transformation on $n$ qubits. There are possibly multiple unitary matrices that can perform this transformation. In that case it becomes an important problem to find out which candidate unitary matrix can be decomposed using the minimal number of elementary gates. Furthermore, the number of elementary gates needed to implement a well defined Quantum Circuit also varies with the architecture of the actual Quantum Computer. The architectures of current generation Quantum Computers do not allow for CNOT gates to be implemented between any two arbitrary qubits. This CNOT constraint may further increase the total number of CNOT and single qubit gates needed to implement a Quantum Circuit on a specific Quantum Architecture. Against this backdrop, let us draw out the organization of the rest of the paper along with our contributions. Organization and Contribution ----------------------------- In Section \[sec:2\] we first describe the preliminaries needed to support our work. We first define the concept of maximally partial unitary transformation. We then describe the the circuit in [@dicke] for preparing Dicke States. We denote the circuit described in [@dicke] for preparing $\ket{D^n_k}$ as ${\mathcal{C}}_{n,k}$. We start Section \[sec:3\] by showing that a transformation implemented in ${\mathcal{C}}_{n,k}$ is in fact a partially defined construction. We then show that the unitary matrix used to represent the transformation is not optimal in terms of number of elementary gates needed to decompose it. We propose a different construction that indeed requires lesser number of elementary gates and we also argue its optimality w.r.t the Universal gate set. In Section \[sec:4\] we use the construction to improve the gate count of the circuit ${\mathcal{C}}_{n,k}$ in a generalized manner. We remove the redundant gates in the circuit and analyze the different partially defined transformations implemented in the circuit to further reduce the gate counts of the circuit. We denote the improved circuit for preparing any Dicke State $\ket{D^n_k}$ as ${\mathcal{\widehat{C}}}_{n,k}$. To the best of our knowledge this is the most optimal implementation of a deterministic Dicke state preparation circuit for $\ket{D^n_k},~2 \leq k \leq n-2$. Next in Section \[sec:5\] we discuss the architectural constraints posed by the current generation Quantum Computers that are available for public use through different cloud services. We discuss the restrictions in terms of implementing CNOT gates between two qubits in an architecture and how it increases the number of CNOT gates needed to implement a circuit in an architecture. In this regard we show that the improvements described by us in Section \[sec:4\] not only reduces gate counts but also reduces architectural constraints. We implement the circuits ${\mathcal{C}}_{4,2}$ and ${\mathcal{\widehat{C}}}_{4,2}$ on the IBM-QX machine “ibmqx2”[@ibmq] and calculate the deviation in each case from ideal measurement statistics using a simple error measure. Next we show how two circuits with the same number of CNOT gates and the same architectural restrictions can lead to different expected error due to different CNOT distribution across the qubits. We analyze this by proposing modifications in the circuit ${\mathcal{\widehat{C}}}_{4,2}$ possible because partial nature of certain transformations and how it reduces the number of CNOT gates functioning erroneously on expectation in a fairly generalized error model. We finish this section by drawing out the general CNOT map of ${\mathcal{\widehat{C}}}_{n,k}$, shown as the graph $G^{n,k}$ and observing that there in fact exists $n-k-1$ independent modifications each leading to a different CNOT distribution. We conclude the paper in Section \[sec:6\] by describing the future direction of work in this domain and also note down open problems in this area that we feel will improve our understanding both in the domains of partially defined transformations and architectural constraints. Preliminaries {#sec:2} ============= We first define some terminologies that we frequently use before moving onto some definitions and the preliminaries. Notations --------- 1. $\ket{v_2}$: If we look at a system with $n$ qubits then all the $2^n$ orthogonal states in the computational basis can be expressed as $\ket{b_1b_2\ldots b_n},~ b_i \in \{0,1\}, 1 \leq i \leq n$. In that case for representing the state $\ket{b_1b_2\ldots b_n}$ we treat it as a binary string and express it as $\ket{v_2}$ where $v=\displaystyle \sum_{i=1}^n b_i2^{n-i}$. 2. $R_y(\theta)$: The $R_y$ gate is a single qubit gate defined as follows. $R_y(\theta) \equiv e^{-\theta Y}= \begin{bmatrix} {\ensuremath{\cos(\frac{\theta}{2})}} & -{\ensuremath{\sin(\frac{\theta}{2})}} \\ {\ensuremath{\sin(\frac{\theta}{2})}} & {\ensuremath{\cos(\frac{\theta}{2})}} \end{bmatrix} $. 3. $X$: This is a single qubit gate defined as $X= \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} $. 4. $CU^i_j$: While implementing a controlled unitary on a two qubit subsystem we use the following notations. Let there be a $n$-qubit system. $CU^i_j$ represents a two qubit controlled unitary operation where the $i$-th qubit is the control qubit and the $j$-th qubit is the target qubit. Maximally Partial Unitary Transformation ---------------------------------------- Let there be a unitary transformation that acts on $n$ qubits. To perform this transformation we have to create a corresponding unitary matrix. If the transformation is defined for all $2^n$ states of some orthonormal basis then the unitary matrix is completely defined. On the other hand if the transformation is defined for a single state belonging to the computational basis, only a single column of the corresponding $2^n \times 2^n$ matrix is filled. The rest can be filled up conveniently, provided its unitary property is satisfied. In this regard we call a unitary transformation on $n$ qubits to be maximally partial if it is defined for $2^n-1$ states of some orthonormal basis. That implies only a column of the matrix is not defined. In this paper we observe how corresponding to a maximally partial unitary transformation there can be multiple unitary matrices and how the minimal number of elementary gates needed to implement these matrices may vary. We end this section by describing the structure of Dicke states and a circuit designed for its preparation. The Dicke State Preparation Circuit ${\mathcal{C}}_{n,k}$ --------------------------------------------------------- The circuit ${\mathcal{C}}_{n,k}$ as described in [@dicke] works on the $n$ qubit system $\ket{q_1q_2\ldots q_n}$. The circuit ${\mathcal{C}}_{n,k}$ is broken into $n-1$ blocks of the form $SCS^x_y$ of which the first $n-k$ blocks are of the form $SCS^{n-t}_k,~n-t>k$ which is then followed by $k-1$ blocks of the form $SCS^i_{i-1},k \geq i \geq 2$. A block $SCS^n_k$ consists of a two qubit transformation and $k-1$ three qubit transformations. The two qubit transformation works on the $n-1$ and $n$-th qubits and we denote it as $\mu_n$. We describe the overall structure of the circuit again in Section \[sec:5\]. The three qubit transformations are of the form ${\mathcal{M}}_n^l, n-1 \leq i \leq n-k+1$ where ${\mathcal{M}}_l^n$ works on the qubits $l-1,l$ and $n$. This construction is interesting in how the transformations $\mu$ and ${\mathcal{M}}$ are partially defined which raises different implementation choices, with possibly different number of gates needed for elemental decomposition. We now describe these two transformations for reference. We denote by $\ket{ab}_x$ the qubits in the $x-1$ and $x$-th position in a system. $$\begin{aligned} \mu_n: \quad & \ket{00}_n \rightarrow \ket{00}_n \nonumber \\ & \ket{11}_n \rightarrow \ket{11}_n \nonumber \\ & \ket{01}_n \rightarrow \sqrt{\frac{1}{n}}\ket{01}_n+\sqrt{\frac{n-1}{n}}\ket{10}_n \nonumber\end{aligned}$$ @C=1em @R=2em [ & & & & & & & & &\ & & & & & && &&\ ]{} $$\begin{aligned} {\mathcal{M}}^l_n: \quad & \ket{00}_l\ket{0}_n \rightarrow \ket{00}_l\ket{0}_n \nonumber \\ & \ket{01}_l\ket{0}_n \rightarrow \ket{01}_l\ket{0}_n \nonumber \\ & \ket{00}_l\ket{1}_n \rightarrow \ket{00}_l\ket{1}_n \nonumber \\ & \ket{11}_l\ket{1}_n \rightarrow \ket{11}_l\ket{1}_n \nonumber \\ & \ket{01}_l\ket{1}_n \rightarrow \sqrt{\frac{n-l+1}{n}}\ket{01}_l\ket{1}_n \\ & ~~~~~~~~~~~~~~~~~~~~~~~+\sqrt{\frac{l-1}{n}}\ket{11}_l\ket{0}_n \nonumber\end{aligned}$$ @C=1em @R=2em [ & & & & & & & & &\ & & & & & & & & &\ & & & & & & & &&\ ]{} The implementations of these transformations in [@dicke] is shown in Figure \[fig:c3\] and \[fig:c4\] respectively. The first transformation, $\mu_n$ is in fact a maximally partial unitary transform. Because of the partially defined nature of the transformation the $CR_y$ and $CCR_y$ gates are also not fed all possible inputs. Instead the input to the $CR_y$ gates is only from the subspace spanned by the computational basis states $\ket{00},\ket{10}$ and $\ket{01}$. Similarly the input to the $CCR_y$ gate is only from the subspace spanned by the states $\ket{000}, \ket{010}, \ket{001}, \ket{011},\text{ and } \ket{110}$. Next in Section \[sec:3\] we look how partially defined transformations can be implemented more efficiently, and argue the optimality of this improvement with respect to this particular building block. Then in Section \[sec:4\] we reduce the gate count of the circuit ${\mathcal{C}}_{n,k}$ by removing redundancies and analyzing how the $\mu$ and ${\mathcal{M}}$ transformations act only on a subset of the defined computational basis states in specific cases. Example of Optimality for a Maximally Partial Unitary Transformation {#sec:3} ==================================================================== We have described the two partially defined unitary transformations used in the circuit ${\mathcal{C}}_{n,k}$. The implementation of the first transformation, $\mu_n$ is done using a controlled $R_y$ gate and two CNOT gates. This $CR_y$ gate only acts on the states $\ket{00}, \ket{10}, \ket{01}$ and their superpositions and the transformation never acts on the $\ket{11}$ state. If we take $\theta=2\cos^{-1}\big(\sqrt{\frac{1}{n}}\big)$. We denote the transformation implemented by the $CR_y(\theta)$ gate on the defined basis states as $T_1(\theta)$, and the corresponding transformation is as follows: $$\begin{aligned} \label{eq:tr1} T_1(\theta): \quad &\ket{00} \rightarrow \ket{00} \\ \nonumber &\ket{10} \rightarrow \ket{10} \\ \nonumber &\ket{01} \rightarrow \big( \cos(\frac{\theta}{2})\ket{0}+ \sin(\frac{\theta}{2})\ket{1} \big ) \ket{1} \nonumber\end{aligned}$$ This is in fact a maximally defined partial unitary transformation. While the gate $CR_Y(\theta)$ can perform this transformation, it needs at least $4$ elementary gates to implement. We first prove this necessary requirement using an important result from [@song Theorem B], which we note down for reference. [@song] \[th:opt\] 1. For a controlled gate $CU$ if $tr (UX)=0,~ tr(U) \neq 0,~ \det U=1,~ U \neq \pm I$ then the minimal number of elementary gates needed to implement $CU$ is $4$. 2. For a controlled gate $CU$ if $tr(U)=0,~ \det U=-1, ~ U \neq \pm X$ then the minimal number of elementary gates needed to implement $CU$ is $3$. 3. For a controlled gate $CU$ the minimal number of number of elementary gates needed to implement $CU$ is less than three $3$ iff $U \in \{e^{i\phi}I,e^{i\phi}X,e^{i\phi}Z \},~0 \leq \phi \leq 2\pi$. Our lemma follows immediately. \[th:cry4\] It takes minimum $4$ elementary gates to implement the $CR_y(\theta)$ gate. We calculate the values of $\det R_y(\theta)$ and $tr(R_y(\theta)X)$ to confirm the minimal number of gates needed to decompose $CR_y(\theta)$. $$\begin{aligned} & \det R_y(\theta)= \sin^2(\frac{\theta}{2})+\cos^2(\frac{\theta}{2})=1 \\ & R_y(\theta)X= \begin{bmatrix} -{\ensuremath{\sin(\frac{\theta}{2})}} & {\ensuremath{\cos(\frac{\theta}{2})}} \\ {\ensuremath{\cos(\frac{\theta}{2})}} & {\ensuremath{\sin(\frac{\theta}{2})}} \end{bmatrix} \implies tr(R_y(\theta)X)=0 \end{aligned}$$ The result ($1$) of Theorem \[th:opt\] concludes the proof. However the transformation $T_1(\theta)$ can in fact be implemented using three elementary gates as follows. $$T_1(\theta) \equiv \Big( R_y( \frac{-\alpha}{2}) \otimes I_2 \Big) {\sf CNOT}^2_1 \Big( R_y( \frac{\alpha}{2}) \otimes I_2 \Big),~ \frac{\alpha}{2}=\frac{\pi}{2}-\frac{\theta}{2}$$ This decomposition has also been used by Cruz et.al [@dn1] in the $W_n$ ($D^n_1$) state preparation algorithm. However, the corresponding transformation is defined only for the states $\ket{00}$ and $\ket{01}$ and no insight into the optimality of the implementation is given. We first derive the underlying $4 \times 4$ unitary matrix $U^0(\alpha)$ that describes this three gate transformation. Next we prove that $U^0(\alpha)$ needs at least three gates to be implemented by verifying the conditions of result (2) of Theorem \[th:opt\]. We end this section by showing that the transformation $T_1(\theta)$ needs at least three elementary gates (including one CNOT) to be implemented, proving the optimality of the $U^0(\alpha)$ implementation. @C=1em @R=2em [ &&&& & && && &\ &&&& & & & & & & ]{} @C=1em @R=2em [ &&&& & & && & &\ &&&& & & & & & &\ ]{} The gate $U^0(\alpha)$ performs the partially defined unitary transformation $T_1(\theta)$ where $\alpha=\pi-\theta$ and needs minimum three elementary gates to be implemented. We first study the transformation carried out by $U^0$ in the subspace of $T_1$. $$\begin{aligned} \ket{00} & \\ &\xrightarrow{R_y \left( \frac{\alpha}{2}\right)\text{ on }q1} \Big ( {\ensuremath{\cos(\frac{\alpha}{4})}}\ket{0}+{\ensuremath{\sin(\frac{\alpha}{4})}}\ket{1} \Big ) \ket{0} \\ &\xrightarrow{{\sf CNOT}_1^2} \Big ( {\ensuremath{\cos(\frac{\alpha}{4})}}\ket{0}+{\ensuremath{\sin(\frac{\alpha}{4})}}\ket{1} \Big ) \ket{0} \\ &\xrightarrow{R_y(-\frac{\alpha}{2})\text{ on }q1} \Big( {\ensuremath{\cos(\frac{\alpha}{4})}}\big({\ensuremath{\cos(\frac{\alpha}{4})}}\ket{0}-{\ensuremath{\sin(\frac{\alpha}{4})}})\ket{1}\big)+\\ &~~~~~~~~~~~~~~~~~{\ensuremath{\sin(\frac{\alpha}{4})}})\big({\ensuremath{\sin(\frac{\alpha}{4})}})\ket{0}+{\ensuremath{\cos(\frac{\alpha}{4})}})\ket{1} \big) \Big) \ket{0} \\ &= \big( \cos^2(\frac{\alpha}{4})+\sin^2(\frac{\alpha}{4}) \big)\ket{00}=\ket{00} \\ \ket{10} &\\ & \xrightarrow{R_y \left( \frac{\alpha}{2}\right)\text{ on }q1} \Big ( -{\ensuremath{\sin(\frac{\alpha}{4})}}\ket{0}+{\ensuremath{\cos(\frac{\alpha}{4})}}\ket{1} \Big ) \ket{0} \\ & \xrightarrow{{\sf CNOT}_1^2} \Big ( -{\ensuremath{\sin(\frac{\alpha}{4})}}\ket{0}+{\ensuremath{\cos(\frac{\alpha}{4})}}\ket{1} \Big ) \ket{0} \\ & \xrightarrow{R_y(-\frac{\alpha}{2})\text{ on }q1} \Big( -{\ensuremath{\sin(\frac{\alpha}{4})}}\big({\ensuremath{\cos(\frac{\alpha}{4})}}\ket{0}-{\ensuremath{\sin(\frac{\alpha}{4})}})\ket{1}\big)+ \\ &~~~~~~~~~~~~~~~~~{\ensuremath{\cos(\frac{\alpha}{4})}})\big({\ensuremath{\sin(\frac{\alpha}{4})}})\ket{0}+{\ensuremath{\cos(\frac{\alpha}{4})}})\ket{1} \big) \Big) \ket{0} \\ &= \big( \cos^2(\frac{\alpha}{4})+\sin^2(\frac{\alpha}{4}) \big)\ket{10}=\ket{10} \\ \ket{01} & \\ & \xrightarrow{R_y \left( \frac{\alpha}{2}\right)\text{ on }q1} \Big ( {\ensuremath{\cos(\frac{\alpha}{4})}}\ket{0}+{\ensuremath{\sin(\frac{\alpha}{4})}}\ket{1} \Big ) \ket{1} \\ & \xrightarrow{{\sf CNOT}_1^2} \Big ( {\ensuremath{\sin(\frac{\alpha}{4})}}\ket{0}+{\ensuremath{\cos(\frac{\alpha}{4})}}\ket{1} \Big ) \ket{1} \\ & \xrightarrow{R_y(-\frac{\alpha}{2})\text{ on }q1} \Big( {\ensuremath{\sin(\frac{\alpha}{4})}}\big({\ensuremath{\cos(\frac{\alpha}{4})}}\ket{0}-{\ensuremath{\sin(\frac{\alpha}{4})}})\ket{1}\big)+ \\ &~~~~~~~~~~~~~~~~~{\ensuremath{\cos(\frac{\alpha}{4})}})\big({\ensuremath{\sin(\frac{\alpha}{4})}})\ket{0}+{\ensuremath{\cos(\frac{\alpha}{4})}})\ket{1} \big) \Big) \ket{1} \\ &= \big( 2{\ensuremath{\cos(\frac{\alpha}{4})}}{\ensuremath{\sin(\frac{\alpha}{4})}} \ket{0}+ (\cos^2(\frac{\alpha}{4})-\sin^2(\frac{\alpha}{4})\ket{1} \big)\ket{1} \\ &=\Big( {\ensuremath{\sin(\frac{\alpha}{2})}}\ket{0}+{\ensuremath{\cos(\frac{\alpha}{2})}}\ket{1} \Big)\ket{1}\end{aligned}$$ Setting $\alpha=\pi-\theta$ gives us the same transformation as defined by $T_1(\theta)$. Now we completely define the gate $U^0$ by studying the transformation acted on the state $\ket{11}$. $$\begin{aligned} \ket{11} &\\ &\xrightarrow{R_y \left( \frac{\alpha}{2}\right)\text{ on }q1} \Big ( -{\ensuremath{\sin(\frac{\alpha}{4})}}\ket{0}+{\ensuremath{\cos(\frac{\alpha}{4})}}\ket{1} \Big ) \ket{0} \\ & \xrightarrow{{\sf CNOT}_1^2} \Big ( {\ensuremath{\cos(\frac{\alpha}{4})}}\ket{0}-{\ensuremath{\sin(\frac{\alpha}{4})}}\ket{1} \Big ) \ket{0} \\ & \xrightarrow{R_y(-\frac{\alpha}{2})\text{ on }q1} \Big( {\ensuremath{\cos(\frac{\alpha}{4})}}\big({\ensuremath{\cos(\frac{\alpha}{4})}}\ket{0}-{\ensuremath{\sin(\frac{\alpha}{4})}})\ket{1}\big)- \\ &~~~~~~~~~~~~~~~~~ {\ensuremath{\sin(\frac{\alpha}{4})}})\big({\ensuremath{\sin(\frac{\alpha}{4})}})\ket{0}+{\ensuremath{\cos(\frac{\alpha}{4})}})\ket{1} \big) \Big) \ket{0} \\ &= \big( (\cos^2(\frac{\alpha}{4})-\sin^2(\frac{\alpha}{4})\ket{0}- 2{\ensuremath{\cos(\frac{\alpha}{4})}}{\ensuremath{\sin(\frac{\alpha}{4})}} \ket{0} \big)\ket{1} \\ &=\Big( {\ensuremath{\cos(\frac{\alpha}{2})}}\ket{0}-{\ensuremath{\sin(\frac{\alpha}{2})}}\ket{1} \Big)\ket{1}\end{aligned}$$ So the overall transformation provided by $U^0(\alpha)$ is: $$\begin{aligned} & \ket{00} \rightarrow \ket{00} \\ & \ket{10} \rightarrow \ket{10} \\ & \ket{01} \rightarrow \Big( {\ensuremath{\sin(\frac{\alpha}{2})}}\ket{0}+{\ensuremath{\cos(\frac{\alpha}{2})}}\ket{1} \Big)\ket{1} \\ & \ket{11} \rightarrow \Big( {\ensuremath{\cos(\frac{\alpha}{2})}}\ket{0}-{\ensuremath{\sin(\frac{\alpha}{2})}}\ket{1} \Big)\ket{1}\end{aligned}$$ Therefore the gate $U^0(\alpha)$ is a two qubit gate which can be expressed as a controlled gate $CU(\alpha)$ gate where $U(\alpha)= \begin{bmatrix} {\ensuremath{\sin(\frac{\alpha}{2})}} & {\ensuremath{\cos(\frac{\alpha}{2})}} \\ {\ensuremath{\cos(\frac{\alpha}{2})}} & -{\ensuremath{\sin(\frac{\alpha}{2})}} \\ \end{bmatrix} $. Now $tr U(\alpha)=0$ and $\det U(\alpha)=-1$ for all $\alpha$. Therefore we can conclude from the result ($2$) in Theorem \[th:opt\] that this gate requires at least three gates to be implemented. We finally show the optimality of this implementation for implementing the two qubit transformation $T_1(\theta)$. The transformation $T_1(\theta)$ needs at least one CNOT and two single qubit gates to be implemented for $0 <\theta <\pi$. The transformation $T_1(\theta)$ is only defined for the basis states $\ket{00},\ket{01}$ and $\ket{10}$. Any matrix $M(\theta)$ that can carry out the transformation is of the form $\begin{bmatrix} 1 & 0 & 0 & a \\ 0 & \cos(\frac{\theta}{2}) & 0 & b \\ 0 & 0 & 1 & c \\ 0 & \sin(\frac{\theta}{2}) & 0 & d \end{bmatrix}$ where $a,b,c,d$ are complex unknowns. However since $M(\theta)$ is unitary we have $M(\theta)M^{\dagger}(\theta)=I$. Therefore $1+aa^*=1 \implies a=0$ and $1+cc^*=1 \implies c=0$. That is the matrix $M_{\theta}$ is a controlled unitary $CM_1(\theta)$ and $M_1(\theta)= \begin{bmatrix} \cos(\frac{\theta}{2}) & b \\ \sin(\frac{\theta}{2}) & d \end{bmatrix}$. Now for $0 < \theta < \pi$ both $\cos(\frac{\theta}{2})$ and $\sin(\frac{\theta}{2})$ are non zero. This implies that the matrix cannot fulfill the necessary conditions defined in result (3) of Theorem \[th:opt\] and therefore cannot be expressed with less than three elementary gates. Now we use our observations to improve the gate count of the circuit ${\mathcal{C}}_{n,k}$. Improved gate counts for Circuits of $\ket{D^n_k}$ {#sec:4} ================================================== We first count the number of CNOT and single qubit gates in ${\mathcal{C}}_{n,k}$ by reviewing the circuit. The circuit is composed of $n-1$ blocks of gates called $SCS$. There are $n-k-1$ blocks of the form $SCS^t_k,~ k<t \leq n$ and $k-1$ blocks of the form $SCS^{i+1}_i,~1 \leq i \leq k-1$. Each block $SCS^t_k$ consists of one two qubit transformation $\mu_t$ which is implemented on the qubits $t-1$ and $t$ and $k-1$ three qubit transformations of the type ${\mathcal{M}}^l_t,~t-1 \leq l \leq t-k-2$. Here $\mu_t$ is implemented on the $t-1$ and $t$-th qubit and ${\mathcal{M}}^l_t$ is implemented on the $l-1$, $l$ and $t$-th qubit, as described in Section \[sec:2\]. Each transformation of type $\mu$ is decomposed into two CNOT and a $T_1(\theta)$ transformation which is implemented as a $CR_y$ gate by adjusting the value of $\theta$. We have shown in Lemma \[th:cry4\] that a $CR_y$ transformation needs minimum $4$ gates to implement. In fact it needs at least two CNOT gates. Therefore each $\mu$ transformation needs four CNOT and two single qubit gate. The number of transformations of type ${\mathcal{M}}^l_n$ is $$\begin{aligned} &(n-k)(k-1)+ \displaystyle \sum_{i=1}^{k-2}i \\ &=nk-n+k-k^2 + \frac{(k-1)(k-2)}{2} \\ &=nk- \frac{k(k+1)}{2}-n+1.\end{aligned}$$ Each ${\mathcal{M}}^l_n$ transformation is shown to require six CNOT and four single qubit gates. However one CNOT gate of for each ${\mathcal{M}}^l_n$ transformation can be canceled by rearranging the first two CNOT gates of the next transformation. The total number of CNOT gates and single qubit gates used to prepare the state $\ket{D^n_k}$ is shown in Table \[tab:cnot1\]. CNOT gates $5(nk- \frac{k(k+1)}{2}-n+1)+4(n-1)$ -------------------- -------------------------------------- single qubit gates $4(nk- \frac{k(k+1)}{2}-n+1)+2(n-1)$ : Gates needed to prepare $\ket{D^n_k}$ as in [@dicke] []{data-label="tab:cnot1"} Figure \[fig:cir63\_0\] shows the circuit ${\mathcal{C}}_{6,3}$ in terms of CNOT, $CR_y$ and $CCR_y$ gates. @C=0.27em @R=0.1em [ & & &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&\ & & &&&&&&&&&&&&&&&&& &&&&&&&&&&&& &&&&&&&&\ & & &&&&&&& & & &&&& & & && &&&&&&&&&&&&&&&&&&&&\ & && &&& & & && && & & &&&&&&&&&&&&&&&&&&&&&&&&&\ & & & && && &&&&&& &&& &&&&&&&&&&&&&&&&&&&&&&&\ & & && &&& &&& &&&&&&&&&&&&&&&&&&&&&&&&&&&&&\ ]{} We show the circuit of $\ket{D^4_2}$ formed according to this construction method in Figure \[fig:cir63\_0\]. We now improve the gate counts of the circuit ${\mathcal{C}}_{n,k}$ in the following ways. Replacing $CR_y$ with $CU$ {#replacing-cr_y-with-cu .unnumbered} -------------------------- We first use the $CU$ gate shown in Section \[sec:3\] to implement the transformation $T_1$ corresponding to each $\mu$ transformation which needs one CNOT and two single qubit gates to be implemented. Therefore each $\mu$ transformation needs three CNOT and two single qubit gates. Since there are $n-1$ $\mu$ transformations this reduces the number of CNOT by $n-1$ for any $\ket{D^n_k}$. Next we observe that some of the $\mu_n$ and ${\mathcal{M}}^l_n$ transformations act as identity transformation, which we count as a function of $k$ for any $\ket{D^n_k}$. The $\mu$ and ${\mathcal{M}}$ transformations that act like Identity {#the-mu-and-mathcalm-transformations-that-act-like-identity .unnumbered} -------------------------------------------------------------------- Let there be a $n$ qubit system in some state $\ket{\phi}$. This state can be uniquely represented as a superposition of all $2^n$ (computational) basis state. The amplitude of a particular basis state may or may not be zero depending on the description of $\ket{\phi}$. We call a basis state with non zero amplitude an active basis state. The affect of a unitary transformation $T$ on this state can be completely described by observing how it transforms the active basis states of $\ket{\phi}$. If the $k$-th qubit is in the zero (one) state in all the active basis states and the transformation $T$ doesn’t act on the $k$-th qubit in non trivial way on any of those basis states then the $k$-th qubit in all the active basis states of $T\ket{\phi}$ will also be in the zero(one) state. Using this simple fact we prove the following theorem using induction. If the $n$ qubit system is expressed as superposition of computational basis states after the block $SCS^{n-t}_k$ has acted then it can be expressed as $$\begin{aligned} \sum\limits_{a=0}^{2^{t+1}-1} ~ \sum\limits_{b=0}^{2^{t+1}-1}\alpha_{a,b}\Big( &\ket{0}^{\otimes n-k-1-t} \big(\bigotimes_{i=1}^{t+1}\ket{a^{bin}_i} \big)\\ &\ket{1}^{\otimes k-1-t} \big( \bigotimes_{j=1}^{t+1}\ket{b^{bin}_j} \big)\Big).\end{aligned}$$ The statement implies that the first $n-k-t-1$ qubits are all in the state $\ket{0}$ and the $(n-k+1)$-th qubit and the next $k-t-1$ qubits are all in the state $\ket{1}$ in all active basis states of the $n$ qubit system after the block $SCS^{n-t}_k$ has acted on it. We first prove the statement for $t=0$. The $n$ qubit system is at first in the state $\ket{\psi_0}=\ket{0}^{\otimes (n-k)}\ket{1}^{\otimes k}$ and the block to be applied is $SCS^n_k$. This block consists of the gates $\mu_n$, ${\mathcal{M}}^{i}_n,n-1 \leq i \leq n-k+1$. We know that the transformation $\mu_n$ affects the $(n-1)$ and the $n$-th qubit and the transformation ${\mathcal{M}}^l_n$ affects the $(l-1)$ and $n$ th qubit. Additionally $\mu_n$ acts as identity on a basis state if the $n-q$th qubit of the basis state is in the $\ket{1}$ state. Similarly ${\mathcal{M}}^l_n$ acts as identity on a basis state if the $(n-l-1)$-th qubit is in the $\ket{1}$ state. The last $k$ qubits of $\ket{\psi_0}$ are in the state $\ket{1}$ and therefore $\mu_n$ and ${\mathcal{M}}^{n-1}_n, {\mathcal{M}}^{n-2}_n, \ldots {\mathcal{M}}^{n-k+2}_n$ act as identity transformations. Finally the transformation ${\mathcal{M}}^{n-k+2}_n$ is applied. The first qubit to this transformation is in the state $\ket{0}$ and therefore this transformation may lead to basis states with either $\ket{0}$ or $\ket{1}$ in the $(n-k)$-th and $n$-th positions. Therefore the resultant state can be written as $$\displaystyle \sum_{a_1,a_2 \in \{0,1\}} \alpha_{a_1a_2} \ket{0}^{\otimes n-k-1}\ket{a_1}\ket{1}^{\otimes k-1}\ket{a_2}.$$ Thus the first $n-k-1$ qubits are all in the state $\ket{0}$ and the $(n-k+1)$-th qubit and the next $k-1$ qubits are all in the $\ket{1}$ state in all active basis states of the system.This concludes the base case. Now assuming that our statement holds true for some $t-1<k-2$ we show that the statement also holds for $t$. The $SCS^{n-t}_k$ block is composed of the transformations $\mu_{n-t}$, ${\mathcal{M}}^{i}_{n-t},n-t-1 \leq t \leq n-t-k+1$. The $n$ qubit system is in the state $$\begin{aligned} \ket{\psi_{t-1}} =\sum\limits_{a=0}^{2^{t}-1} ~ \sum\limits_{b=0}^{2^{t}-1} \alpha_{a,b} &\Big(\ket{0}^{\otimes n-k-t} \big(\bigotimes_{i=1}^{t}\ket{a^{bin}_i} \big) \\ & \ket{1}^{\otimes k-t} \big( \bigotimes_{j=1}^{t}\ket{b^{bin}_j} \big)\Big).\end{aligned}$$ That is, the first $n-k-t$ qubits are in the state $\ket{0}$ in all active basis states and the $n-k+1$ and the next $k-t-1$ qubits are in the state $\ket{1}$. These are the first qubits to the transformations $\mu_{n-t}$,${\mathcal{M}}^{i}_{n-t}, n-t-1 \leq i \leq n-k$. This implies the $\mu$ transformation and the $(k-2-t)$ ${\mathcal{M}}$ transformations act as identity transformations on all active basis states. The next $t$ ${\mathcal{M}}$ transformations may get the $\ket{0}$ state as the first qubit and therefore the $n$-qubit system before the last ${\mathcal{M}}$ has been applied is in the state $$\begin{aligned} \ket{\psi_{t-1}'} =\sum\limits_{a=0}^{2^{t}-1} ~ \sum\limits_{b=0}^{2^{t+1}-1} \alpha_{a,b}\Big( &\ket{0}^{\otimes n-k-t} \big(\bigotimes_{i=1}^{t}\ket{a^{bin}_i} \big) \\ &\ket{1}^{\otimes k-1-t} \big( \bigotimes_{j=1}^{t+1}\ket{b^{bin}_j} \big)\Big).\end{aligned}$$ Finally the last three qubit transformation of the block $SCS^{n-t}_k$ ${\mathcal{M}}^{n-t-k+1}_{n-t}$ acts on the system. Now since the $(n-t-k)$-th qubit is in the state $\ket{0}$ in all active basis states, the ${\mathcal{M}}$ gate may non trivially act on it and the $(n-t)$-th qubit. This results in the state $$\begin{aligned} \ket{\psi_t} =\sum\limits_{a=0}^{2^{t+1}-1} ~ \sum\limits_{b=0}^{2^{t+1}-1} \alpha_{a,b}\Big( &\ket{0}^{\otimes n-k-t-1} \big(\bigotimes_{i=1}^{t+1}\ket{a^{bin}_i} \big) \\ &\ket{1}^{\otimes k-1-t} \big( \bigotimes_{j=1}^{t+1}\ket{b^{bin}_j} \big)\Big).\end{aligned}$$ This completes the proof. It is important to note that there may be many basis states in the expression of $\ket{\psi_t}$ with zero amplitude. However our focus is on qubits that are definitely going to be either in the zero state or in the one state in all active basis states. This proof also shows that the $\mu$ transformation and the $k-2-t$ ${\mathcal{M}}$ transformations in the block $SCS^{n-t}_k,t<k-1$ act as identity transformations and therefore can be removed from the circuit, which is the second improvement. Therefore the number of $\mu$ transformations omitted is $k-1$ and the number of ${\mathcal{M}}$ transformation omitted are $\frac{(k-2)(k-1)}{2}$. This removes $3(k-1)+\frac{5(k-2)(k-1)}{2}$ CNOT and $2(k-1)+\frac{4(k-2)(k-1)}{2}$ single qubit gates. The first non identity ${\mathcal{M}}$ transformation in $SCS^n_k$ {#the-first-non-identity-mathcalm-transformation-in-scsn_k .unnumbered} ------------------------------------------------------------------ Having removed the $\mu$ and ${\mathcal{M}}$ transformations we now look at the first transformation of the block $SCS^{n-t}_k,t<k-1$ which is ${\mathcal{M}}^{n-k+1}_{n-t}$. This transformation depends on the state of the $n-k,n-k+1$ and $(n-t)$-th qubits and affects the state of the $(n-k)$-th and the $(n-t)$-th qubit. At this stage the $n=$ qubit system is at the state $$\sum\limits_{a=0}^{2^{t}-1} ~ \sum\limits_{b=0}^{2^{t}-1} \alpha_{a,b} \ket{0}^{\otimes n-k-t} \Big(\bigotimes_{i=1}^{t}\ket{a^{bin}_i} \Big) \ket{1}^{\otimes k-t} \Big( \bigotimes_{j=1}^{t}\ket{b^{bin}_j} \Big).$$ Therefore in all the active basis states both the $n-k$th and the $n-t$th qubits are in the state $\ket{1}$. Therefore the three qubit transformation applied by ${\mathcal{M}}^{n-k+1}_{n-t}$ can be expressed as follows substituting $l=n-k+1$: $$\begin{aligned} & \ket{11}_{l}\ket{1}_{n-t} \rightarrow \ket{11}_{l}\ket{1}_{n-t} \\ & \ket{01}_{l}\ket{1}_{n-t} \rightarrow \sqrt{\frac{n-t-l+1}{n-t}}\ket{01}_{l}\ket{1}_{n-t} \\ &~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+\sqrt{\frac{l-1}{n-t}}\ket{11}_{l}\ket{0}_{n-t}.\end{aligned}$$ This is in-fact can be implemented as a a two qubit transformation of the type $\mu$. as the $(n-k+1)$-th qubit is in the $\ket{1}$ state in all the active basis states. The transformation acts on the $(n-k)$-th and $(n-t)$-th qubits as ${\mathcal{M}}^{n-k+1}_{n-t} \equiv ({\sf CNOT}^{n-k}_{n-t})(CU^{n-t}_{n-k}(\theta)) ({\sf CNOT}^{n-k}_{n-t})$ where $\theta =2 \cos^{-1}\Big(\sqrt{\frac{n-t-l+1}{n-t}} \Big)$. We know that the $CU$ gate requires one CNOT and two $R_y$ gates to be implemented therefore ${\mathcal{M}}^{n-k+1}_{n-t}$ requires only three CNOT and two $R_y$ gates. This improvement is reflected for all $SCS^{n-t}_k$ such that $n-t \geq n-k+2$ that is for $0 \leq t \leq k-2$. Therefore it reduces the number of CNOT gate in the circuit by further $2(k-1)$ and the number of single qubit gates by $2(k-1)$ as well. Additionally for $\ket{D^n_k},~k>1$ when $SCS^n_k$ is applied the $n$-qubit system is in the state $\ket{0}^{\otimes n-k}\ket{1}^{\otimes k}$ and therefore the transformation ${\mathcal{M}}^{n-k+1}_{n-t}$ only acts on the basis state $\ket{011}$. The corresponding transformation is $$\begin{aligned} \ket{01}_{n-k+1}\ket{1}_{n} \rightarrow & \sqrt{\frac{k}{n}}\ket{01}_{n-k+1}\ket{1}_{n} \\ &~~~~+\sqrt{\frac{n-k}{n}}\ket{11}_{n-k+1}\ket{0}_{n}.\end{aligned}$$ This can be implemented using a $R_y(\cos^{-1}\sqrt{\frac{k}{n}})$ on the $(n-k)$-th qubit followed by a CNOT gate ${\sf CNOT}^{n-k}_n$ which removes further two CNOT and one single qubit gate. @C=0.2em @R=0.2em [ &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&\ &&&&&&&&&&&&&& &&&&&&&&&&&& &&&&&&&&\ &&&&&&&&&& & & && &&&&&&&&&&&&&&&&&&&&\ & & &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&\ & & &&&&&&&&& &&&&&&&&&&&&&&&&&&&&&&&\ & & &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&\ ]{} We denote this circuit by ${\mathcal{\widehat{C}}}_{n,k}$. Figure \[fig:cir63\_1\] shows the structure of ${\mathcal{\widehat{C}}}_{6,3}$. Combining the results we get the following count of CNOT and single qubit gates in the improved circuit. We now calculate the total improvement in the CNOT and single qubit gate counts for the $\ket{D^n_k},k>1$ preparation circuit ${\mathcal{\widehat{C}}}_{n,k}$. - The total number of CNOT gates removed $=n-1+3(k-1)+\frac{5(k-2)(k-1)}{2}+2(k-1)+2$. - The total number of single qubit gates removed $=2(k-1)+\frac{4(k-2)(k-1)}{2}+2(k-1)+1$. Therefore the total number of CNOT gates present in the circuit is $$\begin{aligned} & 5(nk- \frac{k(k+1)}{2})-n+1 \\ &-\Big(n-1+3(k-1)+\frac{5(k-2)(k-1)}{2}+2(k-1) +2 \Big) \\ &=5nk-5k^2-2n\end{aligned}$$ The number of single qubit gates present in the circuit is $$\begin{aligned} & 4(nk- \frac{k(k+1)}{2}-n+1)+2(n-1) \\ &-\Big(2(k-1)+\frac{4(k-2)(k-1)}{2}+2(k-1)+1 \Big) \\ & = 4nk-4k^2-2n+1\end{aligned}$$ For $k=1$ We get the number of CNOT as $3n-3$ (from $n-1$ $\mu$ transformations) and the number of single qubits gate as $2n-2$. However, one CNOT gate can be further removed from each $\mu$ gate as the active basis states in input to the $\mu$ transformations are only $\ket{00}$ and $\ket{01}$. The resultant circuit is identical to the linear $W_n$ preparation circuit in [@dn1] and contains $2n-2$ CNOT and $2n-2$ single qubit gates and thus we don’t elaborate it further. We know that the state $\ket{D^n_k}$ can be prepared by first forming the state $\ket{D^n_{n-k}}$ and then applying a X gate to each qubit. On that note it is interesting to observe that after the improvements the Circuits for $\ket{D^n_k}$ and $\ket{D^n_{n-k}}$ require the same number of CNOT gates. 4 5 6 7 8 --- ------- ------- ------- ------- -------- -- 2 22,12 31,20 40,28 49,36 58,44 3 27,7 41,20 55,33 69,46 83,59 4 46,10 65,28 84,46 103,64 5 70,13 94,36 118,59 6 99,16 128,44 7 133,19 : CNOT gate count of the pair ${\mathcal{C}}_{n,k},{\mathcal{\widehat{C}}}_{n,k}$[]{data-label="tab:cnotcomp"} 4 5 6 7 8 --- ------ ------- ------- ------- ------- -- 2 14,9 20,15 26,21 32,27 38,33 3 18,5 28,15 38,25 48,35 58,45 4 32,7 46,21 60,35 74,49 5 50,9 68,27 86,45 6 72,11 94,33 7 98,13 : Single qubit gate count of the pair ${\mathcal{C}}_{n,k},{\mathcal{\widehat{C}}}_{n,k}$[]{data-label="tab:scomp"} Table \[tab:cnotcomp\] and \[tab:scomp\] show the number of CNOT and single qubit gates needed to implement the states $\ket{D^n_k}$ for $4 \leq n \leq 8, 1 \leq k \leq n-1$, respectively. In the next section we show that our observations not only reduces the gate counts of the circuit but also reduces its architectural constraints. Actual Implementation and architectural constraints {#sec:5} =================================================== Architectural Constraints ------------------------- We are at the stage where quantum circuits can be implemented on actual quantum computers using cloud services, such as IBM Quantum Experience, also known as IQX [@ibmq]. However the architecture of the individual back-end quantum machines pose restrictions to implementation of a particular circuit. The most prominent constraint is that of the CNOT implementation. In this regard we use the terms architectural constraint and CNOT constraint interchangeably. Every quantum system $Q$ with $n$ (physical) qubits has a CNOT map, which we express as $G^Q_A(V^Q,E^Q)$ where $V^Q=\{q_1,q_2,\ldots q_n\}$. In this graph the nodes represent the qubits and the edges represent CNOT implementability. A directed edge $q_i \rightarrow q_j$ implies that a CNOT can be implemented with $q_i$ as control and $q_j$ as target in the system $Q$. The edges in the CNOT maps of all the publicly available IQX machines are bidirectional. Let there be a circuit ${\mathcal{C}}$ on $n$ qubits. We denote the (logical) qubits $c_1,c_2,\ldots c_n$. We also have a CNOT map corresponding to the circuit, which describes the CNOT gates used in the circuit. We describe this as the directed graph $G_{{\mathcal{C}}}(V^{{\mathcal{C}}},E^{{\mathcal{C}}})$ where $V^{{\mathcal{C}}}=\{ c_1,c_2, \ldots c_m \}$ and there is a directed edge $c_i \rightarrow c_j$ if there is one or more CNOT with $c_i$ as control and $c_j$ as target. Therefore if the graph $G_{{\mathcal{C}}}$ can shown to be the subgraph of $G^Q_A$ by some mapping of the logical qubits to the physical qubits then the circuit ${\mathcal{C}}$ can be implemented on the architecture with the same number of CNOT gates. However, if such a map is not possible, then the circuit can be implemented on that architecture either by using swap gates which require additional CNOT gates or changing the construction of the circuit. There are mapping solutions such as the one applied by IQX which dynamically changes the structure of the circuit to implement a circuit in an architecture that does not meet the circuit’s CNOT constraints. Similarly in their paper Zulehner et.al [@efmap] have also proposed an efficient mapping solution. However it is not always possible to avoid an increase in the number of CNOT gates. Given a circuit it is crucial to find it’s minimal architectural needs in terms of the CNOT map without increasing the CNOT gate count. The IBM-Q systems mapping solutions show the modified circuit as the transpiled circuit given any circuit as input, although its solutions are not always optimal. In this paper we first consider the system “${\sf ibmqx2}$” ($Q_1$) of IQX. The CNOT map of $Q_1$ is shown in the Figure \[fig:cmapqx2\]. \(E) [$1$]{}; (A)\[above=of E\] [${2}$]{}; (B) \[right=of E\] [${3}$]{}; (C) \[left=of E\] [$0$]{}; (D) \[below=of E\] [$4$]{}; \(A) edge (E);(E) edge (A); (B) edge (E);(E) edge (B); (C) edge (E);(E) edge (C); (D) edge (E);(E) edge (D); (A) edge (C);(C) edge (A); (D) edge (B);(B) edge (D); Against this backdrop we observe the CNOT constraints of the circuit ${\mathcal{C}}_{4,2}$, implemented to prepare $\ket{D^4_2}$. Then we implement the circuit ${\mathcal{\widehat{C}}}_{4,2}$ which is the result of the improvements shown in Section \[sec:4\]. We observe that the improvement proposed by us not only reduces gate counts but also reduces CNOT constraints. We implement these circuits in the system $Q_1$ and compare the measurement statistics of the two circuits by measuring the deviation from the ideal measurement statistics and find that the results of ${\mathcal{\widehat{C}}}_{4,2}$ is much more closely aligned with the ideal results. We end this section by showing how some changes in the circuit ${\mathcal{\widehat{C}}}_{4,2}$ possible because of partially defined transformations can lower the error in the circuit due to CNOT on expectation without a reduction in number of CNOT gates or change in the CNOT constraints. Implementation and Improvement for $\ket{D^4_2}$ ------------------------------------------------ We start by constructing the circuit ${\mathcal{C}}_{4,2}$. We implement every $CR_y$ gate using two CNOT gates and two $R_y$ gate as we know that the $CR_y$ gate needs at least $4$ gates to be implemented and every three qubit ${\mathcal{M}}^l_n$ transformation using five CNOT and four $R_y$ gates (as given in the description of [@dicke]). The resultant circuit ${\mathcal{C}}_{4,2}$ is shown in Figure \[fig:cir42\_0\]. This circuit contains $22$ CNOT gates. We use the notation $\theta^x_y$ to denote the angle $2\cos^{-1}(\sqrt{\frac{x}{y}})$. The CNOT map of the circuit is shown in Figure \[fig:cmapstage1\]. @C=0.2em @R=1em [ & & & & & & & & & & & & & & & & & & & & & & & && &  &  &   && && & & &&&& & &&\ & & & & & & & & & & & && && && & & && && & & & & & & & & & & && & && && &\ & & & & & & & & & & & & & & & & & &&& && &&&&& &&&& &&& & & & & & &&\ & & & & & & & & & && && & & & & && & & & & & & & & & & & & & & & & & & & &&&\ ]{} \(A) [${q_0}$]{}; (B) \[right=of A\] [${q_1}$]{}; (C) \[below=of A\] [$q_2$]{}; (D) \[below=of B\] [$q_3$]{}; \(A) edge (B);(B) edge (A); (B) edge (C);(C) edge (B); (C) edge (D);(D) edge (C); (D) edge (B);(B) edge (D); (A) edge (C);(C) edge (A); We then implement ${\mathcal{\widehat{C}}}_{4,2}$ by making the following changes to ${\mathcal{C}}_{4,2}$. 1. Implement the $CU^o$ gate instead of $CR_y$ gates. 2. Remove the Redundant $\mu$ and ${\mathcal{M}}$ transformation. 3. Reduce the gate count in implementation of ${\mathcal{M}}^{n-k+1}_{n-t}$ type transformations. This brings the total number of CNOT gates in the circuit to $12$. We name the circuit at this stage ${\mathcal{\widehat{C}}}_{4,2}$. These steps not only reduce the CNOT gates in the circuit but also reduces the CNOT constraints of the circuit. Figure \[fig:cir42\_3\] shows the circuit at this stage and the reduced CNOT map $G_{{\mathcal{\widehat{C}}}_{4,2}}$ as shown in the Figure \[fig:cmapstage3\]. @C=0.2em @R=1em [ &&&&&& & & & & & & & & & & & & && & && && && & & &&&& & &&\ &&&&&& & & & & & & & & && && & & & & & & & & & & &&& && && &\ &&&&&& & & & & & & &&&&& &&&&& &&&& &&& & & & & & &&\ &&&&&& & & & & & & && & & & & & & & & & & & & & & & & & & & &&&\ ]{} \(A) [${q_0}$]{}; (B) \[right=of A\] [${q_1}$]{}; (C) \[below=of A\] [$q_2$]{}; (D) \[below=of B\] [$q_3$]{}; \(A) edge (B);(B) edge (A); (B) edge (C);(C) edge (B); (B) edge (D); (A) edge (C);(C) edge (A); In fact the Graph $G^{{\mathcal{\widehat{C}}}_{4,2}}$ can be shown to be a subgraph of $G^{Q_1}_A$ under several mappings of qubits. Therefore this circuit can be implemented in the [“ibmqx2”]{} ($Q_1$) machine with $12$ CNOT. However the CNOT constraints of the circuits corresponding to even $D^5_k,k>1$ cannot be met by any IBM-Q architecture at this stage. Now we compare the results of the circuits ${\mathcal{C}}_{4,2}$ which is due to [@dicke] and ${\mathcal{\widehat{C}}}_{4,2}$ which is what we obtained after the reductions and modifications. Comparison of Measurement Statistics of ${\mathcal{C}}_{4,2}$ and $Cir_{4,2}^1$ {#comparison-of-measurement-statistics-of-mathcalc_42-and-cir_421 .unnumbered} ------------------------------------------------------------------------------- The output by an ideal Quantum Computer would produce the state $\sqrt{ \frac{1}{{n \choose w}}}\displaystyle \sum_{wt(i)=w} \ket{i_2}$ on a correct $\ket{D^n_w}$ preparation circuit. We first verify the resultant state vectors to of the two circuits to see that they both ideally produce $\sqrt{\frac{1}{6}} \Big(\ket{0011}+\ket{0101}+\ket{0110}+\ket{1100}+\ket{1010}+\ket{1001} \Big)$ and then use a primary error measure based on measurement in computational basis to estimate the closeness of the states formed by the two circuits from the ideal state $\ket{D^4_2}$. We run both the circuits for the maximum possible shots $(8192)$ and use the measurement statistics to estimate the closeness to the desired state using the following error measure. We define our error measure $EM_{n,w}$ for the Dicke state $\ket{D^n_w}$ as follows. Let $p_i$ be the percentage of times the measurement of the circuit ${\mathcal{C}}$ yields the result $i_2$. Then we have $$EM_{n,w}({\mathcal{C}})= \frac{1}{2} \Big( \displaystyle \sum_{j , wt(j)=w} \abs*{p_j-\frac{1}{{n \choose w}}} + \displaystyle \sum_{j , wt(j) \neq w} p_j \Big)$$  An $EM$ value of $0$ signifies that the measurement statistics are exactly aligned with the expected ideal results while the $EM$ value can at maximum be $1$. We have calculated the $EM$ values for results of different mappings for the circuit ${\mathcal{C}}_{4,2}$. It is very interesting to see that under different mapping of logical qubits to physical qubits in $Q_1$ from the user end the IQX mapping solution provided different transpiled circuits. We know that the number of CNOT in ${\mathcal{C}}_{4,2}$ is $22$. The transpiled circuits for ${\mathcal{C}}_{4,2}$ had a minimum of $25$ CNOT gates and were as high as $31$ in some cases. The corresponding transpiled circuit contains $25$ CNOT gates which is the least of all the transpiled circuits. Figure \[fig:D42basic\] shows the measurement statistics corresponding to the circuit with the minimum $EM$ value, which is equal to $0.4088$.  Next we look at the measurement statistics of the circuit ${\mathcal{\widehat{C}}}_{4,2}$. There are many mappings between logical and physical qubits in this case such that the CNOT constraint of the circuit is met. Let such a map be $M:\{q_0,q_2,q_3,q_4\} \rightarrow \{0,1,2,3,4\}$. Then if there is a CNOT between $q_i$ and $q_j$ then there is an edge $M(q_i) \leftrightarrow M(q_j)$ in graph $G^{Q_1}_A$. In such mappings the IQX mapping solution didn’t implement any modification in the transpiled circuit as expected. Here we present the result for the following map $M_1$ $$\begin{aligned} M1:\quad & q_0 \rightarrow 3,\quad q_1 \rightarrow 2,\quad q_2 \rightarrow 4,\quad q_3 \rightarrow 0.\end{aligned}$$ Figure \[fig:D42opt\] shows the measurement statistics corresponding to this mapping and the resultant $EM$ value is $0.282103$. These results show that the circuit ${\mathcal{C}}_{4,2}$ needs more than the specified number of CNOT while being implemented on “ibmqx2” and the measurement statistics of ${\mathcal{\widehat{C}}}_{4,2}$ is much more closely aligned with the ideal measurement statistics compared to ${\mathcal{C}}_{4,2}$. Modifications leading to different CNOT error distributions {#subsec:1} ----------------------------------------------------------- We now discuss how we can in fact use partially defined transformations to further fine tune the circuit ${\mathcal{\widehat{C}}}_{4,2}$ depending on the specifications of the architecture. We consider a four qubit architecture $A_4$ with the same CNOT connectivity as $G^{{\mathcal{\widehat{C}}}_{4,2}}$ and only differs in CNOT error distribution. We then observe how further modifying the circuit ${\mathcal{\widehat{C}}}_{4,2}$ can lead to lower CNOT error on expectation against some error distributions in the architecture $A_4$. We assume every edge in the CNOT map of $A_4$ is bidirectional, as is the case with all currently publicly available IBM-Q machines. The CNOT error rate when applying a CNOT between qubits $i$ and $j$ (such that the edge $i \leftrightarrow j$ is present in $G_A$) is denoted as $e_{ij}$.Figure \[fig:arch4\] shows the CNOT map of the architecture. ### CNOT error model {#cnot-error-model .unnumbered} In this regard we define our error model to calculate CNOT error on expectation of a circuit implemented in the architecture $A_4$. The probability of a CNOT placed between qubits $i$ and $j$ acting erroneously in a circuit is dependent on the error rate of the corresponding CNOT coupling in the architecture. We call this CNOT error. We denote this probability with $f_e:[0,1] \rightarrow [0,1]$. We do not assume the exact nature of $f_e$, but only that it is directly proportional to error rate (i.e. an increasing function) which is by definition. Next we define the following Bernoulli random variables to calculate the the number of CNOT acting erroneously on expectation when a circuit is applied on this architecture. We define a variable $x_k$ corresponding to each CNOT used in a circuit. The variable is assigned zero if the $k$-th CNOT is applied correctly while executing a circuit, and one otherwise. Let us suppose the $k$-th CNOT is applied between qubits $i$ and $j$. Then we have $Pr(x_k=1)=f_e(e_{ij})$ and The expected error while applying the CNOT is $\mathbb{E}(x_k)=f_e(e_{ij})$. Therefore the CNOT error on expectation while implementing a circuit ${\mathcal{C}}$ on the architecture is $$\mathbb{E}({\mathcal{C}})=\displaystyle \sum v_{ij} f_e(e_{ij}).$$ Having described the error model we look at the CNOT distribution of the circuit ${\mathcal{\widehat{C}}}_{4,2}$ as a weighted graph $G_f$. The vertices and the edges of this graph is same as that of $G^{{\mathcal{\widehat{C}}}_{4,2}}$. The weight of an edge $q_i \leftrightarrow q_j$ is the number of CNOT gates applied between the two qubits in the circuit. The graph $G_f$ is shown in Figure \[fig:cmapdis\]. Now we implement the circuit ${\mathcal{\widehat{C}}}_{4,2}$ on the architecture $A_4$ so that all CNOT constraints can be met. We observe that only the qubit $1$ has degree $3$ and therefore $q_1$ is mapped to the physical qubit $1$. Then we can have the following maps which satisfies all the CNOT constraints. 1. $q_1\rightarrow 1$, $q_0\rightarrow 0$, $q_2\rightarrow 2$, $q_3\rightarrow 3$. 2. $q_1\rightarrow 1$, $q_0\rightarrow 2$, $q_2\rightarrow 0$, $q_3\rightarrow 3$. Then the expected CNOT error of the circuit ${\mathcal{\widehat{C}}}_{4,2}$ when applied on the architecture $A_4$ is $$\mathbb{E}({\mathcal{\widehat{C}}}_{4,2})= 5f_e(e_{01})+3f_e(e_{02})+3f_e(e_{12})+ f_e(e_{13}).$$ We now show the circuit ${\mathcal{\widehat{C}}}_{4,2}$ (described in Figure \[fig:cir42\_3\]) can be further modified using partially defined transformations so that the CNOT error in the circuit on this architecture will reduce on expectation under some error distribution conditions. \(A) [${q_0}$]{}; (B) \[right=of A\] [${q_1}$]{}; (C) \[below=of A\] [$q_2$]{}; (D) \[below=of B\] [$q_3$]{}; \(A) edge (B);(B) edge node\[above\] [$5$]{} (A); (B) edge (C);(C) edge node\[left\] [$3$]{} (B); (D) edge (B);(B) edge node\[right\] [$1$]{} (D); (A) edge (C);(C) edge node\[left\] [$3$]{} (A); \(A) [$0$]{}; (B) \[right=of A\] [$1$]{}; (C) \[below=of A\] [$2$]{}; (D) \[below=of B\] [$3$]{}; \(A) edge (B);(B) edge node\[above\] [$e_{01}$]{} (A); (B) edge (C);(C) edge node\[left\] [$e_{12}$]{} (B); (D) edge (B);(B) edge node\[right\] [$e_{13}$]{} (D); (A) edge (C);(C) edge node\[left\] [$e_{02}$]{} (A); \(A) [${q_0}$]{}; (B) \[right=of A\] [${q_1}$]{}; (C) \[below=of A\] [$q_2$]{}; (D) \[below=of B\] [$q_3$]{}; \(A) edge (B);(B) edge node\[above\] [$5$]{} (A); (B) edge (C);(C) edge node\[left\] [$3$]{} (B); (C) edge (D);(D) edge node\[above\] [$1$]{} (C); (A) edge (C);(C) edge node\[left\] [$3$]{}(A); @C=0.2em @R=1em [ & & & && & & && & & & && & & & & && & && && && & & &&&& & &&\ & & & && & & && & & & && & & && & & & & & & & & & & &&& && && &\ & & & && & & && & & & & & && & &&&&& &&&& &&& & & & & & &&\ & & & && & & && & & & & & & & & & & & & & & & & & & & & & & & & &&&\ ]{} The first $R_y$ gate that acts on the second qubit of ${\mathcal{\widehat{C}}}_{4,2}$ is followed by the CNOT gates ${\sf CNOT}^2_3$ and ${\sf CNOT}^2_4$. The combined transformation $T_4$ of these two CNOT is defined only for two basis states on $4$ qubits $\ket{0011} \rightarrow \ket{0011}$ and $\ket{0111} \rightarrow \ket{0100}$. We use the partial nature of the transformation to modify the circuit as follows. Note that transformation $T_4$ is the first transformation that acts on $q_3$. Then if we start the circuit from the state $\ket{0010}$ instead of $\ket{0011}$ then we can define the transformation $T_4^1$ such that $$\begin{aligned} & T_4^1 \equiv (I_2 \otimes {\sf CNOT}^3_2 \otimes I_2) (I_2 \otimes I_2 \otimes {\sf CNOT}^2_1) \\ \implies & T_4^1\ket{0010}=\ket{0011},~~ T_4^1\ket{0110}=\ket{0100}\end{aligned}$$ resulting in the same output states as $T_4$ for all the computational basis states for which $T_4$ is defined. It is important to note that this implementation would not have been possible if the transformation was defined for all the $8$ basis states of the second third and fourth qubits. We denote this circuit as ${\mathcal{\widehat{C}}}_{4,2}'$ and it is drawn in Figure \[fig:cir42\_4\]. We denote the weighted CNOT map of the circuit ${\mathcal{\widehat{C}}}_{4,2}'$ as $G_f'$ and it is shown in Figure \[fig:cmapmod2\]. We now see that in $G_f'$ $q_2$ has degree three and therefore any map that meets all the CNOT constraints will have $q_2 \rightarrow 1$. Therefore we can have the following maps that satisfies all the CNOT constraints. 1. $q_2 \rightarrow 1,~~ q_0 \rightarrow 0,~~q_1 \rightarrow 2,~~q_3 \rightarrow 3 $. 2. $q_2 \rightarrow 1,~~ q_0 \rightarrow 2,~~q_1 \rightarrow 0,~~q_3 \rightarrow 3 $. The CNOT error on expectation for both the circuits is $$\mathbb{E}({\mathcal{\widehat{C}}}_{4,2}')= 5f_e(e_{02})+3f_e(e_{01})+3f_e(e_{12})+ f_e(e_{13}).$$ Now we calculate the conditions when $\mathbb{E}({\mathcal{\widehat{C}}}_{4,2}')$ is less than $\mathbb{E}({\mathcal{\widehat{C}}}_{4,2}')$. $$\begin{aligned} & \mathbb{E}({\mathcal{\widehat{C}}}_{4,2}')<\mathbb{E}({\mathcal{\widehat{C}}}_{4,2}) \\ \implies & 5f_e(e_{02})+3f_e(e_{01})+3f_e(e_{12})+ f_e(e_{13}) \\ &< 5f_e(e_{01})+3f_e(e_{02})+3f_e(e_{12})+ f_e(e_{13}) \\ \implies & f_e(e_{02})< f_e(e_{01}) \\ \implies & e_{02} < e_{01}.\end{aligned}$$ This gives us an insight into how different CNOT distributions in a circuit may lead to better results without reduction in the number of CNOT gates or a reduction in the architectural constraints. We conclude this section by describing the architectural constraint of the circuit ${\mathcal{\widehat{C}}}_{n,k}$. The CNOT map of ${\mathcal{\widehat{C}}}_{n,k}$ ----------------------------------------------- The CNOT gates in the circuit ${\mathcal{\widehat{C}}}_{n,k}$ are due to implementation of the $\mu$ and ${\mathcal{M}}$ transformation of the different $SCS^n_k$ blocks. $\mu_n$ forms an edge in the CNOT map of the form $n-1 \leftrightarrow n$. where as ${\mathcal{M}}^l_n$ forms the edges $(l-1) \leftrightarrow n$ and $l \rightarrow (l-1)$. However in the circuit ${\mathcal{\widehat{C}}}_{n,k}$ the transformations ${\mathcal{M}}^{n-k+1}_t$ do not have a CNOT between the neighboring qubits $n-k$ and $n-k+1$. We divide the edges into two groups. One corresponding to CNOT gates between neighboring qubits and one where the positions of the qubits differ at least by two. We calculate the edges of each of these types. - The neighboring qubits with CNOT connections are the qubits $(n-k+1-i)$ and $(n-k-i)$ where $i$ varies from $0$ to $n-k-1$. The other neighboring qubits do not have CNOT connections due to removal of identity transformations form those qubits. This results in $n-k$ edges. - Now we consider the second kind of connections. These connections are formed between $l-1$ and $t$ th qubit for any ${\mathcal{M}}^l_t$ transformation. There are $n-k$ $SCS^n_K$ blocks with originally $k-1$ ${\mathcal{M}}$ transformations in ${\mathcal{C}}_{n,k}$ which forms the edges: $$(n-t) \leftrightarrow (n-t-2-i),~~0 \leq i \leq k-2, 0 \leq t \leq n-k-1.$$ Then there are $k-1$ blocks of $SCS^{i+1}_{i}$ with $i-1$ ${\mathcal{M}}$ transformations which forms the edges: $$(k-t) \leftrightarrow (k-t-2-i),~~ 0 \leq i \leq k-t-2, 0 \leq t \leq k-2.$$ However in ${\mathcal{\widehat{C}}}_{n,k}$ there are no ${\mathcal{M}}$ transformations of the type ${\mathcal{M}}^{n-k+1+x}_y,~x>0$. Removing such edges $n-k+x \leftrightarrow y$ gives us the complete description of the CNOT map of ${\mathcal{\widehat{C}}}_{n,k}$, which we denote by $G^{n,k}$. Additionally, in the transformation ${\mathcal{M}}^{n-k+1}_n$ the edge $n \rightarrow (n-k)$ is not present. Figure \[fig:cg62\] and \[fig:cg63\] show the CNOT maps $G^{6,2}$ and $G^{6,3}$ respectively. \(A) [$1$]{}; (B)\[below right=of A\] [${2}$]{}; (C) \[below=of B\] [${3}$]{}; (D) \[below left=of C\] [$4$]{}; (E) \[above left=of D\] [$5$]{}; (F) \[above=of E\] [${6}$]{}; \(A) edge (B);(B) edge (A); (B) edge (C);(C) edge (B); (C) edge (D);(D) edge (C); (D) edge (E);(E) edge (D); (D) edge (F); (C) edge (E);(E) edge (C); (D) edge (B);(B) edge (D); (A) edge (C);(C) edge (A); \(A) [$1$]{}; (B)\[below right=of A\] [${2}$]{}; (C) \[below=of B\] [${3}$]{}; (D) \[below left=of C\] [$4$]{}; (E) \[above left=of D\] [$5$]{}; (F) \[above=of E\] [${6}$]{}; \(A) edge (B);(B) edge (A); (B) edge (C);(C) edge (B); (C) edge (D);(D) edge (C); (C) edge (F); (C) edge (E);(E) edge (C); (B) edge (E);(E) edge (B); (D) edge (B);(B) edge (D); (D) edge (A);(A) edge (D); (A) edge (C);(C) edge (A); We now count the number of edges in $G^{n,k}$. The number of edges present due to ${\mathcal{M}}$ transformations is $nk-\frac{k(k+1)}{2}-n+1-\frac{(k-1)(k-2)}{2}=nk-n-k^2+k$. There are further $n-k$ edges due to the $\mu$ transformations. Which brings the total number of edges in $G^{n,k}$ to $nk-k^2$. It is important to note that although the number of edges in $G^{n,k}$ and $G^{n,n-k}$ are same they are not isomorphic. Moreover the Graph $G^{n,i}$ is not a subgraph of $G^{n,i+1}$. Finally we observe that the circuit ${\mathcal{\widehat{C}}}_{n,k}$ can be modified so that the number of CNOT gates between the qubits change for certain cases, although the total number of CNOT gates and the overall CNOT map does not change. We call these different instances as different CNOT distributions of ${\mathcal{\widehat{C}}}_{n,k}$. Different CNOT distributions for ${\mathcal{\widehat{C}}}_{n,k}$ {#different-cnot-distributions-for-mathcalwidehatc_nk .unnumbered} ---------------------------------------------------------------- We know from the description of ${\mathcal{C}}_{n,k}$ [@dicke] that the number of CNOT gates in the three qubit transformation ${\mathcal{M}}$ is reduced from $6$ to $5$ by canceling the last CNOT of every transformation by rearranging the first two CNOT gates of the next transformation. Figure \[fig:mod11\] shows the original layout as per the algorithm and Figure \[fig:mod12\] shows the reduction due to [@dicke]. Now let us consider the last transformation ($k$-th) of each $SCS^{n-i}_k,0 < i<n-k$ block, ${\mathcal{M}}^{n-i-k+1}_{n-i}$. This transformation acts on the qubits $n-i-k, n-i-k+1$ and $(n-i)$. This is in fact the first transformation that affects the qubit $(n-i-k)$ and thus the qubit is in the state $\ket{0}$. If we do not cancel the last CNOT of the preceding ${\mathcal{M}}$ transformation ($CNOT^{n-i-k+1}_{n-i}$) this then enables us to remove of the CNOT gate ($CNOT^{n-i-k}_{n-i}$), changing the CNOT distribution of the circuit without a change in CNOT map or number of CNOT gates. This leads to the implementation shown in Figure \[fig:mod13\]. Since there are $n-k-1$ such transformations, this leads to a total of $2^{n-k-1}$ different CNOT distributions. However, these modifications do not alter the CNOT map of the circuit due to the fact that there are other CNOT gates applied between these qubits, which is evident from the circuit description. As we have observed in Section \[subsec:1\] such different CNOT distributions may lead to different number of CNOT gates acting erroneously on expectation and thus affect the overall error induced in the circuit. @C=1em @R=1em [ &n-k-i&&&& & & & &\ &n-k-i+1&&&& & & & &\ &n-i&&&& & & & &\ ]{}  \ @C=1em @R=1em [ & n-k-i &&&& && &\ & n-k-i+1&&&& & &&\ & n-i &&&& &&&\ ]{}  \ @C=1em @R=1em [ &n-k-i&&&& & & &\ &n-k-i+1&&&& & & &\ &n-i&&&& & & &\ ]{} Conclusion {#sec:6} ========== In this paper we have explored the domain of optimal circuit implementation in terms of CNOT and single qubit gates. In this regard we have concisely realized partially defined unitary transformations to improve the gate count of the most optimal deterministic Dicke state ($\ket{D^n_k}$) preparation circuit (${\mathcal{C}}_{n,k}$). We have improved the implementation of one such transformation and have also proven the optimality of our implementation. We have further improved the Dicke State preparation circuit by removing redundant gates and modifying implementations of certain partially defined unitary transformations depending on the active basis states that that act as input to these transformations. We have then shown that these improvements not only reduce the number of CNOT and single qubit gates but also reduces the architectural constraints of the circuit using the case of $\ket{D^4_2}$. The resultant circuit is the deterministic Dicke State ($\ket{D^n_k},~2 \leq k \leq n-1$) Preparation Circuit with the least number of elementary gates to the best of our knowledge. We have implemented the circuits ${\mathcal{C}}_{4,2}$ and the improved circuit ${\mathcal{\widehat{C}}}_{4,2}$ on the IBM-Q machine “ibmqx2” and observed that the deviation from ideal measurement statistics is significantly lesser in case of ${\mathcal{\widehat{C}}}_{4,2}$. Furthermore, we have shown that how different CNOT distributions can help a circuit without changing the number of gates or the architectural constraints by comparing the expected CNOT error of two such distributions against a fairly generalized error model. We have concluded by describing the CNOT map of the circuit ${\mathcal{\widehat{C}}}_{n,k}$ and observe the exponential number of different CNOT distributions that can be derived by modifying the circuit to complete our generalization. We observed that even the circuits for $\ket{D^5_2}$ could not be implemented in the IBM back end machines without adding further CNOT gates to our description. This is because of incompatibility of the architecture and circuit CNOT maps. Therefore it is of all the more importance to form the circuit for an algorithm in the most concise way possible. Against this backdrop we have shown how optimally realizing partially defined unitary transformations can lead to better implementation results. In conclusion we note down the following optimization problems that will help us implement algorithms more efficiently in the current scenario. 1. Given a maximally partial unitary transformation what is the corresponding unitary matrix that can be decomposed using the least number of elementary gates? 2. Given two circuits corresponding to an algorithm with isomorphic CNOT maps and the same number of CNOT gates, but different CNOT distribution across the qubits, which circuit will produce less erroneous outcome? [9]{} A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. W. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter. 1995. Elementary gates for quantum computation. Phys. Rev. A, 52:3457-3467. A. B[ä]{}rtschi and S. Eidenbenz. 2019. Deterministic Preparation of Dicke States. Fundamentals of Computation Theory, 126-139. K. Chakraborty, B. Choi, A. Maitra and S. Maitra. 2014. Efficient quantum algorithms to construct arbitrary Dicke states. Quantum Inf Process 13, 2049–2069. A. M. Childs, E. Farhi, J. Goldstone, and S. Gutmann. 2002. Finding cliques by quantum adiabatic evolution. Quantum Information & Computation, 2(3):181–191, Apr 2002. D. Cruz, R. Fournier, F. Gremion, A. Jeannerot, K. Komagata, T. Tosic, J. Thiesbrummel, C.L. Chan, N. Macris,M.-A. Dupertuis and C. Javerzac‐Galy. 2019. Efficient Quantum Algorithms for GHZ and W States, and Implementation on the IBM Quantum Computer. Adv. Quantum Technol., 2: 1900015. G. Song and A. Klappenecker. 2003. Optimal realizations of controlled unitary gates. Quantum Info. Comput. 3, 2 (March 2003), 139–156. B. Langenberg, H. Pham and R. Steinwandt. 2020. Reducing the Cost of Implementing the Advanced Encryption Standard as a Quantum Circuit. IEEE Transactions on Quantum Engineering, vol. 1, pp. 1-12, 2020, 2500112. M. Mosca and P. Kaye. 2001. Quantum Networks for Generating Arbitrary Quantum States. Optical Fiber Communication Conference and International Conference on Quantum Information ICQI, page PB28, Jun 2001. M. Möttönen, J. J. Vartiainen, V. Bergholm, and M. M. Salomaa. 2004. Quantum Circuits for General Multiqubit Gates. Phys. Rev. Lett, 93, 130502. A. Zulehner,A. Paler, and R. Wille. 2019. An Efficient Methodology for Mapping Quantum Circuits to the IBM QX Architectures. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 38, 1226-1236. IBM Q Experience Website, <https://quantum-computing.ibm.com>

--- abstract: 'We present joint multi-dimension pruning (named as JointPruning), a new perspective of pruning a network on three crucial aspects: spatial, depth and channel simultaneously. The joint strategy enables to search a better status than previous studies that focused on individual dimension solely, as our method is optimized collaboratively across the three dimensions in a single end-to-end training. Moreover, each dimension that we consider can promote to get better performance through colluding with the other two. Our method is realized by the adapted stochastic gradient estimation. Extensive experiments on large-scale ImageNet dataset across a variety of network architectures MobileNet V1&V2 and ResNet demonstrate the effectiveness of our proposed method. For instance, we achieve significant margins of 2.5% and 2.6% improvement over the state-of-the-art approach on the already compact MobileNet V1&V2 under an extremely large compression ratio.' author: - | Zechun Liu[^1]\ HKUST\ `zliubq@connect.ust.hk`\ Xiangyu Zhang\ MEGVII Technology\ `zhangxiangyu@megvii.com`\ Zhiqiang Shen$^{\dag}$\ CMU\ `zhiqians@andrew.cmu.edu`\ Zhe Li              Yichen Wei\ MEGVII Technology\ `{lizhe,weiyichen}@megvii.com`\ Kwang-Ting Cheng\ HKUST\ `timcheng@ust.hk`\ Jian Sun\ MEGVII Technology\ `sunjian@megvii.com`\ bibliography: - 'my.bib' title: 'Joint Multi-Dimension Pruning' --- Introduction ============ Network pruning has been acknowledged as one of the most effective model compression methods for adapting heavy models to the resource-limited devices [@he2017channelpruning; @slimmable; @networkslimming]. The pruning methods evolved from unstructured weight pruning [@guo2016dynamic; @han2015learning] to structured channel pruning [@ding2019centripetal; @zhuang2018discrimination] with the purpose of more hardware-friendly implementation. With the increasing demand of highly compressed models, merely pruning the channel dimension becomes insufficient to strike a good computation-accuracy trade-off. Some previous studies begin to seek compression schemes besides the channel dimension. For example, OctaveConv [@chen2019drop] proposed to manually reduce the spatial redundancy in feature maps. MobileNet [@howard2017mobilenets] scales down the input image’s spatial size for reducing the computational overhead. Besides the spatial dimension, depth is also a dimension worth exploring [@crowley2018closer]. These three dimensions (i.e., channel, spatial, depth) are inter-related: A reduction in the network depth can enable using larger feature maps or more channels under the same computational constraint; An alteration in the spatial size will affect the optimal channel pruning scheme. To this end, manually set the spatial size and network depth while only adjust the layer-wise channel pruning strategy leads to sub-optimal solutions. It is of high significance to automatically allocate computational resource among three dimensions with a unified consideration. However, jointly pruning channels together with the spatial and depth dimension is challenging and some difficulties are always underestimated. The first challenge is that most of the pruning algorithms are developed based on the unique property of the channel dimension, for instance, removing channels with small weight L1 norms [@networkslimming; @yang2018netadapt]. Such approaches are inflexible and can hardly generalize to other dimensions, especially the spatial dimension. The second challenge stems from the overladen search space within the three integrated dimensions, since the potential choices of these dimensions are numerically consecutive integers. Therefore, the previous enumerate-based methods are always computationally prohibitive or under-optimized. To resolve the aforementioned limitations and derive a decent solution, we propose the JointPruning that considers three dimensions (channel, spatial, depth) simultaneously. In contrast to previous pruning methods that mainly focus on exploring the correlation between channels (e.g., the redundancy), we formulate this problem by defining channel numbers associating with the spatial and depth into an integrated space through a configuration vector. Within this joint space, we can not only mine the relationships inside a single dimension, but also find relatively optimal options across three dimensions by optimizing this configuration vector. That is to say, we consider the global reciprocity between the three aspects and model joint pruning with the defined pruning configuration vector specifying layer-wise channel numbers, spatial size, and the network depth. This circumvents the algorithmic dependency on the exclusive properties in the channel dimension. As there is no explicit function between the loss function of the pruned network and its configuration vector ([*i.e.*]{}, it is non-differentiable to the pruning configuration), also considering the efficiency of optimization, we propose to use stochastic gradient estimation to approximate the gradient *w.r.t.* configuration vectors. We first convert the loss function to an expectation of the loss of configuration vectors sampled from a Gaussian distribution around the original vector. Then we utilize the [*log likelihood trick*]{} [@sutton2000policy] to obtain the gradient of expectation as an approximation to the underlying gradient. We further design weight sharing mechanism for evaluating the loss of configuration vectors without training each corresponding pruned network from scratch. With alternatively updating weights and configuration vector, the proposed JointPruning can automatically learn the global pruning configuration across three dimensions. We show the effectiveness of JointPruning on MobileNet V1/V2 and ResNet structures. It achieves up to $9.1\%/6.3\%$ higher accuracy than MobileNet V1/V2 baselines under FLOPs constraint. Moreover, JointPruning is friendly to tackle channels in shortcuts and also supports multiple resource constraints. It can automatically adapt the pruning scheme according to the underlying hardware specialty and different circumstances due to its larger potential options. Under CPU latency constraint, JointPruning achieves comparable or higher results with less searching time than advanced adaptation based ChamNet [@dai2018chamnet]. When targeting at GPU latency, it surpasses MobileNet V1/V2 baselines with more than 3.9% and 2.4% improvement under an extremely large compression ratio. Contributions: $\bullet$ We reveal the perspective that joint pruning neural network in three dimensions (*i.e.*, channel, spatial and depth) is crucial for striking a good accuracy-computation trade-off. We define pruning configuration as a vector specifying the spatial size, network depth and layer-wise channel numbers, and formulate joint pruning as optimizing the configuration vector. $\bullet$ We propose to use stochastic gradient estimation *w.r.t.* the configuration vector for efficient optimization. This strategy inherits the high efficiency advantage of the gradient descent algorithm meanwhile overcomes the non-differentiability in the configuration vectors. $\bullet$ We design weight sharing strategy to avoid training individual pruned network from scratch, and an alternative updating strategy to jointly optimize weights and configuration vectors, which further facilitates optimization. $\bullet$ The proposed JointPruning can flexibly incorporate different resource constraints including FLOPs and hardware latency. It outperforms traditional pruning algorithms with less human efforts and achieved higher results than state-of-the-art model adaptation methods with less optimization time. Related Work ============ Model compression is recognized as an effective approach for efficient deep learning [@han2015deep; @wu2019efficient]. The categories of model compression expands from pruning [@he2017channelpruning; @chin2019legr; @huang2018data], quantization [@zhuang2019effective; @zhuang2019training], to compact network architecture design [@ma2018shufflenet; @zhang2018shufflenet; @howard2017mobilenets; @sandler2018mobilenetv2]. Our approach is most related to pruning. **Pruning and Model Adaptation** Early works [@lecun1990optimal; @hassibi1993optimal; @guo2016dynamic] prune individual redundant weights and recent studies focus on removing the entire kernel [@networkslimming; @luo2017thinet] to produce structured pruned networks. While traditional pruning involves massive human participation in determining the pruning ratio, AutoML-based methods with reinforcement learning [@he2018amc], a feedback loop [@yang2018netadapt] or a meta network [@liu2019metapruning] can automatically decide the best pruning ratio. In those pruning methods, the optimization space is confined to channels only. Intuitively, a higher compression ratio and better trade-off can be obtained through jointly reducing channel, spatial and depth dimensions. So far, only a few works address the joint adaptation across three dimensions. EfficientNet [@tan2019efficientnet] focuses on scales all dimensions of depth/width/resolution with a single compound coefficient. The coefficient is obtained by grid search. ChamNet [@dai2018chamnet] uses the Gaussian process to predict the accuracy of compression configurations in three dimensions. Despite their high accuracy, grid search or Gaussian process requires training hundreds of networks from scratch, which are with high computational cost. Instead, we propose to model the joint pruning as optimizing the numerical values of architecture configurations (i.e., channel, depth and spatial), which enables to use a *stochastic gradient estimation* *w.r.t.* the configuration vector in optimization, and further improves the efficiency greatly. **Gradient Estimation** Gradient estimation is initially proposed for optimizing non-differentiable objectives [@nesterov2017random]. Reinforcement learning (RL) algorithms often utilize gradient estimation to update the policy [@williams1992simple; @o2016combining; @gu2017interpolated; @lillicrap2015continuous; @beyer2002evolution; @salimans2017evolution; @back1991survey]. We customize a relaxation of the objective function and use the [*log likelihood trick*]{} to calculate the estimated gradient [@silver2014deterministic]. Different from these algorithms designed for the typical RL tasks, such as Atari and MuJoCo, our approach focuses on optimizing the network configuration and assumes the policy function to be a Gaussian distribution, which can better approximate the actual gradient and is free of hyper-parameters. Furthermore, we utilize the Lipchitz continuous property of the neural networks *w.r.t.* the configuration vectors. Such that we adopt a progressively shrinking Gaussian window for obtaining more accurate gradient estimations, and use an alternative paradigm for updating configuration vectors and the weight parameters. **Neural Architecture Search and Parameter Sharing** Different from pruning tasks, which aims to find a compressed model through adjusting the numerical values in architecture configurations, NAS are targeting at choosing the best options [@zoph2018learning; @real2018regularized; @tan2018mnasnet; @xie2018snas] and/or connections [@xie2019exploring]. Although a few NAS studies include two or three channel number choices in their search space [@wu2018fbnet; @cai2018proxylessnas], they enumerate these choice as independent nodes. The gradient-based neural architecture search (NAS) adopts Straight Through Estimation [@cai2018proxylessnas] or probabilistic relaxation [@liu2018darts], which models choices as independent and is essentially unsuitable for continuous numerical choices in the pruning configuration. While proposed JointPruning tailored for this task utilizes the consecutiveness in the numerical values, and is thus more effective. Parameter sharing is studied in NAS. Our parameter-sharing scheme is inspired by the practice in one-shot architecture search [@guo2019single; @stamoulis2019single; @bender2018understanding; @cai2019once]. Different from the operation-level sharing mechanism designed for independent operations, proposed weight sharing adopts a matrix-level sharing, which is specialized for a more efficient sharing among the consecutive channel/spatial/depth choices. The details of our algorithm are described in Sec. \[sec:method\]. Methodology {#sec:method} =========== To solve the pruning problem in three dimensions, we define the pruned networks with the corresponding unique architecture configuration vectors: $$\begin{aligned} v_{conf} = \{c,s,d\},\end{aligned}$$ where $c$ denotes the number of channels in each layer, c = $\{c_1,c_2,...,c_n\}$, $n$ is the total number of layers, $s$ denotes the spatial size of the input image to the network and $d$ denotes the network depth. We formulate the JointPruning as a constrained optimization problem over the configuration vector. The objective is to minimize the loss under a given resource constraint: $$\begin{aligned} & {\rm minimize} \ \mathcal{L}(v,w) \\ & {\rm subject \ to} \ \ \mathcal{C} (v) < constraint\end{aligned}$$ where $\mathcal{L}$ refers to the loss function of the neural network, $w$ is the weight parameters, $\mathcal{C}$ is the computational cost given the network configuration $v$ and $constraint$ denotes the target resource constraint (i.e., FLOPs or latency). We merge the resource constraint as a regularization term in the error function $\mathcal{E}$ as follows: $$\begin{aligned} \mathcal{E}(v,w) = \mathcal{L}(v,w) + \rho ||\mathcal{C}(v)-constraint||^2,\end{aligned}$$ where $\rho$ is the positive regularization coefficient. Consequently, the goal is to find the architecture configuration vector $v$ that minimizes the error function $\mathcal{E}$: $$\begin{aligned} v = \mathop{argmin}_{v}(\mathcal{E})\end{aligned}$$ However, it is computationally prohibitive to obtain the error $\mathcal{E}$ through enumerating the configuration vectors and training each corresponding pruned network from scratch. To tackle this, we propose the gradient estimation method to approximate the steepest gradient descent direction of the configuration vector that minimizes the error, and also customize a weight sharing mechanism to further enhance the optimization efficiency. We detail our gradient estimation method and the weight sharing mechanism in Section \[sec:gradient\_estimation\] and Section \[sec:weight\_sharing\], respectively. Stochastic Gradient Estimation for Configuration Vectors {#sec:gradient_estimation} -------------------------------------------------------- Since the error function $\mathcal{E}$ is naturally non-differentiable with respect to the configuration vectors, we propose gradient estimation to approximate the underlying gradients. Inspired by [@rechenberg1994evolutionsstrategie; @yi2009stochastic; @sehnke2010parameter], we conduct the gradient estimation by relaxing the objective error function $\mathcal{E}$ to the expectation of $\mathcal{E}$ with configuration vector $v$ obeying a distribution $p_{\theta}(v)$ . $$\begin{aligned} \mathcal{E}(v) \approx \mathbb{E}_{v \sim p_{\theta}(v)}\mathcal{E}(v),\end{aligned}$$ where the distribution $p_{\theta}(v)$ is parameterized by $\theta$. To simplify the expression, we omit the $w$ in the term $\mathcal{E}(v,w)$. In previous works, $p_{\theta}(v)$ is chosen to be a function of $v$. Instead, and considering the relaxation purpose is for obtaining precise gradient estimation *w.r.t.* the configuration vector $v$, we simplify the probability distribution $p_{\theta}(v)$ to be an isotropic multivariate gaussian distribution: $$\begin{aligned} p_{\mu}(v)\sim \mathcal{N}(\mu,\sigma),\end{aligned}$$ with the mean $\mu$ being the current configuration vector and the deviation set to $\sigma$. Consequently, the approximation to the objective function can be written as: $$\begin{aligned} \mathcal{E}(\mathbf{\mu}) \approx \mathbb{E}_{v \sim \mathcal{N}(\mu,\sigma)}\mathcal{E}(v) = \mathbb{E}_{n \sim \mathcal{N}(0,\sigma)}\mathcal{E}(\mu+n),\end{aligned}$$ where $n$ denotes the random multidimensional Gaussian noise added to the current configuration vector. This approximation holds because the neural network is *Lipschitz* continuous with respect to $v$, i.e., there exists a positive real constant $K$ such that: $$\begin{aligned} ||\mathcal{E}(\mu+n) - \mathcal{E}(\mu)|| < K||n||.\end{aligned}$$ In configuration vector optimization, $n$ is confined to small variation values, i.e., $||n|| < \epsilon$. Therefore, the alteration in the error function is bounded within $K\epsilon$. In experiments, we observe the constant $K$ to be diminutive within a local region around the current configuration vector and thus the variation is tolerable. With this approximation to the error function, we can derive the estimated gradient following the commonly adopted [*log likelihood trick*]{} in reinforcement learning: $$\begin{aligned} \nabla_{\mu} \mathbb{E}_{v \sim \mathcal{N}(\mu,\sigma)}\mathcal{E}(v) & = \mathbb{E}_{v \sim \mathcal{N}(\mu,\sigma)} [\mathcal{E}(v) \nabla_{\mu} \log(p(v;\mu))] \\ & = \mathbb{E}_{v \sim \mathcal{N}(\mu,\sigma)} [\mathcal{E}(v) \nabla_{\mu} (\log(\frac{1}{\sigma \sqrt{2\pi}}e^{-\frac{1}{2}(\frac{v-\mu}{\sigma})^2}))]\\ %& = \mathbb{E}_{v \sim \mathcal{N}(\mu,\sigma)} [\mathcal{E}(v) \nabla_{\mu} (-\frac{1}{2}(\frac{v-\mu}{\sigma})^2)] \\ & = \mathbb{E}_{v \sim \mathcal{N}(\mu,\sigma)} [\mathcal{E}(v)\frac{v-\mu}{\sigma^2}] \\ & = \frac{1}{\sigma^2} \mathbb{E}_{n \sim \mathcal{N}(0,\sigma)} [\mathcal{E}(\mu+n)n].\end{aligned}$$ Thus, the gradient of $\mathcal{E}(v)$ can be estimated by the configuration vector $v$ stochastically sampled around the current vector $\mu$ under a gaussian distribution $n\sim \mathcal{N}(0,\sigma)$: $$\begin{aligned} %\nabla_{v} \mathcal{E}(v) \approx \nabla_{\mu} \mathbb{E}_{v \sim \mathcal{N}(\mu,\sigma)}\mathcal{E}(v) \approx \frac{1}{\sigma^2} \sum_{i=1}^M \mathcal{E}(\mu+n_i)n_i,\end{aligned}$$ where $M$ is the total number of samples. An intuitive interpretation of this gradient estimation approach is: by weighting the variation directions with the corresponding error, the gradient direction can be approximated with the variation direction that has the lower expected error. Then the configuration vector is updated with the estimated gradient: $$\begin{aligned} \mu' = \mu - \alpha \nabla_{\mu} \mathcal{E}(v),\end{aligned}$$ where $\alpha$ denotes the update rate, and the geometric meaning of $\mu$ is the center configuration vector for adding Gaussian variations. The resulting algorithm iteratively executes the following two steps: (1) stochastic sampling of the architecture configuration vectors and evaluate the corresponding errors; (2) integrating the evaluations to estimate the gradient and update the configuration vector. Weight Sharing {#sec:weight_sharing} -------------- In the aforementioned stochastic gradient estimation paradigm, one crucial issue that remains unsolved is how to obtain the error evaluation of the configuration vector. Recall that the error $\mathcal{E}$ is a function of $v$ and $w$: $\mathcal{E} = \mathcal{E}(v,w)$. To evaluate the configuration vector($v$), the weights($w$) in the corresponding pruned network need to be trained. However, it would be too computationally prohibitive to train each pruned network from scratch. To address this, we customize a parameter-sharing mechanism to share the weights for different architectures varying in the channel, spatial and depth configurations. ![The weight-sharing mechanism.[]{data-label="fig:weight_share"}](weight_share.pdf){width="1\linewidth"} The parameter-sharing mechanism is illustrated in Figure \[fig:weight\_share\]. For spatial dimension, decreasing the input image resolution will reduce the feature map size in the network, as Figure \[fig:weight\_share\](a). This does not require any modification in the weight kernels and thus all the network weights are shared when adjusting the spatial resolution. To prune the channels, we follow [@liu2019metapruning] to keep the first $c$ channels of feature maps in the original network and crop the weight tensors correspondingly, as shown in Figure \[fig:weight\_share\](b). To reduce depth, we adopt block dropping. That is keeping the first $d$ blocks and skipping the rest of the blocks, as Figure \[fig:weight\_share\](c). Combining these three dimensions, we come up with the joint parameter-sharing mechanism as illustrated in Figure \[fig:weight\_share\](d). With this mechanism, we only need to train one set of weights in the weight-shared network and evaluate the error of different pruning configuration vectors with corresponding weights cropped from this network. Alternatively Update Weights and Configuration Vectors {#sec:update} ------------------------------------------------------ For training the weights in the shared network, the optimization goal is set to minimize the error concerning the expectation of a bunch of configuration vectors. The objective is defined as: $$\begin{aligned} \min_w \mathbb{E}_{v \sim \mathcal{N}(\mu,\sigma)}\mathcal{E}(v, w) \end{aligned}$$ [r]{}[0.45]{} \[alg:1\] **Input:** Learning rate $\alpha$, standard deviation of gaussian distribution $\sigma$, initial mean $\mu_0$, configuration vector $v$, initial weight: $w$, error function: $\mathcal{E}$, number of total iterations: $K$, number of weight update iterations: $N$, number of configuration samples: $M$.\ **Output**: Optimized configuration $\mu^*$\ $ \min_w \mathbb{E}_{v \sim \mathcal{N}(\mu,\sigma)}\mathcal{E}(v, w)$ $\mathcal{E}_j = \mathcal{E}(\mu_t+n_j)$ $\mu_{t+1} = \mu_t - \alpha \sum_{i=1}^M \mathcal{E}_j n_j$ return $\mu^*$ To optimize this objective function, we inject noise to the configuration vectors during weight training. That is, in each weight training iteration, the configuration vectors $v$ is randomly sampled from $\mathcal{N}(\mu,\sigma)$. After training the weights for one epoch, we use the stochastic gradient estimation to estimate the gradient *w.r.t.* the $\mu$ and take a step in the $\mu$ variable space to decline the error: $$\begin{aligned} \min_{\mu} \mathbb{E}_{v \sim \mathcal{N}(\mu,\sigma)}\mathcal{E}(v, w) \end{aligned}$$ We alternatively optimize the weights and the architecture configuration vectors and meanwhile decrease the deviation $\sigma$ till convergence. With the error function $\mathcal{E}$ well approximated by its smoothed version $\mathbb{E}_{v \sim \mathcal{N}(\mu,\sigma)}\mathcal{E}(v, w)$, we can obtain the optimized configuration vector to be the mean of the configuration vector distributions (i.e., $\mu$) of the final model. The optimization pipeline is detailed in Algorithm \[alg:1\]. Experiments =========== In this section, we begin by introducing our experiment settings in Sec. \[sec:exp\_setting\]. Then, we explain the configuration vector settings in Sec. \[sec:configuration\_setting\]. Thirdly, we show the comparison between our method and the state-of-the-arts on various architectures and under different resource constraint metrics in Sec. \[sec:sota\]. Lastly, we visualize two of the pruned networks obtained by our algorithm in Sec. \[sec:visualization\]. Experiment Settings {#sec:exp_setting} ------------------- **Dataset** All experiments are conducted on the ImageNet 2012 classification dataset [@russakovsky2015imagenet]. For optimizing the configuration vector, we randomly split the original training set into two subsets: 50,000 images for validation (50 images for each class) and use the rest as the training set. After the architecture hyperparameters are optimized, we train the corresponding architecture from scratch using the original training/validation dataset split, following the practice in [@cai2018proxylessnas; @wu2018fbnet]. **Training** We alternatively optimize the configuration vector and weights with an outer loop of 75 iterations. In each inner loop, we train the weights for 2000 iterations for MobileNet V1/V2 and 1000 iterations for ResNet with a batch size of 256. We use the SGD optimizer with momentum for weight training. Then we update the configuration vector for 20 iterations with the estimated gradients. In gradient estimation, the number of samples $M$ is set to 100 for each iteration, the deviation $\sigma$ and the update rate $\alpha$ are initialized with 1.25 and 5 respectively, and linearly decay to 0.25 and 0, respectively, with the outer loop. **Constraints** The experiments are carried out under the FLOPs, GPU latency as well as the CPU latency constraints. For the latency constraints, we follow the practice in FBNet [@wu2018fbnet] to build a latency look-up-table for a layer with different configurations and obtain the total latency of the network by summing up the latency of all layers. In our experiments, we estimate GPU latency on the GTX 1080Ti with a batch size of 256. For CPU latency, we use the look-up-table released by ChamNet [@dai2018chamnet] for a fair comparison. As ChamNet has a very sparse look-up-table, we use the Gaussian process to predict the missing latency values as [@dai2018chamnet] did for energy estimation. Details in configuration vector settings {#sec:configuration_setting} ---------------------------------------- For MobileNet V1, we set the optimization vector($v$) as $v=\{c_1,c_2,…c_n, s\}$, where $c_i$ are the numbers of output channels in layer $i$, and $n$ denotes total number of layers. $s$ is the spatial size of the input image. As MobileNet V1 is a network without shortcuts, reducing the depth will result in channel mismatch. With this structure restriction, we only optimize the channel and spatial dimensions. We initiate the configuration vector as the configuration of the original MobileNet V1. After obtaining the optimal configuration, the pruned network is constructed through pruning each layer’s input channel equaling to previous layer’s output channel, specified by $c_i$, and then crop the weight kernels correspondingly as shown in Figure \[fig:weight\_share\](b). Similar to MobileNet V1, the optimization vector($v$) for MobileNet V2 is defined as $v=\{c_1,c_2,…c_n, s, d\}$. Since MobileNet V2 contains the shortcut structure, we confine the layers connected with identity shortcuts to have the same number of output channels. For pruning depth dimension, we drop the last block in the stage that contains the most number of blocks. For ResNet, we adopt similar settings of MobileNet V2, and confine the configuration vectors to layer-wise channel numbers only, i.e. $v=\{c_1,c_2,…c_n\}$ for a direct comparison to the traditional channel pruning methods. Comparisons with state-of-the-arts {#sec:sota} ---------------------------------- To verify the effectiveness of proposed JointPruning, we compare our method with the uniform baselines, traditional pruning methods as well as the AutoML-based model adaptation methods. Our experiments are conducted on MobileNet V1, MobileNet V2 [@howard2017mobilenets; @sandler2018mobilenetv2] and ResNet [@he2016deep] backbones. **MobileNet V1&V2.** \[subsec:mbv1\] MobileNets [@howard2017mobilenets; @sandler2018mobilenetv2] are already compact networks, many recent pruning algorithms focus on these structures for the practical significance in pruned models. The baseline group in Table \[table:mb\_flops\] shows that rescaling channel and spatial dimension (C+S) by the same ratio can achieve higher accuracy than merely rescale channel (C) or spatial (S), suggesting the importance of jointly dealing with multi-dimensions. Then, in this joint pruning space, the proposed JointPruning can automatically capture the delicate linkage between spatial pruning and channel-wise pruning ratio adjustments, which outputs optimized pruned networks with higher accuracy than state-of-the-art pruning methods focusing on channels only. When further considering the depth dimension, JointPruning obtains even better accuracy, especially under the extreme small FLOPs constraint (45M), which demonstrates the superiority to others. Besides the FLOPs constraint, we extensively study the behavior of JointPruning under direct metrics like GPU latency and CPU latency. Given various latency constraints, JointPruning consistently improves the accuracy by a remarkable margin especially for highly paralleled GPU devices. As shown in Table \[table:mb\_gpu\_latency\], on MobileNet V1 JointPruning with the channel and spatial dimension (C+S) achieves more than 3.9% accuracy enhancements. On MobileNet V2, JointPruning comprehensively considers channel, spatial and depth (C+S+D) dimensions, and automatically capture and utilize the underlying hardware specialty, which further boosts the accuracy. Compared to the state-of-the-art model adaptation method ChamNet building atop the Gaussian process, JointPurning with stochastic gradient estimation enjoys high optimization efficiency. As shown in Table \[table:mbv2\_cpu\_latency\], JointPruning achieves comparable or higher accuracy compared to ChamNet, but with much lower optimization time cost. \[table:mb\_flops\] \[table:mb\_gpu\_latency\] **ResNet.** \[subsec:resnet\] Despite that JointPruning is designed for multi-dimensional pruning, we show that it is also effective when targeting at channel dimension only. In comparison to the traditional channel pruning methods, JointPruning achieves better performance with less human participation. The accuracy enhancement mainly comes from JointPruning’s ability in pruning shortcuts. For traditional pruning with a layer-by-layer scheme, shortcuts will affect more than one layer which is tough to deal with. While JointPruning can effortlessly prune the shortcut by modeling the shortcut pruning as updating the numerical channel numbers. [**Random search**]{} Compared to the random search which determines the configuration in three dimensions randomly, Table \[table:random\_search\] shows that JointPurning is able to end up at an optimized minimum with high accuracy, and significantly outperforms random search scheme. Optimization results visualization {#sec:visualization} ---------------------------------- In this section, we visualize two sets of pruning configurations for MobileNetV2 focusing on optimizing CPU and GPU latency, respectively. By incorporating the latency into the error function $\mathcal{E}$, JointPruning is able to take advantage of hardware characteristics without knowing the implementation details. The final network optimized under the CPU latency constraint is deep with smaller spatial resolution. While for the GPU, the corresponding architecture discovered by JointPruning adopts large spatial resolution, more channels with fewer layers to fully utilize the parallel computing capability of GPUs. ![This figure shows the architecture hyper-parameters (input image size, layer-wise channel numbers and depth) of MobileNet V2 network structure optimized under (a) CPU latency constraint with latency = 10ms and (b) GPU latency constraint with latency = 5.62ms. []{data-label="fig:visualization"}](visualization_cropped.pdf){width="\linewidth"} Conclusion ========== In this work, we focused on the joint multi-dimension pruning which naturally has broader pruning space on spatial, depth and channel for digging out better configurations than the isolated single dimension solution. We have proposed to use stochastic gradient estimation to optimize this problem and further introduced a weight sharing strategy to avoid repeatedly training multiple models from scratch. Our results on large-scale ImageNet dataset with MobileNet V1&V2 and ResNet outperformed previous state-of-the-art pruning methods with significant margins. In the future, we will conduct more deep analyses about how multi-dimension pruning helps to find better optimum. It will also be interesting to apply our method to other tasks like unsupervised pruning to explore the upper limit of the proposed method. Acknowledgement =============== The authors would like to acknowledge National Key R&D Program of China (No. 2017YFA0700800), Beijing Academy of Artificial Intelligence (BAAI) and HKSAR RGC’s funding support under grant GRF-16203918. [^1]: This work is done when Zechun Liu interns at MEGVII Technology. $^{\dag}$Corresponding author.

--- abstract: 'In recent years, with the trend of applying deep learning (DL) in high performance scientific computing, the unique characteristics of emerging DL workloads in HPC raise great challenges in designing, implementing HPC AI systems. The community needs a new yard stick for evaluating the future HPC systems. In this paper, we propose HPC AI500 — a benchmark suite for evaluating HPC systems that running scientific DL workloads. Covering the most representative scientific fields, each workload from HPC AI500 is based on real-world scientific DL applications. Currently, we choose 14 scientific DL benchmarks from perspectives of application scenarios, data sets, and software stack. We propose a set of metrics for comprehensively evaluating the HPC AI systems, considering both accuracy, performance as well as power and cost. We provide a scalable reference implementation of HPC AI500. The specification and source code are publicly available from <http://www.benchcouncil.org/HPCAI500/index.html>. Meanwhile, the AI benchmark suites for datacenter, IoT, Edge are also released on the BenchCouncil web site.' author: - Zihan Jiang - Wanling Gao - Lei Wang - Xingwang Xiong - Yuchen Zhang - Xu Wen - Chunjie Luo - Hainan Ye - Xiaoyi Lu - Yunquan Zhang - Shengzhong Feng - Kenli Li - Weijia Xu - 'Jianfeng Zhan [^1]' title: 'HPC AI500: A Benchmark Suite for HPC AI Systems' ---  \ Introduction ============  \[Introduction\] The huge success of AlexNet [@alexnet] in the ImageNet [@imagenetweb] competition marks that deep learning(DL) is leading the renaissance of Artificial Intelligence (AI). Since then, a wide range of application areas have started using DL and achieved unprecedented results, such as image recognition, natural language processing, and even autonomous driving. In the commercial fields, many DL-based novel applications have emerged, creating huge economic benefits. In the fields of high performance scientific computing, similar classes of problems are faced, i.e., predicting extreme weather [@weather], finding signals of new particles [@high-energy], and estimating cosmological parameters [@cosmology]. These scientific fields are essentially solving the common class of problems that exist in commercial fields such as classifying images, predicting classes labels, or regressing a numerical quantity. In several scientific computing fields, DL has replaced traditional scientific computing methods and becomes a promising tool [@deeplearninginscience]. As an emerging workload in high performance scientific computing, DL has many unique features compared to traditional high performance computing. First, training a DL model depends on massive data that are represented by high-dimensional matrices. Second, leveraging deep learning frameworks such as Tensorflow [@tensorflow] and caffe [@caffe] aggravates the difficulty of the software and hardware co-design. Last but not least, the heterogeneous computing platform of DL is far more complicated than traditional scientific workloads, including CPU, GPU, and various domain-specific processor (e.g. Cambricon Diannao [@diannao] or Google TPU [@tpu]). Consequently, the community requires a new yardstick for evaluating future HPC AI systems. However, the diversity of scientific DL workloads raise great challenges in HPC AI benchmarking. 1. Dataset: Scientific data is often more complex than MINST or ImageNet data sets. First, the shape of scientific data can be 2D images or higher-dimension structures. Second, there are hundreds of channels in a scientific image, while the popular image data often consists of only RGB. Third, Scientific datasets are always terabytes or even petabytes in size. 2. Workloads: Modern scientific DL doesn’t adopt off-the-shelf models, instead builds more complex model with domain scientific principles (e.g. energy conservation) [@weather]. 3. Metrics: Due to the importance of accuracy, using a single performance metric such as FLOPS leads to insufficient evaluation. For a comprehensively evaluation, the selected metrics should not only consider the performance of the system, but also consider the accuracy of the DL model [@dawnbench]. 4. Scalability: Since the scientific DL workloads always run on the supercomputers, which are equipped with tens of thousands nodes, the benchmark program must be highly scalable. Most of the existing AI benchmarks [@dawnbench; @MLPerf; @TBD; @fathom; @BenchNN; @ccbdbench] are based on commercial scenarios. Deep500 [@deep500] is a benchmarking framework aiming to evaluate high-performance deep learning. However, its reference implementation uses commercial open source data sets and simple DL models, hence cannot reflect real-world HPC AI workloads. We summary these major benchmarking efforts for AI and compare them with HPC AI500 as shown in the table below. ----------- ----------------- -------------- -------------- -------------- -------------- -------------- -------------- EWA Cos HEP HPC AI500 Scientific data $\checkmark$ $\checkmark$ $\checkmark$ $\times$ $\checkmark$ $\checkmark$ TBD Commercial data $\times$ $\times$ $\times$ $\checkmark$ $\checkmark$ $\times$ MLPerf Commercial data $\times$ $\times$ $\times$ $\checkmark$ $\checkmark$ $\times$ DAWNBench Commercial data $\times$ $\times$ $\times$ $\checkmark$ $\checkmark$ $\times$ Fathom Commercial data $\times$ $\times$ $\times$ $\checkmark$ $\checkmark$ $\times$ Deep500 Commercial data $\checkmark$ $\checkmark$ ----------- ----------------- -------------- -------------- -------------- -------------- -------------- -------------- : Comparison of AI Benchmarking Efforts. Extreme Weather Analysis Cosmology High Energy Physics Consequently, targeting above challenges, we propose HPC AI500—a benchmark suite for HPC AI systems. Our major contributions are as follows: 1. We create a new benchmark suite that covers the major areas of high performance scientific computing. The benchmark suite consists of micro benchmarks and component benchmarks. The workloads from component benchmarks use the state-of-the-art models and representative scientific data sets to reflect the real-world performance results. In addition, we select several DL kernels as the micro benchmarks for evaluating the upper bound performance of the systems. 2. We propose a set of metrics for comprehensively evaluating the HPC AI systems. Our metrics for component benchmarks include both accuracy and performance. For micro benchmarks, we provide metrics such as FLOPS to reflect the upper bound performance of the system. Coordinated by BenchCouncil (<http://www.benchcouncil.org>), we also release the datacenter AI benchmarks [@datacenter; @aibench], the IoT AI benchmarks [@IoT], edge AI benchmarks [@edge], and big data benchmarks [@bdb; @dcbench], which are publicly available from <http://www.benchcouncil.org/HPCAI500/index.html>. Deep Learning in Scientific Computing {#ScientificDL} ===================================== In order to benchmark HPC AI systems, the first step is to figure out how DL works in scientific fields. Although it is an emerging field, several scientific fields have applied DL to solve many important problems, such as extreme weather analysis  [@climatedataset; @Globenet; @extremeweather; @weather], high energy physics  [@trackfinding; @cellularHEP; @jet-images; @jetdiscrimination; @high-energy], and cosmology  [@cosmology; @cosmology2; @galaxyimage; @universerGAN; @cosmologicalmodel]. Extreme Weather Analysis ------------------------ Extreme Weather Analysis (EWA) poses a great challenge to human society. It brings severe damage to people health and economy every single year. For instance, the heatwaves in 2018 caused over 1600 deaths according to the UN report [@heatwave]. And the landfall of hurricane Florence and Michael caused about 40 billion dollars worth of damage to US economy [@hurricane]. In this context, understanding extreme weather life cycle and even predicting its future trend become a significant scientific goal. Achieving this goal always requires accurately identifying the weather patterns to acquire the insight of climate change based on massive climate data analysis. Traditional climate data analysis methods are built upon human expertise in defining multi-variate thresholds of extreme weather events. However, it has a major drawback: there is no commonly held set of criteria that can define a weather event due to the man-made subjectivism, which leads to inaccurate pattern extraction. Therefore, DL has become another option for climate scientists. Liu et al. (2016) [@climatedataset] develop a relatively simple CNN model with two convolutional layers to classify three typical extreme weather events and achieve up to 99% accuracy. Racah et al. (2017) [@extremeweather] implement a multichannel spatiotemporal CNN architecture for semi-supervised prediction and exploratory extreme weather data analysis. GlobeNet [@Globenet] is a CNN model with inception units for typhoon eye tracking. Kurth et al. (2018) [@weather] use variants of Tiramisu and DeepLabv3+ neural networks which are both built on Residual Network (ResNet) [@resnet]. They deployed these two networks on Summit and firstly achieved exascale deep learning for climate analysis. High Energy Physics ------------------- Particle collision is the most important experiment approach in High Energy Physics (HEP). Detecting the signal of new particle is the major goal in experimental HEP. Today’s HEP experimental facility such as LHC creates particle signals with hundreds of millions channels with a high data rate. The signal data from different channels in every collision usually are represented as a sparse 2d image, so called a jet-image. In fact, accurately classifying these jet-images is the key to find signals of new particles. In recent years, due to the excellent performance in pattern recognition, DL has become the focus of the data scientists in HEP community and has a tendency to go mainstream. Oliveira et al. (2016) [@jet-images] use a CNN model with 3 convolutional layers to tag jet-images. They firstly demonstrated that using DL not only improve the discrimination power, but also gain new insights compared to designing physics-inspired features. Komiske et al. (2017) [@jetdiscrimination] adopt a CNN model to discriminate quark and gluon jet-image. Kurth et al.(2017) [@high-energy] successfully deploy CNN to analyze massive HEP data on the HPC system and achieve petaflops performance. Their work is the first attempt at scaling DL on large-scale HPC systems. Cosmology --------- Cosmology is a branch of astronomy concerned with the studies of the origin and evolution of the universe, from the Big Bang to today and on into the future [@cosmologydef]. In 21st century, the most fundamental problem in cosmology is the nature of dark energy. However, this mysterious energy greatly affects the distribution of matter in the universe that is described by cosmological parameters. Thus, accurately estimating these parameters is the key to understand the insight of the dark energy. For solving this problem, Ravanbakhsh et al. (2017) [@cosmology2] firstly propose a 3D CNN model with 6 convolutional layers and 3 fully-connected layers and opens the way to estimating the parameters with high accuracy. Mathuriya et al. (2018) propose CosmoFlow [@cosmology], which is a project aiming to process large 3D cosmology dataset on HPC systems. They extend the CNN model designed by Ravanbakhsh et al. (2017) [@cosmology2]. Meanwhile, in order to guarantee the high fidelity numerical simulations and avoid the use of expensive instruments, generating high quality cosmological data is also important. Ravanbakhsh et al. (2017) [@galaxyimage] propose a deep generative model for acquiring high quality galaxy images. Their results show a reliable alternative for generating the calibration data of cosmological surveys. Summary ------- After investigating the above representative scientific fields, we have identified the representative DL applications and abstracted these DL applications into classical AI tasks. As shown in Table \[table:scientificDL\], almost all the applications are essentially using CNN to extract the patterns of various scientific image data. From this perspective, *image recognition*, *image generation*, and *object detection* are the most important tasks in modern scientific DL. In our benchmark methodology (Section \[meth\]), we use these three classic AI tasks as the component workloads of the HPC AI500 Benchmark. [|c|c|c|c|]{} **Scientific Fields** & **DL Applications** & **Classical DL Tasks** & **Model Type**\ Extreme Weather Analysis & Identify weather patterns & Object Detection & CNN\ High Energy Physics & Jet-images discrimination & Image Recognition & CNN\ Cosmology & ------------------------- Estimate parameters Galaxy image generation ------------------------- : Modern Scientific Deep Learning. & ------------------- Image Recognition Image Generation ------------------- : Modern Scientific Deep Learning. & CNN\ \[table:scientificDL\] Benchmarking Methodology and Decisions {#Benchmark} ====================================== Methodology {#meth} ----------- Our benchmarking methodology is shown in Figure \[methodology\], similar to that [@bdb]. As HPC AI is an emerging and evolving domain, we take an incremental and iterative approach. First of all, we investigate the scientific fields that use DL widely. As mentioned in Section \[ScientificDL\], *extreme weather analysis, high energy physics*, and *cosmology* are the most representative fields. Then, we pay attention to the typical DL workloads and data sets in these three application fields. In order to cover the diversity of workloads, we focus on the critical tasks that DL has performed in the aforementioned fields. Based on our analysis in Section \[ScientificDL\], we extracts three important component benchmarks that can represent modern scientific DL, namely *image recognition*, *image generation*, and *object detection*. This shows that CNN models play an important role. In each component, we choose the state-of-the-art model and software stack from the applications. We also select the hotspot DL operators as the micro benchmark for evaluating upper bound performance of the system. We chose three real-world scientific data sets from aforementioned scientific fields and consider their diversity from the perspective of data formats. In modern DL, the raw data is always transformed into matrix for downstream processing. Therefore, we classify these matrices into three kinds of formats: 2D sparse matrix, 2D dense matrix, and 3 dimensional matrix. In each matrix format, we also consider the unique characteristics (e.g., multichannel that more than RGB, high resolution) in the scientific data. ![HPCAI500 Methodology[]{data-label="methodology"}](HPCAI500_Meth.eps){width="\textwidth"} The Selected Datasets --------------------- We investigate the representative data sets in our selected scientific fields and collect three data sets as shown in Table \[table:datasets\]. Our selection guidelines follow the aforementioned benchmarking methodology. **The Extreme Weather Data set** [@weatherdata] is made up of 26-year of climate data. The data of every year is available as one HDF5 file. Each HDF5 file contains two data sets: images and boxes. *Images data set has 1460 example dense images* (4 per day, 365 days per year) with 16 channels. Each channel is 768 \* 1152 corresponding to one measurement per 25 square km on earth. Boxes dataset records the coordinates of the four extreme weather events in the corresponding images: tropical depression, tropical cyclone, extratropical cyclone and the atmospheric river. **The HEP Data set** [@smalljetimages] is divided into two classes: the RPV-Susy signal and the most prevalent background. The training data set is composed of around 400 k jet-images. Each jet-image is represented as a 64\*64 sparse matrix and has 3 channels. It also provides validation and test data. All the data are generated by using the Pythia event generator [@pythia] interfaced to *the Delphes fast detector simulation* [@jet-images]. **The Cosmology Data set** [@cosmology] aims to predict the parameters of cosmology. It is based on dark matter N-body simulations produced using the MUSIC [@MUSIC] and pycola [@pycola] packages. Each simulation covers the volumes of $512h^{-1}Mpc^3$ and contains $512^3$ dark matter particles. **Dataset** **Data Format** **Scientific Features** ------------------------- ------------------ ------------------------------- Extreme Weather Dataset 2D dense matrix high resolution, multichannel HEP Dataeset 2D sparse matrix multichannel Cosmology Dataset 3D matrix multidimensional : The Chosen Datasets \[table:datasets\] The Selected Workloads ---------------------- ### Component Benchmarks Since object detection, image recognition, and image generation are the most representative DL tasks in modern scientific DL. We choose the following state-of-the-art models as the HPC AI500 component benchmarks. #### Faster-RCNN  [@fasterRCNN] targets real-time object detection. Unlike the previous object detection model  [@RCNN; @fastRCNN], it replaces the selective search by a region proposal network that achieves nearly cost-free region proposals. Further more, Faster-RCNN combines the advanced CNN model as their base network for extracting features and is the foundation of the 1st-place winning entries in ILSVRC’15 (ImageNet Large Scale Visual Recognition Competition). #### ResNet  [@ResNet] is a milestone in Image Recognition, marking the ability of AI to identify images beyond humans. It solves the degradation problem, which means in the very deep neural network the gradient will gradually disappear in the process of propagation, leading to poor performance. Due to the idea of ResNet, researchers successfully build a 152-layer deep CNN. This ultra deep model won all the awards in ILSVRC’15. #### DCGAN  [@DCGAN] is one of the popular and successful neural network for GAN [@GAN]. Its fundamental idea is replacing fully connected layers with convolutions and using transposed convolution for upsampling. The proposal of DCGAN helps bride the gap between CNNs for supervised learning and unsupervised learning. ### Micro Benchmarks We choose the following primary operators in CNN as our micro benchmarks. #### Convolution In mathematics, convolution is a mathematical operation on two functions to produce a third function that expresses how the shape of one is modified by the other [@wiki_conv]. In a CNN, convolution is the operation occupying the largest proportion, which is the multiply accumulate of the input matrix and the convolution kernel, and then produces feature maps. There are many convolution kernels distributed in different layers responsible for learning different level features. #### Full-connected The full-connected layer can be seen as the classifier of a CNN, which is essentially matrix multiplication. It is also the cause of the explosion of CNN parameters. For example, in AlexNet [@alexnet], the number of training parameters of fully-connected layers reaches about 59 million and accounts for 94 percent of the total. #### Pooling Pooling is a sample-based discretization process. In a CNN, the objective of pooling is to down-sample the inputs (e.g., feature maps), which leads to the reduction of dimensionality and training parameters. In addition, it enhances the robustness of the whole network. The commonly used pooling operations including max-pooling and average-pooling. [c|c|c|c|c|c]{} **App Scenarios** & **Workloads** & **Fields** & **Datasets** & **Data Format** & **Software Stack**\ Micro Benchmarks & ----------------- -- Convolution Pooling Fully-Connected ----------------- -- : The Summary of HPC AI500 Benchmark. & ----- -- HEP EWA Cos ----- -- : The Summary of HPC AI500 Benchmark. & Matrix & ------------------ -- Sparse 2D Matrix Dense 2D Matrix 3D Matrix ------------------ -- : The Summary of HPC AI500 Benchmark. & ------ -- CUDA MKL ------ -- : The Summary of HPC AI500 Benchmark. \ Image Recognition & ResNet & ----- HEP Cos ----- : The Summary of HPC AI500 Benchmark. & ------------- HEP Dataset Cos Dataset ------------- : The Summary of HPC AI500 Benchmark. & ------------------ Sparse 2D matrix 3D matrix ------------------ : The Summary of HPC AI500 Benchmark. & ------------ -- TensorFlow Pytorch ------------ -- : The Summary of HPC AI500 Benchmark. \ Object Detection& Faster-RCNN & EWA & EWA Dataset & Dense 2D Matrix & ------------ -- TensorFlow Pytorch ------------ -- : The Summary of HPC AI500 Benchmark. \ Image Generation & DCGAN & Cos & Cos Dataset& 3D Matrix & ------------ -- TensorFlow Pytorch ------------ -- : The Summary of HPC AI500 Benchmark. \ High Energy Physics Extreme Weather Analysis Cosmology \[table:summary\] Metrics ------- ### Metrics for Component Benchmarks At present, time-to-accuracy is the most well-received solution [@dawnbench; @MLPerf]. For comprehensive evaluate, the training accuracy and validation accuracy are both provided. The former is used to measure the training effect of the model, and the latter is used to measure the generalization ability of the model. The threshold of target accuracy is defined as a value according to the requirement of corresponding application domains. Each application domain needs to define its own target accuracy. In addition, cost-to-accuracy and power-to-accuracy are provided to measure the money and power spending of training the model to the target accuracy. ### Metrics for Micro Benchmarks The metrics of the micro benchmarks is simple since we only measure the performance without considering accuracy. we adopt FLOPS and images per second(images/s) as two main metrics. We also consider power and cost related metrics. Reference Implementation ======================== Component Benchmarks -------------------- According to the survey [@survey] of NERSC (National Energy Research Scientific Computing Center, the most representative DL framework is TensorFlow, and the proportion of which is increasing year by year. Consequently, we adopt TensorFlow for preferred framework. In order to evaluate large-scale HPC systems running scientific DL, scalability is the fundamental requirement. In modern distributed DL, synchronized training through data parallelism is the mainstream. In this training scheme, each training process gets a different portion of the full dataset but has a complete copy of the neural network model. At the end of each batch computation, all processes will synchronize the model parameters by *all\_reduce* operation to ensure they are training a consistent model. TensorFlow implements *all\_reduce* through a parameter server [@ps] and use the GRPC protocol for communication by default. The master-slave architecture and socket-based communication can not extend to large-scale clusters [@tfnotscale]. Horovod [@horovod] iirrespective a library originally designed for scalable distributed deep learning using TensorFlow. It implements *all\_reduce* operation using ring-based algorithm [@ring-based] and MPI (Message Passing Interface) for communication. Due to the decentralized design and high effective protocol, the combination of TensorFlow and Horovod has successfully scaled to 27360 GPUs on Summit [@weather]. Therefore, we leverage Horovod to improve the scalability. Micro Benchmarks ---------------- The goal of micro benchmarks is to determine the upper bound performance of the system. To do so, we implement it with succinct software stack. Every DL operator is written in C++ or call the low-level neural networks library (e.g. CuDNN) without any other dependencies. Conclusion ========== In this paper, we propose HPC AI500—a benchmark suite for evaluating HPC system that running scientific deep learning workloads. Our benchmarks model real-world scientific deep learning applications, including extreme weather analysis, high energy physics, and cosmology. We propose a set of metrics for comprehensively evaluating the HPC AI systems, considering both accuracy, performance as well as power and cost. We provide a scalable reference implementation of HPC AI500. The specification and source code of HPC AI500 are publicly available from <http://www.benchcouncil.org/HPCAI500/index.html>. Acknowledgment {#acknowledgment .unnumbered} ============== This work is supported by the Standardization Research Project of Chinese Academy of Sciences No.BZ201800001. [00]{} Krizhevsky, Alex, Ilya Sutskever, and Geoffrey E. Hinton. “Imagenet classification with deep convolutional neural networks.” Advances in neural information processing systems. 2012. http://www.image-net.org/ Abadi, Martín, et al. “Tensorflow: a system for large-scale machine learning.” OSDI. Vol. 16. 2016. Jia, Yangqing, et al. “Caffe: Convolutional architecture for fast feature embedding.” Proceedings of the 22nd ACM international conference on Multimedia. ACM, 2014. Chen, Yunji, et al. “DianNao family: energy-efficient hardware accelerators for machine learning.” Communications of the ACM 59.11 (2016): 105-112. Jouppi, Norman P., et al. “In-datacenter performance analysis of a tensor processing unit.” Computer Architecture (ISCA), 2017 ACM/IEEE 44th Annual International Symposium on. IEEE, 2017. Adolf, Robert, et al. “Fathom: Reference workloads for modern deep learning methods.” Workload Characterization (IISWC), 2016 IEEE International Symposium on. IEEE, 2016. Coleman, Cody, et al. “DAWNBench: An End-to-End Deep Learning Benchmark and Competition.” Training 100.101 (2017): 102. Zhu, Hongyu, et al. “TBD: Benchmarking and Analyzing Deep Neural Network Training.” arXiv preprint arXiv:1803.06905 (2018). Shi, Shaohuai, et al. “Benchmarking state-of-the-art deep learning software tools.” Cloud Computing and Big Data (CCBD), 2016 7th International Conference on. IEEE, 2016. Hennessy, John L., and David A. Patterson. Computer architecture: a quantitative approach. Elsevier, 2011. Wang, Lei, et al. “Bigdatabench: A big data benchmark suite from internet services.” 2014 IEEE 20th International Symposium on High Performance Computer Architecture (HPCA). IEEE, 2014. Jia Z, Wang L, Zhan J, et al. Characterizing data analysis workloads in data centers\[C\]//2013 IEEE International Symposium on Workload Characterization (IISWC). IEEE, 2013: 66-76. Hao T, Huang Y, Wen X, et al. “Edge AIBench: Towards comprehensive end-to-end edge computing benchmarking.” 2018 BenchCouncil International Symposium on Benchmarking, Measuring and Optimizing (Bench18), 2018. Luo C, Zhang F, Huang C, Xiong X, J. Chen, et al. “AIoT Bench: Towards comprehensive benchmarking mobile and embedded device intelligence.” 2018 BenchCouncil International Symposium on Benchmarking, Measuring and Optimizing (Bench18), 2018. Gao W, Tang F, Wang L, Zhan J, et al. “AIBench: An Industry Standard Internet Service AI Benchmark Suite.” Technical Report 2019. Gao W, Luo C, Wang L, Xiong X, et al. “AIBench: Towards Scalable and Comprehensive Datacenter AI Benchmarking.” 2018 BenchCouncil International Symposium on Benchmarking, Measuring and Optimizing (Bench18), 2018. J. Dean. Keynote: Large Scale Deep Learning. Collobert, Ronan, Samy Bengio, and Johnny Mariéthoz. Torch: a modular machine learning software library. No. EPFL-REPORT-82802. Idiap, 2002. He, Kaiming, et al. “Deep residual learning for image recognition.” Proceedings of the IEEE conference on computer vision and pattern recognition. 2016. Kurth T, Treichler S, Romero J, et al. Exascale deep learning for climate analytics\[C\]//Proceedings of the International Conference for High Performance Computing, Networking, Storage, and Analysis. IEEE Press, 2018: 51. Kurth T, Zhang J, Satish N, et al. Deep learning at 15pf: supervised and semi-supervised classification for scientific data\[C\]//Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis. ACM, 2017: 7. Mathuriya A, Bard D, Mendygral P, et al. CosmoFlow: using deep learning to learn the universe at scale\[C\]//Proceedings of the International Conference for High Performance Computing, Networking, Storage, and Analysis. IEEE Press, 2018: 65. https://www.oreilly.com/ideas/a-look-at-deep-learning-for-science Bhimji W, Farrell S A, Kurth T, et al. Deep Neural Networks for Physics Analysis on low-level whole-detector data at the LHC\[C\]//Journal of Physics: Conference Series. IOP Publishing, 2018, 1085(4): 042034. Ravanbakhsh S, Oliva J B, Fromenteau S, et al. Estimating Cosmological Parameters from the Dark Matter Distribution\[C\]//ICML. 2016: 2407-2416. He K, Zhang X, Ren S, et al. Deep residual learning for image recognition\[C\]//Proceedings of the IEEE conference on computer vision and pattern recognition. 2016: 770-778. Chen T, Chen Y, Duranton M, et al. BenchNN: On the broad potential application scope of hardware neural network accelerators\[C\]//2012 IEEE International Symposium on Workload Characterization (IISWC). IEEE, 2012: 36-45. https://mlperf.org/ Ben-Nun T, Besta M, Huber S, et al. A Modular Benchmarking Infrastructure for High-Performance and Reproducible Deep Learning\[J\]. arXiv preprint arXiv:1901.10183, 2019. Patton R M, Johnston J T, Young S R, et al. 167-PFlops deep learning for electron microscopy: from learning physics to atomic manipulation\[C\]//Proceedings of the International Conference for High Performance Computing, Networking, Storage, and Analysis. IEEE Press, 2018: 50. Li M, Andersen D G, Park J W, et al. Scaling distributed machine learning with the parameter server\[C\]//11th [USENIX]{} Symposium on Operating Systems Design and Implementation ([OSDI]{} 14). 2014: 583-598. Ravanbakhsh S, Lanusse F, Mandelbaum R, et al. Enabling dark energy with deep generative models of galaxy images\[C\]//Thirty-First AAAI Conference on Artificial Intelligence. 2017. Mustafa M, Bard D, Bhimji W, et al. Creating virtual universes using generative adversarial networks\[J\]. arXiv preprint arXiv:1706.02390, 2017. Schmelzle J, Lucchi A, Kacprzak T, et al. Cosmological model discrimination with Deep Learning\[J\]. arXiv preprint arXiv:1707.05167, 2017. Peterson C. Track finding with neural networks\[J\]. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 1989, 279(3): 537-545. Denby B. Neural networks and cellular automata in experimental high energy physics\[J\]. Computer Physics Communications, 1988, 49(3): 429-448. de Oliveira L, Kagan M, Mackey L, et al. Jet-images-deep learning edition\[J\]. Journal of High Energy Physics, 2016, 2016(7): 69. Komiske P T, Metodiev E M, Schwartz M D. Deep learning in color: towards automated quark/gluon jet discrimination\[J\]. Journal of High Energy Physics, 2017, 2017(1): 110. Liu Y, Racah E, Correa J, et al. Application of deep convolutional neural networks for detecting extreme weather in climate datasets\[J\]. arXiv preprint arXiv:1605.01156, 2016. Hong S, Kim S, Joh M, et al. Globenet: Convolutional neural networks for typhoon eye tracking from remote sensing imagery\[J\]. arXiv preprint arXiv:1708.03417, 2017. Racah E, Beckham C, Maharaj T, et al. ExtremeWeather: A large-scale climate dataset for semi-supervised detection, localization, and understanding of extreme weather events\[C\]//Advances in Neural Information Processing Systems. 2017: 3402-3413. Gómez-Bombarelli R, Wei J N, Duvenaud D, et al. Automatic chemical design using a data-driven continuous representation of molecules\[J\]. ACS central science, 2018, 4(2): 268-276. https://www.ecowatch.com/un-extreme-weather-climate-change-2633131018.html https://www.cbsnews.com/news/extreme-weather-events-2018-top-3-most-expensive-climate-driven-events-took-place-in-us/ https://extremeweatherdataset.github.io/ http://stanford.edu/group/stanford\_atlas/ Spira M, Djouadi A, Graudenz D, et al. Higgs boson production at the LHC\[J\]. Nuclear Physics B, 1995, 453(1-2): 17-82. https://en.wikipedia.org/wiki/Cosmology Goodfellow I, Pouget-Abadie J, Mirza M, et al. Generative adversarial nets\[C\]//Advances in neural information processing systems. 2014: 2672-2680. Bhimji W, Farrell S A, Kurth T, et al. Deep Neural Networks for Physics Analysis on low-level whole-detector data at the LHC\[C\]//Journal of Physics: Conference Series. IOP Publishing, 2018, 1085(4): 042034. Sjöstrand T, Mrenna S, Skands P. PYTHIA 6.4 physics and manual\[J\]. Journal of High Energy Physics, 2006, 2006(05): 026. https://www-n.oca.eu/ohahn/MUSIC/ https://bitbucket.org/tassev/pycola/ https://en.wikipedia.org/wiki/Convolution Mathuriya A, Kurth T, Rane V, et al. Scaling grpc tensorflow on 512 nodes of cori supercomputer\[J\]. arXiv preprint arXiv:1712.09388, 2017. Sergeev A, Del Balso M. Horovod: fast and easy distributed deep learning in TensorFlow\[J\]. arXiv preprint arXiv:1802.05799, 2018. Andrew Gibiansky. Bringing HPC techniques to deep learning. http://research.baidu. com/bringing-hpc-techniques-deep-learning, 2017. \[Online; accessed 6-December2017\]. https://www.open-mpi.org/ https://www.jlab.org/indico/event/247/session/8/contribution/30/material/slides/0.pdf Ren, Shaoqing, et al. “Faster r-cnn: Towards real-time object detection with region proposal networks.” Advances in neural information processing systems. 2015. Girshick, Ross, et al. “Rich feature hierarchies for accurate object detection and semantic segmentation.” Proceedings of the IEEE conference on computer vision and pattern recognition. 2014. Girshick, Ross. “Fast r-cnn.” Proceedings of the IEEE international conference on computer vision. 2015. Radford, Alec, Luke Metz, and Soumith Chintala. “Unsupervised representation learning with deep convolutional generative adversarial networks.” arXiv preprint arXiv:1511.06434 (2015). Goodfellow, Ian, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. “Generative adversarial nets.” In Advances in neural information processing systems, pp. 2672-2680. 2014. [^1]: Jianfeng Zhan is the corresponding author.

--- abstract: 'Topic models are evaluated based on their ability to describe documents well (i.e. low perplexity) and to produce topics that carry coherent semantic meaning. In topic modeling so far, perplexity is a direct optimization target. However, topic coherence, owing to its challenging computation, is not optimized for and is only evaluated after training. In this work, under a neural variational inference framework, we propose methods to incorporate a topic coherence objective into the training process. We demonstrate that such a coherence-aware topic model exhibits a similar level of perplexity as baseline models but achieves substantially higher topic coherence.' author: - | Ran Ding, Ramesh Nallapati, Bing Xiang\ Amazon Web Services\ [{rding, rnallapa, bxiang}@amazon.com]{} bibliography: - 'ref.bib' title: 'Coherence-Aware Neural Topic Modeling' --- Introduction ============ In the setting of a topic model [@blei2012probabilistic], perplexity measures the model’s capability to describe documents according to a generative process based on the learned set of topics. In addition to describing documents well (i.e. achieving low perplexity), it is desirable to have topics (represented by top-$N$ most probable words) that are human-interpretable. Topic interpretability or coherence can be measured by *normalized point-wise mutual information* (NPMI) [@aletras2013evaluating; @lau2014machine]. The calculation of NPMI however is based on look-up operations in a large reference corpus and therefore is non-differentiable and computationally intensive. Likely due to these reasons, topic models so far have been solely optimizing for perplexity, and topic coherence is only evaluated after training. As has been noted in several publications [@chang2009reading], optimization for perplexity alone tends to negatively impact topic coherence. Thus, without introducing topic coherence as a training objective, topic modeling likely produces sub-optimal results. Compared to classical methods, such as mean-field approximation [@hoffman2010online] and collapsed Gibbs sampling [@griffiths2004finding] for the latent Dirichlet allocation (LDA) [@blei2003latent] model, neural variational inference [@kingma2013auto; @rezende2014stochastic] offers a flexible framework to accommodate more expressive topic models. We build upon the line of work on topic modeling using neural variational inference [@miao2016neural; @miao2017discovering; @srivastava2017autoencoding] and incorporate topic coherence awareness into topic modeling. Our approaches of constructing topic coherence training objective leverage pre-trained word embeddings [@mikolov2013efficient; @pennington2014glove; @salle2016matrix; @joulin2016bag]. The main motivation is that word embeddings carry contextual similarity information that is highly related to the mutual information terms involved in the calculation of NPMI. In this paper, we explore two methods: (1) we explicitly construct a differentiable surrogate topic coherence regularization term; (2) we use word embedding matrix as a factorization constraint on the topical word distribution matrix that implicitly encourages topic coherence. Models ====== Baseline: Neural Topic Model (NTM) ---------------------------------- The model architecture shown in Figure \[fig:ntm\] is a variant of the Neural Variational Document Model (NVDM) [@miao2016neural]. Let $x \in \mathbb{R}^{|V| \times 1}$ be the bag-of-words (BOW) representation of a document, where $|V|$ is the size of the vocabulary and let $z \in \mathbb{R}^{K \times 1}$ be the latent topic variable, where $K$ is the number of topics. In the encoder $q_\phi(z|x)$, we have $\pi = f_{MLP}(x)$, $\mu(x) = l_1(\pi)$, $\log \sigma(x) = l_2(\pi)$, $h(x, \epsilon) = \mu + \sigma \odot \epsilon$, where $\epsilon \sim \mathcal{N}(0,I)$, and finally $z=f(h)=\mathrm{ReLU}(h)$. The functions $l_1$ and $l_2$ are linear transformations with bias. We choose the multi-layer perceptron (MLP) in the encoder to have two hidden layers with $3 \times K$ and $2 \times K$ hidden units respectively, and we use the sigmoid activation function. The decoder network $p_\theta(x|z)$ first maps $z$ to the predicted probability of each of the word in the vocabulary $y \in \mathbb{R}^{|V| \times 1}$ through $y=\text{softmax}(Wz+b)$, where $W \in \mathbb{R}^{|V| \times K}$. The log-likelihood of the document can be written as $\log p_\theta(x|z)=\sum_{i=1}^{|V|} \{ x \odot \log y \}$. We name this model Neural Topic Model (NTM) and use it as our baseline. We use the same encoder MLP configuration for our NVDM implementation and all variants of NTM models used in Section \[sec:exp\]. In NTM, the objective function to maximize is the usual *evidence lower bound* (ELBO) which can be expressed as $$\begin{split} & \mathcal{L}_{ELBO}(x^i) \\ & \approx \frac{1}{L}\sum_{l=1}^{L} \log p_\theta(x^i|z^{i,l}) - D_{KL}(q_\phi(h|x)|| p_\theta(h) \end{split}$$ where $z^{i,l}=\mathrm{ReLU}(h(x^i, \epsilon^{l}))$, $\epsilon^l \sim \mathcal{N}(0,I)$. We approximate $\mathbb{E}_{z\sim q(z|x)}[\log p_\theta(x|z)]$ with Monte Carlo integration and calculate the Kullback-Liebler (KL) divergence analytically using the fact $D_{KL}(q_\phi(z|x)|| p_\theta(z)) = D_{KL}(q_\phi(h|x)|| p_\theta(h))$ due to the invariance of KL divergence under deterministic mapping between $h$ and $z$. ![Model architecture[]{data-label="fig:ntm"}](figure1.pdf){width="45.00000%"} Compared to NTM, NVDM uses different activation functions and has $z=h$. Miao proposed a modification to NVDM called Gaussian Softmax Model (GSM) corresponding to having $z=\text{softmax}(h)$. Srivastava proposed a model called ProdLDA, which uses a Dirichlet prior instead of Gaussian prior for the latent variable $h$. Given a learned $W$, the practice to extract top-$N$ most probable words for each topic is to take the most positive entries in each column of $W$ [@miao2016neural; @miao2017discovering; @srivastava2017autoencoding]. This is an intuitive choice, provided that $z$ is non-negative, which is indeed the case for NTM, GSM and ProdLDA. NVDM, GSM, and ProdLDA are state-of-the-art neural topic models which we will use for comparison in Section \[sec:exp\]. Topic Coherence Regularization: NTM-R ------------------------------------- The topic coherence metric NPMI [@aletras2013evaluating; @lau2014machine] is defined as $$\begin{split} &\mathrm{NPMI}(\boldsymbol{w}) \\&= \frac{1}{N(N-1)}\sum_{j=2}^{N} \sum_{i=1}^{j-1} \frac{\log\frac{P(w_i, w_j)}{P(w_i)P(w_j)}}{-\log P(w_i, w_j)} \end{split}$$ where $\boldsymbol{w}$ is the list of top-$N$ words for a topic. $N$ is usually set to 10. For a model generating $K$ topics, the overall NPMI score is an average over all topics. The computational overhead and non-differentiability originate from extracting the co-occurrence frequency from a large corpus[^1]. From the NPMI formula, it is clear that word-pairs that co-occur often would score high, unless they are rare word-pairs – which would be normalized out by the denominator. The NPMI scoring bears remarkable resemblance to the contextual similarity produced by the inner product of word embedding vectors. Along this line of reasoning, we construct a differentiable, computation-efficient word embedding based topic coherence (WETC). Let $E$ be the row-normalized word embedding matrix for a list of $N$ words, such that $E \in \mathbb{R}^{N \times D}$ and ${\left\lVertE_{i,:}\right\rVert}=1$, where $D$ is the dimension of the embedding space. We can define *pair-wise* word embedding topic coherence in a similar spirit as NPMI: $$\begin{split} \mathrm{WETC}_{PW}(E) &= \frac{1}{N(N-1)}\sum_{j=2}^{N} \sum_{i=1}^{j-1} \langle E_{i,:}, E_{j,:}\rangle \\ &= \frac{\sum \{E^TE\} - N}{2N(N-1)} \end{split}$$ where $\langle\cdot,\cdot\rangle$ denotes inner product. Alternatively, we can define *centroid* word embedding topic coherence $$\mathrm{WETC}_{C}(E) = \frac{1}{N} \sum \{Et^T\}$$ where vector $t \in \mathbb{R}^{1 \times D}$ is the centroid of $E$, normalized to have ${\left\lVertt\right\rVert}=1$. Empirically, we found that the two WETC formulations behave very similarly. In addition, both $\mathrm{WETC}_{PW}$ and $\mathrm{WETC}_{C}$ correlate to human judgement almost equally well as NPMI when using `GloVe` [@pennington2014glove] vectors[^2]. With the above observations, we propose the following procedure to construct a WETC-based surrogate topic coherence regularization term: (1) let $E \in \mathbb{R}^{|V| \times D}$ be the pre-trained word embedding matrix for the vocabulary, rows aligned with $W$; (2) form the $W$-weighted centroid (topic) vectors $T \in \mathbb{R}^{D \times K}$ by $T=E^TW$; (3) calculate the cosine similarity matrix $S \in \mathbb{R}^{|V| \times K}$ between word vectors and topic vectors by $S = ET$; (4) calculate the $W$-weighted sum of word-to-topic cosine similarities for each topic $C \in \mathbb{R}^{1 \times K}$ as $C=\sum_i (S\odot W)_{i,:}$. Compared to $\mathrm{WETC}_{C}$, in calculating $C$ we do not perform top-$N$ operation in $W$, but directly use $W$ for weighted sum. Specifically, we use $W$-weighted topic vector construction in Step-2 and $W$-weighted sum of the cosine similarities between word vectors and topic vectors in Step-4. To avoid unbounded optimization, we normalize the rows of $E$ and the columns of $W$ before Step-2, and normalize the columns of $T$ after Step-2. The overall maximization objective function becomes $\mathcal{L}_{R}(x; \theta, \phi) = \mathcal{L}_{ELBO} + \lambda \sum_i{C_i}$, where $\lambda$ is a hyper-parameter with positive values controlling the strength of topic coherence regularization. We name this model NTM-R. Word Embedding as a Factorization Constraint: NTM-F and NTM-FR {#sec:ntm-f} -------------------------------------------------------------- Instead of allowing all the elements in $W$ to be freely optimized, we can impose a factorization constraint of $W = E \hat T$, where $E$ is the pre-trained word embedding matrix that is *fixed*, and only $\hat T$ is allowed to be learned through training. Under this configuration, $\hat T$ lives in the embedding space, and each entry in $W$ is the dot product similarity between a topic vector $\hat{T_i}$ and a word vector $E_j$. As one can imagine, similar words would have similar vector representations in $E$ and would have similar weights in each column of $W$. Therefore the factorization constraint encourages words with similar meaning to be selected or de-selected together thus potentially improving topic coherence. We name the NTM model with factorization constraint enabled as NTM-F. In addition, we can apply the regularization discussed in the previous section on the resulting matrix $W$ and we name the resulting model NTM-FR. Experiments and Discussions {#sec:exp} =========================== {width="80.00000%"} Results on *20NewsGroup* ------------------------ First, we compare the proposed models to state-of-the-art neural variational inference based topic models in the literature (NVDM, GSM, and ProdLDA) as well as LDA benchmarks, on the *20NewsGroup* dataset[^3]. In training NVDM and all NTM models, we used `Adadelta` optimizer [@zeiler2012adadelta]. We set the learning rate to 0.01 and train with a batch size of 256. For NTM-R, NTM-F and NTM-FR, the word embedding we used is `GloVe` [@pennington2014glove] vectors pre-trained on Wikipedia and Gigaword with 400,000 vocabulary size and 50 embedding dimensions[^4]. The topic coherence regularization coefficient $\lambda$ is set to 50. The results are presented in Table \[table:20ng\_table\]. Overall we can see that LDA trained with collapsed Gibbs sampling achieves the best perplexity, while NTM-F and NTM-FR models achieve the best topic coherence (in NPMI). Clearly, there is a trade-off between perplexity and NPMI as identified by other papers. So we constructed Figure \[fig:npmi\_vs\_ppx\], which shows the two metrics from various models. For the models we implemented, we additionally show the full evolution of these two metrics over training epochs. From Figure \[fig:npmi\_vs\_ppx\], it becomes clear that although ProdLDA exhibits good performance on NPMI, it is achieved at a steep cost of perplexity, while NTM-R achieves similar or better NPMI at much lower perplexity levels. At the other end of the spectrum, if we look for low perplexity, the best numbers among neural variational models are between 750 and 800. In this neighborhood, NTM-R substantially outperforms the GSM, NVDM and NTM baseline models. Therefore, we consider NTM-R the best model overall. Different downstream applications may require different tradeoff points between NPMI and perplexity. However, the proposed NTM-R model does appear to provide tradeoff points on a Pareto front compared to other models across most of the range of perplexity. Comments on NTM-F and NTM-FR ---------------------------- It is worth noting that although NTM-F and NTM-FR exhibit high NPMI early on, they fail to maintain it during the training process. In addition, both models converged to fairly high perplexity levels. Our hypothesis is that this is caused by NTM-F and NTM-FR’s substantially reduced parameter space - from $|V|\times K$ to $D\times K$, where $|V|$ ranges from 1,000 to 150,000 in a typical dataset, while $D$ is on the order of 100. Some form of relaxation could alleviate this problem. For example, we can let $W=E\hat T + A$, where $A$ is of size $|V|\times K$ but is heavily regularized, or let $W=EQ\hat T$ where $Q$ is allowed as additional free parameters. We leave fully addressing this to future work. Validation on other Datasets ---------------------------- To further validate the performance improvement from using WETC-based regularization in NTM-R, we compare NTM-R with the NTM baseline model on a few more datasets: DailyKOS, NIPS, and NYTimes[^5] [@asuncion2007uci]. These datasets offer a wide range of document length (ranging from $\sim$100 to $\sim$1000 words), vocabulary size (ranging from $\sim$7,000 to $\sim$140,000), and type of documents (from news articles to long-form scientific writing). In this set of experiments, we used the same settings and hyperparameter $\lambda$ as before and did not fine-tune for each dataset. The results are presented in Figure \[fig:cross\_ds\]. In a similar style as Figure \[fig:npmi\_vs\_ppx\], we show the evolution of NPMI and WETC versus perplexity over epochs until convergence. Among all datasets, we observed improved NPMI at the same perplexity level, validating the effectiveness of the topic coherence regularization. However, on the NYTimes dataset, the improvement is quite marginal even though WETC improvements are very noticeable. One particularity about the NYTimes dataset is that approximately 58,000 words in the 140,000-word vocabulary are named entities. It appears that the large number of named entities resulted in a divergence between NPMI and WETC scoring, which is an issue to address in the future. [0.3]{} {width="\textwidth"} [0.3]{} {width="\textwidth"} [0.3]{} {width="\textwidth"} [0.3]{} {width="\textwidth"} [0.3]{} {width="\textwidth"} [0.3]{} {width="\textwidth"} Conclusions =========== In this work, we proposed regularization and factorization constraints based approaches to incorporate awareness of topic coherence into the formulation of topic models: NTM-R and NTM-F respectively. We observed that NTM-R substantially improves topic coherence with minimal sacrifice in perplexity. To our best knowledge, NTM-R is the first topic model that is trained with an objective towards topic coherence – a feature directly contributing to its superior performance. We further showed that the proposed WETC-based regularization method is applicable to a wide range of text datasets. Word Embedding Topic Coherence {#sec:wetc} ============================== As studied in [@aletras2013evaluating] and [@lau2014machine], the NPMI metric for assessing topic coherence over a list of words $\boldsymbol{w}$ is defined in Eq. \[eq:npmi\]. $$\label{eq:npmi} \begin{split} &\mathrm{NPMI}(\boldsymbol{w}) \\&= \frac{1}{N(N-1)}\sum_{j=2}^{N} \sum_{i=1}^{j-1} \frac{\log\frac{P(w_i, w_j)}{P(w_i)P(w_j)}}{-\log P(w_i, w_j)} \end{split}$$ where $P(w_i)$ and $P(w_i, w_j)$ are the probability of words and word pairs, calculated based on a reference corpus. $N$ is usually set to 10, by convention, so that NPMI is evaluated over the topic-10 words for each topic. For a model generating $K$ topics, the overall NPMI score is an average over all the topics. The computational overhead comes from extracting the relevant co-occurrence frequency from a large corpus. This problem is exacerbated when the look-up also requires a small sliding window as the authors of [@lau2014machine] suggested. A typical calculation of 50 topics based on a few million documents from the Wikipedia corpus takes $\sim$20 minutes[^6]. For a list of words $\boldsymbol{w}$ of length $N$, we can assemble a corresponding word embedding matrix $E \in \mathbb{R}^{N \times D}$ with each row corresponding to a word in the list. $D$ is the dimension of the embedding space. Averaging across the rows, we can obtain vector $t \in \mathbb{R}^{1 \times D}$ as the centroid of all the word vectors. It may be regarded as a “topic” vector. In addition, we assume that each row of $E$ and $t$ is normalized, i.e. ${\left\lVertt\right\rVert}=1$ and ${\left\lVertE_{i,:}\right\rVert}=1$. With these, we define *pair-wise* and *centroid* word embedding topic coherence $\mathrm{WETC}_{PW}$ and $\mathrm{WETC}_C$ as follows: $$\label{eq:pw_wetc} \begin{split} \mathrm{WETC}_{PW}(E) &= \frac{1}{N(N-1)}\sum_{j=2}^{N} \sum_{i=1}^{j-1} \langle E_{i,:}, E_{j,:}\rangle \\ &= \frac{\sum \{E^TE\} - N}{2N(N-1)} \end{split}$$ $$\label{eq:centroid_wetc} \mathrm{WETC}_{C}(E) = \frac{1}{N} \sum \{Et^T\}$$ where $\langle\cdot,\cdot\rangle$ denotes inner product. The simplification in Eq. \[eq:pw\_wetc\] is due to the row normalization of $E$. In this setting, we have the flexibility to use any pre-trained word embeddings to construct $E$. To experiment, we compared several recently developed variants [^7]. The dataset from [@aletras2013evaluating] provides human ratings for 300 topics coming from 3 corpora: 20NewsGroup (20NG), New York Times (NYT) and genomics scientific articles (Genomics), which we use as the human gold standard. We use Pearson and Spearman correlations to compare NPMI and WETC scores against human ratings. The results are shown in Table \[table:wetc\_table\]. From Table \[table:wetc\_table\] we observed a minimal difference between pair-wise and centroid based WETC in general. Overall, `GloVe` appears to perform the best across different types of corpora and its correlation with human ratings is very comparable to NPMI-based scores. Our NPMI calculation is based on the Wikipedia corpus and should serve as a fair comparison. In addition to the good correlation exhibited by WETC, the evaluation of WETC only involves matrix multiplications and summations and thus is fully differentiable and several orders of magnitude faster than NPMI calculations. WETC opens the door of incorporating topic coherence as a training objective, which is the key idea we will investigate in the subsequent sections. It is worth mentioning that, for `GloVe`, the low dimensional embedding (50d) appears to perform almost equally well as high dimensional embedding (300d). Therefore, we will use Glove-400k-50d in all subsequent experiments. While the WETC metric on its own might be of interest to the topic modeling research community, we leave the task of formally establishing it as a standard metric in place of NPMI to future work. In this work, we still use the widely accepted NPMI as the objective topic coherence metric for model comparisons. [^1]: A typical calculation of NPMI over 50 topics based on the Wikipedia corpus takes $\sim$20 minutes, using code provided by [@lau2014machine] at <https://github.com/jhlau/topic_interpretability>. [^2]: See Appendix A for details on an empirical study of human judgement of topic coherence, NPMI and WETC with various types of word embeddings. [^3]: We use the exact dataset from [@srivastava2017autoencoding] to avoid subtle differences in pre-processing [^4]: Obtained from <https://nlp.stanford.edu/projects/glove/> [^5]: <https://archive.ics.uci.edu/ml/datasets/Bag+of+Words> [^6]: Using code provided by [@lau2014machine] at <https://github.com/jhlau/topic_interpretability>. Running parallel processes on 8 Intel Xeon E5-2686 CPUs. [^7]: Details of pre-trained word embeddings used in Table \[table:wetc\_table\] - `Word2Vec` [@mikolov2013efficient]: pre-trained on GoogleNews, with 3 million vocabulary size and 300 embedding dimension. Obtained from <https://code.google.com/archive/p/word2vec/>. - `GloVe` [@pennington2014glove]: pre-trained on Wikipedia and Gigaword, with 400,000 vocabulary size and 50 and 300 embedding dimension. Obtained from <https://nlp.stanford.edu/projects/glove/>. - `FastText` [@joulin2016bag]: pre-trained on Wikipedia with 2.5 million vocabulary size and 300 embedding dimension. Obtained from <https://github.com/facebookresearch/fastText>. - `LexVec` [@salle2016matrix]: pre-trained on Wikipedia with 370,000 vocabulary size and 300 embedding dimension. Obtained from <https://github.com/alexandres/lexvec>.

--- abstract: 'The main source of various religious teachings is their sacred texts which varies from religion to religion based on different factors like the geographical region or time of birth of particular religion. Despites these differences there could be similarities between the sacred texts based on what lessons it teaches to it’s followers. This paper attempts to find the similarity using text mining techniques. The corpus consisting of Asian (Tao Te Ching, Buddhism, Yogasutra, Upanishad) and non Asian (four Bible texts) is used to explore findings of similarity measures like Euclidean, Manhattan, Jaccard and Cosine on raw Document Term Frequency \[DTM\], normalized DTM which reveals similarity based on word usage. The performance of Supervised learning algorithms like K-Nearest Neighbor \[KNN\], Support Vector Machine \[SVM\] and Random Forest is measured based on it’s accuracy to predict correct scared text for any given chapter in the corpus. The K-means clustering visualizations on Euclidean distances of raw DTM reveals that there exists a pattern of similarity among these sacred texts with Upanishads and Tao Te Ching being the most similar text in the corpus.' address: - 'College of Computing and Information Sciences, Rochester Institute of Technology, 85 Lomb Memorial Drive, Rochester, New York 14623' - 'School of Mathematical Sciences, Rochester Institute of Technology, 85 Lomb Memorial Drive, Rochester, New York 14623' author: - Preeti Sah - 'Ernest Fokoué\' title: | What do Asian Religions Have in Common?\ An Unsupervised Text Analytics Exploration --- Introduction ============ The purpose of religion is to facilitate love, compassion, patience, tolerance, humility and forgiveness. The sacred texts are cornerstone of religion and medium to instill the religious teachings in the people. Every part of the world follow different sacred texts to learn and preach about their religion. The following scripts were collected for different religions which is followed in different countries: - Hinduism (India): Yogasutras, Upanishads - Buddhism (Tibet): Four Noble Truth of Buddhism - Taoism (China): Tao Te Ching - Christianity (Central Asia/America): Book of Proverb, Book of Ecclesiastes, Book of Ecclesiasticus, Book of Wisdom All the data collected was English translations of the original language in which it was written.This was done to make sure that we have uniformity of texts collected from different sources. The sources of the data were: - Yogasutras: Project Gutenberg’s The Yoga Sutras of Patanjali, by Charles Johnston - Upanishads: The Project Gutenberg EBook of The Upanishads, by Swami Paramananda - Four Noble Truth of Buddhism: https://www.accesstoinsight.org/lib/study/truths.html - Tao Te Ching: Tao Te Ching - Translated by J. Legge - Book of Proverb: Project Gutenberg EBook The Bible, Douay-Rheims, Book 22: Proverbs - Book of Ecclesiastes: Project Gutenberg EBook The Bible, Douay-Rheims, Book 23: Ecclesiastes - Book of Ecclesiasticus: Project Gutenberg EBook The Bible, Douay-Rheims, Book 26: Ecclesiasticus - Book of Wisdom: Project Gutenberg EBook The Bible, Douay-Rheims, Book 25: Wisdom Buddhism teaches about four noble truth. Each of these truths entails a duty: stress is to be comprehended, the origination of stress abandoned, the cessation of stress realized, and the path to the cessation of stress developed. When all of these duties have been fully performed, the mind gains total release [@Budd]. Tao Te Ching teaches that Tao is The Way, Not ‘Your Way’about. The chapters talk about staying detached, letting go and keeping things simple [@Tao]. Yogasutra contains essence of wisdom. We think of ourselves as living a purely physical life, in these material bodies of ours. In reality, we have gone far indeed from pure physical life; for ages, our life has been psychical, we have been centred and immersed in the psychic nature [@Yoga]. The Upanishads represent the loftiest heights of ancient Indo-Aryan thought and culture. They form the wisdom portion or Gnana-Kanda of the Vedas, as contrasted with the Karma-Kanda or sacrificial portion. In each of the four great Vedas–known as Rik, Yajur, Sama and Atharva–there is a large portion which deals predominantly with rituals and ceremonials, and which has for its aim to show man how by the path of right action he may prepare himself for higher attainment [@Upanishad]. Book of Proverbs consists of wise and weighty sentences: regulating the morals of men: and directing them to wisdom and virtue [@Proverb]. Book of Ecclesiastes or The Preacher, (in Hebrew, Coheleth,) because in it, Solomon, as an excellent preacher, setteth forth the vanity of the things of this world: to withdraw the hearts and affections of men from such empty toys [@Ecclesiastes]. Book of Ecclesiasticus gives admirable lessons of all virtues [@Ecclesiasticus]. Book of Wisdom abounds with instructions and exhortations to kings and all magistrates to minister justice in the commonwealth, teaching all kinds of virtues under the general names of justice and wisdom [@Wisdom]. Buddhism :[*And what are fabrications? There are these six classes of intention: intention aimed at sights, sounds, aromas, tastes, tactile sensations, ideas. These are called fabrications.*]{} Tao Te Ching: [*Heaven and earth do not act from (the impulse of) any wish to be benevolent; they deal with all things as the dogs of grass are dealt with. The sages do not act from (any wish to be) benevolent; they deal with the people as the dogs of grass are dealt with. May not the space between heaven and earth be compared to a bellows? ’Tis emptied, yet it loses not its power; ’Tis moved again, and sends forth air the more. Much speech to swift exhaustion lead we see; Your inner being guard, and keep it free [@Budd].* ]{} Tao Te Ching: [*Heaven and earth do not act from (the impulse of) any wish to be benevolent; they deal with all things as the dogs of grass are dealt with. The sages do not act from (any wish to be) benevolent; they deal with the people as the dogs of grass are dealt with. May not the space between heaven and earth be compared to a bellows? ’Tis emptied, yet it loses not its power; ’Tis moved again, and sends forth air the more. Much speech to swift exhaustion lead we see; Your inner being guard, and keep it free [@Tao].*]{} Upanishad : [*The Brahman once won a victory for the Devas. Through that victory of the Brahman, the Devas became elated. They thought, “This victory is ours. This glory is ours.” Brahman here does not mean a personal Deity. There is a Brahma, the first person of the Hindu Trinity; but Brahman is the Absolute, the One without a second, the essence of all. There are different names and forms which represent certain personal aspects of Divinity, such as Brahma the Creator, Vishnu the Preserver and Siva the Transformer; but no one of these can fully represent the Whole. Brahman is the vast ocean of being, on which rise numberless ripples and waves of manifestation. From the smallest atomic form to a Deva or an angel, all spring from that limitless ocean of Brahman, the inexhaustible Source of life. No manifested form of life can be independent of its source, just as no wave, however mighty, can be independent of the ocean. Nothing moves without that Power. He is the only Doer. But the Devas thought: “This victory is ours, this glory is ours.” [@Upanishad]* ]{} Yogasutra : [*perception of the true nature of things. When the object is not truly perceived, when the observation is inaccurate and faulty, thought or reasoning based on that mistaken perception is of necessity false and unsound [@Yoga].*]{} Book of Proverb : [*Doth not wisdom cry aloud, and prudence put forth her voice? 8:2. Standing in the top of the highest places by the way, in the midst of the paths, 8:3. Beside the gates of the city, in the very doors she speaketh, saying: 8:4. O ye men, to you I call, and my voice is to the sons of men. 8:5. O little ones understand subtlety, and ye unwise, take notice. 8:6. Hear, for I will speak of great things: and my lips shall be opened to preach right things. 8:7. My mouth shall meditate truth, and my lips shall hate wickedness [@Proverb].*]{} Book of Ecclesiastes : [*Speak not any thing rashly, and let not thy heart be hasty to utter a word before God. For God is in heaven, and thou upon earth: therefore let thy words be few. 5:2. Dreams follow many cares: and in many words shall be found folly. 5:3. If thou hast vowed any thing to God, defer not to pay it: for an unfaithful and foolish promise displeaseth him: but whatsoever thou hast vowed, pay it. 5:4. And it is much better not to vow, than after a vow not to perform the things promised. 5:5. Give not thy mouth to cause thy flesh to sin: and say not before the angel: There is no providence: lest God be angry at thy words, and destroy all the works of thy hands. 5:6. Where there are many dreams, there are many vanities, and words without number: but do thou fear God [@Ecclesiastes].*]{} Book of Ecclesiasticus : [*Then Nathan the prophet arose in the days of David. 47:2. And as the fat taken away from the flesh, so was David chosen from among the children of Israel. 47:3. He played with lions as with lambs: and with bears he did in like manner as with the lambs of the flock, in his youth. 47:4. Did not he kill the giant, and take away reproach from his people? 47:5. In lifting up his hand, with the stone in the sling he beat down the boasting of Goliath: 47:6. For he called upon the Lord the Almighty, and he gave strength in his right hand, to take away the mighty warrior, and to set up the horn of his nation. 47:7. So in ten thousand did he glorify him, and praised him in the blessings of the Lord, in offering to him a crown of glory: 47:8. For he destroyed the enemies on every side, and extirpated the Philistines the adversaries unto this day: he broke their horn for ever. 47:9. In all his works he gave thanks to the holy one, and to the most High, with words of glory. 47:10. With his whole heart he praised the Lord, and loved God that made him: and he gave him power against his enemies: 47:11. And he set singers before the altar, and by their voices he made sweet melody [@Ecclesiasticus].*]{} Book of Wisdom : [*Love justice, you that are the judges of the earth. Think of the Lord in goodness, and seek him in simplicity of heart: 1:2. For he is found by them that tempt him not: and he sheweth himself to them that have faith in him. 1:3. For perverse thoughts separate from God: and his power, when it is tried, reproveth the unwise: 1:4. For wisdom will not enter into a malicious soul, nor dwell in a body subject to sins. 1:5. For the Holy Spirit of discipline will flee from the deceitful, and will withdraw himself from thoughts that are without understanding, and he shall not abide when iniquity cometh in. 1:6. For the spirit of wisdom is benevolent, and will not acquit the evil speaker from his lips: for God is witness of his reins, and he is a true searcher of his heart, and a hearer of his tongue [@Wisdom].*]{} These texts from sacred scripts originated in different geographical locations and at different historic time-line. The question arises is there are any similarity between them in terms what these texts want to teach and how they are teaching various religious lessons. Text Mining using machine learning and feature extraction is helpful in finding patterns of words in document collections [@Qahl]. Using text mining the aim of this research is to find if various sacred texts are strongly connected. The similarity measures such as Euclidean, Manhattan, Jaccard and Cosine is firstly applied to word frequency matrix of the raw corpus to find similarities based on word usage. The distance matrices on Document Term Matrix formed by LDA was calculated to find the similarities between texts based on probabilistic models [@Cao] [@Romain] by selecting k topics [@Rajkumar][@Thomas]. The unsupervised learning algorithm such as K mean clustering on raw frequency DTM reveals the strong similarity between sacred texts [@Bjornar]. Also supervised learning techniques like K-Nearest Neighbor, Support Vector Machine and Random Forest on labeled corpus was implemented to find if these algorithms can predict accurately if any chapter belongs to which sacred text. Methodology {#motiv} =========== Overview -------- The Figure  \[fig:method\] shows overview of the steps and algorithms used to find similarity between religious scripts. ![Various steps involved in finding similarity between scriptures[]{data-label="fig:method"}](method.PNG) Bag of Words assumes that each document is the fragment of text from a sacred book. The distinction between sacred books is supervised via the creation of corresponding label. The closeness of sacred books is found in terms of document distances calculated using various similarity measures. Similarity Measures ------------------- Throughout the rest of this paper, we will use the $p$-dimensional vector ${\mathbf{x}}_l = ({\mathrm{x}}_{l1}, {\mathrm{x}}_{l2}, \cdots, {\mathrm{x}}_{lp})^\top$ to denote the entries of the $l$th row of the term document matrix ${\bf X}$. Given two rows ${\mathbf{x}}_l$ and ${\mathbf{x}}_m$ of ${\bf X}$, we use the generic notation $d({\mathbf{x}}_l, {\mathbf{x}}_m)$ to denote the distance between the two rows, which is essentially the distance between two chosen chapters of the whole corpus regardless of which sacred book each belongs to. The chapter here is our basic document. In this section we introduce different mathematical distances grouped mathematically and we empirically evaluate their performance. Each distance family has specific mathematical properties that differentiates one another from each other. The effectiveness of applying the similarity measure is believed to be related to the mathematical properties of each family.\ The various similarity measures helps us to understand the similarity of various chapters within the same book and also similarity between different book in the corpus.\ The different measures used for the corpus: - The very commonly known Euclidean distance belongs to Minkowski Family. The Euclidean distance between two chapters in the corpus is calculated as: $$\begin{aligned} d_E({\mathbf{x}}_l, {\mathbf{x}}_m) = \left(\sum_{j=1}^p{({\mathrm{x}}_{lj}-{\mathrm{x}}_{mj})^2}\right)^{\frac{1}{2}}\end{aligned}$$ - Manhattan distance belongs to Minkowski Family and distance between two chapters of the corpus is defined as: $$\begin{aligned} d_M({\mathbf{x}}_l, {\mathbf{x}}_m) = \sum_{j=1}^p{|{\mathrm{x}}_{lj}-{\mathrm{x}}_{mj}|}\end{aligned}$$ - Cosine Similarity measure is the normalized inner product between two documents on the vector space that measures the cosine of the angle between them. The formula to find cosine similarity between two chapters can be written as: $$\begin{aligned} d_C({\mathbf{x}}_l, {\mathbf{x}}_m) = \frac{{\mathbf{x}}_l^\top{\mathbf{x}}_m}{({\mathbf{x}}_l^\top{\mathbf{x}}_l)^{\frac{1}{2}}({\mathbf{x}}_m^\top{\mathbf{x}}_m)^{\frac{1}{2}}}\end{aligned}$$ - The Jaccard similarity measures the intersection between two chapters. Jaccard coefficient is calculated using the formula: $$\begin{aligned} {\sf sim}({\mathbf{x}}_l,{\mathbf{x}}_m)= \frac{\displaystyle \sum_{j=1}^{p}{\min\{{\mathrm{x}}_{lj},{\mathrm{x}}_{mj}\}}}{\displaystyle \sum_{k=1}^{p}{\max\{{\mathrm{x}}_{lk},{\mathrm{x}}_{mk}\}}}\end{aligned}$$ The Jaccard distance between two chapters is defined as: $$\begin{aligned} d_J({\mathbf{x}}_l,{\mathbf{x}}_m)= 1 - {\sf sim}({\mathbf{x}}_l,{\mathbf{x}}_m)\end{aligned}$$ Using the above defined similarity measures on given books $X_a$ and $X_b$ we are trying to: - study $X_a$ or $X_b$ separately\ $d({\mathbf{X}}_l^{(a)}, {\mathbf{X}}_m^{(a)})\equiv$ distance between two chapters of same book $X^{(a)}$ This helps to discover the relationship of various chapters within the same book\ - study relationship between $X_a$ or $X_b$\ $$\begin{aligned} d({\mathbf{X}}^{(a)}, {\mathbf{X}}^{(b)}) = \underset{\substack{{{\mathbf{x}}_l \in {\mathbf{X}}^{(a)}}\\{{\mathbf{x}}_m \in {\mathbf{X}}^{(b)}}}}{\min}{\big\{d({\mathbf{x}}_l, {\mathbf{x}}_m)\big\}}\end{aligned}$$ We are calculating the mean, median, minimum and maximum distances between chapters of different books to discover the relationship between the books. $$d(X_a,X_b) = \begin{cases} \min \limits_{i \epsilon (1,..,n_a), j\epsilon (1,..,n_b)}\hspace{0.1 cm}{d(X_{ai},X_{bj})}\\ \max \limits_{i \epsilon (1,..,n_a), j\epsilon (1,..,n_b)}\hspace{0.1 cm} {d(X_{ai},X_{bj})}\\ \underset{i \epsilon (1,..,n_a), j\epsilon (1,..,n_b)}{\operatorname{average}} \hspace{0.1 cm}{d(X_{ai},X_{bj})}\\ \underset{i \epsilon (1,..,n_a), j\epsilon (1,..,n_b)}{\operatorname{median}} \hspace{0.1 cm}{d(X_{ai},X_{bj})} \\ \end{cases}$$ Within the book distance matrix helps to cluster the chapter in the same book and is represented as: $$D_X = \begin{bmatrix} d_{11} & d_{12} & d_{13} & \dots & d_{1n} \\ d_{21} & d_{22} & d_{23} & \dots & d_{2n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ d_{n1} & d_{n2} & d_{n3} & \dots & d_{nn} \end{bmatrix}$$ Distance between n chapters of script X\ Distance matrix between eight books helps to cluster books across the corpus and is represented as: $$\Delta = \begin{bmatrix} X_{11} & X_{12} & X_{13} & \dots & X_{18} \\ X_{21} & X_{22} & X_{23} & \dots & X_{28} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ X_{81} & X_{82} & X_{83} & \dots & X_{88} \end{bmatrix}$$ Supervised Learning Algorithms ------------------------------ Predictive aspects helps in prediction of the origin of fragments of spiritual literature. How well can we predict which sacred text a fragment of spiritual literature comes from? Three supervised algorithms: K-Nearest Neighbor, Support Vector Machine and Random Forest was applied on the labeled corpus. The supervised machines were trained on 70% of the corpus and tested on remaining 30%. The algorithm providing the maximum accuracy will be best in predicting the sacred text for a given chapter. Data Analysis ------------- Our goals with the data are - Create a corpus where document is smallest unit of data - Create Bag of Words DTM after data cleaning - Attempt to confirm or discover some of the closeness among scared texts using similarity measures - Measure the performance of supervised learning in identifying the book label for any document There are several challenges with the data: non uniform structure data in each sacred book, initial preprocessing reveals large amount of stop words data which can mislead the similarity measures. Through this paper, document analysis assumes that (a) document is the smallest unit of data being used for finding similarity (b) within the bag of words (BOW) assumption/approach, each document is represented by the words. Using the BOW assumption, our basic data structure after pre-processing, is the term document matrix (tdm) also known as the document term matrix (dtm), which can be written in the following $n \times p$ matrix $${\bf X} = \left[ \begin{array}{ccccccccc} X_{11} & X_{12} & \cdots & \cdots & \cdots & \cdots & X_{1j} & \cdots & X_{1p}\\ \vdots & \vdots & \ddots & \ddots & \cdots & \cdots & \cdots &\cdots & \vdots\\ X_{i1} & X_{i2} & \cdots & \cdots & \cdots & \cdots & X_{ij} & \cdots & X_{ip}\\ \vdots & \vdots & \ddots & \ddots & \cdots & \cdots & \cdots & \cdots & \vdots\\ X_{n1} & X_{n2} & \cdots & \cdots & \cdots & \cdots & X_{nj} & \cdots & X_{np}\\ \end{array} \right] \label{eq:dtm:1}$$ Each column $X_j$ of ${\bf X}$ represents a atomic word like [*truth*]{}, [*diligent*]{}, [*sense*]{}, [*power*]{}, [*right*]{}. In most document analysis tasks, the term document matrix ${\bf X}$ is typically very sparse, with $90\%$ of zeroes not unusual. Besides, except in rare cases, ${\bf X}$ tends to be ultra-high dimensional, meaning that $p \gg n$ as depicted in the matrix, since the number of words tends be much much higher than the number of documents to be text-analyzed. Depending on the analysis, the entries $X_{ij}$ of ${\bf X}$ can be of one of the following types: - $X_{ij}\equiv$ [Frequency of word $j$ in document $i$.]{} - $X_{ij}\equiv$ [logarithmized relative frequency of word $j$ in document $i$.]{} As indicated earlier, one of the most interesting questions one may seek to answer in the presence of a collection of documents dealing with the different sacred texts: [*are there any similarity between the various sacred texts ? If so, can we measure that?*]{} As we shall see later we will tackle this question using methods like [*K-means clustering*]{}. Specifically, if we anticipate $k$ groups of sacred texts, and denote by $P_k = C_1 \cup \cdots \cup C_k$, the partitioning of the data into $k$ groups/clusters, then we seek the optimum clustering. $$\begin{aligned} P_k^* = \underset{P_k}{\tt argmin}\left\{\sum_{j=1}^k{\sum_{i=1}^{n}{z_{ij}d({\mathbf{x}}_i,{\mathbf{x}}_j^*)}}\right\}, \label{eq:clustering:1}\end{aligned}$$ where $z_{ij}=\mathbb{L}({\mathbf{x}}_i \in C_j)$ and $d(\cdot)$ could be any distance like the Euclidean $d({\mathbf{x}}_i,{\mathbf{x}}_j^*)=\|{\mathbf{x}}_i-{\mathbf{x}}_j^*\|^2$ or the Manhattan distance $d({\mathbf{x}}_i,{\mathbf{x}}_j^*)=\|{\mathbf{x}}_i-{\mathbf{x}}_j^*\|_1$, or any other suitable distance. Section $3$ of this paper is dedicated to the exploration of the clustering of the documents in our corpus. The other question that naturally arises from such a corpus of documents is: [*For any given document can we predict which sacred text it belongs to?*]{} - [Data Processing]{} The unstructured nature of text data adds an extra layer of complexity in the feature extraction task, and the inherently sparse nature of the corresponding data matrices makes text mining a distinctly difficult task. To deal with this problem it was required to process that data. There was a need to clean the noise using Natural Language processing (NLP). - [Data Cleaning]{} The data cleaning involved removing of stop words using NLTK library. Apart from stop words present in library it was observed that the data required further cleaning. This was done by removing unnecessary punctuation marks, special characters and ancient English words which were not recognized as stop words by NLTK library. - [Data Sampling]{} The organization of the text was: - Books: Collection of entire script data - Paragraphs: Division of script based on the topic being explained - Chapters: Division of paragraph based on subtopic within each topic Unit of Sampling: Chapter was taken as smallest unit of sampling. Each religious scripts was fragmented to chapters and stored for further process of finding the similarities.These units existed in the text such as Tao Te Ching while in other books it was approximated from texts headings.\ Corpus $\equiv$ Various sacred texts Chapter $\equiv$ Collection of V words from corpus $Chapter_d$ $\equiv$ $x_d$ $\equiv$ ($x_{d1},x_{d2},..., x_{dv}$) $\equiv$ Input Vector $\equiv$ 1 chapter in a book\ - [Document Term Matrix (DTM) on Raw Text]{} The first input of similarity measures done using the raw corpus. The raw corpus in this case refers to corpora after applying data cleaning and processing. We are interested to handle the big corpus without any possible modification to test distance measures performance. Hence, the term document matrix of the raw texts was used as: The rows of our term document matrix refer to a fragment of text from one of the sacred books, which is a chapter in the sense adopted in this paper. The sacred book to which a document belongs is traced in a supervised manner with a variable $Y$ from the set of labels of all the books considered here namely $\mathcal{Y} = \{g_1, g_2, \cdots, g_8\}$ where - $g_1$ is Book $1$ containing chapters on [the teachings of the Buddha]{} - $g_2$ is Book 2 referring to the [Tao Te Ching]{} - $g_3$ is Book 3 referring to the [Upanishads]{} - $g_4$ is Book 4 referring to [YogaSutra]{} - $g_5$ is Book 5 referring to the [Book of Proverb]{} - $g_6$ is Book 6 referring to [Book of Ecclesiastes]{} - $g_7$ is Book 7 referring to [Book of Ecclesiasticus]{} - $g_8$ is Book 8 referring to [Book of Wisdom]{} Results ======= The minimum, maximum and average distances might contain outliers i.e chapters which are very similar to each other or quite dissimilar. To deal with this problem median of all distances of each chapter with every other chapter was used. The median distances was able to capture the similarities which do not take outliers into consideration.\ The Euclidean distance was able to separate the distances amongst different scripts while Cosine, Manhattan and Jaccard were unable to distinguish that. Figure \[fig:eucldist\] shows the Euclidean median distance of chapters within the same scripts and across the script. Between the scripts, distance is minimum between Upanishads and Tao Te Ching. Within the same script the distance of chapters within Upanishads is minimum (considering the diagonal). ![Euclidean median distance between different scripts[]{data-label="fig:eucldist"}](graph1_new.PNG) Figure \[fig:Buddhism\],  \[fig:Tao\],  \[fig:Upanishads\] and  \[fig:Yogasutra\] the Euclidean distance of chapters within the Asian scriptures which helps to find most similar chapters within the same script. [0.49]{} ![Euclidean distance between different chapters of Asian Religious scriptures[]{data-label="fig:AsianScripts"}](Buddhist_baseline_Euclidean_Distance_-_Copy.png "fig:"){width="\textwidth"} [0.49]{} ![Euclidean distance between different chapters of Asian Religious scriptures[]{data-label="fig:AsianScripts"}](TaoTeChing_baseline_Euclidean_Distance_-_Copy.png "fig:"){width="\textwidth"} [0.49]{} ![Euclidean distance between different chapters of Asian Religious scriptures[]{data-label="fig:AsianScripts"}](Upnishad_baseline_Euclidean_Distance.png "fig:"){width="\textwidth"} [0.49]{} ![Euclidean distance between different chapters of Asian Religious scriptures[]{data-label="fig:AsianScripts"}](YogaSutra_baseline_Euclidean_Distance_-_Copy.png "fig:"){width="\textwidth"} Figure \[fig:Proverb\],  \[fig:Ecc\],  \[fig:Eccle\] and  \[fig:wisdomchp\] shows the euclidean distance of chapters within the Bible texts which helps to find most similar chapters within the same book. [0.49]{} ![Euclidean distance between different chapters of Bible texts[]{data-label="fig:BibleScripts"}](BookProverb_baseline_Euclidean_Distance_-_Copy.png "fig:"){width="\textwidth"} [0.49]{} ![Euclidean distance between different chapters of Bible texts[]{data-label="fig:BibleScripts"}](BookEcclesiastes_baseline_Euclidean_Distance_-_Copy.png "fig:"){width="\textwidth"} [0.49]{} ![Euclidean distance between different chapters of Bible texts[]{data-label="fig:BibleScripts"}](BookEccleasiasticus_baseline_Euclidean_Distance_-_Copy.png "fig:"){width="\textwidth"} [0.49]{} ![Euclidean distance between different chapters of Bible texts[]{data-label="fig:BibleScripts"}](BookWisdom_baseline_Euclidean_Distance_-_Copy.png "fig:"){width="\textwidth"} Amongst all scripts, chapters within Upanishads were most similar to themselves which is shown in figure \[fig:Upanishads\]. The diagonals represent the distance of a chapter to itself thus resulting in minimum distance of 0. The strength of similarity between different scripts can be found by visualizing k-means clustering results calculated from Euclidean distances in figure \[fig:multiscale2\],  \[fig:multiscale3\]  \[fig:multiscale4\],  \[fig:multiscale5\],  \[fig:multiscale6\] and  \[fig:multiscale7\]. Each node is the network graph represents a script and strength between two scripts is proportional to the width and brightness of edge. The cluster number(k) varies from two to seven and each figure represents groups of similarity for different k. \[Nodes : Bdd = Buddhism / Tao = TaoTeChing/ Upd = Upanishad/ Yoga = YogaSutra/ Prv = Proverb/ Ecc = Ecclesiastes/ Ecs = Ecclesiasticus/ Wsd = Wisdom\] [0.49]{} ![Clustering with k = 2[]{data-label="fig:multiscale2"}](2cluster_Graph.png "fig:"){width="\textwidth"} [0.49]{} ![Clustering with k = 2[]{data-label="fig:multiscale2"}](2cluster_Tree.png "fig:"){width="\textwidth"} [0.49]{} ![Clustering with k = 3[]{data-label="fig:multiscale3"}](3Cluster_Euclidean_Median_Graph.png "fig:"){width="\textwidth"} [0.49]{} ![Clustering with k = 3[]{data-label="fig:multiscale3"}](3cluster_Tree.png "fig:"){width="\textwidth"} [0.49]{} ![Clustering with k = 4[]{data-label="fig:multiscale4"}](4Cluster_Graph.png "fig:"){width="\textwidth"} [0.49]{} ![Clustering with k = 4[]{data-label="fig:multiscale4"}](4cluster_Tree.png "fig:"){width="\textwidth"} [0.49]{} ![Clustering with k = 5[]{data-label="fig:multiscale5"}](5Cluster_Graph.png "fig:"){width="\textwidth"} [0.49]{} ![Clustering with k = 5[]{data-label="fig:multiscale5"}](5cluster_Tree.png "fig:"){width="\textwidth"} [0.49]{} ![Clustering with k = 6[]{data-label="fig:multiscale6"}](6cluster_Graph.png "fig:"){width="\textwidth"} [0.49]{} ![Clustering with k = 6[]{data-label="fig:multiscale6"}](6cluster_Tree.png "fig:"){width="\textwidth"} [0.49]{} ![Clustering with k = 7[]{data-label="fig:multiscale7"}](7cluster_Graph.png "fig:"){width="\textwidth"} [0.49]{} ![Clustering with k = 7[]{data-label="fig:multiscale7"}](7cluster_Tree.png "fig:"){width="\textwidth"} The figure  \[fig:multiscale3\] represents that Asian texts are more similar to themselves as compared to the Biblical texts. As we increase the number of cluster from 2 to 7 we can visualize the similarities amongst Asian scripts. While moving from k=3 to 5 all biblical texts belong to different clusters which means even 4 biblical texts are quite different from each other. At the end Upanishads and Tao Te Ching are the most similar scripts as they belong to same cluster when k=7. The performance of different supervised algorithms in predicting sacred text for any chapter in the scripture is shown in Table  \[table:KNN\],  \[table:SVM\] and  \[table:RF\]. Buddhism Ecclesiastes Ecclesiasticus Proverb Tao Upanishad Wisdom Yoga ---------------- ---------- -------------- ---------------- --------- ------- ----------- -------- ------- Buddhism **4** 0 0 0 0 0 0 0 Ecclesiastes 0 **0** 0 0 0 0 0 0 Ecclesiasticus 0 0 **0** 0 0 0 1 0 Proverb 0 0 4 **4** 0 0 0 0 Tao 0 0 0 0 **0** 0 0 0 Upanishad 10 3 7 3 23 **43** 3 61 Wisdom 0 0 1 0 0 0 **1** 0 Yoga 1 0 0 0 0 0 0 **5** : \[table:KNN\] Confusion matrix generated by KNN having accuracy = 0.339 Buddhism Ecclesiastes Ecclesiasticus Proverb Tao Upanishad Wisdom Yoga ---------------- ---------- -------------- ---------------- --------- ------- ----------- -------- -------- Buddhism **1** 0 0 0 0 0 0 0 Ecclesiastes 0 **0** 0 0 0 0 0 0 Ecclesiasticus 0 1 **10** 6 0 0 2 0 Proverb 0 0 0 **0** 0 0 0 0 Tao 0 0 0 0 **0** 0 0 0 Upanishad 0 0 1 1 0 **0** 0 0 Wisdom 0 0 0 0 0 0 **0** 0 Yoga 14 2 4 0 23 43 3 **66** : \[table:SVM\] Confusion matrix generated by SVM having accuracy = 0.435 Buddhism Ecclesiastes Ecclesiasticus Proverb Tao Upanishad Wisdom Yoga ---------------- ---------- -------------- ---------------- --------- -------- ----------- -------- -------- Buddhism **8** 0 0 0 0 0 0 0 Ecclesiastes 0 **0** 0 0 0 0 0 0 Ecclesiasticus 0 1 **14** 0 0 0 5 0 Proverb 0 0 1 **7** 0 0 0 0 Tao 0 0 0 0 **14** 0 0 0 Upanishad 7 0 0 0 8 **43** 0 8 Wisdom 0 0 0 0 0 0 **0** 0 Yoga 0 2 0 0 1 0 0 **58** : \[table:RF\] Confusion matrix generated by Random Forest having accuracy = 0.8136 Amongst all three supervised algorithms Random Forest has highest accuracy of predicting which sacred text a fragment of spiritual literature comes from, as shown in Table \[table:RF\]. The Upnaishads and Yogasutra have the largest number of chapters in the corpus and random forest is accurately able to predict most of the chapters for these two sacred texts which SVM and KNN fail to identify. Conclusions =========== After projecting Euclidean distances on various DTM (raw data DTM and normalized log DTM) we can conclude that the pattern of strong closeness exists among the different religious scripts. The similarity is driven by geography of origin of the religions. Bag of words is powerful to find the pattern of strong closeness between the four Asian religious scripts: Buddhism, Tao Te Ching, Upanishad and Yogasutra whose place of origin are geographical close. The two most similar scripts Tao Te Ching and Upanishad depicts the influence of two neighbouring countries China and India on their common religious teachings. An interesting potential work in this direction would be extracting main sematics features of the texts. Also, k-medoids using PAM can be implemented to observe the similarity between scripts. Using k-medoids ensures that the centers of clusters are actual points in the DTM and can give better results. This work also initiates the conversation about interesting results that be obtained from Markov models. [10]{} The Four Noble Truths: A Study Guide. Available at: https://www.accesstoinsight.org/lib/study/truths.html \[Accessed 25 Mar. 2018\]. 2010. Data. Three Things to Learn from Tao Te Ching - ’The’ Way, Not ’Your’ Way. \[online\] Tao Te Ching. Available at: http://tao-in-you.com/three-things-about-tao-te-ching/ \[Accessed 25 Mar. 2018\]. Data. The Yoga Sutras of Patanjali. Available at: http://www.gutenberg.org/files/2526/2526.txt \[Accessed 25 Mar. 2018\]. 2010. Data. The Upanishads. Available at: http://www.gutenberg.org/cache/epub/3283/pg3283.txt \[Accessed 25 Mar. 2018\]. 2014. Data. The Bible, Douay-Rheims, Book 22: Proverbs The Challoner Revision. Available at: http://www.gutenberg.org/cache/epub/8322/pg8322.txt \[Accessed 25 Mar. 2018\]. 2014. Data. The Bible, Douay-Rheims, Book 23: Ecclesiastes The Challoner Revision. Available at: http://www.gutenberg.org/cache/epub/8323/pg8323.txt \[Accessed 25 Mar. 2018\]. 2005. Data. The Bible, Douay-Rheims, Book 26: Ecclesiasticus The Challoner Revision. Available at: http://www.gutenberg.org/cache/epub/8326/pg8326.txt \[Accessed 25 Mar. 2018\]. 2005. Data. The Bible, Douay-Rheims, Book 25: Wisdom The Challoner Revision. Available at: http://www.gutenberg.org/cache/epub/8325/pg8325.txt \[Accessed 25 Mar. 2018\]. 2005. Data. An Automatic Similarity Detection Engine Between Sacred Texts Using Text Mining and Similarity Measures" (2014). Thesis. Rochester Institute of Technology. Fast and effective text mining using linear-time document clustering. In Proceedings of the fifth ACM SIGKDD international conference on Knowledge discovery and data mining (KDD ’99). ACM, New York, NY, USA, 16-22. On finding the natural number of topics with latent dirichlet allocation: Some observations. In Advances in knowledge discovery and data mining, Mohammed J. Zaki, Jeffrey Xu Yu, Balaraman Ravindran and Vikram Pudi (eds.). Springer Berlin Heidelberg, 391–402. 2010. A density-based method for adaptive lDA model selection. Neurocomputing — 16th European Symposium on Artificial Neural Networks 2008 72, 7–9: 1775–1781. 2009. Accurate and effective latent concept modeling for ad hoc information retrieval. Document numérique 17, 1: 61–84. 2014. Finding scientific topics. Proceedings of the National Academy of Sciences 101, suppl 1: 5228–5235. 2004.

--- abstract: 'Although there is a nearly universal agreement that type Ia supernovae are associated with the thermonuclear disruption of a CO white dwarf, the exact nature of their progenitors is still unknown. The single degenerate scenario envisages a white dwarf accreting matter from a non-degenerate companion in a binary system. Nuclear energy of the accreted matter is released in the form of electromagnetic radiation or gives rise to numerous classical nova explosions [*prior*]{} to the supernova event. We show that combined X-ray output of supernova progenitors and statistics of classical novae predicted in the single degenerate scenario are inconsistent with X-ray and optical observations of nearby early type galaxies and galaxy bulges. White dwarfs accreting from a donor star in a binary system and detonating at the Chandrasekhar mass limit can account for no more than $\sim 5\%$ of type Ia supernovae observed in old stellar populations.' author: - 'M.Gilfanov' - 'Á. Bogdán' title: Progenitors of type Ia supernovae in elliptical galaxies --- [ address=[Max-Planck-Institut für Astrophysik, Garching bei München, Germany]{} ,altaddress=[Space Research Institute, Russian Academy of Sciences, Moscow, Russia]{} ]{} [ address=[Harvard-Smithsonian Center for Astrophysics, Cambridge, MA, USA]{} ,altaddress=[Max-Planck-Institut für Astrophysik, Garching bei München, Germany]{} ]{} A carbon-oxygen white dwarf (WD) formed through standard stellar evolution can not be more massive than $\approx 1.1-1.2M_\odot$ [@weidemann]. Sub-Chandrasekhar models have been unsuccessful so far in reproducing observed properties of type Ia supernovae (SNe Ia) [@hoeflich96; @nugent97], although the effort continues [@sim2010]. In order to reach the Chandrasekhar mass of $\approx 1.38 M_\odot$ at least $\Delta M\ge 0.2M_\odot$ of matter needs to be accreted. Accretion of hydrogen-rich material onto the white dwarf is accompanied by hydrogen fusion on its surface, which is known to be unstable at small values of the mass accretion rate, giving rise to Classical Nova events [@nomoto07]. Because most of the accreted envelope and some of the original WD material is likely to be lost lost in the nova explosion [@prialnik], it is believed that the WD does not grow in mass if nuclear burning is unstable. For this reason steady burning regime is strongly favored by the accretion scenario [e.g. @livio_rev]. X-ray constraints ----------------- In steady nuclear burning regime, corresponding to the mass accretion rates $\dot{M}\ge 10^{-7}~M_\odot$/yr, energy of hydrogen fusion is liberated in the form of electromagnetic radiation, with luminosity of $$L_{nuc}=\epsilon_H X \dot{M} \sim 10^{37} {\rm ~ erg/s}$$ where $\epsilon_H\approx 6\cdot 10^{18}$ erg/g is energy release per unit mass and $X$ – hydrogen mass fraction (the solar value of $X=0.72$ is assumed). The nuclear luminosity exceeds by more than an order of magnitude the gravitational energy of accretion, $L_{grav}=GM\dot{M}/R$, and maintains the effective temperature of the WD surface at the level, defined by the Stefan-Boltzmann law: $$T_{eff}\approx 67 \left(\frac{\dot{M}}{5\cdot 10^{-7} M_\odot/yr}\right)^{1/4} \left(\frac{R_{WD}}{10^{-2}R_\odot}\right)^{-1/2} {\rm~eV}$$ Such a soft spectrum is prone to absorption by interstellar gas and dust, especially at smaller temperatures. Because the WD radius $R_{WD}$ decreases with its mass [@panei], the $T_{eff}$ increases as the WD approaches the Chandrasekhar limit – the signal, detectable at X-ray wavelengths, will be dominated by the most massive WDs (Fig.\[fig:kt\]) [@nat2010; @distefano2010apj]. The type Ia supernova rate scales with near-infrared luminosity of the host galaxy and for E/S0 galaxies $\dot{N}_{SNIa}\approx 3.5\cdot 10^{-4}$ yr$^{-1}$ per $10^{10}~L_{K,\odot}$ [@mannucci]. If the WD mass grows at a rate $\dot{M}$, a population of $$N_{WD}\sim \frac{\Delta M}{\dot{M}\left<\Delta t\right>}\sim \frac{\Delta M}{\dot{M}}\dot{N}_{SNIa} \label{eq:nwd}$$ accreting WDs is needed in order for one supernova to explode on average every $\left<\Delta t\right>=\dot{N}_{SNIa}^{-1}$ years. Thus, for a typical galaxy, the accretion scenario predicts a numerous population of accreting white dwarfs, $N_{WD}\sim {\rm few} \times (10^2 - 10^3)$, much more than numbers of soft X-ray sources actually observed [@distefano2010an]. Therefore, although the brightest and hottest of them may indeed reveal themselves as super-soft sources, the vast majority of SN Ia progenitors must remain unresolved or hidden from the observer, for example by interstellar absorption, in order for the accretion scenario to work. Their combined luminosity is $$L_{tot}=L_{nuc}\times N_{WD}=\epsilon X\Delta M\dot{N}_{SNIa} \label{eq:ltot}$$ where $\Delta M$ is the difference between the Chandrasekhar mass and the initial WD mass. Predicted luminosity can be compared with observations, after absorption and bolometric corrections are accounted for. --------------- ------------------------- ----------------- -------------------- -------------------- Name          $L_K$ \[$L_{K,\odot}$\]    $N_{WD}$      observed     predicted     observed     predicted   M32 $8.5\cdot 10^{ 8}$ $25$ $1.5\cdot 10^{36}$ $7.1\cdot 10^{37}$ NGC3377 $2.0\cdot 10^{10}$ $5.8\cdot 10^2$ $4.7\cdot 10^{37}$ $2.7\cdot 10^{39}$ M31 bulge $3.7\cdot 10^{10}$ $1.1\cdot 10^3$ $6.3\cdot 10^{37}$ $2.3\cdot 10^{39}$ M105 $4.1\cdot 10^{10}$ $1.2\cdot 10^3$ $8.3\cdot 10^{37}$ $5.5\cdot 10^{39}$ NGC4278 $5.5\cdot 10^{10}$ $1.6\cdot 10^3$ $1.5\cdot 10^{38}$ $7.6\cdot 10^{39}$ NGC3585 $1.5\cdot 10^{11}$ $4.4\cdot 10^3$ $3.8\cdot 10^{38}$ $1.4\cdot 10^{40}$ --------------- ------------------------- ----------------- -------------------- -------------------- : Comparison of the accretion scenario with observations [@nat2010]. X-ray luminosities refer to the soft (0.3–0.7 keV) band. The columns marked “predicted” display total number and combined X-ray luminosity (absorption applied) of accreting WDs in the galaxy predicted in case the single degenerate scenario would produce all SNe Ia. In computing predicted numbers we halved the SN Ia rates as discussed in [@nat2010], other parameters are: $\dot{M}=10^{-7}~M_\odot$/yr, initial WD mass $1.2M_\odot$. \[tab:lx\] To this end we collected archival data of X-ray (Chandra) and near-infrared (Spitzer and 2MASS) observations of several nearby gas-poor elliptical galaxies and for the bulge of M31 ([@bogdan2010; @nat2010], Table \[tab:lx\]). Using K-band measurements to predict the SN Ia rates, we computed combined X-ray luminosities of SN Ia progenitors expected in the accretion scenario and compared them with Chandra observations. Obviously, the observed values present upper limits on the luminosity of the hypothetical population of accreting WD, as there may be other types of X-ray sources contributing to the observed emission. As is clear from the Table \[tab:lx\], predicted luminosities surpass observed ones by a factor of $\sim 30-50$, i.e. the accretion scenario is inconsistent with observations by a large margin. Statistics of recurrent and classical novae ------------------------------------------- Unstable nuclear burning at low $\dot{M}$ is not generally believed to allow accumulation of mass, sufficient to lead to supernova explosion. However, just below the stable burning limit a considerable fraction of the envelope mass can be retained by the WD during the nova explosion [@prialnik]. This motivated some authors to propose recurrent novae as supernova progenitors [e.g. @hachisu2001]. We demonstrate below that such systems will overproduce nova explosions in galaxies. Assuming that classical/recurrent nova sources are the main type of SN Ia progenitors, one can relate the nova and supernova rates: $$\Delta M_{acc}\, \dot{N}_{CN} \sim \Delta M_{SN}\, \dot{N}_{SN}$$ where $\Delta M_{acc}\le 10^{-7}-10^{-4}$ M$_\odot$ is the mass accumulated by the WD per one nova outburst cycle, $\Delta M_{SN}\sim 0.5$ M$_\odot$ is the mass needed for the WD to reach the Chandrasekhar limit. As $\Delta M_{acc}$ depends on the $\dot{M}$ and WD mass (Fig.\[fig:cn\]), we write more precisely: $$\frac{\dot{N}_{CN}}{\dot{N}_{SNIa}}=\int\frac{dM_{WD}}{\Delta M_{acc}(M_{WD},\dot{M})}\ge \int\frac{dM_{WD}}{\Delta M_{CN}(M_{WD},\dot{M})} \label{eq:cn2sn}$$ where $\Delta M_{CN}$ is the mass of the hydrogen shell required for the nova outburst to start. The inequality in the eq.(\[eq:cn2sn\]) follows from the fact that $\Delta M_{acc}\le \Delta M_{CN}$, due to the possible mass loss during the nova outburst. As the $\Delta M_{CN}$ decreases steeply with the WD mass (Fig.\[fig:cn\]), the main contribution to the predicted CN rate is made by the most massive WDs, similar to X-ray emission in the steady nuclear burning regime. They will be producing frequent outbursts with relatively short decay times [@prialnik], thus resulting in a large population of fast (some of them recurrent) novae. This will contradict to the statistics of CNe, as illustrated by the example of M31 shown in Fig.\[fig:cn\]. Indeed, the observed rate of CN with decay time shorter than 20 days in this galaxy is $\approx 5.2\pm1.1$ yr$^{-1}$ [@capaccioli], while eq.(\[eq:cn2sn\]) predicts $\sim 300$ yr$^{-1}$ for the mass accretion rates relevant to the recurrent novae-based progenitors models, $\dot{M}\sim10^{-8}$ M$_\odot$/yr. As $\Delta M_{CN}$ is larger at small $\dot{M}$ (Fig.\[fig:cn\]), the contradiction between observed and predicted nova frequencies becomes less dramatic at smaller $\dot{M}$. However, very low values of $\dot{M}\ll 10^{-10}$ M$_\odot$/yr are not feasible in the context of SN Ia progenitors. More realistic models with $\dot{M}\ge 10^{-8}$ M$_\odot$/yr do not produce more that $\sim 2\%$ of type Ia supernovae. Conclusion ---------- Thus, no more than $\sim$few per cent of SNe Ia in early type galaxies can be produced by white dwarfs accreting from a donor star in a binary system and exploding at the Chandrasekhar mass limit. In the steady nuclear burning regime the supernova progenitors would emit too much of soft X-ray emission, while if nuclear burning is unstable they would overproduce classical nova explosions. At very high $\dot{M}$ the white dwarf could grow in mass without conflicting X-ray constraints or nova statistics, but would do this rather inefficiently, because a significant fraction of the transferred mass is lost in the wind [@hachisu; @vdh97]. Therefore a relatively massive ([*at least*]{} $M\ge 1.3-1.7$ M$_\odot$) donor star is required for the white dwarf to reach the Chandrasekhar limit in this regime. Because the lifetimes of such stars do not exceed $\sim$few Gyr, this mechanism may work only in late-type and in the youngest of early-type galaxies. As relevance of sub-Chandraskhar models is still debated [@hoeflich96; @nugent97; @sim2010], the only currently viable alternative are WDs mergers [@iben; @webbink]. This mechanism may be the main formation channel for SN Ia in early type galaxies. In late-type galaxies, on the contrary, massive donor stars are available, making the mass budget less prohibitive, so that WDs can grow to the Chandrasekhar mass entirely inside an optically thick wind or, via accretion of He-rich material from a He donor star [@iben94]. In addition, a star-forming environment is usually characterized by large amounts of neutral gas and dust, leading to increased absorption obscuring soft X-ray radiation from accreting WDs. Therefore in late-type galaxies the accretion scenario may play a significant role, explaining, for example, the population of prompt supernovae [@bogdan2010a]. [9]{} Bogdán Á & Gilfanov M., 2010, A&A, 512, 168 Bogdán Á & Gilfanov M., 2011, MNRAS, 412, 401 Capaccioli, M., della Valle, M., Rosino, L. & D’Onofrio, M., 1989, Astron. J., 97, 1622 Di Stefano R. et al., 2010, Astronomische Nachrichten, 331, 205 Di Stefano R. et al., 2010, ApJ, 712, 728 Gilfanov M. & Bogdán Á, 2010, Nature, 463, 924 Hachisu, I., Kato, M. & Nomoto, K., 1996, Astrophys. J. Lett., 470, L97 Hachisu I. & Kato M. 2001, ApJ 558, 323 Hoeflich P. & Khokhlov A., 1996, Astrophys. J., 457, 500 Iben I. Jr. & Tutukov A. V., 1984, Astrophys. J. Suppl., 54, 335 Iben, I. Jr. & Tutukov, A. V., 1994, Astrophys. J., 431, 264 Kahabka, P. & van den Heuvel, E. P. J., 1997, Annu. Rev. Astron. Astrophys., 35, 69 Li, X.-D. & van den Heuvel, E. P. J., 1997, Astron. Astrophys., 322, L9 Livio M., 2000, Type Ia Supernovae, Theory and Cosmology. Eds.: J. C. Niemeyer and J. W. Truran. Cambridge University Press, 2000, 33 Mannucci, F. et al., 2005, Astron. Astrophys., 433, 807 Nugent, P., Baron, E., Branch, D. et al., 1997, Astrophys. J., 485, 812 Nomoto, K., Saio, H., Kato, M. & Hachisu, I., 2007, Astrophys. J., 663, 1269 Panei, J. A., Althaus, L. G. & Benvenuto, O. G., 2000, Astron. Astrophys., 353, 970 Sim S. et al., ApJ, 2010, 714, L52 Yaron, O., Prialnik, D., Shara, M. M. & Kovetz, A., 2005, Astrophys. J., 623, 398 Webbink R., 1984, Astrophys. J., 277, 355 Weidemann W., 2000, Astron. Astrophys., 363, 647

--- abstract: 'We consider the Landau-Zener problem for a multilevel quantum system that is coupled to an external environment. In particular, we consider a number of cases of three-level systems coupled to a harmonic oscillator that represents the external environment. We find that, similarly to the case of the Landau-Zener problem with a two-level system, when the quantum system and the environment are both initially in their ground states the probability that the system remains in the same quantum state is not affected by the coupling to the environment. The final occupation probabilities of the other states are well described by a common general principle: the coupling to the environment turns each Landau-Zener transition process in the closed system into a sequence of smaller transitions in the combined Hilbert space of the system and environment, and this sequence of transitions lasts a total duration that increases with increasing system-environment coupling strength. These results provide an intuitive understanding of Landau-Zener transitions in open multilevel quantum systems.' author: - 'S. Ashhab' title: 'Landau-Zener transitions in an open multilevel quantum system' --- Introduction {#Sec:Introduction} ============ The Landau-Zener (LZ) problem is one of the basic paradigms in the physics of quantum systems under the influence of time-dependent Hamiltonians. Specifically, the LZ problem relates to the evolution of the system when two or more energy levels experience an avoided crossing as the external parameters are varied in time. The basic problem with two energy levels and a linear sweep of the external parameter turns out to be simple enough that an analytic solution for the dynamics can be obtained [@Landau; @Zener; @Stueckelberg; @Majorana]. While the two-level problem is extremely valuable in understanding the dynamics of quantum systems at avoided crossings of energy levels, in many realistic problems there are more than two energy levels that (nearly) intersect each other in a certain region in parameter space. One example of such systems is nanomagnets, where experiments have shown the need to go beyond the two-level LZ model [@Wernsdorfer]. Recent experiments on Landau-Zener-Stückelberg interferometry in superconducting circuits have also involved multiple energy levels [@Berns; @Sun; @Shevchenko]. The field of adiabatic quantum computation (AQC) [@Farhi; @Johnson; @Lanting] is another area where recent experiments have shown that a large number of energy levels come close to each other at the most crucial point in the time evolution. Problems related to conical intersections in molecules, relevant to many photo-chemical reactions, also often involve multiple electronic states. The multilevel LZ problem has been studied quite extensively in the literature [@Demkov1968; @Carrol; @Brundobler; @Ostrovsky; @Usuki; @Demkov2000; @Sinitsyn2002; @Shytov; @Vasilev; @Kenmoe; @Kiselev; @Sinitsyn2014; @Sinitsyn2015a; @Sinitsyn2015b; @Patra]. Most studies have focused on finding special cases that allow analytic solutions. The approach in these studies is generally to identify special cases of the multilevel problem where the equations of motion can be reduced to those of the two-level LZ problem, and as a result the transition probabilities of the generalized models generally turn out to be given by products of the usual LZ transition probability $P_{\rm LZ}=\exp\{-2\pi\delta\}$, where the adiabaticity parameter $\delta=\Delta^2/(4v)$, $\Delta$ is the minimum gap at the center of the avoided crossing, and $v$ is the sweep rate (possibly rescaled to take into account the slopes of the energy levels). As mentioned above, the two-level LZ problem can be solved analytically [@Landau; @Zener; @Stueckelberg; @Majorana]. When one introduces a thermal environment, the problem becomes more complex and in general does not allow an analytical solution. Instead, numerous studies have tackled the problem using numerical calculations under a variety of assumptions and approximations [@Kayanuma; @Gefen; @Ao; @Shimshony; @Nishino; @Pokrovsky2003; @Sarandy; @Ashhab2006; @Lacour; @Pokrovsky2007; @Amin; @Nalbach2009; @Nalbach2013; @Dodin; @Xu; @Haikka; @Nalbach2014; @Ashhab2014; @Javanbakht; @Wild]. One notable exception is the special case where the system and environment are initially in their ground state, in which case the problem can be solved analytically [@Wubs]. One interesting and important result that one finds in this case is the fact that for longitudinal system-environment coupling the final occupation probabilities of the system’s two quantum states are unaffected by the coupling to the environment, even though the environment could end up in a highly excited state depending on the various details of the problem. Results such as this one raise the question of whether there are similar results in the case of multilevel systems. Here we treat a number of three-level LZ problems where the system of interest is coupled to a harmonic oscillator that models the external environment. Following the approach of Ref. [@Ashhab2014], we numerically solve the time-dependent Schrödinger equation of the large system comprising the three-level system that experiences the avoided crossing in addition to the harmonic oscillator. We find that an extension of the result concerning the environment independence of the system’s final state is obtained in this case as well, and we find some general principles governing the dynamics in the open-system multilevel LZ problem. The remainder of this paper is organized as follows: In Sec. \[Sec:Hamiltonian\] we describe the basic setup and introduce the corresponding Hamiltonian. In Sec. \[Sec:NumericalCalculations\] we describe our numerical calculations. In Sec. \[Sec:Results\] we present the results of these calculations and discuss the interpretation of the results. Section \[Sec:Conclusion\] contains some concluding remarks. Model system and Hamiltonian {#Sec:Hamiltonian} ============================ We consider a multilevel quantum system with a linearly changing Hamiltonian. In other words, the Hamiltonian is given by $$H_S = \hat{A} + \hat{B} t, \label{Eq:HamiltonianClosedSystem}$$ where $\hat{A}$ and $\hat{B}$ are time-independent operators, and the time variable $t$ goes from $-\infty$ at the initial time to $+\infty$ at the final time. At both extremes of the time variable, i.e. when $|t|\rightarrow\infty$, the second term dominates and the energy eigenstates of the system are the eigenstates of $\hat{B}$ [@DegeneracyFootnote]. As a result, the energy eigenstates at the initial and final times are the same, except for the fact that their order in the energy level ladder changes. In the case of a two-level system (TLS), the Hamiltonian can always be expressed in the form $$H_{\rm TLS} = -\frac{vt}{2} \hat{\sigma}_{z}-\frac{\Delta}{2} \hat{\sigma}_{x}. \label{Eq:TwoLevelLZHamiltonianClosedSystem}$$ The simple form of this Hamiltonian leads to the result that the probability of making a transition between the quantum states is determined by a single parameter, i.e. the parameter $\delta$ defined in Sec. \[Sec:Introduction\] or equivalent combinations of $\Delta$ and $v$. ![Energy level diagrams of the three-level LZ problems considered in this paper. The three models analyzed here are the equal-slope model (a), the bow-tie model (c) and the triangle model (d). The time and energy axes, as well as the magnified avoided crossing, are not shown in Panels (c) and (d) to avoid unnecessary duplication. Panel (b) shows the energy level diagram for the equal-slope model (from Panel a) when the environment degree of freedom is included. The first number in each state label represents the state of the three-level system, while the second number represents the state of the environment starting from zero and going up to infinity. It should be noted that the environment states generally depend on the system state. For example the environment component of the state ${\left| 1,0 \right\rangle}$ is generally different from that of the state ${\left| 2,0 \right\rangle}$.[]{data-label="Fig:EnergyLevelDiagram"}](DissipativeLandauZenerWithMultipleLevelsFigEnergyLevelDiagram.eps){width="14.0cm"} In the general case with more than two energy levels, the intermediate region can in general be a complex network of avoided crossings between all the energy levels. Even for a three-level system, the number of parameters becomes large enough that one has to consider special cases of the problem. Here we shall focus on the three-level problem and analyze the three representative cases shown in Fig. \[Fig:EnergyLevelDiagram\] and described in Table \[Table:DifferentCases\]. First, we shall consider the equal-slope model, where the energy levels are divided into two groups and all the energy levels in each group have the same slope. In the present case with only three energy levels, one group will have two energy levels and the other group will have a single energy level. We shall show results only for the special case where the nonzero off-diagonal matrix elements in $\hat{A}$ are equal to each other and to the energy difference between the two parallel energy levels, but we have verified that our main results are unaffected by this specific choice, and the conclusions that we shall draw from our results should apply in the general case with any choice of parameters. Next we shall consider the bow-tie model, where the diagonal matrix elements of $\hat{A}$ are all equal to zero and all the energy levels would meet at a single point if it were not for the off-diagonal matrix elements in $\hat{A}$, which create the avoided-crossing structure. We note here that in general the bow-tie model allows quite a bit of freedom in choosing the parameters: the only requirement is that the off-diagonal matrix elements in $\hat{A}$ couple only one of the $N$ quantum states to all the other states with no direct coupling between any of these $N-1$ states. The asymptotic slopes for example do not need to satisfy any special relation. As such, we consider one special case of the bow-tie model, but we have verified that our main results are independent of this choice. Finally we shall consider the triangle model, which is in some sense the most general among three-level LZ models, because it does not have any of the symmetries contained in the other two models. The lack of any symmetry also leads to the result that there is no analytic solution for the transition probabilities in this model. As with the other two models, we consider a specific (and somewhat arbitrary) choice of parameters with the assumption that the results will contain the essential physics that is generally expected in this model. [cccc]{} Name & $\hat{A}\times 2/\Delta$ & $\hat{B}/v$\ \ Equal slope & $\left( \begin{array}{ccc} 0 & 1 & 1 \\ 1 & 0 & 0 \\ 1 & 0 & 1 \end{array} \right)$ & $\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{array} \right)$\ \ Bow tie & $\left( \begin{array}{ccc} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{array} \right)$ & $\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{array} \right)$\ \ Triangle & $\left( \begin{array}{ccc} 0 & 1 & 0.8 \\ 1 & -2 & 0.55 \\ 0.8 & 0.55 & 0 \end{array} \right)$ & $\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -\frac{1}{2} \end{array} \right)$ As mentioned above, the LZ problem with a three-level system generally contains several parameters, and defining an adiabaticity parameter is not as straightforward as it is in the case of a two-level system. However, once we specify the operators $\hat{A}$ and $\hat{B}$ with overall coefficients $\Delta$ and $v$ as shown in Table \[Table:DifferentCases\], we can define an overall adiabaticity parameter $\delta=\Delta^2/(4v)$. We shall use this parameter when we present our results below. As explained in Ref. [@Ashhab2014], each method for modeling the environment in the study of the LZ problem with an open quantum system has its strengths and weaknesses. Here we use the same method as in Ref. [@Ashhab2014], i.e. we include a harmonic oscillator that represents the external environment. With this model, we can numerically integrate the Schrödinger equation and obtain results whose correctness does not require making any [*a-priori*]{} assumptions about the dynamics. On the other hand, we might have to use logical arguments at the end to infer from our results how the system would behave under the influence of a large environment, e.g. composed of a continuum of harmonic oscillators. The Hamiltonian describing the multilevel LZ problem, a harmonic oscillator and coupling between the two (with the common assumption that the coupling is longitudinal and linear in the oscillator’s degree of freedom) is given by: $$H = \tilde{A} + \hat{B} t + \hbar \omega \hat{a}^{\dagger} \hat{a} + \hat{C} \otimes \left( \hat{a} + \hat{a}^{\dagger} \right), \label{Eq:HamiltonianSystemPlusHarmonicOscillator}$$ where $\omega$ is the characteristic frequency of the harmonic oscillator, and $\hat{a}$ and $\hat{a}^{\dagger}$ are, respectively, the oscillator’s annihilation and creation operators. Here we assume longitudinal coupling because it leads to the intuitively natural property that away from the avoided crossings the environment only causes dephasing between the energy eigenstates of the system. It should be noted, however, that transverse coupling leads to interesting results as well, as discussed in Refs. [@Wubs; @Javanbakht; @Demkov2000; @Zueco]. The reason why we use the modified operator $\tilde{A}$ here is that the system-environment coupling causes all the energy levels corresponding to the same eigenvalue (${\left| b_i \right\rangle}$) of $\hat{B}$ to be asymptotically shifted by $$\Delta E_i = - \frac{{\left\langle b_i \right|} \hat{C} {\left| b_i \right\rangle}^2}{\hbar\omega}.$$ The appearance of this shift can be understood by considering the last two terms in Eq. (\[Eq:HamiltonianSystemPlusHarmonicOscillator\]): depending on the state of the three-level system, the environment’s ground state energy is shifted down by $\Delta E_i$, which in turn acts as an effective shift in the energy levels of the three-level system. In order to correct for this shift, we define the operator $$\tilde{A} = \hat{A} + \sum_i {\left| b_i \right\rangle} \frac{{\left\langle b_i \right|} \hat{C} {\left| b_i \right\rangle}^2}{\hbar\omega} {\left\langle b_i \right|}$$ and use it in the total Hamiltonian instead of using the original operator $\hat{A}$. If we did not make this change, the bow-tie model for example would in general turn into a triangle model because of the different shifts in the energy levels. In order to cover several possibilities for the decoherence rates between the different quantum states, we shall use five different system-environment coupling operators in our analysis. One of these is the rather generic $$\hat{C}_{1:3} = g \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & \frac{1}{3} & 0 \\ 0 & 0 & 1 \end{array} \right),$$ where the coefficient $g$ quantifies the overall strength of the system-environment coupling. The fact that all the diagonal matrix elements are different from each other means that the environment causes decoherence between all three quantum states of the system. The number $1/3$ is somewhat arbitrary, with the only considerations that we have taken in choosing it being that (1) we would like it to be well inside the interval (0,1) in order to cause decoherence between all three states and (2) we do not want to choose the value 1/2 in order to avoid accidental symmetries associated with the fact that 1/2 is exactly in the middle between 0 and 1. For comparison, we also perform calculations using the operator $$\hat{C}_{3:1} = g \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & \frac{1}{3} \end{array} \right).$$ In the case of the equal-slope model, given that the externally tuned parameter does not affect the energetic separation between the states ${\left| 2 \right\rangle}$ and ${\left| 3 \right\rangle}$, it is quite possible that the environment might similarly not cause any fluctuations in their energetic separation and therefore not cause any significant decoherence in the subspace spanned by these two states. To treat this case, we use the operator $$\hat{C}_{1:1} = g \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right).$$ In order to gain further insight into how the environment affects the dynamics, it is also useful to consider the two other alternatives where the environment does not decohere superpositions of two out of the three quantum states: $$\begin{aligned} \hat{C}_{0:1} = g \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{array} \right), \nonumber \\ \hat{C}_{1:0} = g \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array} \right).\end{aligned}$$ Having described the different combinations of operators and parameters that we use in our analysis, next we describe further details about the numerical calculations. Numerical calculations {#Sec:NumericalCalculations} ====================== We solve the time-dependent Schrödinger (or Liouville-von Neumann) equation using numerical integration with the Hamiltonian given in Eq. (\[Eq:HamiltonianSystemPlusHarmonicOscillator\]). We use a Hilbert space constructed from product states of the system states and the lowest fifty Fock states in the harmonic oscillator. Hence we use a 150-dimensional Hilbert space in our calculations. We have verified that increasing the size of the Hilbert space to 210 does not affect the results presented here. We have set $\hbar\omega$ to different values around $\Delta$ in different calculations, and we find that the results are generally unaffected by the exact value. All the results that presented below were obtained with $\hbar\omega=1.2\Delta$. It should be noted here that the effects of the environment on the LZ problem can be divided into two different types depending on the frequency of the environment or external noise. This distinction is generally similar to that encountered in the study of decoherence in an undriven quantum system, namely low-frequency noise causing dephasing and noise components that are resonant with specific transitions causing relaxation. More detailed discussions of these questions can be found in Refs. [@Ashhab2006; @Nalbach2013; @Nalbach2014; @Ashhab2014]. With this consideration in mind, our calculations are well suited to capture the effects of resonant frequency components but could miss some low-frequency-related effects. As discussed in Ref. [@Ashhab2014] setting $\hbar\omega$ to a small value would require us to keep a large Hilbert space, which could lead to extremely long computation times. We shall therefore not perform such calculations here. Our simulations are all started with the system and environment initialized in their ground states and the time variable set to $vt=-100\Delta$, which is sufficiently large that the system and environment barely experience any effect of the avoided crossings in the initial stages of the evolution. We evolve the time-dependent Schrödinger equation from this initial time to the final time given by $vt=100\Delta$. This latter value is sufficiently large that the occupation probabilities of the different quantum states will be very close to their asymptotic values. At the final time we calculate the occupation probabilities of the three system states. Because we choose a large value of the final time, each energy eigenstate of the large system comprising the three-level system and the harmonic oscillator lies almost completely in the subspace that corresponds to one of the eigenstates of the operator $\hat{B}$, making the classification of the energy eigenstates based on the $\hat{B}$ eigenstates straightforward. It should be noted that there is no such simple correspondence at intermediate times when the energy levels are undergoing avoided crossings. Results {#Sec:Results} ======= In this section we present the results of the numerical simulations described in Sec. \[Sec:NumericalCalculations\] for the three models described in Sec. \[Sec:Hamiltonian\] and a variety of system-environment coupling operators. Equal-slope model ----------------- ![Final occupation probabilities of the three quantum states of the system as functions of the adiabaticity parameter $\delta$ and the system-environment coupling strength $g$ for the equal-slope model. The different rows correspond to different choices of $\hat{C}$ while the different columns correspond to the three system states.[]{data-label="Fig:OccupationProbabilitiesEqualSlope"}](DissipativeLandauZenerWithMultipleLevelsFigColorEqualSlope.eps){width="17cm"} We start with the equal-slope model. In Fig. \[Fig:OccupationProbabilitiesEqualSlope\] we plot the final occupation probabilities of the different quantum states of the three-level system as functions of the adiabaticity parameter $\delta$ and the coupling strength $g$ for the three cases defined by $\hat{C}_{1:3}$, $\hat{C}_{1:1}$ and $\hat{C}_{3:1}$. In the absence of coupling to the environment, i.e. when $g=0$, the final occupation probabilities of the states ${\left| 1 \right\rangle}$ and ${\left| 2 \right\rangle}$ generally follow the dependence seen in the two-level LZ problem: in the fast-sweep limit ($\delta\rightarrow 0$) the quantum system remains in the state ${\left| 1 \right\rangle}$, while in the adiabatic limit ($\delta\rightarrow\infty$) the system adiabatically follows the ground state and therefore ends up in the state ${\left| 2 \right\rangle}$ at the final time. As already discussed in the literature, the occupation probability of the state ${\left| 3 \right\rangle}$ vanishes in both limits, but it has a maximum value of 0.25 attained at $4\pi\delta=4\ln 2=2.77$. When we now consider the effect of the environment, the first observation that we make from Fig. \[Fig:OccupationProbabilitiesEqualSlope\] is that the probability to remain in the state ${\left| 1 \right\rangle}$ is not affected by the coupling to the (zero-temperature) environment. While this result might be counter-intuitive, it is consistent with similar results that are well established in the literature, as can be seen in Refs. [@Wubs; @Sinitsyn2002]. We note here that to our knowledge there is no simple intuitive explanation for this result, even in the two-level case. The dependence of the occupation probabilities for the two other quantum states also contains seemingly surprising results. In the case of $\hat{C}_{1:3}$, the occupation probabilities of the states ${\left| 2 \right\rangle}$ and ${\left| 3 \right\rangle}$ are almost independent of the coupling to the environment. In fact, if we use the operator $\hat{C}_{0:1}$, which does not decohere superpositions of the states ${\left| 1 \right\rangle}$ and ${\left| 2 \right\rangle}$, the occupation probabilities become completely independent of $g$. In the case of $\hat{C}_{1:1}$, i.e. when the environment does not decohere quantum superpositions of the states ${\left| 2 \right\rangle}$ and ${\left| 3 \right\rangle}$, the final occupation probabilities of the states ${\left| 2 \right\rangle}$ and ${\left| 3 \right\rangle}$ steadily approach each other and seem to asymptotically coincide with each other as $g$ is increased. In the case of $\hat{C}_{3:1}$ (or $\hat{C}_{1:0}$), the probability of the state ${\left| 2 \right\rangle}$ decreases and that of the state ${\left| 3 \right\rangle}$ increases with increasing $g$. Furthermore, when $g$ becomes larger than both $\Delta$ and $\hbar\omega$, the probability of the state ${\left| 3 \right\rangle}$ becomes larger than that of the state ${\left| 2 \right\rangle}$. ![Schematic diagram showing how the energy level structure with a single avoided crossing in the two-level LZ problem turns into a structure with an infinite number of energy levels and avoided crossings when the environment degrees of freedom are included. In particular, if the combined system is initially in the state ${\left| 1,0 \right\rangle}$, its final occupation probabilities are governed by the infinite sequence of avoided crossings between the state ${\left| 1,0 \right\rangle}$ and the states ${\left| 2,0 \right\rangle},{\left| 2,1 \right\rangle},{\left| 2,2 \right\rangle},\cdots$.[]{data-label="Fig:TwoLevelLandauZenerWithEnvironment"}](DissipativeLandauZenerWithMultipleLevelsFigEnergyLevelDiagramForTLS.eps){width="8cm"} The three different types of behavior described above can be understood using a single principle that can be deduced from considering the LZ problem with a two-level system in the presence of an environment. In that case, as is illustrated in Fig. \[Fig:TwoLevelLandauZenerWithEnvironment\], the single avoided crossing between the system states ${\left| 1 \right\rangle}$ and ${\left| 2 \right\rangle}$ turns into an infinite sequence of avoided crossings between the states ${\left| 1,0 \right\rangle}$ and ${\left| 2,0 \right\rangle}$, ${\left| 2,1 \right\rangle}$, ${\left| 2,2 \right\rangle}$, ..., where the second index describes the state of the environment (with the label 0 denoting the ground state of the environment, including any system-state-dependent corrections). The energy gaps of these avoided crossings follow the function $\alpha^n\exp\{-\alpha^2/2\}L_0^n(\alpha^2)/\sqrt{n!}$, where $\alpha\propto g/(\hbar\omega)$ (with the exact relation determined by the details of $\hat{C}$), $n$ is the environment state index, and $L_0^n$ are associated Laguerre polynomials, which gives a distribution peaked around $n=\alpha^2$ and with width proportional to $\alpha$ [@Ashhab2010]. As the system-environment coupling strength $g$ increases, the final occupation probability spreads among a larger number of final states and the population also shifts up to higher excited states in the environment. This principle can be translated to the LZ problem with a three-level system. If we consider the dynamics in the limit where the decoherence between the states ${\left| 1 \right\rangle}$ and ${\left| 2 \right\rangle}$ vanishes, some probability is transferred from the state ${\left| 1,0 \right\rangle}$ to the state ${\left| 2,0 \right\rangle}$ at their avoided crossing, and all subsequent dynamics involves transfers of probability from the state ${\left| 1,0 \right\rangle}$ to states of the form ${\left| 3,n \right\rangle}$ with increasing values of $n$. The total probability transferred to the states ${\left| 3,n \right\rangle}$ will be independent of $g$, as occurs in the case of a two-level system. In the opposite case where the environment does not decohere superpositions between the states ${\left| 1 \right\rangle}$ and ${\left| 3 \right\rangle}$, the single avoided crossing between these states is preserved, while an increasing value of $g$ splits the ${\left| 1 \right\rangle}$-${\left| 2 \right\rangle}$ avoided crossing into a large number of avoided crossings most of which occur later in time than the ${\left| 1 \right\rangle}$-${\left| 3 \right\rangle}$ avoided crossing. As a result, after a tiny transfer from the state ${\left| 1 \right\rangle}$ to the state ${\left| 2,0 \right\rangle}$, some probability is transferred from the state ${\left| 1 \right\rangle}$ to the state ${\left| 3 \right\rangle}$ according to the adiabaticity parameter of that avoided crossing traversal, and the significant part of the transfer of probability from the state ${\left| 1 \right\rangle}$ to states of the form ${\left| 2,n \right\rangle}$ starts later in time. One consequence of this picture is that for sufficiently large values of $\delta$, and assuming a sufficiently large value of $g$, essentially all the probability will end up in the state ${\left| 3 \right\rangle}$ before any transfer from ${\left| 1 \right\rangle}$ to ${\left| 2 \right\rangle}$ has a chance to occur. In the strict adiabatic limit ($\delta\rightarrow\infty$, or more specifically $\delta\times\exp\{-(g/\hbar\omega)^2\}\gg 1$), the combined system will follow its ground state and end up in the state ${\left| 2,0 \right\rangle}$, but the exponential function in the inequality makes achieving this limit very difficult for strong system-environment coupling. In the case of $\hat{C}_{1:1}$, and assuming a large value of $g$, the occupation probability experiences a large number of small transfers to states that alternate between being in the manifold of the state ${\left| 2 \right\rangle}$ and being in the manifold of the state ${\left| 3 \right\rangle}$. This picture explains the slow dependence on $g$ and the result that at very large values of $g$ the states ${\left| 2 \right\rangle}$ and ${\left| 3 \right\rangle}$ have equal final probabilities. Obviously, if the ${\left| 1 \right\rangle}$-${\left| 2 \right\rangle}$ and ${\left| 1 \right\rangle}$-${\left| 3 \right\rangle}$ coupling strengths in the matrix $\hat{A}$ were unequal, the states ${\left| 2 \right\rangle}$ and ${\left| 3 \right\rangle}$ would end up with probabilities that are proportionate with the respective coupling strengths as described by the LZ formula. We note here that if we change the distance between the energies of the states ${\left| 2 \right\rangle}$ and ${\left| 3 \right\rangle}$, the above picture remains largely unchanged. One difference is that a larger energy difference will require a larger value of $g$ for any given feature to appear. Bow-tie model ------------- ![Same as Fig. \[Fig:OccupationProbabilitiesEqualSlope\], but for the bow-tie model.[]{data-label="Fig:OccupationProbabilitiesBowTie"}](DissipativeLandauZenerWithMultipleLevelsFigColorBowTie.eps){width="17cm"} ![Final occupation probabilities of the states ${\left| 1 \right\rangle}$ (red squares), ${\left| 2 \right\rangle}$ (green circles) and ${\left| 3 \right\rangle}$ (blue triangles) for the bow-tie model with coupling to the environment described by the operator $\hat{C}_{0:1}$ and strong decoherence ($g/\Delta=4$). The solid, dashed and dotted lines are the respective theoretical fits obtained by assuming that the ${\left| 1 \right\rangle}$-${\left| 2 \right\rangle}$ avoided crossing is traversed first and is then followed by the ${\left| 2 \right\rangle}$-${\left| 3 \right\rangle}$ avoided crossing. The theoretical formulae clearly give very good fits to the simulation data.[]{data-label="Fig:OccupationProbabilitiesBowTie01StrongDecoherence"}](DissipativeLandauZenerWithMultipleLevelsFigBowTie01StrongDecoherence.eps){width="8cm"} We now turn to the bow-tie model. As mentioned above, we set all the non-zero matrix elements in $\hat{A}$ to the same value (as described in Sec. \[Sec:Hamiltonian\]), but our main results are independent of this specific choice. The final occupation probabilities for three choices of $\hat{C}$ are plotted in Fig. \[Fig:OccupationProbabilitiesBowTie\]. In the absence of the coupling to the environment, it is now the state ${\left| 2 \right\rangle}$ whose population vanishes in both the limits $\delta\rightarrow 0$ and $\delta\rightarrow\infty$, with a maximum value of 0.5 attained when $4\pi\delta=4\ln 2=2.77$. In the fast-sweep limit, the system remains in the state ${\left| 1 \right\rangle}$, while in the adiabatic limit the system ends up in the state ${\left| 3 \right\rangle}$, which is the ground state at $t\rightarrow\infty$. If we now include coupling to the environment using the operator $\hat{C}_{0:1}$, the results are rather simple and match intuitive expectations based on the picture described in Sec. \[Sec:Results\].A. If $g$ is very large, the avoided-crossing structure separates into two stages, first an avoided crossing between the states ${\left| 1,0 \right\rangle}$ and ${\left| 2,0 \right\rangle}$ (with transition probability $P_{{\left| 1,0 \right\rangle}\rightarrow{\left| 2,0 \right\rangle}}=1-e^{-2\pi\delta}$) followed by a sequence of avoided crossings between the state ${\left| 2,0 \right\rangle}$ and states of the form ${\left| 3,n \right\rangle}$ with all integer values for $n$. As we have mentioned above, it is established in the literature that a sequence of this kind (i.e. the sequence of avoided crossings between ${\left| 2,0 \right\rangle}$ and ${\left| 3,n \right\rangle}$) gives final probabilities that are independent of the coupling to the environment, i.e. $P_{{\left| 2,0 \right\rangle}\rightarrow{\left| 2,0 \right\rangle}}=e^{-2\pi\delta}$. As a result, the final occupation probabilities are given by $P_1=e^{-2\pi\delta}$, $P_2=(1-e^{-2\pi\delta})\times e^{-2\pi\delta}$ and $P_3=(1-e^{-2\pi\delta})\times (1-e^{-2\pi\delta})$. As shown in Fig. \[Fig:OccupationProbabilitiesBowTie01StrongDecoherence\], the simulation results agree very well with the analytical formulae. The situation becomes more complicated for other choices of $\hat{C}$. For the choices $\hat{C}_{1:3}$, $\hat{C}_{3:1}$ and $\hat{C}_{1:0}$, we find interference patterns as a result of the (generally many) different possible paths that the system can take to reach any given final state. These paths differ by the number of environmental excitations, involving states of the form ${\left| 2,n \right\rangle}$ and ${\left| 3,m \right\rangle}$ with all the possible values for $n$ and $m$. The interference generally results in non-monotonic dependence on both $\delta$ and $g$. In the case of $\hat{C}_{1:1}$, the dependence is monotonic, giving a suppression of the final occupation probability of the state ${\left| 3 \right\rangle}$ with increasing system-environment coupling strength $g$. Looking at the different panels in Fig. \[Fig:OccupationProbabilitiesBowTie\], we can clearly see that in spite of this highly nontrivial dependence on the system parameters the probability of the state ${\left| 1 \right\rangle}$ remains unaffected by the coupling to the environment. The bow-tie model involves direct coupling between only one quantum state and each one of the other quantum states, because this simplification allows one to obtain analytic expressions for the final occupation probabilities. We do not need to restrict ourselves to this constraint, and we can consider a generalized bow-tie model where there are off-diagonal matrix elements in $\hat{A}$ that directly couple all the eigenstates of $\hat{B}$ to each other. We have performed calculations for this case. The overall conclusions are similar to those presented above, and we do not show them in detail here. There are only two main differences that we mention: (1) an interference pattern is obtained even in the case of $\hat{C}_{0:1}$, because the direct coupling between the states ${\left| 1 \right\rangle}$ and ${\left| 3 \right\rangle}$ results in the possibility of multiple paths leading to the same final state even in this case, and (2) we do not have the suppression of state ${\left| 3 \right\rangle}$’s final occupation probability in the case of $\hat{C}_{1:1}$ anymore, because that phenomenon was the result of an interference specific to the exact parameters of the symmetric bow-tie model. Triangle model -------------- ![Same as Fig. \[Fig:OccupationProbabilitiesEqualSlope\], but for the triangle model.[]{data-label="Fig:OccupationProbabilitiesTriangle"}](DissipativeLandauZenerWithMultipleLevelsFigColorTriangle.eps){width="17cm"} ![Final occupation probabilities of the states ${\left| 1 \right\rangle}$ (red squares), ${\left| 2 \right\rangle}$ (green open symbols) and ${\left| 3 \right\rangle}$ (blue closed symbols) for the triangle model with strong decoherence ($g/\Delta=4$). More specifically the symbols are: circles for $\hat{C}_{0:1}$, triangles for $\hat{C}_{1:3}$, inverted triangles for $\hat{C}_{3:1}$ and diamonds for $\hat{C}_{1:0}$. For comparison we also show the occupation probabilities for the case $g=0$: green x symbols for the state ${\left| 2 \right\rangle}$, blue stars for the state ${\left| 3 \right\rangle}$ and none for the state ${\left| 1 \right\rangle}$ because the probabilities coincide with the those shown by the red squares. The solid, dashed and dotted lines show the respective theoretical predictions obtained by assuming that the final occupation probabilities are determined by the sequence of three LZ processes at the three avoided crossings with no contribution from quantum interference terms. The blue symbols and the green symbols both exhibit rather irregular oscillations as functions of $\delta$, and these oscillations also vary depending on the choice of $\hat{C}$. While the details of these features cannot be explained easily without a quantitative analysis, the points to note in this figure are that the theoretical formulae (dashed and dotted lines) give good overall fits to the respective sets of simulation data and in particular that decoherence suppresses the quantum-interference-induced deviations from the results of the probability-based calculation. This behavior confirms that the main effect of the coupling to the environment is to suppress the quantum-interference effects between the three LZ processes.[]{data-label="Fig:OccupationProbabilitiesTriangleStrongDecoherence"}](DissipativeLandauZenerWithMultipleLevelsFigTriangleStrongDecoherence.eps){width="8cm"} Next we consider the triangle model with the parameters given in Table \[Table:DifferentCases\]. The results are plotted in Fig. \[Fig:OccupationProbabilitiesTriangle\]. In the absence of the coupling to the environment, a clear interference pattern is seen in the final occupation probabilities. The reason is obvious: for two of the three quantum states there are two different paths to reach the same final state, and the relative phase between the two paths depends on the different system parameters. When we include coupling to the environment, we obtain different patterns for the different choices of the operator $\hat{C}$. The general trend, however, is the same in all cases. As $g$ increases, the effects of interference between different possible paths diminishes, and the final occupation probabilities slowly approach values that are independent of $g$ or the specific choice for $\hat{C}$. In that limit, the final occupation probabilities are given by the sums of probabilities corresponding to different paths (as given by the LZ formula) without any quantum interference terms. In other words, the final occupation probabilities are given by $$\begin{aligned} P_1 & = & e^{-2\pi\delta}\times e^{-2\pi\delta\times 0.8^2/1.5}, \nonumber \\ P_2 & = & (1-e^{-2\pi\delta})\times e^{-2\pi\delta\times 0.55^2/0.5} + e^{-2\pi\delta}\times (1-e^{-2\pi\delta\times 0.8^2/1.5})\times (1-e^{-2\pi\delta\times 0.55^2/0.5}), \nonumber \\ P_3 & = & e^{-2\pi\delta}\times (1-e^{-2\pi\delta\times 0.8^2/1.5})\times e^{-2\pi\delta\times 0.55^2/0.5} + (1-e^{-2\pi\delta})\times (1-e^{-2\pi\delta\times 0.55^2/0.5}),\end{aligned}$$ where the factors 0.8, 0.55, 1.5 and 0.5 are taken from Table \[Table:DifferentCases\]. This result is illustrated in Fig. \[Fig:OccupationProbabilitiesTriangleStrongDecoherence\] and agrees with the intuitive picture that with strong decoherence one can think of the dynamics in terms of probability transfers between the different quantum states with no interference terms. Both Figs. \[Fig:OccupationProbabilitiesTriangle\] and \[Fig:OccupationProbabilitiesTriangleStrongDecoherence\] show that, once again, the final occupation probability of the state ${\left| 1 \right\rangle}$ is independent of the coupling to the environment. Conclusion {#Sec:Conclusion} ========== We have analyzed several instances of the LZ problem with a three-level system coupled to a harmonic oscillator that represents an uncontrolled environment. Our results all support the picture that coupling to the environment spreads out any given LZ process into a sequence of smaller LZ processes that take a longer time to be completed, albeit with a certain law that preserves the net probability transfer as long as there are no other avoided crossings being traversed at the same time. The spreading of each LZ process into multiple smaller processes means that the coupling to the environment can give rise to additional quantum interference effects in the problem, at the same time that this coupling gradually suppresses interference effects related to the parameters of the system alone. Our results also show that, assuming that the system starts in its ground state and the environment is at zero temperature, the final occupation probability of the state ${\left| 1 \right\rangle}$ is independent of the coupling to the environment. These results help enhance our understanding of the mechanisms governing multilevel LZ processes in open quantum systems and could therefore be relevant to practical applications such as adiabatic quantum computing and energy storage. We would like to thank K. Saito, S. Shevchenko and M. Wubs for useful discussions. [99]{} L. D. Landau, Phys. Z. Sowjetunion [**1**]{}, 88 (1932); [*ibid.*]{} [**2**]{}, 46 (1932). C. Zener, Proc. R. Soc. A. [**137**]{}, 696 (1932). E. C. G. Stueckelberg, Helv. Phys. Acta [**5**]{}, 369 (1932). E. Majorana, Nuovo Cimento [**9**]{}, 43 (1932). W. Wernsdorfer and R. Sessoli, Science [**284**]{}, 133 (1999). D. M. Berns, M. S. Rudner, S. O. Valenzuela, K. K. Berggren, W. D. Oliver, L. S. Levitov, and T. P. Orlando, Nature [**455**]{}, 51 (2008). G. Sun, X. Wen, Y. Wang, S. Cong, J. Chen, L. Kang, W. Xu, Y. Yu, S. Han and P. Wu, Appl. Phys. Lett. [**94**]{}, 102502 (2009). S. N. Shevchenko, S. Ashhab, and F. Nori, Phys. Rep. [**492**]{}, 1 (2010); see also S. Ashhab, J. R. Johansson, A. M. Zagoskin, and F. Nori, Phys. Rev. A [**75**]{}, 063414 (2007). E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser, arXiv:quant ph/0001106; E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren, and D. Preda, Science [**292**]{}, 472 (2001). M. W. Johnson, M. H. S. Amin, S. Gildert, T. Lanting, F. Hamze, N. Dickson, R. Harris, A. J. Berkley, J. Johansson, P. Bunyk, E. M. Chapple, C. Enderud, J. P. Hilton, K. Karimi, E. Ladizinsky, N. Ladizinsky, T. Oh, I. Perminov, C. Rich, M. C. Thom, E. Tolkacheva, C. J. S. Truncik, S. Uchaikin, J. Wang, B. Wilson, and G. Rose, Quantum annealing with manufactured spins, Nature [**473**]{}, 194 (2011). T. Lanting, A. J. Przybysz, A. Yu. Smirnov, F. M. Spedalieri, M. H. Amin, A. J. Berkley, R. Harris, F. Altomare, S. Boixo, P. Bunyk, N. Dickson, C. Enderud, J. P. Hilton, E. Hoskinson, M. W. Johnson, E. Ladizinsky, N. Ladizinsky, R. Neufeld, T. Oh, I. Perminov, C. Rich, M. C. Thom, E. Tolkacheva, S. Uchaikin, A. B. Wilson, G. Rose, Phys. Rev. X [**4**]{}, 021041 (2014). Yu. N. Demkov and V. I. Osherov, Soviet Phys. JETP [**26**]{}, 916 (1968). C. E. Carroll and F. T. Hioe, J. Phys. A [**19**]{}, 1151 (1986); [*ibid.*]{} [**19**]{}, 2061 (1986). S. Brundobler and V. Elser, J. Phys. A: Math. Gen. [**26**]{}, 1211 (1993). V. N. Ostrovsky and H. Nakamura, J. Phys. A [**30**]{}, 6939 (1997). T. Usuki, Phys. Rev. B [**56**]{}, 13360 (1997). Y. N. Demkov and V. N. Ostrovsky, Phys. Rev. A [**61**]{}, 032705 (2000); Y. N. Demkov and V. N. Ostrovsky, J. Phys. B [**34**]{}, 2419 (2001). N. A. Sinitsyn, Phys. Rev. B [**66**]{}, 205303 (2002). A. V. Shytov, Phys. Rev. A [**70**]{}, 052708 (2004). G. S. Vasilev, S. S. Ivanov, and N. V. Vitanov, Phys. Rev. A [**75**]{}, 013417 (2007). M. B. Kenmoe, H. N. Phien, M. N. Kiselev, and L. C. Fai, Phys. Rev. B [**87**]{}, 224301 (2013). M. N. Kiselev, K. Kikoin, and M. B. Kenmoe, EPL [**104**]{}, 57004 (2013). N. A. Sinitsyn, Phys. Rev. A [**90**]{}, 062509 (2014). N. A. Sinitsyn, J. Phys. A: Math. Theor. [**48**]{}, 195305 (2015). N. A. Sinitsyn, Phys. Rev. B [**92**]{}, 205431 (2015). A. Patra and E. A. Yuzbashyan, J. Phys. A: Math. Theor. [**48**]{}, 245303 (2015). Y. Kayanuma, J. Phys. Soc. Jpn. [**53**]{}, 108 (1984); [*ibid.*]{} 53, 118 (1984); [*ibid.*]{} [**54**]{}, 2037 (1985). Y. Gefen, E. Ben-Jacob, and A. O. Caldeira, Phys. Rev. B [**36**]{}, 2770 (1987). P. Ao and J. Rammer, Phys. Rev. B [**43**]{}, 5397 (1991). E. Shimshoni and A. Stern, Phys. Rev. B [**47**]{}, 9523 (1993). M. Nishino, K. Saito, and S. Miyashita, Phys. Rev. B [**65**]{}, 014403 (2001). V. L. Pokrovsky and N. A. Sinitsyn, Phys. Rev. B [**67**]{}, 144303 (2003). M. S. Sarandy and D. A. Lidar, Phys. Rev. Lett. [**95**]{}, 250503 (2005). S. Ashhab, J. R. Johansson, and F. Nori, Phys. Rev. A [**74**]{}, 052330 (2006). X. Lacour, S. Guérin, L. P. Yatsenko, N. V. Vitanov, and H. R. Jauslin, Phys. Rev. A [**75**]{}, 033417 (2007). V.L. Pokrovsky and D. Sun, Phys. Rev. B [**76**]{}, 024310 (2007). M. H. S. Amin, P. J. Love, and C. J. S. Truncik, Phys. Rev. Lett. [**100**]{}, 060503 (2008); M. H. S. Amin, D. V. Averin, and J. A. Nesteroff, Phys. Rev. A [**79**]{}, 022107 (2009); M. H. S. Amin, C. J. S. Truncik, and D. V. Averin, Phys. Rev. A [**80**]{}, 022303 (2009). P. Nalbach and M. Thorwart, Phys. Rev. Lett. [**103**]{}, 220401 (2009); Chem. Phys. [**375**]{}, 234 (2010). P. Nalbach, J. Knörzer, and S. Ludwig, Phys. Rev. B [**87**]{}, 165425 (2013). A. Dodin, S. Garmon, L. Simine, and D. Segal, J. Chem. Phys. [**140**]{}, 124709 (2014). C. Xu, A. Poudel, and M. G. Vavilov, Phys. Rev. A [**89**]{}, 052102 (2014). P. Haikka and K. Mølmer, Phys. Rev. A [**89**]{}, 052114 (2014). P. Nalbach, Phys. Rev. A [**90**]{}, 042112 (2014). S. Ashhab, Phys. Rev. A [**90**]{}, 062120 (2014). S. Javanbakht, P. Nalbach, and M. Thorwart, Phys. Rev. A [**91**]{}, 052103 (2015). D. S. Wild, S. Gopalakrishnan, M. Knap, N. Y. Yao, and M. D. Lukin, Phys. Rev. Lett. [**117**]{}, 150501 (2016). M. Wubs, K. Saito, S. Kohler, P. Hänggi, and Y. Kayanuma, Phys. Rev. Lett. [**97**]{}, 200404 (2006); K. Saito, M. Wubs, S. Kohler, Y. Kayanuma, and P. Hänggi, Phys. Rev. B [**75**]{}, 214308 (2007). Two points should be noted here: (1) If $\hat{B}$ is not diagonal, it can be diagonalized in order to put it in simple and intuitive form at infinite times. (2) If $\hat{B}$ happens to have degenerate eigenvalues, the degeneracy in this subset of the Hilbert space can generally be lifted by defining new basis states from superpositions of the degenerate eigenstates such that the new states diagonalize $\hat{A}$ within the degenerate subspace. D. Zueco, P. Hänggi, and S. Kohler, New J. Phys. [**10**]{}, 115012 (2008). See e.g. S. Ashhab and F. Nori, Phys. Rev. A [**81**]{}, 042311 (2010).

--- abstract: 'We associate an integrable generalized complex structure to each $2$-dimensional symplectic Monge-Ampère equation of divergent type and, using the Gualtieri $\overline{\partial}$ operator, we characterize the conservation laws and the generating function of such equation as generalized holomorphic objects.' address: 'Université de Bretagne Occidentale, 6 Avenue Victor Le Gorgeu BP 809, 29 285 Brest\' author: - Bertrand Banos title: | **Monge-Ampère equations and generalized complex geometry.\ The two-dimensional case.** --- Introduction {#introduction .unnumbered} ============ A general approach to the study of non-linear partial differential equations, which goes back to Sophus Lie, is to see a $k$-order equation on a $n$-dimensional manifold $N^n$ as a closed subset in the manifold of $k$-jets $J^kN$. In particular, a second-order differential equation lives in the space $J^2N$. Neverthess, as it was noticed by Lychagin in his seminal paper “Contact geometry and non-linear second-order differential equations” ([@L]), it is sometimes possible to decrease one dimension and to work on the contact space $J^1N$. The idea is to define for any differential form $\omega\in \Omega^n(J^1N)$, a second order differential operator $\Delta_\omega:C^\infty(N)\rightarrow \Omega^n(N)$ acting according to the rule $$\Delta_\omega(f)=j_1(f)^*\omega,$$ where $j_1(f): N\rightarrow J^1N$ is the section corresponding to the function $f$. The differential equations of the form $\Delta_\omega=0$ are said to be of Monge-Ampère type because of their “hessian - like” non-linearity. Despite its very simple description, this classical class of differential equations attends much interest due to its appearence in different problems of geometry or mathematical physics. We refer to the very rich book *Contact geometry and Non-linear Differential Equations* ([@KLR]) for a complete exposition of the theory and for numerous examples. A Monge-Ampère equation $\Delta_\omega=0$ is said to be symplectic if the Monge-Ampère operator $\Delta_\omega$ is invariant with respect to the Reeb vector field. In other words, the $n$-form $\omega$ lives actually on the cotangent bundle $T^*N$, and symplectic geometry takes place of contact geometry. The Monge-Ampère operator is then defined by $$\Delta_\omega(f)=(df)^*\omega.$$ This partial case is in some sense quite generic because of the beautiful result of Lychagin which says that any Monge-Ampère equation admitting a contact symmetry is equivalent (by a Legendre transform on $J^1N$) to a symplectic one. We are interested here in symplectic Monge-Ampère equations in two variables. These equations are written as : $${\label{MA}} A\frac{\partial^2 f}{\partial q_1^2} + 2B \frac{\partial^2f}{\partial q_1\partial q_2}+ C\frac{\partial^2 f}{\partial q_2^2}+ D\Big(\frac{\partial^2 f}{\partial q_1^2}\frac{\partial^2 f}{\partial q_2^2} - \big(\frac{\partial^2 f}{\partial q_1\partial q_2}\big)^2\Big)+E=0,$$ with $A$, $B$, $C$, $D$ and $E$ smooth functions of $(q,\frac{\partial f}{\partial q})$. These equations correspond to $2$-form on $T^*\mathbb{R}^2$, or equivalently to tensors on $T^*\mathbb{R}^2$ using the correspondence $$\omega(\cdot,\cdot)=\Omega(A\cdot,\cdot),$$ $\Omega$ being the symplectic form on $T^*N$. In the non-degenerate case, the traceless part of this tensor $A$ defines either an almost complex structure or an almost product structure and it is integrable if and only the corresponding Monge-Ampère equation is equivalent to the Laplace equation or the wave equation. This elegant result of Lychagin and Roubtsov ([@LR]) is quite frustrating: which kind of integrable geometry could we define for more general Monge-Ampère equations ? It has been noticed in [@Cr] that such a pair of forms $(\omega,\Omega)$ defines an almost generalized complex structure, a very rich concept defined recently by Hitchin ([@H1]) and developed by Gualtieri ([@G1]), which interpolates between complex and symplectic geometry. It is easy to see that this almost generalized complex structure is integrable for a very large class of $2D$-Monge-Ampère equations, the equations of *divergent type*. This observation is the starting point for the approach proposed in this paper: the aim is to present these differential equations as “generalized Laplace equations”. In the first part, we write down this correspondence between Monge-Ampère equations in two variable and $4$-dimensional generalized complex geometry. In the second part we study the ${\overline{\partial}}$-operator associated with a Monge-Ampère equation of divergent type and we show how the corresponding conservation laws and generating functions can be seen as “holomorphic objects”. Monge-Ampère equations and Hitchin pairs ======================================== In what follows $M$ is the smooth symplectic space $T^*\mathbb{R}^2$ endowed with the canonical symplectic form $\Omega$. Our point of view is local (in particulary we do not make any distinction between closed and exact forms) but most of the results presented here have a global version. A primitive $2$-form is a differential form $\omega\in \Omega^2(M)$ such that $\omega\wedge\Omega=0$. We denote by $\bot: \Omega^k(M)\rightarrow\Omega^{k-2}(M)$ the operator $\theta\mapsto \iota_{X_\Omega}(\theta)$, the bivector $X_\Omega$ being the bivecor dual to $\Omega$. It is straightforward to check that in dimension $4$, a $2$-form $\omega$ is primitive if and only if $\bot\omega=0$. Monge-Ampère operators ---------------------- Let $\omega$ be a $2$-form on $M$. A $2$-dimensional submanifold $L$ is a generalized solution of the equation $\Delta_\omega=0$ if it is bilagrangian with respect to $\Omega$ and $\omega$. Note that a lagrangian submanifold of $T^*\mathbb{R}^2$ which projects isomorphically on $\mathbb{R}^2$ is a graph of a closed $1$-form $df:\mathbb{R}^2\rightarrow T^*\mathbb{R}^2$. A generalized solution can be thought as a smooth patching of classical solutions of the Monge-Ampère equation $\Delta_\omega=0$ on $\mathbb{R}^2$. Consider the $2D$-Laplace equation $$f_{q_1q_1}+f_{q_2q_2}=0.$$ It corresponds to the form $\omega=dq_1\wedge dp_2-dq_2\wedge dp_1$, while the symplectic form is $\Omega=dq_1\wedge dp_1+ dq_2\wedge dp_2$. Introducing the complex coordinates $z_1=q_1+iq_2$ and $z_2=p_2+ip_1$, we get $\omega+i\Omega= dz_1\wedge dz_2$. Generalized solution of the $2D$-Laplace equation appear then as the complex curves of $\mathbb{C}^2$. The following theorem (so called Hodge-Lepage-Lychagin, see [@L]) establishes the $1-1$ correspondence between Monge-Ampère operators and primitive $2$-forms: i) Any $2$-form admits the unique decomposition $ \omega=\omega_0 + \lambda\omega,$ with $\omega_0$ primitive. ii) If two primitive forms vanish on the same lagrangian subspaces, then there are proportional. A Monge-Ampère operator $\Delta_\omega$ is therefore uniquely defined by the primitive part $\omega_0$ of $\omega$, since $\lambda\Omega$ vanish on any lagrangian submanifold. The function $\lambda$ can be arbitrarily chosen. Let $\omega=\omega_0+\lambda\Omega$ be a $2$-form. We define the tensor $A$ by $\omega=\Omega(A\cdot,\cdot)$. One has $A=A_{0}+\lambda Id$ and $$A_0^2=-\operatorname{pf}(\omega_0)Id,$$ where the function $\operatorname{pf}(\omega_0)$ is the pfaffian of $\omega_0$ defined by $$\omega_0\wedge\omega_0= \operatorname{pf}(\omega_0)\Omega\wedge\Omega.$$ Therefore, $$A^2=2\lambda A -(\lambda^2+\operatorname{pf}(\omega_0))Id.$$ The equation $\Delta_\omega=0$ is said to be elliptic if $\operatorname{pf}(\omega_0)>0$, hyperbolic if $\operatorname{pf}(\omega_0)<0$, parabolic if $\operatorname{pf}(\omega_0)=0$. In the elliptic/hyperbolic case, one can define the tensor $$J_0=\frac{A_{0}}{\sqrt{|\operatorname{pf}(\omega_0)|}}$$ which is either an almost complex structure or an almost product structure. The following assertions are equivalent i) The tensor $J_{0}$ is integrable. ii) The form $\omega_0/\sqrt{|\operatorname{pf}(\omega_0)|}$ is closed. iii) The Monge-Ampère equation $\Delta_{\omega}=0$ is equivalent (with respect to the action of local symplectomorphisms) to the (elliptic) Laplace equation $f_{q_1q_1}+f_{q_2q_2}=0$ or the (hyperbolic) wave equation $f_{q_1q_1}-f_{q_2q_2}=0$. Let us introduce now the Euler operator and the notion of Monge-Ampère equation of divergent type (see [@L]). The Euler operator is the second order differential operator $\mathcal{E}: \Omega^2(M)\rightarrow \Omega^2(M)$ defined by $$\mathcal{E}(\omega)=d \bot d\omega.$$ A Monge-Ampère equation $\Delta_\omega=0$ is said to be of divergent type if $\mathcal{E}(\omega)=0$. The Born-Infeld equation is $$(1-f_t)^2 f_{xx}+2f_tf_xf_{tx} - (1+f_x^2)f_{tt}=0.$$ The corresponding primitive form is $$\omega_0=(1-p_1^2)dq_1\wedge dp_2+ p_1p_2(dq_1\wedge dp_1) + (1+p_2^2)dq_2\wedge dp_1.$$ with $q_1=t$ and $q_2=x$. A direct computation gives $$d\omega_0=3(p_1dp_2 - p_2dp_1)\wedge \Omega,$$ and then the Born - Infeld equation is not of divergent type. The Tricomi equation is $$v_{xx} xv_{yy}+\alpha v_x + \beta v_y + \gamma(x,y).$$ The corresponding primitive form is $$\omega_0=(\alpha p _1 + \beta p_2 +\gamma(q))dq_1\wedge dq_2+ dq_1\wedge dp_2-q_2dq_2\wedge dp_1,$$ with $x=q_1$ and $y=q_2$. Since $$d\omega_0=(-\alpha dq_2 + \beta dq_1)\wedge \Omega,$$ we conclude that the Tricomi equation is of divergent type. A Monge-Ampère equation $\Delta_{\omega}=0$ is of divergent type if and only if it exists a function $\mu$ on $M$ such that the form $\omega + \mu \Omega$ is closed. Since the exterior product by $\Omega$ is an isomorphism from $\Omega^1(M)$ to $\Omega^3(M)$, for any $2$-form $\omega$, there exists a $1$-form $\alpha_\omega$ such that $$d\omega=\alpha_\omega\wedge\Omega.$$ Since $\bot(\alpha_\omega\wedge\Omega)=\alpha_\omega$ we deduce that $\mathcal{E}(\omega)=0$ if and only if $d\alpha_\omega=0$, that is $d(\omega+\mu\Omega)=0$ with $d\mu=-\alpha_\omega$. Hence, if $\Delta_\omega=0$ is of divergent type, one can choose $\omega$ being closed. The point is that it is not primitive in general . Hitchin pairs ------------- Let us denote by $T$ the tangent bundle of $M$ and by $T^*$ its cotangent bundle. The natural indefinite interior product on $T\oplus T^*$ is $$(X+\xi,Y+\eta)=\frac{1}{2}(\xi(Y)+\eta(X)),$$ and the Courant bracket on sections of $T\oplus T^*$ is $$[X+\xi,Y+\eta]=[X,Y]+L_X\eta-L_Y\xi -\frac{1}{2}d(\iota_X\eta- \iota_Y\xi).$$ An almost generalized complex structure is a bundle map $\mathbb{J}: T\oplus T^*\rightarrow T\oplus T^*$ satisfying $$\mathbb{J}^2=-1,$$ and $$(\mathbb{J}\cdot,\cdot)=-(\cdot,\mathbb{J}\cdot).$$ Such an almost generalized complex structure is said to be integrable if the spaces of sections of its two eigenspaces are closed under the Courant bracket. The standard examples are $$\mathbb{J}_1=\begin{pmatrix} J&0\\0&-J^*\end{pmatrix}$$ and $$\mathbb{J}_2=\begin{pmatrix} 0&\Omega^{-1}\\ -\Omega & 0\end{pmatrix}$$ with $J$ a complex structure and $\Omega$ a symplectic form. Let $\Omega$ be a symplectic form and $\omega$ any $2$-form. Define the tensor $A$ by $\omega=\Omega(A\cdot,\cdot)$ and the form $\tilde{\omega}$ by $\tilde{\omega}=-\Omega(1+A^2\cdot,\cdot)$. The almost generalized complex structure $${\label{C}} \mathbb{J}=\begin{pmatrix} A& \Omega^{-1}\\ \tilde{\omega} & -A^* \end{pmatrix}$$ is integrable if and only if $\omega$ is closed. Such a pair $(\omega,\Omega)$ with $d\omega=0$ is called a Hitchin pair We get then immediatly the following: To any $2$-dimensional symplectic Monge-Ampère equation of divergent type $\Delta_\omega=0$ corresponds a Hitchin pair $(\omega,\Omega)$ and therefore a $4$-dimensional generalized complex structure. Let $L^2\subset M^4$ be a $2$-dimensional submanifold. Let $T_L\subset T$ be its tangent bundle and $T_L^0\subset T^*$ its annihilator. $L$ is a generalized complex submanifold (according to the terminology of [@G1]) or a generalized lagrangian submanifold (according to the terminology of [@BB]) if $T_L\oplus T^0_L$ is closed under $\mathbb{J}$. When $\mathbb{J}$ is defined by $\eqref{C}$, this is equivalent to saying that $L$ is lagrangian with respect to $\Omega$ and closed under $A$, that is, $L$ is a generalized solution of $\Delta_\omega=0$. Systems of first order partial differential equations ----------------------------------------------------- On $2n$-dimensional manifold, a generalized complex structure write as $$\mathbb{J} =\begin{pmatrix} A& \pi\\ \sigma& -A^*\\ \end{pmatrix}$$ with different relations detailed in [@Cr] between the tensor $A$, the bivector $\pi$ and the $2$-form $\sigma$. The most oustanding being $[\pi,\pi]=0$, that is $\pi$ is a Poisson bivector. In [@Cr], a generalized complex structures is said to be non-degenerate if the Poisson bivector $\pi$ is non-degenerate, that is, if the two eigenspaces $E=\operatorname{Ker}(\mathbb{J}-i)$ and $\overline{E}=\operatorname{Ker}(\mathbb{J}+i)$ are transverse to $T^*$. This leads to our symplectic form $\Omega=\pi^{-1}$ and to our $2$-form $\omega=\Omega(A\cdot,\cdot)$. One could also take the dual point of view and study generalized complex structure transverse to $T$. In this situation, the eigenspace $E$ writes as $$E=\big\{\xi + \iota_\xi P, \xi\in T^*\otimes \mathbb{C}\big\},$$ with $P=\pi+i\Pi$ a complex bivector. This space defines a generalized complex structure if and only if it is a Dirac subbundle of $(T\oplus T^*)\otimes \mathbb{C}$ and if it is transverse to its conjugate $\overline{E}$. According to the Maurer-Cartan type equation described in the famous paper *Manin Triple for Lie bialgebroids* ([@LWX], the first condition is $$[\pi+i\Pi,\pi+i\Pi]=0.$$ The second condition says that $\Pi$ is non-degenerate. Hence, we obtain some analog of the Crainic’s result: A Hitchin pair of bivectors is a pair consisting of two bivectors $\pi$ and $\Pi$, $\Pi$ being non-degenerate, and satisfying $${\label{bihamilt}} \begin{cases} [\Pi,\Pi]=[\pi,\pi]&\\ [\Pi,\pi]=0.&\\ \end{cases}$$ There is a 1-1 correspondence between Generalized complex structure $$\mathbb{J}=\begin{pmatrix} A & \pi_A\\ \sigma& -A^*\end{pmatrix}$$ with $\sigma$ non degenerate and Hitchin pairs of bivector $(\pi,\Pi)$. In this correspondence, we have $$\begin{cases} \sigma=\Pi^{-1}\\ A=\pi\circ\Pi^{-1}\\ \pi_A= -(1+A^2)\Pi \end{cases}$$ If $\pi+i\Pi$ is non-degenerate, it defines a $2$-form $\omega+i\Omega$ which is necessarily closed (this is the complex version of the classical result which says that a non-degenerate Poisson bivector is actually symplectic). We find again an Hitchin pair. So new examples occur only in the degenerate case. Note that $\pi+i\Pi= (A+i)\Pi$, so $\det(\pi+i\Pi)=0$ if and only if $-i$ is an eigenvalue for $A$. In dimension $4$, this implies that $A^2=-1$ but this is not any more true in greater dimensions (see for example the classification of pair of $2$-forms on $6$-dimensional manifolds in [@LR]). Nevertheless, the case $A^2=-1$ is interesting by itself. It corresponds to generalized complex structure of the form $$\mathbb{J}=\begin{pmatrix} J&0\\ \sigma&-J^* \end{pmatrix}$$ with $J$ an integrable complex structure and $\sigma$ a $2$-form satisfying $J^*\sigma= -\sigma$ and $$d\sigma_J=d\sigma(J\cdot,\cdot,\cdot)+ d\sigma(\cdot,J\cdot,\cdot)+d\sigma(\cdot,\cdot,J\cdot).$$ where $\sigma_J=\sigma(J\cdot,\cdot)$ (see [@Cr]). Or equivalently $\sigma+i\sigma_J$ is a $(2,0)$-form satisfying $$\partial(\sigma + i\sigma_J)=0.$$ One typical example of such geometry is the so called HyperKähler geometry with torsion which is an elegant generalization of HyperKähler geometry ([@GP]). Unlike the HyperKäler case, such geometry are always generated by potentials ([@BS]). Let us consider now an Hitchin pair of bivectors $(\pi,\Pi)$ in dimension $4$. Since $\Pi$ is non-degenerate, it defines two $2$-forms $\omega$ and $\Omega$, which are not necessarily closed, and related by the tensor $A$. A generalized lagrangian surface is a surface closed under $A$, or equivalently, bilagrangian: $\omega|_L=\Omega|_L=0$. Locally, $L$ is defined by two functions $u$ and $v$ satisfying a first order system $$\begin{cases} a+b\frac{\partial u}{\partial x} +c\frac{\partial u}{\partial y}+ d\frac{\partial v}{\partial x}+ e\frac{\partial v}{\partial y} + f\det J_{u,v}\\ A+B\frac{\partial u}{\partial x} +C\frac{\partial u}{\partial y}+ D\frac{\partial v}{\partial x}+ E\frac{\partial v}{\partial y} + E\det J_{u,v}\\ \end{cases}$$ with $$J_{u,v}=\begin{pmatrix}\frac{\partial u}{\partial x} & \frac{\partial u}{\partial y}\\ \frac{\partial v}{\partial x}& \frac{\partial v}{\partial y}\\ \end{pmatrix}$$ Such a system generalizes both Monge-Ampère equations and Cauchy-Riemann systems and is called Jacobi-system (see [@KLR]). With the help of Hitchin’s formalism, we understand now the integrability condition as a “divergent type” condition for Jacobi equations. The $\overline{\partial}$-operator ================================== Let us fix now a $2D$- symplectic Monge-Ampère equation of divergent type $\Delta_\omega=0$, the $2$-form $\omega=\omega_0+\lambda\Omega$ being closed. We still denote by $A=A_0+\lambda$ the associated tensor. For any $1$-form $\alpha$, the following relation holds: $${\label{B}} \alpha\wedge\omega - B^*\alpha\wedge \Omega=0$$ with $B=\lambda-A_0$. Let $\alpha=\iota_X\Omega$ be a $1$-form. Since $\omega_0$ is primitive, we get $$0=\iota_X(\omega_0\wedge\Omega)=(\iota_X\omega_0)\wedge\Omega+(\iota_X\Omega)\wedge \omega_0= A_0^*\alpha\wedge\Omega+\alpha\wedge\omega_0.$$ Therefore, $$\alpha\wedge\omega=\alpha\wedge\omega_0+\lambda \alpha\wedge\Omega= (-A_0+\lambda)^*\alpha \wedge\Omega.$$ We denote by $\mathbb{J}$ the generalized complex structure associated with the Hitchin pair $(\omega,\Omega)$. We also define $$\Theta=\omega-i\Omega$$ and $$\Phi=\exp(\Theta)=1+\Theta+\frac{\Theta^2}{2}.$$ Decomposition of forms ---------------------- Using the tensor $\mathbb{J}$, Gualtieri defines a decomposition $$\Lambda^*(T^*)\otimes \mathbb{C}= U_{2}\oplus U_{-1}\oplus U_{0}\oplus U_{1}\oplus U_2$$ which generalizes the Dolbeault decomposition for a complex structure ([@G1]). Let us introduce some notations to understand this decomposition. The space $T\oplus T^*$ acts on $\Lambda^*(T^*)$ by $$\rho(X+\xi)(\theta)=\iota_X\theta + \xi\wedge \theta,$$ and this action extends to an isomorphism (the standard spin representation) between the Clifford algebra $CL(T\oplus T^*)$ and the space of linear endomorphisms $End(\Lambda^*(T^*))$. With these notations, the eigenspace $E=\operatorname{Ker}(\mathbb{J}-i)$ is also defined by $$E=\big\{ X+\xi\in T\oplus T^*, \rho(X+\xi)(\Phi)=0\big\},$$ The space $U_k$ is defined by $$U_k=\rho\big(\Lambda^{2-k}\overline{E}\big)\big(\Phi\big).$$ Note that $\mathbb{J}$ identifyed with the $2$-form $(\mathbb{J}\cdot,\cdot)$ lives in $\Lambda^2(T\oplus T^*)\subset CL(T\oplus T^*)$. We get then an infinitesimal action of $\mathbb{J}$ on $\Lambda^*(T^*)$. $U_k$ is the $ik$-eigenspace of $\mathbb{J}$. We see then immediatly that $U_{-k}=\overline{U_k}$, since $\mathbb{J}$ is a real tensor. i) $U_2=\mathbb{C}\Phi$. ii) $U_1=\big\{\alpha\wedge\Phi, \alpha\in \Lambda^1(T^*)\otimes \mathbb{C}\big\}.$ iii) $U_0=\big\{(\theta-\frac{i}{2}\bot\theta)\wedge\Phi, \theta\in \Lambda^2(T^*)\otimes \mathbb{C}\big\}$. The eigenspace $\overline{E}$ is $$\overline{E}=\big\{X-\iota_X\overline{\Theta}, X\in T\otimes \mathbb{C}\big\}.$$ Now, $$\rho(X-\iota_X \overline{\Theta})(\Phi)=\iota_X\Theta+\iota_X\Theta\wedge \Theta-\iota_X\overline{\Theta}-\iota_X\overline{\Theta}\wedge\Theta=\iota_X(\Theta-\overline{\Theta})\wedge (1+\Theta).$$ Since $\Theta-\overline{\Theta}=-2i\Omega$ and $X\mapsto\iota_X\Omega$ is an isomorphism between $T$ and $T^*$, we get then the description of $U_1$. Choose now two complex vectors $X$ and $Y$ and define $\alpha=\iota_X\Omega$ and $\beta=\iota_Y\Omega$: $$\begin{aligned} \rho\big((X-\iota_X\overline{\Theta}) & \wedge (Y-\iota_Y\overline{\Theta})\big)\big(\Phi\big)\\ &=\rho\big(X-\iota_X\overline{\Theta}\big)\big(-2i\beta\wedge\Phi\big)\\ &=-2i\rho\big(X-\iota_X\overline{\Theta}\big)\big(\beta+\beta\wedge\Theta\big)\\ &=-2i\big(\beta(X)(1+\Theta)-\beta\wedge\iota_X\Theta-\iota_X\overline{\Theta}\wedge\beta -\iota_X\overline{\Theta}\wedge\beta\wedge\Theta\big)\\ &=-2i\big(\beta(X)(1+\Theta) + \iota_X(\Theta-\overline{\Theta})\wedge\beta\wedge (1+\Theta)-\iota_X\Theta\wedge\beta\wedge\Theta\big)\\ &=-2i\big(\beta(X)(1+\Theta) - 2i\alpha\wedge\beta\wedge(1+\Theta)+ \beta\wedge\iota_X\frac{\Theta^2}{2}\big)\\ \end{aligned}$$ Moreover, since $\beta\wedge\Theta^2=0$, we have $\beta(X)\Theta^2= \beta\wedge \iota_X\Theta^2$ and then $$\rho\big((X-\iota_X\overline{\Theta})\wedge (Y-\iota_Y\overline{\Theta})\big)\big(\Phi\big)= -2i(\beta(X)-2i\alpha\wedge\beta)\wedge\Phi.$$ But $\bot(\alpha\wedge\beta)=-\beta(X)=\alpha(Y)$. We obtain then the description of $U_0$. The next proposition describes the space $U_0^\mathbb{R}$ of *real* forms in $U_0$. It is a direct consequence of the proposition above. [\[Ureel\]]{} Let $\Lambda^2_0$ be the space of (real) primitive $2$-forms. Then $$U_0^{\mathbb{R}}=\big\{[\theta+a(i\Omega+1)]\wedge\Phi,\text{ $ \theta\in \Lambda^2_0$ and $a\in \mathbb{R}$}\big\}.$$ We have actually $$(\Lambda^1\oplus \Lambda^3)\otimes\mathbb{C} = U_{-1}\oplus U_{1}$$ and $$(\Lambda^0\oplus \Lambda^2\oplus\Lambda^4)\otimes \mathbb{C}=U_{-2}\oplus U_0\oplus U_2.$$ For example, the decomposition of a $1$-form $\alpha\in \Lambda^1(T^*)$ is $$\alpha=\frac{\alpha-iB\alpha}{2}\wedge\Phi+ \frac{\alpha+iB\alpha}{2}\wedge\overline{\Phi}.$$ This decomposition is a pointwise decomposition. Denote now by $\mathcal{U}_k$ the space of smooth sections of the bundle $U_k$. The Gualtieri decomposition is now $$\Omega^*(M)\otimes \mathbb{C}=\mathcal{U}_{-2}\oplus \mathcal{U}_{-1}\oplus \mathcal{U}_0\oplus \mathcal{U}_1 \oplus \mathcal{U}_2.$$ The operator $\overline{\partial}:\mathcal{U}_k\rightarrow \mathcal{U}_{k+1}$ is simply $\overline{\partial}=\pi_{k+1}\circ d$ The next theorem is completely analogous to the corresponding statement involving an almost complex structure and the Dolbeault operator $\overline{\partial}$. The almost generalized complex structure $\mathbb{J}$ is integrable if and only if $$d=\partial + \overline{\partial}.$$ Let $\alpha\in \Omega^1(M)$ be a $1$-form. From $d(\alpha\wedge\Phi)=d\alpha\wedge\Phi$ we get $$\begin{cases} \overline{\partial}(\alpha\wedge\Phi)=\frac{i}{2}(\bot d\alpha)\Phi&\\ \partial(\alpha\wedge\Phi)=(d\alpha-\frac{i}{2}\bot d\alpha)\wedge\Phi. \end{cases}$$ It is worth mentionning that one can also define the real differential operator $d^\mathbb{J}=[d,\mathbb{J}]$, or equivalently (see [@C]) $$d^{\mathbb{J}}= -i(\partial-\overline{\partial}).$$ Cavalcanty establishes in [@C], for the particular case $\omega=0$, an isomorphism $\Xi: \Omega^*(M)\otimes\mathbb{C}\rightarrow \Omega^*(M)\otimes\mathbb{C}$ satisfying $$\Xi(d\theta)=\partial\Xi(\theta),\;\;\; \Xi(\delta\theta)=\overline{\partial} \Xi(\theta)$$ with $\delta=[d,\bot]$ the symplectic codifferential. Since $d\delta$ is the Euler operator, Monge-Ampère equations of divergent type write as $\Delta_{\omega}=0$ with $\Xi(\omega)$ pluriharmonic on the generalized complex manifold $\big(M^4,\exp(i\Omega)\big)$. Conservation laws and Generating functions ------------------------------------------ The notion of conservation laws is a natural generalization to partial differential equations of the notion of first integrals. A $1$-form $\alpha$ is a conservation law for the equation $\Delta_\omega=0$ if the restriction of $\alpha$ to any generalized solution is closed. Note that conservations laws are actually well defined up closed forms. Let us consider the Laplace equation and the complex structure $J$ associated with. The $2$-form $d\alpha$ vanish on any complex curve if and only if $[d\alpha]_{1,1}=0$, that is $$\overline{\partial}\alpha_{1,0} + \partial \alpha_{0,1}=0$$ or equivalently $$\overline{\partial}\alpha_{1,0}= \overline{\partial}\partial \psi$$ for some real function $\psi$. (Here $\overline{\partial}$ is the usual Dolbeault operator defined by the integrable complex structure $J$.) We deduce that $\alpha -d\psi = \beta_{1,0}+ \beta_{0,1}$ with $\beta_{1,0}=\alpha_{1,0}-\partial\psi$ is a holomorphic $(1,0)$-form. Hence, the conservation laws of the $2D$-Laplace equation are (up exact forms) real part of $(1,0)$-holomorphic forms. According to the Hodge-Lepage-Lychagin theorem, $\alpha$ is a conservation law if and only if there exist two functions $f$ and $g$ such that $d\alpha=f\omega+ g\Omega$. The function $f$ is called a generating function of the Monge-Ampère equation $\Delta_\omega=0$. By analogy with the Laplace equation, we will say that the function $g$ is the conjugate function to the generating function $f$. A function $f$ is a generating function if and only if $$dBdf=0.$$ $f$ is a generating function if and only if there exists a function $g$ such that $$0=d(f\omega+g\Omega)=df\wedge\omega + dg\wedge\Omega=(dg+Bdf)\wedge\Omega,$$ and therefore $g$ exists if and only if $dBdf=0$. If $f$ is a generating function and $g$ is its conjugate then for any $c\in \mathbb{C}$, $L_c=(f+ig)^{-1}(c)$ is a generalized solution of the Monge-Ampère equation $\Delta_\omega=0$. The tangent space $T_aL_c$ is generated by the hamiltonian vector fields $X_f$ and $X_g$. Since $$\Omega(BX_f,Y)=\Omega(X_f,BY)=df(BY)=Bdf(Y)=dg(Y),$$ we deduce that $X_g=BX_f$ and therefore $L_c$ is closed under $B=\lambda-A_0$. $L_c$ is then closed under $A_0$ and so bilagrangian with respect to $\Omega$ and $\omega$. A generating function of the $2D$-Laplace equation satisfies $dJdf=0$, and hence it is the real part of a holomorphic function. The above lemma has a nice interpretation in the Hitchin/Gualtieri formalism: A function $f$ is a generating function of the Monge-Ampère equation $\Delta_\omega=0$ if and only if $f$ is a pluriharmonic function on the generalized complex manifold $(M^4,\exp(\omega-i\Omega))$, that is $$\partial\overline{\partial} f =0.$$ The spaces $U_1$ and $U_{-1}$ are respectively the $i$ and $-i$ eigenspaces for the infinitesimal action of $\mathbb{J}$. So $$\begin{aligned} \mathbb{J}df&=\mathbb{J}\big (\frac{df - i Bdf}{2}\wedge \Phi + \frac{df + i Bdf}{2}\wedge \overline{\Phi}\big)\\ & =i\big (\frac{df - i Bdf}{2}\wedge \Phi - \frac{df + i Bdf}{2}\wedge \overline{\Phi}\big)\\ & = Bdf + (B^2+1)df\wedge\Omega. \end{aligned}$$ Moreover, $$d\big((B^2+1)df\wedge \Omega\big)=d(B^2df\wedge\Omega)=d(Bdf\wedge\omega)=(dBdf)\wedge\omega.$$ We deduce that $d\mathbb{J}df=0$ if and only if $dBdf=0$. Since $d\mathbb{J}df= 2i\partial\overline{\partial} f$, the proposition is proved. Decompose the function $f$ as $f=f_{-2}+f_0+f_2$. Since $\partial f_{-2}=0$ and $\overline{\partial} f_2=0$, $f$ is pluriharmonic if and only if $f_0$ is so. Assume that the $\partial\overline{\partial}$-lemma holds (see [@C] and [@G2]). Then it exists $\psi\in \mathcal{U}_{1}$ such that $$\overline{\partial} f_0= \overline{\partial}\partial \psi.$$ Define then $G_0\in \mathcal{U}_0$ by $G_0=i(\partial \psi-\overline{\partial}\overline{\psi})$. We obtain $$\overline{\partial}(f_0+iG_0)=0$$ and $f_0$ appears as the real part of an “holomorphic object”. Nevertheless, this assumption is not really clear. Does the $\partial\overline{\partial}$-lemma always hold locally ? The following proposition gives an alternative “holomorphic object” when the closed form $\omega$ is primitive (that is $\lambda=0$). Assume that the closed form $\omega$ is primitive and consider the real forms $U=\omega\wedge\Phi$ and $V=(i\Omega+1)\wedge\Phi$. A function $f$ is a generating function of the Monge-Ampère equation $\Delta_\omega=0$ with conjugate function $g$ if and only $$\overline{\partial}(fU-igV)=0.$$ According to proposition \[Ureel\], the closed forms $U$ and $V$ live in $\mathcal{U}_0^{\mathbb{R}}$. Therefore, $d^\mathbb{J} (fU)=-\mathbb{J}d(fU)$ and $d^\mathbb{J} (gV)=-\mathbb{J}d(gV)$. Since $\mathbb{J}^2=-1$ on $U_{-1}\oplus U_{1}$, we get $$2\overline{\partial}(fU-igV)=(d-id^\mathbb{J})(fU-igV)=(1+i\mathbb{J})(dfU-d^\mathbb{J}gV).$$ But, $$dfU=df\wedge \omega\wedge\Phi= df\wedge\omega,$$ and $$\begin{aligned} d^\mathbb{J}gV&=-\mathbb{J}dg\wedge V\\ & = -\mathbb{J}(idg\wedge\Omega+ dg\wedge\Phi)\\ & = -\frac{1}{2}\mathbb{J}(dg\wedge\Phi + dg\wedge\overline{\Phi})\\ & = -\frac{i}{2}(dg\wedge\Phi - dg\wedge\overline{\Phi})\\ &= -dg\wedge\Omega. \end{aligned}$$ We obtain finally $$2\overline{\partial}(fU-igV) = df\wedge\omega + dg\wedge\Omega.$$ The $2D$-Von Karman equation is $$v_xv_{xx}-v_{yy}=0.$$ The corresponding primitive form is $$\omega=p_1dq_2\wedge dp_1+dq_1\wedge dp_2,$$ which is obviously closed. The form $U$ and $V$ are $$\begin{cases} U=p_1dq_2\wedge dp_1+dq_1\wedge dp_2 + 2p_1dq_1\wedge dq_2\wedge dp_1\wedge dp_2&\\ V=1+p_1dq_2\wedge dp_1+dq_1\wedge dp_2 + (p_1-1)dq_1\wedge dq_2\wedge dp_1\wedge dp_2&\\ \end{cases}$$ Generalized Kähler partners --------------------------- Gualtieri has also introduced the notion of Generalized Kähler structure. This is a pair of commuting generalized complex structure such that the symmetric product $(\mathbb{J}_1\mathbb{J}_2)$ is definite positive. The remarkable fact in this theory is that such a structure gives for free two integrable complex structures and a compatible metric (see [@G1]). This theory has been used to construct explicit examples of bihermtian structures on $4$-dimensional compact manifolds (see [@H2]). The idea is that the $+1$-eigenspace $V_+$ of $\mathbb{J}_1\mathbb{J}_2$ is closed under $\mathbb{J}_1$ and $\mathbb{J}_2$ and that the restriction of $(\cdot,\cdot)$ to it is definite positive. The complex structures and the metric come then from the natural isomorphism $V_+\rightarrow T$. From our point of view, this approach gives us the possibility to associate to a given partial differential equation, natural integrable complex structures and inner products. Nevertheless, at least for hyperbolic equations, such inner product should have a signature, and we have may be to a relax a little bit the definition of generalized Kähler structure: Let $\Delta_\omega=0$ be a $2D$-symplectic Monge-Ampère equation of divergent type and let $\mathbb{J}$ be the generalized complex structure associated with. We will say that this Monge-Ampère equation admits a generalized Kähler partner if it exists a generalized complex structure $\mathbb{K}$ commuting with $\mathbb{J}$ such that the two eigenspaces of $\mathbb{J}\mathbb{K}$ are transverse to $T$ and $T^*$. Note that a powerful tool has been done in [@H2] to construct such structures: Let $\exp{\beta_1}$ and $\exp{\beta_2}$ be two complex closed form defining generalized complex struture $\mathbb{J}_1$ and $\mathbb{J}_2$ on $4$-dimensional manifold. Suppose that $$(\beta_1-\beta_2)^2=0=(\beta_1-\overline{\beta_2})^2$$ then $\mathbb{J}_1$ and $\mathbb{J}_2$ commute. Let us see now on a particular case how one can use this tool. Consider an elliptic Monge-Ampère equation $\Delta_\omega=0$ with $d\omega=0$ and $\Omega\wedge\omega=0$. Assume moreover it exists a closed $2$-form $\Theta$ such that $$\Omega\wedge\Theta=\omega\wedge\Theta=0$$ and $$4\omega=\Omega^2+\Theta^2.$$ Note that $\exp(\omega-i\Omega)$ and $\exp(-\omega-i\Theta)$ satisfy the conditions of the above lemma. We suppose also that $\Theta^2=\lambda^2\Omega$ with $\lambda$ a non vanishing function. This implies that $\omega^2=\mu^2\Omega^2$ with $$\mu=\frac{\sqrt{1+\lambda^2}}{2}.$$ The triple $(\omega,\Omega,\Theta)$ defines a metric $G$ and an almost hypercomplex structure $(I,J,K)$ such that $$\omega=\mu G(I\cdot,\cdot),\;\;\; \Omega=G(J\cdot,\cdot),\;\;\; \Theta=\lambda G(K\cdot,\cdot).$$ Define now the two almost complex structures $$I_+=\frac{K+\lambda J}{\mu},\;\;\;I_-=\frac{K-\lambda J}{\mu}.$$ From $$\omega=\frac{\Omega+\Theta}{2}(I_-\cdot,\cdot)$$ and $$\omega=\frac{\Omega-\Theta}{2}(I_+\cdot,\cdot)$$ we deduce that $I_+$ and $I_-$ are integrable. A function $g$ is the conjugate of a generating function $f$ of the Monge-Ampère equation $\Delta_\omega=0$ if and only if $$dI_+dg=-dI_-dg.$$ $f$ is a generating function with conjugate $g$ if and only if $$0=df\wedge\omega+ dg\wedge\Omega = (-\mu Kdf + dg)\wedge\Omega$$ that is if and only if $d\frac{K}{\mu}dg=0$. Consider again the Von Karman equation $$v_xv_{xx}-v_{yy}=0.$$ with corresponding primitive and closed form $$\omega=p_1dq_2\wedge dp_1+dq_1\wedge dp_2.$$ Define then $\Theta$ by $$\Theta=dp_1\wedge dp_2+(1+4p_1)dq_1\wedge dq_2.$$ With the triple $(\omega,\Omega,\Theta)$ we construct $I_+$ and $I_-$ defined by $$I_+=\frac{1}{2}\begin{pmatrix} 0&-1&1&0\\-1/p_1&0&0&-1/p_1\\ -(1+4p_1)/p_1&0&0&-1/p_1\\ 0&1+4p_1&-1&0\\ \end{pmatrix}$$ $$I_-=\frac{1}{2}\begin{pmatrix} 0&-1&-1&0\\-1/p_1&0&0&1/p_1\\ (1+4p_1)/p_1&0&0&-1/p_1\\ 0&-(1+4p_1)&-1&0\\ \end{pmatrix}$$ It is worth mentioning that $I_+$ and $I_-$ are well defined for all $p_1\neq 0$. But the metric $G$ is definite positive only for $p_1<-\frac{1}{4}$. It would be very interesting to understand the behaviour of generating functions and generalized solution of this kind of Monge-Ampère equations with respect to the Gualtieri metric. In particulary, Gualtieri has introduced a scheming generalized Laplacian $dd^*+d^*d$ (see [@G2]) and to know if generating functions (which are pluriharmonic as we have seen) are actually harmonic would give important informations on the global nature of the solutions. This will be the object of further investigations. [99]{} B. Banos, A. Swann, *Potentials for hyper-Kähler metrics with torsion*, Class. Quantum Grav 21, 2004, 3127-3135 O. Ben-Bassat, M. Boyarchenko, *Submanifolds of generalized complex manifolds*, J. Symplectic Geom. 2, 2004, No 3, 309–355 G. R. Cavalcanti, *New aspects of the $dd^c$-lemma*, 2005, math.DG/0501406 M. Crainic, *Generalized complex structures and Lie brackets*, 2004, math.DG/0412097 G. Grantcharov, Y. S. Poon, *Geometry of hyper-Kähler connections with torsion* Comm. Math. Phys. 213 , 2000, No 1, 19–37 M. Gualtieri, *Generalized complex geometry*, 2004, math.DG/0401221 M. Gualtieri, *Generalized geometry and the Hodge decomposition*, 2004, math.DG/04090903 N. J. Hitchin, *Generalized Calabi - Yau manifolds*, Q. J. Math. 54, 2003, 281-308 N. J. Hitchin, *Instantons, Poisson structures and generalized Kähler geometry*, 2005, math.DG/0503432 A. Kushner, V. Lychagin and V. Roubtsov, *Contact geometry and Non-linear Differential Equations*, to appear Z.J. Liu, A. Weinstein, P. Xu, *Manin triples for Lie bialgebroids*, J. Differential Geom. 45, 1997, No 3, 547–574 Lychagin (V.V.): *Contact geometry and non-linear second order differential equations*, Uspèkhi Mat. Nauk, vol 34, 1979, 137-165 (in Russian); english transl. in Russian Math. Surveys, vol 34, 1979 Lychagin (V.V.), Roubtsov (V.N.) and Chekalov (I.V.): *A classification of Monge-Ampère equations*, Ann. scient. Ec. Norm. Sup, 4 ème série, t.26, 1993, 281-308

--- abstract: | We study spectral properties for $H_{K,\Omega}$, the Krein–von Neumann extension of the perturbed Laplacian $-\Delta+V$ defined on $C^\infty_0(\Omega)$, where $V$ is measurable, bounded and nonnegative, in a bounded open set $\Omega\subset\mathbb{R}^n$ belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class $C^{1,r}$, $r>1/2$. In particular, in the aforementioned context we establish the Weyl asymptotic formula $$\#\{j\in\mathbb{N}\,|\,\lambda_{K,\Omega,j}\leq\lambda\} = (2\pi)^{-n} v_n |\Omega|\,\lambda^{n/2}+O\big(\lambda^{(n-(1/2))/2}\big) \, \mbox{ as }\, \lambda\to\infty,$$ where $v_n=\pi^{n/2}/ \Gamma((n/2)+1)$ denotes the volume of the unit ball in $\mathbb{R}^n$, and $\lambda_{K,\Omega,j}$, $j\in\mathbb{N}$, are the non-zero eigenvalues of $H_{K,\Omega}$, listed in increasing order according to their multiplicities. We prove this formula by showing that the perturbed Krein Laplacian (i.e., the Krein–von Neumann extension of $-\Delta+V$ defined on $C^\infty_0(\Omega)$) is spectrally equivalent to the buckling of a clamped plate problem, and using an abstract result of Kozlov from the mid 1980’s. Our work builds on that of Grubb in the early 1980’s, who has considered similar issues for elliptic operators in smooth domains, and shows that the question posed by Alonso and Simon in 1980 pertaining to the validity of the above Weyl asymptotic formula continues to have an affirmative answer in this nonsmooth setting. We also study certain exterior-type domains $\Omega = \mathbb{R}^n\backslash K$, $n\geq 3$, with $K\subset \mathbb{R}^n$ compact and vanishing Bessel capacity $B_{2,2} (K) = 0$, to prove equality of Friedrichs and Krein Laplacians in $L^2(\Omega; d^n x)$, that is, $-\Delta|_{C_0^\infty(\Omega)}$ has a unique nonnegative self-adjoint extension in $L^2(\Omega; d^n x)$. address: - 'Department of Mathematics, University of Missouri, Columbia, MO 65211, USA' - 'Department of Mathematics, University of Missouri, Columbia, MO 65211, USA' - 'Department of Mathematics, University of Missouri, Columbia, MO 65211, USA' - | Faculty of Mathematics\ University of Vienna\ Nordbergstrasse 15\ 1090 Wien\ Austria\ and International Erwin Schrödinger Institute for Mathematical Physics\ Boltzmanngasse 9\ 1090 Wien\ Austria author: - 'Mark S. Ashbaugh' - Fritz Gesztesy - Marius Mitrea - Gerald Teschl title: | Spectral Theory for Perturbed Krein Laplacians\ in Nonsmooth Domains --- [^1] [^2] Introduction {#s1} ============ Let $-\Delta_{D,{\Omega}}$ be the Dirichlet Laplacian associated with an open set $\Omega\subset{{\mathbb{R}}}^n$, and denote by $N_{D,{\Omega}}(\lambda)$ the corresponding spectral distribution function (i.e., the number of eigenvalues of $-\Delta_{D,{\Omega}}$ not exceeding $\lambda$). The study of the asymptotic behavior of $N_{D,{\Omega}}(\lambda)$ as $\lambda\to\infty$ has been initiated by Weyl in 1911–1913 (cf. [@We12a], [@We12], and the references in [@We50]), in response to a question posed in 1908 by the physicist Lorentz, pertaining to the equipartition of energy in statistical mechanics. When $n=2$ and $\Omega$ is a bounded domain with a piecewise smooth boundary, Weyl has shown that $$\label{WA-1} N_{D,{\Omega}}(\lambda)=\frac{{\rm area}\,({\Omega})}{4\pi}\lambda+o(\lambda) \, \mbox{ as }\, \lambda\to\infty,$$ along with the three-dimensional analogue of . In particular, this allowed him to complete a partial proof of Rayleigh, going back to 1903. This ground-breaking work has stimulated a great deal of activity in the intervening years, in which a large number of authors have provided sharper estimates for the remainder, and considered more general elliptic operators equipped with a variety of boundary conditions. For a general elliptic differential operator ${{\mathcal A}}$ of order $2m$ ($m\in{{\mathbb{N}}}$), with smooth coefficients, acting on a smooth subdomain ${\Omega}$ of an $n$-dimensional smooth manifold, spectral asymptotics of the form $$\label{WA-2} N_{D,{\Omega}}({{\mathcal A}};\lambda)=(2\pi)^{-n}\biggl(\int_{\Omega}dx\int_{a^0(x,\xi)<1}d\xi\biggr) \lambda^{n/(2m)}+O\big(\lambda^{(n-1)/(2m)}\big) \, \mbox{ as }\, \lambda\to\infty,$$ where $a^0(x,\xi)$ denotes the principal symbol of ${{\mathcal A}}$, have then been subsequently established in increasing generality (a nice exposition can be found in [@Ag97]). At the same time, it has been realized that, as the smoothness of the domain ${\Omega}$ and the coefficients of ${{\mathcal A}}$ deteriorate, the degree of detail with which the remainder can be described decreases accordingly. Indeed, the smoothness of the boundary of the underlying domain $\Omega$ affects both the nature of the remainder in , as well as the types of differential operators and boundary conditions for which such an asymptotic formula holds. Understanding this correlation then became a central theme of research. For example, in the case of the Laplacian in an arbitrary bounded, open subset ${\Omega}$ of ${{\mathbb{R}}}^n$, Birman and Solomyak have shown in [@BS70] (see also [@BS71], [@BS72], [@BS73], [@BS79]) that the following Weyl asymptotic formula holds $$\label{Wey-1} N_{D,{\Omega}}(\lambda)=(2\pi)^{-n}v_n|\Omega|\,\lambda^{n/2}+o\big(\lambda^{n/2}\big) \, \mbox{ as }\, \lambda\to\infty,$$ where $v_n$ denotes the volume of the unit ball in ${{\mathbb{R}}}^n$, and $|\Omega|$ stands for the $n$-dimensional Euclidean volume of $\Omega$. On the other hand, it is known that may fail for the Neumann Laplacian $-\Delta_{N,{\Omega}}$. Furthermore, if $\alpha\in(0,1)$ then Netrusov and Safarov have proved that $$\label{Wey-2} \Omega\in{\rm Lip}_{\alpha}\,\text{ implies }\, N_{D,{\Omega}}(\lambda)=(2\pi)^{-n}v_n|\Omega|\,\lambda^{n/2} + O\big(\lambda^{(n-\alpha)/2}\big) \, \mbox{ as }\, \lambda\to\infty,$$ where ${\rm Lip}_{\alpha}$ is the class of bounded domains whose boundaries can be locally described by means of graphs of functions satisfying a Hölder condition of order $\alpha$; this result is sharp. See [@NS05] where this intriguing result (along with others, similar in spirit) has been obtained. Surprising connections between Weyl’s asymptotic formula and geometric measure theory have been explored in [@Cae95], [@HL97], [@LF06] for fractal domains. Collectively, this body of work shows that the nature of the Weyl asymptotic formula is intimately related not only to the geometrical properties of the domain (as well as the type of boundary conditions), but also to the smoothness properties of its boundary (the monograph by Safarov and Vassiliev [@SV97] contains a wealth of information on this circle of ideas). These considerations are by no means limited to the Laplacian; see [@Cae98] for the case of the Stokes operator, and [@BF07], [@BS87] for the case the Maxwell system in nonsmooth domains. However, even in the case of the Laplace operator, besides $-\Delta_{D,{\Omega}}$ and $-\Delta_{N,{\Omega}}$ there is a multitude of other concrete extensions of the Laplacian $-\Delta$ on $C^\infty_0({\Omega})$ as a nonnegative, self-adjoint operator in $L^2({\Omega};d^nx)$. The smallest (in the operator theoretic order sense) such realization has been introduced, in an abstract setting, by M. Krein [@Kr47]. Later it was realized that in the case where the symmetric operator, whose self-adjoint extensions are sought, has a strictly positive lower bound, Krein’s construction coincides with one that von Neumann had discussed in his seminal paper [@Ne29] in 1929. For the purpose of this introduction we now briefly recall the construction of the Krein–von Neumann extension of appropriate $L^2({\Omega}; d^n x)$-realizations of the differential operator ${{\mathcal A}}$ of order $2m$, $m\in{{\mathbb{N}}}$, $$\begin{aligned} & {{\mathcal A}}= \sum_{0 \leq |\alpha| \leq 2m} a_{\alpha}(\cdot) D^{\alpha}, {\label}{Wey-3} \\ & D^{\alpha} = (-i \partial/\partial x_1)^{\alpha_1} \cdots (-i\partial/\partial x_n)^{\alpha_n}, \quad \alpha =(\alpha_1,\dots,\alpha_n) \in {{\mathbb{N}}}_0^n, {\label}{Wey-3A} \\ & a_{\alpha} (\cdot) \in C^\infty({\overline}{\Omega}), \quad C^\infty({\overline}{\Omega}) = \bigcap_{k\in{{\mathbb{N}}}_0} C^k({\overline}{\Omega}), {\label}{Weyl-3B}\end{aligned}$$ where $\Omega\subset{{\mathbb{R}}}^n$ is a bounded $C^\infty$ domain. Introducing the particular $L^2({\Omega}; d^n x)$-realization $A_{c,{\Omega}}$ of ${{\mathcal A}}$ defined by $$A_{c,{\Omega}} u = {{\mathcal A}}u, \quad u \in \operatorname{dom}(A_{c,{\Omega}}):=C^\infty_0({\Omega}), {\label}{Wey-3a}$$ we assume the coefficients $a_\alpha$ in ${{\mathcal A}}$ are chosen such that $A_{c,{\Omega}}$ is symmetric, $$\label{Wey-4} (u, A_{c,{\Omega}} v)_{L^2({\Omega};d^nx)}=(A_{c,{\Omega}} u, v)_{L^2({\Omega};d^nx)}, \quad u,v\in C^\infty_0({\Omega}),$$ has a (strictly) positive lower bound, that is, there exists $\kappa_0>0$ such that $$\label{Wey-4bis} (u, A_{c,{\Omega}} u)_{L^2({\Omega};d^nx)}\geq\kappa_0\,\|u\|^2_{L^2({\Omega};d^nx)}, \quad u\in C^\infty_0({\Omega}),$$ and is strongly elliptic, that is, there exists $\kappa_1>0$ such that $$\label{Wey-5} a^0(x,\xi):= {\mathop\mathrm{Re}}\bigg(\sum_{|\alpha|=2m} a_{\alpha}(x) \xi^{\alpha}\bigg) \geq \kappa_1\,|\xi|^{2m}, \quad x\in{\overline}{{\Omega}}, \; \xi \in{{\mathbb{R}}}^n.$$ Next, let $A_{min,{\Omega}}$ and $A_{max,{\Omega}}$ be the $L^2({\Omega};d^nx)$-realizations of ${{\mathcal A}}$ with domains (cf. [@Ag97], [@Gr09]) $$\begin{aligned} \label{Wey-6} \operatorname{dom}(A_{min,{\Omega}})&:=H^{2m}_0({\Omega}), \\ \operatorname{dom}(A_{max,{\Omega}})&:=\big\{u\in L^2(\Omega;d^nx)\,\big|\, {{\mathcal A}}u\in L^2(\Omega;d^nx)\big\}. \end{aligned}$$ Throughout this manuscript, $H^s({\Omega})$ denotes the $L^2$-based Sobolev space of order $s\in{{\mathbb{R}}}$ in ${\Omega}$, and $H_0^{s}(\Omega)$ is the subspace of $H^{s}({{\mathbb{R}}}^n)$ consisting of distributions supported in ${\overline}{{\Omega}}$ (for $s>\frac{1}{2}$, $\big(s-\frac{1}{2}\big)\notin{{\mathbb{N}}}$, the space $H_0^{s}(\Omega)$ can be alternatively described as the closure of $C^\infty_0({\Omega})$ in $H^s({\Omega})$). Given that the domain ${\Omega}$ is smooth, elliptic regularity implies $$\label{Kre-DefY} (A_{min,{\Omega}})^*=A_{max,{\Omega}}\, \mbox{ and }\, {\overline}{A_{c,{\Omega}}}=A_{min,{\Omega}}.$$ Functional analytic considerations (cf. the discussion in Section \[s2\]) dictate that the Krein–von Neumann (sometimes also called the “soft”) extension $A_{K,{\Omega}}$ of $A_{c,{\Omega}}$ on $C^\infty_0({\Omega})$ is the $L^2({\Omega};d^nx)$-realization of $A_{c,{\Omega}}$ with domain (cf.  derived abstractly by Krein) $$\label{Kre-DefX} \operatorname{dom}(A_{K,{\Omega}})=\operatorname{dom}\big({\overline}{A_{c,{\Omega}}}\big)\, \dot{+}\ker\big((A_{c,{\Omega}})^*\big).$$ Above and elsewhere, $X\dot{+}Y$ denotes the direct sum of two subspaces, $X$ and $Y$, of a larger space $Z$, with the property that $X\cap Y=\{0\}$. Thus, granted , we have $$\begin{aligned} \label{Gr-r2} \begin{split} \operatorname{dom}(A_{K,{\Omega}}) &= \operatorname{dom}(A_{min,{\Omega}})\,\dot{+}\ker(A_{max,{\Omega}}) \\ &= H^{2m}_0({\Omega})\,\dot{+}\, \big\{u\in L^2({\Omega};d^nx)\,\big|\,Au=0\mbox{ in }\Omega\big\}. \end{split}\end{aligned}$$ In summary, for domains with smooth boundaries, $A_{K,{\Omega}}$ is the self-adjoint realization of $A_{c,{\Omega}}$ with domain given by . Denote by $\gamma^{m}_D u:= \bigl(\gamma_N^ju\bigr)_{0\leq j\leq m-1}$ the Dirichlet trace operator of order $m\in{{\mathbb{N}}}$ (where $\nu$ denotes the outward unit normal to ${\Omega}$ and $\gamma_N u:=\partial_{\nu}u$ stands for the normal derivative, or Neumann trace), and let $A_{D,{\Omega}}$ be the Dirichlet (sometimes also called the “hard”) realization of $A_{c,{\Omega}}$ in $L^2(\Omega;d^nx)$ with domain $$\label{Wey-8} \operatorname{dom}(A_{D,{\Omega}}):=\big\{u\in H^{2m}(\Omega)\,\big|\,\gamma^{m}_D u=0\big\}.$$ Then $A_{K,{\Omega}}$, $A_{D,{\Omega}}$ are “extremal” in the following sense: Any nonnegative self-adjoint extension $\widetilde{A}$ in $L^2({\Omega};d^nx)$ of $A_{c,{\Omega}}$ (cf. ), necessarily satisfies $$\label{Wey-10} A_{K,{\Omega}} \leq \widetilde{A} \leq A_{D,{\Omega}}$$ in the sense of quadratic forms (cf. the discussion surrounding ). Returning to the case where $A_{c,{\Omega}}=-\Delta|_{C^\infty_0({\Omega})}$, for a bounded domain ${\Omega}$ with a $C^\infty$-smooth boundary, $\partial{\Omega}$, the corresponding Krein–von Neumann extension admits the following description $$\begin{aligned} \begin{split} & -\Delta_{K,{\Omega}} u:= -\Delta u, \\ & \; u\in \operatorname{dom}(-\Delta_{K,{\Omega}}):=\{v\in\operatorname{dom}(-\Delta_{max,{\Omega}})\,|\, \gamma_N v +M_{D,N,{\Omega}}(\gamma_D v)=0\}, \label{Wey-11} \end{split}\end{aligned}$$ where $M_{D,N,{\Omega}}$ is (up to a minus sign) an energy-dependent Dirichlet-to-Neumann map, or Weyl–Titchmarsh operator for the Laplacian. Compared with , the description has the advantage of making explicit the boundary condition implicit in the definition of membership to $\operatorname{dom}(-\Delta_{K,{\Omega}})$. Nonetheless, as opposed to the classical Dirichlet and Neumann boundary condition, this turns out to be [*nonlocal*]{} in nature, as it involves $M_{D,N,{\Omega}}$ which, when ${\Omega}$ is smooth, is a boundary pseudodifferential operator of order $1$. Thus, informally speaking, is the realization of the Laplacian with the boundary condition $$\label{B.A-1} \partial_\nu u=\partial_{\nu}H(u)\, \mbox{ on }\, \partial\Omega,$$ where, given a reasonable function $w$ in ${\Omega}$, $H(w)$ is the harmonic extension of the Dirichlet boundary trace $\gamma^0_D w$ to $\Omega$ (cf. ). While at first sight the nonlocal boundary condition $\gamma_N v +M_{D,N,{\Omega}}(\gamma_D v)=0$ in for the Krein Laplacian $-\Delta_{K,{\Omega}}$ may seem familiar from the abstract approach to self-adjoint extensions of semibounded symmetric operators within the theory of boundary value spaces, there are some crucial distinctions in the concrete case of Laplacians on (nonsmooth) domains which will be delineated at the end of Section \[s8\]. For rough domains, matters are more delicate as the nature of the boundary trace operators and the standard elliptic regularity theory are both fundamentally affected. Following work in [@GM10], here we shall consider the class of [*quasi-convex domains*]{}. The latter is the subclass of bounded, Lipschitz domains in ${{\mathbb{R}}}^n$ characterized by the demand that 1. there exists a sequence of relatively compact, $C^2$-subdomains exhausting the original domain, and whose second fundamental forms are bounded from below in a uniform fashion (for a precise formulation see Definition \[Def-AC\]), or 1. near every boundary point there exists a suitably small $\delta>0$, such that the boundary is given by the graph of a function $\varphi:{{\mathbb{R}}}^{n-1}\to{{\mathbb{R}}}$ (suitably rotated and translated) which is Lipschitz and whose derivative satisfy the pointwise $H^{1/2}$-multiplier condition $$\label{MaS-T4} \sum_{k=1}^{n-1}\|f_k\,\partial_k\varphi_j\|_{H^{1/2}({{\mathbb{R}}}^{n-1})}\leq\delta \sum_{k=1}^{n-1}\|f_k\|_{H^{1/2}({{\mathbb{R}}}^{n-1})}, \quad f_1,...f_{n-1}\in H^{1/2}({{\mathbb{R}}}^{n-1}).$$ See Hypothesis \[h.Conv\] for a precise formulation. In particular, is automatically satisfied when $\omega(\nabla\varphi,t)$, the modulus of continuity of $\nabla\varphi$ at scale $t$, satisfies the square-Dini condition (compare to [@MS85], [@MS05], where this type of domain was introduced and studied), $$\label{MaS-T7} \int_0^1\Bigl(\frac{\omega(\nabla\varphi;t)}{t^{1/2}}\Bigr)^2\,\frac{dt}{t} <\infty.$$ In turn, is automatically satisfied if the Lipschitz function $\varphi$ is of class $C^{1,r}$ for some $r>1/2$. As a result, examples of quasi-convex domains include: 1. All bounded (geometrically) convex domains. 2. All bounded Lipschitz domains satisfying a uniform exterior ball condition (which, informally speaking, means that a ball of fixed radius can be “rolled” along the boundary). 3. All open sets which are the image of a domain as in $(i),(ii)$ above under a $C^{1,1}$-diffeomorphism. 4. All bounded domains of class $C^{1,r}$ for some $r>1/2$. We note that being quasi-convex is a local property of the boundary. The philosophy behind this concept is that Lipschitz-type singularities are allowed in the boundary as long as they are directed outwardly (see Figure 1 on p. ). The key feature of this class of domains is the fact that the classical elliptic regularity property $$\label{Df-H1} \operatorname{dom}(-\Delta_{D,{\Omega}})\subset H^2({\Omega}),\quad \operatorname{dom}(-\Delta_{N,{\Omega}})\subset H^2({\Omega})$$ remains valid. In this vein, it is worth recalling that the presence of a single re-entrant corner for the domain $\Omega$ invalidates . All our results in this paper are actually valid for the class of bounded Lipschitz domains for which holds. Condition is, however, a regularity assumption on the boundary of the Lipschitz domain ${\Omega}$ and the class of quasi-convex domains is the largest one for which we know to hold. Under the hypothesis of quasi-convexity, it has been shown in [@GM10] that the Krein Laplacian $-\Delta_{K,{\Omega}}$ (i.e., the Krein–von Neumann extension of the Laplacian $-\Delta$ defined on $C^\infty_0(\Omega)$) in is a well-defined self-adjoint operator which agrees with the operator constructed using the recipe in . The main issue of the current paper is the study of the spectral properties of $H_{K,{\Omega}}$, the Krein–von Neumann extension of the perturbed Laplacian $$\label{Per-D} -\Delta+V\, \mbox{ on }\, C^\infty_0(\Omega),$$ in the case where both the potential $V$ and the domain ${\Omega}$ are nonsmooth. As regards the former, we shall assume that $0\leq V\in L^\infty(\Omega;d^nx)$, and we shall assume that $\Omega\subset{{\mathbb{R}}}^n$ is a quasi-convex domain (more on this shortly). In particular, we wish to clarify the extent to which a Weyl asymptotic formula continues to hold for this operator. For us, this undertaking was originally inspired by the discussion by Alonso and Simon in [@AS80]. At the end of that paper, the authors comment to the effect that [*“It seems to us that the Krein extension of $-\Delta$, i.e., $-\Delta$ with the boundary condition $\eqref{B.A-1}$, is a natural object and therefore worthy of further study. For example: Are the asymptotics of its nonzero eigenvalues given by Weyl’s formula?”*]{} Subsequently we have learned that when $\Omega$ is $C^\infty$-smooth this has been shown to be the case by Grubb in [@Gr83]. More specifically, in that paper Grubb has proved that if $N_{K,{\Omega}}({{\mathcal A}};\lambda)$ denotes the number of nonzero eigenvalues of $A_{K,{\Omega}}$ (defined as in ) not exceeding $\lambda$, then $$\label{Df-H2} \Omega\in C^\infty\,\text{ implies }\, N_{K,{\Omega}}({{\mathcal A}};\lambda)=C_{A,n}\lambda^{n/(2m)}+O\big(\lambda^{(n-\theta)/(2m)}\big) \, \mbox{ as }\, \lambda\rightarrow\infty,$$ where, with $a^0(x,\xi)$ as in , $$\label{Df-H3} C_{A,n}:=(2\pi)^{-n}\int_\Omega d^nx \int_{a^0(x,\xi)<1} d^n \xi$$ and $$\label{Df-H4} \theta:=\max\,\Bigl\{\frac{1}{2}-\varepsilon\,,\,\frac{2m}{2m+n-1}\Bigl\}, \, \mbox{ with $\varepsilon>0$ arbitrary}.$$ See also [@Mik94] where the author announces a sharpening of the remainder in to any $\theta<1$ (but no proof is provided). To show –, Grubb has reduced the eigenvalue problem $$\label{Df-H5} {{\mathcal A}}u=\lambda\,u,\quad u\in\operatorname{dom}(A_{K,{\Omega}}),\; \lambda>0,$$ to the higher-order, elliptic system $$\begin{cases} {{\mathcal A}}^2 v=\lambda\, {{\mathcal A}}v\,\mbox{ in }\,{\Omega}, \\ \gamma^{2m}_D v =0\,\mbox{ on }\,{{\partial\Omega}}, \\ v\in C^\infty({\overline}{\Omega}). \end{cases} \label{Df-H6}$$ Then the strategy is to use known asymptotics for the spectral distribution function of regular elliptic boundary problems, along with perturbation results due to Birman, Solomyak, and Grubb (see the literature cited in [@Gr83] for precise references). It should be noted that the fact that the boundary of $\Omega$ and the coefficients of ${{\mathcal A}}$ are smooth plays an important role in Grubb’s proof. First, this is used to ensure that holds which, in turn, allows for the concrete representation (a formula which in effect lies at the start of the entire theory, as Grubb adopts this as the [*definition*]{} of the domains of the Krein–von Neumann extension). In addition, at a more technical level, Lemma 3 in [@Gr83] is justified by making appeal to the theory of pseudo-differential operators on $\partial\Omega$, assumed to be an $(n-1)$-dimensional $C^\infty$ manifold. In our case, that is, when dealing with the Krein–von Neumann extension of the perturbed Laplacian , we establish the following theorem: \[Th-InM\] Let ${\Omega}\subset{{\mathbb{R}}}^n$ be a quasi-convex domain, assume that $0 \leq V\in L^\infty({\Omega}; d^nx)$, and denote by $H_{K,{\Omega}}$ the Krein–von Neumann extension of the perturbed Laplacian . Then there exists a sequence of numbers $$\label{Xmam-1} 0<\lambda_{K,{\Omega},1} \leq \lambda_{K,{\Omega},2}\leq\cdots\leq\lambda_{K,{\Omega},j} \leq\lambda_{K,{\Omega},j+1} \leq\cdots$$ converging to infinity, with the following properties. 1. The spectrum of $H_{K,{\Omega}}$ is given by $$\label{XMi-7} \sigma(H_{K,{\Omega}})=\{0\}\cup\{\lambda_{K,{\Omega},j}\}_{j\in{{\mathbb{N}}}},$$ and each number $\lambda_{K,{\Omega},j}$, $j\in{{\mathbb{N}}}$, is an eigenvalue for $H_{K,{\Omega}}$ of finite multiplicity. 2. There exists a countable family of orthonormal eigenfunctions for $H_{K,{\Omega}}$ which span the orthogonal complement of the kernel of this operator. More precisely, there exists a collection of functions $\{w_j\}_{j\in{{\mathbb{N}}}}$ with the following properties: $$\begin{aligned} \label{Xmam-21} & w_j\in\operatorname{dom}(H_{K,{\Omega}})\, \mbox{ and }\, H_{K,{\Omega}}w_j=\lambda_{K,{\Omega},j}w_j, \; j\in{{\mathbb{N}}}, \\ & (w_j,w_k)_{L^2({\Omega};d^nx)}=\delta_{j,k}, \; j,k\in{{\mathbb{N}}}, \label{Xmam-22}\\ & L^2(\Omega;d^nx)=\ker(H_{K,{\Omega}})\,\oplus\, {\overline}{{\rm lin. \, span} \{w_j\}_{j\in{{\mathbb{N}}}}},\, \mbox{ $($orthogonal direct sum$)$.} \label{Xmam-23}\end{aligned}$$ If $V$ is Lipschitz then $w_j\in H^{1/2}(\Omega)$ for every $j$ and, in fact, $w_j\in C^\infty(\overline{\Omega})$ for every $j$ if $\Omega$ is $C^\infty$ and $V\in C^\infty(\overline{\Omega})$. 3. The following min-max principle holds: $$\begin{aligned} \label{Xmam-26} \hspace*{6mm} & \lambda_{K,{\Omega},j} =\min_{\stackrel{W_j\text{ subspace of }H^2_0({\Omega})}{\dim(W_j)=j}} \bigg(\max_{0\not=u\in W_j} \bigg(\frac{\|(-\Delta+V)u\|^2_{L^2({\Omega};d^nx)}} {\|\nabla u\|^2_{(L^2({\Omega};d^nx))^n}+\|V^{1/2}u\|^2_{L^2({\Omega};d^nx)}}\bigg)\bigg), {\notag}\\ & \hspace*{10.85cm} j\in{{\mathbb{N}}}.\end{aligned}$$ 4. If $$\label{Ymam-1} 0<\lambda_{D,{\Omega},1} \leq\lambda_{D,{\Omega},2} \leq\cdots\leq\lambda_{D,{\Omega},j} \leq\lambda_{D,{\Omega},j+1} \leq\cdots$$ are the eigenvalues of the perturbed Dirichlet Laplacian $-\Delta_{D,{\Omega}}$ $($i.e., the Friedrichs extension of in $L^2({\Omega};d^nx)$$)$, listed according to their multiplicities, then $$\label{Xmam-39} 0< \lambda_{D,{\Omega},j} \leq\lambda_{K,{\Omega},j}, \quad j\in{{\mathbb{N}}},$$ Consequently introducing the spectral distribution functions $$\label{Xmam-44} N_{X,{\Omega}}(\lambda):=\#\{j\in{{\mathbb{N}}}\,|\,\lambda_{X,{\Omega},j} \leq\lambda\}, \quad X\in\{D,K\},$$ one has $$\label{Xmam-45} N_{K,{\Omega}}(\lambda)\leq N_{D,{\Omega}}(\lambda).$$ 5. Corresponding to the case $V\equiv 0$, the first nonzero eigenvalue $\lambda^{(0)}_{K,{\Omega},1}$ of $-\Delta_{K,{\Omega}}$ satisfies $$\label{Xmx-4} \lambda^{(0)}_{D,{\Omega},2} \leq \lambda^{(0)}_{K,{\Omega},1}\, \mbox{ and }\, \lambda^{(0)}_{K,{\Omega},2} \leq\frac{n^2+8n+20}{(n+2)^2}\lambda^{(0)}_{K,{\Omega},1}.$$ In addition, $$\label{Xmx-1} \sum_{j=1}^n\lambda^{(0)}_{K,{\Omega},j+1} <(n+4)\lambda^{(0)}_{K,{\Omega},1} -\frac{4}{n+4}(\lambda^{(0)}_{K,{\Omega},2}-\lambda^{(0)}_{K,{\Omega},1}) {\leqslant}(n+4)\lambda^{(0)}_{K,{\Omega},1},$$ and $$\label{Xmx-3} \sum_{j=1}^k \big(\lambda^{(0)}_{K,{\Omega},k+1}-\lambda^{(0)}_{K,{\Omega},j}\big)^2 \leq\frac{4(n+2)}{n^2} \sum_{j=1}^k \big(\lambda^{(0)}_{K,{\Omega},k+1}-\lambda^{(0)}_{K,{\Omega},j}\big) \lambda^{(0)}_{K,{\Omega},j} \quad k\in{{\mathbb{N}}}.$$ Moreover, if ${\Omega}$ is a bounded, convex domain in ${{\mathbb{R}}}^n$, then the first two Dirichlet eigenvalues and the first nonzero eigenvalue of the Krein Laplacian in ${\Omega}$ satisfy $$\label{A-P.1U} \lambda^{(0)}_{D,{\Omega},2} \leq \lambda^{(0)}_{K,{\Omega},1} \leq 4\,\lambda^{(0)}_{D,{\Omega},1}.$$ 6. The following Weyl asymptotic formula holds: $$\label{Xkko-12} N_{K,{\Omega}}(\lambda) =(2\pi)^{-n}v_n|\Omega|\,\lambda^{n/2}+O\big(\lambda^{(n-(1/2))/2}\big) \, \mbox{ as }\, \lambda\to\infty,$$ where, as before, $v_n$ denotes the volume of the unit ball in ${{\mathbb{R}}}^n$, and $|\Omega|$ stands for the $n$-dimensional Euclidean volume of $\Omega$. This theorem answers the question posed by Alonso and Simon in [@AS80] (which corresponds to $V\equiv 0$), and further extends the work by Grubb in [@Gr83] in the sense that we allow nonsmooth domains and coefficients. To prove this result, we adopt Grubb’s strategy and show that the eigenvalue problem $$\label{Df-H7} (-\Delta+V)u=\lambda\,u,\quad u\in\operatorname{dom}(H_{K,{\Omega}}),\; \lambda>0,$$ is equivalent to the following fourth-order problem $$\label{Df-H8} \begin{cases} (-\Delta+V)^2w=\lambda\,(-\Delta+V)w\, \mbox{ in }\, {\Omega}, \\\ \gamma_D w=\gamma_N w=0\, \mbox{ on } \,{{\partial\Omega}}, \\ w\in\operatorname{dom}(-\Delta_{max}). \end{cases}$$ This is closely related to the so-called problem of the [*buckling of a clamped plate*]{}, $$\label{Df-H8F} \begin{cases} -\Delta^2 w=\lambda\,\Delta w \,\mbox{ in }\, {\Omega}, \\ \gamma_D w =\gamma_N w =0\, \mbox{ on }\, {{\partial\Omega}}, \\ w\in\operatorname{dom}(-\Delta_{max}), \end{cases}$$ to which reduces when $V\equiv 0$. From a physical point of view, the nature of the later boundary value problem can be described as follows. In the two-dimensional setting, the bifurcation problem for a clamped, homogeneous plate in the shape of $\Omega$, with uniform lateral compression on its edges has the eigenvalues $\lambda$ of the problem as its critical points. In particular, the first eigenvalue of is proportional to the load compression at which the plate buckles. One of the upshots of our work in this paper is establishing a definite connection between the Krein–von Neumann extension of the Laplacian and the buckling problem . In contrast to the smooth case, since in our setting the solution $w$ of does not exhibit any extra regularity on the Sobolev scale $H^s({\Omega})$, $s\geq 0$, other than membership to $L^2({\Omega};d^nx)$, a suitable interpretation of the boundary conditions in should be adopted. (Here we shall rely on the recent progress from [@GM10] where this issue has been resolved by introducing certain novel boundary Sobolev spaces, well-adapted to the class of Lipschitz domains.) We nonetheless find this trade-off, between the $2$nd-order boundary problem which has nonlocal boundary conditions, and the boundary problem which has local boundary conditions, but is of fourth-order, very useful. The reason is that can be rephrased, in view of and related regularity results developed in [@GM10], in the form of $$\label{XMM-2} (-\Delta+V)^2 u=\lambda\,(-\Delta+V)u \,\mbox{ in } \,\Omega,\quad u\in H^2_0(\Omega).$$ In principle, this opens the door to bringing onto the stage the theory of generalized eigenvalue problems, that is, operator pencil problems of the form $$\label{XMM-3} Tu=\lambda\,Su,$$ where $T$ and $S$ are certain linear operators in a Hilbert space. Abstract results of this nature can be found for instance, in [@LM08], [@Pe68], [@Tr00] (see also [@Le83], [@Le85], where this is applied to the asymptotic distribution of eigenvalues). We, however, find it more convenient to appeal to a version of which emphasizes the role of the symmetric forms $$\begin{aligned} \label{Xkko-13} & a(u,v):=\int_{{\Omega}}d^nx\,{\overline}{(-\Delta+V)u}\,(-\Delta+V)v, \quad u,v\in H^2_0({\Omega}), \\ & b(u,v):=\int_{{\Omega}}d^nx\,{\overline}{\nabla u}\cdot\nabla v +\int_{{\Omega}}d^nx\,{\overline}{V^{1/2}u}\,V^{1/2}v,\quad u,v\in H^2_0({\Omega}), \label{Xkko-13F}\end{aligned}$$ and reformulate as the problem of finding $u\in H^2_0({\Omega})$ which satisfies $$\label{Xkko-14} a(u,v)=\lambda\,b(u,v) \quad v\in H^2_0({\Omega}).$$ This type of eigenvalue problem, in the language of bilinear forms associated with differential operators, has been studied by Kozlov in a series of papers [@Ko79], [@Ko83], [@Ko84]. In particular, in [@Ko84], Kozlov has obtained Weyl asymptotic formulas in the case where the underlying domain $\Omega$ in is merely Lipschitz, and the lower-order coefficients of the quadratic forms – are only measurable and bounded (see Theorem \[T-Koz\] for a precise formulation). Our demand that the potential $V$ is in $L^\infty({\Omega};d^nx)$ is therefore inherited from Kozlov’s theorem. Based on this result and the fact that the problems – and are spectral-equivalent, we can then conclude that holds. Formulas – are also a byproduct of the connection between and and known spectral estimates for the buckling plate problem from [@As04], [@As09], [@AL96], [@CY06], [@HY84], [@Pa55], [@Pa67], [@Pa91]. Similarly, for convex domains is based on the connection between and and the eigenvalue inequality relating the first eigenvalue of a fixed membrane and that of the buckling problem for the clamped plate as proven in [@Pa60] (see also [@Pa67], [@Pa91]). In closing, we wish to point out that in the $C^\infty$-smooth setting, Grubb’s remainder in could, in principle, be sharper than that in . However, the main novel feature of our Theorem \[Th-InM\] is the low regularity assumptions on the underlying domain $\Omega$, and the fact that we allow a nonsmooth potential $V$. As was the case with the Weyl asymptotic formula for the classical Dirichlet and Neumann Laplacians (briefly reviewed at the beginning of this section), the issue of regularity (or lack thereof) has always been of considerable importance in this line of work (as early as 1970, Birman and Solomyak noted in [@BS70] that “[*there has been recently some interest in obtaining the classical asymptotic spectral formulas under the weakest possible hypotheses*]{}.”). The interested reader may consult the paper [@BS79] by Birman and Solomyak (see also [@BS72], [@BS73]), as well as the article [@Da97] by Davies for some very readable, highly informative surveys underscoring this point (collectively, these papers also contain more than 500 references concerning this circle of ideas). Finally, a notational comment: For obvious reasons in connection with quantum mechanical applications, we will, with a slight abuse of notation, dub $-\Delta$ (rather than $\Delta$) as the “Laplacian” in this paper. The Abstract Krein–von Neumann Extension {#s2} ======================================== To get started, we briefly elaborate on the notational conventions used throughout this paper and especially throughout this section which collects abstract material on the Krein–von Neumann extension. Let ${{\mathcal H}}$ be a separable complex Hilbert space, $({\,\cdot\,},{\,\cdot\,})_{{{\mathcal H}}}$ the scalar product in ${{\mathcal H}}$ (linear in the second factor), and $I_{{{\mathcal H}}}$ the identity operator in ${{\mathcal H}}$. Next, let $T$ be a linear operator mapping (a subspace of) a Banach space into another, with $\operatorname{dom}(T)$ and $\operatorname{ran}(T)$ denoting the domain and range of $T$. The closure of a closable operator $S$ is denoted by ${\overline}S$. The kernel (null space) of $T$ is denoted by $\ker(T)$. The spectrum, essential spectrum, and resolvent set of a closed linear operator in ${{\mathcal H}}$ will be denoted by $\sigma(\cdot)$, $\sigma_{\rm ess}(\cdot)$, and $\rho(\cdot)$, respectively. The Banach spaces of bounded and compact linear operators on ${{\mathcal H}}$ are denoted by ${{\mathcal B}}({{\mathcal H}})$ and ${{\mathcal B}}_\infty({{\mathcal H}})$, respectively. Similarly, the Schatten–von Neumann (trace) ideals will subsequently be denoted by ${{\mathcal B}}_p({{\mathcal H}})$, $p\in (0,\infty)$. The analogous notation ${{\mathcal B}}({{\mathcal X}}_1,{{\mathcal X}}_2)$, ${{\mathcal B}}_\infty ({{\mathcal X}}_1,{{\mathcal X}}_2)$, etc., will be used for bounded, compact, etc., operators between two Banach spaces ${{\mathcal X}}_1$ and ${{\mathcal X}}_2$. Moreover, ${{\mathcal X}}_1\hookrightarrow {{\mathcal X}}_2$ denotes the continuous embedding of the Banach space ${{\mathcal X}}_1$ into the Banach space ${{\mathcal X}}_2$. In addition, $U_1 \dotplus U_2$ denotes the direct sum of the subspaces $U_1$ and $U_2$ of a Banach space ${{\mathcal X}}$; and $V_1 \oplus V_2$ represents the orthogonal direct sum of the subspaces $V_j$, $j=1,2$, of a Hilbert space ${{\mathcal H}}$. Throughout this manuscript, if $X$ denotes a Banach space, $X^*$ denotes the [*adjoint space*]{} of continuous conjugate linear functionals on $X$, that is, the [*conjugate dual space*]{} of $X$ (rather than the usual dual space of continuous linear functionals on $X$). This avoids the well-known awkward distinction between adjoint operators in Banach and Hilbert spaces (cf., e.g., the pertinent discussion in [@EE89 p. 3, 4]). Given a reflexive Banach space ${{\mathcal V}}$ and $T \in{{\mathcal B}}({{\mathcal V}},{{\mathcal V}}^*)$, the fact that $T$ is self-adjoint is defined by the requirement that $$\label{B.5} {}_{{{\mathcal V}}}\langle u,T v \rangle_{{{\mathcal V}}^*} = {}_{{{\mathcal V}}^*}\langle T u, v \rangle_{{{\mathcal V}}} = {\overline}{{}_{{{\mathcal V}}}\langle v, T u \rangle_{{{\mathcal V}}^*}}, \quad u, v \in {{\mathcal V}},$$ where in this context bar denotes complex conjugation, ${{\mathcal V}}^*$ is the conjugate dual of ${{\mathcal V}}$, and ${}_{{{\mathcal V}}}\langle{\,\cdot\,},{\,\cdot\,}\rangle_{{{\mathcal V}}^*}$ stands for the ${{\mathcal V}}, {{\mathcal V}}^*$ pairing. A linear operator $S:\operatorname{dom}(S)\subseteq{{\mathcal H}}\to{{\mathcal H}}$, is called [*symmetric*]{}, if $$\label{Pos-2} (u,Sv)_{{\mathcal H}}=(Su,v)_{{\mathcal H}}, \quad u,v\in \operatorname{dom}(S).$$ If $\operatorname{dom}(S)={{\mathcal H}}$, the classical Hellinger–Toeplitz theorem guarantees that $S\in{{\mathcal B}}({{\mathcal H}})$, in which situation $S$ is readily seen to be self-adjoint. In general, however, symmetry is a considerably weaker property than self-adjointness and a classical problem in functional analysis is that of determining all self-adjoint extensions in ${{\mathcal H}}$ of a given unbounded symmetric operator of equal and nonzero deficiency indices. (Here self-adjointness of an operator ${\widetilde}S$ in ${{\mathcal H}}$, is of course defined as usual by $\big({\widetilde}S\big)^* = {\widetilde}S$.) In this manuscript we will be particularly interested in this question within the class of densely defined (i.e., ${\overline}{\operatorname{dom}(S)}={{\mathcal H}}$), nonnegative operators (in fact, in most instances $S$ will even turn out to be strictly positive) and we focus almost exclusively on self-adjoint extensions that are nonnegative operators. In the latter scenario, there are two distinguished constructions which we will briefly review next. To set the stage, we recall that a linear operator $S:\operatorname{dom}(S)\subseteq{{\mathcal H}}\to {{\mathcal H}}$ is called [*nonnegative*]{} provided $$\label{Pos-1} (u,Su)_{{\mathcal H}}\geq 0, \quad u\in \operatorname{dom}(S).$$ (In particular, $S$ is symmetric in this case.) $S$ is called [*strictly positive*]{}, if for some $\varepsilon >0$, $(u,Su)_{{\mathcal H}}\geq \varepsilon \|u\|_{{{\mathcal H}}}^2$, $u\in \operatorname{dom}(S)$. Next, we recall that $A \leq B$ for two self-adjoint operators in ${{\mathcal H}}$ if $$\begin{aligned} \begin{split} & \operatorname{dom}\big(|A|^{1/2}\big) \supseteq \operatorname{dom}\big(|B|^{1/2}\big) \, \text{ and } \\ & \big(|A|^{1/2}u, U_A |A|^{1/2}u\big)_{{{\mathcal H}}} \leq \big(|B|^{1/2}u, U_B |B|^{1/2}u\big)_{{{\mathcal H}}}, \quad u \in \operatorname{dom}\big(|B|^{1/2}\big), {\label}{AleqB} \end{split}\end{aligned}$$ where $U_C$ denotes the partial isometry in ${{\mathcal H}}$ in the polar decomposition of a densely defined closed operator $C$ in ${{\mathcal H}}$, $C=U_C |C|$, $|C|=(C^* C)^{1/2}$. (If in addition, $C$ is self-adjoint, then $U_C$ and $|C|$ commute.) We also recall ([@Fa75 Part II], [@Ka80 Theorem VI.2.21]) that if $A$ and $B$ are both self-adjoint and nonnegative in ${{\mathcal H}}$, then $$\begin{aligned} \begin{split} & 0 \leq A\leq B \, \text{ if and only if } \, 0 \leq A^{1/2 }\leq B^{1/2}, \label{PPa-1} \\ & \quad \text{equivalently, if and only if } \, (B + a I_{{\mathcal H}})^{-1} \leq (A + a I_{{\mathcal H}})^{-1} \, \text{ for all $a>0$,} \end{split}\end{aligned}$$ and $$\ker(A) =\ker\big(A^{1/2}\big)$$ (with $C^{1/2}$ the unique nonnegative square root of a nonnegative self-adjoint operator $C$ in ${{\mathcal H}}$). For simplicity we will always adhere to the conventions that $S$ is a linear, unbounded, densely defined, nonnegative (i.e., $S\geq 0$) operator in ${{\mathcal H}}$, and that $S$ has nonzero deficiency indices. In particular, $${\rm def} (S) = \dim (\ker(S^*-z I_{{{\mathcal H}}})) \in {{\mathbb{N}}}\cup\{\infty\}, \quad z\in {{\mathbb{C}}}\backslash [0,\infty), {\label}{DEF}$$ is well-known to be independent of $z$. Moreover, since $S$ and its closure ${\overline}{S}$ have the same self-adjoint extensions in ${{\mathcal H}}$, we will without loss of generality assume that $S$ is closed in the remainder of this section. The following is a fundamental result to be found in M. Krein’s celebrated 1947 paper [@Kr47] (cf. also Theorems 2 and 5–7 in the English summary on page 492): \[T-kkrr\] Assume that $S$ is a densely defined, closed, nonnegative operator in ${{\mathcal H}}$. Then, among all nonnegative self-adjoint extensions of $S$, there exist two distinguished ones, $S_K$ and $S_F$, which are, respectively, the smallest and largest $($in the sense of order between self-adjoint operators, cf. $)$ such extension. Furthermore, a nonnegative self-adjoint operator $\widetilde{S}$ is a self-adjoint extension of $S$ if and only if $\widetilde{S}$ satisfies $$\label{Fr-Sa} S_K\leq\widetilde{S}\leq S_F.$$ In particular, determines $S_K$ and $S_F$ uniquely.\ In addition, if $S\geq \varepsilon I_{{{\mathcal H}}}$ for some $\varepsilon >0$, one has $S_F \geq \varepsilon I_{{{\mathcal H}}}$, and $$\begin{aligned} \operatorname{dom}(S_F) &= \operatorname{dom}(S) \dotplus (S_F)^{-1} \ker (S^*), {\label}{SF} \\ \operatorname{dom}(S_K) & = \operatorname{dom}(S) \dotplus \ker (S^*), {\label}{SK} \\ \operatorname{dom}(S^*) & = \operatorname{dom}(S) \dotplus (S_F)^{-1} \ker (S^*) \dotplus \ker (S^*) {\notag}\\ & = \operatorname{dom}(S_F) \dotplus \ker (S^*), {\label}{S*} \end{aligned}$$ in particular, $$\label{Fr-4Tf} \ker(S_K)= \ker\big((S_K)^{1/2}\big)= \ker(S^*) = \operatorname{ran}(S)^{\bot}.$$ Here the operator inequalities in are understood in the sense of and hence they can equivalently be written as $$(S_F + a I_{{{\mathcal H}}})^{-1} {\leqslant}\big({\widetilde}S + a I_{{{\mathcal H}}}\big)^{-1} {\leqslant}(S_K + a I_{{{\mathcal H}}})^{-1} \, \text{ for some (and hence for all\,) $a > 0$.} {\label}{Res}$$ We will call the operator $S_K$ the [*Krein–von Neumann extension*]{} of $S$. See [@Kr47] and also the discussion in [@AS80], [@AT03], [@AT05]. It should be noted that the Krein–von Neumann extension was first considered by von Neumann [@Ne29] in 1929 in the case where $S$ is strictly positive, that is, if $S \geq \varepsilon I_{{{\mathcal H}}}$ for some $\varepsilon >0$. (His construction appears in the proof of Theorem 42 on pages 102–103.) However, von Neumann did not isolate the extremal property of this extension as described in and . M. Krein [@Kr47], [@Kr47a] was the first to systematically treat the general case $S\geq 0$ and to study all nonnegative self-adjoint extensions of $S$, illustrating the special role of the [*Friedrichs extension*]{} (i.e., the “hard” extension) $S_F$ of $S$ and the Krein–von Neumann (i.e., the “soft”) extension $S_K$ of $S$ as extremal cases when considering all nonnegative extensions of $S$. For a recent exhaustive treatment of self-adjoint extensions of semibounded operators we refer to [@AT02]–[@AT09]. For classical references on the subject of self-adjoint extensions of semibounded operators (not necessarily restricted to the Krein–von Neumann extension) we refer to Birman [@Bi56], [@Bi08], Friedrichs [@Fr34], Freudenthal [@Fr36], Grubb [@Gr68], [@Gr70], Krein [@Kr47a], [S]{}traus [@St73], and Vi[s]{}ik [@Vi63] (see also the monographs by Akhiezer and Glazman [@AG81a Sect. 109], Faris [@Fa75 Part III], and the recent book by Grubb [@Gr09 Sect. 13.2]). An intrinsic description of the Friedrichs extension $S_F$ of $S\geq 0$ due to Freudenthal [@Fr36] in 1936 describes $S_F$ as the operator $S_F:\operatorname{dom}(S_F)\subset{{\mathcal H}}\to{{\mathcal H}}$ given by $$\begin{aligned} & S_F u:=S^*u, {\notag}\\ & u \in \operatorname{dom}(S_F):=\big\{v\in\operatorname{dom}(S^*)\,\big|\, \mbox{there exists} \, \{v_j\}_{j\in{{\mathbb{N}}}}\subset \operatorname{dom}(S), \label{Fr-2} \\ & \quad \mbox{with} \, \lim_{j\to\infty}\|v_j-v\|_{{{\mathcal H}}}=0 \mbox{ and } ((v_j-v_k),S(v_j-v_k))_{{\mathcal H}}\to 0 \mbox{ as } j,k\to\infty\big\}. {\notag}\end{aligned}$$ Then, as is well-known, $$\begin{aligned} & S_F \geq 0, \label{Fr-4} \\ & \operatorname{dom}\big((S_F)^{1/2}\big)=\big\{v\in{{\mathcal H}}\,\big|\, \mbox{there exists} \, \{v_j\}_{j\in{{\mathbb{N}}}}\subset \operatorname{dom}(S), \label{Fr-4J} \\ & \quad \mbox{with} \lim_{j\to\infty}\|v_j-v\|_{{{\mathcal H}}}=0 \mbox{ and } ((v_j-v_k),S(v_j-v_k))_{{\mathcal H}}\to 0\mbox{ as } j,k\to\infty\big\}, {\notag}\end{aligned}$$ and $$\label{Fr-4H} S_F=S^*|_{\operatorname{dom}(S^*)\cap\operatorname{dom}((S_{F})^{1/2})}.$$ Equations and are intimately related to the definition of $S_F$ via (the closure of) the sesquilinear form generated by $S$ as follows: One introduces the sesquilinear form $$q_S(f,g)=(f,Sg)_{{{\mathcal H}}}, \quad f, g \in \operatorname{dom}(q_S)=\operatorname{dom}(S).$$ Since $S\geq 0$, the form $q_S$ is closable and we denote by $Q_S$ the closure of $q_S$. Then $Q_S\geq 0$ is densely defined and closed. By the first and second representation theorem for forms (cf., e.g., [@Ka80 Sect. 6.2]), $Q_S$ is uniquely associated with a nonnegative, self-adjoint operator in ${{\mathcal H}}$. This operator is precisely the Friedrichs extension, $S_F \geq 0$, of $S$, and hence, $$\begin{aligned} \begin{split} & Q_S(f,g)=(f,S_F g)_{{{\mathcal H}}}, \quad f \in \operatorname{dom}(Q_S), \, g \in \operatorname{dom}(S_F), {\label}{Fr-Q} \\ & \operatorname{dom}(Q_S) = \operatorname{dom}\big((S_F)^{1/2}\big). \end{split} \end{aligned}$$ An intrinsic description of the Krein–von Neumann extension $S_K$ of $S\geq 0$ has been given by Ando and Nishio [@AN70] in 1970, where $S_K$ has been characterized as the operator $S_K:\operatorname{dom}(S_K)\subset{{\mathcal H}}\to{{\mathcal H}}$ given by $$\begin{aligned} & S_Ku:=S^*u, {\notag}\\ & u \in \operatorname{dom}(S_K):=\big\{v\in\operatorname{dom}(S^*)\,\big|\,\mbox{there exists} \, \{v_j\}_{j\in{{\mathbb{N}}}}\subset \operatorname{dom}(S), \label{Fr-2X} \\ & \quad \mbox{with} \, \lim_{j\to\infty} \|Sv_j-S^*v\|_{{{\mathcal H}}}=0 \mbox{ and } ((v_j-v_k),S(v_j-v_k))_{{\mathcal H}}\to 0 \mbox{ as } j,k\to\infty\big\}. {\notag}\end{aligned}$$ By one observes that shifting $S$ by a constant commutes with the operation of taking the Friedrichs extension of $S$, that is, for any $c\in{{\mathbb{R}}}$, $$(S + c I_{{{\mathcal H}}})_{F} = S_F + c I_{{{\mathcal H}}}, {\label}{Fr-c}$$ but by , the analog of for the Krein–von Neumann extension $S_K$ fails. At this point we recall a result due to Makarov and Tsekanovskii [@MT07], concerning symmetries (e.g., the rotational symmetry exploited in Section \[s1vi\]), and more generally, a scale invariance, shared by $S$, $S^*$, $S_F$, and $S_K$ (see also [@HK09]). Actually, we will prove a slight extension of the principal result in [@MT07]: [p2.2a]{} Let $\mu > 0$, suppose that $V, V^{-1} \in {{\mathcal B}}({{\mathcal H}})$, and assume $S$ to be a densely defined, closed, nonnegative operator in ${{\mathcal H}}$ satisfying $$V S V^{-1} = \mu S, {\label}{VS}$$ and $$V S V^{-1} = (V^*)^{-1} S V^* \, \text{ $($or equivalently, $(V^* V)^{-1} S (V^* V) = S$\,$)$.}$$ Then also $S^*$, $S_F$, and $S_K$ satisfy $$\begin{aligned} (V^* V)^{-1} S^* (V^* V) &= S^*, \,\,\,\quad V S^* V^{-1} = \mu S^*, {\label}{VS*} \\ (V^* V)^{-1} S_F (V^* V) &= S_F, \, \quad V S_F V^{-1} = \mu S_F, {\label}{VSF}\\ (V^* V)^{-1} S_K (V^* V) &= S_K, \quad V S_K V^{-1} = \mu S_K. {\label}{VSK} \end{aligned}$$ Applying [@We80 p. 73, 74], yields $V S V^{-1} = (V^*)^{-1} S V^*$. The latter relation is equivalent to $(V^* V)^{-1} S (V^* V) = S$ and hence also equivalent to $(V^* V) S (V^* V)^{-1} = S$. Taking adjoints (and applying [@We80 p. 73, 74] again) then yields $(V^*)^{-1} S^* V^* = V S^* V^{-1}$; the latter is equivalent to $(V^* V)^{-1} S^* (V^* V) = S^*$ and hence also equivalent to $(V^* V) S^* (V^* V)^{-1} = S$. Replacing $S$ and $S^*$ by $(V^* V)^{-1} S (V^* V)$ and $(V^* V)^{-1} S^* (V^* V)$, respectively, in , and subsequently, in , then yields that $$(V^* V)^{-1} S_F (V^* V) = S_F \, \text{ and } \, (V^* V)^{-1} S_K (V^* V) = S_K.$$ The latter are of course equivalent to $$(V^* V) S_F (V^* V)^{-1} = S_F \, \text{ and } \, (V^* V) S_K (V^* V)^{-1} = S_K.$$ Finally, replacing $S$ by $V S V^{-1}$ and $S^*$ by $V S^* V^{-1}$ in then proves $V S_F V^{-1} = \mu S_F$. Performing the same replacement in then yields $V S_K V^{-1} = \mu S_K$. If in addition, $V$ is unitary (implying $V^* V = I_{{{\mathcal H}}}$), Proposition \[p2.2a\] immediately reduces to [@MT07 Theorem 2.2]. In this special case one can also provide a quick alternative proof by directly invoking the inequalities and the fact that they are preserved under unitary equivalence. Similarly to Proposition \[p2.2a\], the following results also immediately follows from the characterizations and of $S_F$ and $S_K$, respectively: [p2.3]{} Let $U\colon{{\mathcal H}}_1\to{{\mathcal H}}_2$ be unitary from ${{\mathcal H}}_1$ onto ${{\mathcal H}}_2$ and assume $S$ to be a densely defined, closed, nonnegative operator in ${{\mathcal H}}_1$ with adjoint $S^*$, Friedrichs extension $S_F$, and Krein–von Neumann extension $S_K$ in ${{\mathcal H}}_1$, respectively. Then the adjoint, Friedrichs extension, and Krein–von Neumann extension of the nonnegative, closed, densely defined, symmetric operator $USU^{-1}$ in ${{\mathcal H}}_2$ are given by $$\begin{aligned} [USU^{-1}]^* = US^*U^{-1}, \quad [USU^{-1}]_F = US_F U^{-1}, \quad [USU^{-1}]_K = US_K U^{-1}\end{aligned}$$ in ${{\mathcal H}}_2$, respectively. [p2.4]{} Let $J\subseteq {{\mathbb{N}}}$ be some countable index set and consider ${{\mathcal H}}= \bigoplus_{j\in J} {{\mathcal H}}_j$ and $S=\bigoplus_{j\in J} S_j$, where each $S_j$ is a densely defined, closed, nonnegative operator in ${{\mathcal H}}_j$, $j\in J$. Denoting by $(S_j)_F$ and $(S_j)_K$ the Friedrichs and Krein–von Neumann extension of $S_j$ in ${{\mathcal H}}_j$, $j\in J$, one infers $$S^*=\bigoplus_{j\in J} \; (S_j)^*, \quad S_F=\bigoplus_{j\in J} \; (S_j)_F, \quad S_K = \bigoplus_{j\in J} \; (S_j)_K.$$ The following is a consequence of a slightly more general result formulated in [@AN70 Theorem 1]: \[Pr-an\] Let $S$ be a densely defined, closed, nonnegative operator in ${{\mathcal H}}$. Then $S_K$, the Krein–von Neumann extension of $S$, has the property that $$\label{an-T1} \operatorname{dom}\big((S_K)^{1/2}\big)=\biggl\{u\in{{\mathcal H}}\,\bigg|\,\sup_{v\in\operatorname{dom}(S)} \frac{|(u,Sv)_{{\mathcal H}}|^2}{(v,Sv)_{{\mathcal H}}} <+\infty\biggr\},$$ and $$\label{an-T2} \big\|(S_K)^{1/2}u\big\|^2_{{\mathcal H}}=\sup_{v\in\operatorname{dom}(S)} \frac{|(u,Sv)_{{\mathcal H}}|^2}{(v,Sv)_{{\mathcal H}}}, \quad u\in\operatorname{dom}\big((S_K)^{1/2}\big).$$ A word of explanation is in order here: Given $S\geq 0$ as in the statement of Proposition \[Pr-an\], the Cauchy-Schwarz-type inequality $$\label{CS-I.1} |(u,Sv)_{{{\mathcal H}}}|^2\leq (u,Su)_{{{\mathcal H}}} (v,Sv)_{{{\mathcal H}}}, \quad u,v\in\operatorname{dom}(S),$$ shows (due to the fact that $\operatorname{dom}(S)\hookrightarrow{{\mathcal H}}$ densely) that $$\label{CS-I.2} u\in\operatorname{dom}(S) \,\mbox{ and } \, (u,S u)_{{{\mathcal H}}} =0 \, \text{ imply }\,Su=0.$$ Thus, whenever the denominator of the fractions appearing in , vanishes, so does the numerator, and one interprets $0/0$ as being zero in , . We continue by recording an abstract result regarding the parametrization of all nonnegative self-adjoint extensions of a given strictly positive, densely defined, symmetric operator. The following results were developed from Krein [@Kr47], Vi[s]{}ik [@Vi63], and Birman [@Bi56], by Grubb [@Gr68], [@Gr70]. Subsequent expositions are due to Faris [@Fa75 Sect. 15], Alonso and Simon [@AS80] (in the present form, the next theorem appears in [@GM10]), and Derkach and Malamud [@DM91], [@Ma92]. We start by collecting our basic assumptions: [h2.6]{} Suppose that $S$ is a densely defined, symmetric, closed operator with nonzero deficiency indices in ${{\mathcal H}}$ that satisfies $$S\geq \varepsilon I_{{{\mathcal H}}} \, \text{ for some $\varepsilon >0$.} {\label}{3.1}$$ \[AS-th\] Suppose Hypothesis \[h2.6\]. Then there exists a one-to-one correspondence between nonnegative self-adjoint operators $0 \leq B:\operatorname{dom}(B)\subseteq {{\mathcal W}}\to {{\mathcal W}}$, ${\overline}{\operatorname{dom}(B)}={{\mathcal W}}$, where ${{\mathcal W}}$ is a closed subspace of ${{\mathcal N}}_0 :=\ker(S^*)$, and nonnegative self-adjoint extensions $S_{B,{{\mathcal W}}}\geq 0$ of $S$. More specifically, $S_F$ is invertible, $S_F\geq \varepsilon I_{{{\mathcal H}}}$, and one has $$\begin{aligned} & \operatorname{dom}(S_{B,{{\mathcal W}}}) =\big\{f + (S_F)^{-1}(Bw + \eta)+ w \,\big|\, f\in\operatorname{dom}(S),\, w\in\operatorname{dom}(B),\, \eta\in {{\mathcal N}}_0 \cap {{\mathcal W}}^{\bot}\big\}, {\notag}\\ & S_{B,{{\mathcal W}}} = S^*|_{\operatorname{dom}(S_{B,{{\mathcal W}}})}, \label{AS-2} \end{aligned}$$ where ${{\mathcal W}}^{\bot}$ denotes the orthogonal complement of ${{\mathcal W}}$ in ${{\mathcal N}}_0$. In addition, $$\begin{aligned} &\operatorname{dom}\big((S_{B,{{\mathcal W}}})^{1/2}\big) = \operatorname{dom}\big((S_F)^{1/2}\big) \dotplus \operatorname{dom}\big(B^{1/2}\big), \\ &\big\|(S_{B,{{\mathcal W}}})^{1/2}(u+g)\big\|_{{{\mathcal H}}}^2 =\big\|(S_F)^{1/2} u\big\|_{{{\mathcal H}}}^2 + \big\|B^{1/2} g\big\|_{{{\mathcal H}}}^2, \\ & \hspace*{2.3cm} u \in \operatorname{dom}\big((S_F)^{1/2}\big), \; g \in \operatorname{dom}\big(B^{1/2}\big), {\notag}\end{aligned}$$ implying, $$\label{K-ee} \ker(S_{B,{{\mathcal W}}})=\ker(B).$$ Moreover, $$B \leq {\widetilde}B \, \text{ implies } \, S_{B,{{\mathcal W}}} \leq S_{{\widetilde}B,{\widetilde}{{\mathcal W}}},$$ where $$\begin{aligned} \begin{split} & B\colon \operatorname{dom}(B) \subseteq {{\mathcal W}}\to {{\mathcal W}}, \quad {\widetilde}B\colon \operatorname{dom}\big({\widetilde}B\big) \subseteq {\widetilde}{{\mathcal W}}\to {\widetilde}{{\mathcal W}}, \\ & {\overline}{\operatorname{dom}\big({\widetilde}B\big)} = {\widetilde}{{\mathcal W}}\subseteq {{\mathcal W}}= {\overline}{\operatorname{dom}(B)}. \end{split}\end{aligned}$$ In the above scheme, the Krein–von Neumann extension $S_K$ of $S$ corresponds to the choice ${{\mathcal W}}={{\mathcal N}}_0$ and $B=0$ $($with $\operatorname{dom}(B)=\operatorname{dom}\big(B^{1/2}\big)={{\mathcal N}}_0=\ker (S^*)$$)$. In particular, one thus recovers , and , and also obtains $$\begin{aligned} &\operatorname{dom}\big((S_{K})^{1/2}\big) = \operatorname{dom}\big((S_F)^{1/2}\big) \dotplus \ker (S^*), {\label}{SKform1} \\ &\big\|(S_{K})^{1/2}(u+g)\big\|_{{{\mathcal H}}}^2 =\big\|(S_F)^{1/2} u\big\|_{{{\mathcal H}}}^2, \quad u \in \operatorname{dom}\big((S_F)^{1/2}\big), \; g \in \ker (S^*). {\label}{SKform2}\end{aligned}$$ Finally, the Friedrichs extension $S_F$ corresponds to the choice $\operatorname{dom}(B)=\{0\}$ $($i.e., formally, $B\equiv\infty$$)$, in which case one recovers . The relation $B \leq {\widetilde}B$ in the case where ${\widetilde}{{\mathcal W}}\subsetneqq {{\mathcal W}}$ requires an explanation: In analogy to we mean $$\big(|B|^{1/2}u, U_B |B|^{1/2}u\big)_{{{\mathcal W}}} \leq \big(|{\widetilde}B|^{1/2}u, U_{{\widetilde}B} |{\widetilde}B|^{1/2}u\big)_{{{\mathcal W}}}, \quad u \in \operatorname{dom}\big(|{\widetilde}B|^{1/2}\big)$$ and (following [@AS80]) we put $$\big(|{\widetilde}B|^{1/2}u, U_{{\widetilde}B} |{\widetilde}B|^{1/2}u)_{{{\mathcal W}}} = \infty \, \text{ for } \, u \in {{\mathcal W}}\backslash \operatorname{dom}\big(|{\widetilde}B|^{1/2}\big).$$ For subsequent purposes we also note that under the assumptions on $S$ in Hypothesis \[h2.6\], one has $$\dim(\ker (S^*-z I_{{{\mathcal H}}})) = \dim(\ker(S^*)) = \dim ({{\mathcal N}}_0) = {\rm def} (S), \quad z\in {{\mathbb{C}}}\backslash [\varepsilon,\infty). {\label}{dim}$$ The following result is a simple consequence of , , and , but since it seems not to have been explicitly stated in [@Kr47], we provide the short proof for completeness (see also [@Ma92 Remark 3]). First we recall that two self-adjoint extensions $S_1$ and $S_2$ of $S$ are called [*relatively prime*]{} if $\operatorname{dom}(S_1) \cap \operatorname{dom}(S_2) = \operatorname{dom}(S)$. [lKF]{} Suppose Hypothesis \[h2.6\]. Then $S_F$ and $S_K$ are relatively prime, that is, $$\operatorname{dom}(S_F) \cap \operatorname{dom}(S_K) = \operatorname{dom}(S). {\label}{RP}$$ By and it suffices to prove that $\ker (S^*) \cap (S_F)^{-1}\ker (S^*) = \{0\}$. Let $f_0 \in \ker (S^*) \cap (S_F)^{-1}\ker (S^*)$. Then $S^* f_0 =0$ and $f_0=(S_F)^{-1}g_0$ for some $g_0 \in \ker (S^*)$. Thus one concludes that $f_0 \in\operatorname{dom}(S_F)$ and $S_F f_0 =g_0$. But $S_F = S^*|_{\operatorname{dom}(S_F)}$ and hence $g_0 =S_F f_0 = S^* f_0 = 0$. Since $g_0 =0$ one finally obtains $f_0 =0$. Next, we consider a self-adjoint operator $$\label{Barr-1} T:\operatorname{dom}(T)\subseteq {{\mathcal H}}\to{{\mathcal H}},\quad T=T^*,$$ which is bounded from below, that is, there exists $\alpha\in{{\mathbb{R}}}$ such that $$\label{Barr-2} T\geq \alpha I_{{{\mathcal H}}}.$$ We denote by $\{E_T(\lambda)\}_{\lambda\in{{\mathbb{R}}}}$ the family of strongly right-continuous spectral projections of $T$, and introduce, as usual, $E_T((a,b))=E_T(b_-) - E_T(a)$, $E_T(b_-) = \operatorname*{s-lim}_{\varepsilon\downarrow 0}E_T(b-\varepsilon)$, $-\infty \leq a < b$. In addition, we set $$\label{Barr-3} \mu_{T,j}:=\inf\,\bigl\{\lambda\in{{\mathbb{R}}}\,|\, \dim (\operatorname{ran}(E_T((-\infty,\lambda)))) \geq j\bigr\},\quad j\in{{\mathbb{N}}}.$$ Then, for fixed $k\in{{\mathbb{N}}}$, either:\ $(i)$ $\mu_{T,k}$ is the $k$th eigenvalue of $T$ counting multiplicity below the bottom of the essential spectrum, $\sigma_{\rm ess}(T)$, of $T$,\ or,\ $(ii)$ $\mu_{T,k}$ is the bottom of the essential spectrum of $T$, $$\mu_{T,k} = \inf \{\lambda \in {{\mathbb{R}}}\,|\, \lambda \in \sigma_{\rm ess}(T)\},$$ and in that case $\mu_{T,k+\ell} = \mu_{T,k}$, $\ell\in{{\mathbb{N}}}$, and there are at most $k-1$ eigenvalues (counting multiplicity) of $T$ below $\mu_{T,k}$. We now record a basic result of M. Krein [@Kr47] with an important extension due to Alonso and Simon [@AS80] and some additional results recently derived in [@AGMST09]. For this purpose we introduce the [*reduced Krein–von Neumann operator*]{} ${\widehat}S_K$ in the Hilbert space (cf. ) $${\widehat}{{\mathcal H}}= [\ker (S^*)]^{\bot} = \big[I_{{{\mathcal H}}} - P_{\ker(S^*)}\big] {{\mathcal H}}= \big[I_{{{\mathcal H}}} - P_{\ker(S_K)}\big] {{\mathcal H}}= [\ker (S_K)]^{\bot}, {\label}{hattH}$$ by $$\begin{aligned} {\widehat}S_K:&=S_K|_{[\ker(S_K)]^{\bot}} \label{Barr-4} \\ \begin{split} & = S_K[I_{{{\mathcal H}}} - P_{\ker(S_K)}] {\label}{SKP} \, \text{ in $[I_{{{\mathcal H}}} - P_{\ker(S_K)}]{{\mathcal H}}$} \\ &= [I_{{{\mathcal H}}} - P_{\ker(S_K)}]S_K[I_{{{\mathcal H}}} - P_{\ker(S_K)}] \, \text{ in $[I_{{{\mathcal H}}} - P_{\ker(S_K)}]{{\mathcal H}}$}, \end{split}\end{aligned}$$ where $P_{\ker(S_K)}$ denotes the orthogonal projection onto $\ker(S_K)$ and we are alluding to the orthogonal direct sum decomposition of ${{\mathcal H}}$ into $${{\mathcal H}}= P_{\ker(S_K)}{{\mathcal H}}\oplus [I_{{{\mathcal H}}} - P_{\ker(S_K)}]{{\mathcal H}}.$$ Assuming Hypothesis \[h2.6\], we recall that Krein [@Kr47] (see also [@Ma92 Corollary 5] for a generalization to the case $S\geq 0$) proved the formula $$\big({\widehat}S_K\big)^{-1} = [I_{{{\mathcal H}}} - P_{\ker(S_K)}] (S_F)^{-1} [I_{{{\mathcal H}}} - P_{\ker(S_K)}]. {\label}{SKinv}$$ [AS-thK]{} Suppose Hypothesis \[h2.6\]. Then, $$\label{Barr-5} \varepsilon \leq \mu_{S_F,j} \leq \mu_{{\widehat}S_K,j}, \quad j\in{{\mathbb{N}}}.$$ In particular, if the Friedrichs extension $S_F$ of $S$ has purely discrete spectrum, then, except possibly for $\lambda=0$, the Krein–von Neumann extension $S_K$ of $S$ also has purely discrete spectrum in $(0,\infty)$, that is, $$\sigma_{\rm ess}(S_F) = \emptyset \, \text{ implies } \, \sigma_{\rm ess}(S_K) \backslash\{0\} = \emptyset. {\label}{ESSK}$$ In addition, let $p\in (0,\infty)\cup\{\infty\}$, then $$\begin{aligned} \begin{split} & (S_F - z_0 I_{{{\mathcal H}}})^{-1} \in {{\mathcal B}}_p({{\mathcal H}}) \, \text{ for some $z_0\in {{\mathbb{C}}}\backslash [\varepsilon,\infty)$} \\ & \quad \text{implies } \, (S_K - zI_{{{\mathcal H}}})^{-1}[I_{{{\mathcal H}}} - P_{\ker(S_K)}] \in {{\mathcal B}}_p({{\mathcal H}}) \, \text{ for all $z\in {{\mathbb{C}}}\backslash [\varepsilon,\infty)$}. {\label}{CPK} \end{split} \end{aligned}$$ In fact, the $\ell^p({{\mathbb{N}}})$-based trace ideals ${{\mathcal B}}_p({{\mathcal H}})$ of ${{\mathcal B}}({{\mathcal H}})$ can be replaced by any two-sided symmetrically normed ideals of ${{\mathcal B}}({{\mathcal H}})$. We note that is a classical result of Krein [@Kr47], the more general fact has not been mentioned explicitly in Krein’s paper [@Kr47], although it immediately follows from the minimax principle and Krein’s formula . On the other hand, in the special case ${\rm def}(S)<\infty$, Krein states an extension of in his Remark 8.1 in the sense that he also considers self-adjoint extensions different from the Krein extension. Apparently, in the context of infinite deficiency indices has first been proven by Alonso and Simon [@AS80] by a somewhat different method. Relation was recently proved in [@AGMST09] for $p \in (0, \infty)$. Finally, we very briefly mention some new results on the Krein–von Neumann extension which were developed when working on this paper. These results exhibit the Krein–von Neumann extension as a natural object in elasticity theory by relating it to an abstract buckling problem as follows: We start by introducing an abstract version of Proposition 1 in Grubb’s paper [@Gr83] devoted to Krein–von Neumann extensions of even order elliptic differential operators on bounded smooth domains. We recall that Proposition 1 in [@Gr83] describes an intimate connection between the nonzero eigenvalues of the Krein–von Neumann extension of an appropriate minimal elliptic differential operator of order $2m$, $m\in{{\mathbb{N}}}$, and nonzero eigenvalues of a suitable higher-order buckling problem (cf. ). The abstract version of this remarkable connection reads as follows: [l3.3]{} Assume Hypothesis \[h2.6\] and let $\lambda \neq 0$. Then there exists $0 \neq v \in \operatorname{dom}(S_K)$ with $$S_K v = \lambda v {\label}{3.1b}$$ if and only if there exists $0 \neq u \in \operatorname{dom}(S^* S)$ such that $$S^* S u = \lambda S u. {\label}{3.1c}$$ In particular, the solutions $v$ of are in one-to-one correspondence with the solutions $u$ of given by the formulas $$\begin{aligned} u & = (S_F)^{-1} S_K v, {\label}{3.1d} \\ v & = \lambda^{-1} S u. {\label}{3.1e}\end{aligned}$$ Of course, since $S_K \geq 0$, any $\lambda \neq 0$ in and necessarily satisfies $\lambda > 0$. We refer to [@AGMST09] for the proof of Lemma \[l3.3\]. Due to the generalized buckling problem , respectively, , we will call the linear pencil eigenvalue problem $S^* Su = \lambda S u$ in the [*abstract buckling problem*]{} associated with the Krein–von Neumann extension $S_K$ of $S$. Next, we turn to a variational formulation of the correspondence between the inverse of the reduced Krein extension ${\widehat}S_K$ and the abstract buckling problem in terms of appropriate sesquilinear forms by following the treatment of Kozlov [@Ko79]–[@Ko84] in the context of elliptic partial differential operators. This will then lead to an even stronger connection between the Krein–von Neumann extension $S_K$ of $S$ and the associated abstract buckling eigenvalue problem , culminating in a unitary equivalence result in Theorem \[t3.3\]. Given the operator $S$, we introduce the following sesquilinear forms in ${{\mathcal H}}$, $$\begin{aligned} a(u,v) & = (Su,Sv)_{{{\mathcal H}}}, \quad u, v \in \operatorname{dom}(a) = \operatorname{dom}(S), {\label}{3.2} \\ b(u,v) & = (u,Sv)_{{{\mathcal H}}}, \quad u, v \in \operatorname{dom}(b) = \operatorname{dom}(S). {\label}{3.3} \end{aligned}$$ Then $S$ being densely defined and closed implies that the sesquilinear form $a$ shares these properties and implies its boundedness from below, $$a(u,u) \geq \varepsilon^2 \|u\|_{{{\mathcal H}}}^2, \quad u \in \operatorname{dom}(S). {\label}{3.4}$$ Thus, one can introduce the Hilbert space ${{\mathcal W}}=(\operatorname{dom}(S), ({\,\cdot\,},{\,\cdot\,})_{{{\mathcal W}}})$ with associated scalar product $$(u,v)_{{{\mathcal W}}}=a(u,v) = (Su,Sv)_{{{\mathcal H}}}, \quad u, v \in \operatorname{dom}(S). {\label}{3.5}$$ In addition, we denote by $\iota_{{{\mathcal W}}}$ the continuous embedding operator of ${{\mathcal W}}$ into ${{\mathcal H}}$, $$\iota_{{{\mathcal W}}} : {{\mathcal W}}\hookrightarrow {{\mathcal H}}. {\label}{3.6}$$ Hence, we will use the notation $$(w_1,w_2)_{{{\mathcal W}}} =a(\iota_{{{\mathcal W}}} w_1,\iota_{{{\mathcal W}}} w_2) = (S\iota_{{{\mathcal W}}} w_1, S\iota_{{{\mathcal W}}} w_2)_{{{\mathcal H}}}, \quad w_1, w_2 \in {{\mathcal W}}, {\label}{3.7}$$ in the following. Given the sesquilinear forms $a$ and $b$ and the Hilbert space ${{\mathcal W}}$, we next define the operator $T$ in ${{\mathcal W}}$ by $$\begin{aligned} \begin{split} (w_1,T w_2)_{{{\mathcal W}}} & = a(\iota_{{{\mathcal W}}} w_1,\iota_{{{\mathcal W}}} T w_2) = (S \iota_{{{\mathcal W}}} w_1,S\iota_{{{\mathcal W}}} T w_2)_{{{\mathcal H}}} \\ & = b(\iota_{{{\mathcal W}}} w_1,\iota_{{{\mathcal W}}} w_2) = (\iota_{{{\mathcal W}}} w_1,S \iota_{{{\mathcal W}}} w_2)_{{{\mathcal H}}}, \quad w_1, w_2 \in {{\mathcal W}}. {\label}{3.8} \end{split}\end{aligned}$$ One verifies that $T$ is well-defined and that $$|(w_1,T w_2)_{{{\mathcal W}}}| \leq \|\iota_{{{\mathcal W}}} w_1\|_{{{\mathcal H}}} \|S \iota_{{{\mathcal W}}} w_2\|_{{{\mathcal H}}} \leq \varepsilon^{-1} \|w_1\|_{{{\mathcal W}}} \|w_2\|_{{{\mathcal W}}}, \quad w_1, w_2 \in {{\mathcal W}}, {\label}{3.9}$$ and hence that $$0 \leq T = T^* \in {{\mathcal B}}({{\mathcal W}}), \quad \|T\|_{{{\mathcal B}}({{\mathcal W}})} \leq \varepsilon^{-1}. {\label}{3.10}$$ For reasons to become clear in connection with –, we called $T$ the [*abstract buckling problem operator*]{} associated with the Krein–von Neumann extension $S_K$ of $S$ in [@AGMST09]. Next, recalling the notation ${\widehat}{{\mathcal H}}= [\ker (S^*)]^{\bot} = \big[I_{{{\mathcal H}}} - P_{\ker(S^*)}\big] {{\mathcal H}}$ (cf. ), we introduce the operator $${\widehat}S: \begin{cases} {{\mathcal W}}\to {\widehat}{{\mathcal H}}, \\ w \mapsto S \iota_{{{\mathcal W}}} w, \end{cases} {\label}{3.12}$$ and note that $$\operatorname{ran}\big({\widehat}S\big) = \operatorname{ran}(S) = {\widehat}{{\mathcal H}}, {\label}{3.12aa}$$ since $S\geq \varepsilon I_{{{\mathcal H}}}$ for some $\varepsilon > 0$ and $S$ is closed in ${{\mathcal H}}$ (see, e.g., [@We80 Theorem 5.32]). In fact, $${\widehat}S\in{{\mathcal B}}({{\mathcal W}},{\widehat}{{\mathcal H}}) \, \text{ maps ${{\mathcal W}}$ unitarily onto ${\widehat}{{\mathcal H}}$.}$$ Continuing, we briefly recall the polar decomposition of $S$, $$S = U_S |S|, {\label}{3.19a}$$ with $$|S| = (S^* S)^{1/2} \geq \varepsilon I_{{{\mathcal H}}}, \; \varepsilon > 0, \quad U_S \in {{\mathcal B}}\big({{\mathcal H}},{\widehat}{{\mathcal H}}\big) \, \text{ unitary,} {\label}{3.19b}$$ and state the principal unitary equivalence result proven in [@AGMST09]: [t3.3]{} Assume Hypothesis \[h2.6\]. Then the inverse of the reduced Krein–von Neumann extension ${\widehat}S_K$ in ${\widehat}{{\mathcal H}}= \big[I_{{{\mathcal H}}} - P_{\ker(S^*)}\big] {{\mathcal H}}$ and the abstract buckling problem operator $T$ in ${{\mathcal W}}$ are unitarily equivalent, in particular, $$\big({\widehat}S_K\big)^{-1} = {\widehat}S T ({\widehat}S)^{-1}. {\label}{3.20}$$ Moreover, one has $$\big({\widehat}S_K\big)^{-1} = U_S \big[|S|^{-1} S |S|^{-1}\big] (U_S)^{-1}, {\label}{3.20a}$$ where $U_S\in {{\mathcal B}}\big({{\mathcal H}},{\widehat}{{\mathcal H}}\big)$ is the unitary operator in the polar decomposition of $S$ and the operator $|S|^{-1} S |S|^{-1}\in{{\mathcal B}}({{\mathcal H}})$ is self-adjoint in ${{\mathcal H}}$. Equation is of course motivated by rewriting the abstract linear pencil buckling eigenvalue problem , $S^* S u = \lambda S u$, $\lambda \neq 0$, in the form $$\lambda^{-1} S^* S u = \lambda^{-1} (S^* S)^{1/2} \big[(S^* S)^{1/2} u\big] = S (S^* S)^{-1/2} \big[(S^* S)^{1/2} u\big] {\label}{3.38}$$ and hence in the form of a standard eigenvalue problem $$|S|^{-1} S |S|^{-1} w = \lambda^{-1} w, \quad \lambda \neq 0, \quad w = |S| u. {\label}{3.39}$$ Concluding this section, we point out that a great variety of additional results for the Krein–von Neumann extension can be found, for instance, in [@AG81a Sect. 109], [@AS80], [@AN70], [@Ar98], [@Ar00], [@AHSD01], [@AT02], [@AT03], [@AT05], [@AT09], [@AGMST09], [@BC05], [@DM91], [@DM95], [@Fa75 Part III], [@FOT94 Sect. 3.3], [@GM10], [@Gr83], [@Ha57], [@HK09], [@HMD04], [@HSDW07], [@KO77], [@KO78], [@Ne83], [@PS96], [@SS03], [@Si98], [@Sk79], [@St96], [@Ts80], [@Ts81], [@Ts92], and the references therein. We also mention the references [@EM05], [@EMP04], [@EMMP07] (these authors, apparently unaware of the work of von Neumann, Krein, Vi[s]{}hik, Birman, Grubb, [S]{}trauss, etc., in this context, introduced the Krein Laplacian and called it the harmonic operator, see also [@Gr06]). Trace Theory in Lipschitz Domains {#s3} ================================= In this section we shall review material pertaining to analysis in Lipschitz domains, starting with Dirichlet and Neumann boundary traces in Subsection \[s3X\], and then continuing with a brief survey of perturbed Dirichlet and Neumann Laplacians in Subsection \[s4X\]. Dirichlet and Neumann Traces in Lipschitz Domains {#s3X} ------------------------------------------------- The goal of this subsection is to introduce the relevant material pertaining to Sobolev spaces $H^s(\Omega)$ and $H^r(\partial\Omega)$ corresponding to subdomains ${\Omega}$ of ${{\mathbb{R}}}^n$, $n\in{{\mathbb{N}}}$, and discuss various trace results. Before we focus primarily on bounded Lipschitz domains, we briefly recall some basic facts in connection with Sobolev spaces corresponding to open sets ${\Omega}\subseteq{{\mathbb{R}}}^n$, $n\in{{\mathbb{N}}}$: For an arbitrary $m\in{{\mathbb{N}}}\cup\{0\}$, we follow the customary way of defining $L^2$-Sobolev spaces of order $\pm m$ in ${\Omega}$ as $$\begin{aligned} \label{hGi-1} H^m({\Omega}) &:=\big\{u\in L^2({\Omega};d^nx)\,\big|\,\partial^\alpha u\in L^2({\Omega};d^nx), \, 0 \leq |\alpha|\leq m\big\}, \\ H^{-m}({\Omega}) &:=\biggl\{u\in{{\mathcal D}}^{\prime}({\Omega})\,\bigg|\,u=\sum_{0 \leq |\alpha|\leq m} \partial^\alpha u_{\alpha}, \mbox{ with }u_\alpha\in L^2({\Omega};d^nx), \, 0 \leq |\alpha|\leq m\biggr\}, \label{hGi-2}\end{aligned}$$ equipped with natural norms (cf., e.g., [@AF03 Ch. 3], [@Ma85 Ch. 1]). Here ${{\mathcal D}}^\prime({\Omega})$ denotes the usual set of distributions on $\Omega\subseteq {{\mathbb{R}}}^n$. Then we set $$\label{hGi-3} H^m_0({\Omega}):=\,\mbox{the closure of $C^\infty_0({\Omega})$ in $H^m({\Omega})$}, \quad m \in {{\mathbb{N}}}\cup\{0\}.$$ As is well-known, all three spaces above are Banach, reflexive and, in addition, $$\label{hGi-4} \bigl(H^m_0({\Omega})\bigr)^*=H^{-m}({\Omega}).$$ Again, see, for instance, [@AF03 Ch. 3], [@Ma85 Ch. 1]. We recall that an open, nonempty set $\Omega\subseteq{{\mathbb{R}}}^n$ is called a [*Lipschitz domain*]{} if the following property holds: There exists an open covering $\{{\mathcal O}_j\}_{1\leq j\leq N}$ of the boundary $\partial\Omega$ of ${\Omega}$ such that for every $j\in\{1,...,N\}$, ${\mathcal O}_j\cap\Omega$ coincides with the portion of ${\mathcal O}_j$ lying in the over-graph of a Lipschitz function $\varphi_j:{{\mathbb{R}}}^{n-1}\to{{\mathbb{R}}}$ $($considered in a new system of coordinates obtained from the original one via a rigid motion$)$. The number $\max\,\{\|\nabla\varphi_j\|_{L^\infty ({{\mathbb{R}}}^{n-1};d^{n-1}x')^{n-1}}\,|\,1\leq j\leq N\}$ is said to represent the [*Lipschitz character*]{} of $\Omega$. The classical theorem of Rademacher on almost everywhere differentiability of Lipschitz functions ensures that for any Lipschitz domain $\Omega$, the surface measure $d^{n-1} \omega$ is well-defined on $\partial\Omega$ and that there exists an outward pointing normal vector $\nu$ at almost every point of $\partial\Omega$. As regards $L^2$-based Sobolev spaces of fractional order $s\in{{\mathbb{R}}}$, on arbitrary Lipschitz domains ${\Omega}\subseteq{{\mathbb{R}}}^n$, we introduce $$\begin{aligned} \label{HH-h1} H^{s}({{\mathbb{R}}}^n) &:=\bigg\{U\in {{\mathcal S}}^\prime({{\mathbb{R}}}^n)\,\bigg|\, {\left\VertU\right\Vert}_{H^{s}({{\mathbb{R}}}^n)}^2 = \int_{{{\mathbb{R}}}^n}d^n\xi\, \big|{\widehat}U(\xi)\big|^2\big(1+{\lvert\xi\rvert}^{2s}\big)<\infty \bigg\}, \\ H^{s}({\Omega}) &:=\big\{u\in {{\mathcal D}}^\prime({\Omega})\,\big|\,u=U|_{\Omega}\text{ for some } U\in H^{s}({{\mathbb{R}}}^n)\big\} = R_{{\Omega}} \, H^s({{\mathbb{R}}}^n), \label{HH-h2}\end{aligned}$$ where $R_{{\Omega}}$ denotes the restriction operator (i.e., $R_{{\Omega}} \, U=U|_{{\Omega}}$, $U\in H^{s}({{\mathbb{R}}}^n)$), ${{\mathcal S}}^\prime({{\mathbb{R}}}^n)$ is the space of tempered distributions on ${{\mathbb{R}}}^n$, and ${\widehat}U$ denotes the Fourier transform of $U\in{{\mathcal S}}^\prime({{\mathbb{R}}}^n)$. These definitions are consistent with , . Next, retaining that ${\Omega}\subseteq {{\mathbb{R}}}^n$ is an arbitrary Lipschitz domain, we introduce $$\label{incl-xxx} H^{s}_0(\Omega):=\big\{u\in H^{s}({{\mathbb{R}}}^n)\,\big|\, {\rm supp} (u)\subseteq{\overline}{\Omega}\big\}, \quad s\in{{\mathbb{R}}},$$ equipped with the natural norm induced by $H^{s}({{\mathbb{R}}}^n)$. The space $H^{s}_0(\Omega)$ is reflexive, being a closed subspace of $H^{s}({{\mathbb{R}}}^n)$. Finally, we introduce for all $s\in{{\mathbb{R}}}$, $$\begin{aligned} \mathring{H}^{s} (\Omega) &= \mbox{the closure of $C^\infty_0(\Omega)$ in $H^s(\Omega)$}, \\ H^{s}_{z} ({\Omega}) &= R_{{\Omega}} \, H^{s}_0(\Omega).\end{aligned}$$ Assuming from now on that ${\Omega}\subset{{\mathbb{R}}}^n$ is a Lipschitz domain with a compact boundary, we recall the existence of a universal linear extension operator $E_{{\Omega}}:{{\mathcal D}}^\prime ({\Omega}) \to {{\mathcal S}}^\prime ({{\mathbb{R}}}^n)$ such that $E_{{\Omega}}: H^s({\Omega}) \to H^s({{\mathbb{R}}}^n)$ is bounded for all $s\in{{\mathbb{R}}}$, and $R_{{\Omega}} E_{{\Omega}}=I_{H^s({\Omega})}$ (cf. [@Ry99]). If $\widetilde{C_0^\infty({\Omega})}$ denotes the set of $C_0^\infty({\Omega})$-functions extended to all of ${{\mathbb{R}}}^n$ by setting functions zero outside of $\Omega$, then for all $s\in{{\mathbb{R}}}$, $\widetilde{C_0^\infty({\Omega})} \hookrightarrow H^s_0({\Omega})$ densely. Moreover, one has $$\label{incl-Ya} \big(H^{s}_0(\Omega)\big)^*=H^{-s}(\Omega), \quad s\in{{\mathbb{R}}}.$$ (cf., e.g., [@JK95]) consistent with , and also, $$\big(H^s({\Omega})\big)^* = H^{-s}_0({\Omega}), \quad s\in{{\mathbb{R}}},$$ in particular, $H^s({\Omega})$ is a reflexive Banach space. We shall also use the fact that for a Lipschitz domain ${\Omega}\subset{{\mathbb{R}}}^n$ with compact boundary, the space $\mathring{H}^{s} (\Omega)$ satisfies $$\label{incl-Yb} \mathring{H}^{s} (\Omega) = H^s_{z}({\Omega}) \, \mbox{ if } \, s > -1/2,\,\,s\notin{\textstyle\big\{{\frac12}}+{{\mathbb{N}}}_0\big\}.$$ For a Lipschitz domain $\Omega\subseteq{{\mathbb{R}}}^n$ with compact boundary it is also known that $$\label{dual-xxx} \bigl(H^{s}(\Omega)\bigr)^*=H^{-s}(\Omega), \quad - 1/2 <s< 1/2.$$ See [@Tr02] for this and other related properties. Throughout this paper, we agree to use the [*adjoint*]{} (rather than the dual) space $X^*$ of a Banach space $X$. From this point on we will always make the following assumption (unless explicitly stated otherwise): \[h2.1\] Let $n\in{{\mathbb{N}}}$, $n\geq 2$, and assume that $\emptyset \neq {\Omega}\subset{{{\mathbb{R}}}}^n$ is a bounded Lipschitz domain. To discuss Sobolev spaces on the boundary of a Lipschitz domains, consider first the case where $\Omega\subset{{\mathbb{R}}}^n$ is the domain lying above the graph of a Lipschitz function $\varphi\colon{{\mathbb{R}}}^{n-1}\to{{\mathbb{R}}}$. In this setting, we define the Sobolev space $H^s(\partial\Omega)$ for $0\leq s\leq 1$, as the space of functions $f\in L^2(\partial\Omega;d^{n-1}\omega)$ with the property that $f(x',\varphi(x'))$, as a function of $x'\in{{\mathbb{R}}}^{n-1}$, belongs to $H^s({{\mathbb{R}}}^{n-1})$. This definition is easily adapted to the case when $\Omega$ is a Lipschitz domain whose boundary is compact, by using a smooth partition of unity. Finally, for $-1\leq s\leq 0$, we set $$\label{A.6} H^s({{\partial\Omega}}) = \big(H^{-s}({{\partial\Omega}})\big)^*, \quad -1 {\leqslant}s {\leqslant}0.$$ From the above characterization of $H^s(\partial\Omega)$ it follows that any property of Sobolev spaces (of order $s\in[-1,1]$) defined in Euclidean domains, which are invariant under multiplication by smooth, compactly supported functions as well as composition by bi-Lipschitz diffeomorphisms, readily extends to the setting of $H^s(\partial\Omega)$ (via localization and pullback). For additional background information in this context we refer, for instance, to [@EE89 Chs. V, VI], [@Gr85 Ch. 1]. Assuming Hypothesis \[h2.1\], we introduce the boundary trace operator ${\gamma}_D^0$ (the Dirichlet trace) by $${\gamma}_D^0\colon C({\overline}{{\Omega}})\to C({{\partial\Omega}}), \quad {\gamma}_D^0 u = u|_{{\partial\Omega}}. \label{2.5}$$ Then there exists a bounded, linear operator $\gamma_D$ $$\begin{aligned} \begin{split} & {\gamma}_D\colon H^{s}({\Omega})\to H^{s-(1/2)}({{\partial\Omega}}) \hookrightarrow {L^2({{\partial\Omega}};d^{n-1} \omega)}, \quad 1/2<s<3/2, \label{2.6} \\ & {\gamma}_D\colon H^{3/2}({\Omega})\to H^{1-\varepsilon}({{\partial\Omega}}) \hookrightarrow {L^2({{\partial\Omega}};d^{n-1} \omega)}, \quad \varepsilon \in (0,1) \end{split}\end{aligned}$$ (cf., e.g., [@Mc00 Theorem 3.38]), whose action is compatible with that of ${\gamma}_D^0$. That is, the two Dirichlet trace operators coincide on the intersection of their domains. Moreover, we recall that $$\label{2.6a} {\gamma}_D\colon H^{s}({\Omega})\to H^{s-(1/2)}({{\partial\Omega}}) \, \text{ is onto for $1/2<s<3/2$}.$$ Next, retaining Hypothesis \[h2.1\], we introduce the operator ${\gamma}_N$ (the strong Neumann trace) by $$\label{2.7} {\gamma}_N = \nu\cdot{\gamma}_D\nabla \colon H^{s+1}({\Omega})\to {L^2({{\partial\Omega}};d^{n-1} \omega)}, \quad 1/2<s<3/2,$$ where $\nu$ denotes the outward pointing normal unit vector to $\partial{\Omega}$. It follows from that ${\gamma}_N$ is also a bounded operator. We seek to extend the action of the Neumann trace operator to other (related) settings. To set the stage, assume Hypothesis \[h2.1\] and observe that the inclusion $$\label{inc-1} \iota:H^{s_0}(\Omega)\hookrightarrow \bigl(H^r(\Omega)\bigr)^*, \quad s_0>-1/2,\; r>1/2,$$ is well-defined and bounded. We then introduce the weak Neumann trace operator $$\label{2.8} {\widetilde}{\gamma}_N\colon\big\{u\in H^{s+1/2}({\Omega})\,\big|\,\Delta u\in H^{s_0}({\Omega})\big\} \to H^{s-1}({{\partial\Omega}}),\quad s\in(0,1),\; s_0>-1/2,$$ as follows: Given $u\in H^{s+1/2}({\Omega})$ with $\Delta u \in H^{s_0}({\Omega})$ for some $s\in(0,1)$ and $s_0>-1/2$, we set (with $\iota$ as in for $r:=3/2-s>1/2$) $$\label{2.9} \langle\phi,{\widetilde}{\gamma}_N u \rangle_{1-s} ={}_{H^{1/2-s}({\Omega})}\langle\nabla\Phi,\nabla u\rangle_{(H^{1/2-s}({\Omega}))^*} + {}_{H^{3/2-s}({\Omega})}\langle\Phi,\iota(\Delta u)\rangle_{(H^{3/2-s}({\Omega}))^*},$$ for all $\phi\in H^{1-s}({{\partial\Omega}})$ and $\Phi\in H^{3/2-s}({\Omega})$ such that ${\gamma}_D\Phi=\phi$. We note that the first pairing in the right-hand side above is meaningful since $$\label{2.9JJ} \bigl(H^{1/2-s}({\Omega})\bigr)^*=H^{s-1/2}({\Omega}),\quad s\in (0,1),$$ that the definition is independent of the particular extension $\Phi$ of $\phi$, and that ${\widetilde}{\gamma}_N$ is a bounded extension of the Neumann trace operator ${\gamma}_N$ defined in . For further reference, let us also point out here that if $\Omega\subset{{\mathbb{R}}}^n$ is a bounded Lipschitz domain then for any $j,k\in\{1,...,n\}$ the (tangential first-order differential) operator $$\label{Pf-2} \partial/\partial\tau_{j,k}:=\nu_j\partial_k-\nu_k\partial_j: H^s(\partial\Omega)\to H^{s-1}(\partial\Omega),\quad 0\leq s\leq 1,$$ is well-defined, linear and bounded. Assuming Hypothesis \[h2.1\], we can then define the tangential gradient operator $$\label{Tan-C1} \nabla_{tan}: \begin{cases}H^1(\partial\Omega)\to \big(L^2(\partial\Omega;d^{n-1}\omega)\big)^n \\ \hspace*{1cm} f \mapsto \nabla_{tan}f:=\Big(\sum_{k=1}^n\nu_k\frac{\partial f}{\partial\tau_{kj}} \Big)_{1\leq j\leq n} \end{cases} ,\quad f\in H^1(\partial\Omega).$$ The following result has been proved in [@MMS05]. \[T-MMS\] Assume Hypothesis \[h2.1\] and denote by $\nu$ the outward unit normal to $\partial\Omega$. Then the operator $$\label{Tan-C2} \gamma_2: \begin{cases} H^2(\Omega)\to \bigl\{(g_0,g_1)\in H^1(\partial\Omega) \times L^2(\partial\Omega;d^{n-1}\omega)\,\big|\, \nabla_{tan}g_0 +g_1\nu\in \bigl(H^{1/2}(\partial\Omega)\bigr)^n\bigl\} \\ \hspace*{8mm} u \mapsto \gamma_2 u=(\gamma_D u\,,\,\gamma_N u), \end{cases}$$ is well-defined, linear, bounded, onto, and has a linear, bounded right-inverse. The space $\bigl\{(g_0,g_1)\in H^1(\partial\Omega) \times L^2(\partial\Omega;d^{n-1}\omega)\,\big|\, \nabla_{tan}g_0+g_1\nu\in \bigl(H^{1/2}(\partial\Omega)\bigr)^n\bigl\}$ in is equipped with the natural norm $$\label{NoRw-1} (g_0,g_1)\mapsto \|g_0\|_{H^1(\partial\Omega)} +\|g_1\|_{L^2(\partial\Omega;d^{n-1}\omega)} +\|\nabla_{tan}g_0+g_1\nu\|_{(H^{1/2}(\partial\Omega))^n}.$$ Furthermore, the null space of the operator is given by $$\label{Tan-C3} \ker(\gamma_2):= \big\{u\in H^2(\Omega)\,\big|\,\gamma_D u =\gamma_N u=0\big\} =H^2_0(\Omega),$$ with the latter space denoting the closure of $C^\infty_0(\Omega)$ in $H^2(\Omega)$. Continuing to assume Hypothesis \[h2.1\], we now introduce $$\label{Tan-C4} N^{1/2}(\partial\Omega):=\big\{g\in L^2(\partial\Omega;d^{n-1}\omega)\,\big|\, g\nu_j\in H^{1/2}(\partial\Omega),\,\,1\leq j\leq n\big\},$$ where the $\nu_j$’s are the components of $\nu$. We equip this space with the natural norm $$\label{Tan-C4B} \|g\|_{N^{1/2}(\partial\Omega)} :=\sum_{j=1}^n\|g\nu_j\|_{H^{1/2}(\partial\Omega)}.$$ Then $N^{1/2}(\partial\Omega)$ is a reflexive Banach space which embeds continuously into $L^2(\partial\Omega;d^{n-1}\omega)$. Furthermore, $$\label{Tan-C5} N^{1/2}(\partial\Omega)=H^{1/2}(\partial\Omega)\, \mbox{ whenever $\Omega$ is a bounded $C^{1,r}$ domain with $r>1/2$}.$$ It should be mentioned that the spaces $H^{1/2}(\partial\Omega)$ and $N^{1/2}(\partial\Omega)$ can be quite different for an arbitrary Lipschitz domain $\Omega$. Our interest in the latter space stems from the fact that this arises naturally when considering the Neumann trace operator acting on $$\label{Tan-C6} \big\{u\in H^2(\Omega)\,\big|\,\gamma_D u =0\big\}=H^2(\Omega)\cap H^1_0(\Omega),$$ considered as a closed subspace of $H^2(\Omega)$ (hence, a Banach space when equipped with the $H^2$-norm). More specifically, we have (cf. [@GM10] for a proof): \[Lo-Tx\] Assume Hypothesis \[h2.1\]. Then the Neumann trace operator $\gamma_N$ considered in the context $$\label{Tan-C7} \gamma_N:H^2(\Omega)\cap H^1_0(\Omega)\to N^{1/2}(\partial\Omega)$$ is well-defined, linear, bounded, onto and with a linear, bounded right-inverse. In addition, the null space of $\gamma_N$ in is precisely $H^2_0(\Omega)$, the closure of $C^\infty_0(\Omega)$ in $H^2(\Omega)$. Most importantly for us here is the fact that one can use the above Neumann trace result in order to extend the action of the Dirichlet trace operator to $\operatorname{dom}(-\Delta_{max,{\Omega}})$, the domain of the maximal Laplacian, that is, $\{u\in L^2(\Omega;d^nx)\,|\,\Delta u\in L^2(\Omega;d^nx)\}$, which we consider equipped with the graph norm $u\mapsto \|u\|_{L^2(\Omega;d^nx)}+\|\Delta u\|_{L^2(\Omega;d^nx)}$. Specifically, with $\bigl(N^{1/2}(\partial\Omega)\bigr)^*$ denoting the conjugate dual space of $N^{1/2}(\partial\Omega)$, we have the following result from [@GM10]: \[New-T-tr\] Assume Hypothesis \[h2.1\]. Then there exists a unique linear, bounded operator $$\label{Tan-C10} \widehat{\gamma}_D:\big\{u\in L^2(\Omega;d^nx)\,\big|\,\Delta u\in L^2(\Omega;d^nx)\big\} \to \bigl(N^{1/2}(\partial\Omega)\bigr)^*$$ which is compatible with the Dirichlet trace introduced in , in the sense that, for each $s>1/2$, one has $$\label{Tan-C11} \widehat{\gamma}_D u =\gamma_D u \, \mbox{ for every $u\in H^s(\Omega)$ with $\Delta u\in L^2(\Omega;d^nx)$}.$$ Furthermore, this extension of the Dirichlet trace operator in allows for the following generalized integration by parts formula $$\label{Tan-C12} {}_{N^{1/2}(\partial\Omega)}\langle\gamma_N w,\widehat{\gamma}_D u \rangle_{(N^{1/2}(\partial\Omega))^*} =(\Delta w,u)_{L^2({\Omega};d^nx)} - (w,\Delta u)_{L^2({\Omega};d^nx)},$$ valid for every $u\in L^2(\Omega;d^nx)$ with $\Delta u\in L^2(\Omega;d^nx)$ and every $w\in H^2(\Omega)\cap H^1_0(\Omega)$. We next review the case of the Neumann trace, whose action is extended to $\operatorname{dom}(-\Delta_{max,{\Omega}})$. To this end, we need to address a number of preliminary matters. First, assuming Hypothesis \[h2.1\], we make the following definition (compare with ): $$\label{3an-C4} N^{3/2}(\partial\Omega):=\bigl\{g\in H^1(\partial\Omega)\,\big|\, \nabla_{tan}g\in \bigl(H^{1/2}(\partial\Omega)\bigr)^n\bigl\},$$ equipped with the natural norm $$\label{3an-C4B} \|g\|_{N^{3/2}(\partial\Omega)} :=\|g\|_{L^2(\partial\Omega;d^{n-1}\omega)}+ \|\nabla_{tan}g\|_{(H^{1/2}(\partial\Omega))^n}.$$ Assuming Hypothesis \[h2.1\], $N^{3/2}(\partial\Omega)$ is a reflexive Banach space which embeds continuously into the space $H^1(\partial\Omega;d^{n-1}\omega)$. In addition, this turns out to be a natural substitute for the more familiar space $H^{3/2}(\partial\Omega)$ in the case where $\Omega$ is sufficiently smooth. Concretely, one has $$\label{3an-C5} N^{3/2}(\partial\Omega)=H^{3/2}(\partial\Omega),$$ (as vector spaces with equivalent norms), whenever $\Omega$ is a bounded $C^{1,r}$ domain with $r>1/2$. The primary reason we are interested in $N^{3/2}(\partial\Omega)$ is that this space arises naturally when considering the Dirichlet trace operator acting on $$\label{3an-C6N} \big\{u\in H^2(\Omega)\,\big|\,\gamma_N u =0\big\},$$ considered as a closed subspace of $H^2(\Omega)$ (thus, a Banach space when equipped with the norm inherited from $H^2(\Omega)$). Concretely, the following result has been established in [@GM10]. \[3o-TxD\] Assume Hypothesis \[h2.1\]. Then the Dirichlet trace operator $\gamma_D$ considered in the context $$\label{3an-C7D} \gamma_D:\big\{u\in H^2(\Omega)\,\big|\,\gamma_N u =0\big\} \to N^{3/2}(\partial\Omega)$$ is well-defined, linear, bounded, onto and with a linear, bounded right-inverse. In addition, the null space of $\gamma_D$ in is precisely $H^2_0(\Omega)$, the closure of $C^\infty_0(\Omega)$ in $H^2(\Omega)$. It is then possible to use the Neumann trace result from Lemma \[3o-TxD\] in order to extend the action of the Neumann trace operator to $\operatorname{dom}(-\Delta_{max,{\Omega}})=\big\{u\in L^2(\Omega;d^nx)\,\big|\, \Delta u\in L^2(\Omega;d^nx)\big\}$. As before, this space is equipped with the natural graph norm. Let $\bigl(N^{3/2}(\partial\Omega)\bigr)^*$ denote the conjugate dual space of $N^{3/2}(\partial\Omega)$. The following result holds: \[3ew-T-tr\] Assume Hypothesis \[h2.1\]. Then there exists a unique linear, bounded operator $$\label{3an-C10} \widehat{\gamma}_N:\big\{u\in L^2(\Omega;d^nx)\,\big|\,\Delta u\in L^2(\Omega;d^nx)\big\} \to \bigl(N^{3/2}(\partial\Omega)\bigr)^*$$ which is compatible with the Neumann trace introduced in , in the sense that, for each $s>3/2$, one has $$\label{3an-C11} \widehat{\gamma}_N u =\gamma_N u \, \mbox{ for every $u\in H^s(\Omega)$ with $\Delta u\in L^2(\Omega;d^nx)$}.$$ Furthermore, this extension of the Neumann trace operator from allows for the following generalized integration by parts formula $$\label{3an-C12} {}_{N^{3/2}(\partial\Omega)}\langle\gamma_D w,\widehat\gamma_N u \rangle_{(N^{3/2}(\partial\Omega))^*} = ( w,\Delta u)_{L^2({\Omega};d^nx)} - (\Delta w,u)_{L^2({\Omega};d^nx)},$$ valid for every $u\in L^2(\Omega;d^nx)$ with $\Delta u\in L^2(\Omega;d^nx)$ and every $w\in H^2(\Omega)$ with $\gamma_N w =0$. A proof of Theorem \[3ew-T-tr\] can be found in [@GM10]. Perturbed Dirichlet and Neumann Laplacians {#s4X} ------------------------------------------ Here we shall discuss operators of the form $-\Delta+V$ equipped with Dirichlet and Neumann boundary conditions. Temporarily, we will employ the following assumptions: \[h.V\] Let $n\in{{\mathbb{N}}}$, $n\geq 2$, assume that ${\Omega}\subset{{{\mathbb{R}}}}^n$ is an open, bounded, nonempty set, and suppose that $$\label{VV-W} V\in L^\infty({\Omega};d^nx) \, \mbox{ and }\, V \,\mbox{is real-valued a.e.\ on } \,\Omega.$$ We start by reviewing the perturbed Dirichlet and Neumann Laplacians $H_{D,{\Omega}}$ and $H_{N,{\Omega}}$ associated with an open set ${\Omega}$ in ${{\mathbb{R}}}^n$ and a potential $V$ satisfying Hypothesis \[h.V\]: Consider the sesquilinear forms in $L^2({\Omega};d^n x)$, $$Q_{D,{\Omega}} (u,v) = (\nabla u, \nabla v) + (u,Vv), \quad u,v \in \operatorname{dom}(Q_{D,{\Omega}}) = H^1_0({\Omega}), {\label}{3.QD}$$ and $$Q_{N,{\Omega}} (u,v) = (\nabla u, \nabla v) + (u,Vv), \quad u,v \in \operatorname{dom}(Q_{N,{\Omega}}) = H^1({\Omega}). {\label}{3.QN}$$ Then both forms in and are densely, defined, closed, and bounded from below in $L^2({\Omega};d^n x)$. Thus, by the first and second representation theorems for forms (cf., e.g., [@Ka80 Sect. VI.2]), one concludes that there exist unique self-adjoint operators $H_{D,{\Omega}}$ and $H_{N,{\Omega}}$ in $L^2({\Omega};d^n x)$, both bounded from below, associated with the forms $Q_{D,{\Omega}}$ and $Q_{N,{\Omega}}$, respectively, which satisfy $$\begin{aligned} & Q_{D,{\Omega}} (u,v) = (u,H_{D,{\Omega}} v), \quad u \in \operatorname{dom}(Q_{D,{\Omega}}), \, v \in \operatorname{dom}(H_{D,{\Omega}}), {\label}{3.QHD} \\ & \operatorname{dom}(H_{D,{\Omega}}) \subset \operatorname{dom}\big(|H_{D,{\Omega}}|^{1/2}\big) = \operatorname{dom}(Q_{D,{\Omega}}) = H^1_0({\Omega}) {\label}{3.HD}\end{aligned}$$ and $$\begin{aligned} & Q_{N,{\Omega}} (u,v) = (u,H_{N,{\Omega}} v), \quad u \in \operatorname{dom}(Q_{N,{\Omega}}), \, v \in \operatorname{dom}(H_{N,{\Omega}}), {\label}{3.QHN} \\ & \operatorname{dom}(H_{N,{\Omega}}) \subset \operatorname{dom}\big(|H_{N,{\Omega}}|^{1/2}\big) = \operatorname{dom}(Q_{N,{\Omega}}) = H^1({\Omega}). {\label}{3.HN}\end{aligned}$$ In the case of the perturbed Dirichlet Laplacian, $H_{D,{\Omega}}$, one actually can say a bit more: Indeed, $H_{D,{\Omega}}$ coincides with the Friedrichs extension of the operator $$H_{c,{\Omega}} u = (-\Delta + V) u, \quad u \in \operatorname{dom}(H_{c,{\Omega}}):=C^\infty_0(\Omega)$$ in $L^2({\Omega};d^nx)$, $$(H_{c,{\Omega}})_F = H_{D,{\Omega}}, {\label}{3.cFD}$$ and one obtains as an immediate consequence of and $$H_{D,{\Omega}}u= (-\Delta+V)u, \quad u\in \operatorname{dom}(H_{D,{\Omega}}) = \big\{v\in H_0^1({\Omega})\,\big|\,\Delta v\in L^2({\Omega};d^n x)\big\}. {\label}{3.HDF}$$ We also refer to [@EE89 Sect. IV.2, Theorem VII.1.4]). In addition, $H_{D,{\Omega}}$ is known to have a compact resolvent and hence purely discrete spectrum bounded from below. In the case of the perturbed Neumann Laplacian, $H_{N,{\Omega}}$, it is not possible to be more specific under this general hypothesis on ${\Omega}$ just being open. However, under the additional assumptions on the domain ${\Omega}$ in Hypothesis \[h2.1\] one can be more explicit about the domain of $H_{N,{\Omega}}$ and also characterize its spectrum as follows. In addition, we also record an improvement of under the additional Lipschitz hypothesis on ${\Omega}$: \[t2.5\] Assume Hypotheses \[h2.1\] and \[h.V\].  Then the perturbed Dirichlet Laplacian, $H_{D,{\Omega}}$, given by $$\begin{aligned} & H_{D,{\Omega}}u= (-\Delta+V)u,{\notag}\\ & u\in \operatorname{dom}(H_{D,{\Omega}}) = \big\{v\in H^1({\Omega})\,\big|\,\Delta v\in L^2({\Omega};d^n x),\, \gamma_D v=0\text{ in $H^{1/2}({{\partial\Omega}})$}\big\} \label{2.39} \\ & \hspace*{2.46cm} =\big\{v\in H_0^1({\Omega})\,\big|\,\Delta v\in L^2({\Omega};d^n x)\big\}, {\notag}\end{aligned}$$ is self-adjoint and bounded from below in $L^2({\Omega};d^nx)$. Moreover, $$\label{2.40} \operatorname{dom}\big(|H_{D,{\Omega}}|^{1/2}\big)=H^1_0({\Omega}),$$ and the spectrum of $H_{D,{\Omega}}$, is purely discrete $($i.e., it consists of eigenvalues of finite multiplicity$)$, $$\sigma_{\rm ess}(H_{D,{\Omega}}) = \emptyset.$$ If, in addition, $V\geq 0$ a.e. in $\Omega$, then $H_{D,{\Omega}}$ is strictly positive in $L^2({\Omega};d^nx)$. The corresponding result for the perturbed Neumann Laplacian $H_{N,{\Omega}}$ reads as follows: \[t2.3\] Assume Hypotheses \[h2.1\] and \[h.V\]. Then the perturbed Neumann Laplacian, $H_{N,{\Omega}}$, given by $$\begin{aligned} \label{2.20} & H_{N,{\Omega}}u = (-\Delta+V)u, \\ & u \in \operatorname{dom}(H_{N,{\Omega}}) = \big\{v\in H^1({\Omega})\,\big|\,\Delta v\in L^2({\Omega};d^nx),\, {\widetilde}\gamma_N v =0\text{ in }H^{-1/2}({{\partial\Omega}})\big\}, {\notag}\end{aligned}$$ is self-adjoint and bounded from below in $L^2({\Omega};d^nx)$. Moreover, $$\label{2.40a} \operatorname{dom}\big(|H_{N,{\Omega}}|^{1/2}\big)=H^1({\Omega}),$$ and the spectrum of $H_{N,{\Omega}}$, is purely discrete $($i.e., it consists of eigenvalues of finite multiplicity$)$, $$\sigma_{\rm ess}(H_{N,{\Omega}}) = \emptyset.$$ If, in addition, $V\geq 0$ a.e. in $\Omega$, then $H_{N,{\Omega}}$ is nonnegative in $L^2({\Omega};d^nx)$. In the sequel, corresponding to the case where $V\equiv 0$, we shall abbreviate $$\label{V-Df1} -\Delta_{D,{\Omega}}\, \mbox{ and }\, -\Delta_{N,{\Omega}},$$ for $H_{D,{\Omega}}$ and $H_{N,{\Omega}}$, respectively, and simply refer to these operators as, the Dirichlet and Neumann Laplacians. The above results have been proved in [@GLMZ05 App. A], [@GMZ07] for considerably more general potentials than assumed in Hypothesis \[h.V\]. Next, we shall now consider the minimal and maximal perturbed Laplacians. Concretely, given an open set $\Omega\subset{\mathbb{R}}^n$ and a potential $0\leq V\in L^\infty({\Omega};d^nx)$, we introduce the maximal perturbed Laplacian in $L^2(\Omega;d^nx)$ $$\begin{aligned} \label{Yan-1} \begin{split} & H_{max,{\Omega}} u:=(-\Delta+V)u, \\ & u\in \operatorname{dom}(H_{max,{\Omega}} ):=\big\{v\in L^2(\Omega;d^nx)\,\big|\, \Delta v \in L^2(\Omega;d^nx)\big\}. \end{split}\end{aligned}$$ We pause for a moment to dwell on the notation used in connection with the symbol $\Delta$: Throughout this manuscript the symbol $\Delta$ alone indicates that the Laplacian acts in the sense of distributions, $$\Delta\colon {{\mathcal D}}^\prime({\Omega}) \to {{\mathcal D}}^\prime({\Omega}). {\label}{DELTA}$$ In some cases, when it is necessary to interpret $\Delta$ as a bounded operator acting between Sobolev spaces, we write $\Delta \in {{\mathcal B}}\big(H^s({\Omega}),H^{s-2}({\Omega})\big)$ for various ranges of $s\in{{\mathbb{R}}}$ (which is of course compatible with ). In addition, as a consequence of standard interior elliptic regularity (cf. Weyl’s classical lemma) it is not difficult to see that if ${\Omega}\subseteq{{\mathbb{R}}}$ is open, $u \in {{\mathcal D}}'({\Omega})$ and $\Delta u \in L^2_{\rm loc}({\Omega}; d^nx)$ then actually $u \in H^2_{\rm loc}({\Omega})$. In particular, this comment applies to $u\in \operatorname{dom}(H_{max,{\Omega}} )$ in . In the remainder of this subsection we shall collect a number of results, originally proved in [@GM10] when $V\equiv 0$, but which are easily seen to hold in the more general setting considered here. \[Max-M1\] Assume Hypotheses \[h2.1\] and \[h.V\]. Then the maximal perturbed Laplacian associated with $\Omega$ and the potential $V$ is a closed, densely defined operator for which $$\label{Yan-2} H^2_0(\Omega)\subseteq \operatorname{dom}((H_{max,{\Omega}})^*) \subseteq \big\{u\in L^2(\Omega;d^nx)\,\big|\, \Delta u\in L^2(\Omega;d^nx), \, \widehat{\gamma}_D u=\widehat{\gamma}_N u =0\big\}.$$ For an open set $\Omega\subset{\mathbb{R}}^n$ and a potential $0\leq V\in L^\infty({\Omega};d^nx)$, we also bring in the minimal perturbed Laplacian in $L^2(\Omega;d^nx)$, that is, $$\label{Yan-6} H_{min,{\Omega}} u:=(-\Delta+V)u,\quad u\in \operatorname{dom}(H_{min,{\Omega}}):=H^2_0(\Omega).$$ \[Max-M2\] Assume Hypotheses \[h2.1\] and \[h.V\]. Then $H_{min,{\Omega}} $ is a densely defined, symmetric operator which satisfies $$\label{Yan-7} H_{min,{\Omega}} \subseteq (H_{max,{\Omega}})^*\, \mbox{ and }\, H_{max,{\Omega}} \subseteq (H_{min,{\Omega}})^*.$$ Equality occurs in one $($and hence, both$)$ inclusions in if and only if $$\label{Yan-8} H^2_0(\Omega) \, \text{ equals } \, \big\{u\in L^2(\Omega;d^nx)\,\big|\,\Delta u\in L^2(\Omega;d^nx), \, \widehat{\gamma}_D u =\widehat{\gamma}_N u =0\big\}.$$ Boundary Value Problems in Quasi-Convex Domains {#s5} =============================================== This section is divided into three parts. In Subsection \[s5X\] we introduce a distinguished category of the family of Lipschitz domains in ${{\mathbb{R}}}^n$, called quasi-convex domains, which is particularly well-suited for the kind of analysis we have in mind. In Subsection \[s6X\] and Subsection \[s7X\], we then proceed to review, respectively, trace operators and boundary problems, and Dirichlet-to-Neumann operators in quasi-convex domains. The Class of Quasi-Convex Domains {#s5X} --------------------------------- In the class of Lipschitz domains, the two spaces appearing in are not necessarily equal (although, obviously, the left-to-right inclusion always holds). The question now arises: What extra properties of the Lipschitz domain will guarantee equality in ? This issue has been addressed in [@GM10], where a class of domains (which is in the nature of best possible) has been identified. To describe this class, we need some preparations. Given $n\geq 1$, denote by $MH^{1/2}({{\mathbb{R}}}^n)$ the class of pointwise multipliers of the Sobolev space $H^{1/2}({{\mathbb{R}}}^n)$. That is, $$\label{MaS-1} MH^{1/2}({{\mathbb{R}}}^n):=\bigl\{f\in L^1_{{\text{\rm{loc}}}}({{\mathbb{R}}}^n)\,\big|\, M_f\in{{\mathcal B}}\bigl(H^{1/2}({{\mathbb{R}}}^n)\bigr)\bigr\},$$ where $M_f$ is the operator of pointwise multiplication by $f$. This space is equipped with the natural norm, that is, $$\label{MaS-2} \|f\|_{MH^{1/2}({{\mathbb{R}}}^n)}:=\|M_f\|_{{{\mathcal B}}(H^{1/2}({{\mathbb{R}}}^n))}.$$ For a comprehensive and systematic treatment of spaces of multipliers, the reader is referred to the 1985 monograph of Maz’ya and Shaposhnikova [@MS85]. Following [@MS85], [@MS05], we now introduce a special class of domains, whose boundary regularity properties are expressed in terms of spaces of multipliers. \[Def-MS\] Given $\delta>0$, call a bounded, Lipschitz domain $\Omega\subset{{\mathbb{R}}}^n$ to be of class $MH^{1/2}_\delta$, and write $$\label{MaS-3} {{\partial\Omega}}\in MH^{1/2}_\delta,$$ provided the following holds: There exists a finite open covering $\{{\mathcal O}_j\}_{1\leq j\leq N}$ of the boundary $\partial\Omega$ of ${\Omega}$ such that for every $j\in\{1,...,N\}$, ${\mathcal O}_j\cap\Omega$ coincides with the portion of ${\mathcal O}_j$ lying in the over-graph of a Lipschitz function $\varphi_j:{{\mathbb{R}}}^{n-1}\to{{\mathbb{R}}}$ $($considered in a new system of coordinates obtained from the original one via a rigid motion$)$ which, additionally, has the property that $$\label{MaS-4} \nabla\varphi_j\in \big(MH^{1/2}({{\mathbb{R}}}^{n-1})\big)^n\, \mbox{ and }\, \|\varphi_j\|_{(MH^{1/2}({{\mathbb{R}}}^{n-1}))^n}\leq\delta.$$ Going further, we consider the classes of domains $$\label{MaS-5} MH^{1/2}_\infty:=\bigcup_{\delta>0}MH^{1/2}_\delta,\quad MH^{1/2}_0:=\bigcap_{\delta>0}MH^{1/2}_\delta,$$ and also introduce the following definition: \[Def-MS2\] We call a bounded Lipschitz domain $\Omega\subset{{\mathbb{R}}}^n$ to be [*square-Dini*]{}, and write $$\label{MaS-6} {{\partial\Omega}}\in {\rm SD},$$ provided the following holds: There exists a finite open covering $\{{\mathcal O}_j\}_{1\leq j\leq N}$ of the boundary $\partial\Omega$ of ${\Omega}$ such that for every $j\in\{1,...,N\}$, ${\mathcal O}_j\cap\Omega$ coincides with the portion of ${\mathcal O}_j$ lying in the over-graph of a Lipschitz function $\varphi_j:{{\mathbb{R}}}^{n-1}\to{{\mathbb{R}}}$ $($considered in a new system of coordinates obtained from the original one via a rigid motion$)$ which, additionally, has the property that the following square-Dini condition holds, $$\label{MaS-7} \int_0^1 \frac{dt}{t} \bigg(\frac{\omega(\nabla\varphi_j;t)}{t^{1/2}}\bigg)^2 <\infty.$$ Here, given a $($possibly vector-valued$)$ function $f$ in ${{\mathbb{R}}}^{n-1}$, $$\label{MaS-8} \omega(f;t):=\sup\,\{|f(x)-f(y)|\,|\,x,y\in{{\mathbb{R}}}^{n-1},\,\,|x-y|\leq t\}, \quad t\in(0,1),$$ is the modulus of continuity of $f$, at scale $t$. From the work of Maz’ya and Shaposhnikova [@MS85] [@MS05], it is known that if $r>1/2$, then $$\label{MaS-9} {\Omega}\in C^{1,r}\Longrightarrow {\Omega}\in{\rm SD}\Longrightarrow {\Omega}\in MH^{1/2}_0\Longrightarrow {\Omega}\in MH^{1/2}_\infty.$$ As pointed out in [@MS05], domains of class $MH^{1/2}_\infty$ can have certain types of vertices and edges when $n\geq 3$. Thus, the domains in this class can be nonsmooth. Next, we recall that a domain is said to satisfy a uniform exterior ball condition (UEBC) provided there exists a number $r>0$ with the property that $$\begin{aligned} \label{UEBC} \begin{split} & \mbox{for every $x\in{{\partial\Omega}}$, there exists $y\in{{\mathbb{R}}}^n$, such that $B(y,r)\cap{\Omega}=\emptyset$} \\ & \quad \mbox{and $x\in\partial B(y,r)\cap{{\partial\Omega}}$}. \end{split} \end{aligned}$$ Heuristically, should be interpreted as a lower bound on the curvature of $\partial\Omega$. Next, we review the class of almost-convex domains introduced in [@MTV]. \[Def-AC\] A bounded Lipschitz domain $\Omega\subset{\mathbb{R}}^n$ is called an almost-convex domain provided there exists a family $\{\Omega_\ell\}_{\ell\in{\mathbb{N}}}$ of open sets in ${\mathbb{R}}^n$ with the following properties: 1. $\partial\Omega_\ell\in C^2$ and $\overline{\Omega_{\ell}}\subset\Omega$ for every $\ell\in{\mathbb{N}}$. 2. $\Omega_\ell\nearrow\Omega$ as $\ell\to\infty$, in the sense that $\overline{\Omega_{\ell}}\subset\Omega_{\ell+1}$ for each $\ell\in{\mathbb{N}}$ and $\bigcup_{\ell\in{\mathbb{N}}}\Omega_{\ell}=\Omega$. 3. There exists a neighborhood $U$ of $\partial\Omega$ and, for each $\ell\in{\mathbb{N}}$, a $C^2$ real-valued function $\rho_{\ell}$ defined in $U$ with the property that $\rho_{\ell}<0$ on $U\cap\Omega_{\ell}$, $\rho_{\ell}>0$ in $U \backslash \overline{\Omega_{\ell}}$, and $\rho_{\ell}$ vanishes on $\partial\Omega_\ell$. In addition, it is assumed that there exists some constant $C_1\in (1,\infty)$ such that $$\begin{aligned} \label{MTV3.1} C_1^{-1}\leq |\nabla\rho_\ell(x)|\leq C_1, \quad x\in \partial\Omega_\ell,\; \ell\in{\mathbb{N}}. \end{aligned}$$ 4. There exists $C_2\geq 0$ such that for every number $\ell\in{\mathbb{N}}$, every point $x\in\partial\Omega_{\ell}$, and every vector $\xi\in{\mathbb{R}}^n$ which is tangent to $\partial\Omega_{\ell}$ at $x$, there holds $$\begin{aligned} \label{MTV3.2} \big\langle{\rm Hess}\,(\rho_\ell)\xi\,,\,\xi\big\rangle\geq -C_2|\xi|^2, \end{aligned}$$ where $\langle{\,\cdot\,},{\,\cdot\,}\rangle$ is the standard Euclidean inner product in ${\mathbb{R}}^n$ and $$\begin{aligned} \label{MTV3.3} {\rm Hess}\,(\rho_\ell):=\left(\frac{\partial^2\rho_\ell} {\partial x_j\partial x_k}\right)_{1\leq j,k \leq n},\end{aligned}$$ is the Hessian of $\rho_{\ell}$. A few remarks are in order: First, it is not difficult to see that ensures that each domain $\Omega_\ell$ is Lipschitz, with Lipschitz constant bounded uniformly in $\ell$. Second, simply says that, as quadratic forms on the tangent bundle $T\partial\Omega_\ell$ to $\partial\Omega_{\ell}$, one has $$\begin{aligned} \label{MT-SR} {\rm Hess}\,(\rho_\ell)\geq -C_2\,I_n,\end{aligned}$$ where $I_n$ is the $n\times n$ identity matrix. Hence, another equivalent formulation of is the following requirement: $$\begin{aligned} \label{MTV3.4} \sum\limits_{j,k=1}^n\frac{\partial^2\rho_\ell}{\partial x_j \partial x_k} \xi_j \xi_k \geq -C_2 \sum\limits_{j=1}^n\xi_j^2, \, \mbox{ whenever }\, \rho_\ell=0\,\mbox{ and }\,\sum\limits_{j=1}^n \frac{\partial\rho_\ell}{\partial x_j}\xi_j =0.\end{aligned}$$ We note that, since the second fundamental form $II_{\ell}$ on $\partial\Omega_{\ell}$ is $II_\ell={{\rm Hess}\,\rho_\ell}/{|\nabla\rho_\ell|}$, almost-convexity is, in view of , equivalent to requiring that $II_\ell$ be bounded from below, uniformly in $\ell$. We now discuss some important special classes of almost-convex domains. \[eu-RF\] A bounded Lipschitz domain $\Omega\subset{\mathbb{R}}^n$ satisfies a local exterior ball condition, henceforth referred to as LEBC, if every boundary point $x_0\in\partial\Omega$ has an open neighborhood ${\mathcal{O}}$ which satisfies the following two conditions: 1. There exists a Lipschitz function $\varphi:{{\mathbb{R}}}^{n-1}\to{{\mathbb{R}}}$ with $\varphi(0)=0$ such that if $D$ is the domain above the graph of $\varphi$, then $D$ satisfies a UEBC. 2. There exists a $C^{1,1}$ diffeomorphism $\Upsilon$ mapping ${\mathcal{O}}$ onto the unit ball $B(0,1)$ in ${{\mathbb{R}}}^n$ such that $\Upsilon(x_0)=0$, $\Upsilon({\mathcal{O}}\cap\Omega)=B(0,1)\cap D$, $\Upsilon({\mathcal{O}} \backslash {{\overline}\Omega})=B(0,1) \backslash \overline{D}$. It is clear from Definition \[eu-RF\] that the class of bounded domains satisfying a LEBC is invariant under $C^{1,1}$ diffeomorphisms. This makes this class of domains amenable to working on manifolds. This is the point of view adopted in [@MTV], where the following result is also proved: \[MTVp3.1\] If the bounded Lipschitz domain $\Omega\subset{\mathbb{R}}^n$ satisfies a LEBC then it is almost-convex. Hence, in the class of bounded Lipschitz domains in ${\mathbb{R}}^n$, we have $$\begin{aligned} \label{ewT-1} \mbox{convex}\,\Longrightarrow\, \mbox{UEBC}\,\Longrightarrow\, \mbox{LEBC}\,\Longrightarrow\, \mbox{almost-convex}.\end{aligned}$$ We are now in a position to specify the class of domains in which most of our subsequent analysis will be carried out. [d.Conv]{} Let $n\in{{\mathbb{N}}}$, $n\geq 2$, and assume that $\Omega\subset{{{\mathbb{R}}}}^n$ is a bounded Lipschitz domain. Then ${\Omega}$ is called a quasi-convex domain if there exists $\delta>0$ sufficiently small $($relative to $n$ and the Lipschitz character of ${\Omega}$$)$, with the following property that for every $x\in{{\partial\Omega}}$ there exists an open subset ${\Omega}_x$ of $\Omega$ such that ${{\partial\Omega}}\cap{{\partial\Omega}}_x$ is an open neighborhood of $x$ in ${{\partial\Omega}}$, and for which one of the following two conditions holds:\ $(i)$ $\Omega_x$ is of class $MH^{1/2}_\delta$ if $n\geq 3$, and of class $C^{1,r}$ for some $1/2<r<1$ if $n=2$.\ $(ii)$ $\Omega_x$ is an almost-convex domain. Given Definition \[d.Conv\], we thus introduce the following basic assumption: [h.Conv]{} Let $n\in{{\mathbb{N}}}$, $n\geq 2$, and assume that $\Omega\subset{{{\mathbb{R}}}}^n$ is a quasi-convex domain. Informally speaking, the above definition ensures that the boundary singularities are directed outwardly. A typical example of such a domain is shown in Fig. \[Pic\] below. (4,4) (2,1.8)[$\Omega$]{} (0.1,2, 1.1,2.2, 1.3,3.1, 0.9,3.6) (0.9,3.6, 1.4,3.5, 1.8,3.1) (1.8,3.1, 2.3,2.9, 2.7,3.1) (2.7,3.1, 3.2,3.45, 3.9,3.5) (3.9,3.5, 3,3.1, 3.05,2.9, 3.2,2.85, 3.3,2.6, 3.4,2.6, 3.6,2.7, 4.1,2.8) (4.1,2.8, 3.5,1.8, 4.2,0.9) (4.2,0.9, 3,0.8, 2.15,0.65, 1.6,0.1) (1.6,0.1, 1.7,0.65, 1.5,0.9, 1.25,1.15, 1.1,1.35, 0.8,1.7, 0.1,2) Being quasi-convex is a certain type of regularity condition of the boundary of a Lipschitz domain. The only way we are going to utilize this property is via the following elliptic regularity result proved in [@GM10]. \[Bjk\] Assume Hypotheses \[h.V\] and \[h.Conv\]. Then $$\label{Yan-9} \operatorname{dom}\big(H_{D,{\Omega}}\big)\subset H^{2}(\Omega), \quad \operatorname{dom}\big(H_{N,{\Omega}}\big)\subset H^{2}(\Omega).$$ In fact, all of our results in this paper hold in the class of Lipschitz domains for which the two inclusions in hold. The following theorem addresses the issue raised at the beginning of this subsection. Its proof is similar to the special case $V\equiv 0$, treated in [@GM10]. \[T-DD1\] Assume Hypotheses \[h.V\] and \[h.Conv\]. Then holds. In particular, $$\begin{aligned} \operatorname{dom}(H_{min,{\Omega}}) & = H^2_0(\Omega) {\notag}\\ & =\big\{u\in L^2(\Omega;d^nx)\,\big|\,\Delta u\in L^2(\Omega;d^nx), \, \widehat{\gamma}_D u =\widehat{\gamma}_N u =0\big\}, {\label}{dmin} \\ \operatorname{dom}(H_{max,{\Omega}} ) & = \big\{u\in L^2(\Omega;d^nx)\,\big|\, \Delta u \in L^2(\Omega;d^nx)\big\}, {\label}{dmax}\end{aligned}$$ and $$\label{Yan-10} H_{min,{\Omega}} = (H_{max,{\Omega}})^*\, \mbox{ and }\, H_{max,{\Omega}} = (H_{min,{\Omega}})^*.$$ We conclude this subsection with the following result which is essentially contained in [@GM10]. \[L-Fri1\] Assume Hypotheses \[h2.1\] and \[h.V\]. Then the Friedrichs extension of $(-\Delta+V)|_{C^\infty_0({\Omega})}$ in $L^2({\Omega};d^nx)$ is precisely the perturbed Dirichlet Laplacian $H_{D,{\Omega}}$. Consequently, if Hypothesis \[h.Conv\] is assumed in place of Hypothesis \[h2.1\], then the Friedrichs extension of $H_{min,{\Omega}} $ in is the perturbed Dirichlet Laplacian $H_{D,{\Omega}}$. Trace Operators and Boundary Problems on Quasi-Convex Domains {#s6X} ------------------------------------------------------------- Here we revisit the issue of traces, originally taken up in Section \[s2\], and extend the scope of this theory. The goal is to extend our earlier results to a context that is well-suited for the treatment of the perturbed Krein Laplacian in quasi-convex domains, later on. All results in this subsection are direct generalizations of similar results proved in the case where $V\equiv 0$ in [@GM10]. \[tH.A\] Assume Hypotheses \[h.V\] and \[h.Conv\], and suppose that $z\in{{\mathbb{C}}}\backslash{\sigma}(H_{D,{\Omega}})$. Then for any functions $f\in L^2({\Omega};d^nx)$ and $g\in (N^{1/2}(\partial\Omega))^*$ the following inhomogeneous Dirichlet boundary value problem $$\label{Yan-14} \begin{cases} (-\Delta+V-z)u=f\text{ in }\,{\Omega}, \\ u\in L^2({\Omega};d^nx), \\ \widehat{\gamma}_D u =g\text{ on }\,{{\partial\Omega}}, \end{cases}$$ has a unique solution $u=u_D$. This solution satisfies $$\label{Hh.3X} \|u_D\|_{L^2({\Omega};d^nx)} +\|\widehat{\gamma}_N u_D\|_{(N^{3/2}(\partial\Omega))^*}\leq C_D (\|f\|_{L^2({\Omega};d^nx)}+\|g\|_{(N^{1/2}(\partial\Omega))^*})$$ for some constant $C_D=C_D(\Omega,V,z)>0$, and the following regularity results hold: $$\begin{aligned} \label{3.3Y} & g\in H^1(\partial\Omega) \,\text{ implies }\, u_D\in H^{3/2}(\Omega), \\ & g\in\gamma_D\bigl(H^2(\Omega)\bigr) \, \text{ implies }\, u_D\in H^2(\Omega). \label{3.3Ys}\end{aligned}$$ In particular, $$\label{3.3Ybis} g=0\, \text{ implies } \, u_D\in H^2(\Omega)\cap H^1_0(\Omega).$$ Natural estimates are valid in each case. Moreover, the solution operator for with $f=0$ $($i.e., $P_{D,{\Omega},V,z}:g\mapsto u_D$$)$ satisfies $$\label{3.34Y} P_{D,{\Omega},V,z}=\big[{\gamma}_N(H_{D,{\Omega}}-{{\overline}z}I_{\Omega})^{-1}\big]^* \in{{\mathcal B}}\big((N^{1/2}(\partial\Omega))^*,L^2({\Omega};d^nx)\big),$$ and the solution of is given by the formula $$\label{3.35Y} u_D=(H_{D,{\Omega}}-zI_{\Omega})^{-1}f -\big[{\gamma}_N(H_{D,{\Omega}}-{\overline}{z}I_{\Omega})^{-1}\big]^*g.$$ \[New-CV22\] Assume Hypotheses \[h.V\] and \[h.Conv\]. Then for every $z\in{{\mathbb{C}}}\backslash {\sigma}(H_{D,{\Omega}})$ the map $$\label{Tan-Bq2} \widehat{\gamma}_D: \big\{u\in L^2(\Omega;d^nx)\,\big|\,(-\Delta+V-z)u=0 \,\mbox{in}\, \Omega\}\to \bigl(N^{1/2}(\partial\Omega)\bigr)^*$$ is an isomorphism $($i.e., bijective and bicontinuous$)$. \[tH.G2\] Assume Hypotheses \[h.V\] and \[h.Conv\] and suppose that $z\in{{\mathbb{C}}}\backslash{\sigma}(H_{N,{\Omega}})$. Then for any functions $f\in L^2({\Omega};d^nx)$ and $g\in (N^{3/2}(\partial\Omega))^*$ the following inhomogeneous Neumann boundary value problem $$\label{n-1H} \begin{cases} (-\Delta+V-z)u=f\text{ in }\,{\Omega}, \\[4pt] u\in L^2({\Omega};d^nx), \\[4pt] \widehat{\gamma}_N u=g\text{ on }\,{{\partial\Omega}}, \end{cases}$$ has a unique solution $u=u_N$. This solution satisfies $$\label{Hh.3f} \|u_N\|_{L^2({\Omega};d^nx)} +\|\widehat{\gamma}_D u_N\|_{(N^{1/2}(\partial\Omega))^*}\leq C_N (\|f\|_{L^2({\Omega};d^nx)}+\|g\|_{(N^{3/2}(\partial\Omega))^*})$$ for some constant $C_N=C_N(\Omega,V,z)>0$, and the following regularity results hold: $$\begin{aligned} \label{3.3f} & g\in L^2(\partial\Omega;d^{n-1}\omega)\, \text{ implies }\, u_N\in H^{3/2}(\Omega), \\ & g\in\gamma_N\bigl(H^2({\Omega})\bigr)\, \text{ implies }\, u_N\in H^2(\Omega). \label{3.3fbis}\end{aligned}$$ Natural estimates are valid in each case. Moreover, the solution operator for with $f=0$ $($i.e., $P_{N,{\Omega},V,z}:g\mapsto u_N$$)$ satisfies $$\label{3.34f} P_{N,{\Omega},V,z}=\big[{\gamma}_D(H_{N,{\Omega}}-{{\overline}z}I_{\Omega})^{-1}\big]^* \in{{\mathcal B}}\big((N^{3/2}(\partial\Omega))^*,L^2({\Omega};d^nx)\big),$$ and the solution of is given by the formula $$\label{3.a5Y} u_N=(H_{N,{\Omega}}-zI_{\Omega})^{-1}f +\big[{\gamma}_D(H_{N,{\Omega}}-{\overline}{z}I_{\Omega})^{-1}\big]^*g.$$ \[New-CV33\] Assume Hypotheses \[h.V\] and \[h.Conv\]. Then, for every $z\in{{\mathbb{C}}}\backslash \sigma(H_{N,{\Omega}})$, the map $$\label{Tan-FFF} \widehat{\gamma}_N:\big\{u\in L^2(\Omega;d^nx)\,\big|\,(-\Delta+V-z)u=0 \,\mbox{ in }\Omega\big\}\to \bigl(N^{3/2}(\partial\Omega)\bigr)^*$$ is an isomorphism $($i.e., bijective and bicontinuous$)$. Dirichlet-to-Neumann Operators on Quasi-Convex Domains {#s7X} ------------------------------------------------------ In this subsection we review spectral parameter dependent Dirichlet-to-Neumann maps, also known in the literature as Weyl–Titchmarsh and Poincaré–Steklov operators. Assuming Hypotheses \[h.V\] and \[h.Conv\], introduce the Dirichlet-to-Neumann map $M_{D,N,{\Omega},V}(z)$ associated with $-\Delta+V-z$ on ${\Omega}$, as follows: $$\label{3.44v} M_{D,N,{\Omega},V}(z) \colon \begin{cases} \bigl(N^{1/2}({{\partial\Omega}})\bigr)^*\to \bigl(N^{3/2}({{\partial\Omega}})\bigr)^*, \\ \hspace*{1.8cm} f\mapsto -\widehat{\gamma}_N u_D, \end{cases} \; z\in{{\mathbb{C}}}\backslash{\sigma}(H_{D,{\Omega}}),$$ where $u_D$ is the unique solution of $$\label{3.45v} (-\Delta+V-z)u=0\,\text{ in }{\Omega},\quad u\in L^2({\Omega};d^nx), \;\; \widehat{\gamma}_D u=f\,\text{ on }{{\partial\Omega}}.$$ Retaining Hypotheses \[h.V\] and \[h.Conv\], we next introduce the Neumann-to-Dirichlet map $M_{N,D,{\Omega},V}(z)$ associated with $-\Delta+V-z$ on ${\Omega}$, as follows: $$\label{3.48v} M_{N,D,{\Omega},V}(z)\colon \begin{cases} \bigl(N^{3/2}({{\partial\Omega}})\bigr)^*\to \bigl(N^{1/2}({{\partial\Omega}})\bigr)^*, \\ \hspace*{1.8cm} g\mapsto\widehat{\gamma}_D u_N, \end{cases} \; z\in{{\mathbb{C}}}\backslash{\sigma}(H_{N,{\Omega}}),$$ where $u_{N}$ is the unique solution of $$\label{3.49v} (-\Delta+V-z)u=0\,\text{ in }{\Omega},\quad u\in L^2({\Omega};d^nx), \;\; \widehat{\gamma}_Nu=g\,\text{ on }{{\partial\Omega}}.$$ As in [@GM10], where the case $V\equiv 0$ has been treated, we then have the following result: \[t3.5v\] Assume Hypotheses \[h.V\] and \[h.Conv\]. Then, with the above notation, $$\label{3.46v} M_{D,N,{\Omega},V}(z)\in{{\mathcal B}}\big((N^{1/2}({{\partial\Omega}}))^*\,,\,(N^{3/2}({{\partial\Omega}}))^*\big), \quad z\in{{\mathbb{C}}}\backslash{\sigma}(H_{D,{\Omega}}),$$ and $$\label{3.47v} M_{D,N,{\Omega},V}(z)=\widehat\gamma_N \big[\gamma_N(H_{D,{\Omega}}-{\overline}{z}I_{\Omega})^{-1}\big]^*, \quad z\in{{\mathbb{C}}}\backslash{\sigma}(H_{D,{\Omega}}).$$ Similarly, $$\label{3.50v} M_{N,D,{\Omega},V}(z)\in{{\mathcal B}}\big((N^{3/2}({{\partial\Omega}}))^*\,,\,(N^{1/2}({{\partial\Omega}}))^*\big), \quad z\in{{\mathbb{C}}}\backslash{\sigma}(H_{N,{\Omega}}),$$ and $$\label{3.52v} M_{N,D,{\Omega},V}(z) = \widehat \gamma_D\big[\gamma_D(H_{N,{\Omega}}-{\overline}{z}I_{\Omega})^{-1}\big]^*, \quad z\in{{\mathbb{C}}}\backslash{\sigma}(H_{N,{\Omega}}).$$ Moreover, $$\label{3.53v} M_{N,D,{\Omega},V}(z)=-M_{D,N,{\Omega},V}(z)^{-1},\quad z\in{{\mathbb{C}}}\backslash({\sigma}(H_{D,{\Omega}})\cup{\sigma}(H_{N,{\Omega}})),$$ and $$\label{NaLa} \big[M_{D,N,{\Omega},V}(z)\big]^*=M_{D,N,{\Omega},V}({\overline}{z}),\quad \big[M_{N,D,{\Omega},V}(z)\big]^*=M_{N,D,{\Omega},V}({\overline}{z}).$$ As a consequence, one also has $$\begin{aligned} \label{3.TTa} & M_{D,N,{\Omega},V}(z)\in{{\mathcal B}}\big(N^{3/2}({{\partial\Omega}})\,,\,N^{1/2}({{\partial\Omega}})\big), \quad z\in{{\mathbb{C}}}\backslash{\sigma}(H_{D,{\Omega}}), \\ & M_{N,D,{\Omega},V}(z)\in{{\mathcal B}}\big(N^{1/2}({{\partial\Omega}})\,,\,N^{3/2}({{\partial\Omega}})\big), \quad z\in{{\mathbb{C}}}\backslash{\sigma}(H_{N,{\Omega}}). \label{3.TTb}\end{aligned}$$ For closely related recent work on Weyl–Titchmarsh operators associated with nonsmooth domains we refer to [@GM08], [@GM09a], [@GM09b], [@GM10], and [@GMZ07]. For an extensive list of references on $z$-dependent Dirichlet-to-Neumann maps we also refer, for instance, to [@Ag03], [@ABMN05], [@AP04], [@BL07], [@BMN02], [@BGW09], [@BHMNW09], [@BMNW08], [@BGP08], [@DM91], [@DM95], [@GLMZ05]–[@GMZ07], [@Gr08a], [@Po08], [@Ry07], [@Ry09], [@Ry10]. Regularized Neumann Traces and Perturbed Krein Laplacians {#s8} ========================================================= This section is structured into two parts dealing, respectively, with the regularized Neumann trace operator (Subsection \[s8X\]), and the perturbed Krein Laplacian in quasi-convex domains (Subsection \[s9X\]). The Regularized Neumann Trace Operator on Quasi-Convex Domains {#s8X} -------------------------------------------------------------- Following earlier work in [@GM10], we now consider a version of the Neumann trace operator which is suitably normalized to permit the familiar version of Green’s formula (cf.  below) to work in the context in which the functions involved are only known to belong to $\operatorname{dom}(-\Delta_{\max,{\Omega}})$. The following theorem is a slight extension of a similar result proved in [@GM10] when $V\equiv 0$. \[LL.w\] Assume Hypotheses \[h.V\] and \[h.Conv\]. Then, for every $z\in{{\mathbb{C}}}\backslash{\sigma}(H_{D,{\Omega}})$, the map $$\label{3.Aw1} \tau_{N,V,z}:\bigl\{u\in L^2({\Omega};d^nx);\,\Delta u\in L^2({\Omega};d^nx)\bigr\} \to N^{1/2}(\partial\Omega)$$ given by $$\label{3.Aw2} \tau_{N,V,z} u:=\widehat\gamma_N u +M_{D,N,{\Omega},V}(z)\bigl(\widehat\gamma_D u \bigr), \quad u\in L^2({\Omega};d^nx),\,\,\Delta u\in L^2({\Omega};d^nx),$$ is well-defined, linear and bounded, where the space $$\big\{u\in L^2({\Omega};d^nx)\,\big|\, \Delta u\in L^2({\Omega};d^nx)\big\}$$ is endowed with the natural graph norm $u\mapsto\|u\|_{L^2({\Omega};d^nx)}+\|\Delta u\|_{L^2({\Omega};d^nx)}$. Moreover, this operator satisfies the following additional properties: 1. The map $\tau_{N,V,z}$ in , is onto $($i.e., $\tau_{N,V,z}(\operatorname{dom}(H_{max,{\Omega}} ))=N^{1/2}(\partial\Omega)$$)$, for each $z\in{{\mathbb{C}}}\backslash{\sigma}(H_{D,{\Omega}})$. In fact, $$\label{3.ON} \tau_{N,V,z}\bigl(H^2({\Omega})\cap H^1_0({\Omega})\bigr)=N^{1/2}(\partial\Omega) \, \mbox{ for each } \, z\in{{\mathbb{C}}}\backslash{\sigma}(H_{D,{\Omega}}).$$ 2. One has $$\label{3.Aw9} \tau_{N,V,z}=\gamma_N(H_{D,{\Omega}}-zI_{{\Omega}})^{-1}(-\Delta-z),\quad z\in{{\mathbb{C}}}\backslash {\sigma}(H_{D,{\Omega}}).$$ 3. For each $z\in{{\mathbb{C}}}\backslash{\sigma}(H_{D,{\Omega}})$, the kernel of the map $\tau_{N,V,z}$ in , is $$\label{3.AKe} \ker(\tau_{N,V,z})=H^2_0(\Omega)\dot{+}\{u\in L^2({\Omega};d^nx)\,|\, (-\Delta+V-z)u=0\,\mbox{ in }\,\Omega\}.$$ In particular, if $z\in{{\mathbb{C}}}\backslash{\sigma}(H_{D,{\Omega}})$, then $$\label{Sim-Gr} \tau_{N,V,z} u =0\, \mbox{ for every }\, u\in\ker(H_{max,{\Omega}} -zI_{{\Omega}}).$$ 4. The following Green formula holds for every $u,v\in\operatorname{dom}(H_{max,{\Omega}} )$ and every complex number $z\in{{\mathbb{C}}}\backslash {\sigma}(H_{D,{\Omega}})$: $$\begin{aligned} \label{T-Green} & ((-\Delta+V-z)u\,,\,v)_{L^2(\Omega;d^nx)} - (u\,,\,(-\Delta+V-{\overline}{z})v)_{L^2(\Omega;d^nx)} \nonumber\\ & \quad =-{}_{N^{1/2}(\partial\Omega)}\langle\tau_{N,V,z} u,\widehat\gamma_D v \rangle_{(N^{1/2}(\partial\Omega))^*} +\,{\overline}{{}_{N^{1/2}(\partial\Omega)}\langle\tau_{N,V,{\overline}{z}} v, \widehat{\gamma}_D u \rangle_{(N^{1/2}(\partial\Omega))^*}}.\end{aligned}$$ The Perturbed Krein Laplacian in Quasi-Convex Domains {#s9X} ----------------------------------------------------- We now discuss the Krein–von Neumann extension of the Laplacian $-\Delta\big|_{C^\infty_0(\Omega)}$ perturbed by a nonnegative, bounded potential $V$ in $L^2({\Omega}; d^n x)$. We will conveniently call this operator the [*perturbed Krein Laplacian*]{} and introduce the following basic assumption: \[h.VK\] $(i)$ Let $n\in{{\mathbb{N}}}$, $n\geq 2$, and assume that $\emptyset \neq {\Omega}\subset{{{\mathbb{R}}}}^n$ is a bounded Lipschitz domain satisfying Hypothesis \[h.Conv\].\ $(ii)$ Assume that $$\label{VV-WW} V\in L^\infty({\Omega};d^nx) \, \mbox{ and }\, V \geq 0 \mbox{ a.e.\ in } \,\Omega.$$ Denoting by ${\overline}{T}$ the closure of a linear operator $T$ in a Hilbert space ${{\mathcal H}}$, we have the following result: \[C-Da\] Assume Hypothesis \[h.VK\]. Then $H_{min,{\Omega}}$ is a densely defined, closed, nonnegative $($in particular, symmetric$)$ operator in $L^2({\Omega}; d^n x)$. Moreover, $$\label{Pos-3} {\overline}{(-\Delta+V)\big|_{C^\infty_0({\Omega})}} = H_{min,{\Omega}}.$$ The first claim in the statement is a direct consequence of Theorem \[T-DD1\]. As for , let us temporarily denote by $H_0$ the closure of $-\Delta+V$ defined on $C^\infty_0({\Omega})$. Then $$\label{Pos-4} u\in\operatorname{dom}(H_0) \, \text{ if and only if } \begin{cases} \mbox{there exist }v\in L^2({\Omega};d^nx)\mbox{ and } u_j\in C^\infty_0({\Omega}),\,j\in{{\mathbb{N}}},\mbox{ such that } \\ u_j\to u \,\mbox{ and } \,(-\Delta+V)u_j\to v \,\mbox{ in }\,L^2({\Omega};d^nx) \,\mbox{ as }\, j\to\infty. \end{cases}$$ Thus, if $u\in \operatorname{dom}(H_0)$ and $v$, $\{u_j\}_{j\in{{\mathbb{N}}}}$ are as in the right-hand side of , then $(-\Delta+V)u=v$ in the sense of distributions in $\Omega$, and $$\begin{aligned} \label{Pos-5} \begin{split} 0&=\widehat\gamma_D u_j \to \widehat\gamma_D u \, \mbox{ in }\, \bigl(N^{1/2}({{\partial\Omega}})\bigr)^* \, \mbox{ as }\, j\to\infty, \\ 0&=\widehat\gamma_N u_j \to \widehat\gamma_N u \, \mbox{ in }\, \bigl(N^{1/2}({{\partial\Omega}})\bigr)^* \, \mbox{ as }\, j\to\infty, \end{split}\end{aligned}$$ by Theorem \[New-T-tr\] and Theorem \[3ew-T-tr\]. Consequently, $u\in\operatorname{dom}(H_{max,{\Omega}} )$ satisfies $\widehat\gamma_D u =0$ and $\widehat\gamma_N u =0$. Hence, $u\in H^2_0({\Omega})=\operatorname{dom}(H_{min,{\Omega}} )$ by Theorem \[T-DD1\] and the current assumptions on $\Omega$. This shows that $H_0\subseteq H_{min,{\Omega}} $. The converse inclusion readily follows from the fact that any $u\in H^2_0({\Omega})$ is the limit in $H^2({\Omega})$ of a sequence of test functions in $\Omega$. \[C-DaW\] Assume Hypothesis \[h.VK\]. Then the Krein–von Neumann extension $H_{K,{\Omega}}$ of $(-\Delta+V)\big|_{C^\infty_0({\Omega})}$ in $L^2({\Omega};d^nx)$ is the $L^2$-realization of $-\Delta+V$ with domain $$\begin{aligned} \label{Kre-Frq1} \begin{split} \operatorname{dom}(H_{K,{\Omega}}) &= \operatorname{dom}(H_{min,{\Omega}})\,\dot{+}\ker(H_{max,{\Omega}}) \\ & =H^2_0({\Omega})\,\dot{+}\, \big\{u\in L^2({\Omega};d^nx)\,\big|\,(-\Delta+V)u=0\mbox{ in }\Omega\big\}. \end{split} \end{aligned}$$ By virtue of , , and the fact that $(-\Delta+V)|_{C^\infty_0({\Omega})}$ and its closure, $H_{min,{\Omega}}$ (cf. ) have the same self-adjoint extensions, one obtains $$\begin{aligned} \label{Kre-Def} \operatorname{dom}(H_{K,{\Omega}}) & = \operatorname{dom}(H_{min,{\Omega}})\,\dot{+}\ker((H_{min,{\Omega}})^*) \nonumber\\ & = \operatorname{dom}(H_{min,{\Omega}} )\,\dot{+}\ker(H_{max,{\Omega}}) {\notag}\\ & = H^2_0({\Omega})\,\dot{+}\, \big\{u\in L^2({\Omega};d^nx)\,\big|\,(-\Delta+V)u=0\mbox{ in }\Omega\big\}, \end{aligned}$$ as desired. Nonetheless, we shall adopt a different point of view which better elucidates the nature of the boundary condition associated with this perturbed Krein Laplacian. More specifically, following the same pattern as in [@GM10], the following result can be proved. \[T-Kr\] Assume Hypothesis \[h.VK\] and fix $z\in{{\mathbb{C}}}\backslash {\sigma}(H_{D,{\Omega}})$. Then $H_{K,{\Omega},z}$ in $L^2(\Omega;d^nx)$, given by $$\begin{aligned} \label{A-zz.1} \begin{split} & H_{K,{\Omega},z} u:=(-\Delta+V-z)u, \\ & u\in \operatorname{dom}(H_{K,{\Omega},z}):=\{v\in\operatorname{dom}(H_{max,{\Omega}} )\,|\, \tau_{N,V,z} v =0\}, \end{split} \end{aligned}$$ satisfies $$\label{A-zz.W} (H_{K,{\Omega},z})^*=H_{K,{\Omega},{\overline}{z}},$$ and agrees with the self-adjoint perturbed Krein Laplacian $H_{K,{\Omega}}=H_{K,{\Omega},0}$ when taking $z=0$. In particular, if $z\in{{\mathbb{R}}}\backslash {\sigma}(H_{D,{\Omega}})$ then $H_{K,{\Omega},z}$ is self-adjoint. Moreover, if $z \leq 0$, then $H_{K,{\Omega},z}$ is nonnegative. Hence, the perturbed Krein Laplacian $H_{K,{\Omega}}$ is a self-adjoint operator in $L^2({\Omega};d^nx)$ which admits the description given in when $z=0$, and which satisfies $$\label{A-zz.b} H_{K,{\Omega}}\geq 0 \,\mbox{ and }\, H_{min,{\Omega}} \subseteq H_{K,{\Omega}}\subseteq H_{max,{\Omega}} .$$ Furthermore, $$\begin{aligned} & \ker(H_{K,{\Omega}})=\big\{u\in L^2({\Omega};d^nx)\,\big|\,(-\Delta+V)u=0\big\}, \\ & \dim(\ker(H_{K,{\Omega}})) = {\rm def} (H_{min,{\Omega}}) = {\rm def} \big({\overline}{(-\Delta+V)\big|_{C^\infty_0({\Omega})}}\big) =\infty, \\ & \operatorname{ran}(H_{K,{\Omega}})=(-\Delta+V) H^2_0({\Omega}), \\ & \text{$H_{K,{\Omega}}$ has a purely discrete spectrum in $(0,\infty)$}, \quad \sigma_{\rm ess}(H_{K,{\Omega}}) = \{0\}, \label{spec-1} \end{aligned}$$ and for any nonnegative self-adjoint extension ${\widetilde}S$ of $(-\Delta+V)|_{C^\infty_0({\Omega})}$ one has $($cf. $)$, $$\label{Ok.1} H_{K,{\Omega}}\leq {\widetilde}S\leq H_{D,{\Omega}}.$$ The nonlocal boundary condition $$\tau_{N,V,0} v = {\widehat}\gamma_N v + M_{D,N,{\Omega},V} (0) v = 0, \quad v \in \operatorname{dom}(H_{K,{\Omega}})$$ (cf. ) in connection with the Krein–von Neumann extension $H_{K,{\Omega}}$, in the special one-dimensional half-line case ${\Omega}= [a,\infty)$ has first been established in [@Ts87]. In terms of abstract boundary conditions in connection with the theory of boundary value spaces, such a condition has been derived in [@DMT88] and [@DMT89]. However, we emphasize that this abstract boundary value space approach, while applicable to ordinary differential operators, is not applicable to partial differential operators even in the case of smooth boundaries $\partial{\Omega}$ (see, e.g., the discussion in [@BL07]). In particular, it does not apply to the nonsmooth domains ${\Omega}$ studied in this paper. In fact, only very recently, appropriate modifications of the theory of boundary value spaces have successfully been applied to partial differential operators in smooth domains in [@BL07], [@BGW09], [@BHMNW09], [@BMNW08], [@Po08], [@PR09], [@Ry07], [@Ry09], and [@Ry10]. With the exception of the following short discussions: Subsection 4.1 in [@BL07] (which treat the special case where ${\Omega}$ equals the unit ball in ${{\mathbb{R}}}^2$), Remark 3.8 in [@BGW09], Section 2 in [@Ry07], Subsection 2.4 in [@Ry09], and Remark 5.12 in [@Ry10], these investigations did not enter a detailed discussion of the Krein-von Neumann extension. In particular, none of these references applies to the case of nonsmooth domains ${\Omega}$. Connections with the Problem of the Buckling of a Clamped Plate {#s10} =============================================================== In this section we proceed to study a fourth-order problem, which is a perturbation of the classical problem for the buckling of a clamped plate, and which turns out to be essentially spectrally equivalent to the perturbed Krein Laplacian $H_{K,{\Omega}}:=H_{K,{\Omega},0}$. For now, let us assume Hypotheses \[h2.1\] and \[h.V\]. Given $\lambda\in{{\mathbb{C}}}$, consider the eigenvalue problem for the generalized buckling of a clamped plate in the domain ${\Omega}\subset{{\mathbb{R}}}^n$ $$\label{MM-1} \begin{cases} u\in\operatorname{dom}(-\Delta_{max,{\Omega}}), \\ (-\Delta+V)^2u=\lambda\,(-\Delta+V)u\,\mbox{ in }\, \Omega, \\ \widehat{\gamma}_D u =0 \,\mbox{ in } \,\big(N^{1/2}({{\partial\Omega}})\big)^*, \\ \widehat{\gamma}_N u =0 \,\mbox{ in } \,\big(N^{3/2}({{\partial\Omega}})\big)^*, \end{cases}$$ where $(-\Delta+V)^2u:=(-\Delta+V)(-\Delta u+Vu)$ in the sense of distributions in ${\Omega}$. Due to the trace theory developed in Sections \[s3\] and \[s5\], this formulation is meaningful. In addition, if Hypothesis \[h.Conv\] is assumed in place of Hypothesis \[h2.1\] then, by , this problem can be equivalently rephrased as $$\label{MM-2} \begin{cases} u\in H^2_0(\Omega), \\ (-\Delta+V)^2u=\lambda\,(-\Delta+V)u \,\mbox{ in } \,\Omega. \end{cases}$$ \[L-MM-1\] Assume Hypothesis \[h.VK\] and suppose that $u\not=0$ solves for some $\lambda\in{{\mathbb{C}}}$. Then necessarily $\lambda\in (0,\infty)$. Let $u,\lambda$ be as in the statement of the lemma. Then, as already pointed out above, $u\in H^2_0(\Omega)$. Based on this, the fact that $\Delta u\in\operatorname{dom}(-\Delta_{max,{\Omega}})$, and the integration by parts formulas and , we may then write (we recall that our $L^2$ pairing is conjugate linear in the [*first*]{} argument): $$\begin{aligned} \label{MM-3} & \lambda\bigl[\|\nabla u\|^2_{(L^2({\Omega};d^nx))^n} +\|V^{1/2}u\|^2_{(L^2({\Omega};d^nx))^n}\bigr] =\lambda\, (u,(-\Delta+V)u)_{L^2({\Omega};d^nx)} \nonumber\\ & \quad =(u\,,\,\lambda\,(-\Delta+V)u)_{L^2({\Omega};d^nx)} = \big(u,(-\Delta+V)^2 u\big)_{L^2({\Omega};d^nx)} \nonumber\\[4pt] & \quad =(u,(-\Delta+V)(-\Delta u+Vu))_{L^2({\Omega};d^nx)} =((-\Delta+V)u,(-\Delta+V)u)_{L^2({\Omega};d^nx)} \nonumber\\[4pt] & \quad =\|(-\Delta+V)u\|^2_{L^2({\Omega};d^nx)}.\end{aligned}$$ Since, according to Theorem \[tH.A\], $L^2({\Omega};d^nx)\ni u\not=0$ and $\widehat{\gamma}_D u =0$ prevent $u$ from being a constant function, entails $$\label{MM-4} \lambda=\frac{\|(-\Delta+V)u\|^2_{L^2({\Omega};d^nx)}} {\|\nabla u\|^2_{(L^2({\Omega};d^nx))^n}+\|V^{1/2}u\|^2_{(L^2({\Omega};d^nx))^n}}>0,$$ as desired. Next, we recall the operator $P_{D,{\Omega},V,z}$ introduced just above and agree to simplify notation by abbreviating $P_{D,{\Omega},V}:=P_{D,{\Omega},V,0}$. That is, $$\label{3.34Yz} P_{D,{\Omega},V}=\big[{\gamma}_N (H_{D,{\Omega}})^{-1}\big]^* \in{{\mathcal B}}\big((N^{1/2}(\partial\Omega))^*,L^2({\Omega};d^nx)\big)$$ is such that if $u:=P_{D,{\Omega},V} g$ for some $g\in\bigl(N^{1/2}(\partial\Omega)\bigr)^*$, then $$\label{Yan-14z} \begin{cases} (-\Delta+V)u=0\text{ in }\,{\Omega}, \\[4pt] u\in L^2({\Omega};d^nx), \\[4pt] \widehat{\gamma}_D u =g\text{ on }\,{{\partial\Omega}}. \end{cases}$$ Hence, $$\begin{aligned} \label{3.Gv} \begin{split} & (-\Delta+V) P_{D,{\Omega},V}=0, \\ & \widehat{\gamma}_N P_{D,{\Omega},V}=-M_{D,N,\Omega,V}(0) \, \mbox{ and }\, \widehat{\gamma}_D P_{D,{\Omega},V}=I_{(N^{1/2}({{\partial\Omega}}))^*}, \end{split} \end{aligned}$$ with $I_{(N^{1/2}({{\partial\Omega}}))^*}$ the identity operator, on $\bigl(N^{1/2}({{\partial\Omega}})\bigr)^*$. \[T-MM-1\] Assume Hypothesis \[h.VK\]. If $0\not=v\in L^2({\Omega};d^nx)$ is an eigenfunction of the perturbed Krein Laplacian $H_{K,{\Omega}}$ corresponding to the eigenvalue $0\not=\lambda\in{{\mathbb{C}}}$ $($hence $\lambda>0$$)$, then $$\label{MM-5} u:=v-P_{D,{\Omega},V}(\widehat{\gamma}_D v)$$ is a nontrivial solution of . Conversely, if $0\not=u\in L^2({\Omega};d^nx)$ solves for some $\lambda\in{{\mathbb{C}}}$ then $\lambda$ is a $($strictly$)$ positive eigenvalue of the perturbed Krein Laplacian $H_{K,{\Omega}}$, and $$\label{MM-6} v:=\lambda^{-1}(-\Delta+V)u$$ is a nonzero eigenfunction of the perturbed Krein Laplacian, corresponding to this eigenvalue. In one direction, assume that $0\not=v\in L^2({\Omega};d^nx)$ is an eigenfunction of the perturbed Krein Laplacian $H_{K,{\Omega}}$ corresponding to the eigenvalue $0\not=\lambda\in{{\mathbb{C}}}$ (since $H_{K,{\Omega}}\geq 0$ –cf. Theorem \[T-Kr\]– it follows that $\lambda>0$). Thus, $v$ satisfies $$\label{MM-7} v\in\operatorname{dom}(H_{max,{\Omega}} ),\quad (-\Delta+V)v=\lambda\,v,\; \tau_{N,V,0} v =0.$$ In particular, $\widehat{\gamma}_D v \in\bigl(N^{1/2}({{\partial\Omega}})\bigr)^*$ by Theorem \[New-T-tr\]. Hence, by , $u$ in is a well-defined function which belongs to $L^2({\Omega};d^nx)$. In fact, since also $(-\Delta+V)u=(-\Delta+V)v\in L^2({\Omega};d^nx)$, it follows that $u\in\operatorname{dom}(H_{max,{\Omega}} )$. Going further, we note that $$\begin{aligned} \label{MM-8} \begin{split} (-\Delta+V)^2u &=(-\Delta+V)(-\Delta+V)u = (-\Delta+V)(-\Delta+V)v \\ &= \lambda\,(-\Delta+V)v=\lambda\,(-\Delta+V)u. \end{split} \end{aligned}$$ Hence, $(-\Delta+V)^2u=\lambda\,(-\Delta+V)u$ in ${\Omega}$. In addition, by , $$\label{MM-9} \widehat{\gamma}_D u =\widehat{\gamma}_D v -\widehat{\gamma}_D(P_{D,{\Omega},V}(\widehat{\gamma}_D v ) =\widehat{\gamma}_D v -\widehat{\gamma}_D v =0,$$ whereas $$\label{MM-10} \widehat{\gamma}_N u =\widehat{\gamma}_N v -\widehat{\gamma}_N(P_{D,{\Omega},V}(\widehat{\gamma}_D v ) =\widehat{\gamma}_N v +M_{D,N,{\Omega},V}(0)(\widehat{\gamma}_D v ) =\tau_{N,V,0} v=0,$$ by the last condition in . Next, to see that $u$ cannot vanish identically, we note that $u=0$ would imply $v=P_{D,{\Omega},V}(\widehat{\gamma}_D v)$ which further entails $\lambda\,v=(-\Delta+V)v =(-\Delta+V)P_{D,{\Omega},V}(\widehat{\gamma}_D v)=0$, that is, $v=0$ (since $\lambda\not=0$). This contradicts the original assumption on $v$ and shows that $u$ is a nontrivial solution of . This completes the proof of the first half of the theorem. Turning to the second half, suppose that $\lambda\in{{\mathbb{C}}}$ and $0\not=u\in L^2({\Omega};d^nx)$ is a solution of . Lemma \[L-MM-1\] then yields $\lambda>0$, so that $v:=\lambda^{-1}(-\Delta+V)u$ is a well-defined function satisfying $$\label{MM-11} v\in\operatorname{dom}(H_{max,{\Omega}} )\, \mbox{ and }\, (-\Delta+V)v=\lambda^{-1}\,(-\Delta+V)^2u=(-\Delta+V)u=\lambda\,v.$$ If we now set $w:=v-u\in L^2(\Omega;d^nx)$ it follows that $$\label{MM-12} (-\Delta+V)w=(-\Delta+V)v-(-\Delta+V)u=\lambda\,v-\lambda\,v=0,$$ and $$\label{MM-13} \widehat{\gamma}_N w =\widehat{\gamma}_N v,\quad \widehat{\gamma}_D w =\widehat{\gamma}_D v.$$ In particular, by the uniqueness in the Dirichlet problem , $$\label{MM-14} w=P_{D,{\Omega},V}(\widehat{\gamma}_D v).$$ Consequently, $$\label{MM-15} \widehat{\gamma}_N v=\widehat{\gamma}_N w =\widehat{\gamma}_N(P_{D,{\Omega},V}(\widehat{\gamma}_D v) =-M_{D,N,{\Omega},V}(0)(\widehat{\gamma}_D v),$$ which shows that $$\label{MM-16} \tau_{N,V,0} v=\widehat{\gamma}_N v +M_{D,N,{\Omega},V}(0)(\widehat{\gamma}_D v)=0.$$ Hence $v\in\operatorname{dom}(H_{K,{\Omega}})$. We note that $v=0$ would entail that the function $u\in H^2_0(\Omega)$ is a null solution of $-\Delta+V$, hence identically zero which, by assumption, is not the case. Therefore, $v$ does not vanish identically. Altogether, the above reasoning shows that $v$ is a nonzero eigenfunction of the perturbed Krein Laplacian, corresponding to the positive eigenvalue $\lambda >0$, completing the proof. [pHKv]{} $(i)$ Assume Hypothesis \[h.VK\] and let $0\neq v$ be any eigenfunction of $H_{K,{\Omega}}$ corresponding to the eigenvalue $0 \neq \lambda \in \sigma(H_{K,{\Omega}})$. In addition suppose that the operator of multiplication by $V$ satisfies $$\label{MUL} M_V\in{{\mathcal B}}\bigl(H^2({\Omega}),H^s({\Omega})\bigr) \, \mbox{ for some } \, 1/2<s\leq 2.$$ Then $u$ defined in satisfies $$u \in H^{5/2}({\Omega}), \, \text{ implying } \, v \in H^{1/2}({\Omega}). {\label}{Kv}$$ $(ii)$ Assume the smooth case, that is, $\partial\Omega$ is $C^\infty$ and $V\in C^\infty({\overline}{\Omega})$, and let $0 \neq v$ be any eigenfunction of $H_{K,{\Omega}}$ corresponding to the eigenvalue $0 \neq \lambda \in \sigma(H_{K,{\Omega}})$. Then $u$ defined in satisfies $$u\in C^\infty({\overline}{\Omega}), \, \text{ implying } \, v\in C^\infty({\overline}{\Omega}). {\label}{KvC}$$ $(i)$ We note that $u\in L^2({\Omega};d^nx)$ satisfies $\widehat\gamma_D(u)=0$, $\widehat\gamma_N(u)=0$, and $(-\Delta+V)u=(-\Delta+V)v=\lambda v\in L^2(\Omega;d^nx)$. Hence, by Theorems \[T-DD1\] and \[tH.A\], we obtain that $u\in H^2_0(\Omega)$. Next, observe that $(-\Delta+V)^2u=\lambda^2 v\in L^2(\Omega;d^nx)$ which therefore entails $\Delta^2u\in H^{s-2}({\Omega})$ by . With this at hand, the regularity results in [@PV95] (cf. also [@AP98] for related results) yield that $u\in H^{5/2}({\Omega})$. $(ii)$ Given the eigenfunction $0\neq v$ of $H_{K,{\Omega}}$, yields that $u$ satisfies the generalized buckling problem , so that by elliptic regularity $u\in C^\infty({\overline}{\Omega})$. By and one thus obtains $$\lambda v = (-\Delta +V) v = (-\Delta + V) u, \, \text{ with } \, u\in C^\infty({\overline}{\Omega}),$$ proving . In passing, we note that the multiplier condition is satisfied, for instance, if $V$ is Lipschitz. We next wish to prove that the perturbed Krein Laplacian has only point spectrum (which, as the previous theorem shows, is directly related to the eigenvalues of the generalized buckling of the clamped plate problem). This requires some preparations, and we proceed by first establishing the following. \[L-MM-2\] Assume Hypothesis \[h.VK\]. Then there exists a discrete subset $\Lambda_{{\Omega}}$ of $(0,\infty)$ without any finite accumulation points which has the following significance: For every $z\in{{\mathbb{C}}}\backslash \Lambda_{{\Omega}}$ and every $f\in H^{-2}({\Omega})$, the problem $$\label{MM-17} \begin{cases} u\in H^2_0(\Omega), \\ (-\Delta+V)(-\Delta+V-z)u=f \,\mbox{ in } \,\Omega, \end{cases}$$ has a unique solution. In addition, there exists $C=C(\Omega,z)>0$ such that the solution satisfies $$\label{MM-18} \|u\|_{H^2(\Omega)}\leq C\|f\|_{H^{-2}(\Omega)}.$$ Finally, if $z\in\Lambda_{{\Omega}}$, then there exists $u\not=0$ satisfying . In fact, the space of solutions for the problem is, in this case, finite-dimensional and nontrivial. In a first stage, fix $z\in{{\mathbb{C}}}$ with ${\mathop\mathrm{Re}}(z)\leq -M$, where $M=M({\Omega},V)>0$ is a large constant to be specified later, and consider the bounded sesquilinear form $$\begin{aligned} \label{MM-19} & a_{V,z}({\,\cdot\,},{\,\cdot\,}):H^2_0({\Omega})\times H^2_0({\Omega})\to {{\mathbb{C}}}, {\notag}\\ & a_{V,z}(u,v):= ((-\Delta+V)u,(-\Delta+V)v)_{L^2({\Omega};d^nx)} + \big(V^{1/2}u,V^{1/2}v\big)_{L^2({\Omega};d^nx)} \\ & \hskip 0.77in -z\, (\nabla u,\nabla v)_{(L^2({\Omega};d^nx))^n},\quad u,v\in H^2_0({\Omega}). {\notag}\end{aligned}$$ Then, since $f\in H^{-2}({\Omega})=\bigl(H^2_0({\Omega})\bigr)^*$, the well-posedness of will follow with the help of the Lax-Milgram lemma as soon as we show that is coercive. To this end, observe that via repeated integrations by parts $$\begin{aligned} \label{MM-20} \begin{split} a_{V,z}(u,u) &= \sum_{j,k=1}^n\int_{\Omega}d^nx\,\Big| \frac{\partial^2 u}{\partial x_j\partial x_k}\Big|^2 -z \sum_{j=1}^n\int_{\Omega}d^nx\, \Big|\frac{\partial u}{\partial x_j}\Big|^2 \\ & \quad +\int_{\Omega}d^nx\,\big|V^{1/2}u\bigr| +2 {\mathop\mathrm{Re}}\bigg(\int_{\Omega}d^nx\,\Delta u\,V{\overline}{u}\bigg), \quad u\in C^\infty_0({\Omega}). \end{split} \end{aligned}$$ We note that the last term is of the order $$\label{MM-20U} O\bigl(\|V\|_{L^\infty({\Omega};d^nx)}\|\Delta u\|_{L^2({\Omega};d^nx)} \|u\|_{L^2({\Omega};d^nx)}\bigr)$$ and hence, can be dominated by $$\label{MM-21U} C\|V\|_{L^\infty({\Omega};d^nx)}\big[\varepsilon\|u\|^2_{H^2({\Omega})} +(4\varepsilon)^{-1}\|u\|^2_{L^2({\Omega};d^nx)}\big],$$ for every $\varepsilon>0$. Thus, based on this and Poincaré’s inequality, we eventually obtain, by taking $\varepsilon>0$ sufficiently small, and $M$ (introduced in the beginning of the proof) sufficiently large, that $$\label{MM-21} {\mathop\mathrm{Re}}(a_{V,z}(u,u)) \geq C\|u\|^2_{H^2({\Omega})},\quad u\in C^\infty_0({\Omega}).$$ Hence, $$\label{MM-22} {\mathop\mathrm{Re}}(a_{V,z}(u,u)) \geq C\|u\|^2_{H^2({\Omega})},\quad u\in H^2_0({\Omega}),$$ by the density of $C^\infty_0({\Omega})$ in $H^2_0({\Omega})$. Thus, the form is coercive and hence, the problem is well-posed whenever $z\in{{\mathbb{C}}}$ has ${\mathop\mathrm{Re}}(z)\leq -M$. We now wish to extend this type of conclusion to a larger set of $z$’s. With this in mind, set $$\label{Mi-1} A_{V,z}:=(-\Delta+V)(-\Delta+V-z I_{{\Omega}})\in {{\mathcal B}}\bigl(H^2_0({\Omega}),H^{-2}({\Omega})\bigr),\quad z\in{{\mathbb{C}}}.$$ The well-posedness of is equivalent to the fact that the above operator is invertible. In this vein, we note that if we fix $z_0 \in{{\mathbb{C}}}$ with ${\mathop\mathrm{Re}}(z_0 )\leq -M$, then, from what we have shown so far, $$\label{Mi-2} A_{V,z_0 }^{-1}\in{{\mathcal B}}\bigl(H^{-2}({\Omega}),H^2_0({\Omega})\bigr)$$ is a well-defined operator. For an arbitrary $z\in{{\mathbb{C}}}$ we then write $$\label{Mi-3} A_{V,z}=A_{V,z_0 }[I_{H^2_0({\Omega})}+B_{V,z}],$$ where $I_{H^2_0({\Omega})}$ is the identity operator on $H^2_0({\Omega})$ and we have set $$\label{Mi-4} B_{V,z}:=A_{V,z_0 }^{-1}(A_{V,z}-A_{V,z_0 }) =(z_0 -z)A_{V,z_0 }^{-1}(-\Delta+V) \in{{\mathcal B}}_\infty\bigl(H^2_0({\Omega})\bigr).$$ Since ${{\mathbb{C}}}\ni z\mapsto B_{V,z}\in{{\mathcal B}}\bigl(H^2_0({\Omega})\bigr)$ is an analytic, compact operator-valued mapping, which vanishes for $z=z_0 $, the Analytic Fredholm Theorem yields the existence of an exceptional, discrete set $\Lambda_{{\Omega}}\subset{{\mathbb{C}}}$, without any finite accumulation points such that $$\label{Mi-5} (I_{H^2_0({\Omega})}+B_{V,z})^{-1}\in{{\mathcal B}}\bigl(H^2_0({\Omega})\bigr),\quad z\in{{\mathbb{C}}}\backslash \Lambda_{{\Omega}}.$$ As a consequence of this, , and , we therefore have $$\label{Mi-6} A_{V,z}^{-1}\in{{\mathcal B}}\bigl(H^{-2}({\Omega}),H^2_0({\Omega})\bigr),\quad z\in{{\mathbb{C}}}\backslash \Lambda_{{\Omega}}.$$ We now proceed to show that, in fact, $\Lambda_{{\Omega}}\subset(0,\infty)$. To justify this inclusion, we observe that $$\label{Mi-6X} \text{$A_{V,z}$ in \eqref{Mi-1} is a Fredholm operator, with Fredholm index zero, for every $z\in{{\mathbb{C}}}$},$$ due to , , and . Thus, if for some $z\in{{\mathbb{C}}}$ the operator $A_{V,z}$ fails to be invertible, then there exists $0\not=u\in L^2({\Omega};d^nx)$ such that $A_{V,z}u=0$. In view of and Lemma \[L-MM-1\], the latter condition forces $z\in(0,\infty)$. Thus, $\Lambda_{{\Omega}}$ consists of positive numbers. At this stage, it remains to justify the very last claim in the statement of the lemma. This, however, readily follows from , completing the proof. \[T-MM-2\] Assume Hypothesis \[h.VK\] and recall the exceptional set $\Lambda_{{\Omega}}\subset(0,\infty)$ from Lemma \[L-MM-2\], which is discrete with only accumulation point at infinity. Then $$\label{Mi-7} \sigma(H_{K,{\Omega}})=\Lambda_{{\Omega}}\cup\{0\}.$$ Furthermore, for every $0\not=z \in{{\mathbb{C}}}\backslash \Lambda_{{\Omega}}$, the action of the resolvent $(H_{K,{\Omega}}-z I_{{\Omega}})^{-1}$ on an arbitrary element $f\in L^2({\Omega};d^nx)$ can be described as follows: Let $v$ solve $$\label{MM-23} \begin{cases} v\in H^2_0(\Omega), \\ (-\Delta+V)(-\Delta+V-z)v=(-\Delta+V)f\in H^{-2}(\Omega), \end{cases}$$ and consider $$\label{MM-24} w:=z^{-1}[(- \Delta+V-z)v-f]\in L^2({\Omega};d^nx).$$ Then $$\label{MM-24X} (H_{K,{\Omega}}-z I_{{\Omega}})^{-1}f=v+w.$$ Finally, every $z \in \Lambda_{{\Omega}}\cup\{0\}$ is actually an eigenvalue $($of finite multiplicity, if nonzero$)$ for the perturbed Krein Laplacian, and the essential spectrum of this operator is given by $$\label{Mi-7S} \sigma_{ess}(H_{K,{\Omega}})=\{0\}.$$ Let $0\not=z \in{{\mathbb{C}}}\backslash \Lambda_{{\Omega}}$, fix $f\in L^2({\Omega};d^nx)$, and assume that $v,w$ are as in the statement of the theorem. That $v$ (hence also $w$) is well-defined follows from Lemma \[L-MM-2\]. Set $$\begin{aligned} \label{MM-26} u := v+w \in & H^2_0({\Omega})\dot{+} \big\{\eta\in L^2({\Omega};d^nx)\,\big|\,(-\Delta+V)\eta=0\mbox{ in }\Omega\big\} \nonumber\\ & \quad = \ker\big(\tau_{N,V,0}\big)\hookrightarrow \operatorname{dom}(H_{max,{\Omega}} ),\end{aligned}$$ by . Thus, $u\in \operatorname{dom}(H_{max,{\Omega}} )$ and $\tau_{N,V,0} u=0$ which force $u\in\operatorname{dom}(H_{K,{\Omega}})$. Furthermore, $$\label{Mi-8} \|u\|_{L^2({\Omega};d^nx)}+\|\Delta u\|_{L^2({\Omega};d^nx)} \leq C\|f\|_{L^2({\Omega};d^nx)},$$ for some $C=C({\Omega},V,z)>0$, and $$\begin{aligned} \label{MM-27} & (-\Delta+V-z)u = (-\Delta+V-z)v+(-\Delta+V-z)w \nonumber\\ & \quad = (-\Delta+V-z)v +z^{-1}(-\Delta+V-z)[(-\Delta+V-z)v-f] \nonumber\\ & \quad = (-\Delta+V-z)v+z^{-1}(-\Delta+V)[(-\Delta+V-z)v-f] -[(-\Delta+V-z)v-f] \nonumber\\ & \quad = f+z^{-1}[(-\Delta+V)(-\Delta+V-z)v-(-\Delta+V)f]=f,\end{aligned}$$ by , . As a consequence of this analysis, we may conclude that the operator $$\label{Mi-9} H_{K,{\Omega}}-z I_{{\Omega}}:\operatorname{dom}(H_{K,{\Omega}})\subset L^2({\Omega};d^nx) \to L^2({\Omega};d^nx)$$ is onto (with norm control), for every $z \in{{\mathbb{C}}}\backslash (\Lambda_{{\Omega}}\cup\{0\})$. When $z \in{{\mathbb{C}}}\backslash (\Lambda_{{\Omega}}\cup\{0\})$ the last part in Lemma \[L-MM-2\] together with Theorem \[T-MM-1\] also yield that the operator is injective. Together, these considerations prove that $$\label{Mi-7X} \sigma(H_{K,{\Omega}})\subseteq\Lambda_{{\Omega}}\cup\{0\}.$$ Since the converse inclusion also follows from the last part in Lemma \[L-MM-2\] together with Theorem \[T-MM-1\], equality follows. Formula , along with the final conclusion in the statement of the theorem, is also implicit in the above analysis plus the fact that $\ker(H_{K,{\Omega}})$ is infinite-dimensional (cf.  and [@MT00]). Eigenvalue Estimates for the Perturbed Krein Laplacian {#s11} ====================================================== The aim of this section is to study in greater detail the nature of the spectrum of the operator $H_{K,{\Omega}}$. We split the discussion into two separate cases, dealing with the situation when the potential $V$ is as in Hypothesis \[h.V\] (Subsection \[s11X\]), and when $V\equiv 0$ (Subsection \[s11Y\]). The Perturbed Case {#s11X} ------------------ Given a domain $\Omega$ as in Hypothesis \[h.Conv\] and a potential $V$ as in Hypothesis \[h.V\], we recall the exceptional set $\Lambda_{{\Omega}}\subset(0,\infty)$ associated with $\Omega$ as in Section \[s10\], consisting of numbers $$\label{mam-1} 0<\lambda_{K,{\Omega},1}\leq\lambda_{K,{\Omega},2}\leq\cdots\leq\lambda_{K,{\Omega},j} \leq\lambda_{K,{\Omega},j+1}\leq\cdots$$ converging to infinity. Above, we have displayed the $\lambda$’s according to their (geometric) multiplicity which equals the dimension of the kernel of the (Fredholm) operator . \[T-MAM-1\] Assume Hypothesis \[h.VK\]. Then there exists a family of functions $\{u_j\}_{j\in{{\mathbb{N}}}}$ with the following properties: $$\begin{aligned} \label{mam-2} & u_j\in H^2_0({\Omega})\, \mbox{ and }\, (-\Delta+V)^2u_j=\lambda_{K,{\Omega},j}(-\Delta+V)u_j, \quad j\in{{\mathbb{N}}}, \\ & ((-\Delta+V)u_j,(-\Delta+V)u_k)_{L^2({\Omega};d^nx)}=\delta_{j,k}, \quad j,k\in{{\mathbb{N}}}, \label{mam-3}\\ & u=\sum_{j=1}^\infty ((-\Delta+V)u,(-\Delta+V)u_j)_{L^2({\Omega};d^nx)}\,u_j, \quad u\in H^2_0({\Omega}), \label{mam-4}\end{aligned}$$ with convergence in $H^2({\Omega})$. Consider the vector space and inner product $$\label{mam-5} {{\mathcal H}}_V:=H^2_0({\Omega}),\quad [u,v]_{{{\mathcal H}}_V}:=\int_{{\Omega}}d^nx\,{\overline}{(-\Delta+V)u}\,(-\Delta+V)v, \quad u,v\in{{\mathcal H}}_V.$$ We claim that $\bigl({{\mathcal H}}_V,[{\,\cdot\,},{\,\cdot\,}]_{{{\mathcal H}}_V}\bigr)$ is a Hilbert space. This readily follows as soon as we show that $$\label{mam-6} \|u\|_{H^2({\Omega})}\leq C\|(-\Delta+V)u\|_{L^2({\Omega};d^nx)},\quad u\in H^2_0({\Omega}),$$ for some finite constant $C=C({\Omega},V)>0$. To justify this, observe that for every $u\in C^\infty_0({\Omega})$ we have $$\begin{aligned} \label{mam-7} \int_{\Omega}d^nx\,|u|^2 &\leq C \sum_{j=1}^n\int_{\Omega}d^nx\, \Big|\frac{\partial u}{\partial x_j}\Big|^2 \nonumber\\ &\leq C\sum_{j,k=1}^n\int_{\Omega}d^nx\,\Big| \frac{\partial^2 u}{\partial x_j\partial x_k}\Big|^2 =\int_{\Omega}d^nx\,|\Delta u|^2,\end{aligned}$$ where we have used Poincaré’s inequality in the first two steps. Based on this, the fact that $V$ is bounded, and the density of $C^\infty_0({\Omega})$ in $H^2_0({\Omega})$ we therefore have $$\label{mam-6Y} \|u\|_{H^2({\Omega})}\leq C\bigl(\|(-\Delta+V)u\|_{L^2({\Omega};d^nx)} +\|u\|_{L^2({\Omega};d^nx)}\bigr),\quad u\in H^2_0({\Omega}),$$ for some finite constant $C=C({\Omega},V)>0$. Hence, the operator $$\label{mam-6YY} -\Delta+V\in{{\mathcal B}}\bigl(H^2_0({\Omega}),L^2({\Omega};d^nx)\bigr)$$ is bounded from below modulo compact operators, since the embedding $H^2_0({\Omega})\hookrightarrow L^2({\Omega};d^nx)$ is compact. Hence, it follows that has closed range. Since this operator is also one-to-one (as $0\not\in\sigma(H_{D,{\Omega}})$), estimate follows from the Open Mapping Theorem. This shows that $$\label{mam-8} {{\mathcal H}}_V=H^2_0({\Omega})\, \mbox{ as Banach spaces, with equivalence of norms}.$$ Next, we recall from the proof of Lemma \[L-MM-2\] that the operator is invertible for $\lambda\in{{\mathbb{C}}}\backslash \Lambda_{{\Omega}}$ (cf. ), and that $\Lambda_{{\Omega}}\subset(0,\infty)$. Taking $\lambda=0$ this shows that $$\label{mam-9} (-\Delta+V)^{-2}:=((-\Delta+V)^2)^{-1}\in{{\mathcal B}}\bigl(H^{-2}({\Omega}),H^2_0({\Omega})\bigr)$$ is well-defined. Furthermore, this operator is self-adjoint (viewed as a linear, bounded operator mapping a Banach space into its dual, cf. ). Consider now $$\label{mam-10} B:=-(-\Delta+V)^{-2}(-\Delta+V).$$ Since $B$ admits the factorization $$\label{mam-11} B:H^2_0({\Omega})\stackrel{-\Delta+V} {-\!\!\!-\!\!\!-\!\!\!\longrightarrow} L^2({\Omega};d^nx)\stackrel{\iota}{\hookrightarrow} H^{-2}({\Omega})\stackrel{-(-\Delta+V)^{-2}} {-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!\longrightarrow}H^2_0({\Omega}),$$ where the middle arrow is a compact inclusion, it follows that $$\label{mam-12} B\in{{\mathcal B}}({{\mathcal H}}_V)\, \mbox{ is compact and injective}.$$ In addition, for every $u,v\in C^\infty_0({\Omega})$ we have via repeated integrations by parts $$\begin{aligned} \label{mam-13} [Bu,v]_{{{\mathcal H}}_V}&= - \big((-\Delta+V)(-\Delta+V)^{-2}(-\Delta+V)u, (-\Delta+V)v\big)_{L^2({\Omega};d^nx)} \nonumber\\ &= -\big((-\Delta+V)^{-2}(-\Delta+V)u,(-\Delta+V)^2v\big)_{L^2({\Omega};d^nx)} \nonumber\\ &= - \big((-\Delta+V)u,(-\Delta+V)^{-2}(-\Delta+V)^2v\big)_{L^2({\Omega};d^nx)} \nonumber\\ &= -((-\Delta+V)u,v)_{L^2({\Omega};d^nx)} \nonumber\\ &= -(\nabla u,\nabla v)_{(L^2({\Omega};d^nx))^n} - \big(V^{1/2}u, V^{1/2}v\big)_{L^2({\Omega};d^nx)}.\end{aligned}$$ Consequently, by symmetry, $[Bu,v]_{{{\mathcal H}}_V}={\overline}{[Bv,u]_{{{\mathcal H}}_V}}$, $u,v\in C^\infty_0({\Omega})$ and hence, $$\label{mam-14} [Bu,v]_{{{\mathcal H}}_V}={\overline}{[Bv,u]_{{{\mathcal H}}_V}} \quad u,v\in{{\mathcal H}}_V,$$ since $C^\infty_0({\Omega})\hookrightarrow{{\mathcal H}}_V$ densely. Thus, $$\label{mam-15} B\in{{\mathcal B}}_\infty ({{\mathcal H}}_V)\, \mbox{ is self-adjoint and injective}.$$ To continue, we recall the operator $A_{V,\lambda}$ from and observe that $$\label{mam-16} (-\Delta+V)^{-2}A_{V,z} =I_{{{\mathcal H}}_V}- z B, \quad z\in{{\mathbb{C}}},$$ as operators in ${{\mathcal B}}\bigl(H^2_0({\Omega})\bigr)$. Thus, the spectrum of $B$ consists (including multiplicities) precisely of the reciprocals of those numbers $z\in{{\mathbb{C}}}$ for which the operator $A_{V,z}\in{{\mathcal B}}\bigl(H^2_0({\Omega}),H^{-2}({\Omega})\bigr)$ fails to be invertible. In other words, the spectrum of $B\in{{\mathcal B}}({{\mathcal H}}_V)$ is given by $$\label{mam-17} \sigma(B)=\{(\lambda_{K,{\Omega},j})^{-1}\}_{j\in{{\mathbb{N}}}}.$$ Now, from the spectral theory of compact, self-adjoint (injective) operators on Hilbert spaces (cf., e.g., [@Mc00 Theorem 2.36]), it follows that there exists a family of functions $\{u_j\}_{j\in{{\mathbb{N}}}}$ for which $$\begin{aligned} \label{mam-18} & u_j\in{{\mathcal H}}_V\, \mbox{ and }\, Bu_j=(\lambda_{K,{\Omega},j})^{-1}u_j, \quad j\in{{\mathbb{N}}}, \\ & [u_j,u_k]_{{{\mathcal H}}_V}=\delta_{j,k}, \quad j,k\in{{\mathbb{N}}}, \label{mam-19}\\ & u=\sum_{j=1}^\infty[u,u_j]_{{{\mathcal H}}_V}\,u_j, \quad u\in{{\mathcal H}}_V, \label{mam-20}\end{aligned}$$ with convergence in ${{\mathcal H}}_V$. Unraveling notation, – then readily follow from –. We note that Lemma \[T-MAM-1\] gives the orthogonality of the eigenfunctions $u_j$ in terms of the inner product for ${{\mathcal H}}_V$ (cf.  and , or see immediately above). Here we remark that the given inner product for ${{\mathcal H}}_V$ does not correspond to the inner product that has traditionally been used in treating the buckling problem for a clamped plate, even after specializing to the case $V \equiv 0$. The traditional inner product in that case is the [*Dirichlet inner product*]{}, defined by $$D(u,v)=\int_\Omega d^n x \, (\nabla u, \nabla v)_{{{\mathbb{C}}}^n}, \quad u, v \in H^1_0({\Omega}),$$ where $(\cdot,\cdot)_{{{\mathbb{C}}}^n}$ denotes the usual inner product for elements of ${{\mathbb{C}}}^n$, conjugate linear in its first entry, linear in its second. When the potential $V {\geqslant}0$ is included, the appropriate generalization of $D(u,v)$ is the inner product $$D_V (u,v) = D(u,v) + \int_\Omega d^nx \, V {\overline}{u} \, v, \quad u, v \in H^1_0({\Omega})$$ (we recall that throughout this paper $V$ is assumed nonnegative, and hence that this inner product gives rise to a well-defined norm). Here we observe that orthogonality of the eigenfunctions of the buckling problem in the sense of ${{\mathcal H}}_V$ is entirely equivalent to their orthogonality in the sense of $D_V (\cdot,\cdot)$: Indeed, starting from the orthogonality in , integrating by parts, and using the eigenvalue equation , one has, for $j \ne k$, $$\begin{aligned} 0& = [u_j,u_k]_{{{\mathcal H}}_V}=\int_\Omega d^nx \, {\overline}{(-\Delta+V)u_j}\,(-\Delta+V)u_k = \int_\Omega d^nx \, {\overline}{u_j}\,(-\Delta+V)^2 u_k {\notag}\\ & = \lambda_k \int_\Omega d^nx \,{\overline}{u_j}\,(-\Delta+V) u_k =\lambda_k \bigg[D(u_j,u_k)+\int_\Omega d^nx \, V {\overline}{u_j} \, u_k\bigg] {\notag}\\ & = \lambda_k \, D_V(u_j,u_k), \quad u, v \in H^2_0({\Omega}),\end{aligned}$$ where $\lambda_k$ is shorthand for $\lambda_{K,\Omega,k}$ of , the eigenvalue corresponding to the eigenfunction $u_k$ (cf. , which exhibits the eigenvalue equation for the eigenpair $(u_j,\lambda_j)$). Since all the $\lambda_j$’s considered here are positive (see ), this shows that the family of eigenfunctions $\{u_j\}_{j \in {{\mathbb{N}}}}$, orthogonal with respect to $[\cdot,\cdot]_{{{\mathcal H}}_V}$, is also orthogonal with respect to the “generalized Dirichlet inner product", $D_V (\cdot,\cdot)$. Clearly, this argument can also be reversed (since all eigenvalues are positive), and one sees that a family of eigenfunctions of the generalized buckling problem orthogonal in the sense of the Dirichlet inner product $D_V (\cdot,\cdot)$ is also orthogonal with respect to the inner product for ${{\mathcal H}}_V$, that is, with respect to $[\cdot,\cdot]_{{{\mathcal H}}_V}$. On the other hand, it should be mentioned that the normalization of each of the $u_k$’s changes if one passes from one of these inner products to the other, due to the factor of $\lambda_k$ encountered above (specifically, one has $[u_k,u_k]_{{{\mathcal H}}_V}=\lambda_k \, D_V (u_k,u_k)$ for each $k$). Next, we recall the following result (which provides a slight variation of the case $V\equiv 0$ treated in [@GM10]). \[th-CL\] Assume Hypothesis \[h.VK\]. Then the subspace $(-\Delta+V)\,H^2_0({\Omega})$ is closed in $L^2(\Omega;d^nx)$ and $$\label{Man-2} L^2(\Omega;d^nx)=\ker(H_{V,\max,{\Omega}}) \oplus \big[(-\Delta+V)\,H^2_0({\Omega})\big],$$ as an orthogonal direct sum. Our next theorem shows that there exists a countable family of orthonormal eigenfunctions for the perturbed Krein Laplacian which span the orthogonal complement of the kernel of this operator: \[TH-Mq1\] Assume Hypothesis \[h.VK\]. Then there exists a family of functions $\{w_j\}_{j\in{{\mathbb{N}}}}$ with the following properties: $$\begin{aligned} \label{mam-21} & w_j\in\operatorname{dom}(H_{K,{\Omega}})\cap H^{1/2}({\Omega}) \, \mbox{ and }\, H_{K,{\Omega}}w_j=\lambda_{K,{\Omega},j} w_j, \;\; \lambda_{K,{\Omega},j}>0, \; j\in{{\mathbb{N}}}, \\ & (w_j,w_k)_{L^2({\Omega};d^nx)}=\delta_{j,k}, \; j,k\in{{\mathbb{N}}}, \label{mam-22}\\ & L^2(\Omega;d^nx)=\ker(H_{K,{\Omega}})\,\oplus\, {\overline}{{\rm lin. \, span} \{w_j\}_{j\in{{\mathbb{N}}}}} \;\, \text{ $($orthogonal direct sum$)$.} \label{mam-23}\end{aligned}$$ That $w_j \in H^{1/2}({\Omega})$, $j\in{{\mathbb{N}}}$, follows from Proposition \[pHKv\]$(i)$. The rest is a direct consequence of Lemma \[th-CL\], the fact that $$\label{mam-24} \ker(H_{V,\max,{\Omega}})=\big\{u\in L^2(\Omega;d^nx)\,\big|\,(-\Delta+V)u=0\big\} =\ker(H_{K,{\Omega}}),$$ the second part of Theorem \[T-MM-1\], and Lemma \[T-MAM-1\] in which we set $w_j:=(-\Delta+V)u_j$, $j\in{{\mathbb{N}}}$. Next, we define the following Rayleigh quotient $$\label{mam-25} R_{K,{\Omega}}[u]:=\frac{\|(-\Delta+V)u\|^2_{L^2({\Omega};d^nx)}} {\|\nabla u\|^2_{(L^2({\Omega};d^nx))^n}+\|V^{1/2}u\|^2_{L^2({\Omega};d^nx)}}, \quad 0\not=u\in H^2_0({\Omega}).$$ Then the following min-max principle holds: \[TH-Mq2\] Assume Hypothesis \[h.VK\]. Then $$\label{mam-26} \lambda_{K,{\Omega},j} =\min_{\stackrel{W_j\text{ subspace of }H^2_0({\Omega})}{\dim (W_j)=j}} \Big(\max_{0\not=u\in W_j}R_{K,{\Omega}}[u]\Big),\quad j\in{{\mathbb{N}}}.$$ As a consequence, given two domains $\Omega$, $\widetilde{{\Omega}}$ as in Hypothesis \[h.Conv\] for which $\Omega\subseteq\widetilde{{\Omega}}$, and given a potential $0\leq \widetilde{V}\in L^\infty(\widetilde{{\Omega}})$, one has $$\label{mam-2S} 0 < {\widetilde}\lambda_{K,\widetilde{{\Omega}},j} \leq \lambda_{K,{\Omega},j}, \quad j\in{{\mathbb{N}}},$$ where $V:=\widetilde{V}|_{{\Omega}}$, and $\lambda_{K,{\Omega},j}$ and ${\widetilde}\lambda_{K,\widetilde{{\Omega}},j}$, $j\in{{\mathbb{N}}}$, are the eigenvalues corresponding to the Krein–von Neumann extensions associated with ${\Omega}, V$ and ${\widetilde}{\Omega}, {\widetilde}V$, respectively. Obviously, is a consequence of , so we will concentrate on the latter. We recall the Hilbert space ${{\mathcal H}}_V$ from and the orthogonal family $\{u_j\}_{j\in{{\mathbb{N}}}}$ in –. Next, consider the following subspaces of ${{\mathcal H}}_V$, $$\label{mam-27} V_0:=\{0\},\quad V_j:={\rm lin. \, span} \{u_i\,|\,1\leq i\leq j\},\quad j\in{{\mathbb{N}}}.$$ Finally, set $$\label{mam-28} V_j^{\bot} :=\{u\in{{\mathcal H}}\,|\,[u,u_i]_{{{\mathcal H}}_V}=0,\,1\leq i\leq j\},\quad j\in{{\mathbb{N}}}.$$ We claim that $$\label{mam-29} \lambda_{K,{\Omega},j}=\min_{0\not=u\in V^{\bot}_{j-1}}R_{K,{\Omega}}[u] =R_{K,{\Omega}}[u_j],\quad j\in{{\mathbb{N}}}.$$ Indeed, if $j\in{{\mathbb{N}}}$ and $u=\sum_{k=1}^\infty c_k u_k \in V^{\bot}_{j-1}$, then $c_k = 0$ whenever $1\leq k \leq j-1$. Consequently, $$\label{mam-30} \|(-\Delta+V)u\|^2_{L^2({\Omega};d^nx)} =\biggl\|\sum_{k=j}^\infty c_k (-\Delta+V)u_k \biggr\|^2_{L^2({\Omega};d^nx)} =\sum_{k=j}^\infty |c_k|^2$$ by , so that $$\begin{aligned} \label{mam-31} & \|\nabla u\|^2_{(L^2({\Omega};d^nx))^n}+\|V^{1/2}u\|^2_{L^2({\Omega};d^nx)} =((-\Delta+V)u,u)_{L^2({\Omega};d^nx)} \nonumber\\ &\quad\quad =\bigg(\sum_{k=j}^\infty c_k (-\Delta+V)u_k, u\bigg)_{L^2({\Omega};d^nx)} {\notag}\\ & \qquad =\bigg(\sum_{k=j}^\infty (\lambda_{K,{\Omega},k})^{-1}c_k (-\Delta+V)^2 u_k, u\bigg)_{L^2({\Omega};d^nx)} \nonumber\\ &\quad\quad =\bigg(\sum_{k=j}^\infty (\lambda_{K,{\Omega},k})^{-1}c_k (-\Delta+V)u_k, (-\Delta+V)u\bigg)_{L^2({\Omega};d^nx)} \nonumber\\ &\quad\quad =\bigg(\sum_{k=j}^\infty (\lambda_{K,{\Omega},k})^{-1}c_k (-\Delta+V)u_k, \sum_{k=j}^\infty c_k (-\Delta+V)u_k \bigg)_{L^2({\Omega};d^nx)} \nonumber\\ &\quad\quad =\sum_{k=j}^\infty(\lambda_{K,{\Omega},k})^{-1}|c_k|^2 \leq (\lambda_{K,{\Omega},j})^{-1}\sum_{k=j}^\infty|c_k|^2 \nonumber\\ &\quad\quad =(\lambda_{K,{\Omega},j})^{-1}\|(-\Delta+V)u\|^2_{L^2({\Omega};d^nx)},\end{aligned}$$ where in the third step we have relied on , and the last step is based on . Thus, $R_{K,{\Omega}}[u]\geq\lambda_{K,{\Omega},j}$ with equality if $u=u_j$ (cf. the calculation leading up to ). This proves . In fact, the same type of argument as the one just performed also shows that $$\label{mam-32} \lambda_{K,{\Omega},j}=\max_{0\not=u\in V_j}R_{K,{\Omega}}[u]=R_{K,{\Omega}}[u_j], \quad j\in{{\mathbb{N}}}.$$ Next, we claim that if $W_j$ is an arbitrary subspace of ${{\mathcal H}}$ of dimension $j$ then $$\label{mam-33} \lambda_{K,{\Omega},j}\leq\max_{0\not=u\in W_j}R_{K,{\Omega}}[u],\quad j\in{{\mathbb{N}}}.$$ To justify this inequality, observe that $W_j\cap V^{\bot}_{j-1}\not=\{0\}$ by dimensional considerations. Hence, if $0\not=v_j\in W_j\cap V^{\bot}_{j-1}$ then $$\label{mam-34} \lambda_{K,{\Omega},j}=\min_{0\not =u\in V^{\bot}_{j-1}}R_{K,{\Omega}}[u]\leq R_{K,{\Omega}}[v_j] \leq \max_{0\not=u\in W_j}R_{K,{\Omega}}[u],$$ establishing . Now formula readily follows from this and . If $\Omega\subset{\mathbb{R}}^n$ is a bounded Lipschitz domain denote by $$\label{mam-35} 0<\lambda_{D,{\Omega},1}\leq\lambda_{D,{\Omega},2}\leq\cdots\leq\lambda_{D,{\Omega},j} \leq\lambda_{D,{\Omega},j+1}\leq\cdots$$ the collection of eigenvalues for the perturbed Dirichlet Laplacian $H_{D,{\Omega}}$ (again, listed according to their multiplicity). Then, if $0\leq V\in L^\infty({\Omega};d^nx)$, we have the well-known formula (cf., e.g., [@DL90] for the case where $V\equiv 0$) $$\label{mam-37} \lambda_{D,{\Omega},j} =\min_{\stackrel{W_j\text{ subspace of }H^1_0({\Omega})}{\dim (W_j)=j}} \Big(\max_{0\not=u\in W_j}R_{D,{\Omega}}[u]\Big),\quad j\in{{\mathbb{N}}},$$ where $R_{D,{\Omega}}[u]$, the Rayleigh quotient for the perturbed Dirichlet Laplacian, is given by $$\label{mam-38} R_{D,{\Omega}}[u]:=\frac{\|\nabla u\|^2_{(L^2({\Omega};d^nx))^n} +\|V^{1/2}u\|^2_{L^2({\Omega};d^nx)}}{\|u\|^2_{L^2({\Omega};d^nx)}}, \quad 0\not=u\in H^1_0({\Omega}).$$ From Theorem \[AS-thK\], Theorem \[t2.5\], and Proposition \[L-Fri1\], we already know that, granted Hypothesis \[h.VK\], the nonzero eigenvalues of the perturbed Krein Laplacian are at least as large as the corresponding eigenvalues of the perturbed Dirichlet Laplacian. It is nonetheless of interest to provide a direct, analytical proof of this result. We do so in the proposition below. \[TH-Mq3\] Assume Hypothesis \[h.VK\]. Then $$\label{mam-39} 0 < \lambda_{D,{\Omega},j}\leq\lambda_{K,{\Omega},j},\quad j\in{{\mathbb{N}}}.$$ By the density of $C^\infty_0({\Omega})$ into $H^2_0({\Omega})$ and $H^1_0({\Omega})$, respectively, we obtain from and that $$\begin{aligned} \label{mam-40} & \lambda_{K,{\Omega},j} =\inf_{\stackrel{W_j\text{ subspace of }C^\infty_0({\Omega})}{\dim(W_j)=j}} \Big(\sup_{0\not=u\in W_j}R_{K,{\Omega}}[u]\Big), \\ & \lambda_{D,{\Omega},j} =\inf_{\stackrel{W_j\text{ subspace of }C^\infty_0({\Omega})}{\dim (W_j)=j}} \Big(\sup_{0\not=u\in W_j}R_{D,{\Omega}}[u]\Big), \label{mam-41}\end{aligned}$$ for every $j\in{{\mathbb{N}}}$. Since, if $u\in C^\infty_0({\Omega})$, $$\begin{aligned} \label{mam-42} & \|\nabla u\|^2_{(L^2({\Omega};d^nx))^n}+\|V^{1/2}u\|^2_{L^2({\Omega};d^nx)} = ((-\Delta+V)u,u)_{L^2({\Omega};d^nx)} \nonumber\\[4pt] & \quad \leq \|(-\Delta+V)u\|_{L^2({\Omega};d^nx)}\|u\|_{L^2({\Omega};d^nx)},\end{aligned}$$ we deduce that $$\label{mam-43} R_{D,{\Omega}}[u]\leq R_{K,{\Omega}}[u], \, \mbox{ whenever }\, 0\not=u\in C^\infty_0({\Omega}).$$ With this at hand, follows from –. \[RRR-em\] Another analytical approach to which highlights the connection between the perturbed Krein Laplacian and a fourth-order boundary problem is as follows. Granted Hypotheses \[h2.1\] and \[h.V\], and given $\lambda\in{{\mathbb{C}}}$, consider the following eigenvalue problem $$\label{MM-1H} \begin{cases} u\in\operatorname{dom}(-\Delta_{max,{\Omega}}),\quad (-\Delta+V)u\in\operatorname{dom}(-\Delta_{max,{\Omega}}), \\ (-\Delta+V)^2u=\lambda\,(-\Delta+V)u \,\mbox{ in } \,\Omega, \\ \widehat{\gamma}_D(u)=0 \,\mbox{ in } \,\Bigl(N^{1/2}({{\partial\Omega}})\Bigr)^*, \\ \widehat{\gamma}_D((-\Delta+V)u)=0 \,\mbox{ in } \,\Bigl(N^{1/2}({{\partial\Omega}})\Bigr)^*. \end{cases}$$ Associated with it is the sesquilinear form $$\label{MM-19X} \begin{cases} \widetilde{a}_{V,\lambda}({\,\cdot\,},{\,\cdot\,}):\widetilde{{{\mathcal H}}}\times\widetilde{{{\mathcal H}}} \longrightarrow{{\mathbb{C}}},\quad\widetilde{{{\mathcal H}}}:=H^2({\Omega})\cap H^1_0({\Omega}), \\ \widetilde{a}_{V,\lambda}(u,v) :=((-\Delta+V)u,(-\Delta+V)v)_{L^2({\Omega};d^nx)} +\big( V^{1/2}u,V^{1/2}v\big)_{L^2({\Omega};d^nx)} \\ \hskip 0.77in -\lambda\,(\nabla u,\nabla v)_{(L^2({\Omega};d^nx))^n},\quad u,v\in\widetilde{{{\mathcal H}}}, \end{cases}$$ which has the property that $$\label{MM-19Y} u\in\widetilde{{{\mathcal H}}} \,\mbox{ satisfies }\,\widetilde{a}_{V,\lambda}(u,v)=0 \,\mbox{ for every } \,v\in \widetilde{{{\mathcal H}}} \, \text{ if and only if } \, \mbox{$u$ solves \eqref{MM-1H}}.$$ We note that since the operator $-\Delta+V:H^2({\Omega})\cap H^1_0({\Omega})\to L^2({\Omega};d^nx)$ is an isomorphism, it follows that $u\mapsto \|(-\Delta+V)u\|_{L^2({\Omega};d^nx)}$ is an equivalent norm on the Banach space $\widetilde{{{\mathcal H}}}$, and the form $\widetilde{a}_{V,\lambda}({\,\cdot\,},{\,\cdot\,})$ is coercive if $\lambda<-M$, where $M=M({\Omega},V)>0$ is a sufficiently large constant. Based on this and proceeding as in Section \[s10\], it can then be shown that the problem has nontrivial solutions if and only if $\lambda$ belongs to an exceptional set $\widetilde{\Lambda}_{{\Omega},V}\subset(0,\infty)$ which is discrete and only accumulates at infinity. Furthermore, $u$ solves if and only if $v:=(-\Delta+V)u$ is an eigenfunction for $H_{D,{\Omega}}$, corresponding to the eigenvalue $\lambda$ and, conversely, if $u$ is an eigenfunction for $H_{D,{\Omega}}$ corresponding to the eigenvalue $\lambda$, then $u$ solves . Consequently, the problem is spectrally equivalent to $H_{D,{\Omega}}$. From this, it follows that the eigenvalues $\{\lambda_{D,{\Omega},j}\}_{j\in{{\mathbb{N}}}}$ of $H_{D,{\Omega}}$ can be expressed as $$\label{mam-26X} \lambda_{D,{\Omega},j} =\min_{\stackrel{W_j\text{ subspace of }\widetilde{{{\mathcal H}}}}{\dim (W_j)=j}} \Big(\max_{0\not=u\in W_j}R_{K,{\Omega}}[u]\Big),\quad j\in{{\mathbb{N}}},$$ where the Rayleigh quotient $R_{K,{\Omega}}[u]$ is as in . The upshot of this representation is that it immediately yields , on account of and the fact that $H^2_0({\Omega})\subset\widetilde{{{\mathcal H}}}$. Next, let $\Omega$ be as in Hypothesis \[h2.1\] and $0\leq V\in L^\infty({\Omega};d^nx)$. For $\lambda\in{{\mathbb{R}}}$ set $$\label{mam-44} N_{X,{\Omega}}(\lambda) :=\#\{j\in{{\mathbb{N}}}\,|\,\lambda_{X,{\Omega},j}\leq\lambda\},\quad X\in\{D,K\},$$ where $\#S$ denotes the cardinality of the set $S$. \[TH-Mq4\] Assume Hypothesis \[h.VK\]. Then $$\label{mam-45} N_{K,{\Omega}}(\lambda)\leq N_{D,{\Omega}}(\lambda),\quad \lambda\in{{\mathbb{R}}}.$$ In particular, $$\label{mam-46} N_{K,{\Omega}}(\lambda)=O(\lambda^{n/2})\, \mbox{ as }\, \lambda\to\infty.$$ Estimate is a trivial consequence of , whereas follows from and Weyl’s asymptotic formula for the Dirichlet Laplacian in a Lipschitz domain (cf. [@BS70] and the references therein for very general results of this nature). The Unperturbed Case {#s11Y} -------------------- What we have proved in Section \[s10\] and Section \[s11X\] shows that all known eigenvalue estimates for the (standard) buckling problem $$\label{MM-1F} u\in H^2_0({\Omega}),\quad \Delta^2 u=-\lambda\,\Delta u \,\mbox{ in } \,\Omega,$$ valid in the class of domains described in Hypothesis \[h.Conv\], automatically hold, in the same format, for the Krein Laplacian (corresponding to $V\equiv 0$). For example, we have the following result with $\lambda^{(0)}_{K,{\Omega},j}$, $j\in{{\mathbb{N}}}$, denoting the nonzero eigenvalues of the Krein Laplacian $-\Delta_{K,{\Omega}}$ and $\lambda^{(0)}_{D,{\Omega},j}$, $j\in{{\mathbb{N}}}$, denoting the eigenvalues of the Dirichlet Laplacian $-\Delta_{D,{\Omega}}$: \[TRm-1\] If $\Omega\subset{{\mathbb{R}}}^n$ is as in Hypothesis \[h.Conv\], the nonzero eigenvalues of the Krein Laplacian $-\Delta_{K,{\Omega}}$ satisfy $$\begin{aligned} \label{mx-1} & \lambda^{(0)}_{K,{\Omega},2}\leq\frac{n^2+8n+20}{(n+2)^2}\lambda^{(0)}_{K,{\Omega},1}, \\ & \sum_{j=1}^n\lambda^{(0)}_{K,{\Omega},j+1}< (n+4)\lambda^{(0)}_{K,{\Omega},1} -\frac{4}{n+4}(\lambda^{(0)}_{K,{\Omega},2}-\lambda^{(0)}_{K,{\Omega},1}) {\leqslant}(n+4)\lambda^{(0)}_{K,{\Omega},1}, \label{mx-2} \\ & \sum_{j=1}^k \big(\lambda^{(0)}_{K,{\Omega},k+1}-\lambda^{(0)}_{K,{\Omega},j}\big)^2 \leq\frac{4(n+2)}{n^2} \sum_{j=1}^k \big(\lambda^{(0)}_{K,{\Omega},k+1}-\lambda_{K,0,j}\big) \lambda^{(0)}_{K,{\Omega},j}, \quad k\in{{\mathbb{N}}}, \label{mx-3}\end{aligned}$$ Furthermore, if $j_{(n-2)/2,1}$ is the first positive zero of the Bessel function of first kind and order $(n-2)/2$ $($cf. [@AS72 Sect. 9.5]$)$, $v_n$ denotes the volume of the unit ball in ${{\mathbb{R}}}^n$, and $|\Omega|$ stands for the $n$-dimensional Euclidean volume of $\Omega$, then $$\label{mx-4} \frac{2^{2/n}j_{(n-2)/2,1}^2v_n^{2/n}}{|\Omega|^{2/n}} < \lambda^{(0)}_{D,{\Omega},2} \leq \lambda^{(0)}_{K,{\Omega},1}.$$ With the eigenvalues of the buckling plate problem replacing the corresponding eigenvalues of the Krein Laplacian, estimates – have been proved in [@As99], [@As04], [@As09], [@CY06], and [@HY84] (indeed, further strengthenings of are detailed in [@As04], [@As09]), whereas the respective parts of are covered by results in [@Kra26] and [@Pa55] (see also [@AL96], [@BP63]). \[Rm-1\] Given the physical interpretation of the first eigenvalue for , it follows that $\lambda^{(0)}_{K,{\Omega},1}$, the first nonzero eigenvalue for the Krein Laplacian $-\Delta_{K,{\Omega}}$, is proportional to the load compression at which the plate $\Omega$ (assumed to be as in Hypothesis \[h.Conv\]) buckles. In this connection, it is worth remembering the long-standing conjecture of Pólya–Szeg[ő]{}, to the effect that amongst all plates of a given area, the circular one will buckle first (assuming all relevant physical parameters being equal). In [@AL96], the authors have given a partial result in this direction which, in terms of the first eigenvalue $\lambda^{(0)}_{K,{\Omega},1}$ for the Krein Laplacian $-\Delta_{K,{\Omega}}$ in a domain ${\Omega}$ as in Hypothesis \[h.Conv\], reads $$\label{A-L.1} \lambda^{(0)}_{K,{\Omega},1}>\frac{2^{2/n}j_{(n-2)/2,1}^2v_n^{2/n}}{|\Omega|^{2/n}} =c_n\lambda^{(0)}_{K,{\Omega}^{\#},1}$$ where ${\Omega}^{\#}$ is the $n$-dimensional ball with the same volume as ${\Omega}$, and $$\label{A-L.2} c_n= 2^{2/n}[j_{(n-2)/2,1}/j_{n/2,1}]^2 =1-(4-{\rm log}\,4)/n+O(n^{-5/3})\to 1 \,\mbox{ as }\, n\to\infty.$$ This result implies an earlier inequality of Bramble and Payne [@BP63] for the two-dimensional case, which reads $$\label{A-L.3} \lambda^{(0)}_{K,{\Omega},1}>\frac{2\pi j_{0,1}^2}{{\rm Area}\,(\Omega)}.$$ Before stating an interesting universal inequality concerning the ratio of the first (nonzero) Dirichlet and Krein Laplacian eigenvalues for a bounded domain with boundary of nonnegative Gaussian mean curvature (which includes, obviously, the case of a bounded convex domain), we recall a well-known result due to Babu[š]{}ka and Výborný [@BV65] concerning domain continuity of Dirichlet eigenvalues (see also [@BL08], [@BLL08], [@Da03], [@Fu99], [@St95], [@We84], and the literature cited therein): \[tDirichletapprox\] Let ${\Omega}\subset {{\mathbb{R}}}^n$ be open and bounded, and suppose that ${\Omega}_m\subset {\Omega}$, $m\in{{\mathbb{N}}}$, are open and monotone increasing toward ${\Omega}$, that is, $${\Omega}_m \subset {\Omega}_{m+1} \subset {\Omega}, \; m\in{{\mathbb{N}}}, \quad \bigcup_{m\in{{\mathbb{N}}}} {\Omega}_m = {\Omega}.$$ In addition, let $-\Delta_{D,{\Omega}_m}$ and $-\Delta_{D,{\Omega}}$ be the Dirichlet Laplacians in $L^2({\Omega}_m;d^n x)$ and $L^2({\Omega};d^n x)$ $($cf. , $)$, and denote their respective spectra by $$\sigma(-\Delta_{D,{\Omega}_m}) = \big\{\lambda^{(0)}_{D,{\Omega}_m,j}\big\}_{j\in{{\mathbb{N}}}}, \; m\in{{\mathbb{N}}}, \, \text{ and } \, \sigma(-\Delta_{D,{\Omega}}) = \big\{\lambda^{(0)}_{D,{\Omega},j}\big\}_{j\in{{\mathbb{N}}}}.$$ Then, for each $j\in{{\mathbb{N}}}$, $$\label{A-P.1a} \lim_{m\to\infty} \lambda^{(0)}_{D,{\Omega}_m,j} = \lambda^{(0)}_{D,{\Omega},j}.$$ \[T-Pay-1\] Assume that ${\Omega}\subset{{\mathbb{R}}}^n$ is a bounded quasi-convex domain. In addition, assume there exists a sequence of $C^\infty$-smooth domains $\{{\Omega}_m\}_{m\in{{\mathbb{N}}}}$ satisfying the following two conditions: 1. The sequence $\{{\Omega}_m\}_{m\in{{\mathbb{N}}}}$ monotonically converges to ${\Omega}$ from inside, that is, $$\label{A-P.2} {\Omega}_m \subset {\Omega}_{m+1}\subset {\Omega}, \; m\in{{\mathbb{N}}}, \quad \bigcup_{m\in{{\mathbb{N}}}} {\Omega}_m = {\Omega}.$$ 2. If ${{\mathcal G}}_m$ denotes the Gaussian mean curvature of $\partial{\Omega}_m$, then $${{\mathcal G}}_m\geq 0 \, \text{ for all $m\in{{\mathbb{N}}}$.} {\label}{A-P.2a}$$ Then the first Dirichlet eigenvalue and the first nonzero eigenvalue for the Krein Laplacian in ${\Omega}$ satisfy $$\label{A-P.1} 1\leq\frac{\lambda^{(0)}_{K,{\Omega},1}}{\lambda^{(0)}_{D,{\Omega},1}}\leq 4.$$ In particular, each bounded convex domain ${\Omega}\subset{{\mathbb{R}}}^n$ satisfies conditions $(i)$ and $(ii)$ and hence holds for such domains. Of course, the lower bound in is contained in , so we will concentrate on establishing the upper bound. To this end, we recall that it is possible to approximate $\Omega$ with a sequence of $C^\infty$-smooth bounded domains satisfying and . By Theorem \[tDirichletapprox\], the Dirichlet eigenvalues are continuous under the domain perturbations described in and one obtains, in particular, $$\label{A-P.3} \lim_{m\to\infty}\lambda^{(0)}_{D,{\Omega}_{m},1} = \lambda^{(0)}_{D,{\Omega},1}.$$ On the other hand, yields that $\lambda^{(0)}_{K,{\Omega},1}\leq \lambda^{(0)}_{K,{\Omega}_m,1}$. Together with , this shows that it suffices to prove that $$\label{A-P.4} \lambda^{(0)}_{K,{\Omega}_m,1}\leq 4 \lambda^{(0)}_{D,{\Omega}_m,1},\quad m\in{{\mathbb{N}}}.$$ Summarizing, it suffices to show that $$\begin{aligned} \label{A-P.5} \begin{split} & \mbox{${\Omega}\subset{{\mathbb{R}}}^n$ a bounded, $C^\infty$-smooth domain, whose Gaussian mean} \\ & \quad \text{curvature ${{\mathcal G}}$ of $\partial{\Omega}$ is nonnegative, implies }\, \lambda^{(0)}_{K,{\Omega},1} \leq 4\,\lambda^{(0)}_{D,{\Omega},1}. \end{split}\end{aligned}$$ Thus, we fix a bounded, $C^\infty$ domain ${\Omega}\subset{{\mathbb{R}}}^n$ with ${{\mathcal G}}\geq 0$ on $\partial{\Omega}$ and denote by $u_1$ the (unique, up to normalization) first eigenfunction for the Dirichlet Laplacian in ${\Omega}$. In the sequel, we abbreviate $\lambda_D:=\lambda^{(0)}_{D,{\Omega},1}$ and $\lambda_K:=\lambda^{(0)}_{K,{\Omega},1}$. Then (cf. [@GT83 Theorems 8.13 and 8.38]), $$\label{A-P.6} u_1\in C^\infty({\overline}{{\Omega}}),\quad u_1|_{{{\partial\Omega}}}=0,\quad u_1>0\mbox{ in }{\Omega},\quad -\Delta u_1=\lambda_D\,u_1 \, \mbox{ in } \, {\Omega},$$ and $$\label{A-P.7} \lambda_D=\frac{\int_{{\Omega}}d^nx\,|\nabla u_1|^2}{\int_{{\Omega}}d^nx\,|u_1|^2}.$$ In addition, (with $j=1$) and $u_1^2$ as a “trial function” yields $$\label{A-P.8} \lambda_K \leq\frac{\int_{{\Omega}}d^nx\,|\Delta(u_1^2)|^2} {\int_{{\Omega}}d^nx\,|\nabla(u_1^2)|^2}.$$ Then follows as soon as one shows that the right-hand side of is less than or equal to the quadruple of the right-hand side of . For bounded, smooth, convex domains in the plane (i.e., for $n=2$), such an estimate was established in [@Pa60]. For the convenience of the reader, below we review Payne’s ingenious proof, primarily to make sure that it continues to hold in much the same format for our more general class of domains and in all space dimensions (in the process, we also shed more light on some less explicit steps in Payne’s original proof, including the realization that the key hypothesis is not convexity of the domain, but rather nonnegativity of the Gaussian mean curvature ${{\mathcal G}}$ of its boundary). To get started, we expand $$\label{A-P.9} (\Delta(u_1^2))^2=4\big[\lambda_D^2u_1^4-2\lambda_D\,u_1^2|\nabla u_1|^2 +|\nabla u_1|^4\big],\quad |\nabla(u_1^2)|^2=4\,u_1^2|\nabla u_1|^2,$$ and use to write $$\label{A-P.10} \lambda_K \leq\lambda_D^2\left(\frac{\int_{{\Omega}}d^nx\,u_1^4} {\int_{{\Omega}}d^nx\,u_1^2|\nabla u_1|^2}\right)-2\lambda_D +\left(\frac{\int_{{\Omega}}d^nx\,|\nabla u_1|^4} {\int_{{\Omega}}d^nx\,u_1^2|\nabla u_1|^2}\right).$$ Next, observe that based on and the Divergence Theorem we may write $$\begin{aligned} \label{A-P.11} \int_{{\Omega}}d^nx\,\big[3u_1^2|\nabla u_1|^2-\lambda_D\,u_1^4\big] &= \int_{{\Omega}}d^nx\,\big[3u_1^2|\nabla u_1|^2+u_1^3\Delta u_1\big] =\int_{{\Omega}}d^nx\, {\rm div} \big(u_1^3\nabla u_1\big) \nonumber\\ &= \int_{\partial{\Omega}}d^{n-1}\omega\,u_1^3\partial_{\nu}u_1=0,\end{aligned}$$ where $\nu$ is the outward unit normal to ${{\partial\Omega}}$, and $d^{n-1}\omega$ denotes the induced surface measure on $\partial\Omega$. This shows that the coefficient of $\lambda_D^2$ in is $3\lambda_D^{-1}$, so that $$\label{A-P.12} \lambda_K \leq\lambda_D +\theta, \, \mbox{ where }\, \theta:=\frac{\int_{{\Omega}}d^nx\,|\nabla u_1|^4}{\int_{{\Omega}}d^nx\,u_1^2|\nabla u_1|^2}.$$ We begin to estimate $\theta$ by writing $$\begin{aligned} \label{A-P.13} \int_{{\Omega}}d^nx\,|\nabla u_1|^4 &= \int_{{\Omega}}d^nx\,(\nabla u_1)\cdot(|\nabla u_1|^2\nabla u_1) =-\int_{{\Omega}}d^nx\,u_1\, {\rm div} (|\nabla u_1|^2\nabla u_1) \nonumber\\ &= -\int_{{\Omega}}d^nx\, \big[(u_1\,\nabla u_1)\cdot(\nabla|\nabla u_1|^2) -\lambda_D\,u_1^2|\nabla u_1|^2\big],\end{aligned}$$ so that $$\label{A-P.14} \frac{\int_{{\Omega}}d^nx\,(u_1\,\nabla u_1)\cdot(\nabla|\nabla u_1|^2)} {\int_{{\Omega}}d^nx\,u_1^2|\nabla u_1|^2}=\lambda_D-\theta.$$ To continue, one observes that because of and the classical Hopf lemma (cf. [@GT83 Lemma 3.4]) one has $\partial_{\nu} u_1 > 0$ on $\partial{\Omega}$. Thus, $|\nabla u_1| \neq 0$ at points in ${\Omega}$ near $\partial {\Omega}$. This allows one to conclude that $$\label{A-P.15} \nu=-\frac{\nabla u_1}{|\nabla u_1|}\, \mbox{ near and on }\, {{\partial\Omega}}.$$ By a standard result from differential geometry (see, for example, [@Ca92 p. 142]) $$\label{A-P.16} {\rm div} (\nu)=(n-1)\,{{\mathcal G}}\, \mbox{ on }\, \partial{\Omega},$$ where ${{\mathcal G}}$ denotes the mean curvature of $\partial{\Omega}$. To proceed further, we introduce the following notations for the second derivative matrix, or [*Hessian*]{}, of $u_1$ and its norm: $$\label{A-P.18} {\rm Hess} (u_1):= \left(\frac{\partial^2 u_1}{\partial x_j\partial x_k}\right)_{1\leq j,k\leq n}, \quad |{\rm Hess} (u_1)|:= \bigg(\sum_{j,k=1}^n|\partial_j\partial_k u_1|^2\bigg)^{1/2}.$$ Relatively brief and straightforward computations (cf. [@KP99 Theorem 2.2.14]) then yield $$\begin{aligned} \label{A-P.19} {\rm div} (\nu) = - \sum_{j=1}^n \partial_j \bigg(\frac{\partial_j u_1}{|\nabla u_1|}\bigg) &=|\nabla u_1|^{-1}[-\Delta u_1 + \langle \nu, {\rm Hess}(u_1) \nu \rangle] \nonumber\\ &=|\nabla u_1|^{-1}\langle \nu, {\rm Hess}(u_1) \nu \rangle \, \mbox{ on } \, \partial{\Omega}\end{aligned}$$ (since $-\Delta u_1 = \lambda u_1 = 0$ on $\partial{\Omega}$), $$\begin{aligned} \label{A-P.20a} \nu \cdot (\partial_{\nu} \nu) &= - \sum_{j,k=1}^n\nu_j \nu_k \partial_k \bigg(\frac{\partial_j u_1}{|\nabla u_1|}\bigg) {\notag}\\ & =-|\nabla u_1|^{-1} \langle \nu,{\rm Hess}(u_1) \nu \rangle + |\nabla u_1|^{-1}|\nu|^2 \langle \nu, {\rm Hess} (u_1) \nu \rangle \nonumber\\ &=0,\end{aligned}$$ and finally, by , $$\begin{aligned} \label{A-P.21a} \partial_\nu(|\nabla u_1|^2) &=\sum_{j,k=1}^n \nu_j \partial_j [(\partial_k u_1)^2] =2 \sum_{j,k=1}^n \nu_j (\partial_k u_1)(\partial_j \partial_k u_1) \nonumber\\ &=-2|\nabla u_1| \langle \nu, {\rm Hess} (u_1) \nu \rangle =-2|\nabla u_1|^2 {\rm div} (\nu) \nonumber\\ &=-2(n-1) {{\mathcal G}}|\nabla u_1|^2 \leq 0 \, \mbox{ on } \, \partial{\Omega}, \end{aligned}$$ given our assumption ${{\mathcal G}}\geq 0$. Next, we compute $$\begin{aligned} \label{A-P.20} & \int_{{\Omega}}d^nx\, \big[|\nabla(|\nabla u_1|^2)|^2-2\lambda_D\,|\nabla u_1|^4 +2|\nabla u_1|^2|{\rm Hess} (u_1)|^2\big] {\notag}\\ & \quad =\int_{{\Omega}}d^nx\, div \big(|\nabla u_1|^2\nabla(|\nabla u_1|^2)\big) =\int_{{{\partial\Omega}}}d^{n-1}\omega\,\nu\cdot \big(|\nabla u_1|^2\nabla(|\nabla u_1|^2)\big) {\notag}\\ & \quad =\int_{{{\partial\Omega}}}d^{n-1}\omega\,|\nabla u_1|^2\partial_{\nu}\big(|\nabla u_1|^2\big)\leq 0,\end{aligned}$$ since $\partial_{\nu}(|\nabla u_1|^2)\leq 0$ on ${{\partial\Omega}}$ by . As a consequence, $$\label{A-P.21} 2\lambda_D\,\int_{{\Omega}}d^nx\,|\nabla u_1|^4\geq \int_{{\Omega}}d^nx\,\big[|\nabla(|\nabla u_1|^2)|^2+2|\nabla u_1|^2|{\rm Hess} (u_1)|^2\big].$$ Now, simple algebra shows that $|\nabla(|\nabla u_1|^2)|^2\leq 4\,|\nabla u_1|^2|{\rm Hess} (u_1)|^2$ which, when combined with , yields $$\label{A-P.22} \frac{4\lambda_D}{3}\,\int_{{\Omega}}d^nx\,|\nabla u_1|^4\geq \int_{{\Omega}}d^nx\,|\nabla(|\nabla u_1|^2)|^2.$$ Let us now return to and rewrite this equality as $$\label{A-P.23} \int_{{\Omega}}d^nx\,|\nabla u_1|^4 = -\int_{{\Omega}}d^nx\,(u_1\,\nabla u_1)\cdot(\nabla|\nabla u_1|^2 -\lambda_D\,u_1\nabla u_1).$$ An application of the Cauchy-Schwarz inequality then yields $$\label{A-P.24} \left(\int_{{\Omega}}d^nx\,|\nabla u_1|^4\right)^2 \leq\left(\int_{{\Omega}}d^nx\,u_1^2\,|\nabla u_1|^2\right) \left(\int_{{\Omega}}d^nx\,|\nabla|\nabla u_1|^2-\lambda_D\,u_1\nabla u_1|^2\right).$$ By expanding the last integrand and recalling the definition of $\theta$ we then arrive at $$\label{A-P.25} \theta^2\leq\lambda_D^2-2\lambda_D\left(\frac{\int_{{\Omega}}d^nx\, (u_1\nabla u_1)\cdot(\nabla|\nabla u_1|^2)}{\int_{{\Omega}}d^nx\,u_1^2|\nabla u_1|^2}\right) +\left(\frac{\int_{{\Omega}}d^nx\,|\nabla(|\nabla u_1|^2)|^2} {\int_{{\Omega}}d^nx\,u_1^2|\nabla u_1|^2}\right).$$ Upon recalling and , this becomes $$\label{A-P.26} \theta^2\leq \lambda_D^2-2\lambda_D(\lambda_D-\theta)+\frac{4\lambda_D}{3}\theta =-\lambda_D^2+\frac{10\lambda_D}{3}\theta.$$ In turn, this forces $\theta\leq 3\lambda_D$ hence, ultimately, $\lambda_K\leq 4\lambda_D$ due to this estimate and . This establishes and completes the proof of the theorem. $(i)$ The upper bound in for two-dimensional smooth, convex $C^{\infty}$ domains ${\Omega}$ is due to Payne [@Pa60] in 1960. He notes that the proof carries over without difficulty to dimensions $n\geq 2$ in [@Pa67 p. 464]. In addition, one can avoid assuming smoothness in his proof by using smooth approximations ${\Omega}_m$, $m\in{{\mathbb{N}}}$, of ${\Omega}$ as discussed in our proof. Of course, Payne did not consider the eigenvalues of the Krein Laplacian $-\Delta_{K,{\Omega}}$, instead, he compared the first eigenvalue of the fixed membrane problem and the first eigenvalue of the problem of the buckling of a clamped plate.\ $(ii)$ By thinking of ${\rm Hess} (u_1)$ represented in terms of an orthonormal basis for $\mathbb{R}^n$ that contains $\nu$, one sees that yields $$\label{A-P.22a} {\rm div} (\nu) = \bigg|\frac{\partial u_1}{\partial \nu}\bigg|^{-1} \, \frac{\partial^2 u_1}{{\partial \nu}^2} =-\bigg(\frac{\partial u_1}{\partial \nu}\bigg)^{-1} \frac{\partial^2 u_1}{{\partial \nu}^2}$$ (the latter because $\partial u_1/\partial \nu < 0$ on $\partial{\Omega}$ by our convention on the sign of $u_1$ (see )), and thus $$\label{A-P.23a} \frac{\partial^2 u_1}{{\partial \nu}^2} = -(n-1) {{\mathcal G}}\frac{\partial u_1}{\partial \nu} \, \mbox{ on } \partial{\Omega}.$$ For a different but related argument leading to this same result, see Ashbaugh and Levine [@AL97 pp. I-8, I-9]. Aviles [@Av86], Payne [@Pa55], [@Pa60], and Levine and Weinberger [@LW86] all use similar arguments as well.\ $(iii)$ We note that Payne’s basic result here, when done in $n$ dimensions, holds for smooth domains having a boundary which is everywhere of nonnegative mean curvature. In addition, Levine and Weinberger [@LW86], in the context of a related problem, consider nonsmooth domains for the nonnegative mean curvature case and a variety of cases intermediate between that and the convex case (including the convex case).\ $(iv)$ Payne’s argument (and the constant 4 in Theorem \[T-Pay-1\]) would appear to be sharp, with any infinite slab in $\mathbb{R}^n$ bounded by parallel hyperplanes being a saturating case (in a limiting sense). We note that such a slab is essentially one-dimensional, and that, up to normalization, the first Dirichlet eigenfunction $u_1$ for the interval $[0,a]$ (with $a>0$) is $$u_1(x)=\sin (\pi x/a) \, \text{ with eigenvalue } \, \lambda=\pi^2/a^2,$$ while the corresponding first buckling eigenfunction and eigenvalue are $$u_1(x)^2=\sin^2 (\pi x/a)=[1-\cos (2\pi x/a)]/2 \, \text{ and } \, 4 \lambda=4\pi^2/a^2.$$ Thus, Payne’s choice of the trial function $u_1^2$, where $u_1$ is the first Dirichlet eigenfunction should be optimal for this limiting case, implying that the bound 4 is best possible. Payne, too, made observations about the equality case of his inequality, and observed that the infinite strip saturates it in 2 dimensions. His supporting arguments are via tracing the case of equality through the inequalities in his proof, which also yields interesting insights. \[Rm-2\] The eigenvalues corresponding to the buckling of a two-dimensional [*square*]{} plate, clamped along its boundary, have been analyzed numerically by several authors (see, e.g., [@AD92], [@AD93], and [@BT99]). All these results can now be naturally reinterpreted in the context of the Krein Laplacian $-\Delta_{K,{\Omega}}$ in the case where ${\Omega}=(0,1)^2\subset{{\mathbb{R}}}^2$. Lower bounds for the first $k$ buckling problem eigenvalues were discussed in [@LP85]. The existence of convex domains ${\Omega}$, for which the first eigenfunction of the problem of a clamped plate and the problem of the buckling of a clamped plate possesses a change of sign, was established in [@KKM90]. Relations between an eigenvalue problem governing the behavior of an elastic medium and the buckling problem were studied in [@Ho91]. Buckling eigenvalues as a function of the elasticity constant are investigated in [@KLV93]. Finally, spectral properties of linear operator pencils $A-\lambda B$ with discrete spectra, and basis properties of the corresponding eigenvectors, applicable to differential operators, were discussed, for instance, in [@Pe68], [@Tr00] (see also the references cited therein). Formula suggests the issue of deriving a Weyl asymptotic formula for the perturbed Krein Laplacian $H_{K,{\Omega}}$. This is the topic of our next section. Weyl Asymptotics for the Perturbed Krein Laplacian in Nonsmooth Domains {#s12} ======================================================================= We begin by recording a very useful result due to V.A. Kozlov which, for the convenience of the reader, we state here in more generality than is actually required for our purposes. To set the stage, let $\Omega\subset{{\mathbb{R}}}^n$, $n{\geqslant}2$, be a bounded Lipschitz domain. In addition, assume that $m>r\geq 0$ are two fixed integers and set $$\label{kko-0} \eta:=2(m-r)>0.$$ Let $W$ be a closed subspace in $H^m(\Omega)$ such that $H^m_0(\Omega)\subseteq W$. On $W$, consider the symmetric forms $$\label{kko-1} a(u,v):=\sum_{0 \leq |\alpha|,|\beta|\leq m} \int_{\Omega}d^nx\,a_{\alpha,\beta}(x){\overline}{(\partial^\beta u)(x)} (\partial^\alpha v)(x), \quad u, v \in W,$$ and $$\label{kko-2} b(u,v):=\sum_{0 \leq |\alpha|,|\beta|{\leqslant}r} \int_{\Omega}d^nx\,b_{\alpha,\beta}(x){\overline}{(\partial^\beta u)(x)} (\partial^\alpha v)(x), \quad u, v \in W.$$ Suppose that the leading coefficients in $a({\,\cdot\,},{\,\cdot\,})$ and $b({\,\cdot\,},{\,\cdot\,})$ are Lipschitz functions, while the coefficients of all lower-order terms are bounded, measurable functions in $\Omega$. Furthermore, assume that the following coercivity, nondegeneracy, and nonnegativity conditions hold: For some $C_0 \in (0,\infty)$, $$\begin{aligned} \label{kko-3} & a(u,u){\geqslant}C_0\|u\|^2_{H^m(\Omega)}, \quad u\in W, \\ & \sum_{|\alpha|=|\beta|=r}b_{\alpha,\beta}(x) \, \xi^{\alpha+\beta}\not=0, \quad x\in{\overline}{{\Omega}}, \; \xi\not=0, \label{kko-4} \\ & b(u,u){\geqslant}0, \quad u\in W. \label{kko-5}\end{aligned}$$ Under the above assumptions, $W$ can be regarded as a Hilbert space when equipped with the inner product $a({\,\cdot\,},{\,\cdot\,})$. Next, consider the operator $T\in{{\mathcal B}}(W)$ uniquely defined by the requirement that $$\label{kko-6} a(u,T v)=b(u,v), \quad u,v\in W.$$ Then the operator $T$ is compact, nonnegative and self-adjoint on $W$ (when the latter is viewed as a Hilbert space). Going further, denote by $$\label{kko-7} 0\leq\cdots\leq\mu_{j+1}(T)\leq\mu_j(T)\leq\cdots\leq\mu_1(T),$$ the eigenvalues of $T$ listed according to their multiplicity, and set $$\label{kko-8} N(\lambda;W,a,b):=\#\,\{j\in{{\mathbb{N}}}\,|\,\mu_j(T)\geq \lambda^{-1}\}, \quad \lambda>0.$$ The following Weyl asymptotic formula is a particular case of a slightly more general result which can be found in [@Ko83]. \[T-Koz\] Assume Hypothesis \[h2.1\] and retain the above notation and assumptions on $a({\,\cdot\,},{\,\cdot\,})$, $b({\,\cdot\,},{\,\cdot\,})$, $W$, and $T$. In addition, we recall . Then the distribution function of the spectrum of $T$ introduced in satisfies the asymptotic formula $$\label{kko-9} N(\lambda;W,a,b) =\omega_{a,b,\Omega}\,\lambda^{n/\eta}+O\big(\lambda^{(n-(1/2))/\eta}\big) \, \mbox{ as }\, \lambda\to\infty,$$ where, with $d\omega_{n-1}$ denoting the surface measure on the unit sphere $S^{n-1}=\{\xi\in{{\mathbb{R}}}^n\,|\,|\xi|=1\}$ in ${{\mathbb{R}}}^n$, $$\label{kko-10} \omega_{a,b,\Omega}:=\frac{1}{n(2\pi)^n} \int_{\Omega}d^nx\,\left(\int_{|\xi|=1}d\omega_{n-1}(\xi)\, \left[\frac{\sum_{|\alpha|=|\beta|=r}b_{\alpha,\beta}(x)\xi^{\alpha+\beta}} {\sum_{|\alpha|=|\beta|=m}a_{\alpha,\beta}(x)\xi^{\alpha+\beta}} \right]^{\frac{n}{\eta}}\right).$$ Various related results can be found in [@Ko79], [@Ko84]. After this preamble, we are in a position to state and prove the main result of this section: \[T-KrWe\] Assume Hypothesis \[h.VK\]. In addition, we recall that $$\label{kko-11} N_{K,{\Omega}}(\lambda)=\#\{j\in{{\mathbb{N}}}\,|\,\lambda_{K,{\Omega},j}\leq\lambda\}, \quad\lambda\in{{\mathbb{R}}},$$ where the $($strictly$)$ positive eigenvalues $\{\lambda_{K,{\Omega},j}\}_{j\in{{\mathbb{N}}}}$ of the perturbed Krein Laplacian $H_{K,{\Omega}}$ are enumerated as in $($according to their multiplicities$)$. Then the following Weyl asymptotic formula holds: $$\label{kko-12} N_{K,{\Omega}}(\lambda) =(2\pi)^{-n}v_n|\Omega|\,\lambda^{n/2}+O\big(\lambda^{(n-(1/2))/2}\big) \, \mbox{ as }\, \lambda\to\infty,$$ where, as before, $v_n$ denotes the volume of the unit ball in ${{\mathbb{R}}}^n$, and $|\Omega|$ stands for the $n$-dimensional Euclidean volume of $\Omega$. Set $W:=H^2_0({\Omega})$ and consider the symmetric forms $$\begin{aligned} \label{kko-13} & a(u,v):=\int_{{\Omega}}d^nx\,{\overline}{(-\Delta+V)u}\,(-\Delta+V)v,\quad u,v\in W, \\ & b(u,v):=\int_{{\Omega}}d^nx\,{\overline}{\nabla u}\cdot\nabla v +\int_{{\Omega}}d^nx\,{\overline}{V^{1/2}u}\,V^{1/2}v,\quad u,v\in W, \label{kko-13a}\end{aligned}$$ for which conditions – (with $m=2$) are verified (cf. ). Next, we recall the operator $(-\Delta+V)^{-2}:= ((-\Delta+V)^2)^{-1}\in{{\mathcal B}}\bigl(H^{-2}({\Omega}),H^2_0({\Omega})\bigr)$ from along with the operator $$\label{kko-14} B\in{{\mathcal B}}_{\infty}(W),\quad Bu:=-(-\Delta+V)^{-2}(-\Delta+V)u,\quad u\in W,$$ from . Then, in the current notation, formula reads $a(Bu,v)=b(u,v)$ for every $u,v\in C^\infty_0({\Omega})$. Hence, by density, $$\label{kko-15} a(Bu,v)=b(u,v),\quad u,v\in W.$$ This shows that actually $B=T$, the operator originally introduced in . In particular, $T$ is one-to-one. Consequently, $Tu=\mu\,u$ for $u\in W$ and $0\not=\mu\in{{\mathbb{C}}}$, if and only if $u\in H^2_0(\Omega)$ satisfies $(-\Delta+V)^{-2}(-\Delta+V)u=\mu\,u$, that is, $(-\Delta+V)^2 u=\mu^{-1}(-\Delta+V)u$. Hence, the eigenvalues of $T$ are precisely the reciprocals of the eigenvalues of the buckling clamped plate problem . Having established this, formula then follows from Theorem \[T-MM-2\] and , upon observing that in our case $m=2$, $r=1$ (hence $\eta=2$) and $\omega_{a,b,\Omega}=(2\pi)^{-n}v_n|{\Omega}|$. Incidentally, Theorem \[T-KrWe\] and Theorem \[T-MM-2\] show that, granted Hypothesis \[h.VK\], a Weyl asymptotic formula holds in the case of the (perturbed) buckling problem . For smoother domains and potentials, this is covered by Grubb’s results in [@Gr83]. In the smooth context, a sharpening of the remainder has been announced in [@Mik94] without proof. In the case where $\Omega\subset{{\mathbb{R}}}^2$ is a bounded domain with a $C^\infty$-boundary and $0\leq V\in C^\infty({\overline}{{\Omega}})$, a more precise form of the error term in was obtained in [@Gr83] where Grubb has shown that $$\label{kko-13x} N_{K,{\Omega}}(\lambda)=\frac{|\Omega|}{4\pi}\,\lambda+O\big(\lambda^{2/3}\big) \, \mbox{ as }\, \lambda\to\infty,$$ In fact, in [@Gr83], Grubb deals with the Weyl asymptotic for the Krein–von Neumann extension of a general strongly elliptic, formally self-adjoint differential operator of arbitrary order, provided both its coefficients as well as the the underlying domain $\Omega\subset{{\mathbb{R}}}^n$ ($n\geq 2$) are $C^\infty$-smooth. In the special case where ${\Omega}$ equals the open ball $B_n(0;R)$, $R>0$, in ${{\mathbb{R}}}^n$, and when $V\equiv 0$, it turns out that , can be further refined to $$\begin{aligned} \label{NN-y} N^{(0)}_{K,B_n(0;R)}(\lambda)&=(2\pi)^{-n}v_n^2 R^n \lambda^{n/2} - (2\pi)^{-(n-1)}v_{n-1}[(n/4)v_n + v_{n-1}] R^{n-1} \lambda^{(n-1)/2} {\notag}\\ & \quad + O\big(\lambda^{(n-2)/2}\big) \mbox{ as }\, \lambda\to\infty, \end{aligned}$$ for every $n\geq 2$. This will be the object of the final Section \[s1vi\] (cf. Proposition \[p10.1\]). A Class of Domains for which the Krein and Dirichlet Laplacians Coincide {#s1v} ======================================================================== Motivated by the special example where $\Omega={{\mathbb{R}}}^2\backslash\{0\}$ and $S={\overline}{-\Delta_{C_0^\infty({{\mathbb{R}}}^2\backslash\{0\})}}$, in which case one can show the interesting fact that $S_F=S_K$ (cf. [@AGHKH87], [@AGHKH88 Ch. I.5], [@GKMT01], and Subsections \[s10.3\] and \[s10.4\]) and hence the nonnegative self-adjoint extension of $S$ is unique, the aim of this section is to present a class of (nonempty, proper) open sets $\Omega={{\mathbb{R}}}^n\backslash K$, $K\subset {{\mathbb{R}}}^n$ compact and subject to a vanishing Bessel capacity condition, with the property that the Friedrichs and Krein–von Neumann extensions of $-\Delta\big|_{C^\infty_0({\Omega})}$ in $L^2({\Omega}; d^n x)$, coincide. To the best of our knowledge, the case where the set $K$ differs from a single point is without precedent and so the following results for more general sets $K$ appear to be new. We start by making some definitions and discussing some preliminary results, of independent interest. Given an arbitrary open set ${\Omega}\subset{{\mathbb{R}}}^n$, $n\geq 2$, we consider three realizations of $-\Delta$ as unbounded operators in $L^2({\Omega};d^nx)$, with domains given by (cf. Subsection \[s4X\]) $$\begin{aligned} \label{YF-1} \operatorname{dom}(-\Delta_{max,{\Omega}})&:=\big\{u\in L^2(\Omega;d^nx)\,\big|\, \Delta u\in L^2(\Omega;d^nx)\big\}, \\ \operatorname{dom}(-\Delta_{D,{\Omega}})&:=\big\{u\in H^1_0(\Omega)\,\big|\, \Delta u\in L^2(\Omega;d^nx)\big\}, \label{YF-2} \\ \operatorname{dom}(-\Delta_{c,{\Omega}})&:=C^\infty_0(\Omega). \label{YF-3}\end{aligned}$$ \[L-ea\] For any open, nonempty subset ${\Omega}\subseteq {{\mathbb{R}}}^n$, $n\geq 2$, the following statements hold: 1. One has $$\label{Fga-2} (-\Delta_{c,{\Omega}})^*=-\Delta_{max,{\Omega}}.$$ 2. The Friedrichs extension of $-\Delta_{c,{\Omega}}$ is given by $$\label{Fga-3} (-\Delta_{c,{\Omega}})_F=-\Delta_{D,{\Omega}}.$$ 3. The Krein–von Neumann extension of $-\Delta_{c,{\Omega}}$ has the domain $$\begin{aligned} & \operatorname{dom}((-\Delta_{c,{\Omega}})_K)= \big\{u\in\operatorname{dom}(-\Delta_{\max,{\Omega}})\,\big|\,\mbox{there exists } \{u_j\}_{j\in{{\mathbb{N}}}} \in C^\infty_0({\Omega}) \label{Gkj-1} \\ & \quad \mbox{with } \lim_{j\to\infty}\|\Delta u_j\ - \Delta u\|_{L^2({\Omega};d^nx)} = 0 \mbox{ and $\{\nabla u_j\}_{j\in{{\mathbb{N}}}}$ Cauchy in $L^2({\Omega};d^nx)^n$}\big\}. {\notag}\end{aligned}$$ 4. One has $$\label{F-2Lb} \ker((-\Delta_{c,{\Omega}})_{K}) = \big\{u\in L^2({\Omega};d^nx)\,\big|\,\Delta\,u=0 \mbox{ in } {\Omega}\big\},$$ and $$\label{F-2La} \ker((-\Delta_{c,{\Omega}})_{F})=\{0\}.$$ Formula follows in a straightforward fashion, by unraveling definitions, whereas is a direct consequence of or (compare also with Proposition \[L-Fri1\]). Next, is readily implied by and . In addition, is easily derived from , and . Finally, consider . In a first stage, and yield that $$\label{F-2Lc} \ker ((-\Delta_{c,{\Omega}})_{F}) = \big\{u\in H^1_0({\Omega})\,\big|\,\Delta\,u=0 \mbox{ in } {\Omega}\big\},$$ so the goal is to show that the latter space is trivial. To this end, pick a function $u\in H^1_0({\Omega})$ which is harmonic in ${\Omega}$, and observe that this forces $\nabla u=0$ in ${\Omega}$. Now, with tilde denoting the extension by zero outside ${\Omega}$, we have $\widetilde{u}\in H^1({{\mathbb{R}}}^n)$ and $\nabla(\widetilde{u})=\widetilde{\nabla u}$. In turn, this entails that $\widetilde{u}$ is a constant function in $L^2({{\mathbb{R}}};d^nx)$ and hence $u\equiv 0$ in ${\Omega}$, establishing . Next, we record some useful capacity results. For an authoritative extensive discussion on this topic see the monographs [@AH96], [@Ma85], [@Ta95], and [@Zi89]. We denote by $B_{\alpha,2}(E)$ the Bessel capacity of order $\alpha>0$ of a set $E\subset{{\mathbb{R}}}^n$. When $K\subset {{\mathbb{R}}}^n$ is a compact set, this is defined by $$\label{cap-1} B_{\alpha,2}(K):=\inf\,\bigl\{\|f\|^2_{L^2({{\mathbb{R}}}^n;d^nx)}\,\big|\, g_\alpha\ast f\geq 1\mbox{ on }K,\,f\geq 0\bigr\},$$ where the Bessel kernel $g_\alpha$ is defined as the function whose Fourier transform is given by $$\label{cap-2} \widehat{g_\alpha}(\xi)=(2\pi)^{-n/2}(1+|\xi|^2)^{-\alpha/2}, \quad\xi\in{{\mathbb{R}}}^n.$$ When ${{\mathcal O}}\subseteq{{\mathbb{R}}}^n$ is open, we define $$\label{cap-1X} B_{\alpha,2}({{\mathcal O}}):= \sup\,\{B_{\alpha,2}(K)\,|\,K\subset{{\mathcal O}},\,K\mbox{ compact}\,\},$$ and, finally, when $E\subseteq{{\mathbb{R}}}^n$ is an arbitrary set, $$\label{cap-1Y} B_{\alpha,2}(E):= \inf\,\{B_{\alpha,2}({{\mathcal O}})\,|\,{{\mathcal O}}\supset E,\,{{\mathcal O}}\mbox{ open}\,\}.$$ In addition, denote by ${{\mathcal H}}^k$ the $k$-dimensional Hausdorff measure on ${{\mathbb{R}}}^n$, $0\leq k\leq n$. Finally, a compact subset $K\subset{{\mathbb{R}}}^n$ is said to be [*$L^2$-removable for the Laplacian*]{} provided every bounded, open neighborhood ${{\mathcal O}}$ of $K$ has the property that $$\begin{aligned} \label{Fga-2L.2} & u\in L^2 ({{\mathcal O}}\backslash K;d^nx)\mbox{ with }\Delta u=0 \mbox{ in }{{\mathcal O}}\backslash K \, \text{ imply } \begin{cases} \mbox{there exists $\widetilde{u}\in L^2 ({{\mathcal O}};d^nx)$ so that} \\ \mbox{$\widetilde{u}\Bigl|_{{{\mathcal O}}\backslash K}=u$ and $\Delta\widetilde{u}=0$ in ${{\mathcal O}}$}. \end{cases}\end{aligned}$$ \[R-Ca.1\] For $\alpha>0$, $k\in{{\mathbb{N}}}$, $n\geq 2$ and $E\subset{{\mathbb{R}}}^n$, the following properties are valid: 1. A compact set $K\subset{{\mathbb{R}}}^n$ is $L^2$-removable for the Laplacian if and only if $B_{2,2}(K)=0$. 2. Assume that ${\Omega}\subset{{\mathbb{R}}}^n$ is an open set and that $K\subset\Omega$ is a closed set. Then the space $C^\infty_0({\Omega}\backslash K)$ is dense in $H^k({\Omega})$ $($i.e., one has the natural identification $H^k_0({\Omega})\equiv H^k_0({\Omega}\backslash K)$$)$, if and only if $B_{k,2}(K)=0$. 3. If $2\alpha\leq n$ and ${{\mathcal H}}^{n-2\alpha}(E)<+\infty$ then $B_{\alpha,2}(E)=0$. Conversely, if $2\alpha\leq n$ and $B_{\alpha,2}(E)=0$ then ${{\mathcal H}}^{n-2\alpha+\varepsilon}(E)=0$ for every $\varepsilon>0$. 4. Whenever $2\alpha>n$ then there exists $C=C(\alpha,n)>0$ such that $B_{\alpha,2}(E)\geq C$ provided $E\not=\emptyset$. See, [@AH96 Corollary 3.3.4], [@Ma85 Theorem 3], [@Zi89 Theorem 2.6.16 and Remark 2.6.15], respectively. For other useful removability criteria the interested reader may wish to consult [@Ca67], [@Ma65], [@RS08], and [@Tr08]. The first main result of this section is then the following: \[L-ea-5\] Assume that $K\subset{{\mathbb{R}}}^n$, $n\geq 3$, is a compact set with the property that $$\label{Ha-z} B_{2,2}(K)=0.$$ Define $\Omega:={{\mathbb{R}}}^n\backslash K$. Then, in the domain ${\Omega}$, the Friedrichs and Krein–von Neumann extensions of $-\Delta$, initially considered on $C^\infty_0({\Omega})$, coincide, that is, $$\label{exa-13} (-\Delta_{c,{\Omega}})_F = (-\Delta_{c,{\Omega}})_K.$$ As a consequence, $-\Delta|_{C^\infty_0({\Omega})}$ has a unique nonnegative self-adjoint extension in $L^2({\Omega};d^nx)$. We note that implies that $K$ has zero $n$-dimensional Lebesgue measure, so that $L^2({\Omega};d^nx)\equiv L^2({{\mathbb{R}}}^n;d^nx)$. In addition, by $(iii)$ in Proposition \[R-Ca.1\], we also have $B_{1,2}(K)=0$. Now, if $u\in\operatorname{dom}(-\Delta_{c,{\Omega}})_K$, entails that $u\in L^2({\Omega};d^nx)$, $\Delta u\in L^2({\Omega};d^nx)$, and that there exists a sequence $u_j\in C^\infty_0({\Omega})$, $j\in{{\mathbb{N}}}$, for which $$\label{exa-14} \Delta u_j\to\Delta u \,\mbox{ in } \,L^2({\Omega};d^nx) \,\mbox{ as } \, j\to\infty, \mbox{ and $\{\nabla u_j\}_{j\in{{\mathbb{N}}}}$ is Cauchy in $L^2({\Omega};d^nx)$}.$$ In view of the well-known estimate (cf. the Corollary on p. 56 of [@Ma85]), $$\label{exa-15} \|v\|_{L^{2^*}({{\mathbb{R}}}^n;d^nx)}\leq C_n\|\nabla v\|_{L^2({{\mathbb{R}}}^n;d^nx)},\quad v\in C^\infty_0({{\mathbb{R}}}^n),$$ where $2^*:=(2n)/(n-2)$, the last condition in implies that there exists $w\in L^{2^*}({{\mathbb{R}}}^n;d^nx)$ with the property that $$\label{exa-16} u_j\to w \,\mbox{ in } \,L^{2^*}({{\mathbb{R}}}^n;d^nx)\, \mbox{ and }\, \nabla u_j\to\nabla w \,\mbox{ in } \,L^2({{\mathbb{R}}}^n;d^nx) \,\mbox{ as } \,j\to\infty.$$ Furthermore, by the first convergence in , we also have that $\Delta w=\Delta u$ in the sense of distributions in ${\Omega}$. In particular, the function $$\label{exa-17} f:=w-u\in L^{2^*}({{\mathbb{R}}}^n;d^nx)+L^2({{\mathbb{R}}}^n;d^nx) \hookrightarrow L^2_{{\text{\rm{loc}}}}({{\mathbb{R}}}^n;d^nx)$$ satisfies $\Delta f=0$ in ${\Omega}={{\mathbb{R}}}^n\backslash K$. Granted , Proposition \[R-Ca.1\] yields that $K$ is $L^2$-removable for the Laplacian, so we may conclude that $\Delta f=0$ in ${{\mathbb{R}}}^n$. With this at hand, Liouville’s theorem then ensures that $f\equiv 0$ in ${{\mathbb{R}}}^n$. This forces $u=w$ as distributions in ${\Omega}$ and hence, $\nabla u=\nabla w$ distributionally in ${\Omega}$. In view of the last condition in we may therefore conclude that $u\in H^1({{\mathbb{R}}}^n)=H^1_0({{\mathbb{R}}}^n)$. With this at hand, Proposition \[R-Ca.1\] yields that $u\in H^1_0({\Omega})$. This proves that $\operatorname{dom}(-\Delta_{c,{\Omega}})_K \subseteq\operatorname{dom}(-\Delta_{c,{\Omega}})_F$ and hence, $(-\Delta_{c,{\Omega}})_K \subseteq (-\Delta_{c,{\Omega}})_F$. Since both operators in question are self-adjoint, follows. We emphasize that equality of the Friedrichs and Krein Laplacians necessarily requires that fact that $\inf (\sigma((-\Delta_{c,{\Omega}})_F)) = \inf(\sigma((-\Delta_{c,{\Omega}})_K)) = 0$, and hence rules out the case of bounded domains $\Omega \subset {{\mathbb{R}}}^n$, $n \in {{\mathbb{N}}}$ (for which $\inf (\sigma((-\Delta_{c,{\Omega}})_F)) > 0$). \[C-capF\] Assume that $K\subset{{\mathbb{R}}}^n$, $n\geq 4$, is a compact set with finite $(n-4)$-dimensional Hausdorff measure, that is, $$\label{Ha-zX} {{\mathcal H}}^{n-4}(K)<+\infty.$$ Then, with $\Omega:={{\mathbb{R}}}^n\backslash K$, one has $(-\Delta_{c,{\Omega}})_F = (-\Delta_{c,{\Omega}})_K$, and hence, $-\Delta|_{C^\infty_0({\Omega})}$ has a unique nonnegative self-adjoint extension in $L^2({\Omega};d^nx)$. This is a direct consequence of Proposition \[R-Ca.1\] and Theorem \[L-ea-5\]. In closing, we wish to remark that, as a trivial particular case of the above corollary, formula holds for the punctured space $$\label{puct-u1} \Omega:={{\mathbb{R}}}^n\backslash \{0\},\quad n\geq 4,$$ however, this fact is also clear from the well-known fact that $-\Delta|_{C_0^\infty ({{\mathbb{R}}}^n\backslash \{0\})}$ is essentially self-adjoint in $L^2({{\mathbb{R}}}^n; d^nx)$ if (and only if) $n\geq 4$ (cf., e.g., [@RS75 p. 161], and our discussion concerning the Bessel operator ). In [@GKMT01 Example 4.9] (see also our discussion in Subsection 10.3), it has been shown (by using different methods) that continues to hold for the choice when $n=2$, but that the Friedrichs and Krein–von Neumann extensions of $-\Delta$, initially considered on $C^\infty_0({\Omega})$ with ${\Omega}$ as in , are different when $n=3$. In light of Theorem \[L-ea-5\], a natural question is whether the coincidence of the Friedrichs and Krein–von Neumann extensions of $-\Delta$, initially defined on $C^\infty_0({\Omega})$ for some open set ${\Omega}\subset{{\mathbb{R}}}^n$, actually implies that the complement of ${\Omega}$ has zero Bessel capacity of order two. Below, under some mild background assumptions on the domain in question, we shall establish this type of converse result. Specifically, we now prove the following fact: \[L-ea-5C\] Assume that $K\subset{{\mathbb{R}}}^n$, $n>4$, is a compact set of zero $n$-dimensional Lebesgue measure, and set $\Omega:={{\mathbb{R}}}^n\backslash K$. Then $$\label{exa-13H} (-\Delta_{c,{\Omega}})_F =(-\Delta_{c,{\Omega}})_K \,\text{ implies } \, B_{2,2}(K)=0.$$ Let $K$ be as in the statement of the theorem. In particular, $L^2({\Omega};d^nx)\equiv L^2({{\mathbb{R}}}^n;d^nx)$. Hence, granted that $(-\Delta_{c,{\Omega}})_K=(-\Delta_{c,{\Omega}})_F$, in view of , this yields $$\label{F-2Ld} \big\{u\in L^2({{\mathbb{R}}}^n;d^nx) \,\big|\, \Delta\,u=0 \mbox{ in } {{\mathbb{R}}}^n\backslash K\big\}=\{0\}.$$ It is useful to think of as a capacitary condition. More precisely, implies that ${\rm Cap} (K)=0$, where $$\label{F-3Ld} {\rm Cap} (K):=\sup\,\bigl\{ \bigl|{}_{{{\mathcal E}}'({{\mathbb{R}}}^n)}\langle\Delta u,1\rangle_{{{\mathcal E}}({{\mathbb{R}}}^n)}\bigr|\,\big|\, \|u\|_{L^2({{\mathbb{R}}}^n;d^nx)}\leq 1\mbox{ and }{\rm supp} (\Delta u)\subseteq K \bigr\}.$$ Above, ${{\mathcal E}}({{\mathbb{R}}}^n)$ is the space of smooth functions in ${{\mathbb{R}}}^n$ equipped with the usual Frechét topology, which ensures that its dual, ${{\mathcal E}}'({{\mathbb{R}}}^n)$, is the space of compactly supported distributions in ${{\mathbb{R}}}^n$. At this stage, we recall the fundamental solution for the Laplacian in ${{\mathbb{R}}}^n$, $n\geq 3$, that is, $$\label{Fr-Ta1} E_n(x):=\frac{\Gamma(n/2)}{2(2-n)\pi^{n/2}|x|^{n-2}},\quad x\in{{\mathbb{R}}}^n\backslash \{0\}$$ ($\Gamma(\cdot)$ the classical Gamma function [@AS72 Sect. 6.1]), and introduce a related capacity, namely $$\begin{aligned} \label{F-4Ld} & {\rm Cap}_{\ast} (K):=\sup\,\bigl\{ \big|{}_{{{\mathcal E}}'({{\mathbb{R}}}^n)}\langle f,1\rangle_{{{\mathcal E}}({{\mathbb{R}}}^n)}\bigr| \,\big|\, f\in {{\mathcal E}}'({{\mathbb{R}}}^n),\,\,{\rm supp} (f) \subseteq K, \|E_n\ast f\|_{L^2({{\mathbb{R}}}^n;d^nx)}\leq 1\big\}. $$ Then $$\label{Fr-Ta2} 0\leq{\rm Cap}_{\ast} (K)\leq {\rm Cap} (K)=0$$ so that ${\rm Cap}_{\ast} (K)=0$. With this at hand, [@HP72 Theorem 1.5(a)] (here we make use of the fact that $n>4$) then allows us to strengthen to $$\label{F-5Ld} \big\{u\in L^2_{{\text{\rm{loc}}}}({{\mathbb{R}}}^n;d^nx) \,\big|\, \Delta\,u=0 \mbox{ in } {{\mathbb{R}}}^n\backslash K\big\}=\{0\}.$$ Next, we follow the argument used in the proof of [@MH73 Lemma 5.5] and [@AH96 Theorem 2.7.4]. Reasoning by contradiction, assume that $B_{2,2}(K)>0$. Then there exists a nonzero, positive measure $\mu$ supported in $K$ such that $g_2\ast\mu\in L^2({{\mathbb{R}}}^n)$. Since $g_2(x)=c_n\,E_n (x)+o(|x|^{2-n})$ as $|x|\to 0$ (cf. the discussion in Section 1.2.4 of [@AH96]) this further implies that $E_n\ast\mu\in L^2_{{\text{\rm{loc}}}}({{\mathbb{R}}}^n;d^nx)$. However, $E_n\ast\mu$ is a harmonic function in ${{\mathbb{R}}}^n\backslash K$, which is not identically zero since $$\label{Niz} \lim_{x\to\infty}|x|^{n-2}(E_n\ast\mu)(x)=c_n\mu(K)>0,$$ so this contradicts . This shows that $B_{2,2}(K)=0$. Theorems \[L-ea-5\]–\[L-ea-5C\] readily generalize to other types of elliptic operators (including higher-order systems). For example, using the polyharmonic operator $(-\Delta)^\ell$, $\ell\in{{\mathbb{N}}}$, as a prototype, we have the following result: \[L-ea-8\] Fix $\ell\in{{\mathbb{N}}}$, $n\geq 2\ell+1$, and assume that $K\subset{{\mathbb{R}}}^n$ is a compact set of zero $n$-dimensional Lebesgue measure. Define $\Omega:={{\mathbb{R}}}^n\backslash K$. Then, in the domain ${\Omega}$, the Friedrichs and Krein–von Neumann extensions of the polyharmonic operator $(-\Delta)^\ell$, initially considered on $C^\infty_0({\Omega})$, coincide if and only if $B_{2\ell,2}(K)=0$. For some related results in the punctured space ${\Omega}:={{\mathbb{R}}}^n\backslash \{0\}$, see also the recent article [@Ad07]. Moreover, we mention that in the case of the Bessel operator $h_\nu = (-d^2/dr^2) + (\nu^2 - (1/4))r^{-2}$ defined on $C_0^\infty((0,\infty))$, equality of the Friedrichs and Krein extension of $h_\nu$ in $L^2((0,\infty); dr)$ if and only if $\nu = 0$ has been established in [@MT07]. (The sufficiency of the condition $\nu = 0$ was established earlier in [@GKMT01].) While this section focused on differential operators, we conclude with a very brief remark on half-line Jacobi, that is, tridiagonal (and hence, second-order finite difference) operators: As discussed in depth by Simon [@Si98], the Friedrichs and Krein–von Neumann extensions of a minimally defined symmetric half-line Jacobi operator (cf. also [@BC05]) coincide, if and only if the associated Stieltjes moment problem is determinate (i.e., has a unique solution) while the corresponding Hamburger moment problem is indeterminate (and hence has uncountably many solutions). Examples {#s1vi} ======== The Case of a Bounded Interval $(a,b)$, $-\infty < a < b < \infty$, $V=0$. -------------------------------------------------------------------------- We briefly recall the essence of the one-dimensional example ${\Omega}=(a,b)$, $-\infty < a < b < \infty$, and $V=0$. This was first discussed in detail by [@AS80] and [@Fu80 Sect. 2.3] (see also [@FOT94 Sect. 3.3]). Consider the minimal operator $-\Delta_{min,(a,b)}$ in $L^2((a,b);dx)$, given by $$\begin{aligned} & -\Delta_{min,(a,b)} u = -u'', {\notag}\\ & \;u \in \operatorname{dom}(-\Delta_{min,(a,b)}) =\big\{v \in L^2((a,b);dx) \,\big|\, v, v' \in AC([a,b]); {\label}{10.1} \\ & \hspace*{1.7cm} v(a)=v'(a)=v(b)=v'(b)=0; \, v'' \in L^2((a,b);dx)\big\}, {\notag}\end{aligned}$$ where $AC([a,b])$ denotes the set of absolutely continuous functions on $[a,b]$. Evidently, $$-\Delta_{min,(a,b)} = {\overline}{- {\frac}{d^2}{dx^2}\bigg|_{C_0^\infty((a,b))}} \, ,$$ and one can show that $$-\Delta_{min,(a,b)} \geq [\pi/(b-a)]^2 I_{L^2((a,b);dx)}.$$ In addition, one infers that $$(-\Delta_{min,(a,b)})^* = -\Delta_{max,(a,b)},$$ where $$\begin{aligned} {\label}{10.5} & -\Delta_{max,(a,b)} u = -u'', {\notag}\\ & \; u \in \operatorname{dom}(-\Delta_{max,(a,b)}) = \big\{v \in L^2((a,b);dx) \,\big|\, v, v' \in AC([a,b]); \, v'' \in L^2((a,b);dx)\big\}. \end{aligned}$$ In particular, $${\rm def} (-\Delta_{min,(a,b)}) = (2,2) \, \text{ and } \, \ker ((-\Delta_{min,(a,b)})^*) = {\rm lin. \, span}\{1, x\}.$$ The Friedrichs (equivalently, the Dirichlet) extension $-\Delta_{D,(a,b)}$ of $-\Delta_{min,(a,b)}$ is then given by $$\begin{aligned} & -\Delta_{D,(a,b)} u = -u'', {\notag}\\ & \; u \in \operatorname{dom}(-\Delta_{D,(a,b)}) =\big\{v \in L^2((a,b);dx) \,\big|\, v, v' \in AC([a,b]); {\label}{10.7} \\ & \hspace*{3.85cm} v(a)=v(b)=0; \, v'' \in L^2((a,b);dx)\big\}. {\notag}\end{aligned}$$ In addition, $$\sigma(-\Delta_{D,(a,b)}) = \{j^2 \pi^2 (b-a)^{-2}\}_{j\in{{\mathbb{N}}}},$$ and $$\begin{aligned} \operatorname{dom}\big((-\Delta_{D,(a,b)})^{1/2}\big) = \big\{v \in L^2((a,b);dx) \,\big|\, v \in AC([a,b]); \, v(a)=v(b)=0; \, v' \in L^2((a,b);dx)\big\}.\end{aligned}$$ By , $$\operatorname{dom}(-\Delta_{K,(a,b)}) = \operatorname{dom}(-\Delta_{min,(a,b)}) \dotplus \ker((-\Delta_{min,(a,b)})^*),$$ and hence any $u \in \operatorname{dom}(-\Delta_{K,(a,b)})$ is of the type $$\begin{aligned} & u = f + \eta, \quad f \in \operatorname{dom}(-\Delta_{min,(a,b)}), \quad \eta (x) = u(a) + [u(b)-u(a)] \bigg({\frac}{x-a}{b-a}\bigg), \; x \in (a,b), \end{aligned}$$ in particular, $f(a)=f'(a)=f(b)=f'(b)=0$. Thus, the Krein–von Neumann extension $-\Delta_{K,(a,b)}$ of $-\Delta_{min,(a,b)}$ is given by $$\begin{aligned} & -\Delta_{K,(a,b)} u = -u'', {\notag}\\ & \; u \in \operatorname{dom}(-\Delta_{K,(a,b)}) =\big\{v \in L^2((a,b);dx) \,\big|\, v, v' \in AC([a,b]); {\label}{10.9} \\ & \hspace*{8mm} v'(a)=v'(b)=[v(b)-v(a)]/(b-a); \, v'' \in L^2((a,b);dx)\big\}. {\notag}\end{aligned}$$ Using the characterization of all self-adjoint extensions of general Sturm–Liouville operators in [@We03 Theorem 13.14], one can also directly verify that $-\Delta_{K,(a,b)}$ as given by is a self-adjoint extension of $-\Delta_{min,(a,b)}$. In connection with , , , and , we also note that the well-known fact that $$v, v'' \in L^2((a,b);dx) \, \text{ implies } \, v' \in L^2((a,b);dx). {\label}{10.10}$$ Utilizing , we briefly consider the quadratic form associated with the Krein Laplacian $-\Delta_{K,(a,b)}$. By and , one infers, $$\begin{aligned} &\operatorname{dom}\big((-\Delta_{K,(a,b)})^{1/2}\big) = \operatorname{dom}\big((-\Delta_{D,(a,b)})^{1/2}\big) \dotplus \ker ((-\Delta_{min,(a,b)})^*), {\label}{SKformab1} \\ &\big\|(-\Delta_{K,(a,b)})^{1/2}(u+g)\big\|_{L^2((a,b);dx)}^2 =\big\|(-\Delta_{D,(a,b)})^{1/2} u\big\|_{L^2((a,b);dx)}^2 {\notag}\\ & \quad = ((u+g)',(u+g)')_{L^2((a,b);dx)} - [{\overline}{g(b)}g'(b) - {\overline}{g(a)} g'(a)] {\notag}\\ & \quad = ((u+g)',(u+g)')_{L^2((a,b);dx)} - |[u(b) + g(b)] - [u(a) + g(a)]|^2/(b-a), {\notag}\\ & \hspace*{3cm} u \in \operatorname{dom}\big((-\Delta_{D,(a,b)})^{1/2}\big), \; g \in \ker ((-\Delta_{min,(a,b)})^*). {\label}{SKformab2}\end{aligned}$$ Finally, we turn to the spectrum of $-\Delta_{K,(a,b)}$. The boundary conditions in lead to two kinds of (nonnormalized) eigenfunctions and eigenvalue equations $$\begin{aligned} \begin{split} & \psi(k,x) = \cos(k(x-[(a+b)/2])), \quad k \sin(k(b-a)/2) = 0, \\ & k_{K,(a,b),j} = (j+1)\pi/(b-a), \; j=-1, 1, 3, 5, \dots, \end{split}\end{aligned}$$ and $$\begin{aligned} & \phi(k,x) = \sin(k(x-[(a+b)/2])), \quad k(b-a)/2 = \tan(k(b-a)/2) , {\notag}\\ & k_{K,(a,b),0} =0, \; j\pi < k_{K,(a,b),j} < (j+1)\pi, \; j=2, 4, 6, 8, \dots, \\ & \lim_{\ell\to\infty} [k_{K,(a,b),2\ell} - ((2\ell +1) \pi/(b-a))] =0. {\notag}\end{aligned}$$ The associated eigenvalues of $-\Delta_{K,(a,b)}$ are thus given by $$\sigma(-\Delta_{K,(a,b)}) = \{0\} \cup \{k_{K,(a,b),j}^2\}_{j\in{{\mathbb{N}}}},$$ where the eigenvalue $0$ of $-\Delta_{K,(a,b)}$ is of multiplicity two, but the remaining nonzero eigenvalues of $-\Delta_{K,(a,b)}$ are all simple. The Case of the Ball $B_n(0;R)$, $R>0$, in ${{\mathbb{R}}}^n$, $n\geq 2$, $V=0$. -------------------------------------------------------------------------------- In this subsection, we consider in great detail the scenario when the domain $\Omega$ equals a ball of radius $R>0$ (for convenience, centered at the origin) in ${{\mathbb{R}}}^n$, $${\Omega}=B_n(0;R)\subset{{\mathbb{R}}}^n, \quad R>0, \, n{\geqslant}2. {\label}{10.19}$$ Since both the domain $B_n(0;R)$ in , as well as the Laplacian $-\Delta$ are invariant under rotations in ${{\mathbb{R}}}^n$ centered at the origin, we will employ the (angular momentum) decomposition of $L^2(B_n(0;R); d^nx)$ into the direct sum of tensor products $$\begin{aligned} & L^2(B_n(0;R); d^nx) = L^2((0,R); r^{n-1}dr) \otimes L^2(S^{n-1}; d\omega_{n-1}) = \bigoplus_{\ell\in{{\mathbb{N}}}_0} {{\mathcal H}}_{n,\ell,(0,R)}, {\label}{10.20} \\ & {{\mathcal H}}_{n,\ell,(0,R)} = L^2((0,R); r^{n-1}dr) \otimes {{\mathcal K}}_{n,\ell}, \quad \ell \in {{\mathbb{N}}}_0, \; n\geq 2, {\label}{10.21}\end{aligned}$$ where $S^{n-1}= \partial B_n(0;1)=\{x\in{{\mathbb{R}}}^n\,|\, |x|=1\}$ denotes the $(n-1)$-dimensional unit sphere in ${{\mathbb{R}}}^n$, $d\omega_{n-1}$ represents the surface measure on $S^{n-1}$, $n\geq 2$, and ${{\mathcal K}}_{n,\ell}$ denoting the eigenspace of the Laplace–Beltrami operator $-\Delta_{S^{n-1}}$ in $L^2(S^{n-1}; d\omega_{n-1})$ corresponding to the $\ell$th eigenvalue $\kappa_{n,\ell}$ of $-\Delta_{S^{n-1}}$ counting multiplicity, $$\begin{aligned} \begin{split} & \kappa_{n,\ell} = \ell(\ell + n-2), \\ & \dim({{\mathcal K}}_{n,\ell}) = {\frac}{(2\ell+n-2)\Gamma(\ell+n-2)}{\Gamma(\ell+1)\Gamma(n-1)} : =d_{n,\ell}, \quad \ell\in{{\mathbb{N}}}_0, \; n\geq 2 {\label}{10.22} \end{split} \end{aligned}$$ (cf. [@Mu66 p. 4]). In other words, ${{\mathcal K}}_{n,\ell}$ is spanned by the $n$-dimensional spherical harmonics of degree $\ell\in{{\mathbb{N}}}_0$. For more details in this connection we refer to [@RS75 App. to Sect. X.1] and [@We03 Ch. 18]. As a result, the minimal Laplacian in $L^2(B_n(0;R);d^n x)$ can be decomposed as follows $$\begin{aligned} \begin{split} & -\Delta_{min,B_n(0;R)}= {\overline}{-\Delta|_{C_0^\infty(B_n(0;R))}} = \bigoplus_{\ell\in{{\mathbb{N}}}_0} H_{n,\ell,min}^{(0)} \otimes I_{{{\mathcal K}}_{n,\ell}}, \\ & \; \operatorname{dom}(-\Delta_{min,B_n(0;R)}) = H^2_0(B_n(0;R)), {\label}{10.23} \end{split}\end{aligned}$$ where $H_{n,\ell,min}^{(0)}$ in $L^2((0,R); r^{n-1}dr)$ are given by $$H_{n,\ell,min}^{(0)} = {\overline}{\bigg(-{\frac}{d^2}{dr^2} - {\frac}{n-1}{r}{\frac}{d}{dr} + {\frac}{\kappa_{n,\ell}}{r^2}\bigg)_{C_0^\infty((0,R))}} \, , \quad \ell \in {{\mathbb{N}}}_0.$$ Using the unitary operator $U_n$ defined by $$U_n \colon \begin{cases} L^2((0,R); r^{n-1}dr) \to L^2((0,R); dr), \\ \hspace*{2.55cm} \phi \mapsto (U_n \phi)(r) = r^{(n-1)/2} \phi(r), \end{cases}$$ it will also be convenient to consider the unitary transformation of $H_{n,\ell,min}^{(0)}$ given by $$h_{n,\ell,min}^{(0)} = U_n H_{n,\ell,min}^{(0)} U_n^{-1}, \quad \ell \in {{\mathbb{N}}}_0,$$ where $$\begin{aligned} & h_{n,0,min}^{(0)} = -{\frac}{d^2}{dr^2} + {\frac}{(n-1)(n-3)}{4 r^2}, \quad 0<r<R, {\notag}\\ & \operatorname{dom}\big(h_{n,0,min}^{(0)}\big) = \big\{f \in L^2((0,R); dr) \,\big|\, f, f' \in AC([\varepsilon,R]) \, \text{for all $\varepsilon>0$}; {\notag}\\ & \hspace*{6.1cm} f(R_-)=f'(R_-)=0, \, f_0=0; \\ & \hspace*{3.1cm} (-f'' +[(n-1)(n-3)/4]r^{-2}f) \in L^2((0,R); dr)\big\} \, \text{ for $n=2,3$}, {\notag}\\ & h_{n,\ell,min}^{(0)} = -{\frac}{d^2}{dr^2} + {\frac}{4 \kappa_{n,\ell} + (n-1)(n-3)}{4 r^2}, \quad 0<r<R, {\notag}\\ & \operatorname{dom}\big(h_{n,\ell,min}^{(0)}\big) = \big\{f \in L^2((0,R); dr) \,|\, f, f' \in AC([\varepsilon,R]) \, \text{for all $\varepsilon>0$}; {\notag}\\ & \hspace*{7.3cm} f(R_-)=f'(R_-)=0; \\ & \hspace*{1.9cm} (-f'' +[\kappa_{n,\ell} + ((n-1)(n-3)/4)]r^{-2}f) \in L^2((0,R); dr)\big\} {\notag}\\ & \hspace*{5.4cm} \text{ for $\ell\in{{\mathbb{N}}}$, $n\geq 2$ and $\ell=0$, $n\geq 4$.} {\notag}\end{aligned}$$ In particular, for $\ell\in{{\mathbb{N}}}$, $n\geq 2$, and $\ell=0$, $n\geq 4$, one obtains $$\begin{aligned} \begin{split} & h_{n,\ell,min}^{(0)} = {\overline}{\bigg(-{\frac}{d^2}{dr^2} + {\frac}{4 \kappa_{n,\ell} + (n-1)(n-3)}{4 r^2}\bigg)\bigg|_{C_0^\infty((0,R))}} \\ & \hspace*{3.2cm} \text{ for $\ell \in {{\mathbb{N}}}$, $n\geq 2$, and $\ell=0$, $n\geq 4$.} \end{split} \end{aligned}$$ On the other hand, for $n=2,3$, the domain of the closure of $h_{n,0,min}^{(0)}\big|_{C_0^\infty((0,R))}$ is strictly contained in that of $\operatorname{dom}\big(h_{n,0,min}^{(0)}\big)$, and in this case one obtains for $${\widehat}h_{n,0, min}^{(0)} = {\overline}{\bigg(-{\frac}{d^2}{dr^2} + {\frac}{(n-1)(n-3)}{4 r^2}\bigg)\bigg|_{C_0^\infty((0,R))}}, \quad n=2,3,$$ that $$\begin{aligned} & {\widehat}h_{n,0, min}^{(0)} = -{\frac}{d^2}{dr^2} + {\frac}{(n-1)(n-3)}{4 r^2}, \quad 0<r<R, {\notag}\\ & \operatorname{dom}\big({\widehat}h_{n,0, min}^{(0)}\big) = \big\{f\in L^2((0,R); dr) \,\big|\, f, f' \in AC([\varepsilon,R]) \, \text{for all $\varepsilon>0$}; {\notag}\\ & \hspace*{5.35cm} f(R_-)=f'(R_-)=0, \, f_0=f'_0=0; \\ & \hspace*{3.3cm} (-f'' +[(n-1)(n-3)/4]r^{-2}f) \in L^2((0,R); dr)\big\}. {\notag}\end{aligned}$$ Here we used the abbreviations (cf. [@BG85] for details) $$\begin{aligned} \begin{split} & f_0 = \begin{cases} \lim_{r\downarrow 0} [-r^{1/2}\ln(r)]^{-1} f(r), & n=2, \\ f(0_+), & n=3, \end{cases} \\ & f'_0 = \begin{cases} \lim_{r\downarrow 0} r^{-1/2} [f(r) + f_0 r^{1/2}\ln(r)], & n=2, \\ f'(0_+), & n=3. \end{cases} {\label}{10.33} \end{split} \end{aligned}$$ We also recall the adjoints of $h_{n,\ell, min}^{(0)}$ which are given by $$\begin{aligned} & \big(h_{n,0,min}^{(0)}\big)^* = -{\frac}{d^2}{dr^2} + {\frac}{(n-1)(n-3)}{4 r^2}, \quad 0<r<R, {\notag}\\ & \operatorname{dom}\big(\big(h_{n,0,min}^{(0)}\big)^*\big) = \big\{f \in L^2((0,R); dr) \,\big|\, f, f' \in AC([\varepsilon,R]) \, \text{for all $\varepsilon>0$}; {\label}{10.34} \\ & \hspace*{.4cm} f_0=0; \, (-f'' +[(n-1)(n-3)/4]r^{-2}f) \in L^2((0,R); dr)\big\} \, \text{ for $n=2,3$}, {\notag}\\ & \big(h_{n,\ell,min}^{(0)}\big)^* = -{\frac}{d^2}{dr^2} + {\frac}{4 \kappa_{n,\ell} + (n-1)(n-3)}{4 r^2}, \quad 0<r<R, {\notag}\\ & \operatorname{dom}\big(\big(h_{n,\ell,min}^{(0)}\big)^*\big) = \big\{f \in L^2((0,R); dr) \,\big|\, f, f' \in AC([\varepsilon,R]) \, \text{for all $\varepsilon>0$}; \\ & \hspace*{2.35cm} (-f'' +[\kappa_{n,\ell} + ((n-1)(n-3)/4)]r^{-2}f) \in L^2((0,R); dr)\big\} {\notag}\\ & \hspace*{5.85cm} \text{ for $\ell\in{{\mathbb{N}}}$, $n\geq 2$ and $\ell=0$, $n\geq 4$.} {\notag}\end{aligned}$$ In particular, $$h_{n,\ell,max}^{(0)} = \big(h_{n,\ell,min}^{(0)}\big)^*, \quad \ell\in{{\mathbb{N}}}_0, \; n\geq 2.$$ All self-adjoint extensions of $h_{n,\ell, min}^{(0)}$ are given by the following one-parameter families $h_{n,\ell, \alpha_{n,\ell}}^{(0)}$, $\alpha_{n,\ell} \in{{\mathbb{R}}}\cup\{\infty\}$, $$\begin{aligned} & h_{n,0,\alpha_{n,0}}^{(0)} = -{\frac}{d^2}{dr^2} + {\frac}{(n-1)(n-3)}{4 r^2}, \quad 0<r<R, {\notag}\\ & \operatorname{dom}\big(h_{n,0,\alpha_{n,0}}^{(0)}\big) = \big\{f \in L^2((0,R); dr) \,\big|\, f, f' \in AC([\varepsilon,R]) \, \text{for all $\varepsilon>0$}; {\notag}\\ & \hspace*{5.6cm} f'(R_-)+\alpha_{n,0}f(R_-)=0, \, f_0=0; {\label}{10.37} \\ & \hspace*{3cm} (-f'' +[(n-1)(n-3)/4]r^{-2}f) \in L^2((0,R); dr)\big\} \, \text{ for $n=2,3$}, {\notag}\\ & h_{n,\ell,\alpha_{n,\ell}}^{(0)} = -{\frac}{d^2}{dr^2} + {\frac}{4 \kappa_{n,\ell} + (n-1)(n-3)}{4 r^2}, \quad 0<r<R, {\notag}\\ & \operatorname{dom}\big(h_{n,\ell,\alpha_{n,\ell}}^{(0)}\big) = \big\{f \in L^2((0,R); dr) \,\big|\, f, f' \in AC([\varepsilon,R]) \, \text{for all $\varepsilon>0$}; {\notag}\\ & \hspace*{6.73cm} f'(R_-)+\alpha_{n,\ell}f(R_-)=0; {\label}{10.38} \\ & \hspace*{1.95cm} (-f'' +[\kappa_{n,\ell} + ((n-1)(n-3)/4)]r^{-2}f) \in L^2((0,R); dr)\big\} {\notag}\\ & \hspace*{5.5cm} \text{ for $\ell\in{{\mathbb{N}}}$, $n\geq 2$ and $\ell=0$, $n\geq 4$.} {\notag}\end{aligned}$$ Here, in obvious notation, the boundary condition for $\alpha_{n,\ell}=\infty$ simply represents the Dirichlet boundary condition $f(R_-)=0$. In particular, the Friedrichs or Dirichlet extension $h_{n,\ell, D}^{(0)}$ of $h_{n,\ell, min}^{(0)}$ is given by $h_{n,\ell, \infty}^{(0)}$, that is, by $$\begin{aligned} & h_{n,0,D}^{(0)} = -{\frac}{d^2}{dr^2} + {\frac}{(n-1)(n-3)}{4 r^2}, \quad 0<r<R, {\notag}\\ & \operatorname{dom}\big(h_{n,0,D}^{(0)}\big) = \big\{f \in L^2((0,R); dr) \,\big|\, f, f' \in AC([\varepsilon,R]) \, \text{for all $\varepsilon>0$}; \, f(R_-)=0, {\notag}\\ & \hspace*{.4cm} f_0=0; \, (-f'' +[(n-1)(n-3)/4]r^{-2}f) \in L^2((0,R); dr)\big\} \, \text{ for $n=2,3$}, {\label}{10.39} \\ & h_{n,\ell,D}^{(0)} = -{\frac}{d^2}{dr^2} + {\frac}{4 \kappa_{n,\ell} + (n-1)(n-3)}{4 r^2}, \quad 0<r<R, {\notag}\\ & \operatorname{dom}\big(h_{n,\ell,D}^{(0)}\big) = \big\{f \in L^2((0,R); dr) \,\big|\, f, f' \in AC([\varepsilon,R]) \, \text{for all $\varepsilon>0$}; \, f(R_-)=0; {\notag}\\ & \hspace*{3.5cm} (-f'' +[\kappa_{n,\ell} + ((n-1)(n-3)/4)]r^{-2}f) \in L^2((0,R); dr)\big\} {\notag}\\ & \hspace*{7.1cm} \text{ for $\ell\in{{\mathbb{N}}}$, $n\geq 2$ and $\ell=0$, $n\geq 4$.} {\label}{10.40}\end{aligned}$$ To find the boundary condition for the Krein–von Neumann extension $h_{n,\ell,K}^{(0)}$ of $h_{n,\ell,min}^{(0)}$, that is, to find the corresponding boundary condition parameter $\alpha_{n,\ell,K}$ in , , we recall , that is, $$\operatorname{dom}\big(h_{n,\ell,K}^{(0)}\big) = \operatorname{dom}\big(h_{n,\ell,min}^{(0)}\big) \dotplus \ker\big(\big(h_{n,\ell,min}^{(0)}\big)^*\big).$$ By inspection, the general solution of $$\bigg(-{\frac}{d^2}{dr^2} + {\frac}{4 \kappa_{n,\ell} + (n-1)(n-3)}{4 r^2}\bigg) \psi(r) = 0, \quad r \in (0,R), {\label}{10.41}$$ is given by $$\psi(r) = A r^{\ell +[(n-1)/2]} + B r^{-\ell - [(n-3)/2]}, \quad A, B \in {{\mathbb{C}}}, \; r \in (0,R). {\label}{10.42}$$ However, for $\ell\geq 1$, $n \geq 2$ and for $\ell=0$, $n\geq 4$, the requirement $\psi \in L^2((0,R); dr)$ requires $B=0$ in . Similarly, also the requirement $\psi_0=0$ (cf. ) for $\ell=0$, $n=2,3$, enforces $B=0$ in . Hence, any $u \in \operatorname{dom}\big(h_{n,\ell,K}^{(0)}\big)$ is of the type $$u = f + \eta, \quad f \in \operatorname{dom}\big(h_{n,\ell,min}^{(0)}\big), \quad \eta (r) = u(R_-)r^{\ell +[(n-1)/2]}, \; r \in [0,R),$$ in particular, $f(R_-)=f'(R_-)=0$. Denoting by $\alpha_{n,\ell,K}$ the boundary condition parameter for $h_{n,\ell,K}^{(0)}$ one thus computes $$-\alpha_{n,\ell,K} = {\frac}{u'(R_-)}{u(R_-)} = {\frac}{\eta'(R_-)}{\eta(R_-)} = [\ell + ((n-1)/2)]/R.$$ Thus, the Krein–von Neumann extension $h_{n,\ell,K}^{(0)}$ of $h_{n,\ell,min}^{(0)}$ is given by $$\begin{aligned} & h_{n,0,K}^{(0)} = -{\frac}{d^2}{dr^2} + {\frac}{(n-1)(n-3)}{4 r^2}, \quad 0<r<R, {\notag}\\ & \operatorname{dom}\big(h_{n,0,K}^{(0)}\big) = \big\{f \in L^2((0,R); dr) \,\big|\, f, f' \in AC([\varepsilon,R]) \, \text{for all $\varepsilon>0$}; {\notag}\\ & \hspace*{3.6cm} f'(R_-)- [(n-1)/2]R^{-1} f(R_-)=0, \, f_0=0; {\label}{10.46} \\ & \hspace*{3cm} (-f'' +[(n-1)(n-3)/4]r^{-2}f) \in L^2((0,R); dr)\big\} \, \text{ for $n=2,3$}, {\notag}\\ & h_{n,\ell,K}^{(0)} = -{\frac}{d^2}{dr^2} + {\frac}{4 \kappa_{n,\ell} + (n-1)(n-3)}{4 r^2}, \quad 0<r<R, {\notag}\\ & \operatorname{dom}\big(h_{n,\ell,K}^{(0)}\big) = \big\{f \in L^2((0,R); dr) \,\big|\, f, f' \in AC([\varepsilon,R]) \, \text{for all $\varepsilon>0$}; {\notag}\\ & \hspace*{3.94cm} f'(R_-)-[\ell +((n-1)/2)] R^{-1} f(R_-)=0; {\label}{10.47} \\ & \hspace*{1.65cm} (-f'' +[\kappa_{n,\ell} + ((n-1)(n-3)/4)]r^{-2}f) \in L^2((0,R); dr)\big\} {\notag}\\ & \hspace*{5.15cm} \text{ for $\ell\in{{\mathbb{N}}}$, $n\geq 2$ and $\ell=0$, $n\geq 4$.} {\notag}\end{aligned}$$ Next we briefly turn to the eigenvalues of $h_{n,\ell,D}^{(0)}$ and $h_{n,\ell,K}^{(0)}$. In analogy to , the solution $\psi$ of $$\bigg(-{\frac}{d^2}{dr^2} + {\frac}{4 \kappa_{n,\ell} + (n-1)(n-3)}{4 r^2} -z \bigg) \psi(r,z) = 0, \quad r \in (0,R), {\label}{10.48}$$ satisfying the condition $\psi(\cdot,z) \in L^2((0,R); dr)$ for $\ell=0$, $n\geq 4$ and $\psi_0(z)=0$ (cf. ) for $\ell=0$, $n=2,3$, yields $$\psi(r,z) = A r^{1/2} J_{l+[(n-2)/2]}(z^{1/2} r), \quad A \in {{\mathbb{C}}}, \; r \in (0,R), {\label}{10.49}$$ Here $J_{\nu}(\cdot)$ denotes the Bessel function of the first kind and order $\nu$ (cf. [@AS72 Sect. 9.1]). Thus, by the boundary condition $f(R_-)=0$ in , , the eigenvalues of the Dirichlet extension $h_{n,\ell,D}^{(0)}$ are determined by the equation $\psi(R_-,z)=0$, and hence by $$J_{l+[(n-2)/2]}(z^{1/2}R) = 0.$$ Following [@AS72 Sect. 9.5], we denote the zeros of $J_{\nu}(\cdot)$ by $j_{\nu,k}$, $k\in{{\mathbb{N}}}$, and hence obtain for the spectrum of $h_{n,\ell,F}^{(0)}$, $$\sigma\big(h_{n,\ell,D}^{(0)}\big) = \big\{\lambda^{(0)}_{n,\ell,D,k}\big\}_{k\in{{\mathbb{N}}}} = \big\{ j_{\ell+[(n-2)/2],k}^2R^{-2}\big\}_{k\in{{\mathbb{N}}}}, \quad \ell \in {{\mathbb{N}}}_0, \; n\geq 2. {\label}{10.51}$$ Each eigenvalue of of $h_{n,\ell,D}^{(0)}$ is simple. Similarly, by the boundary condition $f'(R_-) - [\ell +((n-1)/2)]R^{-1}f(R_-)=0$ in , , the eigenvalues of the Krein–von Neumann extension $h_{n,\ell,K}^{(0)}$ are determined by the equation $$\psi'(R,z) - [\ell+ ((n-1)/2)] \psi(R,z) = - A z^{1/2} R^{1/2} J_{\ell+(n/2)}(z^{1/2}R)=0$$ (cf. [@AS72 eq. (9.1.27)]), and hence by $$z^{1/2} J_{\ell+(n/2)}(z^{1/2}R) = 0.$$ Thus, one obtains for the spectrum of $h_{n,\ell,K}^{(0)}$, $$\sigma\big(h_{n,\ell,K}^{(0)}\big) = \{0\} \cup \big\{\lambda^{(0)}_{n,\ell,K,k}\big\}_{k\in{{\mathbb{N}}}} = \{0\} \cup \big\{ j_{\ell+(n/2),k}^2R^{-2}\big\}_{k\in{{\mathbb{N}}}}, \quad \ell \in {{\mathbb{N}}}_0, \; n\geq 2. {\label}{10.54}$$ Again, each eigenvalue of $h_{n,\ell,K}^{(0)}$ is simple, and $\eta (r) = Cr^{\ell +[(n-1)/2]}$, $C\in{{\mathbb{C}}}$, represents the (unnormalized) eigenfunction of $h_{n,\ell,K}^{(0)}$ corresponding to the eigenvalue $0$. Combining Propositions \[p2.2a\]–\[p2.4\], one then obtains $$\begin{aligned} & -\Delta_{max,B_n(0;R)} = (-\Delta_{min,B_n(0;R)})^* = \bigoplus_{\ell\in{{\mathbb{N}}}_0} \big(H_{n,\ell,min}^{(0)}\big)^* \otimes I_{{{\mathcal K}}_{n,\ell}}, \\ & -\Delta_{D,B_n(0;R)} = \bigoplus_{\ell\in{{\mathbb{N}}}_0} H_{n,\ell,D}^{(0)} \otimes I_{{{\mathcal K}}_{n,\ell}}, \\ & -\Delta_{K,B_n(0;R)} = \bigoplus_{\ell\in{{\mathbb{N}}}_0} H_{n,\ell,K}^{(0)} \otimes I_{{{\mathcal K}}_{n,\ell}}, \end{aligned}$$ where (cf. ) $$\begin{aligned} & H_{n,\ell,max}^{(0)} = \big(H_{n,\ell,min}^{(0)}\big)^* = U_n^{-1} \big(h_{n,\ell,min}^{(0)}\big)^* U_n, \quad \ell \in {{\mathbb{N}}}_0, \\ & H_{n,\ell,D}^{(0)} = U_n^{-1} h_{n,\ell,D}^{(0)} U_n, \quad \ell \in {{\mathbb{N}}}_0, \\ & H_{n,\ell,K}^{(0)} = U_n^{-1} h_{n,\ell,K}^{(0)} U_n, \quad \ell \in {{\mathbb{N}}}_0.\end{aligned}$$ Consequently, $$\begin{aligned} & \sigma( -\Delta_{D,B_n(0;R)}) = \big\{\lambda^{(0)}_{n,\ell,D,k}\big\}_{\ell\in{{\mathbb{N}}}_0, k\in{{\mathbb{N}}}} = \big\{ j_{\ell+[(n-2)/2],k}^2R^{-2}\big\}_{\ell\in{{\mathbb{N}}}_0, k\in{{\mathbb{N}}}}, \\ & \sigma_{\rm ess}( -\Delta_{D,B_n(0;R)}) = \emptyset, \\ & \sigma( -\Delta_{K,B_n(0;R)}) = \{0\} \cup \big\{\lambda^{(0)}_{n,\ell,K,k}\big\}_{\ell\in{{\mathbb{N}}}_0, k\in{{\mathbb{N}}}} = \{0\} \cup \big\{ j_{\ell+(n/2),k}^2R^{-2}\big\}_{\ell\in{{\mathbb{N}}}_0, k\in{{\mathbb{N}}}}, \\ & \dim(\ker( -\Delta_{K,B_n(0;R)})) = \infty, \quad \sigma_{\rm ess}( -\Delta_{K,B_n(0;R)}) = \{0\}. \end{aligned}$$ By , each eigenvalue $\lambda^{(0)}_{n,\ell,D,k}$, $k\in{{\mathbb{N}}}$, of $-\Delta_{D,B_n(0;R)}$ has multiplicity $d_{n,\ell}$ and similarly, again by , each eigenvalue $\lambda^{(0)}_{n,\ell,K,k}$, $k\in{{\mathbb{N}}}$, of $ -\Delta_{K,B_n(0;R)}$ has multiplicity $d_{n,\ell}$. Finally, we briefly turn to the Weyl asymptotics for the eigenvalue counting function associated with the Krein Laplacian $-\Delta_{K,B_n(0;R)}$ for the ball $B_n(0;R)$, $R>0$, in ${{\mathbb{R}}}^n$, $n\geq 2$. We will discuss a direct approach to the Weyl asymptotics that is independent of the general treatment presented in Section \[s12\]. Due to the smooth nature of the ball, we will obtain an improvement in the remainder term of the Weyl asymptotics of the Krein Laplacian. First we recall the well-known fact that in the case of the Dirichlet Laplacian associated with the ball $B_n(0;R)$, $$\begin{aligned} \label{10.65} N^{(0)}_{D,B_n(0;R)}(\lambda) &= (2\pi)^{-n}v_n^2 R^n\lambda^{n/2} - (2\pi)^{-(n-1)} v_{n-1} (n/4) v_n R^{n-1} \lambda^{(n-1)/2} {\notag}\\ & \quad + O\big(\lambda^{(n-2)/2}\big) \, \mbox{ as }\, \lambda\to\infty, \end{aligned}$$ with $v_n=\pi^{n/2}/ \Gamma((n/2)+1)$ the volume of the unit ball in ${{\mathbb{R}}}^n$ (and $n v_n$ representing the surface area of the unit ball in ${{\mathbb{R}}}^n$). \[p10.1\] The strictly positive eigenvalues of the Krein Laplacian associated with the ball of radius $R>0$, $B_n(0;R)\subset {{\mathbb{R}}}^n$, $R>0$, $n\geq 2$, satisfy the following Weyl-type eigenvalue asymptotics, $$\begin{aligned} \label{10.66} N^{(0)}_{K,B_n(0;R)}(\lambda)&=(2\pi)^{-n}v_n^2 R^n \lambda^{n/2} - (2\pi)^{-(n-1)} v_{n-1} [(n/4) v_n + v_{n-1}] R^{n-1} \lambda^{(n-1)/2} {\notag}\\ & \quad + O\big(\lambda^{(n-2)/2}\big) \, \mbox{ as }\, \lambda\to\infty. \end{aligned}$$ From the outset one observes that $$\lambda^{(0)}_{n,\ell,D,k} {\leqslant}\lambda^{(0)}_{n,\ell,K,k} {\leqslant}\lambda^{(0)}_{n,\ell,D,k+1}, \quad \ell\in{{\mathbb{N}}}_0, \; k\in{{\mathbb{N}}},$$ implying $$\label{10.67} N^{(0)}_{K,B_n(0;R)}(\lambda) {\leqslant}N^{(0)}_{D,B_n(0;R)}(\lambda), \quad \lambda\in{{\mathbb{R}}}.$$ Next, introducing $${{\mathcal N}}_\nu (\lambda) :=\begin{cases} \text{the largest $k\in{{\mathbb{N}}}$ such that $j_{\nu,k}^2 R^{-2} \leq \lambda$}, \\ 0, \text{ if no such $k\geq 1$ exists,} \end{cases} \quad \lambda \in{{\mathbb{R}}},$$ we note the well-known monotonicity of $j_{\nu,k}$ with respect to $\nu$ (cf. [@Wa96 Sect. 15.6, p. 508]), implying that for each $\lambda \in {{\mathbb{R}}}$ (and fixed $R>0$), $${{\mathcal N}}_{\nu'} (\lambda) \leq {{\mathcal N}}_{\nu} (\lambda) \, \text{ for } \, \nu' \geq \nu \geq 0.$$ Then one infers $$N^{(0)}_{D,B_n(0;R)}(\lambda) = \sum_{\ell\in{{\mathbb{N}}}_0} d_{n,\ell} \, {{\mathcal N}}_{(n/2)-1+\ell} (\lambda), \quad N^{(0)}_{K,B_n(0;R)}(\lambda) = \sum_{\ell\in{{\mathbb{N}}}_0} d_{n,\ell} \, {{\mathcal N}}_{(n/2)+\ell} (\lambda).$$ Hence, using the fact that $$d_{n,\ell} = d_{n-1,\ell} + d_{n,\ell-1}$$ (cf. ), setting $d_{n, -1} =0$, $n\geq 2$, one computes $$\begin{aligned} N^{(0)}_{D,B_n(0;R)}(\lambda) &= \sum_{\ell\in{{\mathbb{N}}}} d_{n,\ell-1} \, {{\mathcal N}}_{(n/2)-1+\ell} (\lambda) + \sum_{\ell\in{{\mathbb{N}}}_0} d_{n-1,\ell} \, {{\mathcal N}}_{(n/2)-1+\ell} (\lambda) {\notag}\\ & \leq \sum_{\ell\in{{\mathbb{N}}}_0} d_{n,\ell} \, {{\mathcal N}}_{(n/2)+\ell} (\lambda) + \sum_{\ell\in{{\mathbb{N}}}_0} d_{n-1,\ell} \, {{\mathcal N}}_{((n-1)/2)-1+\ell} (\lambda) {\notag}\\ & = N^{(0)}_{K,B_n(0;R)}(\lambda) + N^{(0)}_{D,B_{n-1}(0;R)}(\lambda),\end{aligned}$$ that is, $$N^{(0)}_{D,B_n(0;R)}(\lambda) \leq N^{(0)}_{K,B_n(0;R)}(\lambda) + N^{(0)}_{D,B_{n-1}(0;R)}(\lambda). {\label}{10.74a}$$ Similarly, $$\begin{aligned} N^{(0)}_{D,B_n(0;R)}(\lambda) &= \sum_{\ell\in{{\mathbb{N}}}} d_{n,\ell-1} \, {{\mathcal N}}_{(n/2)-1+\ell} (\lambda) + \sum_{\ell\in{{\mathbb{N}}}_0} d_{n-1,\ell} \, {{\mathcal N}}_{(n/2)-1+\ell} (\lambda) {\notag}\\ & \geq \sum_{\ell\in{{\mathbb{N}}}_0} d_{n,\ell} \, {{\mathcal N}}_{(n/2)+\ell} (\lambda) + \sum_{\ell\in{{\mathbb{N}}}_0} d_{n-1,\ell} \, {{\mathcal N}}_{((n-1)/2)+\ell} (\lambda) {\notag}\\ & = N^{(0)}_{K,B_n(0;R)}(\lambda) + N^{(0)}_{K,B_{n-1}(0;R)}(\lambda), \end{aligned}$$ that is, $$N^{(0)}_{D,B_n(0;R)}(\lambda) \geq N^{(0)}_{K,B_n(0;R)}(\lambda) + N^{(0)}_{K,B_{n-1}(0;R)}(\lambda), {\label}{10.76a}$$ and hence, $$N^{(0)}_{K,B_{n-1}(0;R)}(\lambda) \leq \big[N^{(0)}_{D,B_n(0;R)}(\lambda) - N^{(0)}_{K,B_n(0;R)}(\lambda)\big] \leq N^{(0)}_{D,B_{n-1}(0;R)}(\lambda). {\label}{10.77A}$$ Thus, using $$0 \leq \big[N^{(0)}_{D,B_n(0;R)}(\lambda) - N^{(0)}_{K,B_n(0;R)}(\lambda)\big] \leq N^{(0)}_{D,B_{n-1}(0;R)}(\lambda) = {O}\big(\lambda^{(n-1)/2}\big) \, \text{ as $\lambda\to\infty$,}$$ one first concludes that $\big[N^{(0)}_{D,B_n(0;R)}(\lambda) - N^{(0)}_{K,B_n(0;R)}(\lambda)\big] = {O}\big(\lambda^{(n-1)/2}\big)$ as $\lambda\to\infty$, and hence using , $$N^{(0)}_{K,B_n(0;R)}(\lambda) = (2\pi)^{-n}v_n^2 R^n \lambda^{n/2} + {O}\big(\lambda^{(n-1)/2}\big) \, \mbox{ as }\, \lambda\to\infty.$$ This type of reasoning actually yields a bit more: Dividing by $\lambda^{(n-1)/2}$, and using that both, $N^{(0)}_{D,B_{n-1}(0;R)}(\lambda)$ and $N^{(0)}_{K,B_{n-1}(0;R)}(\lambda)$ have the same leading asymptotics $(2\pi)^{-(n-1)}v_{n-1}^2 R^{n-1}\lambda^{(n-1)/2}$ as $\lambda \to\infty$, one infers, using again, $$\begin{aligned} N^{(0)}_{K,B_n(0;R)}(\lambda) &= N^{(0)}_{D,B_n(0;R)}(\lambda) - \big[N^{(0)}_{D,B_n(0;R)}(\lambda) - N^{(0)}_{K,B_n(0;R)}(\lambda)\big] {\notag}\\ &= N^{(0)}_{D,B_n(0;R)}(\lambda) - (2\pi)^{-(n-1)} v_{n-1}^2 R^{n-1} \lambda^{(n-1)/2} + {o}\big(\lambda^{(n-1)/2}\big) {\notag}\\ &=(2\pi)^{-n}v_n^2 R^n \lambda^{n/2} - (2\pi)^{-(n-1)} v_{n-1} [(n/4) v_n + v_{n-1}] R^{n-1} \lambda^{(n-1)/2} {\notag}\\ & \quad + {o}\big(\lambda^{(n-1)/2}\big) \, \mbox{ as }\, \lambda\to\infty. {\label}{10.79A}\end{aligned}$$ Finally, it is possible to improve the remainder term in from ${o}\big(\lambda^{(n-1)/2}\big)$ to $O\big(\lambda^{(n-2)/2}\big)$ as follows: Replacing $n$ by $n-1$ in yields $$N^{(0)}_{D,B_{n-1}(0;R)}(\lambda) \leq N^{(0)}_{K,B_{n-1}(0;R)}(\lambda) + N^{(0)}_{D,B_{n-2}(0;R)}(\lambda). {\label}{10.80A}$$ Insertion of into permits one to eliminate $N_{K,B_{n-1}(0;R)}^{(0)}$ as follows: $$N^{(0)}_{D,B_n(0;R)}(\lambda) \geq N^{(0)}_{K,B_n(0;R)}(\lambda) + N^{(0)}_{D,B_{n-1}(0;R)}(\lambda) - N^{(0)}_{D,B_{n-2}(0;R)}(\lambda), {\label}{10.81A}$$ implies $$\begin{aligned} \begin{split} & \big[N^{(0)}_{D,B_{n}(0;R)}(\lambda) - N^{(0)}_{D,B_{n-1}(0;R)}(\lambda)\big] \leq N^{(0)}_{K,B_n(0;R)}(\lambda) \\ & \quad \leq \big[N^{(0)}_{D,B_{n}(0;R)}(\lambda) - N^{(0)}_{D,B_{n-1}(0;R)}(\lambda)\big]+ N^{(0)}_{D,B_{n-2}(0;R)}(\lambda), \end{split} \end{aligned}$$ and hence, $$0 \leq N^{(0)}_{K,B_n(0;R)}(\lambda) - \big[N^{(0)}_{D,B_{n}(0;R)}(\lambda) - N^{(0)}_{D,B_{n-1}(0;R)}(\lambda)\big] \leq N^{(0)}_{D,B_{n-2}(0;R)}(\lambda).$$ Thus, $N^{(0)}_{K,B_n(0;R)}(\lambda) - \big[N^{(0)}_{D,B_{n}(0;R)}(\lambda) - N^{(0)}_{D,B_{n-1}(0;R)}(\lambda)\big] = {O}\big(\lambda^{(n-2)/2}\big)$ as $\lambda\to \infty$, proving . Due to the smoothness of the domain $B_n(0;R)$, the remainder terms in represent a marked improvement over the general result for domains ${\Omega}$ satisfying Hypothesis \[h.VK\]. A comparison of the second term in the asymptotic relations and exhibits the difference between Dirichlet and Krein–von Neumann eigenvalues. The Case ${\Omega}={{\mathbb{R}}}^n\backslash\{0\}$, $n=2,3$, $V=0$. -------------------------------------------------------------------- [s10.3]{} In this subsection we consider the following minimal operator $-\Delta_{min, {{\mathbb{R}}}^n\backslash\{0\}}$ in $L^2({{\mathbb{R}}}^n;d^nx)$, $n=2,3$, $$-\Delta_{min, {{\mathbb{R}}}^n\backslash\{0\}} ={\overline}{-\Delta\big|_{C^\infty_0 ({{\mathbb{R}}}^n\backslash\{0\})}}\geq 0, \quad n=2,3. {\label}{10.74}$$ Then $$\begin{aligned} \begin{split} & H_{F,{{\mathbb{R}}}^2\backslash\{0\}}=H_{K,{{\mathbb{R}}}^2\backslash\{0\}} =-\Delta, \\ & \operatorname{dom}(H_{F,{{\mathbb{R}}}^2\backslash\{0\}}) = \operatorname{dom}(H_{K,{{\mathbb{R}}}^2\backslash\{0\}})=H^{2}({{\mathbb{R}}}^2) \, \text{ if } \, n=2 {\label}{10.75} \end{split} \end{aligned}$$ is the unique nonnegative self-adjoint extension of $-\Delta_{min, {{\mathbb{R}}}^2\backslash\{0\}}$ in $L^2({{\mathbb{R}}}^2;d^2x)$ and $$\begin{aligned} \begin{split} & H_{F,{{\mathbb{R}}}^3\backslash\{0\}}=H_{D,{{\mathbb{R}}}^3\backslash\{0\}} =-\Delta, \\ & \operatorname{dom}(H_{F,{{\mathbb{R}}}^3\backslash\{0\}}) = \operatorname{dom}(H_{D,{{\mathbb{R}}}^3\backslash\{0\}}) =H^{2}({{\mathbb{R}}}^3) \, \text{ if } \, n=3, {\label}{10.76} \end{split} \\ &H_{K,{{\mathbb{R}}}^3\backslash\{0\}} = H_{N,{{\mathbb{R}}}^3\backslash\{0\}} =U^{-1}h_{0,N,{{\mathbb{R}}}_+}^{(0)}U \oplus\bigoplus_{\ell\in{{\mathbb{N}}}} U^{-1}h_{\ell,{{\mathbb{R}}}_+}^{(0)} U \, \text{ if } \, n=3, {\label}{10.77}\end{aligned}$$ where $H_{D,{{\mathbb{R}}}^3\backslash\{0\}}$ and $H_{N,{{\mathbb{R}}}^3\backslash\{0\}}$ denote the Dirichlet and Neumann[^3] extension of $-\Delta_{min, {{\mathbb{R}}}^n\backslash\{0\}}$ in $L^2({{\mathbb{R}}}^3; d^3 x)$, respectively. Here we used the angular momentum decomposition (cf. also , ), $$\begin{aligned} & L^2({{\mathbb{R}}}^n; d^nx) = L^2((0,\infty); r^{n-1}dr) \otimes L^2(S^{n-1}; d\omega_{n-1}) = \bigoplus_{\ell\in{{\mathbb{N}}}_0} {{\mathcal H}}_{n,\ell,(0,\infty)}, {\label}{10.77a} \\ & {{\mathcal H}}_{n,\ell,(0,\infty)} = L^2((0,\infty); r^{n-1}dr) \otimes {{\mathcal K}}_{n,\ell}, \quad \ell \in {{\mathbb{N}}}_0, \; n=2,3. {\label}{10.77b}\end{aligned}$$ Moreover, we abbreviated ${{\mathbb{R}}}_+=(0,\infty)$ and introduced $$\begin{aligned} &h_{0,N,{{\mathbb{R}}}_+}^{(0)}=-{\frac}{d^2}{dr^2}, \quad r>0, {\notag}\\ &\operatorname{dom}\big(h_{0,N,{{\mathbb{R}}}_+}^{(0)}\big)= \big\{f\in L^2((0,\infty);dr)\,\big|\, f,f'\in AC([0,R]) \text{ for all } R>0; {\label}{10.78} \\ & \hspace*{5.95cm} f'(0_+)=0; \, f''\in L^2((0,\infty);dr)\big\}, {\notag}\\ &h_{\ell,{{\mathbb{R}}}_+}^{(0)}=-{\frac}{d^2}{dr^2}+{\frac}{\ell(\ell +1)}{r^2}, \quad r>0, {\notag}\\ &\operatorname{dom}\big(h_{\ell,{{\mathbb{R}}}_+}^{(0)}\big)= \big\{f\in L^2((0,\infty);dr)\,\big|\, f,f'\in AC([0,R]) \text{ for all } R>0; {\label}{10.79} \\ &\hspace*{4.65cm} -f''+\ell(\ell +1)r^{-2}f\in L^2((0,\infty);dr)\big\}, \quad \, \ell\in{{\mathbb{N}}}. {\notag}\end{aligned}$$ The operators $h_{\ell,{{\mathbb{R}}}_+}^{(0)}|_{C_0^{\infty}((0,\infty))}$, $\ell\in{{\mathbb{N}}}$, are essentially self-adjoint in $L^2((0,\infty);dr)$ (but we note that $f\in \operatorname{dom}\big(h_{\ell,{{\mathbb{R}}}_+}^{(0)}\big)$ implies that $f(0_+)=0$). In addition, $U$ in denotes the unitary operator, $$U:\begin{cases} L^2((0,\infty); r^2dr)\to L^2((0,\infty);dr), \\ \hspace*{1.81cm} f(r)\mapsto (Uf)(r)= r f(r). \end{cases} {\label}{10.80}$$ As discussed in detail in [@GKMT01 Sects. 4, 5], equations – follow from Corollary 4.8 in [@GKMT01] and the facts that $${\label}{10.81} (u_+,M_{H_{F,{{\mathbb{R}}}^n\backslash\{0\}},{{\mathcal N}}_+}(z)u_+)_{L^2({{\mathbb{R}}}^n;d^nx)} = \begin{cases} -(2/\pi) \ln(z) +2i, & n=2, \\ i(2z)^{1/2} +1, & n=3, \end{cases}$$ and $$(u_+,M_{H_{K,{{\mathbb{R}}}^3\backslash\{0\}},{{\mathcal N}}_+}(z)u_+)_{L^2({{\mathbb{R}}}^3;d^3x)} = i(2/z)^{1/2} - 1. {\label}{10.82}$$ Here $$\begin{aligned} \begin{split} & {{\mathcal N}}_+= {\rm lin. \, span}\{u_+\}, \\ & u_+ (x) = G_0(i,x,0)/\|G_0(i,\cdot,0)\|_{L^2({{\mathbb{R}}}^n;d^nx)}, \; x\in{{\mathbb{R}}}^n\backslash\{0\}, \; n=2,3, {\label}{10.83} \end{split} \end{aligned}$$ and $${\label}{10.84} G_0(z,x,y) = \begin{cases} {\frac}{i}{4}H_0^{(1)}(z^{1/2}|x-y|), &x\neq y, \, n=2, \\ e^{iz^{1/2}|x-y|}/(4\pi |x-y|), &x\neq y, \, n=3 \end{cases}$$ denotes the Green’s function of $-\Delta$ defined on $H^{2}({{\mathbb{R}}}^n),$ $n=2,3$ (i.e., the integral kernel of the resolvent $(-\Delta -z)^{-1}$), and $H_0^{(1)}(\cdot)$ abbreviates the Hankel function of the first kind and order zero (cf., [@AS72 Sect. 9.1]). Here the Donoghue-type Weyl–Titchmarsh operators (cf. [@Do65] in the case where $\dim ({{\mathcal N}}_+) =1$ and [@GKMT01], [@GMT98], and [@GT00] in the general abstract case where $\dim ({{\mathcal N}}_+) \in {{\mathbb{N}}}\cup \{\infty\}$) $M_{H_{F,{{\mathbb{R}}}^n\backslash\{0\}},{{\mathcal N}}_+}$ and $M_{H_{K,{{\mathbb{R}}}^n\backslash\{0\}},{{\mathcal N}}_+}$ are defined according to equation (4.8) in [@GKMT01]: More precisely, given a self-adjoint extension ${\widetilde}S$ of the densely defined closed symmetric operator $S$ in a complex separable Hilbert space ${{\mathcal H}}$, and a closed linear subspace ${{\mathcal N}}$ of ${{\mathcal N}}_+ = \ker({S}^* - i I_{{{\mathcal H}}})$, ${{\mathcal N}}\subseteq {{\mathcal N}}_+$, the Donoghue-type Weyl–Titchmarsh operator $M_{{\widetilde}S,{{\mathcal N}}}(z) \in{{\mathcal B}}({{\mathcal N}})$ associated with the pair $({\widetilde}S,{{\mathcal N}})$ is defined by $$\begin{aligned} \begin{split} M_{{\widetilde}S,{{\mathcal N}}}(z)&=P_{{\mathcal N}}(z{\widetilde}S+I_{{\mathcal H}})({\widetilde}S-z I_{{{\mathcal H}}})^{-1} P_{{\mathcal N}}\big\vert_{{\mathcal N}}\\ &=zI_{{\mathcal N}}+(1+z^2)P_{{\mathcal N}}({\widetilde}S-z I_{{{\mathcal H}}})^{-1} P_{{\mathcal N}}\big\vert_{{\mathcal N}}\,, \quad z\in {{\mathbb{C}}}\backslash {{\mathbb{R}}}, {\label}{10.84a} \end{split}\end{aligned}$$ with $I_{{\mathcal N}}$ the identity operator in ${{\mathcal N}}$ and $P_{{\mathcal N}}$ the orthogonal projection in ${{\mathcal H}}$ onto ${{\mathcal N}}$. Equation then immediately follows from repeated use of the identity (the first resolvent equation), $$\begin{aligned} &\int_{{{\mathbb{R}}}^n} d^nx' G_0(z_1,x,x')G_0(z_2,x',0) = (z_1-z_2)^{-1}[G_0(z_1,x,0)-G_0(z_2,x,0)], {\notag}\\ &\hspace*{7.7cm} x\neq 0, \, z_1\neq z_2, \, n=2,3, {\label}{10.85}\end{aligned}$$ and its limiting case as $x\to 0$. Finally, follows from the following arguments: First one notices that $$\big[-(d^2/dr^2)+ \alpha \, r^{-2}\big]\big|_{C_0^\infty((0,\infty))} {\label}{10.86}$$ is essentially self-adjoint in $L^2({{\mathbb{R}}}_+; dr)$ if and only if $\alpha \geq 3/4$. Hence it suffices to consider the restriction of $H_{min, {{\mathbb{R}}}^3\backslash\{0\}}$ to the centrally symmetric subspace ${{\mathcal H}}_{3,0,(0,\infty)}$ of $L^2({{\mathbb{R}}}^3;d^3x)$ corresponding to angular momentum $\ell=0$ in , . But then it is a well-known fact (cf. [@GKMT01 Sects. 4,5]) that the Donoghue-type Dirichlet $m$-function $(u_+,M_{H_{D,{{\mathbb{R}}}^3\backslash\{0\}},{{\mathcal N}}_+}(z)u_+)_{L^2({{\mathbb{R}}}^3;d^3 x)}$, satisfies $$\begin{aligned} (u_+,M_{H_{D,{{\mathbb{R}}}^3\backslash\{0\}},{{\mathcal N}}_+}(z)u_+)_{L^2({{\mathbb{R}}}^3;d^3 x)} &= (u_{0,+},M_{h_{0,D,{{\mathbb{R}}}_+}^{(0)},{{\mathcal N}}_{0,+}}(z)u_{0,+})_{L^2({{\mathbb{R}}}_+; dr)}, {\notag}\\ & = i (2z)^{1/2} + 1, {\label}{10.89}\end{aligned}$$ where $${{\mathcal N}}_{0,+}= {\rm lin. \, span} \{u_{0,+}\}, \quad u_{0,+} (r)= e^{iz^{1/2}r}/[2 {\mathop\mathrm{Im}}(z^{1/2}]^{1/2}, \; r>0,$$ and $M_{h_{0,D,{{\mathbb{R}}}_+}^{(0)},{{\mathcal N}}_{0,+}}(z)$ denotes the Donoghue-type Dirichlet $m$-function corresponding to the operator $$\begin{aligned} &h_{0,D,{{\mathbb{R}}}_+}^{(0)}=-{\frac}{d^2}{dr^2}, \quad r>0, {\notag}\\ &\operatorname{dom}\big(h_{0,D,{{\mathbb{R}}}_+}^{(0)}\big)= \big\{f\in L^2((0,\infty);dr)\,\big|\, f,f'\in AC([0,R]) \text{ for all } R>0; {\label}{10.91} \\ & \hspace*{6.05cm} \, f(0_+)=0; \, f''\in L^2((0,\infty);dr)\big\}, {\notag}\end{aligned}$$ Next, turning to the Donoghue-type Neumann $m$-function given by $(u_+,M_{H_{N,{{\mathbb{R}}}^3\backslash\{0\}},{{\mathcal N}}_+}(z)u_+)_{L^2({{\mathbb{R}}}^3;d^3 x)}$ one obtains analogously to that $$(u_+,M_{H_{N,{{\mathbb{R}}}^3\backslash\{0\}},{{\mathcal N}}_+}(z)u_+)_{L^2({{\mathbb{R}}}^3;d^3 x)} = (u_{0,+},M_{h_{0,N,{{\mathbb{R}}}_+}^{(0)},{{\mathcal N}}_{0,+}}(z)u_{0,+})_{L^2({{\mathbb{R}}}_+; dr)},$$ where $M_{h_{0,N,{{\mathbb{R}}}_+}^{(0)},{{\mathcal N}}_{0,+}}(z)$ denotes the Donoghue-type Neumann $m$-function corresponding to the operator $h_{0,N,{{\mathbb{R}}}_+}^{(0)}$ in . The well-known linear fractional transformation relating the operators $M_{h_{0,D,{{\mathbb{R}}}_+}^{(0)},{{\mathcal N}}_{0,+}}(z)$ and $M_{h_{0,N,{{\mathbb{R}}}_+}^{(0)},{{\mathcal N}}_{0,+}}(z)$ (cf. [@GKMT01 Lemmas 5.3, 5.4, Theorem 5.5, and Corollary 5.6]) then yields $$(u_{0,+},M_{h_{0,N,{{\mathbb{R}}}_+}^{(0)},{{\mathcal N}}_{0,+}}(z)u_{0,+})_{L^2({{\mathbb{R}}}_+; dr)} = i (2/z)^{1/2} - 1,$$ verifying . The fact that the operator $T=-\Delta$, $\operatorname{dom}(T)=H^{2}({{\mathbb{R}}}^2)$ is the unique nonnegative self-adjoint extension of $-\Delta_{min, {{\mathbb{R}}}^2\backslash\{0\}}$ in $L^2({{\mathbb{R}}}^2;d^2x)$, has been shown in [@AGHKH87] (see also [@AGHKH88 Ch. I.5]). The Case ${\Omega}={{\mathbb{R}}}^n\backslash\{0\}$, $V=-[(n-2)^2/4]|x|^{-2}$, $n\geq 2$. ----------------------------------------------------------------------------------------- [s10.4]{} In our final subsection we briefly consider the following minimal operator $H_{min, {{\mathbb{R}}}^n\backslash\{0\}}$ in $L^2({{\mathbb{R}}}^n;d^nx)$, $n\geq 2$, $$H_{min, {{\mathbb{R}}}^n\backslash\{0\}} ={\overline}{(-\Delta - ((n-2)^2/4)|x|^{-2})\big|_{C^\infty_0 ({{\mathbb{R}}}^n\backslash\{0\})}}\geq 0, \quad n\geq 2. {\label}{10.94}$$ Then, using again the angular momentum decomposition (cf. also , ), $$\begin{aligned} & L^2({{\mathbb{R}}}^n; d^nx) = L^2((0,\infty); r^{n-1}dr) \otimes L^2(S^{n-1}; d\omega_{n-1}) = \bigoplus_{\ell\in{{\mathbb{N}}}_0} {{\mathcal H}}_{n,\ell,(0,\infty)}, {\label}{10.95} \\ & {{\mathcal H}}_{n,\ell,(0,\infty)} = L^2((0,\infty); r^{n-1}dr) \otimes {{\mathcal K}}_{n,\ell}, \quad \ell \in {{\mathbb{N}}}_0, \; n\geq 2, {\label}{10.96}\end{aligned}$$ one finally obtains that $$H_{F,{{\mathbb{R}}}^n\backslash\{0\}}=H_{K,{{\mathbb{R}}}^n\backslash\{0\}} = U^{-1}h_{0,{{\mathbb{R}}}_+} U \oplus\bigoplus_{\ell\in{{\mathbb{N}}}} U^{-1}h_{n,\ell,{{\mathbb{R}}}_+} U, \; n\geq 2, {\label}{10.97}$$ is the unique nonnegative self-adjoint extension of $H_{min, {{\mathbb{R}}}^n\backslash\{0\}}$ in $L^2({{\mathbb{R}}}^n;d^n x)$, where $$\begin{aligned} & h_{0,{{\mathbb{R}}}_+} = -{\frac}{d^2}{dr^2} - {\frac}{1}{4 r^2}, \quad r>0, {\notag}\\ & \operatorname{dom}(h_{0,{{\mathbb{R}}}_+}) = \big\{f \in L^2((0,\infty); dr) \,\big|\, f, f' \in AC([\varepsilon,R]) \, \text{for all $0<\varepsilon<R$}; {\label}{10.98} \\ & \hspace*{4.25cm} f_0=0; \, (-f'' -(1/4)r^{-2}f) \in L^2((0,\infty); dr)\big\}, {\notag}\\ & h_{n,\ell,{{\mathbb{R}}}_+} = -{\frac}{d^2}{dr^2} + {\frac}{4 \kappa_{n,\ell} - 1}{4 r^2}, \quad r>0, {\notag}\\ & \operatorname{dom}(h_{n,\ell,{{\mathbb{R}}}_+}) = \big\{f \in L^2((0,\infty); dr) \,\big|\, f, f' \in AC([\varepsilon,R]) \, \text{for all $0<\varepsilon<R$}; {\notag}\\ & \hspace*{1.2cm} (-f'' +[\kappa_{n,\ell} - (1/4)]r^{-2}f) \in L^2((0,\infty); dr)\big\}, \quad \ell\in{{\mathbb{N}}}, \; n\geq 2. {\label}{10.99}\end{aligned}$$ Here $f_0$ in is defined by (cf. also ) $$f_0 = \lim_{r\downarrow 0} [-r^{1/2}\ln(r)]^{-1} f(r).$$ As in the previous subsection, $h_{n,\ell,{{\mathbb{R}}}_+}|_{C_0^{\infty}((0,\infty))}$, $\ell\in{{\mathbb{N}}}$, $n\geq 2$, are essentially self-adjoint in $L^2((0,\infty);dr)$. In addition, $h_{0,{{\mathbb{R}}}_+}$ is the unique nonnegative self-adjoint extension of $h_{0,{{\mathbb{R}}}_+}|_{C_0^{\infty}((0,\infty))}$ in $L^2((0,\infty);dr)$. We omit further details. [**Acknowledgments.**]{} We are indebted to Yury Arlinskii, Gerd Grubb, John Lewis, Konstantin Makarov, Mark Malamud, Vladimir Maz’ya, Michael Pang, Larry Payne, Barry Simon, Nikolai Tarkhanov, Hans Triebel, and Eduard Tsekanovskii for many helpful discussions and very valuable correspondence on various topics of this paper. One of us (F.G.) gratefully acknowledges the extraordinary hospitality of the Faculty of Mathematics of the University of Vienna, Austria, during his three month visit in the first half of 2008. [99]{} [AS72]{} M. Abramovitz, I. A. Stegun, [*Handbook of Mathematical Functions*]{}, Dover, New York, 1972. [AF03]{} R. A. Adams and J. J. F. Fournier, [*Sobolev Spaces*]{}, second edition, Academic Press, 2003. [AH96]{} D. R. Adams and L. I. Hedberg, [*Function Spaces and Potential Theory*]{}, Grundlehren der Mathematischen Wissenschaften, Vol. 314, Springer, Berlin, 1996. [Ad07]{} V. Adamyan, [*Non-negative perturbations of non-negative self-adjoint operators*]{}, Meth. Funct. Anal. Top.  [**13**]{}, no. 2, 103–109 (2007). [AP98]{} V. Adolfsson and J. Pipher, [*The inhomogeneous Dirichlet problem for $\Delta^2$ in Lipschitz domains*]{}, J. Funct. Anal. [**159**]{}, 137–190 (1998). [Ag97]{} M. S. Agranovich, [*Elliptic Boundary Problems*]{}, Partial Differential Equations, IX, Encyclopaedia Math. Sci., Vol. 79, M. S. Agranovich, Yu. V. Egorov, M. A. Shubin (eds.), Springer, Berlin, 1997, pp. 1–144. [Ag03]{} M. S. Agranovich, [*Spectral problems for second-order strongly elliptic systems in smooth and non-smooth domains*]{}, Russ.  Math. Surv. [**57**]{}:5, 847–920 (2003). [AD92]{} A. Aimi and M. Diligenti, [*Upper and lower bounds of eigenvalues of the classical buckling problem*]{}, Calcolo [**29**]{}, no. 3-4, 313–328 (1992). [AD93]{} A. Aimi and M. Diligenti, [*Error estimation in the computation of eigenvectors in the buckling problem for a plate embedded at the boundary*]{}, Calcolo [**30**]{}, no. 2, 171–187 (1993). [AG81a]{} N. I. Akhiezer and I. M. Glazman, [*Theory of Linear Operators in Hilbert Space, Volume II*]{}, Pitman, Boston, 1981. [ABMN05]{} S. Albeverio, J. F. Brasche, M. M. Malamud, and H. Neidhardt, [*Inverse spectral theory for symmetric operators with several gaps: scalar-type Weyl functions*]{}, J. Funct. Anal. [**228**]{}, 144–188 (2005). [AGHKH87]{} S. Albeverio, F. Gesztesy, R. H[ø]{}egh-Krohn, and H. Holden, [*Point interactions in two dimensions: Basic properties, approximations and applications to solid state physics*]{}, J. reine angew. Math. [**380**]{}, 87–107 (1987). [AGHKH88]{} S. Albeverio, F. Gesztesy, R. H[ø]{}egh-Krohn, and H. Holden, [*Solvable Models in Quantum Mechanics*]{}, Springer, Berlin, 1988. [AS80]{} A. Alonso and B. Simon, [*The Birman-Krein-Vishik theory of selfadjoint extensions of semibounded operators*]{}, J. Operator Th.  [**4**]{}, 251–270 (1980); Addenda: [**6**]{}, 407 (1981). [AP04]{} W. O. Amrein and D. B. Pearson, [*$M$ operators: a generalization of Weyl–Titchmarsh theory*]{}, J. Comp. Appl. Math. [**171**]{}, 1–26 (2004). [AN70]{} T. Ando and K. Nishio, [*Positive selfadjoint extensions of positive symmetric operators*]{}, Tohoku Math. J. (2), [**22**]{}, 65–75 (1970). [Ar98]{} Yu. M. Arlinskii, [*On $m$-accretive extensions and restrictions*]{}, Meth. Funct. Anal. Top. [**4**]{}, 1-26 (1998). [Ar00]{} Yu. M. Arlinskii, [*Abstract boundary conditions for maximal sectorial extensions of sectorial operators*]{}, Math. Nachr. [**209**]{}, 5–36 (2000). [AHSD01]{} Yu. M. Arlinskii, S. Hassi, Z. Sebestyén, and H. S. V.  de Snoo, [*On the class of extremal extensions of a nonnegative operator*]{}, in [*Recent Advances in Operator Theory and Related Topics*]{}, L. Kérchy, C. Foias, I. Gohberg, and H. Langer (eds.), Operator Theory: Advances and Applications, Vol. 127, Birkhäuser, Basel, 2001, pp. 41–81. [AT02]{} Yu. M. Arlinskii and E. R. Tsekanovskii, [*On the theory of nonnegative selfadjoint extensions of a nonnegative symmetric operator*]{}, Rep. Nat. Acad. Sci. Ukraine [**2002**]{}, no. 11, 30–37. [AT03]{} Yu. M. Arlinski[ĭ]{} and E. R. Tsekanovski[ĭ]{}, [*On von Neumann’s problem in extension theory of nonnegative operators*]{}, Proc.  Amer. Math. Soc. [**131**]{}, 3143–3154 (2003). [AT05]{} Yu. M. Arlinski[ĭ]{} and E. R. Tsekanovski[ĭ]{}, [*The von Neumann problem for nonnegative symmetric operators*]{}, Integr. Equ.  Oper. Theory [**51**]{}, 319–356 (2005). [AT09]{} Yu. Arlinski[ĭ]{} and E. Tsekanovski[ĭ]{}, [*M.Kre[ĭ]{}n’s research on semibounded operators, its contemporary developments, and applications*]{}, in [*Modern Analysis and Applications. The Mark Krein Centenary Conference*]{}, Vol. 1, V. Adamyan, Y. M. Berezansky, I. Gohberg, M. L. Gorbachuk, V. Gorbachuk, A. N. Kochubei, H. Langer, and G. Popov (eds.), Operator Theory: Advances and Applications, Vol. 190, Birkhäuser, Basel, 2009, pp. 65–114. [As99]{} M. S. Ashbaugh, [*Isoperimetric and universal inequalities for eigenvalues*]{}, in [*Spectral Theory and Geometry*]{}, B. Davies and Yu. Safarov (eds.), London Mathematical Society Lecture Note Series, Vol. 273, Cambridge University Press, Cambridge, 1999, pp. 95–139. [As04]{} M. S. Ashbaugh, [*On universal inequalities for the low eigenvalues of the buckling problem*]{}, in “Partial Differential Equations and Inverse Problems,” Contemp. Math. [**362**]{}, pp. 13–31, Amer.  Math. Soc., Providence, RI, 2004. [As09]{} M. S. Ashbaugh, [*On universal inequalities for the low eigenvalues of the buckling problem II*]{}, preprint, 2008. [AGMST09]{} M. S. Ashbaugh, F. Gesztesy, M. Mitrea, R. Shterenberg, and G. Teschl, [*The Krein–von Neumann extension and its connection to an abstract buckling problem*]{}, Math. Nachr., to appear. [AL96]{} M. S. Ashbaugh and R. S. Laugesen, [*Fundamental tones and buckling loads of clamped plates*]{}, Ann. Scuola Norm. Sup. Pisa Cl. Sci.  (4), [**23**]{}, no. 2, 383–402 (1996). [AL97]{} M. S. Ashbaugh and H. A. Levine, [*Inequalities for the Dirichlet and Neumann eigenvalues of the Laplacian for domains on spheres*]{}, Journées “Équations aux Dérivées Partielles” (Saint-Jean-de-Monts, 1997), Exp. no. I, 15 pp., École Polytechnique, Palaiseau, 1997. [Av86]{} P. Aviles, [*Symmetry theorems related to Pompeiu’s problem*]{}, Amer. J. Math. [**108**]{}, 1023–1036 (1986). [BV65]{} I. Babu[š]{}ka and R. Výborný, [*Continuous dependence of eigenvalues on the domain*]{}, Czech. Math. J. [**15**]{}, 169–178 (1965). [BL07]{} J. Behrndt and M. Langer, [*Boundary value problems for partial differential operators on bounded domains*]{}, J. Funct. Anal. [**243**]{}, 536–565 (2007). [Bi56]{} M. Sh. Birman, [*On the theory of self-adjoint extensions of positive definite operators*]{}, Mat. Sbornik [**38**]{}, 431–450 (1956). (Russian.) [Bi08]{} M. Sh. Birman, [*Perturbations of the continuous spectrum of a singular elliptic operator by varying the boundary and the boundary conditions*]{}, Vestnik Leningrad Univ. [**17**]{}, no. 1, 22–55 (1962) (Russian); Engl. transl. in [*Spectral Theory of Differential Operators: M. Sh.  Birman 80th Anniversary Collection*]{}, T. Suslina and D. Yafaev (eds.), AMS Translations, Ser.  2, Advances in the Mathematical Sciences, Vol.  225, Amer. Math. Soc., Providence, RI, 2008, pp. 19–53. [BF07]{} M. Š. Birman and N. D. Filonov, [*Weyl asymptotics of the spectrum of the Maxwell operator with non-smooth coefficients in Lipschitz domains*]{}, in “Nonlinear Equations and Spectral Theory”, pp.  27–44, Amer. Math. Soc. Transl. Ser. 2, Vol. 220, Amer. Math. Soc., Providence, RI, 2007. [BS70]{} M. Sh. Birman and M. Z. Solomyak, [*Leading term in the asymptotic spectral formula for “non-smooth" elliptic problems*]{}, Funkcional. Anal. i Priložen [**4**]{}, no. 4, 1–13 (1970) (Russian); Engl. transl. in Funct. Anal. Appl. [**4**]{}, 265–275 (1970). [BS71]{} M. Sh. Birman and M. Z. Solomyak, [*On the asymptotic spectrum of “non-smooth" elliptic equations*]{}, Funkcional. Anal. i Priložen [**5**]{}, no. 1, 69–70 (1971) (Russian); Engl. transl. in Funct. Anal. Appl. [**5**]{}, 56–57 (1971). [BS72]{} M. Š. Birman and M. Z. Solomjak, [*Spectral asymptotics of nonsmooth elliptic operators. I*]{}, Trans. Moscow Math. Soc. [**27**]{}, 1–52 (1972). [BS73]{} M. Š. Birman and M. Z. Solomjak, [*Spectral asymptotics of nonsmooth elliptic operators. I*]{}, Trans. Moscow Math. Soc. [**28**]{}, 1–32 (1973). [BS79]{} M. Sh. Birman and M. Z. Solomyak, [*Asymtotic behavior of the spectrum of differential equations*]{}, Itogi Nauki i Tekhniki, Matematicheskii Analiz., [**14**]{}, 5–58 (1977) (Russian); Engl. transl. in J. Soviet Math. [**12**]{}, no. 3, 247–283 (1979). [BS87]{} M. Š. Birman and M. Z. Solomjak, [*Weyl asymptotics of the spectrum of the Maxwell operator for domains with a Lipschitz boundary*]{}, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. [**127**]{}, no. 3, 23–28 (1987) (Russian). [BT99]{} P. E. Bj[ø]{}rstad and B. P. Tj[ø]{}stheim, [*High precision solutions of two fourth order eigenvalue problems*]{}, Computing [**63**]{}, no. 2, 97–107 (1999). [BP63]{} J. H. Bramble and L. E. Payne, [*Pointwise bounds in the first biharmonic boundary value problem*]{}, J. Math. Phys. [**42**]{}, 278–286 (1963). [BMN02]{} J. F. Brasche, M. M. Malamud, and H. Neidhardt, [*Weyl function and spectral properties of self-adjoint extensions*]{}, Integral Eqs. Operator Theory [**43**]{}, 264–289 (2002). [BC05]{} B. M. Brown and J. S. Christiansen, [*On the Krein and Friedrichs extension of a positive Jacobi operator*]{}, Expo. Math. [**23**]{}, 179–186 (2005). [BGW09]{} B. M. Brown, G. Grubb, and I. G. Wood, [*$M$-functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems*]{}, Math. Nachr. [**282**]{}, 314–347 (2009). [BHMNW09]{} M. Brown, J. Hinchcliffe, M. Marletta, S. Naboko, and I. Wood, [*The abstract Titchmarsh–Weyl $M$-function for adjoint operator pairs and its relation to the spectrum*]{}, Integral Equ. Operator Th.  [**63**]{}, 297–320 (2009). [BMNW08]{} M. Brown, M. Marletta, S. Naboko, and I. Wood, [*Boundary triplets and $M$-functions for non-selfadjoint operators, with applications to elliptic PDEs and block operator matrices*]{}, J. London Math. Soc. (2) [**77**]{}, 700–718 (2008). [BGP08]{} J. Brüning, V. Geyler, and K. Pankrashkin, [*Spectra of self-adjoint extensions and applications to solvable Schrödinger operators*]{}, Rev. Math. Phys. [**20**]{}, 1–70 (2008). [BG85]{} W. Bulla and F. Gesztesy, [*Deficiency indices and singular boundary conditions in quantum mechanics*]{}, J. Math. Phys. [**26**]{}, 2520–2528 (1985). [BL08]{} V. I. Burenkov and P. D. Lamberti, [*Spectral stability of Dirichlet second order uniformly elliptic operators*]{}, J. Diff. Eq.  [**244**]{}, 1712–1740 (2008). [BLL08]{} V. I. Burenkov, P. D. Lamberti, and M. Lanza de Cristoforis, [*Spectral stability of nonnegative self-adjoint operators*]{}, J. Math.  Sci. [**149**]{}, 1417–1452 (2008). [Cae95]{} A. M. Caetano, [*On the search for the asymptotic behaviour of the eigenvalues of the Dirichlet Laplacian for bounded irregular domains*]{}, Some aspects of fractals in mathematics and physics (Toulouse, 1993), Internat. J. Appl. Sci. Comput. [**2**]{}, no. 2, 261–287 (1995). [Cae98]{} A. M. Caetano, [*Eigenvalue asymptotics of the Stokes operator for fractal domains*]{}, Proc. London Math. Soc. (3), [**76**]{}, no. 3, 579–602 (1998). [Ca67]{} L. Carleson, [*Selected Problems on Exceptional Sets*]{}, Van Nostrand Mathematical Studies, no. 13, Princeton, 1967. [CY06]{} Q.-M. Cheng and H. Yang, [*Universal bounds for eigenvalues of a buckling problem*]{}, Comm. Math. Phys. [**262**]{}, 663–675 (2006). [Da03]{} D. Daners, [*Dirichlet problems on varying domains*]{}, J.  Diff. Eq. [**188**]{}, 591–624 (2003). [Da97]{} E. B. Davies, [*$L^p$ spectral theory of higher-order elliptic differential operators*]{}, Bull. London Math. Soc. [**29**]{}, no. 5, 513–546 (1997). [DL90]{} R. Dautray and J.-L. Lions, [*Mathematical Analysis and Numerical Methods for Science and Technology*]{}, Vol. 4, Springer, Berlin, 1990. [DM91]{} V. A. Derkach and M. M. Malamud, [*Generalized resolvents and the boundary value problems for Hermitian operators with gaps*]{}, J.Funct. Anal. [**95**]{}, 1–95 (1991). [DM95]{} V. A. Derkach and M. M. Malamud, [*The extension theory of Hermitian operators and the moment problem*]{}, J. Math. Sci. [**73**]{}, 141–242 (1995). [DMT88]{} V. A. Derkach, M. M. Malamud, and E. R. Tsekanovskii, [ *Sectorial extensions of a positive operator and the characteristic function*]{}, Soviet Math. Dokl. [**37**]{}, 106–110 (1988). [DMT89]{} V. A. Derkach, M. M. Malamud, and E. R. Tsekanovskii, [ *Sectorial extensions of a positive operator and the characteristic function*]{}, Ukrainian Math. J. [**41**]{}, 136–142 (1989). [Ca92]{} M. P. do Carmo, [*Riemannian Geometry*]{} (translated by F.  Flaherty from the second Portuguese edition), Birkhauser, Boston, 1992. [Do65]{} W. F. Donoghue, [*On the perturbation of spectra*]{}, Commun. Pure Appl. Math. [**18**]{}, 559-579 (1965). [EE89]{} D. E. Edmunds and W. D. Evans, [*Spectral Theory and Differential Operators*]{}, Clarendon Press, Oxford, 1989. [EM05]{} W. N. Everitt and L. Markus, [*Complex symplectic spaces and boundary value problems*]{}, Bull. Amer. Math. Soc. [**42**]{}, 461–500 (2005). [EMP04]{} W. N. Everitt, L. Markus, M. Plum, [*An unusual self-adjoint linear partial differential operator*]{}, Trans. Amer. Math. Soc. [**357**]{}, 1303–1324 (2004). [EMMP07]{} W. N. Everitt, L. Markus, M. Muzzulini, M. Plum, [*A continuum of unusual self-adjoint linear partial differential operators*]{}, J. Comp. Appl. Math. [**208**]{}, 164–175 (2007). [Fa75]{} W. G. Faris, [*Self-Adjoint Operators*]{}, Lecture Notes in Mathematics, Vol. 433, Springer, Berlin, 1975. [Fr36]{} H. Freudenthal, [*Über die Friedrichsche Fortsetzung halbbeschränkter Hermitescher Operatoren*]{}, Kon. Akad. Wetensch., Amsterdam, Proc. [**39**]{}, 832–833 (1936). [Fr34]{} K. Friedrichs, [*Spektraltheorie halbeschränkter Operatoren und Anwendung auf die Spektralzerlegung von Differentialoperatoren I, II*]{}, Math. Ann. [**109**]{}, 465–487, 685–713 (1934), corrs. in Math. Ann.  [**110**]{}, 777–779 (1935). [Fu99]{} B. Fuglede, [*Continuous domain dependence of the eigenvalues of the Dirichlet Laplacian and related operators in Hilbert space*]{}, J.  Funct. Anal. [**167**]{}, 183–200 (1999). [Fu80]{} M. Fukushima, [*Dirichlet Forms and Markov Processes*]{}, North-Holland, Amsterdam, Kodansha, Tokyo, 1980. [FOT94]{} M. Fukushima, Y. Oshima, and M. Takeda, [*Dirichlet Forms and Symmetric Markov Processes*]{}, de Gruyter, Berlin, 1994. [GKMT01]{} F. Gesztesy, N.J. Kalton, K.A. Makarov, and E. Tsekanovskii, [*Some applications of operator-valued Herglotz functions*]{}, in “Operator Theory, System Theory and Related Topics,” Oper. Theory Adv. Appl., Vol. 123, Birkhäuser, Basel, 2001, pp. 271–321. [GLMZ05]{} F. Gesztesy, Y. Latushkin, M. Mitrea, and M. Zinchenko, [*Nonselfadjoint operators, infinite determinants, and some applications*]{}, Russ. J. Math. Phys., [**12**]{}, 443–471 (2005). [GMT98]{} F. Gesztesy, K. A. Makarov, and E. Tsekanovskii, [*An Addendum to Krein’s formula*]{}, J. Math. Anal. Appl. [**222**]{}, 594–606 (1998). [GM08]{} F. Gesztesy and M. Mitrea, [*Generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains*]{}, in [*Perspectives in Partial Differential Equations, Harmonic Analysis and Applications: A Volume in Honor of Vladimir G. Maz’ya’s 70th Birthday*]{}, D. Mitrea and M. Mitrea (eds.), Proceedings of Symposia in Pure Mathematics, Vol. 79, Amer. Math. Soc., Providence, RI, 2008, pp. 105–173. [GM09a]{} F. Gesztesy and M. Mitrea, [*Robin-to-Robin maps and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains*]{}, in [*Modern Analysis and Applications. The Mark Krein Centenary Conference*]{}, Vol. 2, V. Adamyan, Y. M. Berezansky, I. Gohberg, M. L. Gorbachuk, V. Gorbachuk, A. N. Kochubei, H. Langer, and G. Popov (eds.), Operator Theory: Advances and Applications, Vol. 191, Birkhäuser, Basel, 2009, pp. 81–113. [GM09b]{} F. Gesztesy and M. Mitrea, [*Generalized Robin Laplacians and some remarks on a paper by Filonov on eigenvalue inequalities*]{}, J. Diff. Eq. [**247**]{}, 2871–2896 (2009). [GM10]{} F. Gesztesy and M. Mitrea, [*Self-adjoint extensions of the Laplacian and Krein-type resolvent formulas in nonsmooth domains*]{}, preprint, 2009, arXiv:0907.1750. [GMZ07]{} F. Gesztesy, M. Mitrea, and M. Zinchenko, [*Variations on a theme of Jost and Pais*]{}, J. Funct. Anal. [**253**]{}, 399–448 (2007). [GT00]{} F. Gesztesy and E. Tsekanovskii, [*On matrix-valued Herglotz functions*]{}, Math. Nachr. [**218**]{}, 61–138 (2000). [GT83]{} D. Gilbarg and N. S. Trudinger, [*Elliptic Partial Differential Equations of Second Order*]{}, Springer, Berlin, 1983. [Gr85]{} P. Grisvard, [*Elliptic Problems in Nonsmooth Domains*]{}, Pitman, Boston, 1985. [Gr68]{} G. Grubb, [*A characterization of the non-local boundary value problems associated with an elliptic operator*]{}, Ann. Scuola Norm.  Sup. Pisa (3), [**22**]{}, 425–513 (1968). [Gr70]{} G. Grubb, [*Les problèmes aux limites généraux d’un opérateur elliptique, provenant de le théorie variationnelle*]{}, Bull. Sci. Math. (2), [**94**]{}, 113–157 (1970). [Gr83]{} G. Grubb, [*Spectral asymptotics for the “soft” selfadjoint extension of a symmetric elliptic differential operator*]{}, J. Operator Th.  [**10**]{}, 9–20 (1983). [Gr06]{} G. Grubb, [*Known and unknown results on elliptic boundary problems*]{}, Bull. Amer. Math. Soc. [**43**]{}, 227–230 (2006). [Gr08a]{} G. Grubb, [*Krein resolvent formulas for elliptic boundary problems in nonsmooth domains*]{}, Rend. Semin. Mat. Univ. Politec. Torino [**66**]{}, 271–297 (2008). [Gr09]{} G. Grubb, [*Distributions and Operators*]{}, Graduate Texts in Mathematics, Vol. 252, Springer, New York, 2009. [Ha57]{} P. Hartman, [Perturbation of spectra and Krein extensions]{}, Rend. Circ. Mat. Palermo (2) [**5**]{}, 341–354 (1957). [HP72]{} R. Harvey and J. C. Polking, [*A notion of capacity which characterizes removable singularities*]{}, Trans. Amer. Math. Soc.  [**169**]{}, 183–195 (1972). [HK09]{} S. Hassi and S. Kuzhel, [*On symmetries in the theory of finite rank singular perturbations*]{}, J. Funct. Anal. [**256**]{}, 777–809 (2009). [HMD04]{} S. Hassi, M. Malamud, and H. de Snoo, [*On Kre[ĭ]{}n’s extension theory of nonnegative operators*]{}, Math. Nachr. [**274–275**]{}, 40–73 (2004). [HSDW07]{} S. Hassi, A. Sandovici, H. de Snoo, and H. Winkler, [*A general factorization approach to the extension theory of nonnegative operators and relations*]{}, J. Operator Th. [**58**]{}, 351–386 (2007). [HL97]{} C. He and M. L. Lapidus, [*Generalized Minkowski Content, Spectrum of Fractal Drums, Fractal Strings and the Riemann Zeta-Function*]{}, Mem. Amer. Math. Soc. [**127**]{}, no. 608 (1997). [HY84]{} G. N. Hile and R. Z. Yeh, [*Inequalities for eigenvalues of the biharmonic operator*]{}, Pacific J. Math. [**112**]{}, no. 1, 115–133 (1984). [Ho91]{} S. M. Hook, [*Bounds for the fundamental frequencies of an elastic medium*]{}, Michigan Math. J. [**38**]{}, 3–12 (1991). [JK95]{} D. Jerison and C. Kenig, [*The inhomogeneous Dirichlet problem in Lipschitz domains*]{}, J. Funct. Anal. [**130**]{}, 161–219 (1995). [Ka80]{} T. Kato, [*Perturbation Theory for Linear Operators*]{}, corr.  printing of the 2nd ed., Springer, Berlin, 1980. [KLV93]{} B. Kawohl, H. A. Levine, and W. Velte, [*Buckling eigenvalue for a clamped plate embedded in an elastic medium and related questions*]{}, SIAM J. Math. Anal. [**24**]{}, 327–340 (1993). [Ko79]{} V. A. Kozlov, [*Estimation of the remainder in a formula for the asymptotic behavior of the spectrum of nonsemibounded elliptic systems*]{}, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. [**1979**]{}, no 4., 112–113, 125 (Russian). [Ko83]{} V. A. Kozlov, [*Estimates of the remainder in formulas for the asymptotic behavior of the spectrum for linear operator bundles*]{}, Funktsional. Anal. i Prilozhen [**17**]{}, no. 2, 80–81 (1983). Engl. transl. in Funct. Anal. Appl. [**17**]{}, no. 2, 147–149 (1983). [Ko84]{} V. A. Kozlov, [*Remainder estimates in spectral asymptotic formulas for linear operator pencils*]{}, Linear and Nonlinear Partial Differential Equations. Spectral Asymptotic Behavior, pp. 34–56, Probl.  Mat. Anal. [**9**]{}, Leningrad Univ., Leningrad, 1984; Engl. transl. in J. Sov. Math. [**35**]{}, 2180–2193 (1986). [KKM90]{} V. A. Kozlov, V. A. Kondrat’ev, and V. G. Maz’ya, [*On sign variation and the absence of “strong” zeros of solutions of elliptic equations*]{}, Math. USSR Izv. [**34**]{}, 337–353 (1990). [Kra26]{} E. Krahn, [*Über Minimaleigenschaften der Kugel in drei und mehr Dimensionen*]{}, Acta Comm. Univ. Tartu (Dorpat) [**A9**]{}, 1–44 (1926). [KP99]{} S. G. Krantz and H. R. Parks, [*The Geometry of Domains in Space*]{}, Birkhauser, Boston, 1999. [Kr47]{} M. G. Krein, [*The theory of self-adjoint extensions of semi-bounded Hermitian transformations and its applications. I*]{}, Mat. Sbornik [**20**]{}, 431–495 (1947). (Russian). [Kr47a]{} M. G. Krein, [*The theory of self-adjoint extensions of semi-bounded Hermitian transformations and its applications. II*]{}, Mat. Sbornik [**21**]{}, 365–404 (1947). (Russian). [KO77]{} M. G. Krein and I. E. Ovcharenko, [*$Q$-functions and $sc$-resolvents of nondensely defined hermitian contractions*]{}, Sib. Math. J. [**18**]{}, 728–746 (1977). [KO78]{} M. G. Krein and I. E. Ovčarenko, [*Inverse problems for $Q$-functions and resolvent matrices of positive hermitian operators*]{}, Sov. Math. Dokl. [**19**]{}, 1131–1134 (1978). [LF06]{} M. L. Lapidus and M. van Frankenhuijsen, [*Fractal Geometry, Complex Dimensions and Zeta Functions. Geometry and Spectra of Fractal Strings*]{}, Springer, New York, 2006. [Le83]{} S. Z. Levendorskii, [*Distribution of eigenvalues of systems of the form $Au=tBu$*]{}, Funct. Anal. Appl. [**17**]{}, 149–151 (1983). [Le85]{} S. Z. Levendorski[ĭ]{}, [*Asymptotics of the spectrum of linear operator pencils*]{}, Math. USSR Sbornik [**52**]{}, 245–266 (1985). [LP85]{} H. A. Levine and M. H. Protter, [*Unrestricted lower bounds for eigenvalues for classes of elliptic equations and systems of equations with applications to problems in elasticity*]{}, Math. Meth.  Appl. Sci. [**7**]{}, 210–222 (1985). [LW86]{} H. A. Levine and H. F. Weinberger, [*Inequalities between Dirichlet and Neumann eigenvalues*]{}, Arch. Rat. Mech. Anal. [**94**]{}, 193–208 (1986). [LM08]{} B. V. Loginov and O. V. Makeeva, [*The pseudoperturbation method in generalized eigenvalue problems*]{}, Dokl. Math. [**77**]{}, 194–197 (2008). [MT07]{} K. A. Makarov and E. Tsekanovskii, [*On $\mu$-scale invariant operators*]{}, Meth. Funct. Anal. Top. [**13**]{}, no. 2, 181–186 (2007). [Ma92]{} M. M. Malamud, [*Certain classes of extensions of a lacunary Hermitian operator*]{}, Ukrainian Math. J. [**44**]{}, No. 2, 190–204 (1992). [Ma65]{} V. G. Maz’ja, Ph.D. Thesis, Leningrad State University, 1965. [Ma85]{} V. G. Maz’ja, [*Sobolev Spaces*]{}, Springer, Berlin, 1985. [MH73]{} V. G. Maz’ja and V. P. Havin, [*Use of $(p,l)$-capacity in problems of the theory of exceptional sets*]{}, Math. USSR Sbornik [**19**]{}, no. 4, 547–580 (1973). [MMS05]{} V. Maz’ya, M. Mitrea, and T. Shaposhnikova, [*The Dirichlet problem in Lipschitz domains with boundary data in Besov spaces for higher order elliptic systems with rough coefficients*]{}, J. Analyse Math., to appear. [MS85]{} V. G. Maz’ya and T. O. Shaposhnikova, [*Theory of Multipliers in Spaces of Differentiable Functions*]{}, Monographs and Studies in Mathematics, Vol. 23, Pitman (Advanced Publishing Program), Boston, MA, 1985. [MS05]{} V. G. Maz’ya and T. O. Shaposhnikova, [*Higher regularity in the layer potential theory for Lipschitz domains*]{}, Indiana Univ. Math.  J. [**54**]{}, no. 1, 99–142 (2005). [Mc00]{} W. McLean, [*Strongly Elliptic Systems and Boundary Integral Equations*]{}, Cambridge University Press, Cambridge, 2000. [Mik94]{} V. A. Mikhaĭlets, [*Distribution of the eigenvalues of finite multiplicity of Neumann extensions of an elliptic operator*]{}, Differentsial’nye Uravneniya [**30**]{}, 178–179 (1994) (Russian); Engl. transl. in Diff. Eq. [**30**]{}, 167–168 (1994). [MT00]{} M. Mitrea and M. Taylor, [*Potential theory on Lipschitz domains in Riemannian manifolds: Sobolev-Besov space results and the Poisson problem*]{}, J. Funct. Anal. [**176**]{}, 1–79 (2000). [MTV]{} M. Mitrea, M. Taylor, and A. Vasy, [*Lipschitz domains, domains with corners, and the Hodge Laplacian*]{}, Commun. Partial Diff.  Eq. [**30**]{}, 1445–1462 (2005). [Mu66]{} K. Müller, [*Spherical Harmonics*]{}, Lecture Notes in Mathematics, Vol. 17, Springer, Berlin, 1966. [Ne83]{} G. Nenciu, [*Applications of the Kre[ĭ]{}n resolvent formula to the theory of self-adjoint extensions of positive symmetric operators*]{}, J. Operator Th. [**10**]{}, 209–218 (1983). [NS05]{} Yu. Netrusov and Yu. Safarov, [*Weyl asymptotic formula for the Laplacian on domains with rough boundaries*]{}, Comm. Math. Phys.  [**253**]{}, no. 2, 481–509 (2005). [Pa55]{} L. E. Payne, [*Inequalities for eigenvalues of membranes and plates*]{}, J. Rat. Mech. Anal. [**4**]{}, 517–529 (1955). [Pa60]{} L. E. Payne, [*A note on inequalities for plate eigenvalues*]{}, J. Math. and Phys. [**39**]{}, 155–159 (1960). [Pa67]{} L. E. Payne, [*Isoperimetric inequalities and their applications*]{}, SIAM Rev. [**9**]{}, 453–488 (1967). [Pa91]{} L. E. Payne, [*Some comments on the past fifty years of isoperimetric inequalities*]{}, in [*Inequalities. Fifty Years on from Hardy, Littlewood and Pólya*]{}, W. N. Everitt (ed.), M. Dekker, New York, 1991, pp. 143–161. [Pe68]{} W. V. Petryshyn, [*On the eigenvalue problem $Tu-\lambda Su=0$ with unbounded and nonsymmetric operators $T$ and $S$*]{}, Philos. Trans.  Roy. Soc. London Ser. A [**262**]{}, 413–458 (1968). [PV95]{} J. Pipher and G. C. Verchota, [*Dilation invariant estimates and the boundary G[å]{}rding inequality for higher order elliptic operators*]{}, Ann. of Math. [**142**]{} (1995), 1–38. [Po08]{} A. Posilicano, [*Self-adjoint extensions of restrictions*]{}, Operators and Matrices [**2**]{}, 483–506 (2008). [PR09]{} A. Posilicano and L. Raimondi, [*KreinÕs resolvent formula for self-adjoint extensions of symmetric second-order elliptic differential operators*]{}, J. Phys. A: Math. Theor. [**42**]{}, 015204 (11pp) (2009). [PS96]{} V. Prokaj and Z. Sebestyén, [*On extremal positive operator extensions*]{}, Acta Sci. Math. (Szeged) [**62**]{}, 485–491 (1996). [RS75]{} M. Reed and B. Simon, [*Methods of Modern Mathematical Physics. II: Fourier Analysis, Self-Adjointness*]{}, Academic Press, New York, 1975. [RS08]{} M. Ruzhansky and M. Sugimoto, [*Criteria for Bochner’s extension problem*]{}, preprint, [arXiv:0802.0565v1]{}, 2008. [Ry99]{} V. S. Rychkov, [*On restrictions and extensions of the Besov and Triebel–Lizorkin spaces with respect to Lipschitz domains*]{}, J. London Math. Soc. (2), [**60**]{}, 237–257 (1999). [Ry07]{} V. Ryzhov, [*A general boundary value problem and its Weyl function*]{}, Opuscula Math. [**27**]{}, 305–331(2007). [Ry09]{} V. Ryzhov, [*WeylÐTitchmarsh function of an abstract boundary value problem, operator colligations, and linear systems with boundary control*]{}, Compl. Anal. Oper. Th. [**3**]{}, 289–322 (2009). [Ry10]{} V. Ryzhov, [*Spectral boundary value problems and their linear operators*]{}, preprint, 2009, arXiv:0904.0276. [SV97]{} Yu. Safarov and D. Vassiliev, [*The Asymptotic Distribution of Eigenvalues of Partial Differential Operators*]{}, Transl. of Math. Monographs, Vol. 155, Amer. Math. Soc., Providence, RI, 1997. [SS03]{} Z. Sebestyén and E. Sikolya, [*On Krein–von Neumann and Friedrichs extensions*]{}, Acta Sci. Math. (Szeged) [**69**]{}, 323–336 (2003). [Si98]{} B. Simon, [*The classical moment problem as a self-adjoint finite difference operator*]{}, Adv. Math. [**137**]{}, 82–203 (1998). [Sk79]{} C. F. Skau, [*Positive self-adjoint extensions of operators affiliated with a von Neumann algebra*]{}, Math. Scand. [**44**]{}, 171-195 (1979). [St95]{} P. Stollmann, [*Convergence of Schrödinger operators on varying domains*]{}, in [*Partial Differential Operators and Mathematical Physics*]{}, M. Demuth and B.-W. Schulze (eds.), Operator Theory: Advances and Applications, Vol. 78, Birkhäuser, Basel, 1995, pp. 369–374. [St96]{} O. G. Storozh, [*On the hard and soft extensions of a nonnegative operator*]{}, J. Math. Sci. [**79**]{}, 1378–1380 (1996). [St73]{} A. V. [S]{}traus, [*On extensions of a semibounded operator*]{}, Sov. Math. Dokl. [**14**]{}, 1075–1079 (1973). [Ta95]{} N. N. Tarkhanov, [*The Cauchy Problem for Solutions of Elliptic Equations*]{}, Mathematical Topics, Vol. 7, Akademie Verlag, Berlin, 1995. [Tr00]{} C. Tretter, [*Linear operator pencils $A-\lambda B$ with discrete spectrum*]{}, Integral Equ. Operator Th. [**37**]{}, 357–373 (2000). [Tr02]{} H. Triebel, [*Function spaces in Lipschitz domains and on Lipschitz manifolds. Characteristic functions as pointwise multipliers*]{}, Rev. Mat. Complut. [**15**]{}, 475–524 (2002). [Tr08]{} H. Triebel, [*The dichotomy between traces on $d$-sets $\Gamma$ in ${\mathbb{R}}^n$ and the density of $D({\mathbb{R}}^n\backslash \Gamma)$ in function spaces*]{}, Acta Math. Sin. (Engl. Ser.) [**24**]{}, no. 4, 539–554 (2008). [Ts80]{} E. R. Tsekanovskii, [*Non-self-adjoint accretive extensions of positive operators and theorems of Friedrichs–Krein–Phillips*]{}, Funct.  Anal. Appl. [**14**]{}, 156–157 (1980). [Ts81]{} E. R. Tsekanovskii, [*Friedrichs and Krein extensions of positive operators and holomorphic contraction semigroups*]{}, Funct. Anal.  Appl. [**15**]{}, 308–309 (1981). [Ts87]{} E. R. Tsekanovskii, [*Characteristic function and sectorial boundary value problems*]{}, Trudy Inst. Mat. (Novosibirsk) [**7**]{}, 180–194, 200 (1987), Issled. Geom. Mat. Anal. (Russian.) [Ts92]{} E. R. Tsekanovskii, [*Accretive extensions and problems on the Stieltjes operator-valued functions realizations*]{}, in [*Operator Theory and Complex Analysis*]{}, T. Ando and I. Gohberg (eds.), Operator Theory: Advances and Applications, Vol. 59, Birkhäuser, Basel, 1992, pp. 328–347. [Vi63]{} M. L. Vi[s]{}ik, [*On general boundary problems for elliptic differential equations*]{}, Trudy Moskov. Mat. Obsc. [**1**]{}, 187–246 (1952) (Russian); Engl. transl. in Amer. Math. Soc.  Transl. (2), [**24**]{}, 107–172 (1963). [Ne29]{} J. von Neumann, [*Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren*]{}, Math. Ann. [**102**]{}, 49–131 (1929-30). [Wa96]{} G. N. Watson, [*A Treatise on the Theory of Bessel Functions*]{}, 2nd ed., Cambridge University Press, Cambridge, 1996. [We80]{} J. Weidmann, [*Linear Operators in Hilbert Spaces*]{}, Graduate Texts in Mathematics, Vol. 68, Springer, New York, 1980. [We84]{} J. Weidmann, [*Stetige Abhängigkeit der Eigenwerte und Eigenfunktionen elliptischer Differentialoperatoren vom Gebiet*]{}, Math.  Scand. [**54**]{}, 51–69 (1984). [We03]{} J. Weidmann, [*Lineare Operatoren in Hilbertr[" a]{}umen.  Teil II: Anwendungen*]{}, Teubner, Wiesbaden, Germany, 2003. (German). [We12]{} H. Weyl, [*Über die Abhängigkeit der Eigenschwingungen einer Membran und deren Begrenzung*]{}, J. reine angew. Math. [**141**]{}, 1–11 (1912). [We12a]{} H. Weyl, [*Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Diferentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung)*]{}, Math. Ann. [**71**]{}, 441–479 (1912). [We50]{} H. Weyl, [*Ramifications, old and new, of the eigenvalue problem*]{}, Bull. Amer. Math. Soc. [**56**]{}, 115–139 (1950). [Zi89]{} W. P. Ziemer, [*Weakly Differentiable Functions*]{}, Graduate Texts in Mathematics, Vol. 120, Springer, New York, 1989. [^1]: Based upon work partially supported by the US National Science Foundation under Grant Nos. DMS-0400639 and FRG-0456306 and the Austrian Science Fund (FWF) under Grant No. Y330. [^2]: Adv. Math. [**223**]{}, 1372–1467 (2010). [^3]: The Neumann extension $H_{N,{{\mathbb{R}}}^3\backslash\{0\}}$ of $-\Delta_{min, {{\mathbb{R}}}^n\backslash\{0\}}$, associated with a Neumann boundary condition, in honor of Carl Gottfried Neumann, should of course not be confused with the Krein–von Neumann extension $H_{K,{{\mathbb{R}}}^3\backslash\{0\}}$ of $-\Delta_{min, {{\mathbb{R}}}^n\backslash\{0\}}$.

--- abstract: 'In this paper, interstitial migration generated by scattering with a mobile breather is investigated numerically in a Frenkel-Kontorova one-dimensional lattice. Consistent with experimental results it is shown that interstitial diffusion is more likely and faster than vacancy diffusion. Our simulations support the hypothesis that a long-range energy transport mechanism involving moving nonlinear vibrational excitations may significantly enhance the mobility of point defects in a crystal lattice.' title: Interaction of moving discrete breathers with interstitial defects --- <span style="font-variant:small-caps;">J. Cuevas, B. Sánchez–Rey</span> <span style="font-variant:small-caps;">J.C. Eilbeck and F.M. Russell</span> Introduction ============ The Frenkel–Kontorova (FK) model, introduced almost 70 years ago [@FK38], is one of the most paradigmatic nonlinear systems, whose dynamics has been widely studied during the last decades (see [@BK98; @BK98b; @FM96; @DP06] and references therein). From the point of view of condensed matter physics, its paramount importance relies on the ability to describe a vast number of phenomena, including different kinds of defects such as vacancies (Schottky defects) and, to some extent, interstitials (Frenkel defects), which can play an important role in the design of new materials [@Wuttig]. As the FK model is basically a one-dimensional lattice of particles subjected to a nonlinear periodic substrate potential and a nearest-neighbour interaction, it contains the basic ingredients to sustain localized excitations such as topological solitons (kinks or antikinks) or breathers. Discrete breathers (DBs), also called intrinsic localized modes (for a very recent review about their properties, existence proofs, computational methods and applications see [@Flach08]), are exact solutions of the dynamical equations whose energy, in contrast with normal extended wave excitations, is not shared among lattice components but extends only over a few lattice sites. In this sense, their spatial profiles resembles localized vibrational modes induced by a defect site in a harmonic lattice [@Sievers75]. However DBs arise only thanks to the interplay between nonlinearity and discreteness and, for that reason, they may occur anywhere in the lattice given sufficient vibrational amplitude. They are also rather universal since they are not specific to Hamiltonians with a particular form and can be found in lattices of arbitrary dimensions. Moreover, theoretical studies have shown DBs are linearly stable [@Aubry], which implies they can persist over very long times on top of a thermalized background [@Ivanchenko]. Their investigation is not restricted to simple toy models. Apart from indirect spectroscopic observations [@Swanson], DBs have been detected and studied experimentally in such different macroscopic systems as waveguide arrays [@Eisenberg], micromechanical cantilevers [@Sievers06], antiferromagnetic structures [@Sato] and Josephson-junctions [@Trias]. In this context, an interesting problem that has attracted much attention in recent years is the interaction between a moving localized excitation and a lattice defect. The problem has been addressed within different frameworks: impurities [@CPAR02; @FPM94], lattice junctions [@BSS02; @ARACL06], bending points of a polymer chain [@TSI02; @CK04; @LCBAG04], but most studies assume that the position of the defect is fixed and is not able to move along the lattice. Of current interest is the interaction between lattice defects and moving localized excitations, which might result in movement of the defect. This is especially true in the case of interactions arising during irradiation of solids by swift particles, which usually involve the creation of DBs of either longitudinal or transverse optical mode type. The possibility of such interactions arose in the study of high energy charged particles passing through crystals of muscovite, when scattering events were postulated to create many moving highly energy DBs. It was suggested that when such DBs (there called quodons) reached the end of a chain, which represents a defect in a chain, it might cause the last atom to be ejected from the surface [@RC95]. This prediction was supported by studies using both mechanical and numerical models [@MER98]. Subsequently, it was verified by experiment using a natural crystal of muscovite [@RE07]. In the experiment one edge of a crystal was bombarded with alpha particles at near grazing incidence to create moving DBs. These propagated in chain directions in the layered crystal and caused a proportionate ejection of atoms from a remote edge of the crystal that was $>10^7$ unit cells distance in a chain direction from the site of bombardment. As this experiment was performed at 300K it not only verified the prediction but also demonstrated the stability of these mobile DBs against thermal motion. Other irradiation studies have provided more empirical signs for the interaction of DBs with defects. For instance, in ref. [@SAR00] the authors provide evidence that, after irradiating a silicon crystal with silver ions, a pileup of lattice defects is accomplished at locations spatially separated from the irradiation site. The evidence indicated that defects could be swept by up to about 1 micron from the irradiated region. This effect was ascribed to the propagation of highly localized packets of vibrational energy, or DBs, created by the bombardment of heavy ions. Another ion-induced, athermal transport process was reported in ref. [@AMM06]. In this case interstitial N diffusion in austenitic stainless steel under Ar ion bombardment was investigated. It was found that N mobility increases in depths several orders of magnitude larger than the ion penetration depth. This irradiation-induced enhancement of N diffusion is consistent with previous observations which show a dependence of the nitriding depth on ion energy [@WDWVWM97] and also on the crystalline orientation [@ARTDPM05], but no conventional mechanism of diffusion can explain them. For this reason it was suggested that diffusion of interstitial atoms might be assisted by highly anharmonic localized excitations which propagate distances well beyond the ion penetration depth. Interstitial atoms reside in potential wells between the lattice atoms. When a breather propagates it strongly disturbs the lattice locally. If it passes near an interstitial these oscillatory motions will distort the potential well confining the interstitial and will affect significantly its mobility. Interstitial motion consists of jumps from one potential well to the next. Since experimental measures deal with concentration depth profiles, interstitial diffusion process can be analyzed in terms of an effective movement along a one-dimensional chain of potential wells. Moreover the presence of an interstitial modifies potentials in adjacent atomic chains, causing the spacing between the two nearest atoms in a chain to the interstitial to increase. Therefore, in a first approximation, an interstitial can be modelled introducing and additional particle in a one-dimensional system and this provides the link to the FK model. In this paper, using a FK model with nonlinear nearest-neighbour interaction, it is shown that migration of the disturbance in a chain caused by an interstitial can be induced by scattering with a mobile longitudinal mode breather. Comparison with previous work on vacancies migration [@CKAER03; @CASR06] also suggests that, according to experimental results, interstitial mobility is more likely and faster than that of vacancy defects. Of course, the specific constraints of a one-dimensional system implies that care is needed when attempting to carry over results to higher dimensional lattices. Nevertheless we think that a one-dimensional study is a necessary and useful first step before approaching the problem with a more realistic and complex two or three-dimensional model. The model ========= As described in the introduction, the F-K model consists of a chain of interacting particles subject to a periodic substrate potential. This system is described by the following Hamiltonian: $$H=\sum_{n=1}^N\frac{1}{2} m\dot x_n^2+V(x_n)+ W(x_n-x_{n-1}) \quad ,$$ where $x_n$ is the absolute coordinate of the $n$-th particle. The corresponding dynamical equations are $$\label{eq:dyn} m \ddot x_n+ V'(x_n)+[W'(x_n-x_{n-1})-W'(x_{n+1}-x_n)]=0, \quad n\in \mathbb{Z}.$$ In order to investigate interstitial mobility we have chosen a cosine potential with the lattice period $a$ $$V(x)=\frac{a^2}{4\pi^2}[1-\cos(2\pi x/a)]\; ,$$ as the simplest, periodic substrate potential, with the linear frequency normalized to unity $\omega_0=\sqrt{V''(0)}=1$. For the interaction between particles, we have selected the Morse potential $$W(x)=\frac{C}{2b^2} [e^{-b(x-a)}-1]^2, x>0.$$ which has a minimum at the lattice period $a$ and a hard part that prevents particles from crossing. The well depth of this potential is $C/2b^2$ while $b^{-1}$ is a measure of the well width. Its curvature at the bottom is given by $C=W''(a)$, so that we can modulate the strength of the interaction without changing its curvature by varying parameter $b$. In this system, an interstitial atom is represented by a doubly occupied well of the periodic potential (see the stable equilibrium configuration in panel (a) of Fig.\[fig:FK\]). The relative coordinate of each particle with respect to its equilibrium position can be written as $u_n=x_n-na$. Using these relative coordinates, the interstitial can be visualized as an antikink [@BK98; @BK98b][^1] as it is shown in panel (b) of Fig. \[fig:FK\]. It is well-known that an antikink can be put into movement as soon as an energy barrier, the so-called Peierls-Nabarro barrier (PNB), is overcome. The PNB can be calculated as the energy difference between the unstable and stable antikink equilibrium configurations (panels (c) and (a) of Fig. \[fig:FK\] respectively) and decreases monotonically with $b$ (see panel (d)). It is worth noting that a vacancy can be visualized as a kink in relative coordinates. Its PNB increases with the parameter $b$ and is always higher than the PNB of an interstitial, except for $b=0$ where both activation energies coincide. This is in accordance with the experimental fact that diffusion of interstitials is faster than that of vacancies, and support the idea that it is necessary to consider a nonlinear interaction potential in order to study diffusion of defects, since $b=0$ represents the linear limit of the Morse potential. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![(a) Scheme of the stable equilibrium state of the Frenkel–Kontorova model with cosine substrate potential and Morse nearest neighbor interaction. The doubly-occupied well represents an interstitial. (b) Antikink corresponding to the stable equilibrium configuration in relative coordinates for $b=1$ and $C=0.5$. (c) Unstable equilibrium configuration. (d) Peierls-Nabarro barrier for the antikink.[]{data-label="fig:FK"}](fk.eps "fig:"){width="\singlefig"} ![(a) Scheme of the stable equilibrium state of the Frenkel–Kontorova model with cosine substrate potential and Morse nearest neighbor interaction. The doubly-occupied well represents an interstitial. (b) Antikink corresponding to the stable equilibrium configuration in relative coordinates for $b=1$ and $C=0.5$. (c) Unstable equilibrium configuration. (d) Peierls-Nabarro barrier for the antikink.[]{data-label="fig:FK"}](kink.eps "fig:"){width="\singlefig"} ![(a) Scheme of the stable equilibrium state of the Frenkel–Kontorova model with cosine substrate potential and Morse nearest neighbor interaction. The doubly-occupied well represents an interstitial. (b) Antikink corresponding to the stable equilibrium configuration in relative coordinates for $b=1$ and $C=0.5$. (c) Unstable equilibrium configuration. (d) Peierls-Nabarro barrier for the antikink.[]{data-label="fig:FK"}](fkpn.eps "fig:"){width="\singlefig"} ![(a) Scheme of the stable equilibrium state of the Frenkel–Kontorova model with cosine substrate potential and Morse nearest neighbor interaction. The doubly-occupied well represents an interstitial. (b) Antikink corresponding to the stable equilibrium configuration in relative coordinates for $b=1$ and $C=0.5$. (c) Unstable equilibrium configuration. (d) Peierls-Nabarro barrier for the antikink.[]{data-label="fig:FK"}](pnantik.eps "fig:"){width="\singlefig"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- In our F-K chain, stationary discrete breathers can be numerically obtained using the standard method of continuation from the anticontinuous limit [@ELS84; @MA96]. Translational motion of discrete breathers can be induced [@CAT96; @AC98] by adding a perturbation $\vec{v}=\lambda (...,0,-1/\sqrt{2},0,1/\sqrt{2},0,...)$ to the velocities of the stationary breather, with the nonzero values at the neighboring sites of the initial breather center. The resulting DB kinetics is very smooth and resembles that of a classical free particle. Therefore, the total energy of a moving discrete breather can be estimated as the sum of its vibrational internal energy, equal to that of the stationary breather, plus its translational energy, which is equal to the energy of the added perturbation $K=\lambda^2/2$. Numerical study =============== In order to investigate interstitial mobility, we have generated a breather centered at site $n=-25$, relatively far from an interstitial whose leftmost particle is located at $n=0$, and then launched that breather towards it following the depinning method mentioned above. Throughout the paper, we have normalized the lattice period $a$ and masses to unity and have taken $C=0.5$ so that moving breathers (MBs) exist in the system for a breather frequency $\omega_b=0.9$. As a result of the scattering the defect can be put into movement leading to long–range transport. We have found three well-differentiated regimes depending on the strength of the interaction potential. Below a critical value $b\approx 0.83$ the result of the scattering is unpredictable. The dynamics is extremely sensitive to initial conditions (value of the perturbation $\lambda$ and initial position of the breather) and the interstitial can travel or make random jumps (backward or forward) or even remain at rest. However, a net backward movement of the defect is only possible if the interaction potential is strong enough. In fact we have observed it only for values of $b\lesssim0.69$. An example of a backwards travelling interstitial is shown in Fig. \[fig:edpbk\], whereas Figs. \[fig:edpbksw\] and \[fig:edpfwsw\] show a backwards and forwards, respectively, hopping interstitial. In this case, the interstitial, after several random jumps, remains pinned on the lattice. These three figures display three panels. Left panel corresponds to an energy density plot where lines join points with the same energy in time while darker color indicates larger energy. Central panel displays the time evolution of the antikink (interstitial) center of mass. This graph helps to visualize more clearly the jumps of the interstitial particle and the final oscillatory state around an equilibrium configuration. Finally, right panel shows a streak plot with the time evolution of the breather and the interstitial. It is noteworthy that in our numerical experiments smaller values of $b$ enhance backward movement and hopping behaviour of the interstitial particle. This latest behaviour is the only observed in the harmonic limit of the interaction potential ($b=0$). Notice that the complexity of the dynamics is linked to the discreteness of the F-K model considered [@Dmitriev1; @Dmitriev2]. In the continuous limit with $b=0$ the breather-antikink interaction is an integrable and well-known case, and the resulting scenario is quite simple: the breather always crosses the antikink which moves backwards during a brief lapse of time [@DEGM]. -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![(Left panel) Energy density plot, showing a backward movement of the interstitial defect after breather scattering. The lines join points with the same energy in time. The darker colour the larger energy. (Central panel) Time evolution of the antikink energy center. (Right panel) Streak plot. Parameters: $K=0.0220$ and $b=0.5$.[]{data-label="fig:edpbk"}](edpbk.eps "fig:"){width="\triplefig"} ![(Left panel) Energy density plot, showing a backward movement of the interstitial defect after breather scattering. The lines join points with the same energy in time. The darker colour the larger energy. (Central panel) Time evolution of the antikink energy center. (Right panel) Streak plot. Parameters: $K=0.0220$ and $b=0.5$.[]{data-label="fig:edpbk"}](xebk.eps "fig:"){width="\triplefig"} ![(Left panel) Energy density plot, showing a backward movement of the interstitial defect after breather scattering. The lines join points with the same energy in time. The darker colour the larger energy. (Central panel) Time evolution of the antikink energy center. (Right panel) Streak plot. Parameters: $K=0.0220$ and $b=0.5$.[]{data-label="fig:edpbk"}](streakbk.eps "fig:"){width="\triplefig"} -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Same as Fig. \[fig:edpbk\] but for a hopping interstitial with net backwards displacement. Parameters: $K=0.0050$ and $b=0.2$.[]{data-label="fig:edpbksw"}](edpsw.eps "fig:"){width="\triplefig"} ![Same as Fig. \[fig:edpbk\] but for a hopping interstitial with net backwards displacement. Parameters: $K=0.0050$ and $b=0.2$.[]{data-label="fig:edpbksw"}](xesw.eps "fig:"){width="\triplefig"} ![Same as Fig. \[fig:edpbk\] but for a hopping interstitial with net backwards displacement. Parameters: $K=0.0050$ and $b=0.2$.[]{data-label="fig:edpbksw"}](streaksw.eps "fig:"){width="\triplefig"} ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Same as Fig. \[fig:edpbk\] but for a hopping interstitial with net forward displacement. Parameters: $K=0.00605$ and $b=0.1$.[]{data-label="fig:edpfwsw"}](edpswfw.eps "fig:"){width="\triplefig"} ![Same as Fig. \[fig:edpbk\] but for a hopping interstitial with net forward displacement. Parameters: $K=0.00605$ and $b=0.1$.[]{data-label="fig:edpfwsw"}](xeswfw.eps "fig:"){width="\triplefig"} ![Same as Fig. \[fig:edpbk\] but for a hopping interstitial with net forward displacement. Parameters: $K=0.00605$ and $b=0.1$.[]{data-label="fig:edpfwsw"}](streakswfw.eps "fig:"){width="\triplefig"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Due to the existence of an activation energy to move an antikink in the discrete case, interstitial motion is only found above a threshold value, $K_c$, of the kinetic energy of the incident breather. In the chaotic regime, $b\lesssim0.83$, this threshold value, plotted in figure \[fig:Kc\], increases monotonically in contrast with the PNB behavior found in the previous section. On the contrary, for $b\gtrsim0.87$ we find the opposite tendency: $K_c$ decreases with $b$ indicating a deep change in the dynamics. Indeed in this parameter regime, for $K>K_c$, the interstitial always moves forward after the scattering and, remarkably, it always moves with approximately constant velocity. In this regime, the Morse potential becomes essentially “flat” with a hard core and the dynamics is dominated by the repulsive part of the interaction potential. An example can be observed in Fig. \[fig:edpfw\]. After the collision with the breather, interstitial motion is clearly linear in time. Its velocity has been computed fitting the points of the central panel with linear regression. In the transition between both regimes, i.e. for $0.83\lesssim b\lesssim0.87$, the interstitial always remains pinned on the lattice, at least for those values of $\lambda$ for which the breather propagates without significant distortion. ![Minimum translational energy ($K_c$) of the incoming breather needed to move an interstitial. In the band $0.83\gtrsim b\lesssim0.87$ the interstitial always remains pinned on the lattice.[]{data-label="fig:Kc"}](lamc.eps){width="\singlefig"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Same as Fig. \[fig:edpbk\] but for a breather with $K=0.020$ and $b=1.5$. The interstitial always moves forward with constant velocity in the parameter region $b\gtrsim0.87$, $K>K_c$ []{data-label="fig:edpfw"}](edpfw.eps "fig:"){width="\triplefig"} ![Same as Fig. \[fig:edpbk\] but for a breather with $K=0.020$ and $b=1.5$. The interstitial always moves forward with constant velocity in the parameter region $b\gtrsim0.87$, $K>K_c$ []{data-label="fig:edpfw"}](xefw.eps "fig:"){width="\triplefig"} ![Same as Fig. \[fig:edpbk\] but for a breather with $K=0.020$ and $b=1.5$. The interstitial always moves forward with constant velocity in the parameter region $b\gtrsim0.87$, $K>K_c$ []{data-label="fig:edpfw"}](streakfw.eps "fig:"){width="\triplefig"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Fig. \[fig:edppin\] shows the evolution of a pinned interstitial for $b=1$. In this case the incident breather possesses a translational energy smaller than the critical value $K_c$ and, consequently, the interstitial acts as a wall which totally reflects the breather. It is observed that, after the collision, part of the breather energy is employed in exciting an internal mode of the interstitial with a frequency smaller than that of the incident breather. This linear localized mode corresponds to the line below the phonon spectrum shown in Fig. \[fig:linear\] for the interstitial stable equilibrium configuration. Note that nonlinear localized modes do not exist close to the interstitial since the interaction potential is soft, and the frequency of the linear localized mode is always below ${\omega_{\mathrm{b}}}=0.9$. -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Same as Fig. \[fig:edpbk\] but for a breather with $K=0.0162<K_c$ and $b=1$. As the kinetic energy of the incident breather is below the threshold value $K_c$, the interstitial remains pinned on the lattice.[]{data-label="fig:edppin"}](edppin.eps "fig:"){width="\triplefig"} ![Same as Fig. \[fig:edpbk\] but for a breather with $K=0.0162<K_c$ and $b=1$. As the kinetic energy of the incident breather is below the threshold value $K_c$, the interstitial remains pinned on the lattice.[]{data-label="fig:edppin"}](xepin.eps "fig:"){width="\triplefig"} ![Same as Fig. \[fig:edpbk\] but for a breather with $K=0.0162<K_c$ and $b=1$. As the kinetic energy of the incident breather is below the threshold value $K_c$, the interstitial remains pinned on the lattice.[]{data-label="fig:edppin"}](streakpin.eps "fig:"){width="\triplefig"} -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Linear modes spectrum of the stable equilibrium configuration for $C=0.5$. ${\omega_{\mathrm{b}}}$ and $2{\omega_{\mathrm{b}}}$ are depicted through dashed lines.[]{data-label="fig:linear"}](linear.eps){width="\singlefig"} As mentioned above, for $b\gtrsim0.87$ a stable interstitial propagating mode appears if the kinetic energy of the incident breather is higher than the threshold value $K_c$. In this parameter region the Morse potential becomes essentially a repulsive potential. In fact, in the limit $b\rightarrow\infty$ it becomes a hard-sphere potential. For this reason, in this dynamical regime interstitial particles move roughly like hard spheres on a wavy energy landscape. After the breather scattering, the interstitial particle surmounts the energy barrier of the on-site potential well and collides with the particle that occupies the following well transferring its energy and momentum to it. In this way the defect propagates at constant velocity forever. In figure \[fig:velint\] we have plotted the dependence of the velocity of this propagating mode on the kinetic energy of the incident breather. One can observe that just above the threshold energy, interstitial velocity increases with the kinetic energy, $K$, of the incoming breather. However for higher values of $K$ the interstitial velocity tends to saturate around a value $0.14$, what means that the interstitial particle moves approximately $0.14\; \frac{2\pi}{\omega_b}\approx 1$ site on the chain per breather period, independently of the coupling strength. This phenomenon is confirmed in figure \[fig:velint2\] where we have plotted the interstitial velocity versus parameter $b$ for a fixed value of $K$. Indeed, for $K=0.045$ (dashed line) well above the energy threshold, interstitial velocity takes roughly the saturation value $0.14$ independently of the coupling strength. Intermediate values of kinetic energy as $K=0.02$ (continuous line) also leads to saturation velocities independently of $b$ but with values lower than 0.14 and less fluctuations. ![Interstitial velocity as a function of the translational energy $K$ of the incident breather for three different values of the coupling strength in the regime ($b\gtrsim 0.87$). In this regime the interstitial always moves forward with constant velocity because the interaction potential reduces essentially to a repulsive hard core.[]{data-label="fig:velint"}](vel1.eps){width="\singlefig"} ![Interstitial velocity versus coupling strength for a fixed translational energy of the incident breather.[]{data-label="fig:velint2"}](vel2.eps){width="\singlefig"} Conclusions =========== We have presented numerical results arising from the interaction between a moving discrete breather and an interstitial defect in a FK chain. The main result is the existence of three differentiated regimes depending on the strength of the interaction potential. When the interaction between neighbors is strong the dynamics is chaotic and the behavior of the interstitial particle is unpredictable: it can jump backwards, forwards or remains at rest. However, if the interaction potential is weak enough, the defect moves forwards along the lattice with constant velocity. This stable propagating mode had not been observed to our knowledge in previous numerical studies concerning the interaction between moving breathers and point defects. The effect is ascribed to the fact that the interaction potential reduces essentially to a repulsive hard core. Between these two dynamical regimes there is an narrow intermediate range of the coupling strength in which the interstitial always remains pinned. Out of that pinned regime, the kinetic energy of the incoming breathers must surpass a threshold in order to move the interstitial. This energy threshold has a non-monotonic behavior. It grows with parameter $b$ in the chaotic regime, but decreases with $b$ when the system losses sensitivity to initial conditions and the propagating mode emerges. With due caution these results can assist in understanding the interaction of mobile discrete breathers with true initially stationary interstitial atoms lying adjacent to a chain in a crystal, which may be of the same or different species from that of the chain. The experiments reported in Refs.[@SAR00] and [@AMM06] are of each type. In these experiments an incident discrete breather must supply both the kinetic energy and the momentum of the interstitial that is put into motion. Moreover, once set in motion the interstitial, and thus its influence on the adjacent chain, is expected to move at the same speed as the discrete breather, thus carrying the defect as opposed to repeated sweeping by subsequent discrete breathers, which is less probable. Our results are also in accordance with the experimental fact that interstitial defects diffuse easily and faster than vacancy ones, and support the hypothesis that scattering with high energy mobile breathers may play an important role for defect diffusion in crystals under ion bombardment. Acknowledgements {#acknowledgements .unnumbered} ================ Two of the authors (JC and BSR) acknowledge sponsorship by the Ministerio de Ciencia e Innovación (Spain), project FIS2008-04848. [99]{} Ya. I. Frenkel and T. Kontorova, *On the theory of plastic deformations and twinning*, J. Phys., **1** (1939), 137–149. O.M. Braun and Yu.S. Kivshar, “The Frenkel-Kontorova Model: Concepts, Methods and Applications”, Springer-Verlag, Berlin-Heidelberg, 2004. O.M. Braun and Yu.S. Kivshar, *Nonlinear dynamics of the Frenkel–Kontorova model*, Phys. Rep., **306** (1998), 1–108. L. M. Floría and J. J. Mazo, *Dissipative dynamics of the Frenkel-Kontorova model*, Adv. Phys., **45** (1996), 505–598. T. Dauxois and M. Peyrard, “Physics of Solitons”, Cambridge University Press, 2006. M. Wuttig, D. Lüsebrink, D. Wamwangi, W. We[ł]{}nic, M. Gille[ß]{}en, and R. Dronskowski, *The role of vacancies and local distortions in the design of new phase-change materials*, Nature Materials, **6** (2007), 122–128. S. Flach and A. Gorbach, *Discrete breathers – Advances in theory and applications*, Phys. Rep., **467** (2008), 1–116. A.S. Barker and A.J. Sievers, *Optical studies of the vibrational properties of disordered solids*, Rev. Mod. Phys., **47** (1975), S1–S179. S. Aubry *Breathers in nonlinear lattices: Existence, linear stability and quantization*, Physica D, [**103**]{} (1997), 201–250. M.V. Ivanchenko, O.I. Kanakov, V.D. Shalfeev and S. Flach, *Discrete breathers in transient processes and thermal equilibrium*, Physica D, **198** (2004), 120–135. B.I. Swanson, J.A. Brozik, S.P. Love, G.O. Strouse, A.P. Shreve, A.R. Bishop, W.Z. Wang and M.I. Salkola, *Observation of intrinsically localized modes in a discrete low-dimensional material*, Phys. Rev. Lett., **82** (1999), 3288–3291. H.S. Eisenberg, Y. Silberberg, R. Morandotti, A.R. Boyd and J.S. Aitchison, *Discrete spatial optical solitons in waveguide arrays*, Phys. Rev. Lett., **81** (1998), 3383–3386. M. Sato, B.E. Hubbard and A.J. Sievers, *Nonlinear energy localization and its manipulation in micromechanical ocillator arrays*, Rev. Mod. Phys., **78** (2006), 137–157. M. Sato, B.E. Hubbard, A.J. Sievers, B. Ilic, D.A. Czaplewski and H.G. Craighead, *Observation of locked intrinsic localized vibrational modes in micromechanical oscillator array*, Phys. Rev. Lett., **90** (2003), 044102. E. Trias, J.J. Mazo and T.P. Orlando, *Discrete breathers in nonlinear lattices: Experimental detection in a Josephson array*, Phys. Rev. Lett., **84** (2000), 741–744. J. Cuevas, F. Palmero, J. F. R. Archilla, and F. R. Romero, *Moving discrete breathers in a Klein-Gordon chain with an impurity*, J. Phys. A: Math. and Gen., **35** (2002), 10519–10530. K. Forinash, M. Peyrard, B.A. Malomed, *Interaction of discrete breathers with impurity modes*, Phys. Rev. E, **49** (1994), 3400–3411. I. Bena, A. Saxena and J. M. Sancho, *Interaction of a discrete breather with a lattice junction*, Phys. rev. E, **65** (2002), 036617. A. Álvarez, J. F. R. Archilla, F. R. Romero, J. Cuevas, and P. V. Larsen, *Breather trapping and breather transmission in a DNA model with an interface*, Eur. Phys. J. B, **51** (2006), 119–130. G. P. Tsironis, J. M. Sancho, M. Ibañes, *Localized energy transport in biopolymer models with rigidity*, Europhys. Lett., **57** (2002), 697–703. J. Cuevas and P. G. Kevrekidis, *Breathers statics and dynamics in Klein-Gordon chains with a bend*, Phys. Rev. E, **69** (2004), 056609. P. V. Larsen, P. L. Christiansen, O. Bang, J. F. R. Archilla, and Yu. B. Gaididei. *Energy funneling in a bent chain of Morse oscillators with long-range coupling*, Phys. Rev. E, **69** (2004), 026603. F. M. Russell and D. R. Collins, *Lattice-solitons in radiation damage*, Nucl. Inst. Meth. Phys. Res. B, **105** (1995), 30–34. J. L. Marín, J.C. Eilbeck, and F.M. Russell, *Localized moving breathers in a 2D hexagonal lattice*, Phys. Lett. A, **248** (1998), 225–229. F. M. Russell and J. C. Eilbeck, *Evidence for moving breathers in a layered crystal insulator at 300K*, Europhys. Lett., **78** (2007), 10004. P. Sen, J. Akhtar, and F. M. Russell. *MeV ion-induced movement of lattice disorder in sigle crystalline silicon*, Europhys. Lett., **51** (2000), 401–406. G. Abrasonis, W. Möller, and X. X. Ma. *Anomalous ion accelerated bulk diffusion of interstitial nitrogen*, Phys. Rev. Lett., **96** (2006), 065901. D. L. Williamson, J. A. Davis, P. J. Wilbur, J. J. Vajo, R. Wei, and J. N. Matossian, *Relative roles of ion energy, ion flux, and sample temperature in low-energy nitrogen ion implantation of Fe—Cr—Ni stainless steel*, Nucl. Inst. Meth. Phys. Res. B, **127** (1997), 930–934. G. Abrasonis, J. P. Rivière, C. Templier, A. Declémy, L. Pranevicius and X. Milhet, *Ion beam nitriding of single and polycrystalline austenitic stainless steel*, J. Appl. Phys., **97** (2005), 083531. J. Cuevas, C. Katerji, J. F. R. Archilla, J. C. Eilbeck, and F. M. Russell, *Influence of moving breathers on vacancies migration*, Phys. Lett. A, **315** (2003), 364–371. J. Cuevas, J. F. R. Archilla, B. Sánchez–Rey, and F. R. Romero, *Interaction of moving discrete breathers with vacancies*, Physica D, **216** (2006), 115–120. J. C. Eilbeck, P. S. Lomdahl, and A. C. Scott, *Soliton structure in crystalline acetanilide*, Phys. Rev. B, **30** (1984), 4703–4712. J. L. Marín and S. Aubry, *Breathers in nonlinear lattices: numerical calculation from the anticontinuous limit*, Nonlinearity, **9** (1996), 1501–1528. D. Chen, S. Aubry, and G. P. Tsironis, *Breather mobility in discrete $\phi^4$ nonlinear lattices*, Phys. Rev. Lett., **77** (1996), 4776–4779. S. Aubry and T. Cretegny, *Mobility and reactivity of discrete breathers*, Physica D, **119** (1998), 34–46. S. V. Dmitriev, T. Shigenari, A. A. Vasiliev and A. E. Miroshnichenko, *Effect of discreteness on a sine-Gordon three-soliton solution*, Phys. Lett. A, **246** (1998), 129. S. V. Dmitriev, T. Miyauchi, K. Abe and T. Shigenari, *Kink-breather solution in the weakly discrete Frenkel-Kontorova model*, Phys. Rev. E, **61** (1998), 5880–5885. R. K. Dodd, J. C. Eilbeck, J. D. Gibbon and H. C. Morris, “Solitons and Nonlinear Wave Equations”, Academic Press, London, 1982. Received xxxx 20xx; revised xxxx 20xx. [^1]: Notice that in Ref. [@BK98b] the terms kink and antikink are interchanged.

--- abstract: 'Dislocations are topological defects known to be crucial in the onset of plasticity and in many properties of crystals. Classical Elasticity still fails to fully explain their dynamics under extreme conditions of high strain gradients and small scales, which can nowadays be scrutinized. By separating conformal and shape deformations, we construct a new formalism for two-dimensional (2D) Elasticity and consider edge dislocations as finite disclination dipoles. This lead us to heuristically obtain that dislocations can be driven by a fundamentally new type of force, which is induced by background density (or hydrostatic strain) gradients. The existence of such mechanism is confirmed through atomistic simulations, where we can move and trap individual dislocations using such configurational force. It depends on a small length parameter, has nonlocal character and can provide ground basis for some phenomenological theories of size effects in plasticity.' author: - 'Paulo César N. Pereira' - 'Sérgio W. S. Apolinário' bibliography: - 'biblio.bib' title: Density Gradients Driving Topological Defects in Crystals --- The idea of dislocation defects was first conceived mathematically [@volterra1907equilibre] and later applied in the context of plasticity[@taylor1934], by considering the movement of defects in a periodic lattice. It soon became a vital feature of investigation in real three-dimensional (3D) crystals [@Frank1952; @hirth1967theory; @kubin2013dislocations]. Since the bubble-raft model [@Bragg1947], two-dimensional (2D) crystals have also been used as simple models to study dislocation dynamics (e.g., using colloids [@vanderMeer15356], complex plasmas [@Nosenko2011] and vortices in superconductors [@Miguel2003]). The individual dislocation movement is generally assumed to be governed by some well-known configurational forces: the Peach-Koehler (PK) driving force due to background stresses [@Peach1950] and the Peierls-Nabarro barrier due to crystal’s discreteness [@Peierls1940; @Nabarro1947] besides other possible motion’s resistance, climb and diffusion mechanisms [@hirth1967theory; @phillips2001crystals; @bulatov2006computer; @kubin2013dislocations]. These forces have been widely used to model plastic deformations in Discrete Dislocation Dynamics (DDD) simulations [@bulatov2006computer; @kubin2013dislocations], where the exact locations of all atoms can be ignored and one only needs to consider the dynamics of dislocation lines, in 3D, or points, in 2D. The validity of such mesoscale approach relies on the forces and mobility law that it considers. The PK interactions between dislocations have power law behavior and the resulting dynamics has no intrinsic length scale (thus leading to a “similitude principle" [@Zaiser2014]). The size effects and length scales emerging from DDD simulations [@El-Awady2015; @Chakravarthy2011] and from rigorous theories [@Valdenarie2016; @Groma2016] based on PK driving forces are usually associated with the obstacle and dislocation densities. They still cannot explain the full range of new plastic phenomena with technological impact observed, for instance, in micron and sub-micron scales [@GREER2011; @kraft2010; @GAO2016; @Voyiadjis2017] (with a “smaller is stronger" trend) and during shock loadings [@MEYERS200991; @Remington2015; @Zepeda-Ruiz2017; @Wehrenberg2017]. Thus, several phenomenological and mechanism-based models have been developed, including corrections to the mobility law [@Gurrutxaga2016], nonlocal Elasticity [@eringen2002nonlocal; @Lazar2005] and strain gradient plasticity [@AIFANTIS1992; @fleck1997; @HUANG2004753; @Fleck2015]. The aim of our work is to broaden current knowledge about dislocation dynamics. By separating shear deformations and variations in density and orientation, a new formalism for 2D Elasticity is constructed. It shows to be suitable in the problem of configurational forces on edge dislocations, which can be described as finite disclination dipoles. We heuristically obtain that dislocation glide can be induced by a background density gradient in the glide direction. This new type of driving force have an intrinsic length parameter and nonlocal behavior. Such mechanism cannot be directly predicted by classical continuum Elasticity and provides a more fundamental motivation for strain gradient theories. Finally, using atomistic simulations, we demonstrate its existence and measure its parameter for some systems. *2D Elasticity formalism.*- In an Eulerian description, classical Elasticity theory of deformation uses the displacement field $\mathbf{u}({\mathbf{r}})={\mathbf{r}}-{\mathbf{R}}({\mathbf{r}})$ to relate the current particles’ positions $\{\mathbf{r}\}$ in a crystal with the ones $\{\mathbf{R}\}$ before deformation. Here we define two convenient fields, ${\mathbf{C}}({\mathbf{r}})$ and ${\mathbf{S}}({\mathbf{r}})$, and operations ${\circ}$ and ${\ast}$ such that $${\mathbf{C}}=\boldsymbol{\nabla}{\circ}\mathbf{u}:=\begin{pmatrix} \nabla_xu_x+\nabla_yu_y\\ \nabla_xu_y-\nabla_yu_x \end{pmatrix}=\begin{pmatrix} \boldsymbol{\nabla}\cdot\mathbf{u}\\ \boldsymbol{\nabla}\wedge\mathbf{u} \end{pmatrix} \label{multcirc}$$ and $${\mathbf{S}}=\boldsymbol{\nabla}{\ast}\mathbf{u}:=\begin{pmatrix} \nabla_xu_x-\nabla_yu_y\\ \nabla_xu_y+\nabla_yu_x \end{pmatrix}=\boldsymbol{\nabla}u_x-\bm \epsilon\cdot\boldsymbol{\nabla}u_y, \label{multast}$$ where $\bm\epsilon=\bigl[\begin{smallmatrix} 0&1\\ -1&0\end{smallmatrix} \bigr]$. Note that ${\circ}$ and ${\ast}$ do not necessarily generate true vectors since, after a rotation of coordinates where normal vectors in the system (such as ${\mathbf{u}}$) are rotated by an angle $\theta$, ${\mathbf{C}}$ remains unchanged and ${\mathbf{S}}$ is rotated by $2\theta$ (see Supplementary Material). For ${\mathbf{S}}=0$, we have Cauchy-Riemann equations for the components of ${\mathbf{u}}$ and then ${\mathbf{C}}$ gives conformal deformations, preserving relative angles. In small deformations (linear Elasticity), the components in Eq. (\[multcirc\]) have well-known interpretations [@chaikin2000]: $C_1=\boldsymbol{\nabla}\cdot\mathbf{u}\approx-[\rho-\rho_0]/\rho_0$ is the hydrostatic strain (density change), where $\rho_0$ is the original particle density and $\rho$ is the one after deformation, while $C_2=\boldsymbol{\nabla}\wedge\mathbf{u}$ is twice the angle of rotation and no strain the crystal. In contrast, ${\mathbf{S}}$ is responsible for pure shear, i.e., deviatoric strain (shape change). For small smooth deformations in triangular and hexagonal crystals with short-range interactions, isotropic linear Hyperelasticity is valid and the interaction energy of deformation is $$\mathcal{U}_{int}=\frac{1}{2}\int\big[BC_1^2({\mathbf{r}})+\mu |{\mathbf{S}}({\mathbf{r}})|^2\big]\ {\mathrm{d}}^2r, \label{energy}$$ where $B$ and $\mu$ are the bulk and shear moduli [@chaikin2000], respectively. The interaction force density within the crystal is $\mathbf{f}_{int}=-\frac{\delta\mathcal{U}_{int}}{\delta{\mathbf{u}}}=\boldsymbol{\nabla}{\ast}\frac{\delta\mathcal{U}_{int}}{\delta{\mathbf{C}}}+\boldsymbol{\nabla}{\circ}\frac{\delta\mathcal{U}_{int}}{\delta{\mathbf{S}}}$. Thus, when the particles are subjected to an external body force field $\mathbf{F}_{ext}({\mathbf{r}})$, the equilibrium condition is given by $$B\boldsymbol{\nabla}C_1+\mu\boldsymbol{\nabla}{\circ}{\mathbf{S}}+\rho_0\mathbf{F}_{ext}=0.\label{equilibrium}$$ Note that ${\mathbf{C}}$ and ${\mathbf{S}}$ are derivatives of the same ${\mathbf{u}}$ and then obey some compatibility conditions. Moreover, in the presence of defects, the possibility of ${\mathbf{u}}$ do not satisfy the commutation of partial derivatives must be taken into account. We define the Burgers vector of a single dislocation $i$ as $\mathbf{b}_i=\oint_i{\mathrm{d}}{\mathbf{u}}$, for small counterclockwise closed curves enclosing it, and obtain $$\boldsymbol{\nabla}{\ast}{\mathbf{C}}-\boldsymbol{\nabla}{\circ}{\mathbf{S}}=2\bm\epsilon\cdot\mathbf{B}\label{compatcond}$$ where $\mathbf{B}({\mathbf{r}})=\sum_i\mathbf{b}_i\delta({\mathbf{r}}-{\mathbf{r}}_i)$ is the Burgers vector density. When $\mathbf{B}$ and boundary conditions are known, Eq. (\[compatcond\]) gives ${\mathbf{S}}$ from (nonlocal values of) ${\mathbf{C}}$ and vice versa. We can then entirely describe the deformation using only shape variations (${\mathbf{S}}$) or, alternatively, using only variations in density ($C_1$) and orientation ($C_2$). This physical duality originates from the mathematical duality between ${\mathbf{C}}$ and ${\mathbf{S}}$. Only the ${\mathbf{C}}$-picture of Elasticity can provide the equilibrium condition (\[equilibrium\]) in terms of local strain gradients. On the other hand, the ${\mathbf{S}}$-picture is needed in a local formulation of the PK force. {width="95.00000%"} *Configurational forces on dislocations.*- For a single dislocation, one can use the compatibility and equilibrium conditions (with $\mathbf{F}_{ext}=0$) to obtain the deformation fields ${\mathbf{S}}^{(disl)}$ and ${\mathbf{C}}^{(disl)}$ that are induced in the crystal due to this defect. For a moving dislocation, we can show, by integrating the variation in $C^{(disl)}_1$, that the total number of particles is preserved only if the movement is in the Burgers vector direction $\hat{{\mathbf{b}}}$, i.e., a glide (see Supplementary Material). A dislocation can climb (i.e., move perpendicularly to its $\hat{{\mathbf{b}}}$) through additional mechanisms [@hirth1967theory], such as annihilating a vacancy. In linear Elasticity, the total deformation is the sum of the dislocation contribution with background ones (representing contributions from other defects, external forces and boundary conditions). The background fields can induce dislocation glide if such movement decreases the total energy. A configurational force can then be obtained from Eq. (\[energy\]) by considering the energy variation in the limit of a small dislocation displacement. This gives the well-known Peach-Koehler force [@Peach1950; @phillips2001crystals; @maugin2016configurational] for glide movement, which can be written in our formalism as $$\hat{{\mathbf{b}}}\cdot\mathbf{F}_{disl}^{(PK)}\ \propto\ S^{(bg)}_{res}=\big(\hat{{\mathbf{b}}}{\ast}\hat{{\mathbf{b}}}\big)\wedge{\mathbf{S}}^{(bg)} \label{pk}$$ where $S^{(bg)}_{res}$ is called the (background) resolved shear strain, evaluated at the dislocation position. For $\hat{{\mathbf{b}}}=\hat{\mathbf{x}}$, a positive resolved shear deforms $|\,|\,|$ into $///$, for example. A formal generalization of Eq. (\[pk\]), considering the total Lagrangian, includes a contribution proportional to the dislocation velocity [@maugin2016configurational] and, if the Lagrangian depends only on local strains, no strain gradient term appears. *Strain gradient influences.*- In general, after a deformation, we expect from Eq. (\[pk\]) that the defects nucleate and move towards minimization of shear [@phillips2001crystals]. Heat treatments, for example, allow the crystal to reach dislocation configurations which further minimize the total shear. Implications of this are readily clarified using our formalism. The resulting configuration has ${\mathbf{S}}\simeq0$, with $\simeq$ meaning equal in a coarse-grained average. Then the components of ${\mathbf{u}}$ nearly satisfies the Cauchy-Riemann equations and, from Eq. (\[compatcond\]), $$2\bm\epsilon\cdot\mathbf{B}\simeq\boldsymbol{\nabla}{\ast}{\mathbf{C}}=\boldsymbol{\nabla}C_1-\bm\epsilon\cdot\boldsymbol{\nabla}C_2. \label{gnd}$$ We suggest that this is well approximated in systems near the ground state, explaining why the so-called conformal and quasi-conformal crystals are so ubiquitous in such cases [@Menezes2017; @Soni2018]. Relation (\[gnd\]) shows how lines of discontinuity on $C_1$ or $C_2$ (separating regions with different density or orientation, respectively) require dislocations concentrated on these lines, giving rise to grain boundaries. A result similar to (\[gnd\]) was previously obtained [@Mughal2007], but it considered only the density gradient term. The averaged $\mathbf{B}$ in Eq. (\[gnd\]) provides a direct illustration of Geometrically Necessary Dislocations [@ashby1970] (GNDs), which strongly affect the plastic properties of the crystal. The GNDs motivated phenomenological theories of strain gradient plasticity [@fleck1997] by considering that dislocation configurations must not only depend on strains but also on their gradients. We intend to study if and how such dependence can happen at a fundamental level, contributing to the emergence of size effects in plasticity. The first clue was obtained by Iyer et al. [@Iyer2015], who found through a certain type of electronic structure calculations that the core energy of edge dislocations in Aluminum could depend on the background strains and feel a force due to strain gradients. But no observation of this force was ever obtained in the literature. Neither did a theoretical model to explain which strain gradients (if any) could actually induce glide by some physical mechanism. Eq. (\[gnd\]) suggests yet a trend to $\mathbf{B}\perp\boldsymbol{\nabla}C_1$, i.e., dislocations hardly equilibrate at positions where $|\hat{{\mathbf{b}}}\cdot\boldsymbol{\nabla}C^{(bg)}_1|$ is high. We then investigate this component of background density gradient as possibly able to directly drive dislocations. Such influence can be probed through atomistic simulations, where we can control the background deformation using external potential fields with 2D symmetry. Fig. \[fig:fig2\] illustrates dislocations at equilibrium resulting from Brownian Dynamics (BD) simulations [@Satoh] at low temperatures. The conservative external forces used here induce no $S^{(bg)}_{res}$ but other background deformations, where $\hat{{\mathbf{b}}}\cdot\boldsymbol{\nabla}C^{(bg)}_1$ is the unique type of induced strain gradient which is positive in all cases. As evaluated in Supplementary Material, the total PK forces acting on the dislocations presented in Fig. \[fig:fig2\] try to drive them to annihilate each other. But the external potential produces another effect that forbids the annihilation to happen and, if the potential is enhanced, can even drive the dislocations apart. This was observed in systems ranging from 2D with (scale-invariant) power-law interactions to 3D with Morse interactions which simulates a copper (Cu) crystal where edge dislocations decompose into Shockley partial dislocations. Although Eq. (\[gnd\]) also suggests a trend to $\mathbf{B}\parallel\boldsymbol{\nabla}C_2$, our trials to confirm through simulations if $\boldsymbol{\nabla}C_2^{(bg)}$ (induced by nonconservative external forces) can directly affect dislocations did not succeed. We try to explain these behaviors by looking at the dislocation as a dipole of regions with high and low densities and associating glide with rotations of this dipole. {width="95.00000%"} *Effective torques on disclination dipoles.*- Real dislocations can be more properly described as finite dipoles of disclinations (e.g., see [@Pretko2018]), which are topological defects for which continuum theory requires conservation of both charge and dipole moment. The disclination charge is $s=\oint{\mathrm{d}}C^{(disc)}_2$, i.e., the orientation $C^{(disc)}_2=\boldsymbol{\nabla}\wedge{\mathbf{u}}^{(disc)}$ is a multivalued quantity. Then, for a disclination at the origin, (see Supplementary Material) $$\boldsymbol{\nabla}\!{\ast}\!{\mathbf{C}}^{(disc)}({\mathbf{r}})\!=\!\frac{B\!\!+\!\!2\mu}{\mu}\boldsymbol{\nabla}C^{(disc)}_1({\mathbf{r}})\!=\!-\frac{s(B\!+\!2\mu)\hat{{\mathbf{r}}}}{2\pi(B\!+\!\mu)|{\mathbf{r}}|}.\label{discc}$$ This is an irrotational source/sink type of field. Therefore, Helmholtz decomposition gives that the disclination fields can couple with $\boldsymbol{\nabla}C_1^{(bg)}$ only (i.e., with the irrotational/longitudinal part of $\boldsymbol{\nabla}{\ast}{\mathbf{C}}^{(bg)}$) unless there are other isolated disclinations (which generate irrotational $\bm\epsilon\cdot\boldsymbol{\nabla}C_2^{(bg)}$). Such ${\mathbf{C}}$-picture of Elasticity reveals the particular relevance of density gradients. The disclination dipole is a source-sink pair of density singularities (with $\boldsymbol{\nabla}C_1$ parallel to $\bm\epsilon\cdot\hat{{\mathbf{b}}}$ in their center) while the real crystal dislocation has, in its core, regions with high and low densities. Such regions can be represented by “disclination particles", defined in triangular lattices as the ones having more or less than 6 neighbors in a Voronoi tessellation. During dislocation glide, these particles change due to exchanges of neighbors. We point out that the dipole of disclination particles is not always perfectly aligned in the direction $\perp\hat{{\mathbf{b}}}$. Therefore, fluctuations of this dipole moment can produce variations in the local density gradient, increasing or decreasing its component parallel to $\hat{{\mathbf{b}}}$ due to clockwise or counterclockwise rotation, respectively. Figs. \[fig:fig3\](a)-(d) show snapshots obtained from simulations of a dislocation glide induced when increasing $\hat{{\mathbf{b}}}\cdot\boldsymbol{\nabla}C^{(bg)}_1$. They show the effective disclination dipole moment (represented by the direction $\perp\hat{{\mathbf{b}}}_{eff}$) rotated due to the background density gradient, analogously to the torque exerted on an electric dipole by a background electric field. The rotation is effectively a local resolved shear deformation (just like the dipole can be rotated by $S_{res}^{(bg)}$) and thus leads to dislocation glide. We then propose a glide force which is proportional to the induced resolved shear $$S_{res}^{(ind)}= L\ \hat{{\mathbf{b}}}\cdot\boldsymbol{\nabla}C^{(bg)}_1\label{indres}$$ where $L$ is an intrinsic length scale for the linear response. Between Figs. \[fig:fig3\](a) and \[fig:fig3\](b) (or between Figs. \[fig:fig3\](c) and \[fig:fig3\](d)) $S_{res}^{(ind)}$ was varied but the disclination particles remained the same due to the Peierls-Nabarro barrier. ![\[fig:fig4\] BD results for the external potential strength $V_0$ versus the equilibrium distance $d$ between dislocations in the system of Fig. \[fig:fig3\], with power-law interactions $V_p^{PL}(r)=\varepsilon(a_0/r)^6$, and in a system with the same external forces but Lennard-Jones interactions $V_p^{LJ}(r)=0.387323\varepsilon\big[(a_0/r)^{12}-(a_0/r)^6\big]$. The theoretical curves contain only one fitting parameter and predict no stability in the dashed regions.](fig3c.pdf){width="45.00000%"} *Numerical investigations.*- We propose the expression $$\hat{{\mathbf{b}}}\cdot\mathbf{F}^{(tot)}_{disl}\propto S^{(tot)}_{res}=(\hat{{\mathbf{b}}}{\ast}\hat{{\mathbf{b}}})\wedge{\mathbf{S}}^{(bg)}+L\ \hat{{\mathbf{b}}}\cdot\boldsymbol{\nabla}C^{(bg)}_1 \label{totalforce}$$ for the total configurational force and use low-temperature BD simulations to probe it. For the system in Fig. \[fig:fig3\], $S^{(tot)}_{res}=0$ provides an analytical expression relating the external potential strength $V_0$ with the equilibrium distance $d$ between the dislocations (see Supplementary Material). Such curve for $V_0$ versus $d$ has a minimum at $d\approx84a_0$ and a multiplicative factor involving $L$ (usable as fitting parameter). For $V_0$ below the minimum or for $d<84a_0$, the theory predicts no stability and the simulations confirm this by resulting in dislocation annihilation. Fig. \[fig:fig4\] presents results from simulations for two different types of short-range pair interactions: power-law and Lennard-Jones. For these systems, we can obtain $L$ from the fitting of the theoretical curve. One can observe in Fig. \[fig:fig4\] that $d$ varies with hops (in steps of $\approx\!a_0$). This occurs because of our way of considering each dislocation position, approximated as the mean position of its disclination particles. In fact, within some finite ranges of $V_0$, the system stay in nearly the same $d$ due to the Peierls-Nabarro barrier. We believe that this barrier and nonlinear effects are the main reasons for the theoretical fits in Fig. \[fig:fig4\] start to fail in the region of large $d$, where these effects are most relevant. The fitting for $d<94a_0$, shown in Fig. \[fig:fig4\], provide $L^{PL}\approx0.683a_0$ and $L^{LJ}\approx4.43a_0$ for the power-law and Lennard-Jones interactions, respectively. The theoretical fit for the 3D Cu crystal fails more since the dislocations are not localized but extended. Still, we can obtain the estimate $L^{Cu}\approx0.32a_0$ for the system shown in Figs. \[fig:fig2\](c-d) (see Supplementary Material). The $\boldsymbol{\nabla}C^{(bg)}_1$ needed to glide a dislocation is the Peierls strain (i.e., the $S^{(tot)}_{res}$ needed to glide, equivalent to the Peierls stress divided by $\mu$) divided by $L$. The experimental value for the Peierls strain in the Cu crystal [@Kamimura2013] is $<7\times10^{-6}$ and our results in the 2D systems estimate it as $<10^{-4}$. Therefore, the wavelengths of density variations can be much larger than $L$ and still drive dislocations. *Concluding remarks.*- The total force (Eq. (\[totalforce\])) can neither be expressed *locally* in the ${\mathbf{S}}$-picture of Elasticity nor in the ${\mathbf{C}}$-picture. While the glide due to PK forces directly decreases the local energy density, the force due to $S^{(ind)}_{res}$ has a nonlocal origin. The induced rotation contributes to a local decrease in $|\boldsymbol{\nabla}C_1|$ and then in the integral of $C_1^2$, decreasing the total energy. The induced shear itself costs energy and is compensated by the dislocation glide, as in PK. In fact, the induced resolved shear appears only near the dislocation core. Figs. \[fig:fig3\](e)-(f) shows how much localized it is, along the direction $\perp\hat{{\mathbf{b}}}$. In this direction, it changes sign and goes to zero within a few lattice spacings $a_0$. Because of this change in sign, we expect that the new force may not be effective in high-angle grain boundaries but can drive low-angle ones, in which the dislocations are sufficiently apart. The proposed mechanism driving dislocations to regions with lower particle density has much to be investigated. This general behavior suggests the consideration of a dislocation core energy that depends on the background density. Our theoretical approach still lacks an expression for $L$ and explicit energetic analysis. The consideration of a rotating disclination dipole could not be made within classical Elasticity, which prevents rotation of the dipole by topologically constraining the direction of $\mathbf{b}$ ($=\oint{\mathrm{d}}{\mathbf{u}}$). It may be possible to construct a generalized continua theory [@altenbach2011mechanics] in which the Eshelbian formalism of configurational forces [@maugin2016configurational] provides a more formal derivation of Eq. (\[totalforce\]). Or maybe it can be obtained by some non-singular treatment of the dislocation core [@Cai2006]. We anticipate that expression (\[totalforce\]) could be readily adapted for forces on line elements of edge dislocations in 3D and it can be used in DDD simulations to obtain more reliable results. (Note that, while PK forces between dislocations decay as $\sim 1/r$, the new contributions decay as $\sim L/r^2$.) Moreover, such fundamental influence of strain gradients must be taken into account in constructing better theoretical models for dislocation phenomena. Finally, we hope that recent experimental advances [@rozaliya2014strain; @Tang; @Goldsche2018] allow one to probe the effects of this new force and measure $L$ in important materials. We thank R. M. Menezes and E. O. Lima for technical support. We would also like to thank the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) for the financial support. **Supplementary Material for “Density Gradients Driving Topological Defects in Crystals"** 2D Elasticity using $ {\mathbf{C}}$ and $ {\mathbf{S}}$ ======================================================= Some properties of $ {\mathbf{C}}$ and $ {\mathbf{S}}$ ------------------------------------------------------ Consider a rotation of coordinates by $\theta$ (i.e., ${\mathbf{r}}'=\bm R(\theta)\cdot{\mathbf{r}}$) in which the true vectors (i.e., spin-1) transform as ${\mathbf{u}}'=\bm R(\theta)\cdot{\mathbf{u}}$, where $\bm R(\theta)=\bigl[\begin{smallmatrix} \cos\theta&-\sin\theta \\ \sin\theta&\cos\theta \end{smallmatrix} \bigr]$ is the rotational matrix. If $\mathbf{v}=\bigl(\begin{smallmatrix} v_1 \\ v_2 \end{smallmatrix} \bigr)$ and $\mathbf{w}=\bigl(\begin{smallmatrix} w_1 \\ w_2 \end{smallmatrix} \bigr)$ are objects with spin-$s_v$ and spin-$s_w$, respectively (i.e., $\mathbf{v}'=\bm R(s_v\theta)\cdot\mathbf{v}$ and $\mathbf{w}'=\bm R(s_w\theta)\cdot\mathbf{w}$), then we have $$\label{key} (\mathbf{v}{\circ}\mathbf{w})'\equiv\left(\begin{matrix} v'_1w'_1+v'_2w'_2 \\ v'_1w'_2-v'_2w'_1 \end{matrix} \right)=\bm R\big((s_w-s_v)\theta\big)\cdot(\mathbf{v}{\circ}\mathbf{w})\quad\textrm{and}\quad(\mathbf{v}{\ast}\mathbf{w})'\equiv\left(\begin{matrix} v'_1w'_1-v'_2w'_2 \\ v'_1w'_2+v'_2w'_1 \end{matrix} \right)=\bm R\big((s_v+s_w)\theta\big)\cdot(\mathbf{v}{\ast}\mathbf{w})$$ Thus, Eqs. (1) and (2) of the main text directly gives ${\mathbf{C}}'={\mathbf{C}}$ (i.e., spin-0/scalar), ${\mathbf{S}}'=\bm R(2\theta)\cdot{\mathbf{S}}$ (i.e., spin-2) and we have the true vector (spin-1) fields $$\label{Sastbol} \boldsymbol{\nabla}{\ast}{\mathbf{C}}=\boldsymbol{\nabla}{\ast}\big[\boldsymbol{\nabla}{\circ}{\mathbf{u}}\big]=\nabla^2{\mathbf{u}}+\bm\epsilon\cdot\big[\big(\nabla_x\nabla_y-\nabla_y\nabla_x\big){\mathbf{u}}\big]$$ and $$\label{Sbolast} \boldsymbol{\nabla}{\circ}{\mathbf{S}}=\boldsymbol{\nabla}{\circ}\big[\boldsymbol{\nabla}{\ast}{\mathbf{u}}\big]=\nabla^2{\mathbf{u}}-\bm\epsilon\cdot\big[\big(\nabla_x\nabla_y-\nabla_y\nabla_x\big){\mathbf{u}}\big].$$ For a single dislocation at $ {\mathbf{r}}_0 $, we define its Burgers vector through the line integral $\mathbf{b}=\oint{\mathrm{d}}{\mathbf{u}}=\oint{\mathrm{d}}{\mathbf{r}}\cdot\boldsymbol{\nabla}{\mathbf{u}}({\mathbf{r}})=\int{\mathrm{d}}^2r\big(\boldsymbol{\nabla}\wedge\boldsymbol{\nabla}\big){\mathbf{u}}({\mathbf{r}})$ for any counterclockwise closed curve enclosing $ {\mathbf{r}}_0 $ and then $ \big(\nabla_x\nabla_y-\nabla_y\nabla_x\big){\mathbf{u}}({\mathbf{r}})\equiv\big(\boldsymbol{\nabla}\wedge\boldsymbol{\nabla}\big){\mathbf{u}}({\mathbf{r}})={\mathbf{b}}\delta({\mathbf{r}}-{\mathbf{r}}_0) $. For general distributions of dislocations, we define the density of Burgers vectors by $\mathbf{B}({\mathbf{r}})=\big(\boldsymbol{\nabla}\wedge\boldsymbol{\nabla}\big){\mathbf{u}}({\mathbf{r}})=\sum_i\mathbf{b}_i\delta({\mathbf{r}}-{\mathbf{r}}_i)$. Therefore, by taking equation (\[Sastbol\]) minus equation (\[Sbolast\]), we find the compatibility conditions $$\boldsymbol{\nabla}\ast{\mathbf{C}}-\boldsymbol{\nabla}{\circ}{\mathbf{S}}=2\bm\epsilon\cdot\mathbf{B}.\label{Scompatcond}$$ We can use the Green’s function for the 2D Laplacian, given by $G({\mathbf{r}}-{\mathbf{r}}')=\ln|{\mathbf{r}}-{\mathbf{r}}'|/2\pi$ with $\nabla^2G({\mathbf{r}}-{\mathbf{r}}')=\delta({\mathbf{r}}-{\mathbf{r}}')$ and $ \big(\boldsymbol{\nabla}\wedge\boldsymbol{\nabla}\big)G({\mathbf{r}}-{\mathbf{r}}')=0 $, to obtain solutions for inhomogeneous differential equations of $ \boldsymbol{\nabla}{\ast}$ and $ \boldsymbol{\nabla}{\circ}$. We have $$\begin{aligned} {\mathbf{C}}({\mathbf{r}})&=&\boldsymbol{\nabla}{\circ}\int G({\mathbf{r}}-{\mathbf{r}}')\big[\boldsymbol{\nabla}'{\ast}{\mathbf{C}}({\mathbf{r}}')\big]{\mathrm{d}}^2r'+{\mathbf{C}}^{(bc)}({\mathbf{r}})\nonumber\\ &=&\int \boldsymbol{\nabla}G({\mathbf{r}}-{\mathbf{r}}'){\circ}\big[\boldsymbol{\nabla}'{\ast}{\mathbf{C}}({\mathbf{r}}')\big]{\mathrm{d}}^2r'+{\mathbf{C}}^{(bc)}({\mathbf{r}})\label{Ssolast1}\\ &=&\frac{1}{2\pi}\int\frac{({\mathbf{r}}-{\mathbf{r}}')}{|{\mathbf{r}}-{\mathbf{r}}'|^2}{\circ}\big[\boldsymbol{\nabla}'{\ast}{\mathbf{C}}({\mathbf{r}}')\big]{\mathrm{d}}^2r'+{\mathbf{C}}^{(bc)}({\mathbf{r}}),\label{Ssolast2}\end{aligned}$$ which is a 2D Helmholtz decomposition since $\boldsymbol{\nabla}{\ast}{\mathbf{C}}({\mathbf{r}})=\boldsymbol{\nabla}C_1({\mathbf{r}})-\bm\epsilon\cdot\boldsymbol{\nabla}C_2({\mathbf{r}})$, and $$\begin{aligned} {\mathbf{S}}({\mathbf{r}})&=&\boldsymbol{\nabla}{\ast}\int G({\mathbf{r}}-{\mathbf{r}}')\big[\boldsymbol{\nabla}'{\circ}{\mathbf{S}}({\mathbf{r}}')\big]{\mathrm{d}}^2r'+{\mathbf{S}}^{(bc)}({\mathbf{r}})\nonumber\\ &=&\int \boldsymbol{\nabla}G({\mathbf{r}}-{\mathbf{r}}'){\ast}\big[\boldsymbol{\nabla}'{\circ}{\mathbf{S}}({\mathbf{r}}')\big]{\mathrm{d}}^2r'+{\mathbf{S}}^{(bc)}({\mathbf{r}})\label{Ssolbol1}\\ &=&\frac{1}{2\pi}\int\frac{({\mathbf{r}}-{\mathbf{r}}')}{|{\mathbf{r}}-{\mathbf{r}}'|^2}{\ast}\big[\boldsymbol{\nabla}'{\circ}{\mathbf{S}}({\mathbf{r}}')\big]{\mathrm{d}}^2r'+{\mathbf{S}}^{(bc)}({\mathbf{r}})\label{Ssolbol2}\end{aligned}$$ where $ {\mathbf{C}}^{(bc)} $ and $ {\mathbf{S}}^{(bc)} $ are solutions to the homogeneous equations $ \boldsymbol{\nabla}{\ast}{\mathbf{C}}^{(bc)}=0 $ and $ \boldsymbol{\nabla}{\circ}{\mathbf{S}}^{(bc)}=0 $, respectively, such that the total fields satisfy the boundary conditions. Since the superposition principle is valid in linear Elasticity, we can separately analyze the deformation by contributions from defects, external forces and boundary conditions. Deformation fields of a point dislocation ----------------------------------------- The deformation fields of defects can be separated in a regular part, with $\nabla_x\nabla_y{\mathbf{u}}^{(reg)}=\nabla_y\nabla_x{\mathbf{u}}^{(reg)}$, and singular one, with $\nabla_x\nabla_y{\mathbf{u}}^{(sing)}\neq\nabla_y\nabla_x{\mathbf{u}}^{(sing)}$. A single dislocation with Burgers vector $ {\mathbf{b}}$ at the origin have, from Eqs. (\[Sastbol\]) and (\[Sbolast\]), singular deformation fields satisfying $ \boldsymbol{\nabla}\ast{\mathbf{C}}^{(sing)}({\mathbf{r}})=-\boldsymbol{\nabla}{\circ}{\mathbf{S}}^{(sing)}({\mathbf{r}})=\bm\epsilon\cdot{\mathbf{b}}\ \delta({\mathbf{r}})={\tilde{\mathbf{b}}}\ \delta({\mathbf{r}}) $, where $ {\tilde{\mathbf{b}}}=\bm\epsilon\cdot{\mathbf{b}}$. For deformation fields going to zero at infinity, we use Eqs. (\[Ssolast1\]) and (\[Ssolbol1\]) to obtain $$\label{Ssing} {\mathbf{C}}^{(sing)}({\mathbf{r}})=\boldsymbol{\nabla}G({\mathbf{r}}){\circ}{\tilde{\mathbf{b}}}=\frac{\hat{{\mathbf{r}}}{\circ}{\tilde{\mathbf{b}}}}{2\pi|{\mathbf{r}}|}\qquad\textrm{and}\qquad{\mathbf{S}}^{(sing)}({\mathbf{r}})=-\boldsymbol{\nabla}G({\mathbf{r}}){\ast}{\tilde{\mathbf{b}}}=-\frac{\hat{{\mathbf{r}}}{\ast}{\tilde{\mathbf{b}}}}{2\pi|{\mathbf{r}}|}$$ where $ \hat{{\mathbf{r}}}={\mathbf{r}}/|{\mathbf{r}}| $. These singular fields alone cannot satisfy the mechanical equilibrium equation without external forces (i.e., $ B\boldsymbol{\nabla}C_1^{(sing)}+\mu\boldsymbol{\nabla}{\circ}{\mathbf{S}}^{(sing)}\neq0 $). Regular fields (i.e., satisfying $ \boldsymbol{\nabla}\ast{\mathbf{C}}^{(reg)}=\boldsymbol{\nabla}{\circ}{\mathbf{S}}^{(reg)} $) are thus induced in such a way that $$\begin{aligned} 0&=&B\boldsymbol{\nabla}\big(C_1^{(sing)}+C_1^{(reg)}\big)+\mu\boldsymbol{\nabla}{\circ}\big({\mathbf{S}}^{(sing)}+{\mathbf{S}}^{(reg)}\big)\nonumber\\ &=&B\boldsymbol{\nabla}\big(C_1^{(sing)}+C_1^{(reg)}\big)+\mu\boldsymbol{\nabla}{\ast}\big(-{\mathbf{C}}^{(sing)}+{\mathbf{C}}^{(reg)}\big)\nonumber\\ &=&B\boldsymbol{\nabla}\big(C_1^{(sing)}+C_1^{(reg)}\big)+\mu\big[\boldsymbol{\nabla}\big(-C_1^{(sing)}+C_1^{(reg)}\big)-\bm\epsilon\cdot\boldsymbol{\nabla}\big(-C_2^{(sing)}+C_2^{(reg)}\big)\big]\nonumber\\ &=&\boldsymbol{\nabla}\big[(B-\mu)C_1^{(sing)}+(B+\mu)C_1^{(reg)}\big]-\mu\bm\epsilon\cdot\boldsymbol{\nabla}\big(-C_2^{(sing)}+C_2^{(reg)}\big).\nonumber\end{aligned}$$ The result above is in the form of a Helmholtz decomposition. Here we consider the fields due to the dislocation only. Boundary conditions contributions are left to be considered later. Then we have $$C_1^{(reg)}=\frac{(\mu-B)}{(B+\mu)}C_1^{(sing)}=\frac{(\mu-B)}{(B+\mu)}\boldsymbol{\nabla}G\cdot{\tilde{\mathbf{b}}}\qquad\textrm{and}\qquad C_2^{(reg)}=C_2^{(sing)}=\boldsymbol{\nabla}G\wedge{\tilde{\mathbf{b}}}$$ or simply $$\label{Sregc} {\mathbf{C}}^{(reg)}=\frac{\mu\ \boldsymbol{\nabla}G{\circ}{\tilde{\mathbf{b}}}-B\ {\tilde{\mathbf{b}}}{\circ}\boldsymbol{\nabla}G}{B+\mu}.$$ One can see that such regular field is derived from a regular displacement field (i.e., $ {\mathbf{C}}^{(reg)}=\boldsymbol{\nabla}{\circ}{\mathbf{u}}^{(reg)} $) given by $$\label{Sregdisp} {\mathbf{u}}^{(reg)}({\mathbf{r}})=\frac{2\mu\ {\tilde{\mathbf{b}}}\ln|{\mathbf{r}}|-B\ {\tilde{\mathbf{b}}}{\circ}(\hat{{\mathbf{r}}}{\ast}\hat{{\mathbf{r}}})}{4\pi(B+\mu)},$$ which gives a regular shear field that can be written as $$\label{Sregs} {\mathbf{S}}^{(reg)}({\mathbf{r}})=\boldsymbol{\nabla}{\ast}{\mathbf{u}}^{(reg)}({\mathbf{r}})=\frac{\mu\ \boldsymbol{\nabla}G({\mathbf{r}}){\ast}{\tilde{\mathbf{b}}}+B\ {\tilde{\mathbf{b}}}{\circ}\big[\hat{{\mathbf{r}}}{\ast}\hat{{\mathbf{r}}}{\ast}\boldsymbol{\nabla}G({\mathbf{r}})\big]}{B+\mu}.$$ Note that the multiplication $ {\ast}$ is commutative and associative. Finally, we can write the total deformation fields $ {\mathbf{C}}^{(disl)}={\mathbf{C}}^{(sing)}+{\mathbf{C}}^{(reg)} $ and $ {\mathbf{S}}^{(disl)}={\mathbf{S}}^{(sing)}+{\mathbf{S}}^{(reg)} $ as $${\mathbf{C}}^{(disl)}({\mathbf{r}})=\frac{(B+2\mu)\ \boldsymbol{\nabla}G({\mathbf{r}}){\circ}{\tilde{\mathbf{b}}}-B\ {\tilde{\mathbf{b}}}{\circ}\boldsymbol{\nabla}G({\mathbf{r}})}{B+\mu}=\frac{\bm\epsilon\cdot\big[(B+2\mu)\ \hat{{\mathbf{r}}}{\circ}{\mathbf{b}}+B\ {\mathbf{b}}{\circ}\hat{{\mathbf{r}}}\big]}{2\pi(B+\mu)\lvert{\mathbf{r}}\rvert}\label{Sdislc}$$ and $${\mathbf{S}}^{(disl)}({\mathbf{r}})=-\frac{B\ \big[\boldsymbol{\nabla}G({\mathbf{r}}){\ast}{\tilde{\mathbf{b}}}-{\tilde{\mathbf{b}}}{\circ}[\hat{{\mathbf{r}}}{\ast}\hat{{\mathbf{r}}}{\ast}\boldsymbol{\nabla}G({\mathbf{r}})]\big]}{B+\mu}=-\frac{B\ \bm\epsilon\cdot\big[\hat{{\mathbf{r}}}{\ast}{\mathbf{b}}+{\mathbf{b}}{\circ}\big(\hat{{\mathbf{r}}}{\ast}\hat{{\mathbf{r}}}{\ast}\hat{{\mathbf{r}}}\big)\big]}{2\pi(B+\mu)|{\mathbf{r}}|}.\label{Sdisls}$$ In particular, $$C_1^{(disl)}({\mathbf{r}})=\frac{\mu\ \hat{{\mathbf{r}}}\wedge{\mathbf{b}}}{\pi(B+\mu)|{\mathbf{r}}|}\qquad\textrm{and}\qquad\boldsymbol{\nabla}C_1^{(disl)}({\mathbf{r}})=-\frac{\mu\ \bm\epsilon\cdot\big[{\mathbf{b}}{\circ}(\hat{{\mathbf{r}}}{\ast}\hat{{\mathbf{r}}})\big]}{\pi(B+\mu)|{\mathbf{r}}|^2}.\label{Sdislcx}$$ Net variation of the number of particles when the dislocation moves ------------------------------------------------------------------- If the dislocation of the previous subsection moves from the origin to $ {\mathbf{r}}_0 $, from Eq. (\[Sdislcx\]) we have $$\Delta C_1^{(disl)}({\mathbf{r}})=\frac{\mu}{\pi(B+\mu)}\left(\frac{{\mathbf{r}}-{\mathbf{r}}_0}{|{\mathbf{r}}-{\mathbf{r}}_0|^2}-\frac{{\mathbf{r}}}{|{\mathbf{r}}|^2}\right)\wedge{\mathbf{b}}.\label{Svcx}$$ We consider the dislocation far from the crystal’s edges. In this case, the integral of $C_1^{(disl)}({\mathbf{r}})$ for a fixed dislocation is conditionally convergent. Still, we can estimate the net variation of the number of particles after the dislocation moves $$\label{Svn} \Delta N=\int\Delta\rho({\mathbf{r}}){\mathrm{d}}^2r\approx-\rho_0\int\Delta C_1^{(disl)}({\mathbf{r}}){\mathrm{d}}^2r.$$ We use $\int_{-\infty}^{\infty}\frac{1}{h+y^2}{\mathrm{d}}y=\frac{\pi}{\sqrt{h}}$ to obtain $$\begin{aligned} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\hat{\mathbf{x}}\cdot\left(\frac{{\mathbf{r}}-{\mathbf{r}}_0}{|{\mathbf{r}}-{\mathbf{r}}_0|^2}-\frac{{\mathbf{r}}}{|{\mathbf{r}}|^2}\right){\mathrm{d}}y{\mathrm{d}}x&=&\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\left(\frac{x-x_0}{(x-x_0)^2+(y-y_0)^2}-\frac{x}{x^2+y^2}\right){\mathrm{d}}y{\mathrm{d}}x\nonumber\\ &=&\pi\int_{-\infty}^{\infty}\left[\textrm{sgn}(x-x_0)-\textrm{sgn}(x)\right]{\mathrm{d}}x=-2\pi x_0\label{Sintx},\end{aligned}$$ where the sign function $\textrm{sgn}(x)$ gives $-1$, $0$ and $1$ when $x<0$, $x=0$ and $x>0$, respectively. Similarly, $$\label{Sinty} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\hat{\mathbf{y}}\cdot\left(\frac{{\mathbf{r}}-{\mathbf{r}}_0}{|{\mathbf{r}}-{\mathbf{r}}_0|^2}-\frac{{\mathbf{r}}}{|{\mathbf{r}}|^2}\right){\mathrm{d}}x{\mathrm{d}}y=-2\pi y_0.$$ Note that we used principal value integrals. Using Eq. (\[Svcx\]) in (\[Svn\]) and then using Eqs. (\[Sintx\]) and (\[Sinty\]), we obtain $$\Delta N\approx\frac{2\rho_0\mu}{B+\mu}{\mathbf{r}}_0\wedge{\mathbf{b}}$$ which is zero only for glide movement, i.e., for ${\mathbf{r}}_0\parallel{\mathbf{b}}$. Deformation fields of a disclination ------------------------------------ Consider a disclination with topological charge $s$ at the origin. In a ${\mathbf{C}}$-picture of deformation, the singular field comes from $\oint{\mathrm{d}}C^{(sing)}_2=s$, which implies in $\big(\nabla_x\nabla_y-\nabla_y\nabla_x\big)C^{(sing)}_2=s\ \delta({\mathbf{r}})$, and the $x$-component is regular (i.e., $C_1^{(sing)}=0$). We can use this and Eq. (\[Sbolast\]) in the evaluation of $\boldsymbol{\nabla}{\circ}\left[\boldsymbol{\nabla}{\ast}{\mathbf{C}}^{(sing)}\right]$ and then use Eq. (\[Ssolbol1\]) to obtain $$\label{Ssingdisc} \boldsymbol{\nabla}{\ast}{\mathbf{C}}^{(sing)}=\bm\epsilon\cdot\boldsymbol{\nabla}C^{(sing)}_2=s\boldsymbol{\nabla}G.$$ The regular fields that are necessary to reach equilibrium must satisfy $$\begin{aligned} 0&=&B\boldsymbol{\nabla}C_1^{(reg)}+\mu\boldsymbol{\nabla}{\circ}\big({\mathbf{S}}^{(reg)}+{\mathbf{S}}^{(sing)}\big)\nonumber\\ &=&B\boldsymbol{\nabla}C_1^{(reg)}+\mu\boldsymbol{\nabla}{\ast}\big({\mathbf{C}}^{(reg)}-{\mathbf{C}}^{(sing)}\big)\nonumber\\ &=&B\boldsymbol{\nabla}C_1^{(reg)}+\mu\big[\boldsymbol{\nabla}C_1^{(reg)}-\bm\epsilon\cdot\boldsymbol{\nabla}\big(C_2^{(reg)}-C_2^{(sing)}\big)\big]\nonumber\\ &=&(B+\mu)\boldsymbol{\nabla}C_1^{(reg)}-\mu\bm\epsilon\cdot\boldsymbol{\nabla}C_2^{(reg)}+\mu\bm\epsilon\cdot\boldsymbol{\nabla}C^{(sing)}_2\nonumber\\ &=&(B+\mu)\boldsymbol{\nabla}C_1^{(reg)}-\mu\bm\epsilon\cdot\boldsymbol{\nabla}C_2^{(reg)}+\mu s\boldsymbol{\nabla}G.\nonumber\end{aligned}$$ Then we have $$\boldsymbol{\nabla}C_1^{(reg)}({\mathbf{r}})=-\frac{\mu s\ \boldsymbol{\nabla}G({\mathbf{r}})}{(B+\mu)}=-\frac{\mu s}{(B+\mu)}\frac{\hat{{\mathbf{r}}}}{2\pi|{\mathbf{r}}|}\qquad\textrm{and}\qquad C_2^{(reg)}=0.$$ The result of Eq. (\[Sdislcx\]) for $\boldsymbol{\nabla}C_1^{(disl)}$ is obtained by adding another disclination with charge $-s$ at ${\mathbf{r}}_0$ and then taking the limits $|{\mathbf{r}}_0|\rightarrow0$ and $s\rightarrow\infty$ with $s{\mathbf{r}}_0=\bm\epsilon\cdot{\mathbf{b}}$ constant. Note that our convention is the same of \[S1\], where sevenfold and fivefold disclinations in a triangular crystal have positive and negative charges, respectively. Finally, for the total disclination deformation field in the ${\mathbf{C}}$-picture, $$\begin{aligned} \boldsymbol{\nabla}{\ast}{\mathbf{C}}^{(disc)}({\mathbf{r}})&=&\boldsymbol{\nabla}C_1^{(disc)}({\mathbf{r}})-\bm\epsilon\cdot\boldsymbol{\nabla}C_2^{(disc)}({\mathbf{r}})\nonumber\\ &=&\boldsymbol{\nabla}C_1^{(reg)}({\mathbf{r}})-\bm\epsilon\cdot\boldsymbol{\nabla}C_2^{(sing)}({\mathbf{r}})=-\frac{(B+2\mu) s}{(B+\mu)}\frac{\hat{{\mathbf{r}}}}{2\pi|{\mathbf{r}}|}.\end{aligned}$$ For the shear field, $\boldsymbol{\nabla}{\circ}{\mathbf{S}}^{(disc)}=\boldsymbol{\nabla}{\ast}{\mathbf{C}}^{(reg)}-\boldsymbol{\nabla}{\ast}{\mathbf{C}}^{(sing)}$ is also a gradient (irrotational) field. Deformation fields due to external forces {#Ie} ----------------------------------------- In the presence of a conservative external potential field $V_{ext}^{(c)}({\mathbf{r}})$ and a nonconservative one $V_{ext}^{(nc)}({\mathbf{r}})$, generating the external body force $\mathbf{F}_{ext}=-\boldsymbol{\nabla}V_{ext}^{(c)}+\bm\epsilon\cdot\boldsymbol{\nabla}V^{(nc)}_{ext}$, regular deformation fields ${\mathbf{C}}^{(ext)}$ and ${\mathbf{S}}^{(ext)}$ appear in order to compensate $\mathbf{F}_{ext}$ and reach the equilibrium $$\begin{aligned} 0&=&B\boldsymbol{\nabla}C_1^{(ext)}+\mu\boldsymbol{\nabla}{\circ}{\mathbf{S}}^{(ext)}+\rho_0\mathbf{F}_{ext}\nonumber\\ &=&B\boldsymbol{\nabla}C_1^{(ext)}+\mu\big(\boldsymbol{\nabla}C_1^{(ext)}-\bm\epsilon\cdot\boldsymbol{\nabla}C_2^{(ext)}\big)-\rho_0\boldsymbol{\nabla}V_{ext}^{(c)}+\rho_0\bm\epsilon\cdot\boldsymbol{\nabla}V^{(nc)}_{ext}.\nonumber\end{aligned}$$ Therefore, Helmholtz decomposition gives $$\boldsymbol{\nabla}C_1^{(ext)}=\frac{\rho_0}{B+\mu}\boldsymbol{\nabla}V_{ext}^{(c)}\qquad\textrm{,}\qquad\boldsymbol{\nabla}C_2^{(ext)}=\frac{\rho_0}{\mu}\boldsymbol{\nabla}V^{(nc)}_{ext}$$ and $$\boldsymbol{\nabla}{\circ}{\mathbf{S}}^{(ext)}=\boldsymbol{\nabla}C_1^{(ext)}-\bm\epsilon\cdot\boldsymbol{\nabla}C_2^{(ext)}=\frac{\rho_0}{B+\mu}\boldsymbol{\nabla}V_{ext}^{(c)}-\frac{\rho_0}{\mu}\bm\epsilon\cdot\boldsymbol{\nabla}V^{(nc)}_{ext}.\label{Ssfor}$$ Solutions for the density and orientation fields are direct, given by $$\label{Scext} C_1^{(ext)}({\mathbf{r}})=\frac{\rho_0V_{ext}^{(c)}({\mathbf{r}})}{B+\mu}\qquad\textrm{and}\qquad C_2^{(ext)}({\mathbf{r}})=\frac{\rho_0}{\mu}V^{(nc)}_{ext}({\mathbf{r}}).$$ Here, the boundary conditions contributions are also left to be considered later. In general, solutions for the shear field are more complicate and can be evaluated from Eq. (\[Ssolbol2\]). In the special case of $V_{ext}^{(c)}({\mathbf{r}})=V_{ext}(x)$ and $V_{ext}^{(nc)}({\mathbf{r}})=V_{ext}^{(+)}(x+y)+V_{ext}^{(-)}(x-y)$, for example, we have $$S_2^{(ext)}=0\qquad\textrm{and}\qquad S_1^{(ext)}({\mathbf{r}})=\frac{\rho_0V_{ext}(x)}{B+\mu}-\frac{\rho_0}{\mu}\big[V^{(+)}_{ext}(x+y)-V_{ext}^{(-)}(x-y)\big].\label{Ss1d}$$ In the case of radial conservative external forces (i.e., $V_{ext}^{(c)}({\mathbf{r}})=V_{ext}(|{\mathbf{r}}|)\equiv V_{ext}(r)$), the solution for the shear field is given by $${\mathbf{S}}^{(ext)}({\mathbf{r}})=\frac{\rho_0}{B+\mu}\left[V_{ext}(r)-\frac{2}{r^2}\int_{0}^{r}r'V_{ext}(r')dr'\right]\hat{{\mathbf{r}}}\ast\hat{{\mathbf{r}}}\label{Ssrad}$$ whose magnitude is radial (isotropic). The above result has the property ${\mathbf{S}}^{(ext)}({\mathbf{r}})={\mathbf{S}}^{(ext)}(-{\mathbf{r}})$. For the gaussian-like external potential $V_{ext}(r)=V_0e^{-r^2/2\sigma^2}$, Eq. (\[Ssrad\]) gives $ {\mathbf{S}}^{(ext)}({\mathbf{r}})\!=\!\frac{\rho_0V_0}{B+\mu}\!\left[e^{-r^2/2\sigma^2}\!-\!\frac{2\sigma^2}{r^2}(1\!-\!e^{-r^2/2\sigma^2})\right]\!\hat{{\mathbf{r}}}\ast\hat{{\mathbf{r}}}\ $ whose magnitude is zero at the origin, reaches a maximum at $r\approx1.89\sigma$ and returns to zero at infinity. Atomistic simulations ===================== 2D simulational methods ----------------------- We perform simulations with identical interacting particles inside a box with $-l_x/2\!<\!x\!\le\!l_x/2$ and $-l_y/2\!<\!y\!\le\!l_y/2$ using periodic boundary conditions, thus simulating an infinite system. The box has $l_x=312a_0$ and $l_y=180\sqrt{3}a_0\approx 311.77a_0$, which is approximately square and can accommodate a perfect triangular lattice with 112,320 particles and spacing $a_0=(2/(\sqrt{3}\rho_0))^{1/2}$, where $\rho_0$ is the mean density. We use soft isotropic pairwise interactions $V_p(r)$ that have an energy scale $\varepsilon$ and are further described in the next subsection. The temperature is fixed to $k_BT=0.0000005\varepsilon$ which equilibrates the system in crystalline configurations. At such low temperature, the particles weakly fluctuate around their equilibrium positions and the theoretical approaches can ignore thermal effects. Without the mobility law for time evolution of dislocation positions, we probe the theoretical configurational forces through systems at mechanical equilibrium. Thus, we consider overdamped evolution for the particles, small changes in conservative external forces acting on them and a small temperature to avoid unstable equilibria. The time evolution is then modeled by overdamped Langevin equations of motion, that is, a Brownian Dynamics (BD). These are integrated via Euler finite difference steps following the algorithm for the position of particle $i$ (see \[S2\]) $$\label{key} {\mathbf{r}}_i(t+\Delta t)={\mathbf{r}}_i(t)+\frac{\mathbf{F}_i(t)\Delta t}{\gamma}+\mathbf{g}_i(t)\sqrt{\frac{2k_BT\Delta t}{\gamma}},$$ where $\mathbf{F}_i=-\boldsymbol{\nabla}_i\big[V_{ext}({\mathbf{r}}_i)+\sum_{j\neq i}V_p(|{\mathbf{r}}_i-{\mathbf{r}}_j|)\big]$ is the total force, $\gamma$ is the friction coefficient, $\Delta t$ is the time step and the components of the vector $\mathbf{g}_i(t)$ are independent random variables with standard normal distribution of zero mean and unity variance which accounts for the Langevin kicks. In general, for our systems, $\Delta t/\gamma\sim10^{-3}a_0^2/\varepsilon$ is sufficiently small. The simulation time required to reach equilibrium varied according to the system. We start with a perfect triangular crystal configuration, with some slip direction (or lines of particles) parallel to $\hat{\mathbf{x}}$. We then apply a localized external shear stress at $(x,y)=(0,0)$ in order to nucleate a pair of dislocations. There are many ways to do so. In the following, we describe a procedure which is an illustrative use of our elasticity formalism. The localized shear can be generated, for example, using a conservative external potential given by $V_{ext}^{(ini)}({\mathbf{r}})=V_0^{(ini)}\big(e^{-({\mathbf{r}}-{\mathbf{r}}_0)^2/2\sigma_0^2}+e^{-({\mathbf{r}}+{\mathbf{r}}_0)^2/2\sigma_0^2}\big)$, where ${\mathbf{r}}_0=1.89\sigma_0\bigl(\begin{smallmatrix} \cos\pi/4 \\ \sin\pi/4 \end{smallmatrix} \bigr)$ and $\sigma_0$ is equal to a few lattice spacings. As we can see from the results in the end of the previous section, this superposition of two gaussians produces a shear with maximum magnitude at the origin. It has only $y$-component there and then the resolved component $\big(\hat{{\mathbf{b}}}{\ast}\hat{{\mathbf{b}}}\big)\wedge{\mathbf{S}}^{(ext)}(0)$ is maximized for $\hat{{\mathbf{b}}}=\pm\hat{\mathbf{x}}$ or $\hat{{\mathbf{b}}}=\pm\hat{\mathbf{y}}$. Since $\hat{\mathbf{x}}$ is a slip direction in our triangular crystal and $\hat{\mathbf{y}}$ is not, a pair of dislocations with ${\mathbf{b}}=\pm a_0\hat{\mathbf{x}}$ is nucleated after the resolved shear reaches a critical value. For positive $V_0^{(ini)}$, the dislocation with ${\mathbf{b}}=a_0\hat{\mathbf{x}}$ goes to the right and the other one (with ${\mathbf{b}}=-a_0\hat{\mathbf{x}}$) goes to the left, equilibrating at $x=\pm d/2$ where $d$ is the equilibrium distance between them. Thereafter, we use other types of external potential to control their equilibrium positions, considering the configurational forces described in the mais text, and turn off $V_{ext}^{(ini)}$. In order to isolate the possible effect of new types of configurational force, we use external body forces that produce $\big(\hat{{\mathbf{b}}}{\ast}\hat{{\mathbf{b}}}\big)\wedge{\mathbf{S}}^{(ext)}({\mathbf{r}})=S_2^{(ext)}({\mathbf{r}})=0$ on the dislocations. In the case of Fig. 2(a) in the main text, we use a conservative gaussian potential $V_{ext}(|{\mathbf{r}}|)=-V_0e^{-|{\mathbf{r}}|^2/2\sigma^2}$ centered between the dislocations, which are kept apart. At their positions, $S_2^{(ext)}({\mathbf{r}})=0$ and $\hat{{\mathbf{b}}}\cdot\boldsymbol{\nabla}C^{(ext)}_1({\mathbf{r}})>0$ as we can see from Eqs. (\[Ssrad\]) and (\[Scext\]). Our attempts to identify effects of other types of strain gradients (namely, $\boldsymbol{\nabla}\big[\big(\hat{{\mathbf{b}}}{\ast}\hat{{\mathbf{b}}}\big)\cdot{\mathbf{S}}^{(ext)}\big]$ and $\boldsymbol{\nabla}C^{(ext)}_2$), by using nonconservative potentials like $V_{ext}^{(nc)}({\mathbf{r}})=V_{ext}^{(+)}(x+y)+V_{ext}^{(-)}(x-y)$, have failed: the simulations resulted in dislocation annihilation (due to PK forces). Upper bounds for these possible effects remains to be set. Elastic constants for systems with interactions that are combinations of power-laws ----------------------------------------------------------------------------------- In general, the system has isotropic elastic response described by high-frequency (instantaneous) bulk and shear moduli \[S3\] which depend on temperature and, for 2D, can be evaluated using \[S4\] $$B_\infty=2\rho_0k_BT-\frac{\pi\rho_0^2}{4}\int_{0}^{\infty}r^2g(r)\big[V_p'(r)-rV_p''(r)\big]{\mathrm{d}}r\label{Binf}$$ and $$\mu_\infty=\rho_0k_BT+\frac{\pi\rho_0^2}{8}\int_{0}^{\infty}r^2g(r)\big[3V_p'(r)+rV_p''(r)\big]{\mathrm{d}}r,\label{muinf}$$ where $g(r)$ is the radial distribution function and the primes indicate derivatives. The simulations were carried out at a very low temperature and we can ignore thermal effects on the crystalline configuration. Within this approximation, we have $$2\pi\rho_0\int rg(r)f(r){\mathrm{d}}r\approx\sum_{i}f(r_i),$$ where $\sum_i$ is a lattice sum. It is convenient to define the Madelung energy (the total interaction potential on a particle of the lattice with spacing $a$) $$\label{key} \Phi(a)=\sum_{i}V_p(r_i)=\sum_{i}V_p(ap_i),$$ where $p_i=r_i/a$ are factors of proportionality which depend on the lattice geometry. We can then use the analytical formulas $$\begin{aligned} B&=&\frac{\rho_0}{8}\sum_{i}\big[-r_iV_p'(r_i)+r_i^2V_p''(r_i)\big]\nonumber\\ &=&\frac{\rho_0}{8}\left[-a_0\sum_{i}p_iV_p'(a_0p_i)+a_0^2\sum_{i}p_i^2V_p''(a_0p_i)\right]\nonumber\\ &=&\frac{\rho_0}{8}\left[-a_0\frac{{\mathrm{d}}\Phi}{{\mathrm{d}}a}(a_0)+a_0^2\frac{{\mathrm{d}}^2\Phi}{{\mathrm{d}}a^2}(a_0)\right]=\rho_0^2\left[\frac{{\mathrm{d}}\Phi}{{\mathrm{d}}\rho}(\rho_0)+\frac{\rho_0}{2}\frac{{\mathrm{d}}^2\Phi}{{\mathrm{d}}\rho^2}(\rho_0)\right],\label{B}\end{aligned}$$ where the last expression with density dependence is general for any crystal in any dimension and can be obtained directly from the definition of $B$, and $$\begin{aligned} \mu&=&\frac{\rho_0}{16}\sum_{i}\big[3r_iV_p'(r_i)+r_i^2V_p''(r_i)\big]\nonumber\\ &=&\frac{\rho_0}{16}\left[3a_0\frac{{\mathrm{d}}\Phi}{{\mathrm{d}}a}(a_0)+a_0^2\frac{{\mathrm{d}}^2\Phi}{{\mathrm{d}}a^2}(a_0)\right].\label{mu}\end{aligned}$$ We simulated systems with power-law type $V_p^{PL}(r)=\varepsilon(a_0/r)^6$ and Lennard-Jones type $V_p^{LJ}(r)=\Lambda\varepsilon\big[(a_0/r)^{12}-(a_0/r)^6\big]$ interactions, where $\Lambda$ is a numerical factor. For these cases, we have the Madelung energies $$\label{key} \Phi^{PL}(a)=\varepsilon M_6\left(\frac{a_0}{a}\right)^6\qquad\textrm{and}\qquad\Phi^{LJ}(a)=\Lambda\varepsilon \left[M_{12}\left(\frac{a_0}{a}\right)^{12}-M_6\left(\frac{a_0}{a}\right)^6\right],$$ where $M_n=\sum_i1/p_i^n$ are lattice constants. The constants can be calculated for the triangular lattice using $p_i=\sqrt{i_1^2+i_2^2+i_1i_2}$ and summing in all integer values of $i_1$ and $i_2$ except when both are zero. We obtain $M_6\approx6.37588$ and $M_{12}\approx6.00981$. Therefore, the bulk and shear moduli for these systems can be readily evaluated. In particular, we have $$\label{key} B^{PL}=6 M_6\rho_0\varepsilon\qquad\textrm{and}\qquad B^{LJ}=3(7M_{12}-2M_6)\Lambda\rho_0\varepsilon.$$ In order to have the Lennard-Jones system with the same $B+\mu$ of the power-law one, we use $$\label{equbeh} B^{PL}+\mu^{PL}=B^{LJ}+\mu^{LJ}\qquad\Rightarrow\qquad\Lambda=\left(\frac{57M_{12}}{15M_6}-1\right)^{-1}\approx0.387323.$$ With such factor, we have two different systems with the same elastic response to conservative external forces (Eq. (\[Ssfor\])). External forces and the induced resolved shear strain ----------------------------------------------------- The system starts as a perfect triangular crystal with lattice spacing $a_0=(2/(\sqrt{3}\rho_0))^{1/2}$. By applying a localized shear stress, a pair of dislocations was nucleated and thereafter they were kept symmetrically separate, with ${\mathbf{b}}=\pm a_0\hat{\mathbf{x}}$ at $x=\pm d/2$, solely by the effect of the external potential field $$\label{Svext} V_{ext}(x)=V_0\big[e^{-(x+D+\sigma)^2/2\sigma^2}-e^{-(x+D-\sigma)^2/2\sigma^2}-e^{-(x-D+\sigma)^2/2\sigma^2}+e^{-(x-D-\sigma)^2/2\sigma^2}\big]$$ acting on the crystal. Fig. \[fig:fig1s\] presents the plots of this potential and of its derivative $$\begin{aligned} \nabla_xV_{ext}(x)=(V_0/\sigma^2)&\big[&-(x+D+\sigma)e^{-(x+D+\sigma)^2/2\sigma^2}+(x+D-\sigma)e^{-(x+D-\sigma)^2/2\sigma^2}\nonumber\\ &&+(x-D+\sigma)e^{-(x-D+\sigma)^2/2\sigma^2}-(x-D-\sigma)e^{-(x-D-\sigma)^2/2\sigma^2}\big]\label{Svextd}\end{aligned}$$ for $\sigma=10a_0$ and $D=40a_0$. In the figure, the regions in gray represent regions where the dislocations can stay in equilibrium. They stay there by the mechanism in which the density gradients (provoked by the external forces) induce resolved shear strains on them. Note that only two of the gaussian exponential terms in Eq. (\[Svextd\]) effectively acts on each dislocation, as it can be seen in Fig. \[fig:fig1s\]. ![\[fig:fig1s\]Plots of the conservative external potential and its derivative, given in Eqs. (\[Svext\]) and (\[Svextd\]), for $\sigma=10a_0$ and $D=40a_0$. The gray regions indicate where the dislocations can be trapped (i.e., kept apart) when the external force field is sufficiently strong.](fig1s.pdf){width="65.00000%"} Besides the external potential contribution, the background deformation fields on each dislocation have contributions due to the other one and the boundary conditions. From Eq. (\[Sdislcx\]), we have that each dislocation produces a density gradient that, in the position of the other one, is perpendicular to $\hat{{\mathbf{b}}}$. The external potential already satisfies the periodicity of boundary conditions (its derivatives are approximately zero at $x=\pm l_x/2$) and the contribution to deformation due to these conditions is only the effect of repeated dislocations, which are more investigated in the next subsection and their contribution also gives a density gradient that is $\perp\hat{{\mathbf{b}}}$. Therefore, the total induced resolved shear on the dislocations at $x=\pm d/2$ is only $$\begin{aligned} S_{res}^{(ind)}&=&L\ \hat{{\mathbf{b}}}\cdot\boldsymbol{\nabla}C^{(bg)}_1= \pm L\nabla_xC^{(ext)}_1\big(\pm d/2\big)=\pm\frac{L\rho_0}{B+\mu}\nabla_xV_{ext}\big(\pm d/2\big)\nonumber\\ &\approx&\frac{L\rho_0V_0}{(B+\mu)\sigma^2}\left[\left(\frac{d}{2}-D+\sigma\right)e^{-(d/2-D+\sigma)^2/2\sigma^2}-\left(\frac{d}{2}-D-\sigma\right)e^{-(d/2-D-\sigma)^2/2\sigma^2}\right].\label{Ssindres}\end{aligned}$$ The approximation is valid for dislocations in the gray regions of Fig. \[fig:fig1s\]. Since $S_{res}^{(ind)}>0$ and $\hat{{\mathbf{b}}}\cdot\mathbf{F}_{disl}\propto S_{res}$, the induced forces on the dislocations contribute to push them apart. Boundary conditions and the background resolved shear strain ------------------------------------------------------------ The external potential contribution to shear, which has the form of Eq. (\[Ss1d\]), do not produce background resolved shear strain on the dislocations, which is simply $S^{(bg)}_{res}({\mathbf{r}})=\big(\hat{{\mathbf{b}}}{\ast}\hat{{\mathbf{b}}}\big)\wedge{\mathbf{S}}^{(bg)}({\mathbf{r}})=S^{(bg)}_2({\mathbf{r}})$ since $\hat{{\mathbf{b}}}=\pm\hat{\mathbf{x}}$. In a first approximation, the background resolved shear that the dislocations directly produce on each other is equally negative and, using Eq. (\[Sdisls\]), given by $$\label{S1oapp} S_{res}^{(bg)}\approx -\frac{Ba_0}{\pi(B+\mu)d}$$ where $d$ is the distance between them. However, as it is shown in Fig. \[fig:fig2s\], the periodic boundary conditions create lattices of dislocations that contribute to deformation. Note that each dislocation do not interact with its own lattice but with the lattice formed by the other one. The lattices are approximately square lattices with spacing $l=l_x\approx l_y$. ![\[fig:fig2s\] The contribution to deformation fields originated from the (periodic) boundary conditions is equivalent to infinitely repeated dislocation pairs. The blue and green dislocations have Burgers vectors ${\mathbf{b}}=a_0\hat{\mathbf{x}}$ and $-{\mathbf{b}}$, respectively.](fig2s.pdf){width="54.00000%"} On the dislocation with ${\mathbf{b}}=a_0\hat{\mathbf{x}}$ at $x=d/2$ (i.e., the central blue dislocation in Fig. \[fig:fig2s\]), the total $S_{res}^{(bg)}$ originates from the lattice produced by the other one (i.e., from all green dislocations). A (green) dislocation at ${\mathbf{r}}_{(n,m)}=\bigl(\begin{smallmatrix} -d/2-nl \\ -ml \end{smallmatrix} \bigr)$ generates the background shear ${\mathbf{S}}^{(disl)}(\Delta{\mathbf{r}}_{(n,m)})$ on the (central blue) one at $\bigl(\begin{smallmatrix} d/2 \\ 0 \end{smallmatrix} \bigr)$, using ${\mathbf{b}}=-a_0\hat{\mathbf{x}}$ in Eq. (\[Sdisls\]) and $\Delta{\mathbf{r}}_{(n,m)}=(d/2)\hat{\mathbf{x}}-{\mathbf{r}}_{(n,m)}$. Then $S_{res}^{(bg)}=\big(\hat{{\mathbf{b}}}{\ast}\hat{{\mathbf{b}}}\big)\wedge{\mathbf{S}}^{(bg)}=\sum_{n,m}\big(\hat{\mathbf{x}}{\ast}\hat{\mathbf{x}}\big)\wedge{\mathbf{S}}^{(disl)}(\Delta{\mathbf{r}}_{(n,m)})$ becomes $$\begin{aligned} S_{res}^{(bg)}&=&\sum_{n,m=-\infty}^{\infty}\frac{Ba_0}{2\pi(B+\mu)}\big(\hat{\mathbf{x}}{\ast}\hat{\mathbf{x}}\big)\wedge\left[\bm\epsilon\cdot\left(\frac{\Delta{\mathbf{r}}_{(n,m)}{\ast}\hat{\mathbf{x}}}{|\Delta{\mathbf{r}}_{(n,m)}|^2}+\frac{\hat{\mathbf{x}}{\circ}\big(\Delta{\mathbf{r}}_{(n,m)}{\ast}\Delta{\mathbf{r}}_{(n,m)}{\ast}\Delta{\mathbf{r}}_{(n,m)}\big)}{|\Delta{\mathbf{r}}_{(n,m)}|^4}\right)\right]\nonumber\\ &=&-\frac{Ba_0}{2\pi(B+\mu)}\sum_{n,m=-\infty}^{\infty}\hat{\mathbf{x}}\cdot\left(\frac{\Delta{\mathbf{r}}_{(n,m)}}{|\Delta{\mathbf{r}}_{(n,m)}|^2}+\frac{\Delta{\mathbf{r}}_{(n,m)}{\ast}\Delta{\mathbf{r}}_{(n,m)}{\ast}\Delta{\mathbf{r}}_{(n,m)}}{|\Delta{\mathbf{r}}_{(n,m)}|^4}\right)\nonumber\\ &=&-\frac{Ba_0}{2\pi(B+\mu)}\sum_{n,m=-\infty}^{\infty}\left(\frac{d+nl}{(d+nl)^2+m^2l^2}+\frac{(d+nl)^3-3(d+nl)m^2l^2}{\big[(d+nl)^2+m^2l^2\big]^2}\right)\nonumber\\ &=&-\frac{Ba_0}{\pi(B+\mu)l}\sum_{n,m=-\infty}^{\infty}\frac{d/l+n}{(d/l+n)^2+m^2}\left(1-\frac{2m^2}{(d/l+n)^2+m^2}\right)\nonumber\\ &=&-\frac{Ba_0}{\pi(B+\mu)l}\sum_{n=-\infty}^{\infty}\left[\frac{1}{d/l+n}+2\sum_{m=1}^{\infty}\left(\frac{d/l+n}{(d/l+n)^2+m^2}-\frac{2(d/l+n)m^2}{\big[(d/l+n)^2+m^2\big]^2}\right)\right],\end{aligned}$$ where one can note that the sum in $m$ is absolutely convergent. Now we use (see the series (886) in \[S5\]) $$\sum_{n=1}^{\infty}\frac{h}{(nz)^2+h^2}=\frac{\pi}{z}\left[\frac{1}{e^{2\pi h/z}-1}+\frac{1}{2}-\frac{z}{2\pi h}\right]=\frac{\pi}{2z}\coth\left(\frac{\pi h}{z}\right)-\frac{1}{2h}$$ and $$\sum_{n=1}^{\infty}\frac{2zhn^2}{[(nz)^2+h^2]^2}=-\frac{\partial}{\partial z}\left[\sum_{n=1}^{\infty}\frac{h}{(nz)^2+h^2}\right]=\frac{\pi}{2z^2}\left[\coth\left(\frac{\pi h}{z}\right)-\frac{\pi h}{z}\textrm{csch}^2\left(\frac{\pi h}{z}\right)\right]$$ to obtain $$\sum_{m=1}^{\infty}\!\left(\!\frac{d/l+n}{(d/l+n)^2+m^2}-\frac{2(d/l+n)m^2}{\big[(d/l+n)^2+m^2\big]^2}\!\right)\!=\!\frac{\pi^2(d/l+n)}{2}\textrm{csch}^2\big[\pi(d/l+n)\big]-\frac{1}{2(d/l+n)}\nonumber$$ and then $$\begin{aligned} S_{res}^{(bg)}&=&-\frac{Ba_0}{\pi(B+\mu)l}\sum_{n=-\infty}^{\infty}\left[\frac{1}{d/l+n}+2\left(\frac{\pi^2(d/l+n)}{2}\textrm{csch}^2\big[\pi(d/l+n)\big]-\frac{1}{2(d/l+n)}\right)\right]\nonumber\\ &=&-\frac{Ba_0\pi}{(B+\mu)l}\sum_{n=-\infty}^{\infty}(d/l+n)\textrm{csch}^2\big[\pi(d/l+n)\big].\label{Ssbgtot}\end{aligned}$$ ![\[fig:fig3s\]Approximations for the total background resolved shear on each dislocation due to the interaction with others. Curves obtained by considering only the nearest dislocation, by adding of the second nearest one and by considering some terms in the total result of Eq. (\[Ssbgtot\]). As in Fig. \[fig:fig1s\], the gray region indicate where the dislocations can be trapped in our systems.](fig3s.pdf){width="55.00000%"} Fig. \[fig:fig3s\] presents a graph which compares some approximations for $S_{res}^{(bg)}$. Within our region of interest, the consideration of just the nearest dislocation (the one in the central box of Fig. \[fig:fig2s\]) as in Eq. (\[S1oapp\]), or even when adding the second nearest one (which is in the box on the right side of Fig. \[fig:fig2s\]), do not give good approximations. The more correct evaluation is given by the infinite series in Eq. (\[Ssbgtot\]) which is rapidly convergent and very well approximated by $$\label{Sbgres} S_{res}^{(bg)}\approx-\frac{Ba_0\pi}{(B+\mu)l}\!\left[\frac{d}{l}\textrm{csch}^2\!\left(\frac{\pi d}{l}\right)\!+\!\left(\frac{d}{l}\!+\!1\right)\!\textrm{csch}^2\!\left(\frac{\pi d}{l}\!+\!\pi\right)\!+\!\left(\frac{d}{l}\!-\!1\right)\!\textrm{csch}^2\!\left(\frac{\pi d}{l}\!-\!\pi\right)\right].$$ Equilibrium positions of the dislocations ----------------------------------------- Finally, with the results of Eqs. (\[Sbgres\]) and (\[Ssindres\]), and using Eq. (\[Svextd\]), we find that the equilibrium condition $\hat{{\mathbf{b}}}\cdot\mathbf{F}^{(tot)}_{disl}\!\propto S^{(tot)}_{res}=S^{(ind)}_{res}+S^{(bg)}_{res}=0$ gives $$\label{Sfineq} V_0\approx\frac{Ba_0\pi\sigma^2}{L\rho_0l}\left[\frac{\frac{d}{l}\textrm{csch}^2\left(\frac{\pi d}{l}\right)+\left(\frac{d}{l}+1\right)\textrm{csch}^2\left(\frac{\pi d}{l}+\pi\right)+\left(\frac{d}{l}-1\right)\textrm{csch}^2\left(\frac{\pi d}{l}-\pi\right)}{\left(\frac{d}{2}-D+\sigma\right)e^{-(d/2-D+\sigma)^2/2\sigma^2}-\left(\frac{d}{2}-D-\sigma\right)e^{-(d/2-D-\sigma)^2/2\sigma^2}}\right]$$ or, substituting our parameter values, $$\label{Sfineq2} V_0\approx\frac{Ba_0}{L\rho_0}\frac{25\pi}{12168}\left[\frac{\frac{d}{a_0}\textrm{csch}^2\left(\frac{\pi d}{312a_0}\right)+\left(\frac{d}{a_0}+312\right)\textrm{csch}^2\left(\frac{\pi d}{312a_0}+\pi\right)+\left(\frac{d}{a_0}-312\right)\textrm{csch}^2\left(\frac{\pi d}{312a_0}-\pi\right)}{\left(\frac{d}{a_0}-60\right)e^{-(d/a_0-60)^2/800}-\left(\frac{d}{a_0}-100\right)e^{-(d/a_0-100)^2/800}}\right].$$ The above equation relates the external potential strength with the distance between the dislocations. It can be directly compared with the simulation results and has only one fitting parameter, given by $Ba_0/(L\rho_0)$ which can be viewed as an energy scale for $V_0$. In the simulations we use positive values of $V_0$. The curve of Eq. (\[Sfineq2\]) for $V_0=V_0(d)$ has a minimum at $d\approx84a_0$, then increases with $d$ and blows up at $d\approx104a_0$. This is our region of interest, since elsewhere $V_0(d)$ is either negative or decreasing. When, in simulation, $V_0$ is decreased bellow that minimum, at which $d\approx84a_0$, the configurational force due to the density gradient can no longer kept the dislocations apart and the PK forces drive them to annihilate each other. Indeed, we observed that $d\approx84a_0$ is the minimum distance that the dislocations can be trapped by the action of the external potential. In Fig. 4 of the main text, we can see that our theoretical predictions for the configurational force has a good agreement with the BD results, mainly for smaller $d$ (possible reasons for this are described in the main text). From the fits, we estimate the values $L^{PL}\approx0.683a_0$ and $L^{LJ}\approx4.43a_0$ for the length parameter in the power-law and Lennard-Jones systems, respectively. The results still allow the interpretation that the new effect is an arresting force which induce glide only when accompanied by a change in the background deformation (when we increase $V_0$, both $\hat{{\mathbf{b}}}\cdot\boldsymbol{\nabla}C^{(bg)}_1$ and $d$ increase together). To probe this, we use the fact that the power-law and the Lennard-Jones systems have nearly the same $C^{(ext)}_1(x)$ for the same $V_0$, since $B^{PL}+\mu^{PL}=B^{LJ}+\mu^{LJ}$ from Eq. (\[equbeh\]). We take the equilibrium configuration of the power-law system when $d\approx84a_0$ as the initial configuration of a simulation with the Lennard-Jones interaction and the same $V_0$. The system then evolves with the dislocations gliding away up to $d\approx100a_0$ and little change in $C^{(ext)}_1(x)$. Therefore, $S^{(ind)}_{res}$ provides a true driving force for glide. 3D simulations of a copper (Cu) crystal --------------------------------------- In order to investigate the effect of density gradients on line dislocations in 3D, we consider straight dislocations in a face-centered cubic (fcc) crystal made by layers of triangular lattices. The linear Elasticity of plane strain in the $\hat{\mathbf{x}}$- and $\hat{\mathbf{y}}$-directions is isotropic and we can use the previous theoretical results for straight edge dislocations in the direction of $\hat{\mathbf{z}}$. We perform simulations inside a box with $-l_x/2\!<\!x\!\le\!l_x/2$, $-l_y/2\!<\!y\!\le\!l_y/2$ and $-l_z/2\!<\!z\!\le\!l_z/2$ using periodic boundary conditions in all directions, where $l_x=104a_0$, $l_y=60\sqrt{3}a_0\approx 103.923a_0$ and $l_z=24\sqrt{6}a_0\approx 58.788a_0$. It can accommodate a perfect fcc lattice with 898,560 particles and nearest-neighbor distance $a_0=(\sqrt{2}/\rho_0)^{1/3}$, where $\rho_0$ is the mean density. The temperature, the time step and the algorithm for time evolution are the same of the 2D simulations. The interactions are given by the Morse potential $$\label{key} V_p(r)=\varepsilon\big[e^{-2\alpha(r-r_0)}-2e^{-\alpha(r-r_0)}\big],$$ where $\alpha a_0=3.479$ and $r_0=1.12a_0$, which adequately represents pair interactions in Cu crystals \[S6-S8\]. ![\[fig:fig4s\] Perspective view of the simulation box and the dislocation lines in our 3D simulation of a Cu crystal, obtainedd using the dislocation extraction algorithm \[S9\] implemented in OVITO \[S10\].](fig4s.pdf){width="54.00000%"} The dislocation pair was nucleated in a similar way but using a smaller fcc system, made by 3 layers of triangular lattices with $l_x=104a_0$ and $l_y=60\sqrt{3}a_0$ forming. Then we put density gradients via an external potential in the form of Eq. (\[Svext\]), with $V_0=2\varepsilon$, $\sigma=6a_0$ and $D=17a_0$, to keep the dislocations apart. They responded similarly to the 2D case. Thereafter, we duplicate the configuration to have 6 layers of triangular lattices and let the resulting bigger system relax, which changed the dislocation positions a little. We increased the system again to have 18 layers, then 36 and finally 72 layers (with $l_z=24\sqrt{6}a_0$). The last three systems are virtually indistinguishable in the projection to the $xy$-plane and we take the last one as truly simulating infinitely straight edge dislocations. As expected in the Cu fcc crystal, these defects decompose into Shockley partial dislocations with stacking faults, as shown in Fig. \[fig:fig4s\] and in Figs. 1(c) and (d) of the main text. The equilibrium mean distance between the centers of these stacking faults, as obtained through OVITO, is $d\approx40.5a_0$. The general behavior here is similar to the 2D case, with $d$ increasing when $V_0$ is increased and with dislocation annihilation when the external potential is turned off. But the relation between $V_0$ and $d$ do not satisfactorily obey equation (\[Sfineq\]) since it considers the dislocations as point objects while the real ones are extended, decomposed into partials. More careful theoretical analysis and simulations about how $\hat{{\mathbf{b}}}\cdot\boldsymbol{\nabla}C^{(bg)}_1$ affects edge dislocations in fcc crystals remain to be done. Still, we can obtain an estimate for the effective $L$ using Eq. (\[Sfineq\]) and to do so we need the value of $B$. The bulk modulus for the Morse potential is given by $$\label{key} B=\rho_0^2\left[\frac{{\mathrm{d}}\Phi}{{\mathrm{d}}\rho}(\rho_0)+\frac{\rho_0}{2}\frac{{\mathrm{d}}^2\Phi}{{\mathrm{d}}\rho^2}(\rho_0)\right]=\frac{\sqrt{2}}{9a_0^2}\left[-\frac{{\mathrm{d}}\Phi}{{\mathrm{d}}a}(a_0)+\frac{a_0}{2}\frac{{\mathrm{d}}^2\Phi}{{\mathrm{d}}a^2}(a_0)\right],$$ where $\Phi(a)=\sum_{i}V_p(ap_i)=\sum_{i}\varepsilon\big[e^{-2\alpha(ap_i-r_0)}-2e^{-\alpha(ap_i-r_0)}\big]$. Then $$\label{key} B=\frac{2\varepsilon\rho_0\alpha a_0e^{\alpha r_0}}{9}\left[e^{\alpha r_0}\sum_{i}p_ie^{-2\alpha ap_i}-\sum_{i}p_ie^{-\alpha ap_i}+\frac{\alpha a_0}{2}\left(2e^{\alpha r_0}\sum_{i}p_i^2e^{-2\alpha ap_i}-\sum_{i}p_i^2e^{-\alpha ap_i}\right)\right],$$ The sums can be calculated for the fcc lattice using $p_i=\sqrt{i_1^2+i_2^2+i_3^2+i_1i_2+i_1i_3+i_2i_3}$ and summing in all integer values of $i_1$, $i_2$ and $i_3$ except when $(i_1,i_2,i_3)=(0,0,0)$. For $\alpha a_0=3.479$ and $r_0=1.12a_0$, we obtain $B\approx28.79\varepsilon\rho_0$. Using this value together with $V_0=2\varepsilon$, $d=40.5a_0$, $l=104a_0$, $\sigma=6a_0$ and $D=17a_0$ in Eq. (\[Sfineq\]), we find the estimate $L^{Cu}\approx0.32a_0$. Supplementary References {#supplementary-references .unnumbered} ======================== \[S1\] E. Kroner and K. H. Anthony, Annu. Rev. Mater. Sci. **5**, 43 (1975). \[S2\] A. Satoh, *Introduction to practice of molecular simulation: Molecular Dynamics, Monte Carlo, Brownian Dynamics, Lattice Boltzmann and Dissipative Particle Dynamics* (Elsevier, Amsterdam, 2011). \[S3\] R. Zwanzig and R. D. Mountain, J. Chem. Phys. **43**, 4464 (1965). \[S4\] S. A. Khrapak, N. P. Kryuchkov, and S. O. Yurchenko, Phys. Rev. E **97**, 022616 (2018). \[S5\] L. Jolley, *Summation of Series* (Dover, New York, 1961). \[S6\] L. A. Girifalco and V. G. Weizer, Phys. Rev. **114**, 687 (1959). \[S7\] R. Komanduri, N. Chandrasekaran, and L. M. Raff, Int. J. Mech. Sci. **43**, 2237 (2001). \[S8\] G. Ziegenhain, A. Hartmaier, and H. M. Urbassek, J. Mech. Phys. Solids **57**, 1514 (2009). \[S9\] A. Stukowski, V. V. Bulatov, and A. Arsenlis, Automated identification and indexing of dislocations in crystal interfaces, Model. Simul. Mater. Sci. Eng. **20**, 085007 (2012). \[S10\] A. Stukowski, Visualization and analysis of atomistic simulation data with OVITO-the Open Visualization Tool, Model. Simul. Mater. Sci. Eng. **18**, 015012 (2010).

--- abstract: | We establish a Liouville comparison principle for entire sub- and super-solutions of the equation $(\ast)$ $w_t-\Delta_p (w) = |w|^{q-1}w$ in the half-space ${\mathbb S}= {\mathbb R}^1_+\times {\mathbb R}^n$, where $n\geq 1$, $q>0$ and $ \Delta_p (w):=\mbox{div}_x\left(|\nabla_x w|^{p-2}\nabla_x w\right)$, $1<p\leq 2$. In our study we impose neither restrictions on the behaviour of entire sub- and super-solutions on the hyper-plane $t=0$, nor any growth conditions on the behavior of them or any of their partial derivatives at infinity. We prove that if $1<q\leq p-1+\frac pn$, and $u$ and $v$ are, respectively, an entire weak super- and an entire weak sub-solution of ($\ast$) in $\Bbb S$ which belong, only locally in $\Bbb S$, to the corresponding Sobolev space and are such that $u\leq v$, then $u\equiv v$. The result is sharp. As direct corollaries we obtain both new and known Fujita-type and Liouville-type results. author: - 'Vasilii V. Kurta.' title: 'A Liouville comparison principle for entire sub- and super-solutions of the equation $u_t-\Delta_p (u) = |u|^{q-1}u$.' --- Introduction and Definitions. ============================= The purpose of this work is to obtain a Liouville comparison principle of elliptic type for entire weak sub- and super-solutions of the equation $$\begin{aligned} w_t-\Delta_p (w) = |w|^{q-1}w\end{aligned}$$ in the half-space ${\mathbb S}= (0,+\infty)\times {\mathbb R}^n$, where $n\geq 1$ is a natural number, $q>0$ is a real number and $ \Delta_p (w):=\mbox{div}_x\left(|\nabla_x w|^{p-2}\nabla_x w\right)$, $1<p\leq 2$, defines the well-known $p$-Laplacian operator. Under entire sub- and super-solutions of (1) we understand sub- and super-solutions of (1) defined in the whole half-space $\mathbb S$ and under Liouville theorems of elliptic type we understand Liouville-type theorems which, in their formulations, have no restrictions on the behaviour of sub- or super-solutions to the parabolic equation (1) on the hyper-plane $t=0$. We would also like to underline that we impose no growth conditions on the behavior of sub- or super-solutions of (1), as well as of all their partial derivatives, at infinity. Let $n\geq 1$, $p>1$ and $q>0$. A function $u=u(t,x)$ defined and measurable in $\mathbb S$ is called an entire weak super-solution of the equation (1) in $\mathbb S$ if it belongs to the function space $ L_{q,loc}(\mathbb S)$, with $u_t \in L_{1, loc}(\mathbb S)$ and $|\nabla_x u|^{p}\in L_{1, loc}(\mathbb S)$, and satisfies the integral inequality $$\begin{aligned} \int\limits_{\mathbb S}\left[u_t\varphi+\sum_{i=1}^n |\nabla_x u|^{p-2}u_{x_i}\varphi_{x_i} -|u|^{q-1}u \varphi \right]dtdx \geq 0\end{aligned}$$ for every non-negative function $\varphi \in C^\infty (\mathbb S)$ with compact support in $\mathbb S$, where $C^{\infty}({\mathbb S})$ is the space of all functions defined and infinitely differentiable in ${\mathbb S}$. A function $v=v(t,x)$ is an entire weak sub-solution of (1) if $u=-v$ is an entire weak super-solution of (1). Results. ======== Let $n\geq 1$, $2\geq p> 1$ and $1<q\leq p-1 +\frac {p}n$, and let $u$ be an entire weak super-solution and $v$ an entire weak sub-solution of (1) in $\mathbb S$ such that $u\geq v$. Then $u \equiv v$ in $ \mathbb S$. The result in Theorem 1, which evidently has a comparison principle character, we term a Liouville-type comparison principle, since, in the particular cases when $u\equiv 0$ or $v\equiv 0$, it becomes a Liouville-type theorem of elliptic type, respectively, for entire sub- or super-solutions of (1). Since in Theorem 1 we impose no conditions on the behaviour of entire sub- or super-solutions of the equation (1) on the hyper-plane $t=0$, we can formulate, as a direct corollary of the result in Theorem 1, the following comparison principle, which in turn one can term a Fujita comparison principle, for entire sub- and super-solutions of the Cauchy problem for the equation (1). It is clear that in the particular cases when $u\equiv 0$ or $v\equiv 0$, it becomes a Fujita-type theorem, respectively, for entire sub- or super-solutions of the Cauchy problem for the equation (1). Let $n\geq 1$, $2\geq p> 1$ and $1<q\leq p-1 +\frac {p}n$, and let $u$ be an entire weak super-solution and $v$ an entire weak sub-solution of the Cauchy problem, with possibly different initial data for $u$ and $v$, for the equation (1) in $\mathbb S$ such that $u\geq v$. Then $u \equiv v$ in $\mathbb S$. Note that the results in Theorems 1 and 2 are sharp. The sharpness of these for $q> p-1 +\frac {p}n\geq 1$ follows, for example, from the existence of non-negative self-similar entire solutions to (1) in $\Bbb S$, that was shown in \[1\]. Also, there one can find a Fujita-type theorem on blow-up of non-negative entire solutions of the Cauchy problem for (1), which was obtained as a very interesting generalization of the famous blow-up result in \[2\] to quasilinear parabolic equations. For $0<q\leq1$, it is evident that the function $u(t,x)=e^t$ is a positive entire classical super-solution of (1) in $\Bbb S$. To prove the results in Theorems 1 and 2 we further develop the approach that was proposed for solving similar problems for semilinear parabolic equations in \[3\]. A new key point in our proof is using the fact that for $1<p\leq 2$ the $p$-Laplacian operator $\Delta_p$ satisfies the $\alpha$-monotonicity condition (see, e.g., \[4\]) with $\alpha=p$. [**Acknowledgments.**]{} This research is financially supported by the Alexander von Humboldt Foundation (AvH). The author is very grateful to AvH for the opportunity to visit the Mathematical Institute of Köln University and to Professor B. Kawohl for his cordial hospitality during this visit. [5]{} V.A. Galaktionov and H.A. Levine, *A general approach to critical Fujita exponents in nonlinear parabolic problems*, Nonlinear Anal. (1998) v. 34, no. 7, 1005–1027. H. Fujita, *On the blowing up of solutions of the Cauchy problem for $u_t=\Delta u + u^{1+\alpha}$*, J. Fac. Sci. Univ. Tokyo, Sect. I (1966), v. 13, 109–124. A.G. Kartsatos and V.V. Kurta, *On a comparison principle and the critical Fujita exponents for solutions of semilinear parabolic inequalities*, J. London Math. Soc. (2) 66 (2002), no. 2, 351–360. V.V. Kurta, *Comparison principle for solutions of parabolic inequalities*, C. R. Acad. Sci. Paris, Série I, 322 (1996), 1175–1180. **Author’s address:** Vasilii V. Kurta Mathematical Reviews 416 Fourth Street, P.O. Box 8604 Ann Arbor, Michigan 48107-8604, USA **e-mail:** vkurta@umich.edu, vvk@ams.org

--- abstract: 'We have performed depolarized Impulsive Stimulated Scattering experiments to observe shear acoustic phonons in supercooled triphenylphosphite (TPP) from $\sim$10 - 500 MHz. These measurements, in tandem with previously performed longitudinal and shear measurements, permit further analyses of the relaxation dynamics of TPP within the framework of the mode coupling theory (MCT). Our results provide evidence of $\alpha$ coupling between the shear and longitudinal degrees of freedom up to a decoupling temperature $T_c = 231~K$. A lower bound length scale of shear wave propagation in liquids verified the exponent predicted by theory in the vicinity of the decoupling temperature.' author: - 'Darius H. Torchinsky' - 'Jeremy A. Johnson' - 'Keith A. Nelson' bibliography: - 'TPPShear.bib' title: '$\alpha$-Scale Decoupling of the Mechanical Relaxation and Diverging Shear Wave Propagation Lengthscale in Triphenylphosphite' --- \[sec:intro\]Introduction ========================= The most prominent characteristic of the frequency relaxation spectrum of glass forming liquids is the $\alpha$ peak, a broad, yet distinct feature whose origin lies in the collective degrees of freedom which undergo kinetic arrest during vitrification [@binderkob; @donth; @angellreview]. As the transition from liquid to glass occurs either upon cooling or the application of high pressure, the characteristic time scale of $\alpha$ relaxation varies by 16 orders of magnitude, from the $\sim$100 fs timescale associated with an attempt frequency in the liquid state [@uli] to the 100 s timescale (arbitrarily) chosen at $T_g$ [@angellreview]. One of the fundamental tasks presented to the spectroscopy of glass forming liquids is thus the complete characterization of the $\alpha$ relaxation across this span of frequencies. To date, this challenge has best been met by dielectric spectroscopy, whose wide dynamical range remains unparalleled in the study of glass forming liquids [@lunkenheimer; @loidl1; @dielmct]. Nevertheless, dielectric spectroscopy of molecular liquids probes mainly orientational relaxation, while the glass transition is more naturally understood in terms of the degrees of freedom associated with flow, i.e. the density, which may be probed by longitudinal acoustic waves, and the transverse current, which couples to shear waves [@balucani; @boonyip]. Despite the importance of both of these degrees of freedom, the bulk of the literature on structural relaxation has focussed on study of the longitudinal modulus due to the relatively limited means by which shear waves may be generated and probed. Acoustic transducer [@ferry; @read], Impulsive Stimulated Scattering (ISS) [@nelson; @yan:6240; @yang1] and high-frequency forced Brillouin Scattering [@deathstar] techniques can continuously cover the longitudinal spectrum up to the mid-GHz regime. However, broadband shear wave generation is more accessible experimentally at lower frequencies and exhibits a gap in the range from the $\sim 10$ MHz limit of transducer methods [@ferry; @read] to the low GHz frequencies probed by depolarized (VH) Brillouin Scattering [@huang; @drey1; @drey2; @SAL2; @wang:617; @tao]. The vast majority of depolarized Brillouin Scattering studies are conducted in a single scattering geometry, permitting observation of only one acoustic wavevector. Thus, relaxation information must be extracted by complex modeling incorporating contributions from the various channels responsible for the depolarized scattering of light. Obtaining spectral information from a collection of acoustic frequencies would allow a direct determination of the shear relaxation spectrum without the need for multiparameter fitting. Recent advancements in the generation of narrowband shear waves [@torchinsky1] using the technique of Impulsive Stimulated Brillouin Scattering (ISBS) in the depolarized geometry [@nelson2] now permit study of the shear $\alpha$ relaxation spectrum across a broad range of frequencies. In this study, we employ ISBS to focus on tests of a first-principles theory of the glass transition, the mode coupling theory (MCT) [@bengt; @goetze1988; @ab]. While the natural variables of the theory are the density fluctuations that constitute the longitudinal modes, MCT suggests relationships among the temperature dependencies of the various relaxing variables (e.g., transverse current fluctuations, density fluctuations, orientational fluctuations, etc.) through the property of $\alpha$-scale coupling [@ab]. This prediction states that if two variables, $A$ and $B$, couple to density fluctuations, their associated relaxation times, $\tau_{A}(T)=C_A \tau(T)$ and $\tau_B(T)=C_B \tau(T)$, are equivalent, up to a temperature independent factor, to a universal relaxation time $\tau(T)$ characteristic of the main $\alpha$ relaxation process. Decoupling may then occur at the MCT critical temperature $T_c$ when the relationship between these relaxation times can break down. Prior experimental studies that have examined $\alpha$-coupling have focused on, e.g., the coupling of dielectric to rheological variables [@zorn], dielectric to shear mechanical relaxation [@niss2], and of translational to rotational diffusion [@lohfink:30]. However, there have been no studies that directly link elastic degrees of freedom with each other. MCT predictions have also been formulated concerning the transverse current fluctuations in their own right, in particular the power-law divergence of an upper bound length scale for shear wave propagation derived for hard spheres in a Percus-Yevick approximation [@das1]. While such upper bound has been established in the literature [@balucani] in terms of a wavelength at which the corresponding shear acoustic frequency goes to zero, this particular scaling relationship with temperature is unique to MCT and, to our knowledge, has not undergone scrutiny in the lab. Below, we provide a test of these assertions by building upon prior work [@sil1; @sil2], and present an expanded study of the shear acoustic behavior of the fragile glass-former triphenylphosphite (TPP) ($T_g$ = 202 K [@TPPDI]). Taken together, these experiments fill in the gap between 10 MHz and 1 GHz in the shear relaxaiton spectrum, and thus constitute the broadest bandwidth shear acoustic measurements performed optically any glass forming system to our knowledge. Our measurement approach thus enables characterization of shear relaxation in glass-forming liquids in this frequency regime. Although supercooled liquid $\alpha$ relaxation spectra are typically broader than two decades at any temperature and, as $T$ is varied, the spectra move across many decades, the 10-1000 MHz range is sufficient to permit assessment of whether shear relaxation dynamics are consistent with relaxation dynamics determined from longitudinal wave measurements covering a comparable spectral width at common temperatures. They are also sufficient to test predictions about the low-frequency limit of shear wave propagation as a function of temperature. We begin with a general overview of depolarized ISBS. This motivates a derivation of the signal generated in an ISBS experiment geared toward measuring shear waves from simplified versions of the hydrodynamic equations of motion. After a brief review of the experimental apparatus used, we present data collected in the temperature range from 220 K to 250 K. Our measured shear acoustic frequencies show significant acoustic dispersion, indicative of the complex relaxation dynamics characteristic of supercooled liquids thus providing a means of testing the MCT predictions described above. In particular, we see evidence of $\alpha$-decoupling, yielding an estimate of the MCT $T_c$ of 231 K. Using this value of the crossover temperature yields a power law for the lower limit of shear wave propagation in line with predictions derived in Ref. [@das1]. These results are discussed within the framework of the MCT. \[sec:gen\]ISBS – General Considerations ======================================== In a typical ISS experiment, conducted in a heterodyned four-wave mixing geometry, light from a pulsed laser is incident on a diffractive optical element, typically a binary phase mask (PM) pattern, and split into two parts ($\pm 1$ diffraction orders; other orders are blocked) that are recombined at an angle $\theta$ as depicted in Fig. \[fig:expt\]. The crossed excitation pulses generate an acoustic wave whose wavelength $\Lambda$ is given by $$\Lambda=\frac{\lambda}{2\sin{\theta/2}}$$ where $\lambda$ is the excitation laser wavelength. Probe light (in the present case from a CW diode laser) is also incident on a phase mask pattern (the same one or another with the same spatial period) and split into two parts that are recombined at the sample to serve as probe and reference beams. The signal arises from diffraction of probe light off the acoustic wave and any other spatially periodic responses induced by the excitation pulses. The diffracted signal field is superposed with the reference field for heterodyned time-resolved detection of the signal, which typically shows damped acoustic oscillations from which the acoustic frequency and damping rate can be determined. ![\[fig:expt\]Schematic illustration of the ISS setup. Both the pump and probe beams are incident on the PM and their $\pm 1$ diffraction orders are recombined at the sample. In the case of a depolarized experiment, half-waveplates in the path of each of the four beams are used to create a polarization grating pattern in the sample plane, as shown in the figure inset.](setup.eps) The polarizations of the beams in the present measurements are all vertical (V) or horizontal (H) relative to the scattering plane. We denote the polarizations of the excitation fields, probe field, and reference/signal field in that order, i.e. VHVH denotes V and H excitation, V probe, and H reference and signal polarizations while VVVV denotes all V polarizations. All the measurements reported herein were conducted in the VHVH configuration. In a depolarized (VHVH) experiment, each of the pump arms carries a different polarization and the ensuing grating is described as an alternating polarization pattern, depicted in the Fig. \[fig:expt\] inset. It is the regions of linear polarization that perform electrostrictive work, deforming the excited region in a fashion that generates counterpropagating shear acoustic waves with a driving force that scales with wavevector magnitude $q=2\pi/\Lambda$ [@nelson2]. The driving force also scales with a geometrical factor of $\cos{\theta/2}$ due to a diminishing of the contrast ratio of the polarization grating with increased angle. In the small $\theta$ limit, the signal from shear waves becomes markedly stronger as the wavevector is increased. In this case, the diffracted signal polarization is rotated $90^{\circ}$ from that of the incident probe light, analogous to depolarized (VH) Brillouin scattering. Finally, we note that the excitation pulses may also induce molecular orientational responses that can contribute to signal, analogous to depolarized quasielastic scattering [@sil1; @fayer; @pick1; @pick2; @pick3; @pick4; @azzimani]. \[sec:thry\]ISBS – Theoretical Background ========================================= A full treatment of signal in an ISS experiment includes longitudinal and transverse acoustic modes, molecular orientations, and other nonoscillatory degrees of freedom [@yang1; @pick1; @pick2; @pick3; @pick4; @azzimani; @glorieuxflow]. Here, we present a slightly simplified derivation that allows us to understand the presence of shear and orientational contributions to the signals. We start with the generalized, linearized hydrodynamic equations of motion [@hansmc]. First is the momentum density conservation law, $$\frac{\partial}{\partial t}\rho \textbf{u}(\textbf{r},t) + \nabla \cdot \Pi (\textbf{r},t) = 0$$ where $\rho$ is the density, $\textbf{u}(\textbf{r},t)$ the velocity, and $\Pi (\textbf{r},t)$ the stress tensor. The expression for the stress tensor reads $$\begin{aligned} \label{eq:bem} \Pi^{\alpha \beta} (\textbf{r},t)&=&\delta_{\alpha \beta}P(\textbf{r},t) - \eta \tau_{\alpha \beta}(\textbf{r},t)\nonumber\\ &&+\delta_{\alpha \beta}(\frac{2}{3}\eta -\zeta)\textbf{$\nabla\cdot$ u }(\textbf{r},t)\nonumber\\ &-&\mu Q_{\alpha \beta}(\textbf{r},t)+F_{\alpha \beta}(\textbf{r},t).\end{aligned}$$ Here, $\delta_{\alpha \beta}$ is the Kronecker delta, $P(\textbf{r},t)$ is the pressure, $\eta$ is the shear viscosity, $\zeta$ is the longitudinal viscosity, $Q_{\alpha \beta}(\textbf{r},t)$ is the orientational variable, $\mu$ expresses the coupling of translational force due to rotational motion, and $\textbf{r}$ is the spatial coordinate. $F_{\alpha \beta}(\textbf{r},t)$ is the external shearing stress of the laser field [@ferry; @wang], assumed to be temporally impulsive and spatially periodic, i.e., $F_{\alpha\beta}=F_0^{xy}\delta(t)\cos{\textbf{q}\cdot\textbf{r}}$, and $\tau_{\alpha \beta}$ is the rate of strain, defined by $$\tau_{\alpha \beta}=\frac{\partial u_\alpha (\textbf{r},t)}{\partial r_\beta}+\frac{\partial u_\beta (\textbf{r},t)}{\partial r_\alpha}.$$ The orientational variable obeys its own equation of motion, given by $$\label{eq:qem} \frac{\partial Q_{\alpha \beta}(\textbf{r},t)}{\partial t} = -\Gamma_R Q_{\alpha \beta}(\textbf{r},t)+\xi \tau_{\alpha \beta}(\textbf{r},t)+Q_0(\textbf{r},t)$$ where $\Gamma_R$ is the orientational relaxation rate, $\xi $ is the torque due to translational motion, and $Q_0(\textbf{r},t)$ is the torque exerted by the laser [@wang; @wang:617]. As with the laser-induced stress, the torque will also be modeled as temporally impulsive and spatially periodic. This equation of motion assumes Debye orientational relaxation as a computational convenience. We set the grating wavevector in the $x$ direction and the transverse direction to be $y$. Since we are interested in shear waves, we select the transverse elements of the above equations, and after a Fourier-Laplace transform defined as $$\label{eq:3fourlap} {\cal FL}\{f(\textbf{r},t)\}=\int_{0}^{\infty}dt\int_{-\infty}^{\infty} d\textbf{r}f(\textbf{r},t)e^{i\textbf{q}\cdot\textbf{r}-st},$$ we are left with $$\begin{aligned} \rho s u_y(\textbf{q},s) &=&-i q \Pi^{xy}(\textbf{q},s)\\ \Pi^{xy}(\textbf{q},s) &=&-\eta_s \tau^{xy}(\textbf{q},s) + F_0^{xy}+\mu Q_{xy}(\textbf{q},s) \\ s Q_{xy}(\textbf{q},s)&=&-\Gamma_R Q_{xy}(\textbf{q},s) +\xi \tau^{xy}(\textbf{q},s)+Q_0.\end{aligned}$$ For simplicity, we assume that the measured signal is proportional to the molecular polarizability anisotropy [@wang], i.e. to $Q_{\alpha\beta}$, so we solve for the orientational variable to yield $$Q_{xy}(\textbf{q},s)=\frac{\xi q^2 F_0^{xy} +Q_0(s\rho +q^2\eta)}{(s+\Gamma_R)(s\rho +q^2\eta)-\xi\mu q^2}.$$ In order to arrive at an analytic solution, we make the approximation that $\xi\mu q^2$ is a coupling of higher order that can be ignored in the solution of the equations of motion. Physically, we are making the approximation that we may ignore the recoupling of the orientational degree of freedom to itself via rotational-translational coupling. In this approximation, the expression for $Q_{xy}$ separates as $$Q_{xy}(\textbf{q},s) = \frac{\xi q^2F_0^{xy}}{(s+\Gamma_R)(s\rho +q^2\eta)}+\frac{Q_0}{s+\Gamma_R}.$$ In order to extract the effect of structural relaxation dynamics on signal, we model the $\alpha$ peak by Debye relaxation as $\frac{c_{\infty}^2 \tau_s}{1+s\tau_s}$, where $c_\infty$ is the infinite frequency speed of sound and $\tau_s$ is the characteristic shear relaxation time. This, in effect, makes $\eta$ complex, which enables an elastic component of the shear response to emerge from the equation of motion \[eq:bem\] [@hansmc]. In this model, the above equations can be solved for $Q_{xy}$ to yield $$\label{eq:eps} Q_{xy}(\textbf{q},s)=\frac{\xi q^2F_0^{xy}}{\rho}\frac{1}{(s+\Gamma_R)(s + \frac{c_\infty^2 \tau_s q^2}{1+s\tau_s})}+\frac{Q_0}{s+\Gamma_R}.$$ Equation (\[eq:eps\]) can be recast into the form: $$\begin{aligned} \label{eq:eps2} Q_{xy}(\textbf{q},s)&=&\frac{\xi q^2F_0^{xy}}{\rho}\frac{2 \Gamma_s +s}{(s+\Gamma_R)(s+\Gamma_s+i\omega_s)(s+\Gamma_s-i\omega_s)}\nonumber\\ &&+\frac{Q_0}{s+\Gamma_R}\end{aligned}$$ where the shear acoustic damping rate $\Gamma_s$ is given by $$\label{eq:ga} \Gamma_s=\frac{1}{2\tau_s}$$ and the frequency of oscillation $\omega_s$ by $$\label{eq:wa} \omega_s = \sqrt{c_\infty^2 q^2-\left(\frac{1}{2\tau_s}\right)^2}.$$ The acoustic damping rate $\Gamma_s$ thus comprises the structural relaxation dynamics. We also note that $\omega_s$ may go to zero for finite wavevector $q_0$ when $$\label{eq:cicond} q_0 = \frac{1}{2\tau_s c_{\infty}}.$$ Separation of equation \[eq:eps2\] by partial fractions yields a time-domain solution $$\begin{aligned} \label{eq:shearsol} Q_{xy}(q,t)&=& \frac{\xi q^2 F_0^{xy}}{\rho}[Ae^{-\Gamma_s t}\sin(\omega_s t)\nonumber\\ &&+B\left(e^{-\Gamma_R t}-e^{-\Gamma_s t}\cos(\omega_s t)\right)]\nonumber\\ &&+Q_0\exp(-\Gamma_R t)\end{aligned}$$ where $$A=\frac{\omega_s^2+\Gamma_R\Gamma_s-\Gamma_s^2}{\omega_s\left(\left(\Gamma_s-\Gamma_R\right)^2+\omega_s^2\right)}$$ and $$B=\frac{(2\Gamma_s-\Gamma_R)}{\left(\left(\Gamma_s-\Gamma_R\right)^2+\omega_s^2\right)}.$$ The solution represented by equation \[eq:shearsol\] comprises two pieces. The term proportional to $q^2 F_0^{xy}$ is due to the shear acoustic response, and the other term proportional $Q_0$ is a decaying exponential independent of $q$. This orientational response is the optical kerr effect (OKE) signal. We also note the presence of an orientational contribution to signal in the acoustic response, due to the rotational-translational coupling. In order to derive the relaxation spectrum from our data, we consider a frequency-dependent modulus $G^{\ast}(s)=G'(s)+iG''(s)$ which obeys the dispersion relation [@silence_thesis] $$\label{eq:gdisp3} \rho s^2+G(s)q^2=0,$$ and yields the following expressions for the real and imaginary parts of the shear modulus from the acoustic frequency and damping rate $$\begin{aligned} \label{realmod} G'(\omega_s)&=&\rho\frac{\omega_s^2-\Gamma_s^2}{q^2}\\ G''(\omega_s)&=&\rho\frac{2\omega_s\Gamma_s}{q^2}.\label{imagmod}\end{aligned}$$ As equation \[eq:gdisp3\] has been derived considering the strain, in order to compare it with the results of the above analysis, we must solve for the strain from the original equations of motion. This gives the dispersion relation $$\label{eq:edisp3} \rho s^2+\eta s q^2 =0.$$ Comparison between equations \[eq:gdisp3\] and \[eq:edisp3\] yields the connection between the elastic modulus and the relaxation spectrum $\eta(q,\omega_s)$ $$\begin{aligned} \frac{G'(\omega_s)}{\rho}&=&-\omega_s{\rm Im}[\eta(q,\omega_s)]\\ \frac{G''(\omega_s)}{\rho}&=&\omega_s{\rm Re}[\eta(q,\omega_s)].\end{aligned}$$ We conclude by noting that orientational responses of anisotropic molecules can be induced not only by the excitation pulses, as in the case of OKE, but also by flow that occurs due to the induced density changes [@glorieuxflow]. Both of these sources lead to signals that can be suppressed by proper selection of probe and signal polarizations. This step was impractical when measuring shear responses, as a choice of polarization which would suppress the orientational response often reduced the already weak shear signal beyond the limit of detection. Eliminating this contribution was also deemed unnecessary, as the aim of this study was to directly probe the shear relaxation spectrum via narrowband acoustic measurements of frequencies and damping rates. ISBS – Experimental Details =========================== The pump laser system used for these studies was an Yb:KWG High-Q FemtoRegen lasing at 1035 nm and producing pulses of 500 $\mu$J at a repetition rate of 1 kHz, although 150 $\mu$J was routinely used to avoid cumulative degradation of the sample. While a 300 fs compressed pulse width FWHM was typical, we bypassed the compressor to retrieve pulses directly from the regen that were 80 ps in duration in order to avoid sample damage at high peak powers, yet remain in the impulsive limit relative to the oscillation period. The excitations beams were cylindrically focussed to a spot that was 2.5 mm in the grating wavevector dimension and 100 $\mu$m in the perpendicular dimension so that the acoustic waves would have many periods and the decay of signal would be due primarily to acoustic damping rather than propagation away from the excitation and probing region of the sample. As a probe, we used a Sanyo DL8032-001 CW diode laser output at 830 nm with 150 mW power focused to a spot of 1 mm in the grating dimension by 50 $\mu$m in the perpendicular dimension. We also used a single phase mask optimized for diffraction into $\pm$1 order at 800 nm for both pump and probe beams, and we utilized two-lens 2:1 imaging with Thorlabs’ NIR achromats to recombine the beams at the sample. The local oscillator was attenuated by a factor of $10^{-3}$. Approximately $30\%$ of the pump power was lost into zero order with this configuration, but the pump intensity still had to be reduced significantly to avoid unwanted nonlinear effects. In order to generate the polarization grating, we inserted $\lambda/2$ waveplates into each of the beams. These waveplates were held in precision rotation mounts to provide accurate alignment of the relative polarizations of the V and H polarized beams. This set the upper limit on the grating spacing that could be achieved in our measurements – for longer wavelengths, the beams came close enough together to be clipped by the rotation mounts. This issue was addressed by imaging with longer focal length optics. The signal was collected in a Cummings Electronics Laboratories model 3031-0003 detector and recorded by a Tektronix Model TDS-7404 oscilloscope. The shear signals were weak and required 10,000 averages, resulting in data acquisition times of a few minutes. TPP at 97% nominal purity was purchased from Alfa Aesar and had both water and volatile impurities removed by heating under vacuum with the drying agent ${\rm MgSO_4}$ immersed in the liquid. The sample was then transferred to a cell with movable windows [@halalay] via filtering through a millipore 0.22 $\mu$m teflon filter. After loading, the cell was placed in a Janis ST-100-H cryostat where the temperature was measured with a Lakeshore model PT-102 platinum resistor immersed directly within the liquid, and monitored and controlled with a Lakeshore 331 temperature controller. The grating spacings examined in this study were 2.33 $\mu$m, 3.65 $\mu$m, 6.70 $\mu$m, 7.61 $\mu$m, 9.14 $\mu$m, 10.2 $\mu$m, 11.7 $\mu$m, 13.7 $\mu$m, 15.7 $\mu$m, 18.3 $\mu$m, 21.3 $\mu$m, 24.9 $\mu$m, 28.5 $\mu$m, 33.0 $\mu$m, 38.1 $\mu$m, 42.6 $\mu$m, and 50.7 $\mu$m, while data for 0.48 $\mu$m, 1.52 $\mu$m, 3.14 $\mu$m, and 4.55 $\mu$m grating spacings were taken from prior reported results [@sil1; @sil2]. The acoustic wavelength was calibrated through ISS measurements in ethylene glycol, for which the speed of sound is known to a high degree of accuracy [@silence_thesis]. When the samples were cooled to the desired temperature, the cooling rate never exceeded 6 K/min, with 2 K/min being typical. Data were taken at fixed wavevector every 2 K from 220 K to 250 K upon warming as we found crystallization was less likely to occur upon warming than cooling. Only a few measurements could be taken without having to thermally cycle the liquid, as it invariably crystallized. We noticed that the tendency toward crystallization was particularly pronounced in the temperature range between 234 K and 242 K. After a few days of use, the sample was observed to develop a slightly cloudy yellowish hue, and so was replaced by a new one. The yellowish samples tended more readily toward photoinduced damage, as well as crystallization, than the original, clear samples. Comparison of the signals obtained from the degraded samples and fresh ones yielded the same frequency and damping rate values, indicating that uncertainties in either of these quantities were due mainly to noise in the data. \[sec:randd\]Results and Discussion =================================== ![\[fig:TPP\_2.4\] Shear waves in TPP with $\Lambda =2.33~\mu$m at (a) 220 K, (b) 234 K, and (c) 246 K. The data are in black and the fits in red. As the temperature is increased, the acoustic wave becomes more heavily damped. At higher temperatures, we also note the presence of the orientational relaxation, which appears to skew the signal such that the oscillations do not occur about the zero baseline.](TPP-2.4-rawfits.eps) The results of several VHVH experiments performed at $\Lambda$=2.33 $\mu$m grating spacing are shown in Fig. \[fig:TPP\_2.4\]. There is an initial spike due to the non-resonant electronic response. Immediately following this hyperpolarizability peak are oscillations about the zero baseline from the counterpropagating shear waves. At a sample temperature of 220 K, these oscillations are seen to disappear on the scale of tens of nanoseconds due to acoustic damping. As the sample is warmed, the frequency is observed to decrease and the damping to increase dramatically. At sufficiently high temperatures, the shear wave becomes overdamped. We also note that at some temperatures, the signature of orientational relaxation is observed as a nonoscillatory decay component in the signal. Another illustration of the influence of relaxation dynamics on the signal may be obtained by examining data from a collection of wavevectors at a common temperature, as depicted in Fig. \[fig:TPP\_220K\] where we provide data recorded with 10.2 $\mu$m, 21.3 $\mu$m, and 44.2 $\mu$m grating spacings at 220 K. Data with a fourth wavelength, 2.33 $\mu$m, are shown in Fig. \[fig:TPP\_2.4\]a. As the wavevector and the frequency are reduced, the acoustic oscillation period increases toward the characteristic relaxation timescale $\tau_s$ and therefore the shear wave is more heavily damped. The signals at larger grating spacings are weaker due to the linear $q$-dependence of the excitation efficiency. ![\[fig:TPP\_220K\]Shear acoustic signal in TPP at 220 K for (a) 10.2 $\mu$m, (b) 21.3 $\mu$m, and (c) 44.2 $\mu$m grating spacings. The data are in black, and the fits are in red. We note that already by 220 K at 44.2 $\mu$m, only a few acoustic cycles are observed, and that the signal intensity is reduced.](TPP-GS-220K.eps) Based on the analysis of Sec. \[sec:thry\], time-domain signals were fit to the function $$\label{eq:6shearsig} I(t)=A'\exp(-\Gamma_s t)\sin(\omega_s t+\phi)+B'\exp(-\Gamma_R t) + C\delta(t)$$ which was convolved with the instrument response function, modelled here by a Gaussian with duration 0.262 ns. The convolution was necessary to determine the true $t=0$ for the experiment. Here, $A'$ is the acoustic amplitude, $\Gamma_s$ is the shear damping rate and $\omega_s$ is the shear frequency. $\phi$ is a phase which accounts for the cosine term in Eq. \[eq:shearsol\], which only becomes important when the damping is strong. In the next term, $B'$ is the optical Kerr effect signal amplitude and $\Gamma_R$ is the orientational relaxation rate, and in the last term $C$ is the strength of the hyperpolarizability spike. As discussed in Sec. \[sec:thry\], it is a simplification to model the orientational behavior by a single decaying exponential [@fayer]. A more accurate description might be in terms of a Kohlrausch-Williams-Watts stretched exponential function $e^{\left(-t/tau_s\right)^\beta}$ ($0<\beta\leq 1$), as is used commonly for time-domain relaxation; however, since the orientational signal contributions are weak, we were able to obtain excellent fits with fewer parameters using a single exponential form. The obtained values of the the shear acoustic velocity $c_s=\omega_s/q$ at a collection of wavevectors are shown in Fig. \[fig:tppcs\], while in Fig. \[fig:tppdamp\] the scaled damping rates are shown. Both figures incorporate the data from Refs. [@sil1] and [@sil2]. Data at other wavevectors were consistent with those shown, but are omitted from this and further plots for clarity. Two features of the data are immediately evident in these plots: first, we observe significant acoustic dispersion for the shear waves across all temperatures studied, and this dispersion increases dramatically as the temperature is raised. The second feature we note is that, as a result of the dispersion and the shear softening it represents, at each temperature above 240 K there is a wavelength above which we are unable to observe the shear wave in our measurements due to its increased damping and reduced signal strength. This wavelength is observed to decrease as temperature increases. ![\[fig:tppcs\]Shear speed of sound in TPP as a function of temperature for a variety of grating spacings. We note that the highest temperature for which we could observe shear waves increases with the decrease of grating spacing. Data at $\Lambda=~1.52~\mu$m are from [@sil1; @sil2].](TPP_c_S_3.eps) ![\[fig:tppdamp\]Scaled shear damping rate in TPP as a function of temperature for a variety of grating spacings. Longer wavelength acoustic waves have a higher scaled damping rate which increases with temperature. Data at $\Lambda=~1.52~\mu$m are from [@sil1; @sil2].](TPP_damp_3.eps) From the fitted values for the shear frequency $\omega_s$ and the damping rate $\Gamma_s$, we may compute a value of the reactive and dissipative shear moduli from equations \[realmod\] and \[imagmod\]. The density values we obtained from data in reference [@TPPDEN] were fit to a quadratic function as $$\label{eq:tpp_den} \rho (T)=1.507~[{\rm g/cm^3}]-1.3\times 10^{-3}~T~[{\rm K}]+6.8 \times 10^{-7}~T^2 ~[{\rm K^2}].$$ Figures \[fig:tpprefogt\] and \[fig:tppimfogt\] show plots of the real and imaginary parts of the shear modulus, respectively, as functions of temperature. These plots are for the same collection of wavevectors for which we have plotted the velocity and damping information. As in the plot of the velocity, we see the softening of the modulus at higher temperatures. The imaginary part shows generally monotonic behavior as a function of temperature as well, except for the 1.52 $\mu$m and 10.2 $\mu$m data, which show a small decrease in the imaginary part at higher temperatures, a feature which is only weakly evident in the damping rate itself represented in Fig. \[fig:tppdamp\]. This is likely due to the oscillation period of our shear wave exceeding the characteristic relaxation time $\tau_s$, permitting observation of a piece of the low-frequency side of the relaxation curve. We generally did not observe this trend at most wavevectors, as the shear wave signal became either too weak or too strongly damped to be observed. ![\[fig:tpprefogt\]$G'(T)$ at a number of grating spacings. As the temperature is increased, the shear modulus is observed to soften considerably. Again, the highest temperature for which we could observe shear waves increases with the decrease of grating spacing. Data at $\Lambda=~1.52~\mu$m are from [@sil1; @sil2].](TPP_RE_GofT_3.eps) ![\[fig:tppimfogt\]$G''(T)$ at a number of grating spacings. For most wavevectors examined in this study, we only observed a rise in the value of $G''(T)$ with temperature. At a handful of wavevectors, we were able to see the low-frequency side of the relaxation peak through, as is visible for $2\pi/q=1.52~\mu$m and $10.2~\mu$m. Data at $\Lambda=1.52~\mu$m are from [@sil1; @sil2].](TPP_IM_GofT_3.eps) The moduli at each temperature were plotted as a function of frequency and then fit to the Havriliak-Negami relaxation function $$\label{eq:Ginf} G^{\ast}(\omega_s)=G_{\infty}\left(1-\frac{1}{(1+(i\omega_s\tau_s)^\alpha)^\beta}\right)$$ in order to extract the shear relaxation spectrum ($G_0=0$ for all temperatures, by definition). Since time-temperature superposition has been observed to hold for $\alpha$ relaxation in many liquids and in TPP (at lower temperatures than measured here) [@olsen_dyre], all spectra were fit simultaneously with the spectral parameters $\alpha$ and $\beta$ acting as temperature-independent global variables and $\tau_s$. The value for the temperature dependent infinite shear modulus $G_{\infty}=\rho c_{\infty}^2$ was taken from depolarized Brillouin scattering measurements performed by Chappell and Kivelson [@chappellkivelson] in a linear extrapolation to colder temperatures as $$c_{\infty}=2620.4~[{\rm m/s}] - 7.93 T~[{\rm K}]$$ where the above expression has been obtained by using polarized (VV) Brillouin scattering data from longitudinal acoustic phonons to derive a temperature dependent refractive index [@TPPDEN] in combination with the shear data of the reference. We note that these data fall on a common line with our lower-$T$ shear data of Fig. \[fig:tpprefogt\] at the highest frequencies, i.e., the shortest wavelengths ($\Lambda$ = 1.52 $\mu$m and 2.33 $\mu$m). ![\[fig:TPP\_GW\]Plots of the real ($G'$) and imaginary ($G''$) moduli of TPP at (a) 224 K, (b) 234 K, and (c) 242 K. At each temperature, a different segment of the shear acoustic relaxation spectrum is present within the experimental frequency window.](TPP_Gw.eps) Three representative plots of the complex shear modulus with corresponding fits are shown in Figs. \[fig:TPP\_GW\]a - \[fig:TPP\_GW\]c. As the temperature is increased, the shear relaxation spectrum moves into the probed region. The fits produced spectral parameters $\alpha=0.61$ and $\beta=0.39$. Although not shown here, the longitudinal data from Refs. [@sil1; @sil2] were refit using the same procedure for purposes of consistency and comparison, yielding spectral parameters $\alpha=0.69$ and $\beta=0.30$. Fig. \[fig:vft\_tpp\] presents the fitted values of $\tau_s$ and $\tau_l$ as a function of temperature along with their respective VFT fits. There is excellent agreement between the two timescales up to the temperature $T=231$ K, where the characteristic relaxation times clearly separate. When combined with the observation that the HN spectral parameters $\alpha$ and $\beta$ differ between the two degrees of freedom, we conclude that the shear acoustic wave dynamics differ significantly, at least at the lower sample temperatures measured, from those obtained from the earlier polarized ISTS data [@sil1; @sil2]. This conclusion, based on a larger data set taken with a broader range of acoustic wavevectors, supercedes the results of the previously published work. ![\[fig:vft\_tpp\]Characteristic relaxation time for shear ($\tau_s$) and longitudinal ($\tau_l$) degrees of freedom plotted versus temperature (left ordinate). Refitted values of the VFT parameters of the original longitudinal data are $\tau_0=2.5~\pm 1$ $\mu$s, $B=230~\pm 40$ K, and $T_0=198~\pm 4$ K while the shear fits produced $\tau_0=0.26~\pm 0.2$ $\mu$s, $B=450~\pm 240$ K, and $T_0=180~\pm 14$ K. Also shown is the decoupling parameter $\log{\tau_s}-\log{\tau_l}$ as a function of temperature (right ordinate).](coupling_b.eps) We now use our data to examine the phenomenon of $\alpha$-scale coupling described briefly in the Introduction. Using the fitted values of $\tau_s$ and $\tau_l$ as a function of temperature, we define the coupling parameter as $\log{\tau_s}-\log{\tau_l}$, which is plotted alongside the relaxation time in Fig. \[fig:vft\_tpp\]. The coupling parameter is constant and essentially zero until it begins to decrease with decreasing temperature, reflecting the differing characteristic timescales of the shear and longitudinal relaxation as the former becomes slower. This provides an estimate of the MCT crossover temperature $T_c=231$. ![\[fig:length\_tpp\]Computed value of length scale limit for shear wave propagation as a function of temperature $\Theta = T-T'$ using the decoupling value for $T'=T_c$ (blue circles) as the critical temperature and the glass transition temperature $T'=T_g$ (green diamonds). For comparison, results from Ref. [@das1] are also shown as a function of packing fraction $\Delta$ (red squares). The exponents are from the data plotted with empty symbols.](length_tpp_b.eps) We may attempt to further understand our results in the mode-coupling framework of Ahluwalia and Das [@das1]. Briefly, when a calculation of a collection of hard spheres in a Percus-Yevick approximation is considered, the critical length scale $L_0=2\pi/q_0$ above which propagation of shear waves becomes overdamped obeys the power law in the vicinity of the critical control parameter $$\label{eq:dasshear} L_0=\frac{A}{(\Delta_c-\Delta)^{\delta}}$$ where $A=1$, $\delta=1.2$, $\Delta$ represents the packing fraction, and $\Delta_c$ is the MCT critical packing fraction beyond which shear wave propagation is allowed for all length scales. Using the theoretical results of Sec. \[sec:thry\], we may attempt to deduce a lower length scale for shear wave propagation as a function of temperature by considering at which wavevector shear wave propagation becomes overdamped. For this analysis, we chose the temperature $T$ to be the independent parameter, yielding a similar power law $L_0=A/(T-T')^{\delta}$, where $T'$ represents a critical temperature. Here we consider two significant temperatures for the liquid, i.e., the glass transition temperature $T_g=202$ K [@TPPDI] and the crossover temperature $T_c$ as determined from the MCT decoupling analysis above. Figure \[fig:length\_tpp\] shows a plot of the derived upper bound, as deduced from Eq. \[eq:cicond\] as a function of the variable $\Theta = (T-T')$. The results of Ref. [@das1] are also shown as a function of packing fraction $\Delta$ for comparison. A power law fit is shown to the four points in the vicinity of the relevant temperatures. Picking $T'=T_c$ as the relevant temperature yields a fitted value of $\delta=1.2$, which is in excellent agreement with the theoretical result. A similar fit to the data using the literature value of $T'=T_g$ produces a significantly higher value of $\delta=13.4$. We remark that the wavelength scales $L_0$ reached as this temperature is approached from above are several orders of magnitude larger than those corresponding to any diverging structural correlation length scale. Conclusions =========== We have used depolarized impulsive stimulated Brillouin scattering to measure shear acoustic waves in supercooled triphenyl phosphite from 220 K to 250 K combined with previously obtained results, we are able to examine a frequency regime from $\sim 10$ MHz to almost $1$ GHz. Our results indicate that the shear and longitudinal spectra do not share the same spectral parameters $\alpha$ and $\beta$. We also observed $\alpha$-decoupling of the longitudinal and shear degrees of freedom, yielding an estimate of the mode-coupling $T_c=231$ K. Using the decoupling result, we verified a power law for the diverging lengthscale of shear wave propagation as $\delta=1.24$. A similar test using the literature value of $T_g$ was not in agreement with the theoretical model, as should be expected since $T_g$ is not a mode-coupling theory parameter. Further work in the study of shear relaxation in triphenyl phosphite and other liquids will center on expanding the dynamic range of the measurements. We also note that a comparison with dielectric data via the model of DiMarzio and Bishop [@dim1] could also be performed if dielectric relaxation measurements at a similar combination of temperatures and frequencies were carried out. Acknowledgments =============== We gratefully acknowledge Professor Shankar P. Das for useful discussions. This work was supported in part by National Science Foundation Grants No. CHE-0616939 and IMR-0414895.

--- author: - 'Noël Malod-Dognin' - 'Nataša Pržulj[^1]' bibliography: - 'document.bib' title: '**Functional geometry of protein-protein interaction networks**' --- Abstract {#abstract .unnumbered} ======== **Motivation:** Protein-protein interactions (PPIs) are usually modelled as networks. These networks have extensively been studied using graphlets, small induced subgraphs capturing the local wiring patterns around nodes in networks. They revealed that proteins involved in similar functions tend to be similarly wired. However, such simple models can only represent pairwise relationships and cannot fully capture the higher-order organization of protein interactions, including protein complexes.\ **Results:** To model the multi-sale organization of these complex biological systems, we utilize simplicial complexes from computational geometry. The question is how to mine these new representations of PPI networks to reveal additional biological information. To address this, we define [*simplets*]{}, a generalization of graphlets to simplicial complexes. By using simplets, we define a sensitive measure of similarity between simplicial complex network representations that allows for clustering them according to their data types better than clustering them by using other state-of-the-art measures, e.g., spectral distance, or facet distribution distance.\ We model human and baker’s yeast PPI networks as simplicial complexes that capture PPIs and protein complexes as simplices. On these models, we show that our newly introduced simplet-based methods cluster proteins by function better than the clustering methods that use the standard PPI networks, uncovering the new underlying functional organization of the cell. We demonstrate the existence of the functional geometry in the PPI data and the superiority of our simplet-based methods to effectively mine for new biological information hidden in the complexity of the higher order organization of PPI networks.\ Introduction ============ Motivation ---------- Genome is the blueprint of a cell. DNA regions called genes are transcribed into messenger RNAs that are translated into proteins. These proteins interact with each other and with other molecules to perform their biological functions. Deciphering the patterns of molecular interactions (also called topology) is fundamental to understanding the functioning of the cell [@ryan2013]. In system biology, molecular interactions are modeled as various molecular interaction networks, in which nodes represent molecules and edges connect molecules that interact in some way. Examples include the well-known protein-protein interaction (PPI) networks in which nodes represent proteins and edges connect proteins that can physically bind. Because exact comparison between networks has long been known to be computationally intractable [@cook1971], the topological analyses of biological networks use approximate comparisons (heuristics), commonly called network properties, such as the degree distribution, to approximately say whether the structures of networks are similar [@newman2010]. Advanced network properties that utilize graphlets (small induced subgraphs) [@przulj2004] have been successfully used to mine biological network datasets. Graphlet-based properties include measures of topological similarities between nodes and between networks [@przulj2004; @przulj2007; @yaveroglu2014], as well as between protein 3D structures represented by networks [@malod2014gr; @faisal2017]. In particular, graphlets have been used to characterize and compare the local wiring patterns around nodes in a PPI network [@milenkovic2008], which revealed that molecules involved in similar functions tend to be similarly wired [@davis2015]. These topological similarities between nodes have also been used to to guide the node mapping process of network alignment methods [@kuchaiev2010; @kuchaiev2011; @malod2015; @vijayan2015], which allowed for transferring of biological annotation between nodes in different networks of well-studied species to less studied ones. Despite significant progress, these simple network (also called graph) models of molecular interaction data can only represent pairwise relationships and cannot fully capture the higher organization of molecular interactions, such as protein complexes and biological pathways [@estrada2005]. Hence, we need to model these data by using new mathematical formalisms capable of capturing their multi-scale organization. Furthermore, we need to design new algorithms capable of extracting new biological information hidden in the wiring patterns of the molecular interaction data modeled by using these mathematical formalisms. This paper addresses these issues. Simplicial complexes basics --------------------------- A candidate model for capturing higher-order molecular organization is a simplicial complex [@munkres1984]. A [*simplicial complex*]{} is a set of [*simplices*]{}, where a 0-dimensional simplex is a node, a 1-dimensional simplex is an edge, a 2-dimensional simplex is a triangle, a 3-dimensional simplex is a tetrahedron and their $n$-dimensional counterparts (illustrated in Figure \[fig:complex\]). The dimension of a simplicial complex is the largest dimension of its simplices. ![[**Illustration of a 3-dimensional simplicial complex.**]{} In the presented simplicial complex, nodes 1, 2 and 3 are only connected by 1-dimensional simplices (edges, in black). Nodes 2, 3 and 4 are connected by a 2-dimensional simplex (triangle, in magenta). Nodes 4, 5 , 6 and 7 are connected by a 3-dimensional simplex (tetrahedron, in blue). [\[fig:complex\]]{}](./Figs/SC){width="6cm"} The $(n\text{-}1)$-dimensional sub-simplices of an $n$-dimensional simplex are called its [*faces*]{} (e.g., a triangle has three faces, the three edges). A simplicial complex, $K$, is required to satisfy two conditions: - For any simplex $\delta \in K$, any face $\delta'$ of $\delta$ is also in $K$. - For any two simplices, $\delta_1, \delta_2 \in K$, $\delta_1 \bigcap \delta_2$ is either $\emptyset$, or a face of both $\delta_1$ and $\delta_2$. In a simplicial complex, a [*facet*]{} is a simplex that is not a face of any higher dimensional simplex. Because of this property, a simplicial complex can be summarized by its set of facets. Note that a network is a 1-dimensional simplicial complex and thus, our proposed methodology is directly applicable to both traditional networks and the higher dimensional simplicial complexes. While simple network statistics, such as degrees, shortest paths and centralities, have been generalized to simplicial complexes [@estrada2018], the lack of more advanced statistics capturing the geometry of simplicial complexes limits their usage in practical applications Contributions ------------- To comprehensively capture the multi-scale organization of complex molecular networks, we propose to model them by using simplicial complexes. To extract the information hidden in the geometric patterns of these models, we generalize graphlets to simplicial complexes, which we call [*simplets*]{}. On large scale real-world and synthetic simplicial complexes, we show that simplets can be used to define a sensitive measure of geometric similarity between simplicial complexes. Then, on simplicial complexes capturing the protein interactomes of human and yeast, we show that simplets can be used to relate the local geometry around proteins in simplicial complexes with their biological functions. Comparison between 1-dimensional protein-protein interaction networks and the higher-dimensional simplicial complex representations of the interactomes formed by protein interactions and protein complexes shows that higher-order modeling enabled by simplicial complexes allows for capturing more biological information, which can efficiently be mined with our proposed simplets. Methods ======= Datasets and their simplicial complex representations ----------------------------------------------------- ### Yeast and human protein interactomes {#sec:models} From BioGRID (v. 3.4.156)[@chatr2017], we collected the experimentally validated protein-protein interaction (PPI) networks of human (H. sapiens) and of yeast (S. cerevisiae). From CORUM [@ruepp2010], we collected collected on the $2^{nd}$ of July, 2017) the experimentally validated protein complexes of human, and from CYC2008 (v.2.0) [@pu2009] the experimentally validated protein complexes of yeast. We consider three different models of an organism’s interactome. - [**The 1-dimensional PPI model**]{}: it is the usual PPI network, in which proteins (nodes) are connected by an edge if they can physically bind. Recall that a network is a 1-dimensional simplicial complexes on which our new simplet methodologies can be applied and are equivalent to the standard graphlet methodologies. - [**The higher-dimensional simplicial complex (SC) model**]{}: starting from the PPI network, we additionally connect by simplicies all the proteins that belong to common complexes. I.e., the proteins belonging to a $k$-protein complex are connected by a $(k\text{-} 1)$ dimensional simplex. For human, the PPI network (1D PPI model) has 16,100 nodes and 212,319 edges. When unifying the lower dimensional protein-protein interaction data and the higher order protein complex data as described above, the resulting SC model is a 140-dimensional simplicial complex having 16,140 nodes (with 40 proteins being part of proteins complexes but not having any reported protein-protein interaction) and 205,192 facets. For yeast, the 1D PPI model has 5,842 nodes and 80,900 edges. When unifying the lower dimensional protein-protein interaction data and the higher order protein complex data as described above, the SC model is a 80-dimensional simplicial complex having 5,842 nodes and 76,790 facets. ### Other real-world datasets {#sec:randoms} We collected real-world higher-dimensional datasets from biology and beyond. - [**1,569 simplicial complexes of protein 3D structures:**]{} Proteins are linear arrangements of amino-acids that in the aqueous environment of the cell fold and acquire specific three-dimensional (3D) shapes called tertiary structures. We collected from Astral-40 (SCOPe v.2.06) [@fox2013] the 3D structures of 1,569 protein domains that are at-least 100 amino-acid long. Each protein domain is modeled as a simplicial complex in which simplices connect together all the amino-acids (nodes) that are less than 7.5 Å apart (as measured by the distances between their $\alpha$-carbons).\ - [**132 simplicial complexes of publication authorships:**]{} From the preprint repository arXiv, we collected all the scientific publications in the “computer science” category over eleven years from 2007 to 2017. For each month, we model the scientific collaborations as a simplicial complex in which simplices are formed by all scientists (nodes) that co-authored a scientific publication.\ - [**60 simplicial complexes of genes’ biological annotations:**]{} The biological functions of genes are described by various ontology terms. We collected pathway annotations from Reactome database (v.63) [@fabregat2017], as well as the experimentally validated Gene Ontology (GO)[@ashburner2000] annotations from NCBI’s entrez web-server (collected in February 2018). For GO, we consider biological process, molecular function, and cellular component annotations separately. For each annotation set, we model the functional annotations of the genes of a given species as a simplicial complex in which simplices are formed by all genes (nodes) that have a common annotation term (restricted to terms annotating up-to 50 genes for computational complexity issues). We only considered simplicial complexes having more than 100 nodes. Following this procedure, we generated 18 pathway simplicial complexes, 13 biological process simplicial complexes, 14 molecular function simplicial complexes and 15 cellular component simplicial complexes.\ - [**14 simplicial complexes of protein-protein interactions:**]{} We collected the experimentally validated protein-protein interactions (PPIs) from BioGRID database (v. 3.4.156)[@chatr2017]. These PPIs are first modeled as networks in which proteins (nodes) are connected by edges if they can interact. The corresponding networks are converted into so-called [*clique complexes*]{}, by creating a simplex between all nodes belonging to a maximal clique in the network. ### Random simplicial complexes {#sec:reals} To test our methods, we considered randomly generated simplicial complexes, which we generate according to eight random models. The first four models are based on randomly generated graphs, which are converted into so-called [*clique complexes*]{}, in which simplices connect nodes that belong to a clique in the graph: - A [*random clique complex*]{} (RCC) is the clique complex of an Erdös-Rènyi random graph [@erdos1959]. An Erdös-Rènyi graph is generated by fixing the number of nodes (detailed below) in the graph, and then by adding edges between uniformly randomly chosen pairs of nodes until a given edge density is reached (also detailed below). - A [*Vietoris-Rips complex*]{} (VRC) [@hausmann1995] is the clique complex of a geometric random graph [@penrose2003]. A geometric random graph represents the proximity relationship between uniformly randomly distributed points in a $d$-dimensional space. We generate geometric graphs by uniformly randomly distributing the desired number of nodes (points) in a 3-dimensional unit cube. Then, two nodes are connected by an edge if the Euclidean distance between the corresponding points is smaller than a distance threshold $r$. The distance threshold is chosen to obtain the desired edge density. - A [*scale-free complex*]{} (SFC) is the clique complex of a Barabàsi-Albert scale-free graph [@barabasi1999]. The scale-free graph model constructed by preferential attachment generates graphs based on the “rich-get-richer” principle and are characterized by power-law degree distributions. We create a scale-free graph using an iterative process, in which the graph is grown by attaching new nodes each with $m$ edges that are preferentially attached to the existing nodes with high degree ($m$ is chosen to obtain the desired edge density). - A [*Watts-Strogatz complex*]{} (WCS) is the clique complex of a small-world graph [@watts1998]. Small-world graphs are characterized by short average path lengths and high clustering. We created a small word graph by constructing a regular ring lattice of $n$ nodes and by connecting each node to its $k$ neighbours, $k/2$ on each side ($k$ is chosen to obtain the desired edge density). Then we uniformly randomly rewire 5% of the edges. The four other models are extensions of the [*Linial-Meshulam*]{} model [@linial2006; @meshulam2009], which originally consists in randomly connecting nodes with $k$-dimensional facets. We extended this model to randomly connect nodes with facets while following the facet distribution of an input simplicial complex. In this way, we can create Linial-Meshulam variant of the four clique complex-based models presented above: - A [*Linial-Meshulam random clique complex*]{} (LM- RCC) is a Linial-Meshulam complex that follows the facet distribution of an input random clique complex. - A [*Linial-Meshulam Vietoris-Rips complex*]{} (LM- VRC) is a Linial-Meshulam complex that follows the facet distribution of an input Vietoris-Rips complex. - A [*Linial-Meshulam scale-free complex*]{} (LM-SFC) is a Linial-Meshulam complex that follows the facet distribution of an input scale-free complex. - A [*Linial-Meshulam Watts-Strogatz complex*]{} (LM- WSC) is a Linial-Meshulam complex that follows the facet distribution of an input Watts-Strogatz complex. For each model we choose three node sizes, 1,000, 2,000, and 3,000 nodes, and three edge densities, 0.5%, 0.75% and 1%. We generated 25 random simplicial complexes for each model and each of these node sizes and edge densities. Hence, in total, we generated $8\times 3\times 3\times 25 =$ 1,800 random simplicial complexes. We chose these node sizes and edge densities to roughtly mimic the sizes and densities of real-world data detailed above. Capturing the local geometry around nodes in a simplicial complex with simplets {#sec:simplets} ------------------------------------------------------------------------------- [*Simplets*]{} are small, connected, non-isomorphic, induced simplicial complexes of a larger simplicial complex. Figure \[fig:simplets\] shows the eighteen 2- to 4-node simplets (denoted by $S_1$ to $S_{18}$). Within each simplet, because of symmetries, some nodes can have identical geometries. Analogous to automorphism orbits in graphlets [@przulj2007], we say that such nodes belong to a common [*simplet orbit group*]{}, or [*orbit*]{} for brevity. Figure \[fig:simplets\] shows the thirty-two orbits of the 2- to 4-node simplets (denoted from 1 to 32). Similar to graphlets, we use simplets to generalize the notion of the node degree: the $i^{th}$ [*simplet degree*]{} of node $v$, denoted by $v_i$, is the number of times node $v$ touches a simplet at orbit $i$. We define the [*simplet degree vector*]{} (SDV) of a node as the 32 dimensional vector containing the simplet degrees of the node in the simplicial complex as its coordinates. Hence, the SDV of a node describes the local geometry around the node in the simplicial complex and comparing the SDVs of two nodes provides a measure of local geometric similarity between them. We define the [*SDV similarity*]{} between two nodes as an extension of the graphlet degree similarity [@milenkovic2008]. It is computed as follows. The distance, $D_{i}(u,v)$, between the $i^{th}$ simplet orbits of nodes $u$ and $v$ is defined as: $$D_i(u,v) = w_i \times \frac{|log(u_i + 1) - log(v_i + 1)|}{log(max\{u_i, v_i\} + 2)},$$ where $w_{i}$ is the weight of orbit $i$ that accounts for dependencies between orbits. Weight, $w_{i}$, is computed as $\displaystyle w_i = 1 - \frac{\log (o_i)}{\log (32)}$, where $o_i$ is the number of orbits that orbit $i$ depends on, including itself. For instance, the count of orbit 2 (the middle of a three node path) of a node depends on its count of orbit 0 (i.e. its node degree) and on itself, so $o_2 = 2$. For orbit 9, $o_{9}$ = 3, since it is affected by orbits 0, 2, and itself. The values of $o_i$ for all 2- to 4-nodes simplet orbits are listed in Table \[Tab:01\]. ![[**Illustration of 2- to 4-nodes simplets.**]{} The 18 2- to 4-nodes simplets are denoted by $S_1$ to $S_{18}$. Within each simplet, geometrically interchangeable nodes, belonging to the same orbit, have the same color. These simplets have 32 different orbits, denoted from 1 to 32. Note that simplets $S_4$, $S_8$, $S_{11}$ and $S_{14}$ have only one 2D face (triangle, in blue), while $S_{12}$ and $S_{15}$ have two triangles, $S_{16}$ has 3 triangles and $S_{17}$ has four triangles. $S_{18}$ has four triangles and one 3D face (tetrahedron, in red). [\[fig:simplets\]]{}](./Figs/Simplets){width="8cm"} Orbit, $i$ Weight, $o_i$ --------------------------------------------------------------- --------------- 1 1 2, 3, 4, 5 3 6, 8, 9, 10, 13, 24, 26, 30, 31, 32 3 7, 11, 12, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 27, 28, 29 4 20 5 : The orbit weights.\[Tab:01\] Finally, the SDV similarity, $S(u,v)$, between nodes $u$ and $v$ is defined as: $$S(u,v) = 1 - \frac{\sum_{i | (u_i \neq 0) \text{ or } (v_i \neq 0)}D_i(u,v)}{\sum_{i | (u_i \neq 0) \text{ or } (v_i \neq 0)}w_i}.$$ $S(u,v)$ is in (0, 1\], where similarity 1 means that the SDVs of nodes $u$ and $v$ are identical.\ Capturing the global geometry of a simplicial complex with simplets {#sec:distances} ------------------------------------------------------------------- To the best of our knowledge, researchers from computational geometry have not considered the problem of comparing two simplicial complexes. However, the comparison of biological networks is a foundational problem of system biology. Instead, computational geometry focus on the comparison of two spaces, each represented by a collection of simplicial complexes, e.g. [@collins2004]. Thus, we build upon network analysis and extend graphlet and non-graphlet based network distance measures to directly compare simplicial complexes as follows. ### Simplet correlation distance {#sec:scd} Simplets are like Lego pieces that assemble with each other to build larger simplicial complexes. We exploit this property to summarize the complex structures of simplicial complexes and to compare them, by generalizing Graphlet Correlation Distance [@yaveroglu2014], which is a sensitive measure of topological similarity between networks. Analogous to graphlets, the statistics of different simplet orbits are not independent of each other. The reason behind this is the fact that smaller simplets are induced sub-simplicial complexes of larger simplets. For 2- to 4-node simplets, there are four non-redundant dependency equations between the simplet degrees of a given node $u$: $${{u_1}\choose{2}} = u_2 + u_4 + u_5,$$ $${{u_2}\choose{1}}{{u_1-2}\choose{1}} = 3u_9 + 2u_{11} + 2u_{14} + u_{18} + u_{20} +u_{23},$$ $${{u_3}\choose{1}}{{u_1-1}\choose{1}} = \begin{array}{l}u_7 + u_{12} + u_{15} + 2u_{16} +2u_{17}\\ + 2u_{19} + 2u_{21} + 2u_{22}\end{array},$$ $${{u_1}\choose{3}} = \begin{array}{l}u_9 + u_{11} + u_{14} + u_{18} + u_{20} + u_{23} + u_{24} + u_{25}\\ + u_{26} + u_{27} + u_{28} + u_{29} + u_{30} + u_{31} + u_{32}\end{array}.$$ We used these equations to assess the correctness of our exhaustive simplet counter. In addition to these redundancies there also exist dependencies between simplets, which are dataset dependent. We use these dataset dependent simplet orbit dependencies to characterize the global geometry of simplicial complexes. We capture the dependencies between simplet orbits by the simplicial complex’s [*Simplet Correlation Matrix*]{} (SCM), which we define as follows. We construct a matrix whose rows are the simplet degree vectors of all nodes of the simplicial complex. We calculate the Spearman’s correlation between each two pairs of columns in the resulting matrix, i.e., correlations between the orbits overl all nodes of the simplicial complex. We present these correlations in a $32 \times 32$ dimensional Simplet Correlation Matrix (SCM): it is symmetric and contains Spearman’s correlation values in \[-1,1\] range. As presented in Figure \[fig:scms\], the SCMs of simplicial complexes from different random simplicial complex models are indeed very different. We exploit these differences in SCMs to compare simplicial complexes. We define the [*Simplet Correlation Distance*]{} (SCD) to measure the distance between two simplicial complexes, $K_1$ and $K_2$, by the Euclidean distance between the upper-triangles of their SCMs: $$SCD(K_1, K_2) = \sqrt{ \sum_{i=1}^{32} \sum_{j=i+1}^{32} (SCM_{K_1}[i][j] - SCM_{K_2}[i][j])^2},$$ where $SCM_{K_1}[i][j]$ is the $(i,j)^{th}$ entry in the SCM of $K_1$ (similar for $K_2$). The ability of SCD to group together simplicial complexes according to their underlying models is demonstrated in section \[sec:res\_clustering\]. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- RCC VRC ![[**SCMs of sample simplicial complexes from four random simplicial complex models.**]{} The four simplicial complexes from RCC, VRC, LM-RCC, and LM-VRC models have been generated with 2,000 nodes and the edge density of 0.75%. Note that SCMs of all networks of this size and density coming from a particular model look similar. Hence, these four SCMs are representative of these models at these sizes and densities. [\[fig:scms\]]{}](./Figs/RCC_2000_0007500_3_HCM "fig:"){width="4cm"} ![[**SCMs of sample simplicial complexes from four random simplicial complex models.**]{} The four simplicial complexes from RCC, VRC, LM-RCC, and LM-VRC models have been generated with 2,000 nodes and the edge density of 0.75%. Note that SCMs of all networks of this size and density coming from a particular model look similar. Hence, these four SCMs are representative of these models at these sizes and densities. [\[fig:scms\]]{}](./Figs/VR_2000_0007500_19_HCM "fig:"){width="4cm"}   LM-RCC LM-VRC ![[**SCMs of sample simplicial complexes from four random simplicial complex models.**]{} The four simplicial complexes from RCC, VRC, LM-RCC, and LM-VRC models have been generated with 2,000 nodes and the edge density of 0.75%. Note that SCMs of all networks of this size and density coming from a particular model look similar. Hence, these four SCMs are representative of these models at these sizes and densities. [\[fig:scms\]]{}](./Figs/LM-RCC_2000_0007500_13_HCM "fig:"){width="4cm"} ![[**SCMs of sample simplicial complexes from four random simplicial complex models.**]{} The four simplicial complexes from RCC, VRC, LM-RCC, and LM-VRC models have been generated with 2,000 nodes and the edge density of 0.75%. Note that SCMs of all networks of this size and density coming from a particular model look similar. Hence, these four SCMs are representative of these models at these sizes and densities. [\[fig:scms\]]{}](./Figs/LM-VR_2000_0007500_19_HCM "fig:"){width="4cm"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ### Facet distribution distance In analogy to degree distribution and graphlet degree distribution [@przulj2007], we define the measure of connectivity of a $k$-dimensional simplicial complex, $K$, as the distribution of its facets, $d_{K}$: it is a $k$-dimensional [*facet distribution vector*]{} whose $i^{th}$ entry is the percentage of the facets in $K$ having dimension $i$. The [*Facet Distribution Distance*]{} (FDD) measures the distance between two simplicial complexes, $K_1$ and $K_2$, by the Euclidean distance between their facet distribution vectors, $d_{K_1}$ and $d_{K_2}$: $$FDD(K_1, K_2) = \sqrt{\sum_{i}{(d_{K_1}[i] -d_{K_2}[i])^2}}. \label{eq:specDist}$$ ### Spectral distance Spectral theory captures the topology of networks and simplicial complexes by using the eigenvalues and eigenvectors of matrices representing them, such as the adjacency matrix, or Laplacian matrix [@wilson08]. Let $H$ be the incidence matrix of a simplicial complex, $K$, having $n$ nodes and $f$ facets: $H$ is a $n\times f$ matrix in which entry $H[i][j] = 1$ if node $i$ is in facet $j$, and 0 otherwise. The corresponding degree matrix, $D$, is a $n \times n$ diagonal matrix in which entry $D[i][i]$ is the number of facets containing node $i$. The adjacency matrix, $A$, of a simplicial complex is: $$A = HH^T - D,$$ where $H^T$ is the transpose of $H$ [@zhou2007]. The corresponding Laplacian matrix, $L$, is defined as [@zhou2007]: $$L =\frac{1}{2} D^{-1/2}AD^{-1/2}.$$ The eigen-decomposition of the Laplacian matrix, $L$, of simplicial complex, $K$, is $L = \phi\lambda_K\phi^{T}$, where $\lambda_K = diag(\lambda_{K}^1, \lambda_{K}^2, ..., \lambda_{K}^n)$ is the diagonal matrix with the ordered eigen-values, $\lambda_{K}^i$ as elements and $\phi = (\phi_{1}|\phi_{2}|...|\phi_{n})$ is the matrix with the ordered eigen-vectors as columns. The spectrum of simplicial complex, $K$, is the set of its eigen-values $S_K = \{\lambda_{K}^1, \lambda_{K}^2, ..., \lambda_{K}^n\}$, which are reordered so that $\lambda_{K}^1 \geq \lambda_{K}^2 \geq ... \geq\lambda_{K}^n$. We define the [*spectral distance*]{} (SD) between two simplicial complexes, $K_1$ and $K_2$, as the Euclidean distance between their spectra [@wilson08]: $$SD(K_1, K_2) = \sqrt{\sum_{i}{(\lambda_{K_1}^i - \lambda_{K_2}^i)^2}}. \label{eq:specDist}$$ When the two spectra are of different sizes, $0$ valued eigenvalues are added at the end of the smaller spectrum. Results and discussion ====================== Comparing simplicial complexes {#sec:res_clustering} ------------------------------ We visually inspect how well our simplet correlation distance (SCD, presented in section \[sec:scd\]) groups simplicial complexes of the same type, by embedding simplicial complexes as points in 3D space according to their pairwise SCDs by using multi-dimensional scaling (MDS, [@borg2005]). As presented in Figure \[fig:scd\_random1\], when the simplicial complexes are embedded into 3D space by using multi-dimensional scaling, they visually form clusters. ![[**Illustration of MDS-based embedding of simplicial complexes from eight random models.**]{} The randomly generated simplicial complexes (color-coded) are embedded into 3D space according to their pairwise SCD distances using multi-dimensional scaling (MDS). The eight models and simplicial complex sizes and densities are described in Section \[sec:randoms\]. As described in Section \[sec:randoms\], 25 simplicial complxes are generated for each model and each of its sizes and densities. The grouping of the same colored nodes correspond to simplicial complexes from the same model, but of different sizes and densities. [\[fig:scd\_random1\]]{}](./Figs/3D_view_SCDs_1_random){width="8cm"} We formally assess the clustering ability of SCD and of other distances between simplicial complexes by using the standard Precision-Recall and Receiver Operating Characteristic (ROC) curves analyses [@fawcett2006]. For small increments of parameter $\epsilon \geq 0$, if the distance between two simplicial complexes is smaller than $\epsilon$, then the pair of simplicial complexes is declared to be similar (belong to the same cluster). For each $\epsilon$, four values are computed: - The true positives (TP) are the numbers of correctly clustered pairs of simplicial complexes (grouping together simplicial complexes from the same model). - The true negatives (TN) are the numbers of correctly non-clustered pairs of simplicial complexes (not grouping together simplicial complexes from different models). - The false positives (FP) are the numbers of incorrectly clustered pairs of simplicial complexes (grouping together simplicial complexes from different models). - the false negatives (FN) are the numbers of incorrectly non-clustered pairs of simplicial complexes (not grouping together simplicial complexes from the same model). In Precision-Recall curves, for each $\epsilon$, the precision ($Pr=\frac{TP}{TP+FP}$) is plotted against the recall, also-called true-positive rate ($Re=\frac{TP}{TP+FN}$). The quality of the grouping is measured with the area under the Precision-Recall curve, which is the [*average precision*]{} (AP) of the distance measure. In ROC curves, for each $\epsilon$, the true positive rate (another name for recall) is plotted against the false positive rate ($FPR=\frac{FP}{FP+TN}$). The quality of the grouping is measured with the area under the ROC curve (AUC), which can be interpreted as the probability that a randomly chosen pair of simplicial complexes coming from the same model will have a distance smaller than a randomly chosen pair of simplicial complexes coming from different models. First, we consider our 1,800 randomly generated simplicial complexes (Section \[sec:randoms\]), and all 1,619,100 pairs of these 1,800 simplicial complexes, to measure the ability of SCD to group together the simplicial complexes from the same model. The precision recall curves presented in Figure \[fig:scd\_random2\] confirm our visual illustration of the ability of SCD to classify simplicial complexes. To assess the performance of SCD, we first apply it to synthetic data. In particular, we apply the three distance measures described in Section \[sec:distances\] to the 1,800 model simplicial complexes described in Section \[sec:randoms\]. We find that SCD achieves the highest classification performance with average precision (AP) of 96.99% and an AUC of 86.03%. It is followed by the facet distribution distance (AP of 94.50% and AUC of 76.95%) and by the spectral distance (AP of 89.25% and AUC of 61.26%). Furthermore, on the easier task of grouping together simplicial complexes that are generated from the same models and the same node sizes and edge densities, SCD achieves an almost perfect clusterings having average precision of 99.99% and AUC of 99.17%. It is followed by facet distribution distance (AP of 99.98% and AUC of 98.50%) and by spectral distance (AP of 99.98% and AUC of 98.23%). ![[**Clustering randomly generated simplicial complexes.**]{}The Precision-Recall curves that are achieved when using the three distance measures (color coded, simplet correlation distance in red, facet distribution distance in blue, spectral distance in green) to cluster together the 1,800 randomly generated simplicial complexes into the models that generated them. [\[fig:scd\_random2\]]{}](./Figs/Precision_Recall_random1){width="8cm"} We further validate our methodology by assessing its ability to correctly group our 1,775 real-world simplicial complexes. We calculate the distances between all pairs of the 1,775 real-world simplicial complexes, which results in distances between $1,775 \choose 2$ = 1,574,425 pairs for each of the three distance measures presented in Section \[sec:distances\]. As illustrated in Figure \[fig:scd\_real1\], when the real-world simplicial complexes are embedded into 3D space based on their SCD distances by using multi-dimensional scaling, the simplicial complexes from the same data type group well together. Indeed, the precision-recall curves presented in Figure \[fig:scd\_real2\] show that SCD achieves the highest classification performances (AP of 98.72% and AUC of 99.58%), followed by spectral distance (AP of 94.93% and AUC of 98.64%) and by facet distribution distance (AP of 76.10% and AUC of 93.11%). ![[**Illustration of MDS-based embedding of real-world simplicial complexes based on their SCDs.**]{} The real-world simplicial complexes (color-coded) are embedded into 3D space according to their pairwise SCD distances using multi-dimensional scaling. [\[fig:scd\_real1\]]{}](./Figs/3D_view_SCDs_259_real){width="8cm"} ![[**Clustering real-world simplicial complexes.**]{} The Precision-Recall curves achieved when using SCD to cluster real-world simplicial complexes. [\[fig:scd\_real2\]]{}](./Figs/Precision_Recall_real){width="8cm"} Taken together, these results demonstrate that SCD is a very sensitive measure of simplicial complex similarity. Uncovering biological information from PPI simplicial complexes {#sec:res_clustering} --------------------------------------------------------------- In the experiments presented above, we measured the ability of simplets to capture global geometric features of simplicial complexes. In this section, we focus on the local geometry around nodes in simplicial complexes. We assess if the local geometries of proteins in PPI simplicial complexes (which we capture with simplet degree vectors, see section \[sec:simplets\]) relate to their functional annotations using two different methodologies: clustering and enrichment analysis of the resulting clusters, and canonical correlation analysis. ### Clustering and enrichment analysis In the first step, we investigate if proteins with similar local geometries (i.e., similar simplet degree vectors) tend to also have similar biological functions. For both human and yeast, we computed the simplet degree similarity of the proteins in each of the two models of their interactomes (PPI network model and SC model, see section \[sec:models\]). We used these pairwise similarities as input for spectral clustering [@von2007], which performs $k$-means clustering on the eigen-vectors of the matrix encoding the pairwise simplet degree similarities between the nodes. Spectral clustering is favored over traditional $k$-means as it does not make strong assumptions on the shape of the clusters. While $k$-means produces clusters corresponding to convex sets, spectral clustering can solve a more general problem such as intertwined spirals [@von2007]. To account for the randomness of the underlying $k$-means, each clustering experiments is repeated 10 times. For human and yeast simplicial complexes, we set the number of clusters, $k$, using the rule of thumb [@kodinariya2013]: $k = \sqrt{\frac{n}{2}}$, where $n$ is the number of nodes in the simplicial complex. I.e., we set $k=90$ for human and $k=54$ for yeast data-sets. These values result in coherent clustering according to both sum of square error and normalized mutual information scores [@ana2003]. Then, we measure the biological coherence of the obtained clustering by the percentage of clusters that are statistically significantly enriched in at least one Gene Ontology (GO) annotation [@ashburner2000]. To this aim, we collected the experimentally validated GO annotations of genes from NCBI’s entrez web portal (collected the 8$^{th}$ of March, 2018). We considered GO biological process (GO-BP), GO molecular function (GO-MF), and GO cellular component (GO-CC) annotations separately. A cluster is statistically significantly enriched in a given annotation if the corresponding enrichment $p$-value is lower or equal to 5% after Benjamini-Hochberg [@benjamini1995] correction for multiple hypothesis testing. To account for variability in cluster sizes, we also measure the biological coherence with the percentage of annotated genes that have at least one annotation enriched in their clusters. As presented in Figures \[fig:clustering1\] and \[fig:clustering2\], the clusters obtained from the SC models are more biologically coherent than the clusters obtained from the PPI network models. Over all ten runs, for both species and for the three GO annotation types, the biological coherence in terms of enriched clusters is 50% larger for the SC models than for the PPI network models, with 74.52% of the clusters being enriched for the SC models and 49.51% for the PPI network models. In terms of the enriched genes in the clusters, the biological coherence is 46% larger for the SC models than for the PPI network models, with 25.83% of the genes being enriched for the SC models and 17.63% for the PPI network models. These results demonstrate that proteins having similar geometries in PPI networks modeled as simplicial complexes indeed have similar biological functions. Using our simplets on the SC model allows for clusterings of proteins that best correspond to the hierarchical functional organization of the cell captured by GO biological process annotations, with about 81% more of enriched proteins in the clusters obtained from the SC model by using simplets than on the PPI network by using graphlets (19.6% for the SC model versus 10.8% for the PPI network). Similarly, the clusterings of proteins in the SC model best correspond to the organization of cell captured by GO cellular component annotations, with about 51.1% more of enriched proteins in the clusters obtained from SC model than in the PPI network ones (32.4% for the SC model versus 21.4% for the PPI network). ![[**Biological relevance of clusters of genes,**]{} as measured by the percentage of clusters having at least one enriched GO annotation. The error bars present minimum, average and maximum enrichment values over 10 runs of spectral clustering. [\[fig:clustering1\]]{}](./Figs/1D-HD_clusters){width="8cm"} ![[**Biological relevance of clusters of genes,**]{} as measured by the percentage of annotated genes having at least one GO annotation that is enriched in the cluster. The error bars present minimum, average and maximum enrichment values over 10 runs of spectral clustering. [\[fig:clustering2\]]{}](./Figs/1D-HD_genes){width="8cm"} For GO molecular function annotations, the results are slightly different. On average, the clusters of proteins on the SC models best group together proteins with similar molecular functions, with about 23.5% more of proteins with enriched functions in the clusters obtained from the SC model than in the clustering obtained from the PPI network (25.5% for the SC model versus 20.6% for the PPI network). But when considering on the human datasets alone, the PPI network achieves a slightly better performance (29.0% for the PPI network versus 27.6% for the SC model). These lower performances on molecular functions are expected, because of the very good performances on biological processes. Recall that molecular functions capture the functions of proteins in isolation from each other, while biological processes correspond to higher-order biological functions that are performed by proteins collectively (in Gene Ontology, biological processes must involve more than one distinct molecular functions [^2]). Thus, maximizing the enrichment of molecular function annotations and of biological process annotations are contradicting goals that cannot be expected to be optimized simultaneously. ### Canonical correlation analysis To further investigate the relationships between the local geometry around proteins in simplicial complexes (captured by their simplet degrees) and their biological functions (captured by their GO biological process annotations), we adapt the canonical correlation analysis (CCA) methodologies from [@yaveroglu2014]. In our canonical correlation analysis (CCA) framework, the local geometry around $n$ proteins in a simplicial complex is captured in an $n\times 32$ matrix, $R$, whose entry $R[v][i]$ is the $i^{th}$ simplet degree of node $v$. Similarly, the biological functions of the proteins is captured in an $n\times f$ matrix, $A$, whose entry $A[v][i]$ is 1 if protein $v$ is annotated by term $i$, and 0 otherwise. For both matrices, we excluded the genes that do not have any GO biological process annotations. CCA is an iterative process that identifies linear relationships between the 32 simplet degrees and the $f$ GO biological process annotations. First, CCA outputs two weight vectors, called [*canonical variates*]{}, so that the weighted sum of $R$ is maximally correlated with the weighted sum of $A$. The correlation between the two weighted sums is called their [*canonical correlation*]{}. After finding the first canonical variates, CCA iterates $min\left \{ 32, f \right \}$ times to find more weight vectors, such that the resulting canonical variates are not correlated with any of the previous canonical variates. We refer the interested reader to [@weenink2003] for the mathematical aspects of CCA. ![[**Canonical correlation analysis for human.**]{} For a given simplicial complex, canonical correlation produces variates, which are linear combinations of go annotations and linear combinations of simplet degrees that best correlate over the nodes of the simplicial complex. For both models of human interactomes (PPI network and SC model), we plotted for each variate the corresponding correlation value (only statistically significantly correlated variates are presented, with canonical correlation $p$-value $\leq 5\%$). [\[fig:cca1\]]{}](./Figs/Human_1D-HD_CCA_BP){width="8cm"} ![[**Canonical correlation analysis for yeast.**]{} For both models of yeast interactomes (PPI network and SC model), we plotted for each variate the corresponding correlation value (only statistically significantly correlated variates are presented, with canonical correlation $p$-value $\leq 5\%$). [\[fig:cca2\]]{}](./Figs/Yeast_1D-HD_CCA_BP){width="8cm"} As presented in Figures \[fig:cca1\] and \[fig:cca2\], the SC model allows for uncovering a larger number of linear relationships that the PPI network model. This is because only 15 out of the 32 simplets can appear in a 1-dimensional simplicial complexes, i.e., a PPI network, which correspond to the 15 2- to 4-node graphlets. Hence, CCA can only produce up-to 15 variates for a PPI network and up-to 32 variates for the SC model. Moreover, these linear relationships have higher canonical correlations. This means that by using simplets on the SC models we can capture more and better quality relationships between local geometry around nodes in simplicial complexes and their biological functions than if we use PPI networks. The same is observed when using GO cellular component and GO molecular function annotations (not shown due to space limitations). Conclusion ========== We demonstrate that by the new way of accounting for multi-scale organization of PPI data both through modeling and new algorithms that we propose, we can uncover substantially more biological information than can be obtained by considering only pairwize interactions between proteins in PPI networks. This pioneering observation can further be utilized to predict biological functions of unnanotated genes, which is a subject of further research. We demonstrate the existence of the functional geometry in the PPI data by capturing the higher-order organization of these molecular networks by using simplicial complexes. To mine the geometry of simplicial complexes, we propose simplets, which generalize graphlets to simplicial complexes. On randomly generated and real-world datasets, we define a sensitive measure of global geometrical similarity between simplicial complexes. Also, we propose a higher-dimensional, simplicial complex-based (SC) model of a species’ interactome, which combines protein-protein-interaction and protein complex data. On human and yeast interactomes, by using clustering based on our new simplet-based measures of geometric similarity and cluster enrichment analysis, we show that our SC models are more biologically coherent than protein-protein interaction networks and that our simplets can efficiently mine this SC model as a new source of biological knowledge. Furthermore, while we focus on simplicial complexes emerging from molecular network organization, our methodology is generic and can be applied to multi-scale datasets from any scientific field, such as the multi-scale network data from physics, social sciences and economy. Funding {#funding .unnumbered} ======= This work was supported by the European Research Council (ERC) Starting Independent Researcher Grant 278212, the European Research Council (ERC) Consolidator Grant 770827, the Serbian Ministry of Education and Science Project III44006, the Slovenian Research Agency project J1-8155 and the awards to establish the Farr Institute of Health Informatics Research, London, from the Medical Research Council, Arthritis Research UK, British Heart Foundation, Cancer Research UK, Chief Scientist Office, Economic and Social Research Council, Engineering and Physical Sciences Research Council, National Institute for Health Research, National Institute for Social Care and Health Research, and Wellcome Trust (grant MR/K006584/1) and UK Medical Research Council (MC\_U12266B). [^1]: natasa@cs.ucl.ac.uk [^2]: <http://geneontology.org/page/ontology-documentation>

--- abstract: 'We consider the problem of optimization of cost functionals on the infinite-dimensional manifold of diffeomorphisms. We present a new class of optimization methods, valid for any optimization problem setup on the space of diffeomorphisms by generalizing Nesterov accelerated optimization to the manifold of diffeomorphisms. While our framework is general for infinite dimensional manifolds, we specifically treat the case of diffeomorphisms, motivated by optical flow problems in computer vision. This is accomplished by building on a recent variational approach to a general class of accelerated optimization methods by Wibisono, Wilson and Jordan [@wibisono2016variational], which applies in finite dimensions. We generalize that approach to infinite dimensional manifolds. We derive the surprisingly simple continuum evolution equations, which are partial differential equations, for accelerated gradient descent, and relate it to simple mechanical principles from fluid mechanics. Our approach has natural connections to the optimal mass transport problem. This is because one can think of our approach as an evolution of an infinite number of particles endowed with mass (represented with a mass density) that moves in an energy landscape. The mass evolves with the optimization variable, and endows the particles with dynamics. This is different than the finite dimensional case where only a single particle moves and hence the dynamics does not depend on the mass. We derive the theory, compute the PDEs for accelerated optimization, and illustrate the behavior of these new accelerated optimization schemes.' author: - 'Ganesh Sundaramoorthi[^1] and Anthony Yezzi[^2]' bibliography: - 'accel.bib' title: 'Accelerated Optimization in the PDE Framework: Formulations for the Manifold of Diffeomorphisms' --- Introduction ============ Accelerated optimization methods have gained wide applicability within the machine learning and optimization communities (e.g., [@Bubeck15; @Flammarion15; @Ghadimi16; @Hu09; @Ji09; @Jojic10; @Krichene15; @Li15; @Nesterov05; @Nesterov08; @Nesterov83]). They are known for leading to optimal convergence rates among schemes that use only gradient (first order) information in the convex case. In the non-convex case, they appear to provide robustness to shallow local minima. The intuitive idea is that by considering a particle with mass that moves in an energy landscape, the particle will gain momentum and surpass shallow local minimum and settle in in more wider, deeper local extrema in the energy landscape. This property has made them (in conjunction with stochastic search algorithms) particularly useful in machine learning, especially in the training of deep networks, where the optimization is a non-convex problem that is riddled with local minima. These methods have so far have only been used in optimization problems that are defined in finite dimensions. In this paper, we consider the generalization of these methods to infinite dimensional manifolds. We are motivated by applications in computer vision, in particular, segmentation, 3D reconstruction, and optical flow. In these problems, the optimization is over infinite dimensional geometric quantities (e.g., curves, surfaces, mappings), and so the problems are formulated on infinite dimensional manifolds. Recently there has been interest within the machine learning community in optimization on finite dimensional manifolds, such as matrix groups, e.g., [@zhang2016riemannian; @liu2017accelerated; @hosseini2017alternative], which have particular structure not available on infinite dimensional manifolds that we consider here. [ ]{} Recent work [@wibisono2016variational] has shown that the continuum limit of accelerated methods, which are discrete optimization algorithms, may be formulated with variational principles. This allows one to derive the continuum limit of accelerated optimization methods (Nesterov’s optimization method [@Nesterov83] and others) as an optimization problem on descent paths. The resulting optimal continuum path is defined by an ODE, which when discretized appropriately yields Nesterov’s method and other accelerated optimization schemes. The optimization problem on paths is an action integral, which integrates the Bregman Lagrangian. The Bregman Lagrangian is a time-explicit Lagrangian (from physics) that consists of kinetic and potential energies. The kinetic energy is defined using the Bregman divergence (see Section \[sec:variational\_accelerated\]); it is designed for finite step sizes, and thus differs from classical action integrals in physics [@arnol2013mathematical; @marsden2013introduction]. The potential energy is the cost function that is to be optimized. We build on the approach of [@wibisono2016variational] by formulating accelerated optimization with an action integral, but we generalize that approach to infinite dimensional manifolds. Our approach is general for infinite dimensional manifolds, but we illustrate the idea here for the case of the infinite dimensional manifold of diffeomorphisms of ${\mathbb{R}}^n$ (the case of the manifold of curves has been recently formulated by the authors [@YezziSun2017]). To do this, we abandon the Bregman Lagrangian framework in [@wibisono2016variational] since that assumes that the variable over which one optimizes is embedded in ${\mathbb{R}}^n$. Instead, we adopt the classical formulation of action integrals in physics [@arnol2013mathematical; @marsden2013introduction], which is already general enough to deal with manifolds, and kinetic energies that are defined through general Riemannian metrics rather than a traditional Euclidean metric, thus by-passing the need for the use of Bregman distances. Our approach requires consideration of additional technicalities beyond that of [@wibisono2016variational] and classical physics. Namely, in finite dimensions in ${\mathbb{R}}^n$, one can think of accelerated optimization as a single particle with mass moving in an energy landscape. Since only a single particle moves, mass is a fixed constant that does not impact the dynamics of the particle. However, in infinite dimensions, one can instead think of an infinite number of particles each moving, and these masses of particles is better modeled with a *mass density*. In the case of the manifold of diffeomorphisms of ${\mathbb{R}}^n$ this mass density exists in ${\mathbb{R}}^n$. As the diffeomorphism evolves to optimize the cost functional, it deforms ${\mathbb{R}}^n$ and redistributes the mass, and so the density changes in time. Since the mass density defines the kinetic energy and the stationary action path depends on the kinetic energy, the dynamics of the evolution to minimize the cost functional depends on the way that mass is distributed in ${\mathbb{R}}^n$. Therefore, in the infinite dimensional case, one also needs to optimize and account for the mass density, which cannot be neglected. Further, our approach, due to the infinite dimensional nature, has evolution equations that are PDEs rather than ODEs in [@wibisono2016variational]. Finally, the discretization of the resulting PDEs requires the use of entropy schemes [@sethian1999level] since our evolution equations are defined as viscosity solutions of PDEs, required to treat shocks and rarefaction fans. These phenomena appear not to be present in the finite dimensional case. Related Work ------------ ### Sobolev Optimization Our work is motivated by Sobolev gradient descent approaches [@Sundaramoorthi05; @beg2005computing; @charpiat2005designing; @sundaramoorthi2007sobolev; @charpiat2007generalized; @sundaramoorthi2008coarse; @sundaramoorthi2009new; @mennucci2008sobolev; @sundaramoorthi2011new; @yang2015shape] for optimization problems on manifolds, which have been used for segmentation and optical flow problems. These approaches are general in that they apply to non-convex problems, and they are derived by computing the gradient of a cost functional with respect to a Sobolev metric rather than an $L^2$ metric typically assumed in variational optimization problems. The resulting gradient flows have been demonstrated to yield coarse-to-fine evolutions, where the optimization automatically transitions from coarse to successively finer scale deformations. This makes the optimization robust to local minimizers that plague $L^2$ gradient descents. We should point out that the Sobolev metric is used beyond optimization problems and have been used extensively in shape analysis (e.g., [@klassen2004analysis; @michor2007metric; @micheli2013sobolev; @bauer2014overview]). While such gradient descents are robust to local minimizers, computing them in general involves an expensive computation of an inverse differential operator at each iteration of the gradient descent. In the case of optimization problems on curves and a very particular form of a Sobolev metric this can be made computationally fast [@sundaramoorthi2007sobolev], but the idea does not generalize beyond curves. In this work, we aim to obtain robustness properties of Sobolev gradient flows, but without the expensive computation of inverse operators. Our accelerated approach involves averaging the gradient across time in the descent process, rather than an averaging across space in the Sobolev case. Despite our goal of avoiding Sobolev gradients for computational speed, we should mention that our framework is general to allow one to consider accelerated Sobolev gradient descents (although we do not demonstrate it here), where there is averaging in both space and time. This can be accomplished by changing the definition of kinetic energy in our approach. This could be useful in applications where added robustness is needed but speed is not a critical factor. ### Optimal Mass Transport Our work relates to the literature on the problem of *optimal mass transportion* (e.g., [@villani2003topics; @gangbo1996geometry; @angenent2003minimizing]), especially the formulation of the problem in [@benamou2000computational]. The modern formulation of the problem, called the Monge-Kantorovich problem, is as follows. One is given two probability densities $\rho_0,\rho_1$ in ${\mathbb{R}}^n$, and the goal is to compute a transformation $M : {\mathbb{R}}^n\to{\mathbb{R}}^n$ so that the pushforward of $\rho_0$ by $M$ results in $\rho_1$ such that $M$ has minimal cost. The cost is defined as the average Euclidean norm of displacement: $\int_{{\mathbb{R}}^n} |M(x)-x|^p \rho_0(x){\,\mathrm{d}}x$ where $p\geq 1$. The value of the minimum cost is a distance (called the $L^p$ Wasserstein distance) on the space of probability measures. In the case that $p=2$, [@benamou2000computational] has shown that mass transport can be formulated as a fluid mechanics problem. In particular, the Wasserstein distance can be formulated as a distance arising from a Riemannian metric on the space of probability densities. The cost can be shown equivalent to the minimum Riemannian path length on the space of probability densities, with the initial and final points on the path being the two densities $\rho_0, \rho_1$. The tangent space is defined to be velocities of the density that infinitesimally displace the density. The Riemannian metric is just the kinetic energy of the mass distribution as it is displaced by the velocity, given by $\int_{{\mathbb{R}}^n} \frac 1 2 \rho(x) |v(x)|^2 {\,\mathrm{d}}x$. Therefore, optimal mass transport computes an optimal *path* on densities that minimizes the integral of kinetic energy along the path. In our work, we seek to minimize a potential on the space of diffeomorphisms, with the use of acceleration. We can imagine that each diffeomorphism is associated with a point on a manifold, and the goal is to move to the bottom of the potential well. To do so, we associate a mass density in ${\mathbb{R}}^n$, which as we optimize the potential, moves in ${\mathbb{R}}^n$ via a push-forward of the evolving diffeomorphism. We regard this evolution as a path in the space of diffeomorphisms that arises from an action integral, where the action is the difference of the kinetic and potential energies. The kinetic energy that we choose, purely to endow the diffeomorphism with acceleration, is the same one used in the fluid mechanics formulation of optimal mass transportation for $p=2$. We have chosen this kinetic energy for simplicity to illustrate our method, but we envision a variety of kinetic energies can be defined to introduce different dynamics. The main difference of our approach to the fluid mechanics formulation of mass transport is in the fact that we do not minimize just the path integral of the kinetic energy, but rather we derive our method by computing stationary paths of the path integral of kinetic minus *potential* energies. Since diffeomorphisms are generated by smooth velocity fields, we equivalently optimize over velocities. We also optimize over the mass distribution. Thus, the main difference between the fluid mechanics formulations of $L^2$ mass transport and our approach is the potential on diffeomorphisms, which is used to define the action integral. ### Diffeomorphic Image Registration Our work relates to the literature on diffeomorphic image registration [@beg2005computing; @miller2006geodesic], where the goal, similar to ours, is to compute a registration between two images as a diffeomorphism. There a diffeomorphism is generated by a path of smooth velocity fields integrated over the path. Rather than formulating an optimization problem directly on the diffeomorphism, the optimization problem is formed on a path of velocity fields. The optimization problem is to minimize $\int_0^1 \| v\|^2 {\,\mathrm{d}}t$ where $v$ is a time varying vector field, $\|\cdot\|$ is a norm on velocity fields, and the optimization is subject to the constraint that the mapping $\phi$ maps one image to the other, i.e., $I_1 = I_0\circ\phi^{-1}$. The minimization can be considered the minimization of an action integral where the action contains only a kinetic energy. The norm is chosen to be a Sobolev norm to ensure that the generated diffeomorphism (by integrating the velocity fields over time) is smooth. The optimization problem is solved in [@beg2005computing] by a Sobolev gradient descent on the *space of paths*. The resulting path is a geodesic with Riemannian metric given by the Sobolev metric $\|v\|$. In [@miller2006geodesic], it is shown these geodesics can be computed by integrating a forward evolution equation, determined from the conservation of momentum, with an initial velocity. Our framework instead uses accelerated gradient descent. Like [@beg2005computing; @miller2006geodesic], it is derived from an action integral, but the action has both a kinetic energy and a *potential* energy, which is the objective functional that is to be optimized. In this current work, our kinetic energy arises naturally from physics rather than a Sobolev norm. One of our motivations in this work is to get regularizing effects of Sobolev norms without using Sobolev norms, since that requires inverting differential operators in the optimization, which is computationally expensive. Our kinetic energy is an $L^2$ metric weighted by *mass*. Our method has acceleration, rather than zero acceleration in [@beg2005computing; @miller2006geodesic], and this is obtained by endowing a diffeomorphism with mass, which is a mass density in ${\mathbb{R}}^n$. This mass allows for the kinetic energy to endow the optimization with dynamics. Our optimization is obtained as the stationary conditions of the action with respect to both velocity and *mass density*. The latter links our approach to optimal mass transport, described earlier. Our physically motivated kinetic energy and in particular the mass consideration allows us to generate diffeomorphisms without the use fo Sobolev norms. We also avoid the inversion of differential operators. ### Optical Flow Although our framework is general in solving any optimization on infinite dimensional manifolds, we demonstrate the framework for optimization of diffeomorphisms and specifically for optical flow problems formulated as variational problems in computer vision (e.g., [@horn1981determining; @black1996robust; @brox2004high; @wedel2009improved; @sun2010secrets; @yang2013modeling; @yang2015self]). Optical flow, i.e., determining pixel-wise correspondence between images, is a fundamental problem in computer vision that remains a challenge to solve, mainly because optical flow is a non-convex optimization problem, and thus few methods exist to optimize such problems. Optical flow was first formulated as a variational problem in [@horn1981determining], which consisted of a data fidelity term and regularization favoring smooth optical flow. Since the problem is non-convex, approaches to solve this problem typically involve the assumption of small displacement between frames, so a linearization of the data fidelity term can be performed, and this results in a problem in which the global optimum of [@horn1981determining] can be solved via the solution of a linear PDE. Although standard gradient descent could be used on the non-linearized problem, it is numerically sensitive, extremely computationally costly, and does not produce meaningful results unless coupled with the strategy described next. Large displacements are treated with two strategies: iterative warping and image pyramids. Iterative warping involves iteration of the linearization around the current accumulated optical flow. By use of image pyramids, a large displacement is converted to a smaller displacement in the downsampled images. While this strategy is successful in many cases, there are also many problems associated with linearization and pyramids, such as computing optical flow of thin structures that undergo large displacements. This basic strategy of linearization, iterative warping and image pyramids have been the dominant approach to many variational optical flow models (e.g., [@horn1981determining; @black1996robust; @brox2004high; @wedel2009improved; @sun2010secrets]), regardless of the regularization that is used (e.g., use of robust norms, total variation, non-local norms, etc). In [@wedel2009improved], the linearized problem with TV regularization has been formulated as a convex optimization problem, in which a primal-dual algorithm can be used. In [@yang2015shape] linearization is avoided and rather a gradient descent with respect to a Sobolev metric is computed, and is shown to have a automatic coarse-to-fine optimization behavior. Despite these works, most optical flow algorithms involve simplification of the problem into a linear problem. In this work, we construct accelerated gradient descent algorithms that are applicable to any variational optical flow algorithm in which we avoid the linearization step and aim to obtain a better optimizer. For illustration, we consider here the case of optical flow modeled as a global diffeomorphism, but in principle this can be generalized to piecewise diffeomorphisms as in [@yang2015self]. Since diffeomorphisms do not form a linear space, rather a infinite-dimensional manifold, we generalize accelerated optimization to that space. Background for Accelerated Optimization on Manifolds ==================================================== Manifolds and Mechanics ----------------------- We briefly summarize the key facts in classical mechanics that are the basis for our accelerated optimization method on manifolds. ### Differential Geometry We review differential geometry (from [@do1992riemannian]), as this will be needed to derive our accelerated optimization scheme on the *manifold* of diffeomorphisms. First a *manifold* $M$ is a space in which every point $p\in M$ has a (invertible) mapping $f_p$ from a neighborhood of $p$ to a *model space* that is a linear normed vector space, and has an additional compatibility condition that if the neighborhoods for $p$ and $q$ overlap then the mapping $f_p\circ f_{q}^{-1}$ is differentiable. Intuitively, a manifold is a space that locally appears flat. The model space may be finite or infinite dimensional when the model spaces are finite or infinite dimensional, respectively. In the latter case the manifold is referred to as an *infinite dimensional manifold* and in the former case a *finite dimensional manifold*. The space of diffeomorphisms of ${\mathbb{R}}^n$, the space of interest in this paper, is an infinite dimensional manifold. The *tangent space* at a point $p\in M$ is the equivalence class, $[\gamma]$, of curves $\gamma : [0,1] \to M$ under the equivalence that $\gamma(0)=p$ and $(f_p\circ \gamma)'(0)$ are the same for each curve $\gamma\in [\gamma]$. Intuitively, these are the set of possible directions of movement at the point $p$ on the manifold. The *tangent bundle*, denoted $TM$, is $TM = \{ (p,v) \,:\, p\in M, v \in T_pM\}$, i.e., the space formed from the collection of all points and tangent spaces. In this paper, we will assume additional structure on the manifold, namely, that an inner product (called the *metric*) exists on each tangent space $T_pM$. Such a manifold is called a *Riemannian manifold*. A Riemannian manifold allows one to formally define the lengths of curves $\gamma : [-1,1]\to M$ on the manifold. This allows one to construct paths of critical length, called *geodesics*, a generalization of a path on constant velocity on the manifold. Note that while existence of geodesics is guaranteed on finite dimensional manifolds, in the infinite dimensional case, there is no such guarantee. The Riemannian metric also allows one to define *gradients* of functions $g : M \to {\mathbb{R}}$ defined on the manifold: the gradient $\nabla g(p) \in T_pM$ is defined to be the vector that satisfies ${\frac{{\,\mathrm{d}}{}}{{\,\mathrm{d}}{\varepsilon}}} \left. g( \gamma(\varepsilon) ) \right|_{ \varepsilon = 0 } = {\left< { \nabla g(p) }, { \gamma'(0) } \right>_{}} $, where $\gamma(0)=p$, the left hand side is the directional derivative and the right hand side is the inner product from the Riemannian structure. ### Mechanics on Manifolds We now briefly review some of the formalism of classical mechanics on manifolds that will be used in this paper. The material is from [@arnol2013mathematical; @marsden2013introduction]. The subject of mechanics describes the principles governing the evolution of a particle that moves on a manifold $M$. The equations governing a particle are Newton’s laws. There are two viewpoints in mechanics, namely the *Lagrangian* and *Hamiltonian* viewpoints, which formulate more general principles to derive Newton’s equations. In this paper, we use the Lagrangian formulation to derive equations of motion for accelerated optimization on the manifold of diffeomorphisms. Lagrangian mechanics obtains equations of motion through *variational principles*, which makes it easier to generalize Newton’s laws beyond simple particle systems in ${\mathbb{R}}^3$, especially to the case of manifolds. In Lagrangian mechanics, we start with a function $L : TM \to {\mathbb{R}}$, called the Lagrangian, from the tangent bundle to the reals. Here we assume that $M$ is a Riemannian manifold. One says that a curve $\gamma : [-1,1] \to M$ is *a motion in a Lagrangian system* with Lagrangian $L$ if it is an extremal of $A = \int L(\gamma(t), \dot{\gamma}(t)) {\,\mathrm{d}}t$. The previous integral is called an *action integral*. *Hamilton’s principle of stationary action* states that the motion in the Lagrangian system satisfies the condition that $\delta A = 0$, where $\delta$ denotes the variation, for *all* variations of $A$ induced by variations of the path $\gamma$ that keep endpoints fixed. The variation is defined as $\delta A := {\frac{{\,\mathrm{d}}{}}{{\,\mathrm{d}}{s}}} \left. A( \tilde \gamma(t,s) ) \right|_{s=0}$ where $\tilde \gamma : [-1,1]^2 \to M$ is a smooth family of curves (a variation of $\gamma$) on the manifold such that $\tilde\gamma(t,0) = \gamma(t)$. The stationary conditions give rise to what is known as *Lagrange’s* equations. A *natural Lagrangian* has the special form $L = T - U$ where $T : TM \to {\mathbb{R}}^{+}$ is the *kinetic energy* and $U : M \to {\mathbb{R}}$ is the *potential energy*. The kinetic energy is defined as $T(v) = \frac 1 2 {\left< {v}, {v} \right>_{}} $ where ${\left< {\cdot}, {\cdot} \right>_{}}$ is the inner product from the Riemannian structure. In the case that one has a particle system in ${\mathbb{R}}^3$, i.e., a collection of particles with masses $m_i$, in a natural Lagrangian system, one can show that Hamilton’s principle of stationary action is equivalent to Newton’s law of motion, i.e., that ${\frac{{\,\mathrm{d}}{}}{{\,\mathrm{d}}{t}}} (m_i \dot r_i) = -{\frac{\partial {U}}{\partial {r_i}}} $ where $r_i$ is the trajectory of the $i^{\text{th}}$ particle, and $\dot{r}_i$ is the velocity. This states that mass times acceleration is the force, which is given by minus the derivative of the potential in a conservative system. Thus, Hamilton’s principle is more general and allows us to more easily derive equations of motion for more general systems, in particular those on manifolds. In this paper, we will consider *Lagrangian non-autonomous systems* where the Lagrangian is also an explicit function of time $t$, i.e., $L : TM \times {\mathbb{R}}\to {\mathbb{R}}$. In particular, the kinetic and potential energies can both be explicit functions of time: $T : TM\times{\mathbb{R}}\to {\mathbb{R}}$ and $U : M \times {\mathbb{R}}\to {\mathbb{R}}$. Autonomous systems have an *energy conservation property* and do not converge; for instance, one can think of a moving pendulum with no friction, which oscillates forever. Since the objective in this paper is to minimize an objective functional, we want the system to eventually converge and Lagrangian non-autonomous systems allow for this possibility. For completness, we present some basic facts of the Hamiltonian perspective to elaborate on the previous point, although we do not use this in the present paper. The generalization of total energy is the *Hamiltonian*, defined as the Legendre transform of the Lagrangian: $H(p,q,t) = {\left< {p}, {\dot{q}} \right>_{}} - L(q,\dot{q}, t)$ where $p = {\frac{{\,\mathrm{d}}{L}}{{\,\mathrm{d}}{\dot{q}}}}$ is the fiber derivative of $L$ with respect to $\dot{q}$, i.e., ${\frac{{\,\mathrm{d}}{L}}{{\,\mathrm{d}}{\dot{q}}}} \cdot w = {\frac{{\,\mathrm{d}}{}}{{\,\mathrm{d}}{\varepsilon}}} \left. L(q, \dot q + \varepsilon w ) \right|_{\varepsilon = 0}$. From the Hamiltonian, one can also obtain a system of equations describing motion on the manifold. It can be shown that if $L=T-U$ then $H=T+U$ and more generally, ${\frac{{\,\mathrm{d}}{H}}{{\,\mathrm{d}}{t}}} = -{\frac{\partial {L}}{\partial {t}}}$ along the stationary path of the action. Thus, if the Lagrangian is natural and autonomous, the total energy is preserved, otherwise energy could be dissipated based on the partial of the Lagrangian with respect to $t$. Variational Approach to Accelerated Optimization in Finite Dimensional Vector Spaces {#sec:variational_accelerated} ------------------------------------------------------------------------------------ Accelerated gradient optimization can be motivated by the desire to make an ordinary gradient descent algorithm 1) more robust to noise and local minimizers, and 2) speed-up the convergence while only using first order (gradient) information. For instance, if one computes a noisy gradient due imperfections in obtaining an accurate gradient, a simple heuristic to make the algorithm more robust is to compute a running average of the gradient over iterations, and use that as the search direction. This also has the advantage, for instance in speeding up optimization in narrow shallow valleys. Gradient descent (with finite step sizes) would bounce back and forth across the valley and slowly descend down, but averaging the gradient could cancel the component across the valley and more quickly optimize the function. Strategic dynamically changing weights on previous gradients can boost the descent rate. Nesterov put forth the following famous scheme [@Nesterov83] which attains an optimal rate of order $\frac{1}{t^{2}}$ in the case of a smooth, convex cost function $f(x)$: $$y_{k+1}=x_{k}-\frac{1}{\beta}\nabla f(x_{k}),\qquad x_{k+1}=(1-\gamma_{k})y_{k+1}+\gamma_{k}y_{k},\qquad\gamma_{k}=\frac{1-\lambda_{k}}{\lambda_{k}+1},\qquad\lambda_{k}=\frac{1+\sqrt{1+4\lambda_{k-1}^{2}}}{2}$$ where $x_{k}$ is the $k$-th iterate of the algorithm, $y_{k}$ is an intermediate sequence, and $\gamma_{k}$ are dynamically updated weights. Recently [@wibisono2016variational] presented a variational generalization of Nesterov’s [@Nesterov83] and other accelerated gradient descent schemes in $\mathbb{R}^{n}$ based on the Bregman divergence of a convex distance generating function $h$: $$d(y,x)=h(y)-h(x)- \nabla h(x)\cdot (y-x) \label{eq:breg-divergence}$$ and careful discretization of the Euler-Lagrange equations for the time integral of the following Bregman Lagrangian $$L(X,V ,t)=e^{a(t)+\gamma(t)}\left[d(X+e^{-a(t)}V,X)-e^{b(t)} U(X)\right]$$ where the potential energy $U$ represents the cost to be minimized. In the Euclidean case where $h(x)=\frac 1 2 |x|^2$ gives $d(y,x)=\frac{1}{2}|y-x|^{2}$, this simplifies to $$L=e^{\gamma(t)}\left[ e^{-a(t)} \frac{1}{2}|V|^{2}-e^{a(t)+b(t)}U(X)\right]$$ where $T=\frac 1 2 |V|^2$ is the kinetic energy of a unit mass particle in $\mathbb{R}^{n}$. Nesterov’s methods [@Nesterov83; @Nesterov14; @Nesterov13; @Nesterov08; @Nesterov06; @Nesterov05] belong to a subfamily of Bregman Lagrangians with the following choice of parameters (indexed by $k>0$) $$a=\log k-\log t,\qquad b=k\,\log t+\log\lambda,\qquad\gamma=k\,\log t$$ which, in the Euclidean case, yields a non-autonomous Lagrangian as follows: $$L=\frac{t^{k+1}}{k}\left( T-\lambda k^{2}t^{k-2} U \right)\label{eq:time-action}$$ In the case of $k=2$, for example, the stationary conditions of the integral of this time-explicit action yield the continuum limit of Nesterov’s accelerated mirror descent [@Nesterov05] derived in both [@Su14; @Krichene15]. Since the Bregman Lagrangian assumes that the underlying manifold is a subset of ${\mathbb{R}}^n$ (in order to define the Bregman distance[^3]), which many manifolds do not have - for instance the manifold of diffeomorphisms that we consider in this paper, we instead use the original classical mechanics formulation, which already provides a formalism for considering general metrics though the Riemannian distance, although not equivalent to the Bregman distance. Accelerated Optimization for Diffeomorphisms ============================================ In this section, we use the mechanics of particles on manifolds developed in the previous section, and apply it to the case of the infinite-dimensional manifold of diffeomorphisms in ${\mathbb{R}}^n$ for general $n$. This allows us to generalize accelerated optimization to infinite dimensional manifolds. Diffeomorphisms are smooth mappings $\phi : {\mathbb{R}}^n \to {\mathbb{R}}^n$ whose inverse exists and is also smooth. Diffeomorphisms form a group under composition. The inverse operator on the group is defined as the inverse of the function, i.e., $\phi^{-1}(\phi(x)) = x$. Here smoothness will mean that two derivatives of the mapping exist. The group of diffeomorphisms will be denoted $\mbox{Diff}({\mathbb{R}}^n)$. Diffeomorphisms relate to image registration and optical flow, where the mappings between two images are often modeled as diffeomorphisms[^4]. Recovering diffeomorphisms from two images will be formulated as an optimization problem $U(\phi)$ where $U$ will correspond to the potential energy. Note we avoid calling $U$ the energy as is customary in computer vision literature, because for us the energy will refer to the total mechanical energy (i.e., the sum of the kinetic and potential energies). We do not make any assumptions on the particular form of the potential in this section, as our goal is to be able to accelerate *any* optimization problem for diffeomorphisms, given that one can compute a gradient of the potential. The formulation here allows any of the numerous cost functionals developed over the past three decades for image registration to be accelerated. In the first sub-section, we give the formulation and evolution equations for the case of acceleration without energy dissipation (Hamiltonian is conserved), since most of the calculations are relevant for the case of energy dissipation, which is needed for the evolution to converge to a diffeomorphism. In the second sub-section, we formulate and compute the evolution equations for the energy dissipation case, which generalizes Nesterov’s method to the infinite dimensional manifold of diffeomorphisms. Finally, in the last sub-section we give an example potential and its gradient calculation for a standard image registration or optical flow problem. Acceleration Without Energy Dissipation --------------------------------------- ### Formulation of the Action Integral Since the potential energy $U$ is assumed given, in order to formulate the action integral in the non-dissipative case, we need to define kinetic energy $T$ on the space of diffeomorphisms. Since diffeomorphisms form a manifold, we can apply the the results in the previous section and note that the kinetic energy will be defined on the tangent space to $\mbox{Diff}({\mathbb{R}}^n)$ at a particular diffeomorphism $\phi$. This will be denoted $T_{\phi}\mbox{Diff}({\mathbb{R}}^n)$. The tangent space at $\phi$ can be roughly thought of as the set of local perturbations $v$ of $\phi$ given for all $\varepsilon$ small that perserve the diffeomorphism property, i.e., $\phi + \varepsilon v$ is a diffeomorphism. One can show that the tangent space is given by $$T_{\phi} \mbox{Diff}({\mathbb{R}}^n) = \{ v : \phi({\mathbb{R}}^n) \to {\mathbb{R}}^n \,:\, v \mbox{ is smooth } \}.$$ In the above, since $\phi$ is a diffeomorphism, we have that $\phi({\mathbb{R}}^n) = {\mathbb{R}}^n$. However, we write $v : \phi({\mathbb{R}}^n) \to {\mathbb{R}}^n$ to emphasize that the velocity fields in the tangent space are defined on the range of $\phi$, so that $v$ is interpreted as a Eulerian velocity. By definition of the tangent space, an infinitesimal perturbation of $\phi$ by a tangent vector, given by $\phi + \varepsilon v$, will be a diffeomorphism for $\varepsilon$ sufficiently small. Note that the previous operation of addition is defined as follows: $$( \phi + \varepsilon v )(x) = \phi(x) + \varepsilon v(\phi(x)).$$ The tangent space is a set of smooth vector fields on $\phi({\mathbb{R}}^n)$ in which the vector field at each point $\phi(x)$, displaces $\phi(x)$ infinitesimally by $v(\phi(x))$ to form another diffeomorphism. We note a classical result from [@ebin1970groups], which will be of utmost importance in our derivation of accelerated optimization on ${\mbox{Diff}({\mathbb{R}}^n)}$. The result is that any (orientable) diffeomorphism may be generated by integrating a time-varying smooth vector field over time, i.e., $$\label{eq:phi_evol} \partial_t \phi_t(x) = v_t( \phi_t(x) ), \quad x\in {\mathbb{R}}^n,$$ where $\partial_t$ denotes partial derivative with respect to $t$, $\phi_t$ denotes a time varying family of diffeomorphisms evaluated at the time $t$, and $v_t$ is a time varying collection of vector fields evaluated at time $t$. The path $t \to \phi_t(x)$ for a fixed $x$ represents a trajectory of a particle starting at $x$ and flowing according to the velocity field. The space on which the kinetic energy is defined is now clear, but one more ingredient is needed before we can define the kinetic energy. Any accelerated method will need a notion of *mass*, otherwise acceleration is not possible, e.g., a mass-less ball will not accelerate. We generalize the concept of mass to the infinite dimensional manifold of diffeomorphisms, where there are infinitely more possibilities than a single particle in the finite dimensional case considered by [@wibisono2016variational]. There optimization is done on a finite dimensional space, the space of a *single* particle, and the possible choices of mass are just different fixed constants. The choice of the constant, given the particle’s mass remains fixed, is irrelevant to the final evolution. This is different in than the case of diffeomorphisms. Here we imagine that an infinite number of particles densely distributed in ${\mathbb{R}}^n$ with mass exist and are displaced by the velocity field $v$ at every point. Since the particles are densely distributed, it is natural to represent the mass of all particles with a *mass density* $\rho : {\mathbb{R}}^n \to {\mathbb{R}}$, similar to a fluid at a fixed time instant. The density $\rho$ is defined as mass divided by volume as the volume shrinks. During the evolution to optimize the potential $U$, the particles are displaced continuously and thus the density of these particles will in general change over time. Note the density will change even if the density at the start is constant except in the case of full translation motion (when $v$ is spatially constant). The latter case is not general enough, as we want to capture general diffeomorphisms. We will assume that the system of particles in ${\mathbb{R}}^n$ is closed and so we impose a *mass preservation constraint*, i.e., $$\label{eq:mass_conserve} \int_{{\mathbb{R}}^n} \rho(x) {\,\mathrm{d}}x = 1,$$ where we assume the total mass is one without loss of generality. Note that the evolution of a time varying density $\rho_t$ as it is deformed in time by a time varying velocity is given by the *continuity equation*, which is a local form of the conservation of mass given by . The continuity equation is defined by the partial differential equation $$\label{eq:continuity_eqn} \partial_t \rho(x) + {\mbox{div}\left( {\rho(x) v(x)} \right)} = 0, \quad x\in {\mathbb{R}}^n$$ where ${\mbox{div}\left( {} \right)}$ denotes the divergence operator acting on a vector field and is ${\mbox{div}\left( {F} \right)} = \sum_{i=}^n \partial_{x_i} F^i$ where $\partial_{x_i}$ is the partial with respect to the $i^{\text{th}}$ coordinate and $F^i$ is the $i^{\text{th}}$ component of the vector field. We will assume that the mass distribution dies down to zero outside a compact set so as to avoid boundary considerations in our derivations. We now have the two ingredients, namely the tangent vectors to ${\mbox{Diff}({\mathbb{R}}^n)}$ and the concept of mass, which allows us to define a natural physical extension of the kinetic energy to the case of an infinite mass distribution. We present one possible kinetic energy to illustrate the idea of accelerated optimization, but this is by no means the only definition of kinetic energy. We envision this to be part of the design process in which one could get a multitude of various different accelerated optimization schemes by defining different kinetic energies. Our definition of kinetic energy is just the kinetic energy arising from fluid mechanics: $$T(v) = \int_{ \phi({\mathbb{R}}^n) } \frac 1 2 \rho(x) |v(x)|^2 {\,\mathrm{d}}x,$$ which is just the integration of single particle’s kinetic energy $\frac 1 2 m |v|^2$ and matches the definition of the kinetic energy of a sum of particles in elementary physics. Note that the kinetic energy is just one-half times the norm squared for the norm arising from the Riemannian metric [@arnol2013mathematical], i.e., an inner product on the tangent space of ${\mbox{Diff}({\mathbb{R}}^n)}$. The Riemannian metric is given by ${\left< {v_1}, {v_2} \right>_{}} = \int_{{\mathbb{R}}^n} \rho(x) v_1(x) \cdot v_2(x) {\,\mathrm{d}}x$, which is just a weighted $\mathbb{L}^2$ inner product. We are now ready to define the action integral for the case of ${\mbox{Diff}({\mathbb{R}}^n)}$, which is defined on *paths* of diffeomorphisms. A path of diffeomorphisms is $\phi : [0,\infty) \times {\mathbb{R}}^n \to {\mathbb{R}}^n$ and we will denote the diffeomorphism at time $t$ along this path as $\phi_t$. Since diffeomorphisms are generated by velocity fields, we may equivalently define the action in terms of *paths* of velocity fields. A path of velocity fields is given by $v : [0,\infty) \times {\mathbb{R}}^n \to {\mathbb{R}}^n$, and the velocity at time $t$ along the path is denoted $v_t$. Notice that the action requires a kinetic energy and the kinetic energy is dependent on the mass density. Thus, a path of densities $\rho : [0,\infty) \times {\mathbb{R}}^n \to {\mathbb{R}}^+$ is required, which represents the mass distribution of the particles in ${\mathbb{R}}^n$ as they are deformed along time by the velocity field $v_t$. This path of densities is subject to the continuity equation. With this, the action integral is then $$\label{eq:action} A = \int \left[ T(v_t) - U(\phi_t) \right] {\,\mathrm{d}}t,$$ where the integral is over time, and we do not specify the limits of integration as it is irrelevant as the endpoints will be fixed and the action will be thus independent of the limits. Note that the action is implicitly a function of three paths, i.e., $v_t,\phi_t$ and $\rho_t$. Further, these paths are not independent of each other as $\phi_t$ depends on $v_t$ through the generator relation , and $\rho_t$ depends on $v_t$ through the continuity equation . ### Stationary Conditions for the Action We now derive the stationary conditions for the action integral , and thus the evolution equation for a path of diffeomorphisms, which is Hamilton’s principle of stationary action, equivalent to a generalization of Newton’s laws of motion extended to diffeomorphisms. As discussed earlier, we would like to find the stationary conditions for the action integral , defined on the path $\phi_t$, under the conditions that it is generated by a path of smooth velocity fields $v_t$, which is also coupled with the mass density $\rho_t$. We treat the computation of the stationary conditions of the action as a constrained optimization problem with respect to the two aforementioned constraints. To do this, it is easier to formulate the action in terms of the path of the inverse diffeomorphisms $\phi^{-1}_t$, which we will call $\psi_t$. This is because the non-linear PDE constraint can be equivalently reformulated as the following linear transport PDE in the inverse mappings: $$\label{eq:transport_backward} \partial_t \psi_t(x) + [D\psi_t(x)]v_t(x) = 0, \quad x\in {\mathbb{R}}^n$$ where $D$ denotes the derivative (Jacobian) operator. To derive the stationary conditions with respect to the constraints, we use the method of Lagrange multipliers. We denote by $\lambda : [0,\infty) \times {\mathbb{R}}^n \to {\mathbb{R}}^n$ the Lagrange multiplier according to . We denote $\mu : [0,\infty) \times {\mathbb{R}}^n \to {\mathbb{R}}$ as the Lagrange multiplier for the continuity equation . Because we would like to be able to have possibly discontinuous solutions of the continuity equation, we formulate it in its weak form by multiplying the constraint by the Lagrange multiplier and integrating by parts thereby removing the derivatives on possibly discontinuous $\rho$: $$\int \int_{{\mathbb{R}}^n} \mu \left[ \partial_t \rho + {\mbox{div}\left( { pv } \right)} \right] {\,\mathrm{d}}x {\,\mathrm{d}}t = -\int \int_{{\mathbb{R}}^n} \left[ \partial_t \mu + \nabla \mu \cdot v \right] \rho {\,\mathrm{d}}x {\,\mathrm{d}}t,$$ where $\nabla$ denotes the spatial gradient operator. Notice that we ignore the boundary terms from integration by parts as we will eventually compute stationary conditions, and we are assuming fixed initial conditions for $\rho_0$ and we assume that $\rho_{\infty}$ converges and thus cannot be perturbed when computing the variation of the action integral. With this, we can formulate the action integral with Lagrange multipliers as $$\begin{aligned} \label{eq:action_lagrange} A &= \int \left[ T(v) - U(\phi) \right] {\,\mathrm{d}}t + \int \int_{{\mathbb{R}}^n} \lambda^T[ \partial_t \psi + (D\psi)v ] {\,\mathrm{d}}x {\,\mathrm{d}}t -\int \int_{{\mathbb{R}}^n} \left[ \partial_t \mu + \nabla \mu \cdot v \right] \rho {\,\mathrm{d}}x {\,\mathrm{d}}t,\end{aligned}$$ where we have omitted the subscripts to avoid cluttering the notation. Notice that the potential $U$ is now a function of $\psi$, and the action depends now on $\rho, \psi, v$ and the Lagrange multipliers $\mu, \lambda$. We now compute variations of $A$ as we perturb the paths by variations $\delta \rho$, $\delta v$ and $\delta \phi$ along the paths. The variation with respect to $\rho$ is defined as $\delta A \cdot \delta \rho = \left. {\frac{{\,\mathrm{d}}{}}{{\,\mathrm{d}}{\varepsilon}}} A(\rho + \varepsilon \delta \rho, v, \psi) \right|_{\varepsilon =0 }$, and the other variations are defined in a similar fashion. By computing these variations, we get the following stationary equations: \[thrm:stationary\_lagrange\_mult\] The stationary conditions of the path for the action are $$\begin{aligned} \partial_t \lambda + (D\lambda )v + \lambda {\mbox{div}\left( {v} \right)} &= (\nabla\psi)^{-1}\nabla U(\phi)\\ \rho v + (\nabla \psi) \lambda - \rho\nabla\mu &= 0 \\ \partial_t \mu + \nabla\mu \cdot v &= \frac 1 2 |v|^2 \end{aligned}$$ where $\nabla U(\phi)\in T_{\phi}{\mbox{Diff}({\mathbb{R}}^n)}$ denotes the functional gradient of $U$ with respect to $\phi$ (see Appendix \[app:funct\_grads\]), and $\nabla \mu, \nabla \psi$ are spatial gradients. The original constraints on the mapping and the continuity equation are part of the stationary conditions. See Appendix \[app:stat\_cond\_nondissip\]. While the previous theorem does give the stationary conditions and evolution of the Lagrange multipliers, in order to define a forward evolution method where the initial conditions for the density, mapping and velocity are given, we would need initial conditions for the Lagrange multipliers, which are not known from the calculation leading to Theorem \[thrm:stationary\_lagrange\_mult\]. Therefore, we will now eliminate the Lagrange multipliers and rewrite the evolution equations in terms of forward equations for the velocity, mapping and density. This leads to the following theorem: \[thrm:evol\_final\_non\_dissip\] The stationary conditions for the path of the action integral subject to the constraints on the mapping and the continuity equation are given by the forward evolution equation $$\label{eq:evol_velocity} \partial_t v = -(Dv)v -\frac{1}{\rho} \nabla U(\phi),$$ which describes the evolution of the velocity. The forward evolution equation for the diffeomorphism is given by , that of its inverse mapping is given by , and the forward evolution of its density is given by . See Appendix \[app:velocity\_evol\]. The left hand side of the equation (along with the continuity equation) is the left hand side of the *compressible Euler Equation* [@marsden2013introduction], which describes the motion of a perfect fluid (i.e., assuming no heat transfer or viscous effects). The difference is that the right hand side in is the gradient of the potential, which we seek to optimize, that depends on the diffeomorphism that is the integral of the velocity over time, rather than the gradient of pressure that is purely a function of density in the Euler equations. With this theorem, it is now possible to numerically compute the stationary path of the action, by starting with initial conditions on the density, mapping and velocity. The velocity is updated by , the mapping is then updated by , and the density is updated by . Note that the density at each time impacts the velocity as seen in . These equations are a set of coupled partial differential equations. They describe the path of stationary action when the action integral does not arise from a system that has dissipative forces. Notice the velocity evolution is a natural analogue of Newton’s equations. Indeed, if we consider the material derivative, which describes the time rate of change of a quantity subjected to a time dependent velocity field, then one can write the velocity evolution as follows. The velocity evolution derived as the critical path of the action integral is $$\label{eq:newton_law} \rho\frac{Dv}{Dt} = -\nabla U(\phi),$$ where $\frac{Df}{Dt} := \partial_t f + (Df)v$ is the material derivative. This is consequence of the definition of material derivative. The material derivative is obtained by taking the time derivative of $f$ along the path $t\to \phi(t,x)$, i.e., ${\frac{{\,\mathrm{d}}{}}{{\,\mathrm{d}}{t}}} f(t,\phi(t,x))$. Therefore, $Dv/Dt$ is the derivative of velocity along the path. The equation says the time rate of change of velocity times density is equation to minus the gradient of the potential, which is Newton’s 2nd law, i.e., the mass times acceleration is equal to the force, which is given by the gradient of the potential in a conservative system. The evolution described by the equations above will not converge. This is because the total energy is conserved, and thus the system will oscillate over a (local) minimum of the potential $U$, forever, unless the initialization is at a stationary point of the potential $U$. In practice, due to discretization of the equations, which require entropy preserving schemes [@sethian1999level], the implementation will dissipate energy and the evolution equations eventually converge. ### Viscosity Solution and Regularity An important question is whether the evolution equations given by Theorem \[thrm:evol\_final\_non\_dissip\] maintain that the mapping $\phi_t$ remains a diffeomorphism given that one starts the evolution with a diffeomorphism. This is of course important since all of the derivations above were done assuming that $\phi$ is a diffeomorphism, moreover for many applications one wants to maintain a diffeomorphic mapping. The answer is affirmative since to define a solution of , we define the solution as the *viscosity solution* (see e.g., [@crandall1983viscosity; @rouy1992viscosity; @sethian1999level]). The viscosity solution is defined as the limit of the equation with a diffusive term of the velocity added to the right hand side, as the diffusive coefficient goes to zero. More precisely $$\label{eq:evol_velocity_visc} \partial_t v_{\varepsilon} = -(Dv_{\varepsilon})v_{\varepsilon} + \varepsilon \Delta v_{\varepsilon} -\frac{1}{\rho} \nabla U(\phi),$$ where $\Delta$ denotes the spatial Laplacian, which is a smoothing operator. This leads to a smooth ($C^{\infty}$) solution due to the known smoothing properties of the Laplacian. The viscosity solution is then $v = \lim_{\varepsilon\to 0} v$. In practice, we do not actually add in the diffusive term, but rather approximate the effects with small $\varepsilon$ by using entropy conditions in our numerical implementation. One may of course add the diffusive term to induce more regularity into the velocity and thus into the mapping $\phi$. Since the velocity is smooth ($C^{\infty}$), the integral of a smooth vector field will result in a diffeomorphism [@ebin1970groups]. ### Discussion An important property of these evolution equations, when compared to virtually all previous image registration and optical flow methods is the lack of need to compute inverses of differential operators, which are global smoothing operations, and are expensive. Typically, in optical flow (such as the classical Horn & Schunck [@horn1981determining]) or LDDMM [@beg2005computing] where one computes Sobolev gradients, one needs to compute inverses of differential operators, which are expensive. Of course one could perform standard gradient descent, which does not typically require computing inverses of differential operators, but gradient descent is known not to be feasible and it is hard to numerically implement without significant pre-processing, and easily gets stuck in what are effectively numerical local minima. The equations in Theorem \[thrm:evol\_final\_non\_dissip\] are all local, and experiments suggest they are not susceptible to the problems that plague gradient descent. ### Constant Density Case We now analyze the case when the density $\rho$ is chosen to be a fixed constant, and we derive the evolution equations. In this case, the kinetic energy simplifies as follows $$\label{eq:kinetic_const_density} T(v) = \frac {\rho}{ 2 }\int_{ \phi({\mathbb{R}}^n) } |v(x)|^2 {\,\mathrm{d}}x.$$ We can define the action integral as before with the previous definition of kinetic energy, and we can derive the stationary conditions by defining the following action integral incorporating the mapping constraint . This gives the modified action integral as $$\begin{aligned} \label{eq:action_lagrange_const_density} A &= \int \left[ T(v) - U(\phi) \right] {\,\mathrm{d}}t + \int \int_{{\mathbb{R}}^n} \lambda^T[ \partial_t \psi + (D\psi)v ] {\,\mathrm{d}}x {\,\mathrm{d}}t.\end{aligned}$$ Note that the continuity equation is no longer imposed as a constraint as the density is treated as a fixed constant. This leads to the following stationary conditions. \[thrm:stationary\_lagrange\_mult\_const\_density\] The stationary conditions of the path for the action are $$\begin{aligned} \partial_t \lambda + (D\lambda )v + \lambda {\mbox{div}\left( {v} \right)} &= (\nabla\psi)^{-1}\nabla U(\phi)\\ \rho v + (\nabla \psi) \lambda &= 0 \end{aligned}$$ where $\nabla U(\phi)\in T_{\phi}{\mbox{Diff}({\mathbb{R}}^n)}$ denotes the functional gradient of $U$ with respect to $\phi$, and $\nabla \psi$ are spatial gradients. The original constraint on the mapping is part of the stationary conditions. The computation is similar to the non-constant density case Appendix \[app:stat\_cond\_nondissip\]. Note that stationary condition with respect to the mapping remains the same as the density constraint in the non-constant density case does not depend on the mapping. The stationary condition with respect to the velocity avoids the variation with respect to the density constraint in the non-constant density case, and remains the same except for the last term. As before, we can solve for the velocity evolution directly. This results in the following result. \[thrm:evol\_final\_non\_dissip\_const\_density\] The stationary conditions for the path of the action integral with kinetic energy subject to the constraint on the mapping is given by the forward evolution equation $$\label{eq:evol_velocity_const_density} \partial_t v = -(Dv)v -(\nabla v)v -v{\mbox{div}\left( {v} \right)} - \frac{1}{\rho} \nabla U(\phi)$$ The forward evolution equation for the diffeomorphism is given by , and that of its inverse mapping is given by . We can apply Lemma \[lem:lambda\_t\_w\_t\] in Appendix \[app:velocity\_evol\] with $w = -\frac{1}{\rho} v$. The former equation (without the potential term) is known as the Euler-Poincaré equation (EPDiff), the geodesic equation for the diffeomorphism group under the $L^2$ metric [@miller2006geodesic]. This shows that one relationship between Euler’s equation and EPDiff is that Euler’s equation is derived by a time-varying density in the kinetic energy, which is optimized over the mass distribution along with the velocity whereas EPDiff assumes a constant mass density in the kinetic energy. The non-constant density model (arising in Euler’s equation) has a natural interpretation in terms of Newton’s equations. Acceleration with Energy Dissipation ------------------------------------ We now present the case of deriving the stationary conditions for a system on the manifold of diffeomorphisms in which total energy dissipates. This is important so the system will converge to a local minima, and not oscillate about a local minimum forever, as the evolution equations from the previous section. To do this, we consider time varying scalar functions $a, b : [0,\infty) \to {\mathbb{R}}^+$, and define the action integral, again defined on paths of diffeomorphisms, as follows: $$\label{eq:action_diss} A = \int \left[ a_t T(v_t) - b_t U(\phi_t) \right] {\,\mathrm{d}}t,$$ where $a_t, b_t$ denote the values of the scalar at time $t$. We may again go through finding the stationary conditions subject to the mapping constraint and the continuity equation constraint , with Lagrange multiplier and then derive the forward evolution equations. The final result is as follows: \[thrm:evol\_final\_dissip\] The stationary conditions for the path of the action integral subject to the constraints on the mapping and the continuity equation are given by the forward evolution equation $$\label{eq:evol_velocity_dissp} a \partial_t v + a(Dv)v + (\partial_ta)v= -\frac{b}{\rho} \nabla U(\phi),$$ which describes the evolution of the velocity. The same evolution equations as Theorem \[thrm:evol\_final\_non\_dissip\] for the mappings and , and density hold . See Appendix \[app:stationary\_dissip\]. If we consider certain forms of $a$ and $b$, then one can arrive at various generalizations of Nesterov’s schemes. In particular, the choice of $a$ and $b$ below are those considered in [@wibisono2016variational] to explain various versions of Nesterov’s schemes, which are optimization schemes in finite dimensions. If we choose $$a_t = e^{\gamma_t-\alpha_t} \quad \mbox{and} \quad b_t = e^{\alpha_t+\beta_t+\gamma_t}$$ where $$\alpha_t = \log p - \log t, \quad \beta_t = p\log t + \log C, \quad \gamma_t = p\log t,$$ $C>0$ is a constant, and $p$ is a positive integer, then we will arrive at the evolution equation $$\partial_t v = -\frac {p+1}{t} v - (Dv)v - \frac{1}{\rho}Cp^2 t^{p-2}\nabla U(\phi).$$ In the case $p=2$ and $C=1/4$ the evolution reduces to $$\label{eq:vel_evol_dissp_nesterov} \partial_t v = -\frac {3}{t} v - (Dv)v - \frac{1}{\rho} \nabla U(\phi).$$ The case $p=2$ was considered in [@wibisono2016variational] as the continuum equivalent to Nesterov’s original scheme in finite dimensions. We can notice that this evolution equation is the same as the evolution equations for the non-dissipative case , except for the term $-(3/t) v$. One can interpret the latter term as a frictional dissipative term, analogous to viscous resistance in fluids. Thus, even in this case the equation has a natural interpretation that arises from Newton’s laws. Second Order PDE for Acceleration --------------------------------- We now convert the system of PDE for the forward mapping and velocity into a second order PDE in the forward mapping itself. Interestingly, this eliminates the non-linearity from the non-potential terms. The accelerated optimization, arising from the stationarity of the action integral , given by the system of PDE defined by and the forward mapping is $$a{\frac{\partial {^2 \phi}}{\partial {t^2}}} + (\partial_t a){\frac{\partial {\phi}}{\partial {t}}} + \frac{b}{\rho_0} \widetilde{\nabla} U(\phi) = 0,$$ where $\rho_0$ is the initial density, $\widetilde{\nabla} U(\phi) = [\nabla U(\phi)\circ\phi]\det{\nabla\phi}$ is the gradient defined on the un-warped domain, i.e., $\delta A \cdot \delta \phi = \int_{{\mathbb{R}}^n} \widetilde{\nabla} U(\phi)(x) \cdot \delta \phi(x) {\,\mathrm{d}}x$ is satisfied for all perturbations $\delta \phi$ of $\phi$. We differentiate the definition of the forward mapping in time to obtain and substituting the velocity evolution : $$\begin{aligned} \partial_{tt} \phi &= (\partial_t v)\circ \phi + [ (Dv)\circ\phi] \partial_t \phi \\ &= -[ (Dv)\circ\phi ] v\circ\phi - \frac{\partial_t a}{a} v\circ\phi - \frac{b}{a} \frac{1}{\rho\circ\phi} \nabla U(\phi)\circ\phi + [ (Dv)\circ\phi ]\partial_t \phi \\ &= - \frac{\partial_t a}{a} \partial_t \phi - \frac{b}{a} \frac{1}{\rho\circ\phi} \nabla U(\phi)\circ\phi \end{aligned}$$ We note the following for any $B\subset {\mathbb{R}}^n$, because of mass preservation, we have that $$\int_{B} \rho_0(x) {\,\mathrm{d}}x = \int_{\phi(B)} \rho_t(y) {\,\mathrm{d}}y = \int_{B} \rho_t(\phi(x)) \det{\nabla\phi(x)} {\,\mathrm{d}}x,$$ where the last equality is obtained by a change of variables. Since we can take $B$ arbitrarily small, $\rho_0(x) = \rho_t(\phi(x)) \det{\nabla\phi(x)}$. Using this last formula, we see that $$\frac{1}{\rho\circ\phi} \nabla U(\phi)\circ\phi = \frac{1}{\rho_0} \widetilde{\nabla} U(\phi),$$ which proves the proposition. Illustrative Potential Energy for Diffeomorphisms ------------------------------------------------- We now consider a standard potential for illustrative purposes in simulations, and derive the gradient. The objective is for the evolution equations in the previous section to minimize the potential, which is a function of the mapping. Our evolution equations in the previous section are general and work with *any* potential; our purpose in this section is not to advocate a particular potential, but to show how the gradient of the potential is computed so that it can be used in the evolution equations in the previous section. We consider the standard Horn & Schunck model for optical flow defined as $$\label{eq:potential_HS} U(\phi) = \frac 1 2 \int_{{\mathbb{R}}^n} |I_1(\phi(x)) - I_0(x)|^2 {\,\mathrm{d}}x + \frac 1 2 \alpha \int_{{\mathbb{R}}^n} |\nabla (\phi(x) - x )|^2 {\,\mathrm{d}}x,$$ where $\alpha>0$ is a weight, and $I_0, I_1$ are images. The first term is the data fidelity which measures how close $\phi$ deforms $I_1$ back to $I_0$ through the squared norm, and the second term penalizes non-smoothness of the displacement field, given by $\phi(x)-x$ at the point $x$. Notice that the potential is a function of only the mapping $\phi$, and not the velocity. We now compute the functional gradient of $U$ with respect to the mapping $\phi$, denoted by the expression $\nabla U(\phi)$. This gradient is defined by the relation (see Appendix \[app:funct\_grads\]) $\delta U \cdot \delta \phi = \int_{\phi({\mathbb{R}}^n)} \nabla U(\phi) \cdot \delta \phi {\,\mathrm{d}}x $, i.e., the functional gradient satisfies the relation that the $\mathbb{L}^2$ inner product of it with any perturbation $\delta \phi$ of $\phi$ is equal to the variation of the potential $U$ with respect to the perturbation $\delta \phi$. With this definition, one can show that (see Appendix \[app:funct\_grads\]) $$\label{eq:potential_illustrative} \nabla U(\phi) = \left[ ( I_1 - I_0\circ\psi ) \nabla I_1 - \alpha (\Delta \phi)\circ\psi \right] \det \nabla \psi,$$ where $\det$ denotes the determinant. We can also see that the gradient defined on the un-warped domain is $$\widetilde{\nabla} U(\phi) = ( I_1\circ\phi - I_0 ) \nabla I_1\circ\phi - \alpha \Delta \phi,$$ therefore, the generalization of Nesterov’s method on the original domain itself, in this case is $${\frac{\partial {^2 \phi}}{\partial {t^2}}} + \frac 3 t {\frac{\partial {\phi}}{\partial {t}}} - \frac{\alpha}{\rho_0} \Delta \phi + \frac{1}{\rho_0} ( I_1\circ\phi - I_0 ) \nabla I_1\circ\phi = 0,$$ which is a damped *wave equation*. Experiments =========== We now show some examples to illustrate the behavior of our generalization of accelerated optimization to the infinite dimensional manifold of diffeomorphisms. We compare to standard (Riemannian $L^2$) gradient descent to illustrate how much one can gain by incorporating acceleration, which requires little additional effort over gradient descent. Over gradient descent, acceleration requires only to update the velocity by the velocity evolution in the previous section, and the density evolution. Both these evolutions are cheap to compute since they only involve local updates. Note the gradient descent of the potential $U$ is given by choosing $v = -\nabla U(\phi)$, the other evolution equation for the mapping $\phi$ and $\psi$ remains the same, and no density evolution is considered. We note that we implement the equations as they are, and there is no additional processing that is now common in optical flow methods (e.g., no smoothing images nor derivatives, no special derivative filters, no multi-scale techniques, no use of robust norms, median filters, etc). Although our equations are for diffeomorphisms on all of ${\mathbb{R}}^n$, in practice be have finite images, and the issue of boundary conditions come up. For simplicity to illustrate our ideas, we choose periodic boundary conditions. We should note that our numerical scheme (see Appendix \[app:discretization\]) for implementing accelerated gradient descent is quite basic and not final, and a number of speed ups and / or refinements to the numerics can be done, which we plan to explore in the near future. Thus, at this point we do not compare the method to current optical flow techniques since our numerics are not finalized. Our intention is to show the promise of acceleration and that simply by using acceleration, one can get an impractical algorithm (gradient descent) to become practical, especially with respect to speed. In all the experiments, we choose the step size to satisfy CFL conditions. For ordinary gradient descent we choose $\Delta t < 1/(4\alpha)$, for accelerated gradient descent we have the additional evolution of the velocity , and our numerical scheme has CFL condition $\Delta t < 1/\max_{x\in\Omega} \{ |v(x)|, |Dv(x)| \}$. Also, because there is a diffusion according to regularity, we found that $\Delta t < 1/(4\alpha \cdot \max_{x\in\Omega} \{ |v(x)|, |Dv(x)| \})$ gives stable results, although we have not done a proper Von-Neumann analysis, and in practice we do see we can choose a higher step size. The step size for accelerated gradient descent is lower in our experiments than accelerated gradient descent. In all experiments, the initialization is $\phi(x) = \psi(x) = x$, $v(x)=0$, and $\rho(x) = 1/|\Omega|$ where $|\Omega|$ is the area of the domain of the image. ### Convergence analysis In this experiment, the images are two white squares against a black background. The sizes of the squares are $50\times 50$ pixels wide, and the square (of size $20\times 20$) in the first image is translated by $10$ pixels to form the second image. Small images are chosen due to the fact gradient descent is too impractically slow for reasonable sized images without multi-scale approaches that even modest sized images (e.g., $256\times 256$) do not converge in a reasonable amount of time, and we will demonstrate this in an experiment later. Figure \[fig:potential\_vs\_iter\] shows the plot of the potential energy of both gradient descent and accelerated gradient descent as the evolution progresses. Here $\alpha = 5$ (images are scaled between 0 and 1). Notice that accelerated gradient descent very quickly accelerates to a global minimum, surpasses the global minimum and then oscillates until the friction term slows it down and then it converges very quickly. Notice that this behavior is expected since accelerated gradient descent is not a strict descent method (it does not necessarily decrease the potential energy each step). Gradient descent very slowly decreases the energy each iteration and eventually converges. ![[**Convergence Comparison**]{}: Two binary images with squares in which the square is translated are registered. The value of the functional (to be minimized) versus the iteration number is shown for both gradient descent (GD) and accelerated gradient descent (AGD).[]{data-label="fig:potential_vs_iter"}](figures/expt5) We now repeat the same experiment, but with different images to show that this behavior is not restricted to the particular choice of images, one a translation of the other. To this end, we choose the images again to be $50\times 50$. The first image has a square that is $17\times 17$ and the second image has a rectangle of size $20\times 14$ and is translated by $8$ pixels. We choose the regularity $\alpha=2$, since the regularity should be chosen smaller to account for the stretching and squeezing, resulting in a non-smooth flow field. A plot of results of this simulation is shown in Figure \[fig:potential\_vs\_iter\_scaling\]. Again accelerated gradient accelerates very quickly at the start, then oscillates and the oscillations die down and then it converges. This time the potential does not go to zero since the final flow is not a translation and thus the regularity term is non-zero. Gradient descent converges faster than the case of translation due to larger $\alpha$ and thus larger step size. However, it appears to be stuck in a higher energy configuration. In fact, gradient descent has not fully converged - gradient descent is slow in adapting to the scale changes and becomes extremely slow in stretching and squeezing in different directions. We verify that gradient descent has not fully converged by plotting just the first term of the potential, i.e., the reconstruction error, which is zero for accelerated gradient descent at convergence, indicating that the flow correctly reconstructs $I_0$ from $I_1$. On the other hand, gradient descent has an error of about $50$, indicating the flow does not fully warp $I_1$ to $I_0$, and therefore it not the correct flow. This does not appear to be a local minimum, just slow convergence. ![[**Convergence Comparison**]{}: Two images are registered, each are binary images. The first is a square and the second image is a translated and non-uniformly scaled version of the square in the first image. \[Left\]: The cost functional to be minimized versus the iteration number is shown for both gradient descent (GD) and accelerated gradient descent (AGD). AGD converges to a lower energy solution quicker. \[Right\]: Note that GD did not fully converge as the convergence is extremely slow in obtaining fine scale details of the non-uniform scaling. This is verified by plotting the image reconstruction error: $\|I_1\circ \phi - I_0\|$, which shows that AGD reconstructs $I_0$ with zero error. []{data-label="fig:potential_vs_iter_scaling"}](figures/expt6a "fig:") ![[**Convergence Comparison**]{}: Two images are registered, each are binary images. The first is a square and the second image is a translated and non-uniformly scaled version of the square in the first image. \[Left\]: The cost functional to be minimized versus the iteration number is shown for both gradient descent (GD) and accelerated gradient descent (AGD). AGD converges to a lower energy solution quicker. \[Right\]: Note that GD did not fully converge as the convergence is extremely slow in obtaining fine scale details of the non-uniform scaling. This is verified by plotting the image reconstruction error: $\|I_1\circ \phi - I_0\|$, which shows that AGD reconstructs $I_0$ with zero error. []{data-label="fig:potential_vs_iter_scaling"}](figures/expt6b "fig:") We again repeat the same experiment, but with real images from a cardiac MRI sequence, in which the heart beats. The transformation relating the images is a general diffeomorphism that is not easily described as in the previous experiments. The images are of size $256\times 256$. We choose $\alpha=0.02$. A plot of the potential versus iteration number for both gradient descent (GD) and accelerated gradient descent (AGD) is shown in the left of Figure \[fig:cardiac\_expt\]. The convergence is quicker for accelerated gradient descent. The right of Figure \[fig:cardiac\_expt\] shows the original images and the images warped under both the result from gradient descent and accelerated gradient descent, and that they both produce a similar correct warp, but accelerated gradient obtains the warp in much fewer iterations. ![[**Convergence Comparison**]{}: Two MR cardiac images from a sequence are registered. The images are related through a general deformation. \[Left\]: A plot of the potential versus the iteration number in the minimization using gradient descent (GD) and accelerated gradient descent (AGD). AGD converges at a quicker rate. \[Right\]: The original images and the back-warped images using the recovered diffeomorphisms. Note that $I_1\circ\phi$ should appear close to $I_0$. Both methods seem to recover a similar transformation, but AGD recovers it faster.[]{data-label="fig:cardiac_expt"}](figures/expt7) ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ $I_1$ $I_0$ ![[**Convergence Comparison**]{}: Two MR cardiac images from a sequence are registered. The images are related through a general deformation. \[Left\]: A plot of the potential versus the iteration number in the minimization using gradient descent (GD) and accelerated gradient descent (AGD). AGD converges at a quicker rate. \[Right\]: The original images and the back-warped images using the recovered diffeomorphisms. Note that $I_1\circ\phi$ should appear close to $I_0$. Both methods seem to recover a similar transformation, but AGD recovers it faster.[]{data-label="fig:cardiac_expt"}](figures/cardiac_expt_I1 "fig:") ![[**Convergence Comparison**]{}: Two MR cardiac images from a sequence are registered. The images are related through a general deformation. \[Left\]: A plot of the potential versus the iteration number in the minimization using gradient descent (GD) and accelerated gradient descent (AGD). AGD converges at a quicker rate. \[Right\]: The original images and the back-warped images using the recovered diffeomorphisms. Note that $I_1\circ\phi$ should appear close to $I_0$. Both methods seem to recover a similar transformation, but AGD recovers it faster.[]{data-label="fig:cardiac_expt"}](figures/cardiac_expt_I0 "fig:") $I_1\circ \phi_{gd}$ $I_1\circ \phi_{agd}$ ![[**Convergence Comparison**]{}: Two MR cardiac images from a sequence are registered. The images are related through a general deformation. \[Left\]: A plot of the potential versus the iteration number in the minimization using gradient descent (GD) and accelerated gradient descent (AGD). AGD converges at a quicker rate. \[Right\]: The original images and the back-warped images using the recovered diffeomorphisms. Note that $I_1\circ\phi$ should appear close to $I_0$. Both methods seem to recover a similar transformation, but AGD recovers it faster.[]{data-label="fig:cardiac_expt"}](figures/cardiac_expt_I1w_gd "fig:") ![[**Convergence Comparison**]{}: Two MR cardiac images from a sequence are registered. The images are related through a general deformation. \[Left\]: A plot of the potential versus the iteration number in the minimization using gradient descent (GD) and accelerated gradient descent (AGD). AGD converges at a quicker rate. \[Right\]: The original images and the back-warped images using the recovered diffeomorphisms. Note that $I_1\circ\phi$ should appear close to $I_0$. Both methods seem to recover a similar transformation, but AGD recovers it faster.[]{data-label="fig:cardiac_expt"}](figures/cardiac_expt_I1w_agd "fig:") ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ [ ]{} ### Convergence analysis versus parameter settings We now analyze the convergence of accelerated gradient descent and gradient descent as a function of the regularity $\alpha$ and the image size. To this end, we first analyze an image pair of size $50\times 50$ in which one image has a square of size $16\times 16$ and the other image is the same square translated by $7$ pixels. We now vary $\alpha$ and analyze the convergence. In the left plot of Figure \[fig:speed\_vs\_alpha\], we show the number of iterations until convergence versus the regularity $\alpha$. As $\alpha$ increases, the number of iterations for both gradient descent and accelerated gradient descent increase as expected since there is a inverse relationship between $\alpha$ and the step size. However, the number of iterations for accelerated gradient descent grows more slowly. In all cases, the algorithm is run until the flow field between successive iterations does not change according to a fixed tolerance. In all cases, the flow achieves the ground truth flow. Next, we analyze the number of convergence iterations versus the image size. To this end, we again consider binary images with squares of size $16\times 16$ and translated by $7$ pixels. However, we vary the image size from $50\times 50$ to $200 \times 200$. We fix $\alpha = 8$. Now we show the number of iterations to convergence versus the image size. This is shown in the right plot of Figure \[fig:speed\_vs\_alpha\]. Gradient descent is impractically slow for all the sizes considered, and the number of iterations quickly increases with image size (it appears to be an exponential growth). Accelerated gradient descent, surprisingly, appears to have very little or no growth with respect to the image size. Of course one could use multi-scaling pyramid approaches to improve gradient descent, but as soon as one goes to finer scales, gradient descent is incredibly slow even when the images are related by small displacements. Simple acceleration makes standard gradient descent scalable with just a few extra local updates. ![\[Left\]: [**Convergence Comparison as a Function of Regularity**]{}: Two binary images (a square and a translated square) are registered with varying amounts of regularization $\alpha$ for gradient descent (GD) and accelerated gradient descent (AGD). \[Right\]: [**Convergence Comparison as a Function of Image Size**]{}: We keep the squares in the images and $\alpha=3$ fixed, but we vary the size (height and width) of the image and compare GD with AGD. Very quickly, gradient descent becomes impractical due to extremely slow convergence.[]{data-label="fig:speed_vs_alpha"}](figures/expt3 "fig:") ![\[Left\]: [**Convergence Comparison as a Function of Regularity**]{}: Two binary images (a square and a translated square) are registered with varying amounts of regularization $\alpha$ for gradient descent (GD) and accelerated gradient descent (AGD). \[Right\]: [**Convergence Comparison as a Function of Image Size**]{}: We keep the squares in the images and $\alpha=3$ fixed, but we vary the size (height and width) of the image and compare GD with AGD. Very quickly, gradient descent becomes impractical due to extremely slow convergence.[]{data-label="fig:speed_vs_alpha"}](figures/expt4 "fig:") [ ]{} ### Analysis of Robustness to Noise We now analyze the robustness of gradient descent and accelerated gradient descent to noise. We do this to simulate robustness to undesirable local minima. We choose to use salt and pepper noise to model possible clutter in the image. We consider images of size $50\times 50$. We fix $\alpha = 1$ in all the simulations and vary the noise level; of course one could increase $\alpha$ to increase robustness to noise. However, we are interested in understanding the robustness to noise of the optimization algorithms themselves rather than changing the potential energy to better cope with noise. First, we consider a square of size $16\times 16$ in the first binary image and the same square translated by $4$ pixels in the second image. We plot the error in the flow (measured as the average endpoint error of the flow returned by the algorithm against ground truth flow) versus the noise level. The result is shown in the left plot of Figure \[fig:noise\_stability\]. This shows that accelerated gradient descent degrades much slower than gradient descent. Figure \[fig:noise\_stab\_trans\_img\] shows visual comparison of the final results where we show $I_1\circ\phi$ and compare it to $I_0$ for both accelerated gradient descent and gradient descent. We repeat the same experiment to show that this trend is not just with this configuration of images. To this end, we experiment with $50\times 50$ images one with a square of size $15\times 15$ and a rectangle that is size $20\times 10$ and translated by $5$ pixels. We again fix the regularity to $\alpha=1$. The result of the experiment is plotted in the right of Figure \[fig:noise\_stability\]. A similar trend of the previous experiment is observed: accelerated gradient descent degrades much less than gradient descent. Note we have measured accuracy as the average reconstruction error with the original (non-noisy) images. This is because the ground truth flow is not known. Figure \[fig:noise\_stab\_dil\_img\] shows visual comparison of the final results. ![[**Analysis of Stability to Noise**]{}: We add salt and pepper noise with varying intensity to binary images and then register the images. We plot the error in the recovered flow of both gradient descent (GD) and accelerated gradient descent (AGD) versus the level of noise. The value of $\alpha$ is kept fixed. The error is measured by the average endpoint error of the flow. \[Left\]: The first image is formed from a square and the second image is the same square but translated. \[Right\]: The first image is a square and the second image is the non-uniformly scaled and translated square. The error is measured as the average image reconstruction error.[]{data-label="fig:noise_stability"}](figures/expt1 "fig:") ![[**Analysis of Stability to Noise**]{}: We add salt and pepper noise with varying intensity to binary images and then register the images. We plot the error in the recovered flow of both gradient descent (GD) and accelerated gradient descent (AGD) versus the level of noise. The value of $\alpha$ is kept fixed. The error is measured by the average endpoint error of the flow. \[Left\]: The first image is formed from a square and the second image is the same square but translated. \[Right\]: The first image is a square and the second image is the non-uniformly scaled and translated square. The error is measured as the average image reconstruction error.[]{data-label="fig:noise_stability"}](figures/expt2 "fig:") 1img[0.75in]{} [ ]{} [ ]{} Conclusion ========== We have generalized accelerated optimization, in particular Nesterov’s scheme, to infinite dimensional manifolds. This method is general and applies to optimizing any functional on an infinite dimensional manifold. We have demonstrated this for the class of diffeomorphisms motivated by variational optical flow problems in computer vision. The main objective of the paper was to introduce the formalism and derive the evolution equations that are PDEs. The evolution equations are natural extensions of mechanical principles from fluid mechanics, and in particular connect to optimal mass transport. They require additional evolution equations over gradient descent, i.e., a velocity evolution and a density evolution, but that does not significantly add to the cost of $L^2$ gradient descent per iteration since the updates are all local, i.e., computation of derivatives. Our numerical scheme to implement these equations used entropy conditions, which were employed to cope with shocks and fans of the underlying PDE. Our numerical scheme is not final and could be improved, and we plan to explore this in future work. Experiments on toy examples nevertheless demonstrated the advantages of speed and robustness to local minima over gradient descent, and illustrated the behavior of accelerated gradient descent. Just by simple acceleration, gradient descent, unusable in practice due to scalability with image size, became usable. One area that should be explored further is the choice of the time-explicit functions $a,b$ in the generalized Lagrangian. These were chosen to coincide with the choices to produce the continuum limit of Nesterov’s scheme finite dimensions, which are designed for the convex case to yield optimal convergence. Since the energies that we consider are non-convex, these may no longer be optimal. Of interest would be a design principle for choosing $a,b$ so as to obtain optimal convergence rates. A follow-up question would then be whether the discretization of the PDEs gives optimal rates in discrete-time. Another issue is that we assumed that the domain of the diffeomorphism was ${\mathbb{R}}^n$, but images are compact; we by-passed this complication by assuming periodic boundary conditions. Future work will look into proper treatment of the boundary. Functional Gradients {#app:funct_grads} -------------------- Let $U : {\mbox{Diff}({\mathbb{R}}^n)}\to {\mathbb{R}}$. The gradient (or functional derivative) of $U$ with respect to $\phi \in {\mbox{Diff}({\mathbb{R}}^n)}$, denoted $\nabla U(\phi)$, is defined as the $\nabla U(\phi) \in T_{\phi} {\mbox{Diff}({\mathbb{R}}^n)}$ that satisfies $$\delta U(\phi) \cdot v = \int_{\phi({\mathbb{R}}^n)} \nabla U(\phi)(x) \cdot v(x) {\,\mathrm{d}}x$$ for all $v\in T_{\phi} {\mbox{Diff}({\mathbb{R}}^n)}$. The left hand side is the directional derivative and is defined as $$\delta U(\phi) \cdot v := \left. {\frac{{\,\mathrm{d}}{}}{{\,\mathrm{d}}{\varepsilon}}} U( \phi + \varepsilon v ) \right|_{\varepsilon= 0}.$$ Note that $(\phi + \varepsilon v)(x) = \phi(x) + \varepsilon v( \phi(x) )$ for $x\in {\mathbb{R}}^n$. We now show the computation of the gradient for the illustrative potential used in this paper. First, let us consider the data term $U_1(\phi) = \int_{{\mathbb{R}}^n} |I_1(\phi(x)) - I_0(x)|^2 {\,\mathrm{d}}x$ then $$\delta U_1(\phi) \cdot \delta\phi = \int_{{\mathbb{R}}^n} 2(I_1(\phi(x)) - I_0(x)) DI_1(\phi(x)) \widehat{\delta \phi}(x) {\,\mathrm{d}}x = \int_{\phi({\mathbb{R}}^n)} 2( I_1(x) - I_0(\psi(x)) ) DI_1(x) \delta \phi(x) \det{\nabla\psi(x)}{\,\mathrm{d}}x,$$ where $\widehat{\delta\phi} = \delta\phi\circ\phi$, $\psi = \phi^{-1}$ and we have performed a change of variables. Thus, $\nabla U_1 = 2\nabla I_1( I_1 - I_0\circ\psi) \det{\nabla\psi}$. Now consider the regularity term $U_2(\phi) = \int_{{\mathbb{R}}^n} |\nabla (\phi(x)-x)|^2 {\,\mathrm{d}}x$, then $$\delta U(\phi) = 2\int_{{\mathbb{R}}^n} {\mbox{tr}\left({ \nabla (\phi(x)-\mbox{id})^T \nabla \widehat{\delta\phi}(x) }\right)} {\,\mathrm{d}}x = -\int_{{\mathbb{R}}^n} \Delta \phi(x)^T \delta\phi(x) {\,\mathrm{d}}x = \int_{\Omega} (\Delta \phi)(\psi(x))^T \delta \phi(x) \det{ \nabla\psi(x) }{\,\mathrm{d}}x.$$ Note that in integration by parts, the boundary term vanishes since we assume that $\phi(x) = x$ as $|x|\to \infty$. Thus, $\nabla U_2 = (\Delta \phi) \circ \psi \det{ \nabla\psi }$. Stationary Conditions {#app:stat_cond_nondissip} --------------------- \[eq:stationary\_mapping\] The stationary condition of the action for the mapping is $$\partial_t \lambda + {\mbox{div}\left( {v\lambda^T} \right)}^T = (\nabla\psi)^{-1} \nabla U(\phi).$$ We compute the variation of $A$ (defined in ) with respect to the mapping $\phi$. The only terms in the action that depend on the mapping are $U$ and the Lagrange multiplier term associated with the mapping. Taking the variation w.r.t the potenial term gives $$-\int \int_{ \phi({\mathbb{R}}^n) } \nabla U(\phi) \cdot \delta \phi {\,\mathrm{d}}x {\,\mathrm{d}}t.$$ Now the variation with respect to the Lagrange multiplier term: $$\int\int_{ \phi({\mathbb{R}}^n) } \lambda^T[ \partial_t \widehat{\delta \psi} + D (\widehat{\delta\psi}) v ] {\,\mathrm{d}}x {\,\mathrm{d}}t = -\int \int_{ \phi({\mathbb{R}}^n) } [ \partial_t \lambda^T + {\mbox{div}\left( {v\lambda^T} \right)} ] \widehat{\delta\psi} {\,\mathrm{d}}x {\,\mathrm{d}}t,$$ where we have integrated by parts, the ${\mbox{div}\left( {\cdot} \right)}$ of a matrix means the divergence of each of the columns, resulting in a row vector, and $\widehat{\delta \psi} = \delta \psi \circ\psi$. Note that we can take the variation of $\psi(\phi(x)) = x$ to obtain $$\widehat{\delta\psi} \circ \phi(x) + [D\psi(\phi(x))] \widehat{\delta\phi}(x) = 0,$$ or $$\widehat{\delta\psi}(y) = -[D\psi(y)] \delta\phi(y).$$ Therefore, $$\label{eq:der_action_mapping} \delta A \cdot \delta\phi = \int\int_{ \phi({\mathbb{R}}^n) } \left\{ (\nabla \psi) \left[ \partial_t \lambda + {\mbox{div}\left( {v\lambda^T} \right)}^T \right] - \nabla U(\phi) \right\} \cdot \delta \phi {\,\mathrm{d}}x {\,\mathrm{d}}t.$$ The stationary condition of the action arising from the velocity is $$\rho v + (\nabla\psi)\lambda - \rho\nabla\mu = 0.$$ We compute the variation w.r.t the kinetic energy: $$\delta T \cdot \delta v = \int_{\phi({\mathbb{R}}^n)} \rho v\cdot \delta v {\,\mathrm{d}}x.$$ The variation of the Lagrange multiplier terms is $$\int\int_{\phi({\mathbb{R}}^n)} \lambda^T (D\psi) \delta v - \rho \nabla\mu \cdot \delta v {\,\mathrm{d}}x {\,\mathrm{d}}t = \int\int_{\phi({\mathbb{R}}^n)} [ (\nabla\psi)\lambda - \rho\nabla\mu ] \cdot \delta v {\,\mathrm{d}}x {\,\mathrm{d}}t.$$ Therefore, $$\label{eq:var_action_vel} \delta A \cdot \delta v = \int\int_{\phi({\mathbb{R}}^n)} [ \rho v + (\nabla\psi)\lambda - \rho\nabla\mu ] \cdot \delta v {\,\mathrm{d}}x {\,\mathrm{d}}t.$$ The stationary condition of the action arising from the velocity is $$\partial_t \mu + (D\mu)v = \frac 1 2 |v|^2.$$ Note that the terms that contain the density in are the kinetic energy and the Lagrange multiplier corresponding to the density. We see that $$\label{eq:var_action_rho} \delta A \cdot \delta\rho = \int\int_{\phi({\mathbb{R}}^n)} \frac 1 2 |v|^2 \delta\rho - (\partial_t\mu + \nabla\mu\cdot v) \delta \rho {\,\mathrm{d}}x{\,\mathrm{d}}t,$$ which yields the lemma. Velocity Evolution {#app:velocity_evol} ------------------ \[lem:lambda\_t\_w\_t\] Given that $(\nabla\psi)\lambda = w$, we have that $$\partial_t\lambda + (D\lambda)v + \lambda {\mbox{div}\left( {v} \right)} = (\nabla\psi)^{-1} [ \partial_t w + (Dw)v + (\nabla v)w + w{\mbox{div}\left( {v} \right)} ]$$ Define the Hessian as follows: $$[ D^2\psi ]_{ijk} = \partial_{x_ix_j}^2 \psi^k, \quad [ D^2\psi(a,b) ]_k = \sum_{ij} \partial_{x_ix_j}^2 \psi^k a_i b_j.$$ We compute $$\{ D[ (\nabla\psi)\lambda ] \}_{ ij } = \partial_{ x_j } [ (\nabla\psi)\lambda ]_{i} = \partial_{ x_j } \sum_{ l } \partial_{ x_i } \psi^l \lambda_l = \sum_l (\partial^2_{ x_j x_i } \psi^l \lambda_l) + \partial_{ x_i } \psi^l\partial_{ x_j } \lambda_l.$$ Therefore, $$D[ (\nabla\psi)\lambda ]= D^2\psi(\cdot, \cdot) \cdot \lambda + (\nabla\psi) (D\lambda)$$ Since $ D[ (\nabla\psi)\lambda ] = Dw $ then solving for $D\lambda$ gives $$D\lambda =(\nabla\psi)^{-1} [ Dw - D^2\psi(\cdot, \cdot) \cdot \lambda ],$$ so $$\label{eq:Dlam_v} (D\lambda)v = (\nabla\psi)^{-1} [ (Dw)v - D^2\psi(\cdot, v) \cdot \lambda ].$$ Now differentiating $(\nabla\psi)\lambda = w$ w.r.t $t$, we have $$(\nabla \partial_t\psi)\lambda + (\nabla\psi)\partial_t \lambda = \partial_t w, \quad \mbox{or} \quad \partial_t \lambda = (\nabla \psi)^{-1} [ \partial_t w - (\nabla \partial_t\psi)\lambda]$$ Note that $\partial_t\psi = -(D\psi)v$ so $$\label{eq:lambda_t_w} \partial_t \lambda = (\nabla \psi)^{-1} \left\{ \partial_t w + \nabla [ (D\psi)v] \lambda\right\}.$$ Now computing $\nabla [ (D\psi)v]$ yields $$\{ \nabla [ (D\psi)v ) ] \}_{lk} = \partial_{ x_l } \sum_{ i } \partial_{ x_i } \psi^k v^i = \sum_{i} \partial_{x_l}\partial_{x_i} \psi^k v^i + \partial_{x_i} \psi^k \partial_{x_l} v^i.$$ Then multiplying the above matrix by $\lambda$ gives $$\{ \nabla [ (D\psi)v ) ] \lambda \}_{l} = \sum_{ik} \partial_{x_l}\partial_{x_i} \psi^k v^i \lambda^k + \partial_{x_i} \psi^k \partial_{x_l} v^i \lambda^k,$$ which in matrix form is $$\nabla [ (D\psi)v ) ] \lambda = D^2\psi( \cdot, v )\cdot \lambda + (\nabla v)(\nabla\psi)\lambda = D^2\psi( \cdot, v )\cdot \lambda + (\nabla v)w$$ Therefore, becomes $$\partial_t\lambda = (\nabla\psi)^{-1} [ \partial_t w + D^2\psi( \cdot, v )\cdot \lambda + (\nabla v)w ].$$ Combining the previous with and noting that $\lambda {\mbox{div}\left( {v} \right)} = (\nabla\psi)^{-1}w {\mbox{div}\left( {v} \right)}$ yields $$\partial_t\lambda + (D\lambda)v + \lambda {\mbox{div}\left( {v} \right)} = (\nabla\psi)^{-1} [ \partial_t w + (Dw)v + (\nabla v)w + w{\mbox{div}\left( {v} \right)} ].$$ \[lem:w\_t\_v\_t\] If $w = \rho( \nabla\mu - v )$, then $$\partial_t w + (Dw)v + (\nabla v)w + w{\mbox{div}\left( {v} \right)} = -\rho[ \partial_t v + (Dv)v ].$$ Differentiating $w = \rho( \nabla\mu - v )$, we have $$\begin{aligned} \partial_t w &= (\partial_t\rho)(\nabla\mu - v) + \rho( \nabla\partial_t\mu - \partial_t v) \\ Dw &= (\nabla\mu - v)(D\rho) + \rho[ D(\nabla\mu) - Dv ]. \end{aligned}$$ Therefore, $$\begin{aligned} \partial_t w + (Dw)v + (\nabla v)w + w{\mbox{div}\left( {v} \right)} &= (\nabla\mu - v)( \partial_t\rho +\nabla\rho\cdot v ) + \rho[ \nabla\partial_t\mu - \partial_t v + D(\nabla\mu)v - (Dv)v ] \\ &+ \rho(\nabla v)(\nabla\mu - v) + \rho(\nabla\mu - v){\mbox{div}\left( {v} \right)} \\ &= (\nabla\mu - v)( \partial_t\rho +\nabla\rho\cdot v + \rho{\mbox{div}\left( {v} \right)} ) \\ &+ \rho[ \nabla\partial_t\mu - \partial_t v + D(\nabla\mu)v - (Dv)v + (\nabla v)(\nabla\mu - v) ]. \end{aligned}$$ Note that $\partial_t\rho +\nabla\rho\cdot v + \rho{\mbox{div}\left( {v} \right)} = \partial_t \rho + {\mbox{div}\left( {\rho v} \right)} = 0$, due to the continuity equation. Therefore, $$\begin{aligned} \partial_t w + (Dw)v + (\nabla v)w + w{\mbox{div}\left( {v} \right)} &= \rho[ -\partial_t v - (Dv)v - (\nabla v)v + \nabla\partial_t\mu + D(\nabla\mu)v + (\nabla v)(\nabla\mu) ] \\ &= \rho\left\{ -\partial_t v - (Dv)v - (\nabla v)v + \nabla[ \partial_t\mu + (D\mu)v ] \right\}. \end{aligned}$$ By the stationary condition for the density, $\partial_t\mu + (D\mu)v = 1/2 |v|^2$, so $\nabla[ \partial_t\mu + (D\mu)v ] = (\nabla v)v$, which gives the lemma. The evolution equation for the velocity arising from the stationarity of the action integral is $$\rho[ \partial_t v + (Dv)v ] = -\nabla U(\phi).$$ This is a combination of Lemmas \[eq:stationary\_mapping\], \[lem:lambda\_t\_w\_t\], and \[lem:w\_t\_v\_t\]. Stationary Conditions for the Dissipative Case {#app:stationary_dissip} ---------------------------------------------- \[thrm:stationary\_lagrange\_mult\_dissip\] The stationary conditions of the path for the action $$\begin{aligned} \label{eq:action_dissip_lagrange} A &= \int \left[ a T(v) - b U(\phi) \right] {\,\mathrm{d}}t + \int \int_{{\mathbb{R}}^n} \lambda^T[ \partial_t \psi_t + (D\psi)v ] {\,\mathrm{d}}x {\,\mathrm{d}}t -\int \int_{{\mathbb{R}}^n} \left[ \partial_t \mu + \nabla \mu \cdot v \right] \rho {\,\mathrm{d}}x {\,\mathrm{d}}t, \end{aligned}$$ are $$\begin{aligned} \partial_t \lambda + (D\lambda )v + \lambda {\mbox{div}\left( {v} \right)} &= b(\nabla\psi)^{-1}\nabla U(\phi) \\ a\rho v + (\nabla \psi) \lambda - \rho\nabla\mu &= 0 \\ \partial_t \mu + \nabla\mu \cdot v &= \frac 1 2 a|v|^2. \end{aligned}$$ Note that $$\begin{aligned} \nabla[ b U ](\phi) &= b\nabla U(\phi)\\ \delta [ aT ] \cdot \delta \rho &= \int_{\phi({\mathbb{R}}^n)} \frac 1 2 a |v|^2\delta \rho {\,\mathrm{d}}x\\ \delta [aT] \cdot \delta v &= \int_{\phi({\mathbb{R}}^n)} a \rho v\cdot\delta v {\,\mathrm{d}}x. \end{aligned}$$ Therefore, using and replacing $\nabla U(\phi)$ with $b\nabla U(\phi)$, we have $$\delta A \cdot \delta\phi = \int\int_{ \phi({\mathbb{R}}^n) } \left\{ (\nabla \psi) \left[ \partial_t \lambda + {\mbox{div}\left( {v\lambda^T} \right)}^T \right] - b\nabla U(\phi) \right\} \cdot \delta \phi {\,\mathrm{d}}x {\,\mathrm{d}}t,$$ which yields the stationary condition on the mapping. Also, updating yields $$\delta A \cdot \delta v = \int\int_{\phi({\mathbb{R}}^n)} [ a \rho v + (\nabla\psi)\lambda - \rho\nabla\mu ] \cdot \delta v {\,\mathrm{d}}x {\,\mathrm{d}}t,$$ which yields the stationary condition for the velocity. Finally, updating yields $$\delta A \cdot \delta\rho = \int\int_{\phi({\mathbb{R}}^n)} \frac 1 2 a|v|^2 \delta\rho - (\partial_t\mu + \nabla\mu\cdot v) \delta \rho {\,\mathrm{d}}x{\,\mathrm{d}}t,$$ and that yields the last stationary condition. The evolution equations for the stationary conditions of the action in is $$\rho[ \partial_t(av) + a(Dv)v ] = -b\nabla U(\phi).$$ Let $w = \rho( \nabla\mu-av)$ then $$\begin{aligned} \partial_t w &= (\partial_t\rho)(\nabla\mu - av) + \rho( \nabla\partial_t\mu - \partial_t (av) ) \\ Dw &= (\nabla\mu - av)(D\rho) + \rho[ D(\nabla\mu) - aDv ]. \end{aligned}$$ Then $$\begin{aligned} \partial_t w + (Dw)v + (\nabla v)w + w{\mbox{div}\left( {v} \right)} &= g (\nabla\mu - av)( \partial_t\rho +\nabla\rho\cdot v ) + \rho[ \nabla\partial_t\mu - \partial_t (av) + D(\nabla\mu)v - a(Dv)v ] \\ &+ \rho(\nabla v)(\nabla\mu - av) + \rho(\nabla\mu - av){\mbox{div}\left( {v} \right)} \\ &= (\nabla\mu - av)( \partial_t\rho +\nabla\rho\cdot v + \rho{\mbox{div}\left( {v} \right)} ) \\ &+ \rho[ \nabla\partial_t\mu - \partial_t (av) + D(\nabla\mu)v - a(Dv)v + (\nabla v)(\nabla\mu - av) ] \\ &= \rho\left\{ -\partial_t(av) - a(Dv)v - a(\nabla v)v + \nabla[ \partial_t\mu + (D\mu)v ] \right\} \\ &= \rho\left\{ -\partial_t(av) - a(Dv)v \right\}. \end{aligned}$$ By Lemma \[lem:lambda\_t\_w\_t\] and the previous expression, we have our result. Discretization {#app:discretization} -------------- We present the discretization of the velocity PDE first. In one dimension, the terms involving $v$ are Burger’s equation, which is known to produce shocks. We thus use an entropy scheme. Writing the PDE component-wise, we get $$\begin{aligned} \label{eq:velocity_ev_comp} \partial_t v_1 &= -\frac 1 2 \partial_{x_1} (v_1)^2 - v_2 \partial_{x_2} v_1 - \frac 3 t v_1 - \frac{1}{\rho} (\nabla U)_1 \\ \partial_t v_2 &= -\frac 1 2 \partial_{x_2} (v_2)^2 - v_1 \partial_{x_1} v_1 - \frac 3 t v_2 - \frac {1}{\rho} (\nabla U)_2,\end{aligned}$$ where the subscript indicates the component of the vector. We use forward Euler for the time derivative, and for the first term on the right hand side, we use an entropy scheme for Burger’s equation which results in the following discretization: $$\partial_{x_1} (v_1)^2(x) \approx \max\{ v_1(x), 0 \}^2 - \min\{ v_1(x), 0 \}^2 + \min\{ v_1(x_1+\Delta x, x_2), 0 \}^2 - \max\{ v_1(x_1+\Delta x, x_2), 0 \}^2,$$ where $\Delta x$ is the spatial sampling size, and the $\partial_{x_2} (v_2)^2$ follows similarly. For the second term on the right hand side of , we follow the discretization of a transport equation using an up-winding scheme, which yields the following discretization: $$v_2(x) \partial_{x_2} v_1(x) \approx v_2(x) \cdot \begin{cases} v_1( x_1, x_2 ) - v_1( x_1, x_2 -\Delta x ) & v_2(x) > 0 \\ v_1( x_1, x_2 +\Delta x ) - v_1( x_1, x_2 ) & v_2(x) < 0 \end{cases}.$$ With regards to the gradient of potential, if we use the potential , then all the derivatives are discretized using central differences, as the key term is a diffusion. The step size $\Delta t / \Delta x < 1/\max_x\{ |v(x)|, |Dv(x)| \}$. The backward map $\psi$ evolves according to a transport PDE , and thus an up-winding scheme similar to the transport term in the velocity term is used. For the discretization of the continuity equation, we use a staggered grid (so that the values of $v$ are defined in between grid points and $\rho$ is defined at the grid points). The discretization is just the sum of the fluxes coming into the point: $$-{\mbox{div}\left( {\rho(x)v(x)} \right)} \approx \sum_{i=1}^2 \left[ -v_i(x) \begin{cases} \rho(x) & v_i(x) > 0 \\ \rho(x +\Delta x_i) & v_i(x) <0 \end{cases} + v_i(x-\Delta x_i) \begin{cases} \rho(x-\Delta x_i) & v_1(x-\Delta x_i) > 0 \\ \rho(x) & v_1(x-\Delta x_i) < 0 \end{cases} \right],$$ where $\Delta x_i$ denotes the vector of the spatial increment $\Delta x$ in the $i^{\text{th}}$ coordinate direction, $v_1(x)$ denotes the velocity defined at the midpoint between $(x_1,x_2)$ and $(x_1+\Delta x, x_2)$, and $v_2(x) $ denotes the velocity defined at the midpoint between $(x_1,x_2)$ and $(x_1, x_2+\Delta x)$. The term $\partial_t\rho(x)$ is discretized with forward Euler. This scheme is guaranteed to preserve mass. [^1]: KAUST (King Abdullah University of Science and Technology), [ganesh.sundaramoorthi@kaust.edu.sa ]{} [^2]: Georgia Institute of Technology, [ayezzi@ece.gatech.edu]{} [^3]: One could in fact generalize such operations as addition and subtraction in manifolds, using the exponential and logarithmic maps. We avoid this since in the types of manifolds that we deal with, computing such maps itself requires solving a PDE or another optimization problem. We avoid all these complications, by going back to the formalism in classical mechanics. [^4]: In medical imaging, the model of diffeomorphisms for registration is fairly accurate since typically full 3D scans are available and thus all points in one image correspond to the other image and vice versa. Of course there are situations (such as growth of tumors) where the diffeomorphic assumption is invalid. In vision, typically images have occlusion phenomena and multiple objects moving in different ways. So a diffeomorphism is not a valid assumption, it is however a good model when restricted to a single object in the un-occluded part.

--- abstract: 'We study the influence that population density and the road network have on each others’ growth and evolution. We use a simple model of formation and evolution of city roads which reproduces the most important empirical features of street networks in cities. Within this framework, we explicitely introduce the topology of the road network and analyze how it evolves and interact with the evolution of population density. We show that accessibility issues -pushing individuals to get closer to high centrality nodes- lead to high density regions and the appearance of densely populated centers. In particular, this model reproduces the empirical fact that the density profile decreases exponentially from a core district. In this simplified model, the size of the core district depends on the relative importance of transportation and rent costs.' author: - Marc Barthélemy - Alessandro Flammini title: 'Co-evolution of density and topology in a simple model of city formation' --- Introduction ============ It has been recently estimated that more than $50\%$ of the world population lives in cities and this figure is bound to increase [@UN]. The migration towards urban areas has dictated a fast and short-term planned urban growth which needs to be understood and modelled in terms of socio-geographical contingencies, and of the general forces that drive the development of cities. Previous studies [@Christaller; @Levinson; @Fujita] about urban morphology have mostly focused on various geographical, historical, and social-economical mechanisms that have shaped distinct urban areas in different ways. A recent example of these studies can be found in [@Levinson], where the authors study the process of self-organization of transportation networks with a model that takes into account revenues, costs and investments. The goal of the present study is to model the coupling between the evolution of the transportation network and the population density. More precisely, the question we aim to answer is the following. Given the pattern of growth of the entire population of a given city, how is the local density of population changing within the boundaries of the city itself, and how the road network’s topology is modified in order to accommodate these changes? There are in principle a huge number of potentially relevant factors that may influence the growth and shape of urban settlements, first and foremost the social, economical and geographical conditions that causes the population of a given city to increase in a particular moment of its history. We neglect in the present study this class of factors and consider the overall growth in the number of inhabitants as an exogenous variable. In order to achieve conclusions that have a good degree of generality, and, at the same time, to maintain the number of assumptions as limited as possible, we focus on two main features only: the local density of population and the structure of the road network. Population density and the topology of the network constitute two different facets of the spatial organization of a city, and from a purely qualitative point of view it is not hard to believe that their evolution is strongly correlated. Indeed, Levinson, in a recent case study [@Levinson2] about the city of London in the $19^{th}$ and $20^{th}$ centuries has demonstrated how the changes in population density and transportation networks deployment are strictly and positively correlated. Obviously, the road network tends to evolve to better serve the changing density of population. In turn, the road network influences the accessibility and governs the attractiveness of different zones and thus, their growth. However, attractiveness leads to an increase in the demand for these zones, which in turn will lead to an increase of prices. High prices will eventually limit the growth of the most desirable areas. It is the mutual interaction between these processes that we aim to model in the present work. Although there are many other economical mechanisms (type of land use, income variations, etc.) which govern the individual choice of a location for a new ‘activity’ (home, business, etc), we limit ourselves to the two antagonist mechanisms of accessibility and housing price. These loosely defined notions can be taken into account when translated in term of transportation and rent costs. We note that in the context of the structure of land use surrounding cities, von Thünen [@vonthunen:1966] already identified the distance to the center (a simple measure of accessibility) and rent prices as being the two main relevant factors. At first we will discuss separately the two mechanisms of road formation and location choice. In particular, we explicitely consider the shape of the network and model its evolution as the result of a local cost-optimization principle [@Barthelemy:2007]. In classical models used in urban economics, transportation costs are usually described in a very simplified fashion in order to avoid the description of a separate transportation industry [@Fujita]. Also, when space is explicitly taken into consideration, the shape of the transportation networks is rarely considered and transportation costs are computed according to the distance to a city center (as it is the case in the classical von Thünen’s [@vonthunen:1966] or Dixit-Stiglitz’s [@Dixit:1977] models). In these approaches transportation networks are absent, and displacements of goods and individuals are assumed to take place in continuous space. On one side this allows for a more detailed description of the economical processes at play during the shaping of a city. On the other side, these approaches often rely on the hypothesis that the processes shaping a city are slow enough to allow the balancing of the different forces that contribute to these processes, allowing as a consequence the achievement of the global minimum of some opportune cost function. The point of view inspiring our work, instead, is that the evolution of a city is inherently an ‘out-of-equilibrium’ process where the city evolves in time to adapt to continuously changing circumstances. If some sort of optimization or ‘planning’ is driving the growth, it has to be continuously redefined in order to take into account the ever-changing economic and social conditions that are ultimately responsible for the evolution of urban areas. We do not, therefore, assume the optimization of a global cost (or utility) function. Finally, we would like to mention that our goal is not to be as realistic as possible but to consistently reproduce a set of coarse grained and very general features of real cities under a minimal set of plausible assumptions. Alternative explanations might also be possible and it would be interesting to compare our results with those produced in the same spirit. We hope that this simplified model could serve as a first step in the direction of designing more elaborated models. This paper is organized in three main parts. In the first part, we briefly establish the framework to describe the model and discuss the empirical evidences that motivated it. In the second part, we address the issue of how the growth of the local density affects the growth of the road network. In the third part we will study how the road network affects the potential for density growth in different areas. We finally integrate all these elements in the fourth section, where we study the full model and discuss our results. The model: empirical evidences and definition ============================================= Framework --------- In our simplified approach, we represent cities as a collection of points scattered on a two dimensional area (a square of linear size $L$ throughout this study), and connected by a urban road network. The description of the street network adopted here consists of a graph whose links represent roads, and vertices represent roads’ intersections and end points. Although the primary interest here is on roads’ networks [@Cardillo1; @Buhl], it is worth mentioning that transportations networks appear in variety of different contexts including plant/leaves morphology [@Rolland], rivers [@Iturbe], mammalian circulatory systems [@West], commodity delivery [@Gastner], and technological infrastructures [@Schwartz]. Indeed, networks are the most natural and possibly simplest representation of a transportation system  [@stevens; @ball]. It would be impossible to review, even schematically, the approaches and the insights gained in the specific fields mentioned above, but it is at least worth to mention a few studies which attempted to connect the evolution of networks to an optimization principle (as it is the case of the present work). Maybe is it not surprising that man-made transportation networks have been designed with the goal to serve efficiently and cost little [@Gastner], but relevant examples occur in natural sciences as well. It is remarkable, for example, how the Kirchoff law, that determines the current in the edges of a resistor network can be derived assuming the minimization of the dissipated energy [@doyle]. More recently, optimization principles have been successfully applied to the study of the transportation of nutrients through mammalian circulatory systems in order to explain the allometric scaling laws in biology [@West; @Banavar]. River networks constitute a further example where relevant features of the network organization can be derived from an optimality principle [@Iturbe; @maritan]. A last example worth mentioning is that of metabolic networks, where it has been found that specific pathways appear if conditions for optimal growth are assumed (see [ *e. g.*]{} [@price]). It is interesting to notice that there have been attempts to put some of the examples discussed above in the context of a single framework (see [@fittest]). In addition, let us mention that there is a huge mathematical literature that studies optimal networks and the flow they support; Minimal Spanning Trees [@mst], Steiner Trees [@steiner], and Minimum Cost Network Flows [@mcfn] are just three examples. Although the present study assumes a notion of optimality, there are some important aspects that differentiate it from the works discussed above. The first is that the principle of optimality is at work only locally: there is no global cost function that our road networks are supposed to minimize. The second, and possibly more important, is that we attempt to establish a connection between the evolution of the network and that of the quantity that such network is supposed to transport, i.e. population. Transportation networks, as shown from the example cited above, can generally display a large variety of patterns. However, recent empirical studies  [@Batty; @Makse1; @Makse2; @Crucitti; @Jiang; @Cardillo1; @Cardillo2; @Lammer; @Porta; @Roswall:2005; @Jiang:2004] have shown that roads’ networks, despite the peculiar geographical, historical, social-economical processes that have shaped distinct urban areas in different ways, exhibit unexpected quantitative similarities, suggesting the possibility to model these systems through quite general and simple mechanisms. In the following subsection we present the evidences that support the previous statement. Empirical results ----------------- The degree distribution of planar networks decays very fast for large degrees as it usually happens for networks embedded in Euclidean space or with strong physical constraints which prohibits the emergence of hubs [@Amaral:2000]. The degree distribution is therefore strongly peaked around its average (over the whole city) $\langle k \rangle=2E/N\equiv 2e$ ($E$ is the number of edges-the roads-and $N$ is the number of nodes-the intersections). Concerning the average degree of random planar networks, little is known: For one- and two-dimensional lattices $e=1$ and $e=2$, respectively, and a classical result shows that for any planar network $\langle k\rangle\leq 6$, implying $e\leq 3$ (see [*e.g.*]{} [@Itzykson]). It has also been recently shown that planar networks obtained from random Erdos-Renyi graphs over a randomly plane-distributed set of points upon rejection of non-planar occurrences, have $e>13/7$ [@Gerke]. ![Left: Numbers of roads versus number of nodes (ie. intersections and centers) for data from [@Cardillo1] (circles) and from [@Buhl] (squares). In the inset, we show a zoom for a small number of nodes. Right: Total length versus the number of nodes. The line is a fit which predicts a growth as $\sqrt{N}$ (data from [@Cardillo1]).[]{data-label="fig:k_cost"}](fig1a.eps "fig:") ![Left: Numbers of roads versus number of nodes (ie. intersections and centers) for data from [@Cardillo1] (circles) and from [@Buhl] (squares). In the inset, we show a zoom for a small number of nodes. Right: Total length versus the number of nodes. The line is a fit which predicts a growth as $\sqrt{N}$ (data from [@Cardillo1]).[]{data-label="fig:k_cost"}](fig1b.eps "fig:") These facts are summarized in Fig. \[fig:k\_cost\] (left), together with empirical data from $20$ cities in different continents [@Cardillo1]. A first important empirical observation is that $1.05\leq e_{emp}\leq 1.69$ , in a range lying between trees and 2d lattices, and the average degree over all these cities is $\langle k\rangle\approx 2.87$. The strongly peaked degree distribution suggests that a quasi-regular lattice could give a fair account of the road network topology. This suggestion is reinforced if one considers the cumulative length $\ell$ of the roads. With this picture in mind, one would expect that for a given average density $\rho=N/L^2$, the typical inter distance between nodes is $\ell_1\sim \frac{1}{\sqrt{\rho}}$. The total length is then the number of edges times the typical inter-distance which leads to $$\ell\sim E\ell_1\sim \frac{\langle k\rangle}{2}L\sqrt{N}$$ This behavior is reproduced in fig. (\[fig:k\_cost\]b), where a fit of the empirical data in [@Cardillo1] with a function of the form $aN^{1/2}$ gives $a_{emp}\approx 1.46\pm 0.04$ (a fit with a function of the form $aN^{\tau}$ leads to $a\approx 1.51\pm 0.24$ and $\tau\approx 0.49\pm 0.03$). The value $a_{emp}=1.46$ has to be compared with the average degree over all cities. One finds $2.87/2\approx 1.44$ and considering statistical errors, it is hard to reject the hypothesis of a slightly perturbed lattice as a model for the road network. The two empirical facts above lend credibility to the simple picture that city streets are described by a quasi-regular lattice with an essentially constant degree (equal to approximately $3$) and constant road length ($\ell_1\sim 1/\sqrt{\rho}$). There is however a further empirical fact which forces us to reconsider this simple picture. The roads’ network define a tessellations of the surface and the authors of [@Lammer] measured the distribution of the area $A$ of the polygons delimited by the edges of the network. Surprisingly, they found a power law behavior of the form $$P(A)\sim A^{-\alpha}$$ with $\alpha\simeq 1.9$ (the standard error is not available in [@Lammer]). This fact contradicts the simple model of an almost regular lattice since the latter would predict a distribution $P(A)$ very peaked around a value of the order of $\ell_1^2$. The authors of  [@Lammer] also measured the distribution of the form factor given by the area of a cell divided by the area of the circumscribed circle (for this value they use the largest distance $D$ between nodes of the cell, a convention that we adopted): $\phi=4A/(\pi D^2)$. They found that most cells have a form factor between $0.3$ and $0.6$ indicating a large variety of cell shapes. A first challenge is therefore to design a model for planar networks that can reproduce quantitatively these featurees and which is based on a plausible (small) set of assumptions. The simple indicators discussed above show that one cannot model the network by either lattices, Voronoi tessellation, random planar Erdos-Renyi graphs, all these networks having a peaked distribution of areas and form factors. Let’s note that the scale-invariant distribution for cell sizes can be obviously reproduced by assuming by the fractal model of  [@Kalapala:2006] which assumes a self-similar process of road generation. The power law distribution for cell sizes automatically follows from this assumption. In the following, we present a model that relies on a simple plausible mechanism, does not assume self-similarity and quantitatively accounts for the empirical facts presented above. The model of road formation =========================== We first discuss the part of our model that describes the evolution of the road network. Our main assumption is that the network grows by trying to connect to a set of points -the ‘centers’- in an efficient and economic way. These centers can represent either homes, offices or businesses. This parameter free model is based on a principle of local optimality and has been proposed in [@Barthelemy:2007]. For the sake of self-consistency and readability, we first describe this model in detail. The application of optimality principles to both natural and artificial transportation networks has a long tradition [@Stevens; @Ball]. The rationale to invoke a local optimality principle in this context is that every new road is built to connect a new location to the existing road network in the most efficient way [@Bejan]. During the evolution of the street network, the rule is implemented locally in time and space. This means that at each time step the road network is grown by looking only at the current existing neighboring sites. This reflects the fact that evolution histories greatly exceed the time-horizon of planners. The self-organized pattern of streets emerges as a consequence of the interplay of the geometrical disorder and the local rules of optimality. In this regard our model is quite different from approaches to transportation networks where an equilibrium situation is assumed and which are based on either (i) minimization of an average quantity ([*e.g.*]{} the total travel time), or (ii) on the inclusion of many different socio-economical factors ( [*e.g.*]{} land use). Network growth -------------- When new centers (such as new homes or businesses) appear, they need to connect to the existing road network. If at a given stage of the evolution a single new center is present, it is reasonable to assume that it will connect to the nearest point of the existing road network. When two or more new centers are present (as in Fig. \[fig:delta\]) and they want to connect to the same point in the network, we assume that economic considerations impose that a single road - from the chosen network’s point - is built to connect both of them. ![The nearest road to the centers $A$ and $B$ is $M$. The road will grow to point M’. The proposed minimum expenditure principle suggests that the next point M’ will be such that the variation of the total distance to the two points A and B is maximal.[]{data-label="fig:delta"}](fig2.eps) In the example of figure \[fig:delta\], the nearest point of the network to both new centers $A$ and $B$ is $M$. We grow a single new portion of road of fixed length $dx$ from $M$ to a new point $M'$ in order to grant the maximum reduction of the cumulative distance of $A$ and $B$ from the network. This translates in the requirement that $$\delta d=d(M,A)+d(M,B)-[d(M',A)+d(M',B)]$$ is maximal ($dx$ being fixed). A simple calculation shows that the maximization of $\delta d$ leads to $$d\overrightarrow{MM'}\propto \vec{u}_A+\vec{u}_B \label{vec_rule}$$ where $\vec{u}_A$ ($\vec{u}_B$) is the unitary vector from $M$ to $A$ ($B$). The procedure described above is iterated until the road from $M$ reaches the the line connecting $A$ and $B$, where a singularity occurs: $d\overrightarrow{MM'}=0$. From there two independent roads to $A$ and $B$ need to be built to connect to the two new centers. The rule Eq. (\[vec\_rule\]) can be easily generalized to the case of $n$ new centers, and, interestingly, was proposed in the context of visualization of leafs’ venation patterns [@Rolland]. The growth scheme described so far leads to tree-like structures and we implement ideas proposed in [@Rolland] in order to create networks with loops. Indeed, even if tree-like structures are economical, they are hardly efficient: the length of the path along a minimum spanning tree network for example, scales as a power $5/4$ of the Euclidean distance between the end-points [@Duco]. Better accessibility is then granted if loops are present. In order to obtain loops, we assume, following [@Rolland], that a center can affect the growth of more than one single portion of road per time step and can stimulate the growth from any point in the network which is in its relative neighborhood, a notion which has been introduced in [@Toussaint]. In the present context a point $P$ in the network is in the relative neighborhood of a center $C$ if the intersection of the circles of radius $d(P,C)$ and centered in $P$ and $C$, respectively, contains no other centers or point of the network [@Toussaint]. This definition rigorously captures the loosely defined requirement that, for $v$ to belong to the relative neighborhood of $s$, the region between $s$ and $v$ must be empty. At a given time step, a generic center $C$ then stimulates the addition of new portions of road (pointing to P) from all points in the network that are in its relative neighborood, naturally creating loops. When more than one center stimulates the same point P the prescription of (\[vec\_rule\]) is applied and the evolution ends when the list of stimulated points is exhausted (We refer the interested reader to [@Barthelemy:2007] for a detailed exposition of the algorithm). The formula above can be straigthforwardly extended to the case of centers with non-uniform weight $\eta$. This leads to a modified version of Eq. (\[vec\_rule\]), where the sum of distances to be minimized is weighted by $\eta$ and leads to $$d\overrightarrow{MM'}\propto \eta_A\vec{u}_A+\eta_B\vec{u}_B . \label{vec_rule2}$$ where $\eta_A$ and $\eta_B$ can be different. Simulations with non-uniform centers weights show that - as far as the location of ‘heavy’ and ‘light’ centers is uniformly distributed in space, uncorrelated and not broad - that the structure of the network is locally modified, but that its large scale properties are virtually unchanged. In the algorithm presented above, once a center is reached by all the roads it stimulates, it becomes inactive. An interesting variant of this model assumes that centers can stay active indefinetively. In this case we expect a larger effect of the weights’ heterogeneity. We will leave this problem for future studies. In the following, we study networks resulting from the growth process described above. We assume that the appearance of new centers is given exogenously and is independent from the existing road network and from the position and number of the centers already present. The model accounts quantitatively for a list of descriptors -the ones discussed above in an empirical context- that characterize at a coarse grained level the topology of street patterns. At a more qualitative level, the model leads to the presence of perpendicular intersections, and also reproduces the tendency to have bended roads even if geographical obstacles are absent. We show in Fig. \[fig:time\] examples of patterns obtained at different times. The model gives information about the time evolution of the road network: at earlier times, the density is low and the typical inter-distance between centers is large (see Fig. \[fig:time\]). As time passes, the density increases and the typical length to connect a center to the existing road network becomes shorter. Since the number of points grows with time, the simple assumption that the typical road length is given by $1/\sqrt{\rho}$ leads to $\ell_1\sim 1/\sqrt{t}$ which is indeed what the model predicts. ![Snapshots of the network at different times of its evolution: for (a) $t=1,000$, (b) $t=2,000$, (c) $t=3,000$, (d) $t=4,000$. At short times, we have a tree structure and loops appear for larger density values obtained at larger times. For $t=4,000$, we have approximately $1,700$ nodes connected by $2,000$ roads.[]{data-label="fig:time"}](fig3a.eps "fig:") ![Snapshots of the network at different times of its evolution: for (a) $t=1,000$, (b) $t=2,000$, (c) $t=3,000$, (d) $t=4,000$. At short times, we have a tree structure and loops appear for larger density values obtained at larger times. For $t=4,000$, we have approximately $1,700$ nodes connected by $2,000$ roads.[]{data-label="fig:time"}](fig3b.eps "fig:") ![Snapshots of the network at different times of its evolution: for (a) $t=1,000$, (b) $t=2,000$, (c) $t=3,000$, (d) $t=4,000$. At short times, we have a tree structure and loops appear for larger density values obtained at larger times. For $t=4,000$, we have approximately $1,700$ nodes connected by $2,000$ roads.[]{data-label="fig:time"}](fig3c.eps "fig:") ![Snapshots of the network at different times of its evolution: for (a) $t=1,000$, (b) $t=2,000$, (c) $t=3,000$, (d) $t=4,000$. At short times, we have a tree structure and loops appear for larger density values obtained at larger times. For $t=4,000$, we have approximately $1,700$ nodes connected by $2,000$ roads.[]{data-label="fig:time"}](fig3d.eps "fig:") Beyond visual similarities, the model allows quantitative comparisons with the empirical findings The ratio $e=E/N$, initially close to $1$ (indicating that the corresponding network is tree-like), increases rapidly with $N$, to reach a value of order $1.25$ which is in the ballpark of empirical findings. The cumulative length of the roads produced by the model (Fig. \[fig:l\_phi\]a) shows a behavior of the form $a\sqrt{N}$ with $a\approx 1.90$, in good agreement with the empirical measurements $a_{emp}\approx 1.87$). The form factor distribution (Fig. \[fig:l\_phi\]b) has an average value $\phi=0.6$ and values essentially contained in the interval $[0.3,0.7]$ in agreement with the results in [@Lammer] for 20 German cities. Effect of the center spatial distribution ----------------------------------------- An important feature of street networks is the large diversity of cell shapes and the broad distribution of cell areas. So far, we have assumed that centers are distributed uniformly across the plane. Within this assumption, the model predicts a cell area distribution following an exponential (with a large cut-off however) as shown in Fig. \[fig:l\_phi\](d) and Fig. \[fig:area\]. ![Upper left plot: Uniform distribution of points ($1000$ centers, $100$ configurations). In this case, the area distribution is exponentially distributed (bottom left). Upper right plot: Exponential distribution of centers ($5000$ centers, $100$ configurations, exponential cut-off $r_c=0.1$). In this case, we observe a power law (bottom right). The line is a power law fit which gives an exponent $\approx 1.9$.[]{data-label="fig:area"}](fig5a.eps "fig:") ![Upper left plot: Uniform distribution of points ($1000$ centers, $100$ configurations). In this case, the area distribution is exponentially distributed (bottom left). Upper right plot: Exponential distribution of centers ($5000$ centers, $100$ configurations, exponential cut-off $r_c=0.1$). In this case, we observe a power law (bottom right). The line is a power law fit which gives an exponent $\approx 1.9$.[]{data-label="fig:area"}](fig5b.eps "fig:") ![Upper left plot: Uniform distribution of points ($1000$ centers, $100$ configurations). In this case, the area distribution is exponentially distributed (bottom left). Upper right plot: Exponential distribution of centers ($5000$ centers, $100$ configurations, exponential cut-off $r_c=0.1$). In this case, we observe a power law (bottom right). The line is a power law fit which gives an exponent $\approx 1.9$.[]{data-label="fig:area"}](fig5c.eps "fig:") ![Upper left plot: Uniform distribution of points ($1000$ centers, $100$ configurations). In this case, the area distribution is exponentially distributed (bottom left). Upper right plot: Exponential distribution of centers ($5000$ centers, $100$ configurations, exponential cut-off $r_c=0.1$). In this case, we observe a power law (bottom right). The line is a power law fit which gives an exponent $\approx 1.9$.[]{data-label="fig:area"}](fig5d.eps "fig:") The empirical distribution of centers in real cities, however, is not accurately described by an uniform distribution but decreases exponentially from the center [@Makse1; @Makse2]. We thus use such an exponential distribution $P(r)=\exp (-|r|/r_c)$ for the center spatial location and measure the areas formed by the resulting network (in the last section of this point is further discussed). Although most quantities (such as the average degree and the total road length) are not very sensitive to the center distribution, the impact on the area distribution is drastic. In Fig. \[fig:area\] a power law with exponent equal to $1.9\pm 0.1$ is found, in remarkable agreement with the empirical facts reported in [@Lammer] for the city of Dresden. Although we cannot claim that this exponent is the same for all cities, the appearance of a power law in good agreement with empirical observations confirms the fact that the simple local optimization principle is a possible candidate for the main process driving the evolution of city street patterns. This result also demonstrates that the centers’ distribution is crucial in the evolution process of a city. The optimization process described above has several interesting consequences on the global pattern of the street network when geographical constraints are imposed, as illustrated by the following example. We simulated the presence of a river assuming that new centers cannot appear on a stripe of given width (and are otherwise uniformly distributed). ![In the presence of an obstacle (here a ‘river’ delimited by the two dotted line) in which the centers are not allowed to be located, the local optimization principle leads to a natural solution with a small number of bridges.[]{data-label="fig:river"}](fig6.eps) The resulting pattern is shown in Fig. \[fig:river\]. The local optimization principle naturally creates a small number of bridges that are roughly equally spaced along the river and organizes the road network. To conclude, it is worth noting that in the present framework we didn’t attempt the modelization of planning efforts. Simulations show that, at the present simplified stage, the presence of a skeleton of “planned” large roads has the effect of partitioning the plane in different regions where the growth of the network is dominated by the mechanism described above, and reproducing on a smaller scale the structures shown in fig. \[fig:time\]. Hierarchical structure of the traffic ------------------------------------- Finally, we discuss now the presence of hierarchy in the network generated by the model. Indeed, geographers have recognized for a long time (see [*e.g.*]{} [@Christaller]) that many systems are organized in a hierarchical fashion. Highways are connected to intermediate roads which in turn dispatch the traffic through smaller roads at smaller spatial scales. In order to test for the existence of such a hierarchy in our model, we use the edge betweenness centrality as a simple proxy for the traffic on the road network. For a generic graph, the betweenness centrality $g(e)$ of an edge $e$ [@Freeman; @Goh:2001; @Barthelemy:2003] is the fraction of shortest paths between any pair of nodes in the network that go through $e$. Allowing the possibility of multiple shortest paths between two points, one has $$g(e)=\sum_{s\neq t}\frac{\sigma_{st}(e)}{\sigma_{st}}$$ where $\sigma_{st}$ is the number of shortest paths going from $s$ to $t$ and $\sigma_{st}(e)$ is the number of shortest paths going from $s$ to $t$ and passing through $e$. Central edges are therefore those that are most frequently visited if shortest paths are chosen to move from and to arbitrary points. We computed this quantity for all edges of the road network generated by our algorithm. It appears that this quantity $g(e)$ is broadly distributed and varies over more than $6$ orders of magnitude. In order to get a simple representation of this quantity we arbitrarily group the edges in three classes: $[1,10^4]$, $[10^4,10^5]$, $[10^5,\infty]$ and plot them with different thicknesses. In particular, we see in Fig \[fig:hierarchy\], that edges with the largest centrality (represented by the thickest line) form almost a tree of large arteries. Proportionality between traffic and edge-centrality, as defined above, is virtually equivalent to assuming: [*i)*]{} a uniform origin-destination matrix, [*ii)*]{} everybody choose the shortest path to reach a destination, and [*iii)*]{} roads are “large” enough to support the traffic generated by [*i)*]{} and [*ii)*]{} without congestion effects. Under these assumption one indeed observes a hierarchy of smaller roads and streets with a decreasing typical length and the existence of a hierarchical structure of arteries, roads and streets. Location of centers: effect of density and accessibility ======================================================== In the simple version of the model presented above, the location of centers are independent from the topology of the road network. In real urban systems, this is however unlikely to happen. There is an extensive spatial economics literature (see [@Fujita] and references therein) that focuses on the several factors that may potentially influence the choice location for new businesses, homes, factories, or offices (see also [@Jensen] and references therein). Empirical evidences suggest a strong correlation between transportation networks and density increase have been recently provided by Levinson [@Levinson2]. Our goal here is to discuss, based on very simple and plausible assumptions, the coupled evolution of the road network and the population density. We first divide the city in square sectors of area $S$, and we assume that the choice of a location for a new center is governed by a probability $P(i)$ that one ($i$) of these sectors is chosen. This probability, which reflects the attractiveness of a location, depends a priori on a large number of factors such as accessibility, renting prices, income distribution, number and quality of schools, shops, etc. A key observation made by previous authors (see for example  [@Brueckner] and references therein) is that commuting cost differences must be balanced by differences in living spaces prices. We will follow this observation and we will thus focus on two factors which are the rent price and the accessibility (which we will reduce to commuting costs). Rent price and accessibility ---------------------------- The housing price of a given location is probably determined by many factors comprising tax policies, demography, etc., and is by itself an important subject of study (see for example [@Goodman]). We will make here the simplifying assumption that the rent price is an increasing function of the local density (for each grid sector) $\rho(i)=N(i)/S$, where $N(i)$ is the number of centers in the sector $(i)$, and in particular that the rent price $C_R$ is directly proportional to the local density of population (which can be seen as the first term of an expansion of the price as a function of the density) $$C_R(i)=A\rho(i)$$ where $A$ is some positive prefactor corresponding to the price per density. We note here that a more general form of the type $C_R=A\rho(i)^\tau$ could be used. A preliminary study suggests that as long as the rent cost is an increasing function of the density, our results remain qualitatively unchanged. It would be however very interesting to measure this function empirically. The second important factor for the choice of a location is its accessibility. Locations which are easily accessible and which allow to reach easily arbitrary destinations are more attractive, all other parameters being equal. Also, for a new commercial activity, high traffic areas can strongly enhance profit opportunities. In terms of the existing network, the best locations are therefore the most central and standard models of city formation (see for example [@Fujita]) indeed integrate the distance to the center and its associated (commuting) cost as a main factor. Euclidean distance, however, can be a poor estimator of the effective accessibility of a given location, if this location is poorly connected to the transportation network. This is why the notion of centrality has to be considered not only in geographical terms, but also from the point of view of the network that grants mobility. The possibility to easily reach an arbitrary location when movement is constrained by a network is nicely captured in quantitative terms by the notion of node betweenness centrality. In the previous section, we defined the [*edge*]{} betweenness centrality and here we need a similar quantity defined for nodes rather than for edges. The node betweenness centrality $g(v)$ [@Freeman; @Goh:2001; @Barthelemy:2003] of a node $v$ is defined as the fraction of shortest paths between any pair of points in the network that go through $v$. The mathematical expression of this quantity is then $$g(v)=\frac{1}{N(N-1)}\sum_{s\neq t}\frac{\sigma_{st}(v)}{\sigma_{st}}$$ where $\sigma_{st}$ is the number of shortest paths going from $s$ to $t$ and $\sigma_{st}(v)$ is the number of shortest paths going from $s$ to $t$ and passing through $v$. Betweenness centrality was initially introduced as a natural substitute for geometric centrality in graphs that are not embedded in Euclidean space (see fig. \[fig:bw\]). ![Illustration of the notion of node betweenness centrality for a non planar graph[]{data-label="fig:bw"}](fig8.eps) Betweenness also naturally serves our purpose to quantify accessibility on planar graphs, especially in our simplified framework where an explicit distinction between resources and users has been sacrified to the sake of simplicity. It is important to note that betweenness centrality, on planar graphs, is strictly correlated to other, more common, measures of centrality. The two first panels of fig \[fig:bw2\] show the contour and the 3D plot of node-betweeness for a Manhattan-like grid of 25 blocks per side and clearly show how central nodes are those that are the most frequently visited if shortest paths are chosen to move from and to arbitrary points. The third panel of the same figure shows the relation between the betweenness of a node and its average distance from all other nodes (along the same square grid). This plot demonstrates that the larger the betweenness of a node is, the shorter is its average distance from a generic node. ![Top panel: Contour plot of node betweenness centrality -not normalized-for a square grid of linear size 25 (A light color indicates a high value of the betweenness). Middle panel: 3D plot of node betweenness centrality (not normalized) for the same square grid. Bottom panel: Plot of the node betweenness of a generic node vs its average distance from all the other nodes.[]{data-label="fig:bw2"}](fig9a.eps "fig:") ![Top panel: Contour plot of node betweenness centrality -not normalized-for a square grid of linear size 25 (A light color indicates a high value of the betweenness). Middle panel: 3D plot of node betweenness centrality (not normalized) for the same square grid. Bottom panel: Plot of the node betweenness of a generic node vs its average distance from all the other nodes.[]{data-label="fig:bw2"}](fig9b.eps "fig:") ![Top panel: Contour plot of node betweenness centrality -not normalized-for a square grid of linear size 25 (A light color indicates a high value of the betweenness). Middle panel: 3D plot of node betweenness centrality (not normalized) for the same square grid. Bottom panel: Plot of the node betweenness of a generic node vs its average distance from all the other nodes.[]{data-label="fig:bw2"}](fig9c.eps) The betweenness thus extends the concept of geographical centrality to networks whose structure is not a lattice or planar. In our simple model, we will assume that accessibility reduces here to the facility of reaching quickly any other location in the network. This can also be seen as the average commuting cost which in previous models [@Fujita] was assumed to be proportional to the distance to the center. The natural extension for a network is then to take the transportation cost depending on the betweenness centrality. For each sector $S_i$ of the grid, we first compute the average betweenness centrality as $$\overline{g}(i)=\frac{1}{N(i)}\sum_{v\in S_i}g(v) .$$ where the bar represents the average over all nodes (centers and intersections) which belong in a given sector. Transportation costs are a decreasing function of the betweenness centrality and we will assume here that the transportation cost $C_T(i)$ for a center in sector $(i)$ is given by $$C_T(i)=B(g_m-\overline{g}(i))$$ where $B$ and $g_m$ are positive constants (other choices, as long as the cost decreases with centrality, linearly or not, give similar qualitative results). Finally, we will assume (as it is frequently done in many models, see for example [@Brueckner] and references therein) that all new centers have the same income $Y(c)=Y$. This assumption is certainly a rough approximation, as demonstrated by effects such as urban segregation, but in order to not overburden our model, we will neglect income disparities in the present study. The net income of a new center $c$ in a sector $(i)$ is then $$K(i)=Y-C_R(i)-C_T(i).$$ The higher the net income $K(i)$ and the more likely the location $(i)$ will be chosen for the implantation of a new business, home, etc. In urban economics the location is usually chosen by minimizing costs, and we relax this assumption by defining the probability that a new center will choose the sector $(i)$ as its new location under the form $$P(i)=\frac{e^{\beta K(i)}}{\sum_je^{\beta K(j)}} .$$ This expression rewritten as $$P(i)=\frac{ e^{\beta ( \lambda \overline{g}(i)-\rho(i)) } } {\sum_j e^{\beta ( \lambda \overline{g}(j)-\rho(j)) } } \label{eq:proba}$$ where $\beta A$ is redefined as $\beta$ and where $\lambda=B/A$. For numerical simulations, the local density is normalized by the global density $N/L^2$, in order to have the density and centrality contributions defined in the same interval $[0,1]$. The relative weight between centrality and density is then described by $\lambda$ and the parameter $\beta$ implicitly describes in an ‘effective’ way all the factors (which could include anything from individual taste to the presence of schools, malls, etc) that have not been explicitly taken into account, and that may potentially influence the choice of location. If $\beta\approx 0$, cost is irrelevant and new centers will appear uniformly distributed across the different sectors: $$P(i)\sim \frac{1}{N(i)} .$$ In the opposite case, $\beta\rightarrow\infty$, the location with the minimal cost will be chosen deterministically. $$\begin{cases} P(i)=1 & \text{for $i$ such that $K(i)$ is minimum} \\ P(i)=0 & \text{for all other sectors} \end{cases}$$ The parameter $\beta$ can thus be used in order to adjust the importance of the cost relative to that of other factors not explicitly included in the model. Co-evolution of the network and the density =========================================== We finally have all the ingredients needed to simulate the simultaneous evolution of the population density and the road network. Before introducing the full model, analogously to what we have done for the first part of the model, it is worth to study this second part separately. To do that, we consider a toy -one dimensional- case, where the network plays no role, since a single path only exists between each pair of nodes. Despite the simplicity of the setting, it is possible to draw some general conclusion. One-dimensional model --------------------- We assume that the centers are located on a one-dimensional segment $[-L,L]$. Since only a single path exists between any two points, the calculation of centrality is trivial. In the continuous limit, and for a generic location $x$ it can be written as the product of the number of points that lie at the right and left of the given location $$g(x)=\int_{-L}^{x}\rho(y,t)dy \left[N-\int_{-L}^{x}\rho(y,t)dy\right]$$ where $\rho(x,t)$ is the density at $x$. The equation for the density therefore reads: $$\partial_{t} \rho(x,t) =e^{\beta \left[ \lambda \frac{\int_{-L}^{x}\rho(y,t)dy}{N} \frac{( N-\int_{-L}^{x}\rho(y,t)dy)}{N}-\frac{\rho(x,t)}{N}\right]} \label{eq:oned}$$ where $N=\int_{-L}^{L}\rho(y,t)dy$. The numerical integration of Eq. (\[eq:oned\]) shows that, after a transient regime, the process locks in a pattern of growth in which the total population grows at a constant rate $$N=\int_{-L}^{L}\rho(y,t)dy \propto t.$$ This suggests that a solution for large $t$ can be found via the separation of variables under the form $$\rho(x,t)=\alpha f(x) t \label{eq:sepvar}$$ where one can set $\int_{-L}^{L}f(x)dx=1$ without loss of generality. Plugging the expression (\[eq:sepvar\]) into Eq. (\[eq:oned\]) one gets $$\alpha f(x)= e^{ \beta \left[ \lambda \int_{-L}^{x}f(y)dy (1-\int_{-L}^{x}f(y)dy)-f(x) \right] } \label{onedeq}$$ where $\alpha$ is an integration constant to be determined. An explicit solution for the inverse $f^{-1}(x)$ can be achieved via the Lambert function (Lambert’s function is the principal branch of the inverse of $z=w \exp^w$), but the expression is not particularly illuminating and it is therefore not presented here. Several facts can however be understood using a direct numerical integration of Eq. (\[eq:oned\]) or the simulation of the relative stochastic process: - [At large times, population in different location grows with a rate $f(x)$ that depends on the location but not explicitely on time. This is a direct consequence of Eq. (\[eq:sepvar\]) and is obviously a different behavior from uniform growth. The ratio of population density in two different locations $x_1$ and $x_2$ stabilizes in the long run to $f(x_1)/f(x_2)$]{} - [Although $\beta$ models the ‘noise’ in the choice of location and $\lambda$ the relative importance of centrality as compared to density, they have similar effects on the expected density in a given location. An increase in $\beta$ and $\lambda$ corresponds a concentration of density in the areas of large centrality and a steeper decay of density towards the periphery, as shown in fig. \[oned\]a. This can intuitively be understood by looking [*e.g.*]{} at the role played by the parameter $\beta$ in Eq. (\[oned\]): as $\beta$ decreases to 0, the differences in rate of growth in different locations becomes negligible ]{} - [Eq. (\[eq:oned\]) describes the average (or expected) behavior of the population density over time. Numerical simulations of the corresponding stochastic process show fluctuations from the above mentioned expected value. Such fluctuations increases as noise increases (ie. when $\beta$ decreases).]{} - [Numerical integration of Eq. (\[eq:oned\]) suggests that, as $\lambda$ increases, the decay of density assumes a power law form whose exponent depends on $\beta$ and $\lambda$ and approaches $-1$ as $\lambda$ gets very large. This can be explained by assuming $f(x) \approx \gamma x^{-r} $, and using Eq. (\[onedeq\]) and its derivative both computed in $L$. The derivative of $f(x)$ is: $$f'(x)=\beta \lambda f(x)\left( 1- 2 \int_{-L}^{x}f(y)dy \right) f(x).$$ The above expression can be computed in $x=L$, taking into account that that $\int_{-L}^{L}f(x)dx=1$ and the assumed algebraic functional form of $f(x)$. This leads to $$\gamma r L^{-1} = \gamma \beta L^{-r}( \lambda \gamma - r \gamma L^{-1}).$$ In the limit of large $L$ one can keep only the leading orders in $L$, and match the power and the coefficient of the leading order on the two sides of the equation above. This gives $r=1$ and $\gamma= 1/(\beta \lambda)$. The validity of this argument can be verified looking at fig. \[oned\]b, where $\ln(f)$ is plotted vs $\ln(x)$ to highlight the power law behavior of $f(x)$ and where the line $1/ (\beta \lambda x)$ has been plotted as a reference for the case $\beta = 10$ and $\lambda=10$.]{} ![The stationary growth rate for different values of the parameters.(a) Large values of $\beta$ and $\lambda$ implies larger degree of centralization and a faster decay of density from center to periphery. (b) At large values of $\lambda$ the decay of density becomes algebraic for location away from the center. The exponent approaches $-1$ and $f(x)$ is approximated in that region by $1/(\beta \lambda /x)$.[]{data-label="oned"}](fig10a.eps "fig:"){width="25.00000%"} ![The stationary growth rate for different values of the parameters.(a) Large values of $\beta$ and $\lambda$ implies larger degree of centralization and a faster decay of density from center to periphery. (b) At large values of $\lambda$ the decay of density becomes algebraic for location away from the center. The exponent approaches $-1$ and $f(x)$ is approximated in that region by $1/(\beta \lambda /x)$.[]{data-label="oned"}](fig10b.eps "fig:"){width="25.00000%"} This simple one-dimensional model thus allowed us to understand some basic features of the model that will be discussed in their full generality in the next section. Two-dimensional case: existence of a localized regime ----------------------------------------------------- We now apply the probability in Eq. (\[eq:proba\]) to the growth model described in the first part of this paper. The process starts with a ‘seed’ population settlement (few centers distributed over a small area) and a small network of roads that connects them. At any stage, the density and the betweenness centrality of all different subareas are computed, and a few new centers are introduced. Their location in the existing subareas is determined according to the probability defined in Eq. (\[eq:proba\]). Roads are then grown until the centers that just entered the scene are connected to the existing network. This process is iterated until the desired number of centers has been introduced and connected. In the two panels of figure \[fig:lambda\] we show the emergent pattern of roads that is obtained when $\lambda$ is small and very large, respectively. ![Networks obtained for different values of $\lambda$ (and for $N=500$ and $\beta=1$). On the left, $\lambda=0$ and only the density plays a role and we obtain a uniform distribution of centers. On the right, we show the network obtained for $\lambda=8$. In this case, the centrality is the most important factor leading to a few dominant areas with high density.[]{data-label="fig:lambda"}](fig11.eps) When $\lambda$ is small the density plays the dominant role in determining the location of new centers. New centers appear preferably where density is small, smoothing out the eventual fluctuations in density that may occur by chance and the resulting density is uniform. On the other hand, when $\lambda$ is very large, centrality plays the key role, leading to a city where all centers are located in the same small area. The centrality has thus an effect opposite to that of density and tends to favor concentration. We will now describe in more details the transition between the two regimes described above. We compute, in the two cases, the following quantity (previously introduced in a different context [@Derrida:1987; @Barthelemy:2003b]): $$Y_2=\sum_i\left[\frac{N(i)}{N}\right]^2$$ where the sum runs over all sectors which number is $N_s$. In the uniform case, all the $N(i)$ are approximately equal and one obtains $Y_2\sim 1/N_s$, which is usually small. In contrast, when most of the population concentrates in just a few sectors which represent a finite fraction of the total population, we obtain $Y_2\sim 1/n$ where $n$ represents the order of magnitude of these highly-populated sectors- the ‘dominating sectors’. The quantity $$\sigma=\frac{1}{Y_2N_s}$$ gives therefore the fraction of dominating sectors. ![Fraction of dominating sectors (obtained for $500$ centers and averaged over $100$ configurations). When $\lambda$ is small, the center distribution is more uniform and $\sigma$ is large (close to $100\%$).When $\lambda$ increases, we see the appearance of a few sectors dominating and concentrating most of the population. This effect is smoothen out for smaller values of $\beta$ corresponding to the possibility of choice. []{data-label="fig:fraction"}](fig12.eps) The behavior of $\sigma$ vs. $\lambda$ is shown in Fig. \[fig:fraction\]. We observe that $\sigma$ decreases very fast when $\lambda$ increases, signaling that a phenomenon of localization sets in as soon as transportation costs are involved. We conclude this section discussing the role played by the parameter $\beta$. Analogously to what happens in the one dimensional case, the concentration effect is weakened by a small values of $\beta$. The parameter $\beta$ describes the overall importance of the cost-factors with respect to other factors that have not been explicitly taken into account, or, equivalently, the possibility of choice. Indeed, when $\beta$ is very large, the location which maximizes the cost is chosen. In contrast, when the parameter $\beta$ is small, the cost differences are smoothen out and a broader range of choices is available for new settlements. Figure \[fig:fraction\] illustrates the importance of choice. In particular, the appearance of large-density zones (controlled by the importance of transportation accessibility) is counterbalanced by the possibility of choice and the resulting pattern is more uniform. Density profile: the appearance of core districts ------------------------------------------------- In this last part, we describe the effect of the interplay of transportation and rent costs on the decay of population density from the city center. In the following, the core district is identified as the sector with the largest density. The whole plane is then divided in concentric shells with internal radius $r$ and width $dr$. The density profile $\rho (r)$ is given by the ratio of the number $\delta n$ of centers in a shell to its surface $\delta S(r)$ $$\rho(r)=\frac{\delta n}{\delta S}$$ For small $\lambda$, the density is uniform, as expected. In figure \[fig:density\] we show the density profile $\rho(r)$ in the case of $\lambda$ large, ![Density profiles for $\lambda=8$ ($N=200$, averaged over $500$ configurations). The decay of the density profile is well fitted by an exponential, signalling the appearance of a well-defined core district (error bars represent one standard deviation). []{data-label="fig:density"}](newfig13.eps) where we observe a fast exponential decay of the form $\exp{-r/r_c}$, in agreement with empirical observations [@Makse1]. This behavior is the signature of the appearance of a well-defined core district of typical size $r_c$, whose typical size $r_c$ decreases with $\lambda$. This simplified model predicts, therefore, the existence of a highly populated central area whose size can be estimated in terms of the relative importance of transport and rent costs. Discussion and perspectives =========================== We presented a basic model that describes the impact of economical mechanisms on the evolution of the population density and the topology of the road network. The interplay between rent costs and demand for accessibility leads to a transition in the population spatial density. When transportation costs are moderate, the density is approximately uniform and the road network is a typical planar network that does not show any strong heterogeneity. In contrast, if transportation costs are higher, we observe the appearance of a very densely populated area around which the density decays exponentially, in agreement with previous empirical findings. The model also predicts that the demand for accessibility easily prevails on the disincentive constituted by high rent costs. A very important ingredient in modeling the evolution of a city is how individuals choose the location for a new business or a new home. We isolated in this work the two important factors of rent price and transportations costs. For these costs, we assume some reasonable forms but it is clear that large scale empirical measures are needed. In particular, it would be interesting to characterize empirically how the rent price varies with the density and how transportation costs varies with the centrality. Possible outcomes to these studies would be to give an idea of the value of the parameter $\lambda$ (and possibly also $\beta$) and thus to determine how much the city is centralized. As it happens in every modeling effort, a satisfactory compromise between realism and feasibility must be found and we opted for sacrificing some important economic considerations in order to be able to explicitly take into account the topology of the road transportation network and not the distance to a center only, as it is usually assumed in most models. Our model predicts so far the appearance of a core center, but it is however known that cities present a large diversity in their structure, ranging from a monocentric organization to different levels of polycentrism. In addition, interesting scaling relations between different parameters (total wages, walking speed, total traveled length, etc) and population size were recently found  [@Bettencourt:2007; @Moses:2007] showing that beyond the apparent diversity, there are some fundamental processes driving the evolution of a city. These different results appear as various facets of the process of city formation and evolution and it is at this stage not clear how to connect the scaling to the structural organization of a city and more generally, how to reconciliate the different existing results in a unified picture. We believe that the present model- modified or generalized- could help for future studies in this direction. Acknowledgments. We thank G. Santoboni for many discussions at various stages of this work, and two anonymous referees for several important comments and suggestions. MB also thanks Indiana University for its warm welcome where part of this work was performed. [99]{} UN Population division. <http://www.unpopulation.org>. W. Christaller, [*Central Places in Southern Germany*]{}. English translation by C. W. Baskin, London: Prentice hall, (1966). D. Levinson and B. Yerra, Transportation Science [**40**]{}, 179-188 (2006). M. Fujita, P. Krugman, A.J. Venables, [*The Spatial Economy: Cities, Regions, and International Trade*]{}, The MIT Press, Cambridge (1999). D. Levinson Journal of Economic Geography [**8**]{}, 55-77, (2008) J.H. von Thünen, [*Von Thünen’s isolated state*]{}, Oxford: Pergmanon Press, 1966. M. Barthélemy and A. Flammini, Physical Review Letters, [**100**]{}, 138702 (2008). A.K. Dixit and J.E. Stiglitz, [*Monopolistic competition and optimum product diversity*]{}, Amer. Eco. Rev. [**67**]{}, 297-308 (1977). V. Kalapala, V. Sanwalani, A. Clauset, and C Moore [*Scale invariance in road networks*]{}, Phys. Rev. E, [**73**]{}, 026130 (2006). A. Cardillo, S. Scellato, V. Latora S. Porta, Phys. Rev. E [**73**]{}, 066107 (2006). S. Scellato, A. Cardillo, V. Latora, S. Porta, arXiv:physics/0511063. J. Buhl, J. Gautrais, N. Reeves, R.V. Sole, S. Valverde, P. Kuntz, G. Theraulaz, Eur. Phys. J. B [**49**]{}, 513-522 (2006). A. Runions, A. M. Fuhrer, B. Lane, P. Federl, A.-G. Rolland-Lagan, and P. Prusinkiewicz, ACM Transactions on Graphics 24(3): 702-711 (2005). B. Duplantier, J. Stat. Phys. [**54**]{}, 581 (1989). A. Coniglio, Phys. Rev. Lett. [**62**]{}, 3054 (1989). G. T. Toussaint, Pattern Recognition, [**12**]{}, 261 (1980). J. W. Jaromczyk, and G. T. Toussaint, Proc. IEEE, [ **80**]{}, 1502 (1992). I. Rodriguez-Iturbe and A. Rinaldo,[*Fractal river basins: chance and self-organization*]{}, Cambridge University Press, Cambridge (1997). G. B. West, J. H. Brown, and B. J. Enquist , Science [**276**]{}, 122, (1997) ; M. T. Gastner, and M. E. J. Newman, Phys. Rev. E, [**74**]{}, 016117, (2006). M. Schwartz, [*Telecommunication networks: protocols, modelling and analysis*]{}, Addison-Wesley Longman Publishing Co., Inc. Boston, MA, USA (1986). P. S. Stevens, [*Patterns in Nature*]{}, Little, Brown, Boston, (1974). P. Ball, [*The Self-Made Tapestry: Pattern Formation in Nature*]{} Oxford University Press, Oxford, (1998). G. Doyle and J. L. Snell, [*Random Walk and Electric Networks*]{}, American Mathematical Society, Providence, (1989). J. R. Banavar, A. Maritan, and A. Rinaldo, Nature [**398**]{}, 130–132 (1999). A. Maritan, F. Colaiori, A. Flammini, M. Cieplak, and J. R. Banavar, Science [**272**]{}, 984–986 (1996). N. D. Price, J. L. Reed , B. O. Palsson , Nat. Rev. Microbiol. [**2**]{}, 886–897, (2004). J. R. Banavar, F. Colaiori, A. Flammini, A. Maritan, and A. Rinaldo, Phys. Rev. Lett. [**84**]{}, 4745, (2000). R. L. Graham, and P. Hell, Ann. History Comput. [**7**]{}, 43-57, (1985). M.W. Bern and R. L. Graham, Sci. Am. [**260**]{}, 66–71, (1989). R. K. Ahuja, T. L. Magnanti, and J. B. Orlin, [*Network Flows*]{}, Prentice Hall, New Jersey, (1993). M. Batty, [*Cities and Complexity*]{}, The MIT press, Cambridge (2005). H. A. Makse, S. Havlin, and H. E. Stanley, Nature [**377**]{}, 608, (2002). H. Makse, J.S. Andrade, M. Batty, S. Havlin, H. E. Stanley, Phys. Rev. E [**58**]{}, 7054 (1998). P. Crucitti, V. Latora, S. Porta, Phys. Rev. E [**73**]{}, 036125 (2006). B. Jiang, Env. Plann. B [**31**]{}, 151-162 (2002). S. Lammer, B. Gehlsen, D. Helbing, Physica A [**363**]{}, 89, (2006). S. Porta, P. Crucitti, V. Latora, ArXiv:physics/0506009 (2005). M. Roswall, A. Trusina, P. Minnhagen, K. Sneppen, arXiv:cond-mat/0407054. B. Jiang, C. Claramunt, Env. Plann. B [**31**]{}, 151-162 (2004). L.A.N. Amaral, A. Scala, M. Barthélemy, H.E. Stanley, Proc. Natl. Acad. Sci. (USA) [**97**]{}, 11149 (2000). C. Itzykson and J.-M. Drouffe, [*Statistical field theory*]{} Vol. 2, Cambridge University Press (1989). S. Gerke, C. McDiarmid, Combinatorics, Probability and Computing [**13**]{}, 165, (2004). P. S. Stevens, [*Patterns in Nature*]{}, Little, Brown, Boston, (1974). P. Ball, [*The self-made tapestry: pattern formation in Nature*]{}, Oxford University Press,Oxford, (1998). A. Bejan and G.A. Ledezma, Physica A [**255**]{}, 211-217 (1998). P. Jensen, Phys. Rev. E (R) [**74**]{}, 035101 (2006). J.K. Brueckner, in E.J. Mills (Ed.) [*Handbook of regional and urban economics*]{}, volume 2, pp. 821-845 (1987). A.C. Goodman, Journal of Urban Economics [**23**]{}, 327-353 (1988). L. C. Freeman, Sociometry [**40**]{}, 35 (1977) . K.-I. Goh, B. Kahng, D. Kim, Phys. Rev. Lett. [ **87**]{}, 278701 (2001). M. Barthélemy, Eur. Phys. J. B [**38**]{}, 163 (2003). B. Derrida, H. Flyvbjerg, J. Phys. A: Math. Gen. [**20**]{}, 5273-5288 (1987). M. Barthélemy, B. Gondran, E. Guichard, Physica A [**319**]{}, 633-642 (2003). L.M. Bettencourt, J. Lobo, D. Helbing, C. Kuhnert, and G.B. West, Proc. Natl. Acad. Sci. (USA) [**104**]{}, 7301-7306 (2007). H. Samaniego, and M.E. Moses “Cities as Organisms: Allometric Scaling as an Optimization Model to Assess Road Networks in the USA,” presented at the Access to Destinations II conference, Minneapolis, August 2007.

--- abstract: 'For one spatial variable, a new kind of nonlinear wave equation for Emden-Fowler type is considered with boundary value null and initial values. Under certain conditions on the initial data and the exponent $p$, we exhibit that the viscoelastic term leads our problem to be dissipative and the global solutions still non-exist in $L^2$ at given finite time.' address: - 'Lakhdar Kassah Laouar Departement de Mathematiquess, Université de Constantine 1, Algerie.' - 'Khaled zennir First address: Department of Mathematics, College of Sciences and Arts, Al-Ras, Qassim University, Kingdom of Saudi Arabia. Second address: Laboratory LAMAHIS, Department of mathematics, University 20 Août 1955- Skikda, 21000, Algeria' - 'Amar Guesmia Laboratoire de LAMAHIS, Departement de mathematiques, Université 20 Août 1955- Skikda, 21000, Algerie' author: - 'Lakhdar Kassah Laouar, Khaled zennir and Amar Guesmia' title: 'Blow up at finite time for wave equation in viscoelasticity: a new kind for one spatial variable Emden-Fowler type ' --- \[section\] \[theorem\][Lemma]{} Introduction ============= We consider a new kind of Emden-Fowler type wave equation in viscoelasticity $$\left\{ \begin{array}{ll}\label{e1} t^2u''- u_{xx} +\int_{1}^{t}\mu\left(s\right) u_{xx}\left(t-s\right) ds =u^p\quad \hbox{ in }[1,T) \times(r_1,r_2),\\ u(1,x) =u_0(x) \in H^2(r_1,r_2) \cap H_0^{1}(r_1,r_2),\\ u'(1,x) =u_1(x) \in H_0^{1}(r_1,r_2) \end{array} \right.$$ where $p>1$, $r_1$ and $r_2$ are real numbers and the scalar function $\mu$ (so-called relaxation kernel) is assumed to only be $\mu: \mathbb{R}^+\rightarrow \mathbb{R}^+$ of $C^1$, nonincreasing and satisfying $$\begin{aligned} \mu(0)>0, 1-\int_{0}^{\infty}e^{s/2}\mu(s)ds=l>0.\label{2.1} \end{aligned}$$ The study of the Emden-Fowler equation originated from earlier theories concerning gaseous dynamics in astrophysics around the turn of the 20-th century. The fundamental problem in the study of stellar structure at that time was to study the equilibrium configuration of the mass of spherical clouds of gas. The Emden-Fowler equation has an impact on many astrophysics evolution phenomena. It has been poorly studied by scientists until now, essentially in the qualitative point of view.\ Under the assumption that the gaseous cloud is under convective equilibrium (first proposed in 1862 by Lord Kelvin [@T1]), Lane studied the equation $$\frac{d}{dt}\Big(t^2\frac{du}{dt}\Big) +t^2u^p=0, \label{e*}$$ for the cases $p=1.5$ and $2.5$. Equation (\[e\*\]) is commonly referred to as the Lane-Emden equation [@C1]. Astrophysicists were interested in the behavior of the solutions of (\[e\*\]) which satisfy the initial condition: $u(0) =1$, $u'(0) =0$. Special cases of (\[e\*\]), namely, when $p=1$ the explicit solution to $$\frac{d}{dt}\big(t^2\frac{du}{dt}\big) +t^2u=0,\quad u(0) =1,\; u'(0)=0$$ is $u=\sin(t)/t$, and when $p=5$, the explicit solution to $$\frac{d}{dt}\big(t^2\frac{du}{dt}\big) +t^2u^{5}=0,\quad u(0) =1,\; u'(0)=0$$ is $u=1/\sqrt{1+t^2/3}$.\ Many properties of solutions to the Lane-Emden equation were studied by Ritter [@R2] in a series of 18 papers published during 1878-1889. The publication of Emden’s treatise Gaskugeln [@E1] marks the end of first epoch in the study of stellar configurations governed by (\[e\*\]). The mathematical foundation for the study of such an equation and also of the more general equation $$\frac{d}{dt}\Big(t^{\rho}\frac{du}{dt}\Big) +t^{\sigma}u^{\gamma}=0,\quad t\geq0, \label{e**}$$ was made by Fowler [@F1; @F2; @F3; @F4] in a series of four papers during 1914-1931.\ The first serious study on the generalized Emden-Fowler equation $$\frac{d^2u}{dt^2}+a(t) | u| ^{\gamma}\operatorname{sgn}u=0,\quad t\geq0,$$ was made by Atkinson and *al*.\ Recently, M.-R. Li in [@L11] considered and studied the blow-up phenomena of solutions to the Emden-Fowler type semilinear wave equation $$t^2u_{tt}-u_{xx}=u^p\quad \hbox{ in }[1,T) \times(a,b)).$$\ The present research aims to extend the study of mden-Fowler type wave equation to the case when the viscoelastic term is injected in domain $[r_1,r_2]$ where there is no result about this topic. Thus, a wider class of phenomena can be modeled.\ The main results here are to exhibit the role of the viscoelasticity, which makes our problem dissipative, in the Blow up of solutions in $L^2$ at finite time given by $$\ln T_1^{\ast}, s.t. T_1^{\ast}=\frac{2}{p-1}T_1^{\ast}=\frac{2}{p-1}\Big( \int_{r_1}^{r_2} \vert u_0\vert dx\Big) \Big(\int_{r_1}^{r_2} u_0u_1dx\Big)^{-1},$$ for Emden-Fowler type wave equation when the energy is null which will be the main results of subsection 3.1. In the subsection 3.2, we will discuss the blow up in finite time $\ln T_2^{\ast}<\ln T_1^{\ast}$ of problem for large class of solution in the case when the associated energy is negative. The questions of local existence and uniqueness will be also considered in the section 2.\ Preliminaries, local Existence of unique solution ================================================= Under some suitable transformations, we can get the local existence of solutions to equation (\[e1\]). Taking the transform $$\tau=\ln t, \qquad v =u, \qquad u_{xx}=v_{xx},$$ then $$u'=t^{-1}v_{\tau},\qquad t^2 u''=-v_{\tau}+v_{\tau\tau},$$ equation (\[e1\]) takes the form $$\begin{aligned} \label{e4} &&v_{\tau\tau}- v_{xx} +\int_{0}^{\tau}\mu(s) v_{xx}(\tau-s) ds =v_{\tau}+v^p\quad \hbox{ in }[0,\ln T) \times(r_1,r_2), \nonumber\\\nonumber\\ &&v(x,0) =u_0(x),\quad u_{\tau}(x,0) =u_1(x), \nonumber\\\nonumber\\ && v(r_1,\tau) = v(r_2,\tau)=0.\end{aligned}$$ Let $$\begin{aligned} &&v(\tau,x) =e^{\tau/2}w(\tau,x),\nonumber \\\nonumber\\ &&v_{\tau}(\tau,x)=e^{\tau/2}w_{\tau}(\tau,x)+\frac{1}{2}e^{\tau/2}w(\tau,x),\nonumber\\\nonumber\\ &&v_{\tau\tau}(\tau,x)=e^{\tau/2}w_{\tau\tau}(\tau,x)+e^{\tau/2}w_{\tau}(\tau,x)+\frac{1}{4}e^{\tau/2}w(\tau,x),\nonumber\end{aligned}$$ then (\[e4\]) can be rewritten as $$\begin{aligned} &&e^{\tau/2}w_{\tau\tau}-e^{\tau/2}w_{xx}+\int_{0}^{\tau}e^{s/2}\mu(s)w_{xx}(\tau-s)ds,\nonumber\\\nonumber\\ && =\frac{1}{4}e^{\tau/2}w+e^{p\tau/2}w^p, \nonumber \end{aligned}$$ then $$\begin{aligned} \label{e5} w_{\tau\tau}- w_{xx} +e^{-\tau/2}\int_{0}^{\tau}e^{s/2}\mu(s) w_{xx}(\tau-s) ds =\frac{1}{4}w+e^{(p-1)\tau/2}w^p.\end{aligned}$$ The following technical Lemma will play an important role. \[lemma0\] For any $w\in C^{1}\left( 0,T,H^{1}(r_1,r_2 )\right)$ we have for any nonincreasing differentiable function $\alpha$ satisfying $\alpha(\tau)>0$ $$\begin{aligned} &&\int_{r_1}^{r_2}\alpha(\tau)\int_{0}^{\tau}e^{s/2}\mu(\tau-s) w_{xx}(s) w'(\tau)dsdx\nonumber\\\nonumber\\ &=&\frac{1}{2}\frac{d}{d\tau}\alpha(\tau) \int_{0}^{\tau} e^{s/2}\mu(\tau-s)\int_{r_1}^{r_2} \vert w_x(\tau)-w_x(s)\vert^2dxds \nonumber\\\nonumber\\ &-&\frac{1}{2}\frac{d}{d\tau}\alpha(\tau) \int_{0}^{\tau}e^{s/2}\mu(s)ds\int_{r_1}^{r_2} \left\vert w_{x}(\tau)\right\vert ^{2}dx \nonumber\\\nonumber\\ &-&\frac{1}{2}\alpha\int_{0}^{\tau} \Big(e^{s/2}\mu(\tau-s)\Big)'\int_{r_1}^{r_2} \vert w_x(\tau)-w_x(s)\vert^2dxds\nonumber\\\nonumber\\ &+&\frac{1}{2}\alpha(\tau)e^{\tau/2}\mu(\tau)\int_{r_1}^{r_2} \left\vert w_{x}(\tau)\right\vert ^{2}dx \nonumber\\\nonumber\\ &&-\frac{1}{2}\alpha'(\tau)\int_{0}^{\tau} e^{s/2}\mu(\tau-s)\int_{r_1}^{r_2} \vert w_x(\tau)-w_x(s)\vert^2dxds\nonumber\\ \nonumber\\ &+&\frac{1}{2}\alpha'(\tau)\int_{0}^{s}e^{s/2}\mu(s)ds\int_{r_1}^{r_2} \left\vert w_{x}(\tau)\right\vert ^{2}dx.\nonumber \end{aligned}$$ It’s not hard to see $$\begin{aligned} &&\int_{r_1}^{r_2}\alpha(\tau)\int_{0}^{\tau}e^{s/2}\mu(\tau-s) w_{xx}(s) w'(\tau)dsdx\nonumber\\\nonumber\\ &=&-\alpha(\tau)\int_{0}^{\tau}e^{s/2}\mu(\tau-s)\int_{r_1}^{r_2} w'_x(\tau) w_{x}(s)dxds \nonumber \\\nonumber\\ &=&-\alpha(\tau)\int_{0}^{\tau}e^{s/2}\mu(\tau-s)\int_{r_1}^{r_2} w'_x(v)\left[ w_{x}(s)- w_{x}(\tau)\right] dxds \nonumber \\\nonumber\\ &&-\alpha(\tau)\int_{0}^{\tau}e^{s/2}\mu(s)\int_{r_1}^{r_2}w'_x(\tau) w_{x}(\tau)dxds.\nonumber \end{aligned}$$ Consequently, $$\begin{aligned} &&\int_{r_1}^{r_2}\alpha(\tau)\int_{0}^{\tau}e^{s/2}\mu(\tau-s) w_{xx}(s) w'(\tau)dsdx\nonumber\\\nonumber\\ &=&\frac{1}{2}\alpha(\tau)\int_{0}^{\tau}e^{s/2}\mu(\tau-s)\frac{d}{d\tau}\int_{r_1}^{r_2} \left\vert w_{x}(s)- w_{x}(\tau)\right\vert ^{2}dxds \nonumber\\\nonumber\\ &&-\alpha(\tau)\int_{0}^{\tau}e^{s/2}\mu(s)\left( \frac{d}{d\tau}\frac{1}{2}\int_{r_1}^{r_2} \left\vert w_{x}(\tau)\right\vert ^{2}dx\right) ds\nonumber \end{aligned}$$ which implies, $$\begin{aligned} &&\int_{r_1}^{r_2}\alpha(\tau)\int_{0}^{\tau}e^{s/2}\mu(\tau-s) w_{xx}(s) w'(\tau)dsdx\nonumber\\\nonumber\\ &=&\frac{1}{2}\frac{d}{d\tau}\left[\alpha(\tau) \int_{0}^{\tau}e^{s/2}\mu(\tau-s)\int_{r_1}^{r_2} \left\vert w_{x}(s)- w_{x}(\tau)\right\vert ^{2}dxds\right] \nonumber\\\nonumber\\ &&-\frac{1}{2}\frac{d}{d\tau}\left[\alpha(\tau)\int_{0}^{\tau}e^{s/2}\mu(s)\int_{r_1}^{r_2} \left\vert w_{x}(v)\right\vert ^{2}dxds\right] \nonumber\\\nonumber\\ &&-\frac{1}{2}\alpha(\tau)\int_{0}^{\tau}\Big(e^{s/2}\mu(\tau-s)\Big)'\int_{r_1}^{r_2} \left\vert w_{x}(s)- w_{x}(\tau)\right\vert ^{2}dxds \nonumber\\\nonumber\\ &&+\frac{1}{2}\alpha(\tau)e^{\tau/2}\mu(\tau)\int_{r_1}^{r_2} \left\vert w_{x}(\tau)\right\vert ^{2}dx.\nonumber\\\nonumber\\ &&-\frac{1}{2}\alpha'(\tau)e^{s/2}\mu(\tau-s)\int_{r_1}^{r_2} \left\vert w_{x}(s)- w_{x}(\tau)\right\vert ^{2}dxds\nonumber\\\nonumber\\ &&+\frac{1}{2}\alpha'(\tau)\int_{0}^{s}e^{s/2}\mu(s)ds\int_{r_1}^{r_2} \left\vert w_{x}(\tau)\right\vert ^{2}dxds.\nonumber \end{aligned}$$ This completes the proof. We introduce the modified energy associated to problem (\[e5\]). $$\begin{aligned} &&2E_{w}(\tau) = \int_{r_1}^{r_2} \vert w_{\tau}\vert^2dx+(1-\int_{0}^{\tau}e^{s/2}\mu(s)ds) \int_{r_1}^{r_2} \vert w_{x}\vert^2dxd \nonumber\\\nonumber\\ &&+\int_{0}^{\tau}e^{s/2}\mu(\tau-s)\int_{r_1}^{r_2} \left\vert w_{x}(s)- w_{x}(\tau)\right\vert ^{2}dxds\nonumber\\\nonumber\\ &&-\frac{1}{4}\int_{r_1}^{r_2} \vert w\vert^2dx-\frac{2}{p+1}e^{\frac{(p-1)\tau}{2}}\int_{r_1}^{r_2}\vert w\vert^{p+1}dx.\label{energy}\end{aligned}$$ and $$\begin{aligned} &&2E_{w}(0) = \int_{r_1}^{r_2} ( u_{1 }-\frac{1}{2}u_0)^2dx+\int_{r_1}^{r_2} \vert u_{0x}\vert^2dx\nonumber\\\nonumber\\ &&+\int_{r_1}^{r_2}u_{0}u_1dx-\frac{2}{p+1}\int_{r_1}^{r_2}\vert u_0\vert^{p+1}dx.\nonumber\end{aligned}$$ Direct differentiation, using (\[2.1\]), (\[e5\]), leads to $$\begin{aligned} E'_{w}(\tau )\leq 0.\nonumber\end{aligned}$$ We now can obtain the next important Lemma. \[lem1\] Suppose that $v\in C^{1}(0,T,H_0^{1}(r_1,r_2) ) \cap C^2(0,T,L^2(r_1,r_2) )$ is a solution of the semi-linear wave equation (\[e5\]). Then for $\tau\geq0$, $$\begin{aligned} E_{w}(\tau) \leq E_{w}(0) -\frac{p-1}{p+1}\int_0^{\tau} e^{\frac{(p-1)s}{2}}\int_{r_1}^{r_2}\vert w\vert^{p+1}dxds, \label{e6}\end{aligned}$$ Taking the $L^2$ product of (\[e5\]) with $w_{\tau}$ yields $$\begin{aligned} &&\int_{r_1}^{r_2}w_{\tau\tau}w_{\tau}dx-\int_{r_1}^{r_2 }\Big( w_{xx} -e^{-\tau/2}\int_{0}^{t}e^{s/2}\mu(s) w_{xx}(t-s) ds\Big) w_{\tau}dx\nonumber\\\nonumber\\ &&=\frac{1}{4}\int_{r_1}^{r_2}w w_{\tau}dx+\int_{r_1}^{r_2} e^{(p-1)\tau/2}w^p w_{\tau}dx. \nonumber \end{aligned}$$ Thus, by Lemma \[lemma0\] with $\alpha(\tau)=e^{-\tau /2}$, we have $$\begin{aligned} &&\frac{1}{2}\frac{d}{d\tau} \Big[\int_{r_1}^{r_2}\vert w_{\tau}\vert^2 dx+ (1-\int_{0}^{t}e^{s/2}\mu(s)ds) \int_{r_1}^{r_2} \vert w_{x}\vert^2 dx \Big]\nonumber\\\nonumber\\ &&+\frac{1}{2}\frac{d}{d\tau}\int_{0}^{\tau}e^{s/2}\mu(\tau-s)\int_{r_1}^{r_2} \left\vert w_{x}(s)- w_{x}(\tau)\right\vert ^{2}dxds\nonumber\\\nonumber\\ &&=\frac{1}{8}\frac{d}{d\tau} \int_{r_1}^{r_2}\vert w\vert ^2dx+\frac{1}{p+1}\frac{d}{d\tau}\int_{r_1}^{r_2} e^{(p-1)\tau/2}w^{p+1} w_{\tau}dx+\frac{2(p-1)}{p+1}\int_{r_1}^{r_2} e^{(p-1)\tau/2}w^{p+1}dx. \nonumber\\\nonumber\\ &&+\frac{1}{2}\alpha(\tau)\int_{0}^{\tau}\Big(e^{s/2}\mu(\tau-s)\Big)'\int_{r_1}^{r_2} \left\vert w_{x}(s)- w_{x}(\tau)\right\vert ^{2}dxds \nonumber\\ \nonumber\\ &&-\frac{1}{2}\mu(\tau)\int_{r_1}^{r_2} \left\vert w_{x}(t)\right\vert ^{2}dx \nonumber\\\nonumber\\ &&+\frac{1}{2}\alpha'(\tau)\int_{0}^{\tau}e^{s/2}\mu(\tau-s)\int_{r_1}^{r_2} \left\vert w_{x}(s)- w_{x}(\tau)\right\vert ^{2}dxds\nonumber\\ \nonumber\\ &&-\frac{1}{2}\alpha'(\tau)\int_{0}^{s}e^{s/2}\mu(s)ds\int_{r_1}^{r_2} \left\vert w_{x}(\tau)\right\vert ^{2}dx.\nonumber \end{aligned}$$ Then, by conditions on $\mu, \alpha$ and (\[energy\]), the assertions (\[e6\]) is proved. Blow up result for $E_u(0)=0$ ============================= Under small amplitude initial data, we prove that $w$ blows up in $L^2$ at finite time $\ln T^{\ast}$ in the following Theorem \[thm2\]. \[thm2\] Suppose that $w \in C^{1}(0,T,H_0^{1}(r_1,r_2) ) \cap C^2(0,T,L^2(r_1,r_2) )$ is a weak solution of equation (\[e5\]) with $$e(0):=\int_{r_1}^{r_2}u_0u_1(x) dx>0,\qquad E_u(0)=0$$ and $0<r_2-r_1\leq 1$. Then there exists $T_1^{\ast}$ such that $$\int_{r_1}^{r_2} \vert u(t,x)\vert ^2dx \to +\infty \quad\hbox{ as } t\to T_1^{\ast},$$ where $$T_1^{\ast}=\frac{2}{p-1}\Big( \int_{r_1}^{r_2} \vert u_0\vert dx\Big) \Big(\int_{r_1}^{r_2} u_0u_1dx\Big)^{-1}.$$ We need to state and prove the next intermediate Lemma. \[Le10\] Suppose that $w $ is a weak solution of equation (\[e5\]). Then $$\begin{aligned} && \int_{r_1}^{r_2}e^{\frac{p-1}{2}s}w^{p+1}(s,x) dx \nonumber\\\nonumber\\ && \geq \frac{p+1}{2}\Big[\int_{r_1}^{r_2} \vert w_{s}\vert ^2dx+(1-\int_{0}^{t}e^{s/2}\mu(s)ds)\int_{r_1}^{r_2} \vert w_{x}\vert^2dx-\frac{1}{4}\int_{r_1}^{r_2}\vert w\vert ^2dx \Big] \nonumber\\\nonumber\\ &&+\int_{0}^{\tau}e^{s/2}\mu(\tau-s)\int_{r_1}^{r_2} \left\vert w_{x}(s)- w_{x}(\tau)\right\vert ^{2}dxds-(p+1) E_{w}( 0) e^{\frac{p-1}{2}s} \nonumber\\\nonumber\\ &&\quad +\frac{p^2-1}{2}\int_0^{s}e^{\frac{p-1}{2}(s-r) } \Big[\int_{r_1}^{r_2} \vert w_{s}\vert ^2dx+(1-\int_{0}^{t}e^{s/2}\mu(s)ds)\int_{r_1}^{r_2} \vert w_{x}\vert^2dx-\frac{1}{4}\int_{r_1}^{r_2}\vert w\vert ^2dx \Big]dr\nonumber\\\nonumber\\ &&+\frac{p^2-1}{2}\int_0^{s}e^{\frac{p-1}{2}(s-r) }\int_{0}^{\tau}e^{s/2}\mu(\tau-s)\int_{r_1}^{r_2} \left\vert w_{x}(s)- w_{x}(\tau)\right\vert ^{2}dxds.\nonumber \end{aligned}$$ Set $$\begin{aligned} L(s) &=&\frac{1}{p+1}\int_0^{s} e^{\frac{p-1}{2}r}\int_{r_1}^{r_2}\vert w\vert^{p+1}dx dr,\nonumber\\\nonumber\\ F(s) &=&\int_{r_1}^{r_2}\vert w_{s}\vert^2dx+(1-\int_{0}^{s}e^{\tau/2}\mu(\tau)d\tau)\int_{r_1}^{r_2} \vert w_{x}\vert ^2dx\nonumber \\\nonumber\\ &-&\frac{1}{4}\int_{r_1}^{r_2}\vert w\vert^2dx+\int_{0}^{\tau}e^{s/2}\mu(\tau-s)\int_{r_1}^{r_2}\left\vert w_{x}(s)- w_{x}(\tau)\right\vert ^{2}dxds,\nonumber\end{aligned}$$ By Lemma \[lemma0\] and Lemma \[lem1\], equation (\[e6\]) can be rewritten as $$\begin{aligned} E_{w}(0)\geq F-2L'+(p-1) L, \label{e8}\end{aligned}$$ therefore, $$\begin{aligned} (e^{\frac{p-1}{-2}s}L) ' &=&e^{\frac{p-1}{-2}s}\Big(L'-\frac{p-1}{2}L\Big)\nonumber \\\nonumber\\ &\geq&\frac{1}{2}e^{\frac{p-1}{-2}s}(F-E_{w}(0) ), \nonumber\end{aligned}$$ and $$\begin{aligned} e^{\frac{p-1}{-2}s}L & \geq&\frac{1}{2}\int_0^{s}e^{\frac{p-1}{-2}r}( F(r) -E_{w}(0) ) dr\nonumber\\ \nonumber \\ & \geq&\frac{1}{2}\int_0^{s}e^{\frac{p-1}{-2}r}F(r) dr-\frac{E_{w}(0) }{p-1}\Big(1-e^{\frac{p-1}{-2}s}\Big),\nonumber\end{aligned}$$ and $$\begin{aligned} L\geq\frac{1}{2}\int_0^{s}e^{\frac{p-1}{2}(s-r) }F( r) dr-\frac{E_{w}(0) }{p-1}\Big(e^{\frac{p-1}{2}s}-1\Big);\nonumber\end{aligned}$$ this implies $$\begin{aligned} && \frac{1}{p+1}\int_0^{s} e^{\frac{p-1}{2}r}\int_{r_1}^{r_2} \vert w\vert ^{p+1}dx \,dr\nonumber\\\nonumber\\ && \geq\frac{1}{2}\int_0^{s}e^{\frac{p-1}{2}(s-r) }\Big[ \int_{r_1}^{r_2}\vert w_{s}\vert^2dx+(1-\int_{0}^{t}e^{s/2}\mu(s)ds)\int_{r_1}^{r_2} \vert w_{x}\vert^2dx-\frac{1}{4}\int_{r_1}^{r_2}\vert w\vert^2dx \Big]\,dr\nonumber\\\nonumber\\ &&-\frac{E_{w}(0) }{p-1}(e^{\frac{p-1}{2}s}-1)+\frac{1}{2}\int_0^{s}e^{\frac{p-1}{2}(s-r) }\int_{0}^{\tau}e^{s/2}\mu(\tau-s)\int_{r_1}^{r_2} \left\vert w_{x}(s)- w_{x}(\tau)\right\vert ^{2}dxds, \nonumber\end{aligned}$$ and $$\begin{aligned} \label{e9} & & \int_0^{s}\int_{r_1}^{r_2}e^{\frac{p-1}{2}r}w^{p+1}(r,x) \,dx\,dr\nonumber\\\nonumber\\ && \geq\frac{p+1}{2}\int_0^{s}e^{\frac{p-1}{2}(s-r) }\Big[ \int_{r_1}^{r_2}\vert w_{s}\vert^2dx+(1-\int_{0}^{t}e^{s/2}\mu(s)ds)\int_{r_1}^{r_2} \vert w_{x}\vert^2dx-\frac{1}{4}\int_{r_1}^{r_2}\vert w\vert^2dx \Big]dr\nonumber\\\nonumber\\ && -\frac{p+1}{p-1}E_{w}(0) (e^{\frac{p-1}{2}s}-1)+\frac{p+1}{2}\int_0^{s}e^{\frac{p-1}{2}(s-r) }\int_{0}^{\tau}e^{s/2}\mu(\tau-s)\int_{r_1}^{r_2} \left\vert w_{x}(s)- w_{x}(\tau)\right\vert ^{2}dxds, \nonumber\end{aligned}$$ and $$\begin{aligned} && \int_{r_1}^{r_2}e^{\frac{p-1}{2}s}w^{p+1}(s,x) dx \nonumber\\\nonumber\\ && \geq\frac{p+1}{2}\Big[\int_{r_1}^{r_2} \vert w_{s}\vert ^2dx+(1-\int_{0}^{t}e^{s/2}\mu(s)ds)\int_{r_1}^{r_2} \vert w_{x}\vert^2dx-\frac{1}{4}\int_{r_1}^{r_2}\vert w\vert ^2dx \Big] \nonumber\\\nonumber\\ &&+\int_{0}^{\tau}e^{s/2}\mu(\tau-s)\int_{r_1}^{r_2} \left\vert w_{x}(s)- w_{x}(\tau)\right\vert ^{2}dxds-(p+1) E_{w}( 0) e^{\frac{p-1}{2}s} \nonumber\\\nonumber\\ &&\quad +\frac{p^2-1}{2}\int_0^{s}e^{\frac{p-1}{2}(s-r) } \Big[\int_{r_1}^{r_2} \vert w_{s}\vert ^2dx+(1-\int_{0}^{t}e^{s/2}\mu(s)ds)\int_{r_1}^{r_2} \vert w_{x}\vert^2dx-\frac{1}{4}\int_{r_1}^{r_2}\vert w\vert ^2dx \Big]dr\nonumber\\\nonumber\\ &&+\frac{p^2-1}{2}\int_0^{s}e^{\frac{p-1}{2}(s-r) }\int_{0}^{\tau}e^{s/2}\mu(\tau-s)\int_{r_1}^{r_2} \left\vert w_{x}(s)- w_{x}(\tau)\right\vert ^{2}dxds.\label{A}\end{aligned}$$ This completes the proof. We are now ready to prove Theorem \[thm2\] (Of Theorem \[thm2\])\ Let $$A(s) :=\int_{r_1}^{r_2}\vert w\vert ^2dx,$$ then we have $$A'(s) =2\int_{r_1}^{r_2}ww_{s}(s,x) dx.$$ and $$\begin{aligned} A''(s) &=&2\int_{r_1}^{r_2}ww_{ss}(s,x) dx+2\int_{r_1}^{r_2}w^2_{s}(s,x) dx\nonumber\\\nonumber\\ &=&2\int_{r_1}^{r_2}(ww_{xx}-we^{-\tau/2}\int_{0}^{t}e^{s/2}\mu(s)w_{xx}(t-s)ds+\frac{1}{4}w^2+w_{s}^2 +e^{\frac{p-1}{2}s}w^{p+1}) dx\nonumber\\\nonumber\\ & =&2\int_{r_1}^{r_2}(-w_{x}^2+w_xe^{-\tau/2}\int_{0}^{t}e^{s/2}\mu(s)w_{x}(t-s)ds+\frac{1}{4}w^2+w_{s} ^2+e^{\frac{p-1}{2}s}w^{p+1})dx.\nonumber\end{aligned}$$ By Lemma\[lemma0\], Lemmad\[Le10\] and (\[A\]), then $$\begin{aligned} A''(s) &\geq&2 \Big((\int_0^te^{s/2}\mu(s)ds-1)\int_{r_1}^{r_2} \vert w_{x}\vert^2dx +\frac{1}{4}\int_{r_1}^{r_2}\vert w\vert^2dx+\int_{r_1}^{r_2}\vert w_{s}\vert^2dx\Big) \nonumber\\ \nonumber\\ &-&2\int_{0}^{\tau}e^{s/2}\mu(\tau-s)\int_{r_1}^{r_2} \left\vert w_{x}(s)- w_{x}(\tau)\right\vert ^{2}dxds\nonumber\\ \nonumber\\ &+&(p+1)2 \Big((\int_0^te^{s/2}\mu(s)ds-1)\int_{r_1}^{r_2} \vert w_{x}\vert^2dx +\frac{1}{4}\int_{r_1}^{r_2}\vert w\vert^2dx+\int_{r_1}^{r_2}\vert w_{s}\vert^2dx\Big) \nonumber\\ \nonumber\\ &-&(p+1)\int_{0}^{\tau}e^{s/2}\mu(\tau-s)\int_{r_1}^{r_2} \left\vert w_{x}(s)- w_{x}(\tau)\right\vert ^{2}dxds\nonumber \\\nonumber \\ &+&(p^2-1) \int_0^{s}e^{\frac{p-1}{2}(s-r)} \Big((\int_0^te^{s/2}\mu(s)ds-1)\int_{r_1}^{r_2} \vert w_{x}\vert^2dx +\frac{1}{4}\int_{r_1}^{r_2}\vert w\vert^2dx+\int_{r_1}^{r_2}\vert w_{s}\vert^2dx\Big) \nonumber\\ \nonumber\\ &-&(p^2-1) \int_0^{s}e^{\frac{p-1}{2}(s-r)}\int_{0}^{\tau}e^{s/2}\mu(\tau-s)\int_{r_1}^{r_2} \left\vert w_{x}(s)- w_{x}(\tau)\right\vert ^{2}dxds \nonumber \\\nonumber \\ &-&2(p+1) E_{w}(0) e^{\frac{p-1}{2}s} \nonumber \\\nonumber \\ & \geq& \big[(p+3) \int_{r_1}^{r_2} \vert w_{s}\vert^2dx+( p-1)(1-\int_0^te^{s/2}\mu(s)ds) \int_{r_1}^{r_2} \vert w_{x}\vert^2dx-\frac{p-1}{4}\int_{r_1}^{r_2}\vert w\vert^2dx\big] \nonumber\\\nonumber\\ &-&2(p+1) E_{w}(0) e^{\frac{p-1}{2}s} +(p-1)\int_{0}^{\tau}e^{s/2}\mu(\tau-s)\int_{r_1}^{r_2} \left\vert w_{x}(s)- w_{x}(\tau)\right\vert ^{2}dxds\\\nonumber\\ \label{e11} &+&(p^2-1) \int_0^{s}e^{\frac{p-1}{2}(s-r) } \Big(\int_{r_1}^{r_2}\vert w_{s}\vert^2dx+(\int_0^te^{s/2}\mu(s)ds-1)\int_{r_1}^{r_2} \vert w_{x}\vert^2dx+\frac{1}{4}\int_{r_1}^{r_2}\vert w\vert^2dx\Big) \,dr\nonumber\\ \nonumber\\ &-&(p^2-1) \int_0^{s}e^{\frac{p-1}{2}(s-r) }\int_{0}^{\tau}e^{s/2}\mu(\tau-s)\int_{r_1}^{r_2} \left\vert w_{x}(s)- w_{x}(\tau)\right\vert ^{2}dxdsdr. \nonumber\end{aligned}$$ As in [@L11], let us setting $$J(s) :=A(s) ^{-k}, \qquad k=\frac{p-1} {4}>0.$$ Then $$J'(s) =-kA(s) ^{-k-1}A'(s),$$ and $$\begin{aligned} \label{e12} J''(s) & =-kA(s) ^{-k-2}[A(s) A''(s) -(k+1) A'(s) ^2] \nonumber\\\nonumber\\ & \leq-kA(s) ^{-k-1}\big[A''(s) -4(k+1) \int_{r_1}^{r_2}w_{s}^2 dx\big].\end{aligned}$$ Since $E_{u}(0)=0$, we have $$\begin{aligned} & & A''(s) -4(k+1) \int_{r_1}^{r_2}\vert w_{s}\vert ^2dx \nonumber\\\nonumber\\ && \geq\Big[(p+3) \int_{r_1}^{r_2}\vert w_{s}\vert^2dx+( p-1)(1-\int_0^te^{s/2}\mu(s)ds) \int_{r_1}^{r_2} \vert w_{x}\vert^2dx-\frac{p-1}{4}\int_{r_1}^{r_2}\vert w\vert^2dx\Big] \nonumber\\\nonumber\\ && +(p^2-1) \int_0^{s}e^{\frac{p-1}{2}(s-r) } \Big(\int_{r_1}^{r_2}\vert w_{s}\vert^2dx+(1-\int_0^te^{s/2}\mu(s)ds)\int_{r_1}^{r_2} \vert w_{x}\vert^2dx-\frac{1}{4}\int_{r_1}^{r_2}\vert w\vert^2dx\Big) \,dr\nonumber\\\nonumber\\ & &\quad -4(k+1) \int_{r_1}^{r_2}\vert w_{s}\vert^2dx+(p-1)\int_{0}^{\tau}e^{s/2}\mu(\tau-s)\int_{r_1}^{r_2} \left\vert w_{x}(s)- w_{x}(\tau)\right\vert ^{2}dxds\nonumber\\ &&+(p^2-1) \int_0^{s}e^{\frac{p-1}{2}(s-r) }\int_{0}^{\tau}e^{s/2}\mu(\tau-s)\int_{r_1}^{r_2} \left\vert w_{x}(s)- w_{x}(\tau)\right\vert ^{2}dxdsdr,\nonumber\end{aligned}$$ then, $$\begin{aligned} && A''(s) -4(k+1) \int_{r_1}^{r_2}\vert w_{s}\vert^2dx\nonumber\\\nonumber\\ && \geq(p-1) \Big[(1-\int_0^te^{s/2}\mu(s)ds)\int_{r_1}^{r_2} \vert w_{x}\vert^2dx-\frac{1}{4}\int_{r_1}^{r_2}\vert w\vert^2dx\Big] \nonumber\\\nonumber\\ &&+(p-1)\int_{0}^{\tau}e^{s/2}\mu(\tau-s)\int_{r_1}^{r_2} \left\vert w_{x}(s)- w_{x}(\tau)\right\vert ^{2}dxds \nonumber\\ & & +(p^2-1) \int_0^{s}e^{\frac{p-1}{2}(s-r)} \Big(\int_{r_1}^{r_2}\vert w_{s}\vert^2dx+(1-\int_0^te^{s/2}\mu(s)ds)\int_{r_1}^{r_2} \vert w_{x}\vert^2dx-\frac{1}{4}\int_{r_1}^{r_2}\vert w\vert^2dx\Big) dr\nonumber\\\nonumber\\ &&+(p^2-1) \int_0^{s}e^{\frac{p-1}{2}(s-r)}\int_{0}^{\tau}e^{s/2}\mu(\tau-s)\int_{r_1}^{r_2} \left\vert w_{x}(s)- w_{x}(\tau)\right\vert ^{2}dxdsdr \nonumber\\ \nonumber\\ & & \geq(p-1) \big(1-(r_2-r_1) ^2\big) \Big(\int_{r_1}^{r_2} \vert w_{x}\vert^2dx+\int_{0}^{\tau}e^{s/2}\mu(\tau-s)\int_{r_1}^{r_2} \left\vert w_{x}(s)- w_{x}(\tau)\right\vert ^{2}dxds\Big)\nonumber\\\nonumber\\ &&+(p+1) \int_0^{s}e^{\frac{p-1}{2}(s-r) } \Big(\int_{r_1}^{r_2} \vert w_{s}\vert ^2dx+(1-\int_0^te^{s/2}\mu(s)ds)\int_{r_1}^{r_2} \vert w_{x}\vert ^2dx\Big)dr \nonumber\\ && +(p+1) \int_0^{s}e^{\frac{p-1}{2}(s-r) }\int_{0}^{\tau}e^{s/2}\mu(\tau-s)\int_{r_1}^{r_2} \left\vert w_{x}(s)- w_{x}(\tau)\right\vert ^{2}dxdsdr>0,\nonumber\end{aligned}$$ where $r_2\leq1+r_1$.\ Therefore, by (\[e12\]) we obtain that for, $r_2 -r_1\leq1$, $J''(s) <0$ for all $s\geq0$.\ $$\begin{aligned} J'(s) \leq J'(0) &=&-\frac {p-1}{4}A(0) ^{-\frac{p+3}{4}}A'(0)\nonumber\\\nonumber\\ &=&-\frac{p-1}{2}e(0)\int_{r_1}^{r_2}\vert u_0\vert ^{-(p+3)}dx,\nonumber\end{aligned}$$ and $$\begin{aligned} J(s) &\leq& J(0) -\frac{p-1}{2}e(0)\int_{r_1}^{r_2}\vert u_0\vert ^{-(p+3)}dxs\nonumber\\ \nonumber\\ &=&\int_{r_1}^{r_2}\| u_0\| ^{-(p-1)}dx-\frac{p-1}{2}e(0)\int_{r_1}^{r_2}\vert u_0\vert ^{-(p+3)}dxs\nonumber\\ \nonumber\\ &=&\int_{r_1}^{r_2}\vert u_0\vert ^{-(p+3)}dx\Big(\int_{r_1}^{r_2}\vert u_0\vert dx-\frac{p-1}{2}e(0) s\Big). \nonumber\end{aligned}$$ Then $$\begin{aligned} J(s) \to 0 \quad \hbox{ as }s\to T^{\ast}=\frac{2}{p-1}\frac{\int_{r_1}^{r_2}\vert u_0\vert dx}{e(0)}.\end{aligned}$$ Thus $w$ solution of (\[e5\]) blows up in $L^2$ at finite time $T^{\ast}$. Blow up result for $E_u(0)<0$ ============================= In the following theorem we shall state and prove our second blowing up result \[thm3\] Suppose that $w\in C^{1}(0,T,H_0^{1}(r_1,r_2) ) \cap C^2(0,T,L^2(r_1,r_2) )$ is a weak solution of equation (\[e1\]) with $$e(0)=\int_{r_1}^{r_2}u_0 u_1(x) dx>0,\qquad E_u(0)<0,$$ and $0<r_2-r_1\leq1$. Then, there exists $T_2^{\ast}$ such that $$\frac{1}{\int_{r_1}^{r_2}\vert u(t,x)\vert ^{2}dx} \to0\quad\hbox{ as } t\to\ln T_2^{\ast}.$$ Further, we have $\ln T_2^{\ast}<\ln T_1^{\ast}$, and the estimate $$\int_{r_1}^{r_2}w^2 dx \geq \int_{r_1}^{r_2}u^2_0 dx -2E_{u}(0) \frac {p+1}{p-1}\big[se^{\frac{p-1}{2}s}-\frac{2}{p-1}(e^{\frac{p-1}{2} s}-1) \big].$$ By (\[e11\]), Lemma\[lemma0\], $E_{u}(0) <0$, $e(0)>0$ and $0<r_2-r_1\leq1$, we have\ $$\begin{aligned} \label{e13} J''(s) &\leq&-k\Big(\int_{r_1}^{r_2}w^2 dx\Big) ^{-k-1}\Big[A''(s) -(p+3) \int_{r_1}^{r_2}w_{s}^2(s,x) dx\Big] \nonumber\\\nonumber\\ & =&-k\Big(\int_{r_1}^{r_2}w^2 dx\Big) ^{-k-1}\Big[ -2(p+1) E_{w}(0) e^{\frac{p-1}{2}s}\nonumber\\\nonumber\\ &+&(p-1) \Big((1-\int_{0}^{t}e^{s/2}\mu(s)ds)\int_{r_1}^{r_2} \vert w_{x}\vert ^2dx-\frac{1}{4}\int_{r_1}^{r_2}\vert w\vert^2dx\Big) \nonumber\\\nonumber\\ &+&(p-1)\int_{0}^{\tau}e^{s/2}\mu(\tau-s)\int_{r_1}^{r_2} \left\vert w_{x}(s)- w_{x}(\tau)\right\vert ^{2}dxds \nonumber\\\nonumber\\ &+&(p^2-1) \int_0^{s}e^{\frac{p-1}{2}(s-r) } \Big(\int_{r_1}^{r_2}\vert w_{s}\vert^2dx+(1-\int_{0}^{t}e^{s/2}\mu(s)ds)\int_{r_1}^{r_2} \vert w_{x}\vert^2dx-\frac{1}{4}\int_{r_1}^{r_2}\vert w\vert^2dx\Big) dr \Big] \nonumber\\\nonumber\\ &+&(p^2-1) \int_0^{s}e^{\frac{p-1}{2}(s-r) }\int_{0}^{\tau}e^{s/2}\mu(\tau-s)\int_{r_1}^{r_2} \left\vert w_{x}(s)- w_{x}(\tau)\right\vert ^{2}dxdsdr\nonumber\\\nonumber\\ & \leq& 2k(p+1) E_{u}(0) e^{\frac{p-1}{2}s}J(s) ^{1+\frac{1}{k}}<0,\end{aligned}$$ where $k=(p-1) /4$, we can obtain the same conclusions as in Theorem \[thm2\].\ By the inequality (\[e13\]) and $J'<0$ we can estimate $J$ further, $$\begin{aligned} J''(s) &\leq&2k(p+1) E_{u}(0) e^{\frac{p-1}{2}s}J(s) ^{1+\frac{1}{k}} \nonumber\\\nonumber\\ &=&\frac{1}{2}(p^2-1) E_{u}(0) e^{\frac{p-1}{2}s}J(s) ^{1+\frac{1}{k}}<0, \nonumber\end{aligned}$$ and $$\begin{aligned} J'(s) &\leq& J'(0) +\frac{s} {2}(p^2-1) E_{u}(0) e^{\frac{p-1}{2}s}J( s) ^{1+\frac{1}{k}}\nonumber\\\nonumber\\ &\leq&\frac{s}{2}(p^2-1) E_{w}( 0) e^{\frac{p-1}{2}s}J(s) ^{1+\frac{1}{k}},\nonumber\end{aligned}$$ and $$\begin{aligned} -k\big(J(s) ^{-\frac{1}{k}}\big)' &=&J(s) ^{-1-\frac{1}{k}}J'(s) \nonumber\\\nonumber\\ &\leq&\frac{E_{u}(0) }{2}(p^2-1) se^{\frac{p-1}{2}s},\nonumber\end{aligned}$$ and $$\begin{aligned} -k(J(s) ^{-\frac{1}{k}}-J(0) ^{-\frac{1}{k}}) & \leq&\frac{E_{u}(0) }{2}(p^2-1) \Big(\frac{2}{p-1}se^{\frac{p-1}{2}s}-(\frac{2}{p-1}) ^2(e^{\frac{p-1}{2}s}-1) \Big) \nonumber \\\nonumber\\ & =&E_{w}(0) (p+1) \big[se^{\frac{p-1}{2} s}-\frac{2}{p-1}(e^{\frac{p-1}{2}s}-1) \big],\nonumber\end{aligned}$$ which implies $$\int_{r_1}^{r_2}w^2 dx \geq \int_{r_1}^{r_2}u_0^2 dx -2\frac{p+1}{p-1}E_{u}( 0) \big[se^{\frac{p-1}{2}s}-\frac{2}{p-1}(e^{\frac{p-1}{2} s}-1) \big]$$Then $u$ solution of our initial problem (\[e1\]) blows up in $L^2$  at finite time $\ln T_2^{\ast}$. This completes the proof. [00]{} A. Benaissa, D. Ouchenane and Kh. Zennir, *Blow up of positive initial-energy solutions to systems of nonlinear wave equations with degenerate damping and source terms*, Nonlinear studies. Vol. 19, No. 4, pp. 523-535, 2012. F. E. Browder; *On non-linear wave equations*. M.Z. 80. pp. 249-264 (1962). M.M. Cavalcanti, L.H. Fatori and T.F. Ma; *Attractors for wave equations with degenerate memory*, J. Diff. Eq., 260 (2016), pp. 56-83. S. Chandrasekhar; *Introduction to the Study of Stellar Structure*, Chap. 4. Dover, New York, 1957 C.M. Dafermos. An abstract Volterra equation with applications to linear viscoelasticity. J. Diff. Equations, 7 (1970), 554-569. Dafermos C. M., On the existence and the asymptotic stability of solution to the equations of linear thermoelasticity. Arch. Ration. Mech. Anal., 29, (1968) 241-271. C. M. Dafermos, H.P. Oquendo, Asymptotic stability in viscoelasticity. Arch. Ration. Mech. Anal. 37(1970), 297-308. Conti, Graffi, G. Sansone; *Qualitative Methods in the Theory of Nonlinear Vibrations*, Proc. Internat. Sympos. Nonlinear Vibrations, vol. II, 1961, pp. 172-189. R. Emden, Gaskugeln; *Anwendungen der mechanischen Warmetheorie auf Kosmologie und meteorologische Probleme*, B. G.Teubner, Leipzig, Germany 1907. R. H. Fowler; *The form near infinity of real, continuous solutions of a certain differential equation of the second order*, Quart. J. Math., 45 (1914), pp. 289-350. R. H. Fowler; *The solution of Emden’s and similar differential equations*, Monthly Notices Roy. Astro. Soc., 91 (1930), pp. 63-91. R. H. Fowler; *Some results on the form near infinity of real continuous solutions of a certain type of second order differential equations*, Proc. London Math. Soc., 13 (1914), pp. 341-371. R. H. Fowler; *Further studies of Emden’s and similar differential equations*, Quart. J. Math., 2 (1931), pp. 259-288. R. Glassey; *Finite-time blow-up for solutions of nonlinear wave equations*. M. Z. 177 (1981), pp. 323-340. F. John; *Blow-up for quasilinear wave equations in three space dimensions*. Comm.Pure. Appl. Math. 36 (1981) pp. 29-51. M. L. J. Hautus; *Uniformly asymptotic formulas for the Emden-Fowler differential equation*, J. Math. Anal. Appl., 30 (1970), pp. 680-694. S. Klainerman; *Global existence for nonlinear wave equations*. Comm.Pure Appl. Math. 33 (1980), pp. 43-101. S. Klainerman, G. Ponce; *Global, small amplitude solutions to nonlinear evolution equations*. Comm. Pure Appl. Math. 36 (1983), pp. 133-141. M. R. Li; *Nonexistence of global solutions of Emden-Fowler type semilinear wave equations with non-positive energy*. Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 93, pp. 1–10.\ M. R. Li; *Estimates for the life-span of the solutions of some semilinear wave equations*. ACPAA. Vol. 7 (2008), No. 2, pp. 417-432. M. R. Li; *Existence and uniqueness of local weak solutions for the Emden–Fowler wave equation in one dimension*, Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 145, pp. 1–10. D. Ouchenane, Kh. Zennir and M. Bayoud, *Global nonexistence of solutions for a system of nonlinear viscoelastic wave equations with degenerate damping and source terms*, Ukrainian Mathematical Journal, Vol. 65, No. 7(2013), 723-739. R. Racke; *Lectures on nonlinear Evolution Equations: Initial Value Problems*. Aspects of Math. Braunschweig Wiesbaden Vieweg(1992). A. Ritter; *Untersuchungen über die Höhe der Atmosphäre und die Konstitution gasformiger Weltkörper, 18 articles*, Wiedemann Annalen der Physik, 5-20, pp. 1878-1883. W. A. Strauss; *Nonlinear Wave Equations*, AMS Providence(1989). Dimensions. J. Differential Equations 52 (1984), pp.378-406. T. Sideris; *Nonexistence of global solutions to semilinear wave equations in high dimensions*. J. Differential Equations 52(1982). pp. 303-345. W. Thompson (Lord Kelvin); *On the convective equilibrium of temperature in the atmosphere*, Manchester Philos. Soc. Proc., 2 (1860-62), pp.170-176; reprint, Math. and Phys. Papers by Lord Kelvin, 3 (1890), pp. 255-260. Kh. Zennir and A. Guesmia; *Existence of solutions to nonlinear kth-order coupled Klein-Gordon equations with nonlinear sources and memory term*. Applied Mathematics E-Notes, 15(2015), 121-136. Kh. Zennir and S. Zitouni; *On the absence of solutions to damped system of nonlinear wave equations of Kirchhoff-type*. Vladikavkaz Mathematical Journal, 17(4), (2015), 44-58.

--- abstract: 'As is well known, when an $SU(2)$ operation acts on a two-level system, its Bloch vector rotates without change of magnitude. Considering a system composed of [*two*]{} two-level systems, it is proven that for a class of nonlocal interactions of the two subsystems including $\sigma_i\otimes \sigma_j$ (with $i,j \in \{x,y,z\}$) and the Heisenberg interaction, the geometric description of the motion is particularly simple: each of the two Bloch vectors follows an elliptical orbit within the Bloch sphere. The utility of this result is demonstrated in two applications, the first of which bears on quantum control via quantum interfaces. By employing nonunitary control operations, we extend the idea of controllability to a set of points which are not necessarily connected by unitary transformations. The second application shows how the orbit of the coherence vector can be used to assess the entangling power of Heisenberg exchange interaction.' author: - 'A. Mandilara and J. W. Clark' - 'M. S. Byrd' title: Elliptical orbits in the Bloch sphere --- Introduction ============ The Bloch vector, or vector of coherence [@Alicki], provides a geometric description of the density matrix of a spin-1/2 particle which is commonly used in nuclear magnetic resonance. Mathematically, the Bloch vector may be viewed as the adjoint representation of an $su(2)$ object in an $so(3)$ basis [@alta3]. Extension of the notion of vector of coherence to two-spin systems [@fano; @quan], and more generally to quantum spin systems of higher dimensions [@byrd], has drawn attention in the contexts of quantum information theory and quantum computation. Specific motivations include the prospects of a useful quantification of entanglement for composite systems [@mahl; @byrd; @alta1] and the quest for equations describing observables in quantum networks [@quan]. In the present work, the extension of the Bloch formalism to two spins is used to obtain a geometric representation of the orbits of the vector of coherence for each spin system in the case that a nonlocal interaction of the form $\sigma_i\otimes\sigma_j$ is introduced. We propose that this geometric picture will be useful in devising schemes for control of a quantum state via quantum interfaces [@lloyd], i.e., through the mediation of an ancillary system. In this vein, we investigate the limits of control of a quantum state $S$, mixed or pure, given a nonlocal interaction and an ancilla $Q$. The simple geometric picture developed below also applies to another special case of nonlocal interaction, namely the Heisenberg exchange Hamiltonian. As a second application of our formal results, we investigate the entanglement power of the Heisenberg interaction. Product of operator basis for a density matrix ============================================== One qubit --------- The density matrix $\rho$ of a two-state system is a positive semi-definite Hermitian $2 \times 2$ matrix having unit trace. It can always be given expression in terms of the three trace-free Pauli matrices $\sigma_i,~i=1,2,3 $, which are generators of $su(2)$, and $I/{\sqrt{2}}$ ($I$ being the unit matrix): $$\rho=\frac{1}{2} I +{\bf v}{\bf \sigma}\,. \label{onequbit}$$ Here $\bf v$ is the vector of coherence, whose magnitude is bounded by $0 \leq\parallel{\bf v}\parallel\leq 1/2$ because $1/2\leq {\rm Tr}(\rho^2)\leq1$. The two limiting values of the norm correspond to maximally mixed and pure states, respectively. The magnitude of the Bloch vector differs by a factor of $1/2$ from that of the vector of coherence, as a matter of convention. Unitary operations rotate the Bloch vector without changing its magnitude: $ SU(2)$ operations on the qubit correspond to $SO(3)$ operations on the Bloch vector. The dynamical evolution of the Bloch vector under non-local operations is considered in the next section. Two qubits and the correlation tensor ------------------------------------- In analogy to the representation (\[onequbit\]), we adopt the generators of ${\mit G}=SU(4)$, i.e., the elements of the algebra ${\mit g}=su(4)$ (together with the unit matrix), as an orthonormal basis for the $4\times4$ density matrix of the two-qubit system. We employ this basis as it appears in Ref. , noting that it differs from the basis used in [@byrd; @mahl] only in the coefficients. The dynamical evolution of the system becomes more transparent if we choose basis elements of the algebra ${\mit g}= su(4)$ in accordance with its Cartan Decomposition ${\mit g}={\mit p}\oplus{\mit e}$ [@Bro; @Zhang]. The algebras ${\mit p}$ and ${\mit e}$ satisfy the commutations relations $$[{\mit e},{\mit e}] \subset {\mit e}\,, \quad [{\mit p},{\mit e}] \subset {\mit p}\,, \quad [{\mit p},{\mit p}] \subset {\mit e}\,.$$ The basis elements, $W_j,~j=1,\ldots,15$ of the orthogonal algebra pair $(e,p)$ are $${\mit e}= {\rm span} \frac{i}{2} \{\sigma_x\otimes1,\sigma_y\otimes1, \sigma_z\otimes1, 1\otimes\sigma_x,1\otimes\sigma_y,1\otimes\sigma_z\}\,, \label{cart1}$$ $${\mit p}= {\rm span} \frac{i}{2} \{\sigma_x\otimes\sigma_x, \sigma_x\otimes \sigma_y,\sigma_x\otimes\sigma_z, \\ \sigma_y\otimes\sigma_x, \sigma_y\otimes \sigma_y,\sigma_y\otimes\sigma_z, \\ \sigma_z\otimes\sigma_x, \sigma_z\otimes \sigma_y,\sigma_z\otimes\sigma_z \}\,. \label{cart2}$$ The basis defined by Eqs. (\[cart1\]) and (\[cart2\]) is used to expand the density matrix as $$\rho=\sum_{j=0}^{15} {\rm Tr}(\rho X_j) X_j =\sum_{j=0}^{15}\rho_j X_j\,,$$ where $X_0=I/\sqrt{4}$, $\rho_0=1/\sqrt{4}$, and $X_j= -i W_j$ ($j=1, \ldots, 15$). In this representation, the density matrix is specified by three objects, namely the vectors of coherence ${\bf r}_1$ and ${\bf r}_2$ for the two subsystems along with the spin-spin correlation tensor $T_j^i$. $${\bf r}_1=\left( \begin{array}{c} \rho_1 \\ \rho_2 \\ \rho_3 \end{array}\right)\,, \qquad {\bf r}_2=\left( \begin{array}{c} \rho_4 \\ \rho_5 \\ \rho_6 \end{array}\right)\,, \qquad T_j^i=\left( \begin{array}{ccc} \rho_7 & \rho_8 & \rho_9 \\ \rho_{10} & \rho_{11} & \rho_{12} \\ \rho_{13} & \rho_{14} & \rho_{15} \end{array}\right)\,.$$ We note that the object $ T_j^i$ has other names: Stokes tensor [@Boston], entanglement tensor [@mahl], and tensor of coherence (when combined with the coherence vectors in one object). Details of the properties of $T_j^i$ can be found in Ref. , where many prior studies are cited. This tensor contains information on the correlations between the two subsystems, of both classical and quantum nature. Necessary and sufficient conditions for separability of a pure state can be stated in terms of its properties, whereas in the case of a mixed state, only necessary conditions for separability can be given [@alta1]. Evolution Under Local and non Local Operations ============================================== As we have seen, the Lie algebra ${\mit g}=su(4)$ possesses a Cartan decomposition $ {\mit g}={\mit e}\oplus {\mit p}$, which informs us that there exists within the Lie group ${\mit G}=SU(4)$ a subgroup of local operations ${\mit G}_L=SU(2)\otimes SU(2)$ generated by ${\mit e}$. All the other operations are nonlocal and members of the coset space $SU(4)/SU(2)\otimes SU(2)$, which does not form a subgroup of $SU(4)$. It is known (see [*Proposition 1*]{} of Ref. ) that any $U\in SU(4)$ can be written as $$U=k_1Ak_2 \label{decom1}$$ with $$A = \exp\left[\frac{i}{2}(c_1\sigma_{x}^{1}\sigma_{x}^{2} +c_2\sigma_{y}^{1}\sigma_{y}^{2}+c_3\sigma_{z}^{1}\sigma_{z}^{2})\right]\,, \label{decom2}$$ where $k_1,~k_2 \in SU(2)\otimes SU(2)$ and $c_1,~c_2,~c_3~\in R$. In the following, we focus on the effect of nonlocal operations generated by a single operator among the possibilities for $\sigma_i\otimes\sigma_j$, where $i,j \in\{x,y,z\}$. (This consideration includes the special case in which two of the parameters $c_1$, $c_2$, and $c_3$ in the decomposition (\[decom1\])-(\[decom2\]) are zero.) Such nonlocal operations will be called one-dimensional. Local operations ---------------- Local operations are operations $g\in SU(2)\otimes SU(2)$ generated by the elements of ${\mit e}$. From the commutation relations $[{\mit e},{\mit e}] \subset {\mit e}$ and $[{\mit p},{\mit e}] \subset {\mit p}$ it is clear that the elements of the vectors ${\bf r}_i$ and tensor $(T_i^j)$ do not mix and do not affect one another. Under local operations, the vectors behave just like ordinary Cartesian vectors. In particular, a vector of coherence is rotated about some vector $\hat n$ as illustrated in Fig. 1, i.e., $$(r')_{1}^{i}=R^{i}_{j}r_1^j\,, \qquad (r')_{2}^{i}=R^{i}_{j}r_2^j\,.$$ On the other hand, the correlation tensor transforms like a mixed Cartesian tensor, $$(T')^{i}_{j}=R^{i}_{m}(R')^{n}_{j} T^{m}_{n}\,.$$ The magnitude of each object remains invariant under local operations. In addition, there exist fifteen more invariants which can be constructed from the vectors and the tensor [@makhl]. {width="12cm"} One-dimensional nonlocal operations ------------------------------------ The nonlocal operations in the coset space $SU(4)/SU(2)\otimes SU(2)$ require, in their construction, exponentiation of at least one of the elements of ${\mit p}$. Hence, under these operations the elements of the tensor and vectors of coherence are mixed, due to the commutation relations $[{\mit p},{\mit e}] \subset {\mit p}$ and $[{\mit p},{\mit p}] \subset {\mit e}$. We shall establish that the one-dimensional nonlocal operations generated by the chosen interaction $\sigma_i\otimes\sigma_j$ give rise to elliptical orbits for the vectors of coherence of the subsystems. The characteristics of these elliptic paths depend on the indices $i$ and $j$, on the initial states of the subsystems, and on the degree of correlations between them. These orbits can be described by non-unitary transformations on each of the individual subsystems when one traces over the other’s degrees of freedom. Accordingly, we take the interaction Hamiltonian between the two spins to be $H_I=\sigma_i\otimes\sigma_j/2$, and, for reasons of simplicity, we suppose that the internal Hamiltonians for the two spins may be ignored. Assuming that the duration of the interaction is $\phi$, and appealing to (i) the commutation relations as summarized in Ref. [@Zhang] and (ii) the identity $$\exp\left[-i(\phi/2)\sigma_i\otimes\sigma_j\right] = \cos(\phi/2)I-i\sin(\phi/2)\sigma_i\otimes\sigma_j\,,$$ we can make the following observations: 1. The components $r^{i}_1$ and $r^{j}_2$ of the vectors of coherence remain unaffected; hence the vectors are confined to planes perpendicular to the $i$-axis and $j$-axis respectively. 2. Of the nine elements of the correlation tensor $T^{k}_{l}$, only four experience changes. The five that are unchanged under the action of $\sigma_i\otimes \sigma_j$ are $T^i_j$ and $T^k_l$ with $k\neq i$ and $l\neq j$. 3. The vectors $r_1^m+T_j^m$ and $r_2^m+T_m^i$ are rotated, without change of magnitude, through an angle $\phi$ about the $i$ and $j$ axes, respectively. (Here $m$ ranges freely over $\{x,y,z\}$). 4. More explicitly, the components of the vectors transform according to $$\begin{array}{l} r_{1}^{i}\rightarrow (r')_{1}^{i}=r_{1}^{i}\,, \nonumber \\ r_{1}^{k}\rightarrow (r')_{1}^{k}=r_1^k \cos \phi- T^{l}_{j}\sin \phi \,,\nonumber \\ r_{1}^{l}\rightarrow (r')_{1}^{l}= T^{k}_{j}\sin \phi + r_{1}^{l}\cos \phi \,, \end{array} \begin{array}{l} r_{2}^{j}\rightarrow (r')_{2}^{j}=r_{2}^{j}\,,\nonumber \\ r_{2}^{m}\rightarrow (r')_{2}^{m}=r_{2}^{m}\cos \phi- T^{i}_{n}\sin \phi\,,\nonumber\\ r_{2}^{n}\rightarrow (r')_{2}^{n}= T^{i}_{m}\sin \phi +r_{2}^{n}\cos \phi \,, \end{array}$$ and the components of the tensor of coherence, according to $$\begin{array}{l} T_{j}^{i}\rightarrow (T')_{j}^{i}=T_{j}^{i}\,,\nonumber \\ T_{j}^{k}\rightarrow (T')_{j}^{k}=T^{k}_{j}\cos \phi-r^{l}_{1}\sin \phi \,,\nonumber\\ T_{j}^{l}\rightarrow (T')_{j}^{l} =r^{k}_{1}\sin \phi + T_{j}^{l} \cos \phi \,, \end{array},~~ \begin{array}{l} T_{j}^{i}\rightarrow (T')_{j}^{i}=T_{j}^{i}\,,\nonumber \\ T_{m}^{i}\rightarrow (T')_{m}^{i}=T^{i}_{m}\cos \phi -r^{n}_{2}\sin \phi \,,\nonumber\\ T_{n}^{i}\rightarrow (T')_{n}^{i} =r^{m}_{2}\sin \phi + T_{n}^{i}\cos \phi \,, \end{array}$$ with no change in the tensor’s other elements. The ordered sets of indices $(i,l,k)$ and $(j,m,n)$ belong to $\{(x,y,z),(y,z,x),(z,x,y)\} $. Given this behavior, it is not difficult to show that [*${\bf r}_1(\phi)$ and ${\bf r}_2(\phi)_2$ follow elliptical orbits*]{}. Since the 1,2 labeling is arbitrary, it suffices to demonstrate this property for the the vector ${\bf r}_1(\phi)$. [*Proof.*]{} Referring to Fig. 2(a), the coordinates for a vector $\bf s$ tracing an ellipse in the $x-y$ plane, with principal axes $a$ and $b$ rotated by an angle $\psi$, are $$\begin{array}{l} s_x(\phi)= a ~{\rm cos}\phi~{\rm cos}\psi+ b~{\rm sin}\phi~{\rm sin}\psi\,,\nonumber\\ s_y(\phi)= - a~ {\rm cos}\phi~{\rm sin}\psi+ b~{\rm sin}\phi~{\rm cos}\psi \,. \end{array}$$ The angle $\phi$ is zero when the vector $\bf s$ is aligned with the principal axis $a$. The coordinates of the vector of coherence ${\bf r}_1$ moving in the $k-l$ plane are given by $$\begin{array}{l} r_1^k(\phi')=r_1^k(0)\cos \phi' - T_j^l(0) \sin \phi'\,,\nonumber \\ r_1^l(\phi')=T_j^k(0)\sin \phi'+r_1^l(0) \cos \phi' \,. \end{array}$$ Of course, for the vector of coherence, $\phi' = 0 $ does not in general correspond to the principal axis $a$ (see Fig. 2(a)). In fact, $\phi'=\phi+\chi$, and the coordinates of ${\bf r}_1 $ can be rewritten as follows in terms of the phase difference $\chi$: $$\begin{array}{l} r_1^k(\phi)=(r_1^k(0)\cos\chi- T_j^l(0)\sin\chi)\cos\phi+ (-T_j^l(0)\cos\chi-r_1^k(0)\sin\chi)\sin\phi \,, \nonumber\\ r_1^l(\phi)=(T_j^k(0)\sin\chi+r_1^l(0)\cos\chi) \cos\phi +(-r_1^l(0)\sin\chi+T_j^k(0)\cos\chi)\sin\phi \,. \end{array}$$ Comparison of the two sets of coordinates $\{s_x(\phi),s_y(\phi)\}$ and $\{r_1^k(\phi),r_1^l(\phi) \}$ shows that a match can always be made, such that the parameters $a$, $b$, $\psi$, and $\chi$ can be determined by solving the system of equations $$\begin{array}{c} a\cos\psi= r_1^k(0)\cos\chi- T_j^l(0)\sin\chi \,, \nonumber \\ b\sin\psi=-T_j^l(0)\cos\chi-r_1^k(0)\sin\chi \,, \nonumber \\ a\sin\psi=-T_j^k(0)\sin\chi-r_1^l(0)\cos\chi \,, \nonumber \\ b\cos\psi=-r_1^l(0)\sin\chi+T_j^k(0)\cos\chi\,. \label{system} \end{array}$$ This completes the proof. It is important to note that the shape of the ellipse depends explicitly on the spin-spin correlation tensor. {width="12cm"} Solving Eqs. (\[system\]) for the angle $\chi$, we find $$\tan(2\chi)= \frac{2\left[r^{k}_{1}(0)T^{l}_{j}(0)-r^{l}_{1}(0)T^{k}_{j}(0))\right]} {-(r^{l}_{1}(0))^2+(T^{k}_{j}(0))^2 -(r^{k}_{1}(0))^2+(T^{l}_{j}(0))^2}\,,$$ which specifies the initial orientation of the coherence vector ${\bf r}_1$ with respect to the principal axis $a$. Suppose now the two-spin system is initially in a [*product state*]{}. For this case it is easy to prove these corollaries to our principal result: - The phase difference $\chi$ is zero. This means that the initial positions of both coherence vectors lie on the $a$ principal axis (as in Fig. 3(a)). It follows that the linear entropy of the state of each of the subsystems (defined by $1 - {\rm Tr}\,\rho^2$) can only decrease, showing it is possible to increase the entanglement of the system with this interaction. (This will depend on initial conditions. See Section \[sec:ent-heis\].) - The length of the semi-minor axis of the ellipse followed by subsystem 1 is given by $ b_1=|r_2^j(0)|[(r_1^l(0))^2+(r_1^k(0))^2]^{1/2}$ (and likewise for subsystem 2 with $1 \rightarrow 2$ and $\{j,k,l\} \rightarrow \{i,n,m\}$). It follows that for an initially pure state and the assumed single interaction $\sigma_i \otimes \sigma_j$, the maximum attainable entanglement is achieved at $\phi$ values of $\pi/2$ and $3 \pi/2$. For the case of a initial state that is not pure but still separable, the phase difference $\chi$ does not vanish, in general (see figure 3(b)). Accordingly, the implied dynamical behavior of a classically correlated system distinguishes it from an uncorrelated system, but not from a system experiencing quantum entanglement. Moreover, the linear entropy of each subsystem can either increase or decrease, showing it is possible to increase or decrease the amount of entanglement in the system. {width="12cm"} General nonlocal operations --------------------------- From [*Proposition 1*]{} of Ref. , any nonlocal operation can be decomposed as a product of two local operations and an operation of the form $$A= \exp\left[ \frac{i}{2}(c_1\sigma_{x}^{1}\sigma_{x}^{2} +c_2\sigma_{y}^{1}\sigma_{y}^{2}+c_3\sigma_{z}^{1}\sigma_{z}^{2})\right] \,. \label{decom3}$$ The operators $\{Y_i\}=\{i\sigma_{x}^{1}\sigma_{x}^{2}/2, i\sigma_{y}^{1}\sigma_{y}^{2}/2,i\sigma_{z}^{1}\sigma_{z}^{2}/2 \}$ span a maximal Abelian subalgebra of ${\mit P}$, and the relations $$[Y_i,Y_j]=0\,, \qquad [Y_i,Y_j]_+ = -i|\epsilon_{ijk}|Y_k -\frac{1}{2}\delta_{ij}$$ hold, where $[\cdot,\cdot]_+$ denotes the anticommutator. Consequently, $A$ of Eq. (\[decom3\]) can be written in product form, $$A =\exp\left[\frac{i}{2}(c_1\sigma_{x}^{1}\sigma_{x}^{2})\right] \exp\left[\frac{i}{2}(c_2\sigma_{y}^{1}\sigma_{y}^{2})\right] \exp\left[\frac{i}{2}(c_3\sigma_{z}^{1}\sigma_{z}^{2})\right]\,. \label{prodform}$$ The property (\[prodform\]) tells us that [*any nonlocal operation can be decomposed into a sequence of operations effecting a succession of circular and elliptic paths in Bloch space*]{}. This result facilitates the calculation of the final states of the subsystems, but gives only limited insight into the geometric characteristics of the coherence vectors’ time orbits. For all $c_i$ distinct, two general observations can be made: - The motion of the vectors of coherence is no longer restricted to a plane, since there is no linear combination of $\{\sigma_x\otimes 1, \sigma_y\otimes 1, \sigma_z\otimes 1\}$ or $\{1\otimes\sigma_x,1\otimes\sigma_y,1\otimes\sigma_z\}$ that is invariant under the action of $A$. - Characteristics of the trajectories such as periodicity depend in detail on the parameters $c_1$, $c_2$, and $c_3$. A trajectory is periodic only if $c_2/c_1$ and $c_3/c_1$ are both rational numbers. We note also that the set of parameters $\{c_1,c_2,c_3\}$ has been used to determine the equivalence classes of nonlocal interactions [@Zhang] as well as the invariants of the nonlocal interactions [@makhl]. Special case of the Heisenberg Hamiltonian ------------------------------------------ The Heisenberg exchange Hamiltonian, corresponding to $c_1=c_2=c_3=-c/2$, is not included in our general observations on nonlocal interactions (made for all $c_i$ distinct), but like the one-dimensional Hamiltonians, it admits a simple geometric picture. This interaction is the primary two-qubit interaction in several experimental proposals for quantum-dot qubits [@Loss:98; @Kane:98; @Vrijen:00]. It can also be used for universal quantum computing on encoded qubits of several types [@Bacon:00; @Kempe:01; @DiVincenzo:00a; @Lidar:02; @Wu:02; @Byrd:02]. For these reasons, it warrants special attention. Introducing the time parameter $\phi$, the operator $A$ of Eq. (\[prodform\]) now takes the form $$\begin{aligned} A(\phi) &=& \exp[-i(c\phi/2)(\sigma_x\otimes \sigma_x + \sigma_y\otimes \sigma_y + \sigma_z\otimes \sigma_z)] \nonumber \\ &=& \left[\cos^3(c\phi/2) -i\sin^3(c\phi/2)\right]I\otimes I \nonumber \\ && -(i/2)e^{ic\phi/2}\sin(c\phi)(\sigma_x\otimes \sigma_x + \sigma_y\otimes \sigma_y + \sigma_z\otimes \sigma_z ). \end{aligned}$$ The time development of the density matrix under the operator $A$ is given $\rho(\phi)= A(\phi)\rho(0)A^{\dagger}(\phi)$ and the corresponding coherence vectors change according to $$\label{eq:ipart} r_1^i(\phi) = \frac{1}{2}[r^i_1(0)+r^i_2(0)+(r_1^i(0)-r_2^i(0))\cos(2c\phi) +(T_{k}^{j}(0)-T_{j}^{k}(0))\sin(2c\phi)] \,,$$ where $i,j,k =1,2,3$ and cyclic permutations are implied. Similarly, for the coherence tensor we have $$\label{eq:tpart} T_j^i(\phi) = \frac{1}{2}[T^{i}_{j}(0)+T^{j}_{i}(0) +(T^{i}_{j}(0)-T^{j}_{i}(0)) \cos(2c\phi) +(r_1^k(0)-r_2^k(0)) \sin(2c\phi)] \,.$$ The elements of ${\bf r}_2(\phi)$ are found by symmetry $1\leftrightarrow 2$. The quantities $r_1^i+r_2^i$, $T^{i}_{j}+T^{j}_{i}$, and $T^{i}_{i}$ are unchanged by the operation, and the form of the one-parameter set that describes the time-evolving coherence vector is $${\bf r}_1(\phi) = {\bf R} + {\bf S}\cos(2c\phi) + {\bf V}\sin(2c\phi)\,,$$ where ${\bf R}={\bf r}_1(0)+{\bf r}_2(0)$, ${\bf S} = {\bf r_1}(0)-{\bf r}_2(0)$, and $${\bf V} = \left(\begin{array}{c} T^{3}_{2}(0)-T^{2}_{3}(0) \\ T^{1}_{3}(0)-T^{3}_{1}(0) \\ T^{2}_{1}(0)-T^{1}_{2}(0) \end{array}\right)\,.$$ Clearly the vector traces out an ellipse lying in the plane spanned by ${\bf S}$ and ${\bf V}$, defined by ${\bf S}\times{\bf V}$, and passing through the point ${\bf R}$. Applications ============ We shall now illustrate some of the results of Section III with two examples. The first provides a controllability result for nonlocal unitary interactions and the second demonstrates how the orbit of the coherence vector can be used to describe the entangling power of the Heisenberg exchange interaction. Quantum control via quantum controllers and one-dimensional nonlocal interactions --------------------------------------------------------------------------------- Let us now consider the implications of the findings of the preceding sections for the problem of quantum control. To this end, we adopt the nomenclature of Lloyd [@lloyd] and identify spin 1 with the system $S$ whose quantum state we wish to control, and spin 2 with the quantum controller or interface $Q$. It is assumed that (i) only one interaction Hamiltonian $H_I$ is in play between $S$ and $Q$ and (i) system $Q$ is completely controllable via control Hamiltonians $\{H_Q^{m}\}=\{1\otimes \sigma_x, 1\otimes \sigma_y,1\otimes \sigma_z\} $ that span the $su(2)$ algebra. The initial state of the bipartite system is taken to be a product state in the ensuing analysis. Suppose that the interaction Hamiltonian is nonlocal, but takes the one-dimensional form $H_I=\sigma_i\otimes \sigma_j $. Then the set $\{\{H_Q^{m}\},H_I\}$ $=\{1\otimes \sigma_x, 1\otimes \sigma_y, 1\otimes \sigma_z, \sigma_i\otimes \sigma_x, \sigma_i\otimes \sigma_y,\sigma_i\otimes \sigma_z\} $ comprises a closed six-element subalgebra ${\mit G}_6$ of ${\mit G} $. Given this set of operations, the vector of coherence ${\bf r}_S$ of system $S$ remains in the plane perpendicular to the $i$-axis. It has been established in Sec. III that when $H_I=\sigma_i\otimes \sigma_j $ is the only element of the algebra $su(4)$ affecting the two-spin system, the vectors of coherence ${\bf r}_1$ and ${\bf r}_2$ are constrained to move in elliptical orbits. Now, with the six-element subalgebra ${\mit G}_6$ available to the two-spin system $S+Q$, the reachable set of the system $S$ is enlarged to an [*elliptical disk*]{} (see Fig. 4). The principle axis of the disk coincides with the initial coherence vector ${\bf r}_S(0)$ of the $S$ system, while the length of its semiminor axis is given by $b=[(r_S^k(0))^2+(r_S^l(0))^2]^{1/2}|{\bf r}_Q(0)|$, where ${\bf r}_Q(0)$ is the initial coherence vector of system $Q$. [*Proof.*]{} First, if one implements the two-step sequence of a local operation $\in 1\otimes su(2)$ on system $Q$ followed by the nonlocal operation $H_I=\sigma_i\otimes \sigma_j$ on $S+Q$, the orbit of ${\bf r}_S$ is necessarily an ellipse whose $a$ principle axis lies along the initial coherence vector ${\bf r}_S(0)$ and whose semimajor axis $b$ is restricted by $0\le b\le [(r_S^k(0))^2+(r_S^j(0))^2]^{1/2}|{\bf r}_Q(0)|$. Hence the set reachable by this two-step procedure is the elliptic disk in question. Second, using the Baker-Campbell-Hausdorff formula one can show that all the elements of the six-element subalgebra ${\mit G}_6$ can be constructed by this two step sequence, so their reachable sets are the same. From this result we infer that [*the entropy of system $S$ cannot be decreased by intervention of the quantum interface $Q$ if the interaction Hamiltonian is limited to the form $H_I=\sigma_i\otimes \sigma_j$*]{}. Noting that $|{\bf r}_Q|\le 1/2$, it follows that $a\ge b$, where $a$ and $b$ are respectively the magnitudes of the semimajor and semiminor axes of the elliptical reachable set. Furthermore, it is seen that the systems $S$ and $Q$ become maximally entangled if the initial state of the system $S$ is situated on the equatorial plane perpendicular to $i$-axis. ![The gray area is the set of reachable states for the system $S$ if one has full control of the controller $Q$ and the interaction $\sigma_i\otimes\sigma_j$ is available. This elliptical disk is characterized by a semimajor axis coincident with the initial vector of coherence for $S$ and a semiminor axis with $b= [(r_S^k(0))^2+(r_S^j(0))^2]^{1/2}|{\bf r}_Q(0)|$.](fig4.eps){width="12cm"} Entanglement power of Heisenberg interaction {#sec:ent-heis} -------------------------------------------- Upon examining Eq. (\[eq:ipart\]), we see that the maximum entanglement, realized in a maximally entangled pure state, can be achieved if ${\bf {r}}_1(0) = - {\bf {r}}_2(0)$, $|{\bf r}_1|=1/2$, and $c\phi = \pm\pi/4$. Otherwise, the state is not perfectly entangled since the linear entropy 1-Tr($\rho^2$) is not minimized. This conclusion agrees with the result of Zhang [*et al.*]{} [@Zhang] that the only perfect entanglers that can be achieved with the Heisenberg Hamiltonian are the square-root of swap and its inverse. However, suppose that the initial state of the two-spin system is represented by $$\rho(0) = \frac{1}{4}(I + \sigma_z)\otimes (I + \sigma_z) \,,$$ which is a pure-state density matrix for which $r_1^z = 1/2 = r_2^z$ and $T^{z}_{z} = 1/2$, all other elements of the coherence vectors and coherence tensor being zero. Then $$r_1^{x}(\phi) = r_1^z(0)\cos^2(c\phi)+r_2^z(0)\sin^2(c\phi)\,,$$ while all other components of ${\bf r}_1$ and ${\bf r}_2$ vanish at time $\phi$, and all other $r_1^\alpha(0) =0$. In this case the ellipse collapses to a line and the coherence vector simply oscillates between two values along that line. The only element of the correlation tensor that changes is $$T^{1}_{2} = \frac{1}{2}\sin(2c\phi)(r_1^z(0)-r_2^z(0))\,,$$ which vanishes for an initial tensor product of pure states for which the subsystems are polarized in the $+z$ direction. Therefore one cannot create maximally entangled states with these initial conditions. Conclusions =========== In this paper we have developed a geometric representation for the orbits of the coherence vectors of a two-qubit system. In various circumstances we have shown that their evolution is described by elliptical orbits lying within the surface of the Bloch sphere. Importantly, every two-qubit unitary operation can be expressed as a combination of one of the evolutions we have considered, together with “pre” and “post” local single-qubit rotations. We anticipate that this geometric picture will be helpful in devising schemes for control of a quantum state via quantum interfaces, and we have obtained a controllability result appropriate for such applications. Given the utility of the coherence-vector picture for modeling quantum systems and describing their entanglement, further studies along similar lines may be fruitful. Such work could include analysis of the orbits of higher-dimensional quantum states, as well as consideration of the effects of measurement operations on controllability. Acknowledgments {#acknowledgments .unnumbered} =============== This research was supported by the U. S. National Science Foundation under Grant No. PHY-0140316 (JWC and AM) and by the Nipher Fund. [5]{} R. Alicki and K. Lendi, [*Quantum Dynamical Semigroups and Applications*]{} (Springer-Verlag, Berlin, 1987). C. Altafini, J. Math. Phys. [**44**]{}, 2357 (2003). U. Fano, Rev. Mod. Phys. [**55**]{}, 855 (1983). G. Mahler and V. A. Weberru[ß]{}, [*Quantum Networks*]{} (Springer, 1998). M. S. Byrd and N. Khaneja, Phys. Rev. A [**68**]{}, 062322 (2003); G. Kimura, Phys. Lett. [**A314**]{}, 339 (2003); G. Kimura and A. Kossakowski, quant-ph/0408014. C. Altafini, Phys. Rev. A [**69**]{}, 012311 (2004). J. Schlienz and G. Mahler, Phys. Rev. A [**52**]{}, 4396 (1995). S. Lloyd, A. J. Landhahl, and J. E. Slotine, Phys. Rev. A [**69**]{}, 012305 (2004); D. Janzing, F. Armknecht, R. Zeier and Th. Beth, Phys. Rev. A [**65**]{}, 022104 (2004). N. Khaneja, R. Brockett, and S. J. Glaser, Phys. Rev. A [**63**]{}, 032308 (2001). J. Zhang, J. Vala, S. Sastry, and K. B. Whaley, Phys. Rev. A [**67**]{}, 042313 (2003). G. Jaeger, M. Teodorescu-Frumosu, A. Sergienko, B. E. A. Saleh, and M. C. Teich, Phys. Rev. A [**67**]{}, 032307 (2003). Y. Makhlin, Quant. Inf. Proc. [**1**]{}, 243 (2002). R. Vrijen, E. Yablonovitch, K. Wang, H. W. Jiang, A. Balandin, V. Roychowdhury, T. Mor, and D. DiVincenzo, Phys. Rev. A [**62**]{}, 012306 (2000). B. E. Kane, Nature [**393**]{}, 133 (1998). D. Loss and D. P. DiVincenzo, Phys. Rev. A [**57**]{}, 120 (1998). D. Bacon, J. Kempe, D. A. Lidar and K. B. Whaley, Phys. Rev. Lett. [**85**]{}, 1758 (2000). J. Kempe, D. Bacon, D. A. Lidar, and K. B. Whaley, Phys. Rev. A [**63**]{}, 042307 (2001). D. A. Lidar and L.-A. Wu, Phys. Rev. Lett. [**88**]{}, 017905 (2002). D. P. DiVincenzo, D. Bacon, J. Kempe, G. Burkard, and K. B. Whaley, Nature [**408**]{}, 339 (2000). L.-A. Wu and D. A. Lidar, Phys. Rev. Lett. [**88**]{}, 207902 (2002). M. S. Byrd and D. A. Lidar, Phys. Rev. Lett. [**89**]{}, 047901 (2002).